diff --git "a/magnetization damping/3.json" "b/magnetization damping/3.json" new file mode 100644--- /dev/null +++ "b/magnetization damping/3.json" @@ -0,0 +1 @@ +[ { "title": "1806.04881v1.Low_magnetic_damping_of_ferrimagnetic_GdFeCo_alloys.pdf", "content": "1 \n Low magnetic damping of ferrimagnetic GdFeCo alloys \nDuck-Ho Kim1†*, Takaya Okuno1†, Se Kwon Kim2, Se-Hyeok Oh3, Tomoe Nishimura1, \nYuushou Hirata1, Yasuhiro Futakawa4, Hiroki Yoshikawa4, Arata Tsukamoto4, Yaroslav \nTserkovnyak2, Yoichi Shiota1, Takahiro Moriyama1, Kab-Jin Kim5, Kyung-Jin Lee3,6,7, and \nTeruo Ono1,8* \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 6 11-0011, Japan \n2Department of Physics and Astronomy, University of California, Los Angeles, California \n90095, USA \n3Department of Nano-Semiconductor and Engineering, Korea Univers ity, Seoul 02841, \nRepublic of Korea \n4College of Science and Technology, Nihon University, Funabashi, Chiba 274-8501, Japan \n5Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea \n6Department of Materials Science & Engineering, Korea University , Seoul 02841, Republic \nof Korea \n7KU-KIST Graduate School of Converging Science and Technology, K orea University, Seoul \n02841, Republic of Korea \n8Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, \nOsaka University, Osaka 560-8531, Japan \n \n† These authors contributed equally to this work. \n* E-mail: kim.duckho.23z@st.kyoto-u.ac.jp , ono@scl.kyoto-u.ac.jp 2 \n We investigate the Gilbert damping parameter for rare earth (RE)–\ntransition metal (TM) ferrimagnets over a wide temperature rang e. Extracted from the \nfield-driven magnetic domain-wall mobility, was as low as 7.2 × 10-3 and was almost \nconstant across the angular momentum compensation temperature 𝑻𝐀, starkly \ncontrasting previous predictions that should diverge at 𝑻𝐀 due to vanishing total \nangular momentum. Thus, magnetic damping of RE-TM ferrimagnets is not related to \nthe total angular momentum but is dominated by electron scatter ing at the Fermi level \nwhere the TM has a dominant damping role. \n 3 \n Magnetic damping, commonly described by the Gilbert damping par ameter, \nrepresents the magnetization relaxation phenomenon, describing how quickly magnetization \nspins reach equilibrium [1–3]. Understanding the fundamental or igin of the damping as well \nas searching for low damping materials has been a central theme of magnetism research. \nSeveral theoretical models for magnetic damping have been propo sed [4–11] and compared \nwith experiments [12–20]. Ultra-low damping was predicted in fe rromagnetic alloys using a \nlinear response damping model [11] and was demonstrated experim entally for CoFe alloys \n[20]. However, the majority of these studies have focused only on ferromagnetic systems. \nAntiferromagnets, which have alt ernating orientations of their neighboring magnetic \nmoments, have recently received considerable attention because of their potential importance \nfor spintronic applications [21– 30]. Antiferromagnetic spin sys tems can have much faster \nspin dynamics than their ferromagnetic counterparts, which is a dvantageous in spintronic \napplications [21, 25, 31–39]. However, the manipulation and con trol of antiferromagnets is \nchallenging because the net magnetic moment is effectively zero . Recently, antiferromagnetic \nspin dynamics have been successfully demonstrated using the mag netic domain-wall (DW) \ndynamics in ferrimagnets with finite magnetization in the vicin ity of the angular momentum \ncompensation temperature, at which the net angular momentum van ishes [38]. This field-\ndriven antiferromagnetic spin dyn amics is possible because the time evolution of the \nmagnetization is governed by the commutation relation of the an gular momentum rather than \nthe commutation relation of the magnetic moment. \nMotivated by the aforementioned result, in this letter, we inve stigate the magnetic \ndamping of ferrimagnets across th e angular momentum compensatio n temperature, which \nwill allow us to understand magnetic damping in antiferromagnet ically coupled system. We 4 \n selected rare earth (RE)–transition metal (TM) ferrimagnets for the material platforms \nbecause they have an angular momentum compensation temperature 𝑇 w h e r e \nantiferromagnetic spin dynamics are achieved [38, 40, 41]. The magnetic-field-driven DW \nmotion was explored over a wide range of temperatures including 𝑇, and the Gilbert \ndamping parameter was extracted from the measured DW mobility a t each temperature by \nemploying the collective coordina te model initially developed f or ferrimagnetic spin \ndynamics [38]. Contrary to the previous prediction that the Gil bert damping parameter would \ndiverge at 𝑇 due to the vanishing of the total angular momentum [42, 43], w e found that the \nGilbert damping parameter remained nearly constant over a wide range of temperatures \nacross 𝑇 with the estimated value as low as 7.2 × 10-3, which was similar to the reported \nvalues of TM-only ferromagnets [20]. These results suggested th at Gilbert damping was \nmainly governed by electron scattering at the Fermi level, and hence, the 4f electron of the \nR E e l e m e n t , w h i c h l i e s f a r b e l o w t h e F e r m i l e v e l , d i d n o t p l a y an important role in the \nmagnetic damping of RE–TM ferrimagnets. \nFor this study, we prepared perpendicularly magnetized ferrimag netic GdFeCo films \nin which the Gd and FeCo moments were coupled antiferromagnetic ally. Specifically, the \nfilms were 5-nm SiN/30-nm Gd 23.5Fe66.9Co9.6/100-nm SiN on an intrinsic Si substrate. The \nGdFeCo films were then patterned into 5-µm-wide and 500-µm-long microwires with a Hall \ncross structure using electron beam lithography and Ar ion mill ing. For current injection, \n100-nm Au/5-nm Ti electrodes were stacked on the wire. A Hall b ar was designed to detect \nthe DW velocity via the anomalous Hall effect (AHE). \nWe measured the magnetic DW motion using a real-time DW detecti on technique [38, \n40, 41, 44, 45] [see Fig. 1(a) for a schematic]. We first appli ed a magnetic field of –200 mT 5 \n to saturate the magnetization al ong the –z direction. Subsequen tly, a constant perpendicular \nmagnetic field 𝜇𝐻, which was lower than the coercive field, was applied along +z direction. \nNext, a d.c. current was applied along the wire to measure the anomalous Hall voltage. Then, \na current pulse (12 V , 100 ns) was injected through the writing line to nucleate the DW in the \nwire. The created DW was moved along the wire and passed throug h the Hall bar because of \nthe presence of 𝜇𝐻. The DW arrival time was detected by monitoring the change in the Hall \nvoltage using a real-time oscillo scope. The DW velocity could t hen be calculated from the \narrival time and the travel dis tance between the writing line a nd Hall bar (500 µm). \nFigure 1(b) shows the averaged DW velocity 〈𝑣〉 as a function of the perpendicular \nmagnetic field 𝜇𝐻 for several temperatures 𝑇∗. Here, we used the d.c. current density of \n|𝐽|ൌ1.3×1010 A / m2 to measure the AHE change due to DW motion. Note that 𝑇∗ i s a n \nelevated temperature that considers Joule heating by d.c. curre nt [46]. To eliminate the \nundesired current-induced spin-transfer-torque effect, we avera ged the DW velocity for 𝐽 \nand –𝐽, i.e., 〈𝑣〉ൌሾ𝑣ሺ𝐽ሻ𝑣ሺെ𝐽ሻሿ/2. Figure 1(b) shows that 〈𝑣〉 increases linearly with \n𝜇𝐻 for all 𝑇∗. Such linear behavior can be described by 〈𝑣〉ൌ𝜇ሾ𝜇𝐻െ𝜇 𝐻ሿ, where 𝜇 \nis the DW mobility and 𝜇𝐻 is the correction field, which generally arises from \nimperfections in the sample or complexities of the internal DW structure [47, 48]. We note \nthat 𝜇𝐻 can also depend on the temperature dependence of the magnetic properties of \nferrimagnets [45]. Figure 1(c) shows 𝜇 as a function of 𝑇∗ at several current densities \n(|𝐽|ൌ1.3, 1.7, and 2.0 ×1010 A / m2). A sharp peak clearly occurs for 𝜇 a t 𝑇∗ൌ241.5 K \nirrespective of |𝐽|. The drastic increase of 𝜇 is evidence of antiferromagnetic spin dynamics \nat 𝑇, as demonstrated in our pre vious report [38, 40, 41]. \nThe obtained DW mobility was theoretically analyzed as follows. The DW velocity 6 \n of ferrimagnets in the precessional regime is given by [38, 39] \n 𝑉 ൌ 𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ\nሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶ𝜇𝐻, ሺ1ሻ \nwhere 𝑉 is the DW velocity, 𝜆 is the DW width, 𝜇𝐻 is the perpendicular magnetic field, \n𝛼 is the Gilbert damping parameter, 𝑀 and 𝑠 are the magnetization and the spin angular \nmomentum of one sublattice, respectively. The spin angular mome ntum densities are given \nby 𝑠ൌ𝑀 /𝛾 [49], where 𝛾ൌ𝑔 𝜇/ℏ is the gyromagnetic ratio of lattice 𝑖, 𝑔 i s t h e \nLandé g factor of lattice 𝑖, 𝜇 is the Bohr magneton, and ℏ is the reduced Plank’s constant. \nThe Gilbert damping is in principle different for two sublattic e s , b u t f o r s i m p l i c i t y , w e \nassume that it is the same, which can be considered as the aver age value of the damping \nparameters for the two sublattices weighted by the spin angular momentum density. We note \nthat this assumption does not alter our main conclusion: low da mping and its insensitivity to \nthe temperature. Equation (1) gives the DW mobility 𝜇 a s 𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ/\nሼሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶሽ, which can be rearranged as \n 𝜇 ሺ𝑠ଵ𝑠 ଶሻଶ𝛼ଶെ𝜆ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ𝛼𝜇 ሺ𝑠ଵെ𝑠 ଶሻଶൌ 0 ሺ2ሻ \nUsing Eq. (2) to find the solution of 𝛼, we find \n 𝛼 േൌ𝜆ሺ𝑀ଵെ𝑀 ଶሻേඥሾ𝜆ଶሺ𝑀ଵെ𝑀 ଶሻଶെ4 𝜇ଶሺ𝑠ଵെ𝑠 ଶሻଶሿ\n2𝜇ሺ𝑠ଵ𝑠 ଶሻ. ሺ3ሻ \nEquation (3) allows us to estimate 𝛼 for the given 𝜇. We note that for each value of 𝜇, 𝛼 \nca n h av e t w o v a lu e s, 𝛼ା and 𝛼ି because of the quadratic nature of Eq. (2). Only one of \nthese two solutions is physically sound, which can be obtained using the following energy \ndissipation analysis. 7 \n The energy dissipation (per unit cross section) through the DW dynamics is given by \n𝑃ൌ2 𝛼 ሺ 𝑠 ଵ𝑠 ଶሻ𝑉ଶ/𝜆 2𝛼ሺ𝑠 ଵ𝑠 ଶሻ 𝜆Ωଶ [38, 39], where Ω is the angular velocity of the \nDW. The first and the second terms represent the energy dissipa tion through the translational \nand angular motion of the DW, respectively. In the precessional regime, the angular velocity \nis proportional to the translational velocity: Ωൌ ሺ𝑠ଵെ𝑠 ଶሻ𝑉/𝛼ሺ𝑠 ଵ𝑠 ଶሻ𝜆. Replacing Ω b y \nthe previous expression yields 𝑃ൌ𝜂 𝑉ଶ w h e r e 𝜂ൌ2 ሺ 𝑀 ଵെ𝑀 ଶሻ/𝜇 is the viscous \ncoefficient for the DW motion: \n 𝜂 ൌ2\n𝜆ቊ𝛼ሺ𝑠ଵ𝑠 ଶሻ ሺ𝑠ଵെ𝑠 ଶሻଶ\n𝛼ሺ𝑠ଵ𝑠 ଶሻቋ . ሺ4ሻ \nThe first and the second terms in parenthesis capture the contr ibutions to the energy \ndissipation from the translational and angular dynamics of the DW, respectively. The two \nsolutions for the Gilbert damping parameter, 𝛼ା and 𝛼ି, can yield the same viscous \ncoefficient 𝜂. The case of the equal solutions, 𝛼ାൌ𝛼 ି, corresponds to the situation when \nthe two contributions are identical: 𝛼േൌሺ 𝑠 ଵെ𝑠 ଶሻ/ሺ𝑠ଵ𝑠 ଶሻ. For the larger solution 𝛼ൌ\n𝛼ା, the energy dissipation is dominated by the first term, i.e., through the translational DW \nmotion, which should be the case in the vicinity of 𝑇 where the net spin density ሺ𝑠ଵെ𝑠 ଶሻ \nis small and thus the angular velocity is negligible. For examp le, at exact 𝑇, the larger \nsolution 𝛼ା is the only possible solution because the smaller solution is zero, 𝛼ିൌ0, and \nthus unphysical. For the smaller solution 𝛼ൌ𝛼 ି, the dissipation is dominated by the second \nterm, i.e., through the precessional motion, which should descr ibe cases away from 𝑇. \nTherefore, in the subsequent analysis, we chose the larger solu tion 𝛼ା in the vicinity of 𝑇 \nand the smaller solution 𝛼ି far away from 𝑇 and connected the solution continuously in \nbetween. 8 \n The other material parameters such as 𝑀ଵ, 𝑀ଶ, 𝑠ଵ, and 𝑠ଶ a r e e s t i m a t e d b y \nmeasuring the net magnetic moment of GdFeCo film, |𝑀୬ୣ୲|, for various temperatures. \nBecause 𝑀୬ୣ୲ includes contributions from both the Gd and FeCo sub-moments, the sub-\nmagnetic moments, 𝑀ଵ a n d 𝑀ଶ, could be decoupled based on the power law criticality [see \ndetails in refs. 38, 40]. The spin angular momentums, 𝑠ଵ and 𝑠ଶ, were calculated using the \nknown Landé g factor of FeCo and Gd (the Landé g factor of FeCo is 2.2 and that of Gd is \n2.0) [50–52]. \nFigures 2(a)–(c) show the temperature-dependent DW mobility 𝜇, sub-magnetic \nmoment 𝑀, and sub-angular momentum 𝑠, respectively. Here, we used the relative \ntemperature defined as ∆𝑇 ൌ 𝑇∗െ𝑇 to investigate the Gilbert damping near 𝑇. The \nGilbert damping parameter 𝛼 was obtained based on Eq. (3) and the information in Fig. \n2(a)–(c). Figure 2(d) shows the resulting values of 𝛼േ as a function of ∆𝑇. For ∆𝑇ଵ൏\n∆𝑇 ൏ ∆𝑇 ଶ, 𝛼ା is nearly constant, while 𝛼ି varies significantly. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇 \n∆𝑇ଶ, on the other hand, 𝛼ି is almost constant, while 𝛼ା varies significantly. At ∆𝑇 ൌ ∆𝑇 ଵ \nand ∆𝑇 ൌ ∆𝑇 ଶ, the two solutions are equal, corresponding to the aforementio ned case when \nthe energy dissipation through the translational and angular mo tion of the DW are identical. \nThe proper damping solution can be selected by following the gu ideline obtained \nfrom the above analysis. For ∆𝑇ଵ൏∆ 𝑇൏∆ 𝑇 ଶ, which includes 𝑇, the energy dissipation \nshould be dominated by the translational motion, and thus 𝛼ା is a physical solution. Note \nalso that 𝛼ି becomes zero at 𝑇, which results in infinite DW mobility in contradiction with \nthe experimental observation. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇 ∆𝑇 ଶ, where the energy dissipation is \ndominated by the angular motion of the DW, 𝛼ି is the physical solution. 9 \n Figure 3 shows the resultant Gil bert damping parameter in all t ested temperature \nranges. The Gilbert damping parameter was almost constant acros s 𝑇 with 𝛼ൌ7.2 × 10-3 \n(see the dotted line in Fig. 3). This result is in stark contra st to the previous prediction. In ref. \n[42], Stanciu et al. investigated the temperature dependence of the effective Gilb ert damping \nparameter based on a ferromagnet-based model and found that the damping diverged at 𝑇. \nBecause they analyzed the magnetic resonance in ferrimagnetic m aterials based on a \nferromagnet-based model, which cannot describe the antiferromag netic dynamics at 𝑇 a t \nwhich the angular momentum vanis hes, it exhibits unphysical res ults. However, our \ntheoretical analysis for field-driven ferromagnetic DW motion b ased on the collective \ncoordinate approach can properly describe both the antiferromag netic dynamics in the \nvicinity of 𝑇 and the ferromagnetic dynamics away from 𝑇 [38]. Therefore, the \nunphysical divergence of the Gilbert damping parameter at 𝑇 is absent in our analysis. \nOur results, namely the insensitivity of damping to the compens ation condition and \nits low value, have important implications not only for fundame ntal physics but also for \ntechnological applications. From the viewpoint of fundamental p hysics, nearly constant \ndamping across 𝑇 indicates that the damping is almost independent of the total angular \nmomentum and is mostly determined by electron spin scattering n ear the Fermi level. \nSpecifically, our results suggest that the 4f electrons of RE e lements, which lie in a band far \nbelow the Fermi level, do not play an important role in the mag netic damping of RE-TM \nferrimagnets, whereas the 3d and 4s bands of TM elements have a governing role in magnetic \ndamping. This result is consistent with the recently reported t heoretical and experimental \nresults in FeCo alloys [20]. From the viewpoint of practical ap plication, we note that the \nestimated damping of 𝛼ൌ7.2 × 10-3 is the upper limit, as the damping estimated from DW 10 \n dynamics is usually overestimated due to disorders [53]. The ob tained value of the Gilbert \ndamping parameter is consistent with our preliminary ferromagne t i c r e s o n a n c e ( F M R ) \nmeasurements. The experimental results from FMR measurements an d the corresponding \ntheoretical analysis will be publ ished elsewhere. This low valu e of the Gilbert damping \nparameter suggests that ferrimagne ts can serve as versatile pla t f o r m s f o r l o w - d i s s i p a t i o n \nhigh-speed magnetic devices such as spin-transfer-torque magnet ic random-access memory \nand terahertz magnetic oscillators. \nIn conclusion, we investigated the field-driven magnetic DW mot ion in ferrimagnetic \nG d F e C o a l l o y s o v e r a w i d e r a n g e o f t e m p e r a t u r e s a c r o s s 𝑇 and extracted the Gilbert \ndamping parameter from the DW mobility. The estimated Gilbert d amping parameter was as \nlow as 7.2 × 10-3 and almost constant over the temperature range including 𝑇, which is in \nstark contrast to the previous prediction in that the Gilbert d amping parameter would diverge \nat 𝑇 due to the vanishing total angular momentum. 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Shinjo, Science 284, 468 \n(1999). \n[49] In this Letter , the parameters such as the spin angular mo mentum density 𝑠 r e p r e s e n t \nthe magnitudes of the quantities. Their directions are separate ly handled through the signs in \nthe equations of motion. \n[50] C. Kittel, Phys. Rev. 76, 743 (1949). \n[51] G. G. Scott, Rev. Mod. Phys. 34, 102 (1962). \n[52] B. I. Min and Y.-R. Jang, J. Phys. Condens. Matter 3, 5131 (1991). \n[53] H. Min, R. D. McMichael, M. J. Donahue, J. Miltat, and M. D. Stiles, Phys. Rev. Lett. \n104, 217201 (2010). \n 16 \n Figure Captions \nFigure 1(a) Schematic illustration of the GdFeCo microwire devi ce. (b) The averaged DW \nvelocity 〈𝑣〉 as a function of the perpendicular magnetic field 𝜇𝐻 for several temperatures \n𝑇∗ (202, 222, 242, 262, and 282 K). The dots indicate the best li n e a r f i t s . ( c ) T h e D W \nmobility 𝜇 as a function of 𝑇∗ at several current densities ( |𝐽|ൌ1.3, 1.7, and 2.0 ×1010 \nA/m2). \nFigure 2 The temperature-dependent (a) DW mobility 𝜇, (b) sub-magnetic moment 𝑀, and \n(c) sub-angular momentum 𝑠. Here, we use the relative temperature defined as ∆𝑇 ൌ 𝑇∗െ\n𝑇. (d) The Gilbert damping parameter 𝛼േ as a function of ∆𝑇. Here, we use 𝜆ൌ15 nm for \nproper solutions of Eq. (3). \nFigure 3 The resultant Gil bert damping parameter 𝛼 in all tested temperature ranges. \n 17 \n Acknowledgements \nThis work was supported by the JSPS KAKENHI (Grant Numbers 15H0 5702, 26103002, and \n26103004), Collaborative Research Program of the Institute for Chemical Research, Kyoto \nUniversity, and R & D project for ICT Key Technology of MEXT fr om the Japan Society for \nthe Promotion of Science (JSPS). This work was partly supported by The Cooperative \nResearch Project Program of the Research Institute of Electrica l Communication, Tohoku \nUniversity. D.H.K. was supported as an Overseas Researcher unde r the Postdoctoral \nFellowship of JSPS (Grant Number P16314). S.H.O. and K.J.L. wer e supported by the \nNational Research Foundation of Korea (NRF-2015M3D1A1070465, 20 17R1A2B2006119) \nand the KIST Institutional Program (Project No. 2V05750). S.K.K . was supported by the \nArmy Research Office under Contract No. W911NF-14-1-0016. K.J.K . was supported by the \nNational Research Foundation of Korea (NRF) grant funded by the Korea Government \n(MSIP) (No. 2017R1C1B2009686). \nCompeting financial interests \nThe authors declare no competing financial interests. 200 225 250 275 3000.00.51.01.52.0\n 1.3\n1.7\n2.0\n [104 m/sT]\nT* [K]J [1010 A/m2]0 50 100 1500.00.51.01.5\n 202\n 222\n 242\n 262\n 282 [km/s]\n0H [mT]T* [K]\nFigure 1b\nca\nWriting line\n\tܫ\nܸ\nߤܪ\ny xz-60 -40 -20 0 20 40 60 801.52.02.53.0 s1\n s2s [10-6 Js/m3]\nT [K]-60 -40 -20 0 20 40 60 800.00.51.01.52.0\n [104 m/sT]\nT [K]\n-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \n +\n -\nT [K]T1T2-60 -40 -20 0 20 40 60 800.30.40.50.6 M1\n M2M [MA/m]\nT [K]a\nb\nc\nd\nFigure 2Figure 3-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \nT [K]" }, { "title": "0705.1515v1.Magnetization_oscillations_induced_by_a_spin_polarized_current_in_a_point_contact_geometry__mode_hopping_and_non_linear_damping_effects.pdf", "content": "1Magnetization oscillations induced by a spin-polarized current in a point-contact\ngeometry: mode hopping and non-linear damping effects\nD.V. Berkov, N.L.Gorn\nInnovent Technology Development e.V.,\nPruessingstr. 27B, D-07745 Jena, Germany\nABSTRACT\nIn this paper we study magnetization excitations induced in a thin extended film by a spin-\npolarized dc-current injected through a point contact in the current-perpendicular-to-plane\n(CPP) geometry. Using full-scale micromagnetic simulations, we demonstrate that in addition\nto the oscillations of the propagating wave type, there exist also two localized oscillation\nmodes. The first localized mode has a relatively homogeneous magnetization structure of its\nkernel and corresponds to the so called 'bullet' predicted analytically by Slavin and\nTiberkevich (Phys. Rev. Lett., 95 (2005) 237201). Magnetization pattern of the second\nlocalized mode kernel is highly inhomogeneous, leading to a much smaller power of\nmagnetoresistance oscillations caused by this mode. We have also studied the influence of a\nnon-linear damping for this system and have found the following main qualitative effects: (i)\nthe appearance of frequency jumps within the existence region of the propagating wave mode\nand (ii) the narrowing of the current region where the 'bullet' mode exists, until this mode\ncompletely disappears for a sufficiently strong non-linear damping.2I. INTRODUCTION\nSince Slonczewski [1] and Berger [2] have predicted that a spin-polarized current flowing\nthrough a thin magnetic film can induce magnetization excitations (up to a complete\nmagnetization switching) and first experimental confirmations of this effect [3] were\nobtained, the spin-torque induced magnetization dynamics belongs to the most intensively\nstudied topics of the solid state magnetism [4].\nMost experiments in this area have been performed in the so called columnar geometry, i.e.,\nfor a system where an electric current flows through a 'sandwich' structure consisting of two\nor more thin ferromagnetic layers with lateral sizes in the region ~ 50 - 200 nm separated by\nnon-magnetic spacers. Here a substantial degree of understanding has been achieved due to\nhigh-quality experimental studies [5], detailed theoretical analysis [6] and extensive\nnumerical simulations [7] (citations given above are by no means exhaustive).\nIn contrast to this situation, the interpretation of experimental results for the magnetization\ndynamics in the so called point-contact geometry remains controversial [8, 9, 10, 11, 12, 13].\nOn the one hand, this is due to a much more complicated nature of point-contact systems\ncompared to nanopillars. In particular, the macrospin approximation, being of some help for a\nsmall nanoelement, is absolutely invalid in the point-contact setup due to a strong exchange\ninteraction between the oscillating area under the contact with the rest of a thin film. On the\nother hand, the accumulated body of experimental results is much smaller than for the\ncolumnar geometry, so that a quite limited amount of data is available for a comparison with\nanalytical theories and numerical simulations.\nEven the type of spin wave excitations induced in the point-contact experiments remains a\nsubject of an intensive discussion. In particular, experimental threshold of the magnetization\noscillations onset for the in-plane external field is significantly lower than the value predicted\nby Slonczewski for the linear spin wave mode [8], as it was pointed out in [10] (when the\nexternal field is applied normally to the film plane and is strong enough to ensure the out-of-\nplane magnetization of a film, the result of Slonczewski shows a good agreement with\nexperimental data). Another important point is that the oscillation frequency observed, e.g., in\nthe pioneering experiment of Rippard [9] (for an in-plane applied field) is smaller than the\nhomogeneous FMR-frequency for the thin layer with magnetic parameters reported in [9], so\nthat an extended spin-wave with such a frequency could not exist. Basing on these indications,\nSlavin et al. suggested that the spin-wave mode excited under conditions used in [9] is a non-\nlinear localized mode [10, 11] for which the excitation threshold and oscillation frequency\nturned out to be significantly smaller than for the linear propagating wave mode.\nThe analytical theory of Slavin et al. employs several approximations, unavoidable in such\nsituations (see [10, 11] for details) and as such should be verified by rigorous numerical\nsimulations. However, as we have already pointed out in [13], numerical simulations of\nmagnetization dynamics in the point-contact setup encounter several serious methodical\ndifficulties, what partly explains why corresponding studies are extremely rare. In [13] we\nhave explained how these difficulties can be overcome and made an attempt to reproduce\nexperimental results from [9] by simulating the complete trilayer system\nNi80Fe20(5nm)/Co(5nm)/Co90Fe10(20nm) studied in [9] including also the Oersted field effect.\nWe have found that the magnetization dynamics in such a system is extremely complicated.\nMoreover, we have observed qualitative disagreement between our simulations and\nexperimental data. For these reasons and taking into account also that (i) magnetic parameters\nand especially the polycrystalline structure of the 'fixed' Co90Fe10 layer are not known well\nenough and (ii) reliable evaluation of the Oersted field effect without the exact knowledge of\nthe current distribution is not possible, we decided to perform systematic studies of the3simplest possible point-contact setup where the onset of the magnetization dynamics due to\nthe spin torque can be expected: a single magnetic layer subject to a spin-polarized current\nwithin the point contact area (with the Oersted field neglected). As we shall demonstrate\nbelow (Sec. IIIA), despite these simplifications, results of our study reveal many important\naspects of the spin wave dynamics in such systems (including the existence of several\nexcitation modes) and allow a meaningful comparison both with analytical theories [10, 11]\nand experimental data [9].\nIn addition, we have studied the effect of a non-linear damping (damping depending on the\nmagnetization change rate, as suggested in [14]), because for the large-angle spin-torque-\ndriven magnetization oscillations such effects are expected to be especially strong.\nCorresponding results are presented in Sec. IIIB.\nII. SIMULATED SYSTEM AND SIMULATION METHODOLOGY\nAs explained above, to make the system under study as simple as possible, but still retaining\nthe whole non-trivial physics, we have simulated the dynamics of a system consisting of a\n‘free’ layer only with magnetic parameters corresponding to Ni80Fe20 (Permalloy) layer in the\nexperiments described in [9]: saturation magnetization MS = 640 G, exchange stiffness\nconstant A = 1\u000110-6 erg/cm, negligible magnetocrystalline anisotropy, layer thickness h = 5\nnm. For all results presented below the external field Hext = 1000 Oe was applied in the film\nplane. The x-axis of our coordinate system is directed along the external field, z-axis is the\nsecond Cartesian axis in the film plane, so that y-axis is perpendicular to the film. Simulated\narea with the in-plane size 900 x 900 nm2 was discretized into Nx x Nz x Ny = 360 x 360 x 1\ncells, so that the cell size was 2.5 x 2.5 x 5 nm3; periodic boundary conditions (PBC) were\nused. Diameter of the current flooded area D = 40 nm also corresponds to the nominal point\ncontact diameter reported by Rippard et al. [9]. It is important to note that due to such a small\ndiameter discretization cell size larger than used by us (see above) led to inadequate\ndiscretization of this area and caused substantial artefacts by simulating the magnetization\ndynamics - especially when the 2nd localized mode with a fine spatial structure was excited.\nThe Oersted field of the spin-polarized current was neglected for three reasons. First, the pre-\nsence of this field lead to a much more complicated magnetization dynamics and we intended\nto study the minimal non-trivial model to emphasize the effects of the mode localization and\nnon-linear damping as clear as possible. Second, the proportionality coefficient between the\nspin-torque amplitude used in simulations and the current strength (which determines the\nmagnitude of the external field) is not known exactly. Third, the electric current distribution\nin the experimental setup is also known relatively poor, adding another uncertainty to the\nOersted field evaluation.\nTo suppress artificial interference between the ‘original’ spin wave emitted from the point\ncontact area (placed in the center of the simulated square) and waves coming from PBC\nreplica of the initial system, we have employed our method based on the spatially dependent\ndamping coefficient (see [13] for details). In addition, we have smoothed the spatial current\ndistribution also in the same manner as in [13] to avoid artificial generation of the\nmagnetization pattern with the wave length twice the cell size inside the point contact area\n(which appear otherwise due to the abrupt jump of the current density at the border of the area\nbelow the point contact).\nMagnetization dynamics excited by a spin-polarized current was simulated using basically the\nsame software package as in [13] (which is the extension of our commercially available\nMicroMagus package - see [15] for implementation details). Thermal fluctuations were\nneglected (T = 0). Spin torque acting on the magnetization M was included into the Landau-4Lifshitz-Gilbert (LLG) equation of motion in a meanwhile standard way via the additional\nterm st J[ [ ]] a G= ´ ´ M Mp, where p denotes the spin polarization direction of electrons in the\ndc-current through the device and aJ is proportional to the current strength I. In our\nsimulations p was chosen to be opposite to the applied field direction Hext, because in real\npoint-contact experiments the magnetization dynamics is supposed to be driven by spin-\npolarized electrons reflected from the 'fixed' magnetic layer (of a FM/NM/FM trilayer system)\ntowards the 'free' one. The package was extended further in order to take into account non-\nlinear (magnetization rate dependent) damping as suggested in [14]; details of this implemen-\ntation will be presented elsewhere.\nIII. SIMULATION RESULTS AND DISCUSSION\nA. Magnetization dynamics for the linear Gilbert damping\nIn this section we present main features of the magnetization dynamics for the standard linear\nGilbert damping dis S (/)[ ( /)] M d dt l G= ×´M M , where the dissipation parameter l is cons-\ntant; in simulations performed here l = 0.02.\nDependence of the magnetization oscillation frequency f on the spin torque magnitude aJ is\nshown in Fig. 1a with open circles; typical snapshots of magnetization configurations during\nthese oscillations for various oscillation modes (see below) are shown in the same figure as\ngrey-scale maps of the mz(r)-projection (in-plane projection perpendicular to the applied field\ndirection).\nAlready for the linear damping the system demonstrates a fairly rich magnetization dynamics.\nOscillations start at the threshold value J, th3.55 (0.02) a » ± with the extended wave mode at\nthe frequency f \u0002 14.3 GHz. This frequency is much higher than the homogeneous FMR\nfrequency for this system (fFMR \u0002 8.42 GHz) because already at the threshold the wave with a\nlarge wave vector k ~ 1/Rc is exited due to the small radius Rc of the point contact. When the\ncurrent strength (aJ value) is increased, the oscillation amplitude rapidly growths (see Fig.\n1b), leading to the non-linear downward frequency shift as explained theoretically [10, 11, 20]\nand observed experimentally in several papers (see, e.g., papers of Kiselev et al. and\nKrivorotov et al. from [5]).\nWhen aJ exceeds the first critical value (1)\ncr4.45 a» , the first frequency jump occurs: transition\nfrom the extended wave mode (W) to the localized mode of the first type (L1) takes place.\n(first observations of this mode in the point-contact geometry was reported by us in [12, 16]).\nThe oscillation frequency drops below the fFMR-value - which is an immanent feature of a\nlocalized oscillation mode - and decreases linearly and very slowly from f(aJ = 4.5) \u0002 7.45\nGHz to f(aJ = 6.3) \u0002 7.23 GHz (see inset in Fig. 1a).\nDynamics of the transition from the W-mode to the L1-mode for (1)\nJ cr a a> is shown in Fig. 2b.\nHere the time-dependencies of the magnetization projection mx(t) (along the external field)\naveraged over the point contact area are plotted. It can be seen that after the spin-current is\nswitched on, for aJ values slightly larger than (1)\ncra the extended wave mode W is excited first\n(left inset on Fig. 2b). Then the amplitude of magnetization oscillations increases, until the\nmagnetization switching takes place, so that the magnetization begins to oscillate around the\ndirection opposite to the applied field. The frequency jump occurring by this switching can be\nseen very clearly from the comparison of the two insets on Fig. 2b.5At the second critical value (2)\ncr6.3 a» the next frequency jump accompanying the transition\nto the second type (L2) of the localized mode occurs. Typical dynamics of this transitions is\nshown in Fig. 2c. The most interesting feature of this dynamics is that the formation of the\nsecond localized mode L2 from the initial equilibrium magnetization state takes place via the\nintermediate formation of the first localized mode L1. For the current values not much higher\nthan (2)\ncra the first localized mode can still exist for times much larger than its oscillation\nperiod. We shall return to discussion of this metastability below.\nA snapshot of the spatial magnetization distribution within this second spatially localized\nmode L2 is shown as the grey-scale map of mz(r) in Fig. 1a. This mode was found by us\npreviously in the double-layer system [13], where it was the only type of the stable localized\nmode. It was shown that the core magnetization structure of this mode consists of two vortex-\nantivortex pairs. The frequency of the L2-mode f \u0002 4.6 GHz remains nearly constant when the\ncurrent strength is increased further. We are not aware of any analytical predictions\nconcerning this kind of a localized mode.\nIt is highly instructive to analyze dynamical transition between different mode types from the\n‘energetical' point of view.\nCorresponding time dependencies of the system energy for the transition W \u0003 L1 from the\nextended wave mode to the first localized mode are shown in Fig. 3. Here we display time\ndependencies of the energy differences DE = E(t) – E0 between the energy values E(t) and the\nenergy in the initial state E0. We plot these differences for the total system energy Etot and the\nstandard micromagnetic contributions to Etot, namely, the energy in the external field Eext (Fig.\n3b), exchange energy Eexch (Fig. 3c) and stray field energy Edem (Fig. 3d); we remind that the\nsmall magnetocrystalline anisotropy of Permalloy was neglected, so that the anisotropy\nenergy Ean = 0. All energies are evaluated for the magnetization configuration of the total\nsimulation area (i.e. not only for the point contact area !). Plots shown at Fig. 3b-d prove that\nby the transition W \u0003 L1 all partial energy contributions decrease, although both the oscil-\nlation amplitude and the magnetization gradient within the point contact area for the L1-mode\nare much larger than for the W-mode. However, strong localization of the magnetization\noscillations for the L1-mode compensates for these increments, because for the extended mode\nW the whole simulation area contributes to the total system energy.\nThe picture for the second transition L1 \u0003 L2 is qualitatively different (see Fig. 4), because\nhere both modes are localized. By this transition the exchange energy Eexch (Fig. 4c) increases\n- due to the steeper spatial variation of the magnetization within the point contact area for the\nmode L2 compared to L1. This exchange energy increase is compensated, first, by the decrease\nof the stray field (demagnetizing) energy Edem due the formation of a magnetic charge\n‘quadrupole’ by the two vortex-antivortex pairs characterising the L2-type. The stray field\nenergy of such a quadrupole is significantly lower than Edem of the approximately\nhomogeneous magnetization configuration of the L1-mode kernel. Second, the average mode\nenergy in the external field Eext also decreases, due to a much smaller magnitude of the\naverage magnetization avm\u0001\u0002 of the mode kernel for the L2-type. Again, the reason for this\nsmaller magnitude of avm\u0001\u0002 is a more complicated inhomogeneous structure of the L2- kernel.\nThe analysis of the spatial distribution of the oscillation power within the mode kernels and\nthe power emission of the localized modes is presented in Fig. 5. The power spectrum shown\nin these figures is computed by averaging the magnetization oscillation spectra obtained for\neach discretization cell over all cells (see [17] for details). The oscillation power maps\ndisplayed as insets for each spectral peak present the spatial in-plane distribution of the\noscillation power for the my(r)-component at corresponding frequencies.6In accordance with the snapshots shown in Fig. 1, the spatial power distribution of the magne-\ntization oscillations in the L1-kernel shown in Fig. 5a has the elliptical symmetry with respect\nto the point contact center. The power dependence P(r) on the distance from this center is\nnon-monotonous; we shall return to this observation later (see discussion below). Spatial\nmaps for spectral peaks above the basic FMR frequency give the pattern of the energy\nemission out of the point contact area for this mode. One can see that the angular distribution\nof this energy emission is highly anisotropic, whereby for different frequencies the energy is\nemitted in different directions. This radiation anisotropy should be explicitly taken into\naccount in experiments and technical applications where the synchronization of the\nnanocontact oscillators in the in-plane geometry is aimed. In addition, our results demonstrate\nthat numerical simulations where the spin torque effect in nanocontacts is simulated via an\nartificial 'localized magnetic field' (like those presented in [18]), lead to a qualitatively\nincorrect picture of the power emission out of the point contact area. Hence such oversimpli-\nfied simulations can by no means be used for the analysis of spin-current induced\nmagnetization excitations in such systems, not to mention the optimization of the nanocontact\noscillators synchronization.\nThe spatial structure of the oscillation power distribution for the L2-mode (see Fig. 5b) is\nhighly complicated already for the mode kernel (leftmost inset in Fig. 5b). The energy\nemission pattern is also more complicated than for the L1-mode (see middle and right insets in\nFig. 5b); in addition, the preferred direction of the energy emission changes with the current\nstrength aJ (results not shown). We also remind that the average oscillation power at the basic\nfrequency of this mode is much lower than the corresponding power for the L1-mode (see Fig.\n1b), which is also due to a more inhomogeneous magnetization configuration within the\nmode kernel (region under the point contact area).\nWe begin the discussion of our results with their comparison to known analytical theories.\nThe extended wave mode. The propagating wave mode W which in our simulations is excited\nfirst when the current is increased, corresponds to the linear mode studied by Slonczewski in\n[8]. The threshold current Ith for the excitation of this mode can be written for our purposes in\nthe form proposed in [11] as\n0\nth 0 2\nc() 11.86 ()DHI H\nR s\u0003 \u0004» +G \u0005 \u0006\u0005 \u0006\u0007 \b(1)\nwhere the common factor s depends on the magnetic layer thickness d, point contact area\n2\nc c S Rp= , material saturation magnetization MS and the spin polarization degree P of the\ncurrent as B S c /(2|| ) gP eMdS s m= ××. The first term on the right-hand side of (1) describes\nthe energy loss due to energy flow carried out of the point contact area by the extended\ncircular (elliptical) wave. This term is proportional to the spin wave dispersion D(H0), which\nfor the field-in-plane geometry has the form\n0 S\n0\nS 0 0 S2 2()\n( 4 )H M ADHM HH Mp g\np+= ×\n+(2)\nThe 'normal' energy dissipation within the point contact area is given by the second term on\nthe right-hand side of (1), which for the same geometry is 0 0 S () ( 2 ) H H M gl p G =× + . Taking\ninto account that the product of the coefficient s and the current strength I is related to our\nspin torque magnitude 2\nJ S c /(2|| ) a IP eM dS =×× ×× \u0001 [19] via S J I Ma s g= ×, it is easy to\nderive the analytical threshold value an\nJ, tha for the onset of the Slonczewski mode:7an 0 S\n,th 2\nS S c 0 0 S2 1.862 1\n( 4 )JH M AaM M R HH Mpl\np\t \n += × × × + \u000b \f+ \u000b \f \r \u000e(3)\nSubstituting all the values which have been used in our simulations (H0 = 1000 Oe, MS = 640\nG, Rc = 20 nm, A = 1.0\u000110–6 erg/cm, l = 0.02), we obtain an\n,th3.94Ja » . The good agreement of\nthis value with the threshold obtained in simulations num\nJ, th3.55 (0.02) a » ± can be viewed as\nan evidence for the good quality of either the simulation software or approximations used by\nthe derivation of (1); the latter involve mainly the neglect of the group velocity anisotropy in\nthe field-in-plane geometry.\nThe localized mode L1. This first type of the localized mode which existence was demonstra-\nted by us for the first time in [12, 16] corresponds most probably to the localized non-linear\n'bullet', which was predicted and thoroughly analyzed using the non-linear dynamics methods\nby Slavin et al. [11]. First of all, we point out that the frequencies of our mode (f(L1) \u0002 7.25 -\n7.45 GHz) are very close to those predicted in [11] for the same set of magnetic parameters\n(fbull \u0002 7.4 - 7.8 GHz). A very important feature also is that these frequencies are definitely\nbelow the homogeneous FMR frequency fFMR \u0002 8.42 GHz for the studied thin film.\nSecond, our mode L1 is also localized, as it is the bullet mode of Slavin et al. Comparison of\nthe power dependencies on the distance to the contact center P(r) for the propagating wave\nmode W and the L1-mode is shown in Fig. 6. It can be seen, that although the normalized os-\ncillation power of the L1-mode near the point contact center is larger than that of the W-mode,\nfor c rR\u0002 the oscillation power decays for the L1-mode much faster than for the W-mode.\nFrom the left inset to Fig. 6 it can be seen that the W-mode power decays as 1/r (we note in\npassing that due to the Gilbert dissipation, the decay law of the W-mode\n()\ndec ()(1/)exp(/ )WPr r rr-\u0003 contains in principle also an exponential factor; however,\nanalytical estimates give for the corresponding decay radius the value () 4\noscgr dec / 10Wr Tvl\u0003\u0003\nnm, so that this exponent can not be seen for the simulated area size used here). In contrast to\nthe W-mode, the localized bullet mode should decay exponentially with the decay radius\n1()\nc decLr R\u0003 [11]. Indeed, the exponential fit for the P(r)-dependence of the L1-mode (right inset\nin Fig. 6) results in the value 1()\ndec12(1)Lr= ± nm (we remind that Rc = 20 nm).\nHowever, there exist also important discrepancies between our simulation results and analyti-\ncal predictions of Slavin et al. First of all, in simulations the localized mode discussed above\nis excited after the propagating wave mode, whereas according to the analytical theory, the\nexcitation threshold of the 'bullet' mode should be much smaller than for the linear\n(Slonczewski) mode. We attribute this difference to the circumstance, that, first, we start from\nthe homogeneous magnetization state increasing the current value, and, second, we do not\ninclude thermal fluctuations. Hence the mode which is topologically the closest one to the\nhomogeneous state, i.e., the linear mode is excited first. Thus we suppose that if we would be\nable to perform simulations during a sufficiently long time including thermal fluctuations, the\nlocalized mode would be excited for aJ-values below the threshold for the W-mode.\nUnfortunately, such simulations are out of the time range for the state-of-the-art\nmicromagnetic codes, in particular, because thermal fluctuations reduce the accuracy of the\nnumerical integration method by at least one order of magnitude (thus requiring the\ncorresponding decrease of the time step for the prescribed accuracy). This effect is\npronounced especially strong in systems with small discretization cells, as it is the case for the\nproblem under study.8A strong support for the latter argument follows from the fact that the average system energy\nstrongly decrease by the dynamical transition W \u0003 L1 (see Fig. 3 and the discussion above).\nHence we argue that in presence of thermal fluctuations which allow to surmount the energy\nbarrier required to excite the localized mode this mode should be excited first (see discussion\nin [11] about the finite mode amplitude of the 'bullet' at its excitation threshold).\nAnother important difference between our simulations and analytical results of Slavin et al. is\nthe non-monotonous mode profile of the L1-mode immediately after this mode appears (see\nFig. 6), whereby analytical calculations predict the monotonous power decay of the 'bullet'\nmode with increasing distance to the contact center. Again, this discrepancy can be easily\nunderstood assuming that the localized mode observed in our simulations corresponds to the\n'bullet' mode far above its excitation threshold. Thus analytical prediction for the mode profile\ngiven in [11] for current near the excitation threshold, would be invalid for such large\ncurrents, where a more complicated mode profile should exist. We also point out that the non-\nmonotonous mode profile observed in our simulations is the necessary precursor for the\ntransition to the second type of the localized mode L2 with a highly complicated kernel mag-\nnetization configuration.\nComparison to the experimental data. Before we proceed with the comparison of our data\nwith experimental results, we would like to establish a relation between the current strength\nused in a real experiment and the value of the spin torque magnitude aJ employed in our\nsimulations. As mentioned above, in the simplest theoretical approximation the corresponding\nrelation in Gaussian units is 2\nJ S c /(2|| ) a IP eM dS =×× ×× \u0001 , where the only unknown parame-\nter is the current polarization degree P. By adopting, e.g., the value P =0.3 (see corresponding\nestimations in [6] and taking into account the relation between current units in Gaussian\nsystem and SI, we obtain that for the geometry using here aJ = 1 corresponds to I \u0002 2.6 mA.\nWe remind that by simulating the magnetization dynamics we have made here several\napproximations: neglected the interaction with the 'fixed' magnetic layer, the influence of the\nOersted field and thermal fluctuations. Nevertheless, comparing our data with the results of\nRippard et all [9] we can explain some important experimental findings. First of all, it is clear\nthat Rippard and co-workers observed the localized mode of the first type (L1-mode), as it was\nalso suggested in [11]. This conclusion is based, first, on the perfect agreement of frequencies\nbetween our L1-mode with f(L1) \u0002 7.3 GHz and the mode observed in [9] (for H0 = 1000 Oe\nthis frequency for the maximal microwave power was fexp \u0002 7.2 GHz, see Fig. 1 in [9]); we\nremind that our simulations do not contain any adjustable parameters. Additional strong\nindication for the L1-mode is, that the frequency of our simulated L1-mode decays linearly\nwith increasing the spin torque magnitude aJ over the wide range of aJ-values, in a qualitative\nagreement with the linear decrease of the oscillation frequency measured experimentally (see\ninset to Fig. 1b in [9]). We point out that the frequency of the propagating wave mode decays\nwith the current strength non-linearly, as it can be clearly seen from our Fig. 1a.\nAnother important experimental fact that we can explain in our simulations is the abrupt dis-\nappearance of the microwave signal observed in [9] when the current was increased beyond ~\n8.5 mA. Namely, according to our results, at the second critical current value (2)\ncra the L1-\nmode decays to the second type of the localized mode L2. For this L2-mode the oscillation\npower predicted by our simulations is more than 5 times smaller than for the L1-mode.\nOscillations with such a small power, additionally masked by thermal noise, could probably\nnot be detected in a real experiment.\nThe most important disagreement with simulations remains the absence of the propagating W-\nmode in the real experiment [9]. The reason why such a mode was not observed, could be the\nsame as discussed above when we compared our results with analytical predictions9concerning the 'bullet' mode [11]. Namely, if the excitation threshold for the localized mode is\nlower than that for the propagating wave mode, this mode could be experimentally excited\nfirst due to thermal fluctuations, which could assist in overcoming the energy barrier required\nto excite the localized mode. If this our statement is correct, it might be possible to observe\nthe propagating wave mode in the field-in-plane setup by performing experiments at low\ntemperature and taking special precautions against the Joule heating of the nanocontact area.\nB. Effects of a non-linear damping\nTaking into account that the constant damping coefficient l for the standard Gilbert damping\nconsidered above leads to the unphysical decrease of the dissipated power with the growing\namplitude of magnetization oscillations, Tiberkevich and Slavin [14] proposed to introduce a\nphenomenological dependence of this damping coefficient on the magnetization change rate\ndM/dt. As shown in [14], the first non-trivial term in such a dependence should have the form\nG 1 (1 )q ll x = + , where the quantity 2 2\nS (/)/(4 ) d dt M x pg =m includes the dependence of\ndamping on the magnetization change rate (m = M/MS). The value of the non-linear\ncoefficient q1 should be calculated from the microscopic theory, so in the present study we\nconsider it as a phenomenological quantity and explore the dependence of the system\nproperties on the value of q1.\nResults of our simulations of SPC-driven magnetization oscillations in the same point contact\nsetup as described above for various non-linear coefficients q1 are shown in Fig. 7 (weak non-\nlinearity q1 = 1) and Fig. 9 (strong non-linearity q1 = 10 and q1 = 20) in comparison to the\nlinear damping case discussed in the previous Section.\nThe influence of a small non-linearity (q1 = 1) on the magnetization dynamics in the extended\nwave mode W is weak: non-linear damping slightly reduces the oscillation power and\namplitude (see Fig. 7b), thus increasing - also slightly - the oscillation frequency when\ncompared to the linear case (Fig. 7a). The aJ-threshold where the W-mode loses its stability is\nalso getting slightly larger. However, the existence region of the localized modes changes in a\nqualitative way: immediately above the first critical value (1)\ncra the system exhibits a transition\nto a second type of the localized mode L2, and not to L1, as it was the case for the linear\ndamping. For still larger currents the system oscillation mode changes to L1, and by further\nincrease of aJ, the transition back to L2 takes place, so that the current region aJ(L1) where the\nlocalized mode of the 1st kind L1 exists, is limited both from below and from above by the\nregions where the second localized L2 mode determines the magnetization dynamics:\nmin max\nJ 1 J1 J 1 () () () a L aL a L < < . When the value of the non-linear coefficient q1 is getting larger,\nthe lower border of this interval min\nJ 1() a L increases, whereby the upper border max\nJ 1() a L dec-\nreases, so that for a sufficiently strong non-linearity q1 the first type of the localized mode\ndoes not exist anymore (see Fig. 9).\nFor large values of the non-linear parameter q1 also another qualitative effect has been obser-\nved: frequency jumps occur within one and the same mode type - extended wave mode W, as\nshown in Fig. 9a. The number and magnitude of these jumps are determined by the concrete\nvalue of q1: we have found, e.g., one jump within the W-mode for q1 = 10 and 2 such jumps\nfor q1 = 20). These jumps are, of course, accompanied by a corresponding (relatively small)\njumps in the wave vector magnitude of the spin wave emitted from the contact area and by\nsmall kinks on the power dependence P(aJ) (Fig. 9b).\nThe relatively narrow stability region of the first localized mode (which additionally narrows\nby increasing the non-linear parameter q1) results in a natural question whether this mode is\nstable (at least for T = 0) - or it is only dynamically metastable, having the life time larger10than the numerical simulation reach. Namely, every point on f(aJ)-dependencies in Fig. 1, 7\nand 9 is the result of a simulation run which corresponds to the physical time tmax = 25 ns, so\nthat if the L1-mode is unstable for all aJ-values, but for (2)\nJ cr a a< its lifetime is \n1 max Lt t> , then\nthis instability would not be discovered in our simulations. To clarify this question, we have\nplotted the life times of the L1-mode as the function of aJ for (2)\nJ cr a a> , i.e., where the L1-\nmode decays to L2 within the accessible simulation time. The obtained dependence \n1J()Lat is\nshown in Fig. 8 together with the fit \n1J J 0 () /La Aaabt = - , which assumes that the life time\nas the function of aJ diverges when aJ approaches some critical value a0. It can be seen that\nthis functional form fits the obtained \n1J()Lat very good. We consider this observation as an\nindirect evidence that the stability region for the first localized mode really exists, although\nthe limited number of data for \n1J()Lat in the immediate vicinity of a0 does not allow to draw\nthe final inference. Concluding this discussion, we would like to mention that, according to\nour preliminary studies [13] carried out on a double-layer system, the first localized mode\nmay lose its stability also due to the magnetodipolar interaction with the ‘fixed’ magnetic\nlayer in the point-contact geometry.\nIV. CONCLUSION\nIn this paper we have studied systematically (using micromagnetic simulations) magnetization\nexcitations driven by the spin-polarized current injected into a thin film via a point contact.\nWe could show that for an external field in the film plane such excitations may exist in form\nof qualitatively different modes. The possible mode types include a propagating wave mode\nW as suggested by Slonczewski [8] and two strongly localized modes. The kernel of the 1st\ntype of these localized modes (L1) has an approximately homogeneous magnetization state,\nwhereby the magnetization configuration of the 2nd type mode kernel (L2) is strongly inhomo-\ngeneous. The localized mode L1 corresponds most probably to the spin wave 'bullet'\ninvestigated analytically by Slavin et al. [11]. Comparison with the experimental results of\nRippard et al. [9] leads to the conclusion that the microwave resistance oscillations observed\nin [9] are caused by the localized mode of type L1, whereas the disappearance of oscillations\nby increasing current corresponds to the transition L1 \u0003 L2. Further, we have shown that the\npower emission by both types of the localized modes is strongly anisotropic, what is very\nimportant for potential applications of the point-contact devices including the synchronization\nof various point-contact oscillators.\nWe have also demonstrated, that a weak non-linearity of the damping included into the LLG-\nequation of motion shifts the current threshold values corresponding to the transition between\ndifferent mode types. Strong non-linearity causes frequency jumps already within a propaga-\nting wave mode and leads to a complete disappearance of the first localized mode L1.\nACKNOWLEDGEMENT. We greatly acknowledge many useful discussions with A. Slavin.\nFinancial support of the Deutsche Forschungsgemeinschaft (research grant BE 2464/4-1 in\nframes of the Priority Program SPP 1133 \"Ultrafast magnetization dynamics\") is also\nacknowledged.11References:\n[1] J.C. Slonczewski, J. Magn. Magn. Mat., 159 (1996) L1\n[2] L. Berger, Phys. Rev., B54 (1996) 9353\n[3] J.Z. Sun, J. Magn. Magn. Mat., 202 (1999) 157; M. Tsoi, A.G.M. Jansen, J. Bass, W.-C. Chiang, M.\nSeck, V. Tsoi, P. Wyder, Phys. Rev. Lett., 80 (1998) 4281\n[4] Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, B.I. Halperin, Rev. Mod. Phys., 77 (2005) 1375; M.D. Stiles,\nJ. Miltat, Spin Transfer Torque and Dynamics, in: Spin Dynamics in Confined Magnetic Structures III,\nSpringer Series Topics in Applied Physics 101, Springer-Verlag Berlin, Heidelberg 2006.\n[5] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, D. C. Ralph,\nNature, 425 (2003) 380; S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, A. G. F. Garcia, R. A.\nBuhrman, D. C. Ralph, Phys. Rev., B72 (2005) 064430; I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I.\nKiselev, D. C. Ralph, R. A. Buhrman, Science 307 (2005) 228; Q. Mistral, J.-V. Kim, T. Devolder, P. Crozat,\nC. Chappert, J. A. Katine, M. J. Carey, K. Ito, Appl. Phys. Lett., 88 (2006) 192507; Y. Acremann, J.P.\nStrachan, V. Chembrolu, S.D. Andrews, T. Tyliszczak, J.A. Katine, M. J. Carey, B.M. Clemens, H.C.\nSiegmann, J. Stöhr , Phys. Rev. Lett., 96 (2006) 217202; J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N.\nKrivorotov, R. A. Buhrman, D. C. Ralph, Phys. Rev. Lett., 96 (2006) 227601\n[6] X. Waintal et al, Phys. Rev., B62 (2000) 12317, M. D. Stiles, A. Zangwill, Phys. Rev., B66 (2002)\n014407; J.C. Slonczewski, J. Magn. Magn. Mat., 247 (2002) 324; J. Xiao, A. Zangwill, M. D. Stiles, Phys.\nRev., B70 (2004) 172405; L. Berger, Phys. Rev., B72 (2005) R-100402; J.-V. Kim, Phys. Rev., B73 (2006)\n174412;\n[7] J. Z. Sun, Phys. Rev., B62 (2000) 570; J. Miltat, G. Albuquerque, A. Thiaville, C. Vouille, J. Appl. Phys.,\n89 (2001) 6982, Z. Li, S. Zhang, Phys. Rev., B68 (2003) 024404; X. Zhu, J.-G. Zhu, R.M. White, J. Appl.\nPhys., 95 (2004) 6630; J.-G. Zhu, X. Zhu, IEEE Trans. Magn., 40 (2004) 182; K.J. Lee, A. Deac, O. Redon,\nJ.P. Nozieres, B. Dieny, Nature Materials, 3 (2004) 877; B. Montigny, J. Miltat, J. Appl. Phys., 97 (2005)\n10C708; J. Xiao, A. Zangwill, M. D. Stiles, Phys. Rev., B72 (2005) 014446\n[8] J.C. Slonczewski, J. Magn. Magn. Mat., 195 (1999) 261\n [9] W.H. Rippard, M.R. Pufall, S. Kaka, S.E. Russek, T.J. Silva, Phys. Rev. Lett., 92 (2004) 027201\n[10] A.N. Slavin, P. Kabos, IEEE Trans. Magn., MAG-41 (2005) 1264\n[11] A. Slavin, V. Tiberkevich, Phys. Rev. Lett., 95 (2005) 237201\n[12] D.V. Berkov, J. Magn. Magn. Mat., 300 (2006) 159-163\n[13] D.V. Berkov, N.L. Gorn, J. Appl. Phys., 99 (2006) 08Q701\n[14] V. Tiberkevich, A. Slavin, Phys. Rev., B75 (2007) 014440\n[15] D.V. Berkov, N.L. Gorn, MicroMagus - package for micromagnetic simulations,\nhttp:\\\\www.micromagus.de\n[16] D.V. Berkov, N.L.Gorn, invited talk 053-HP on the Joint European Magnetic Symposium, Dresden,\nGermany, Sept. 2004; D.V. Berkov, N.L.Gorn, invited talk 28TL-D-6 on the Moscow International\nSymposium on Magnetism, June 2005, Moscow, Russia\n[17] D.V. Berkov, N.L. Gorn, Phys. Rev., B72 (2005) 094401\n[18] S. Choi, S.-K. Kim, V.E. Demidov, S.O. Demokritov, Appl. Phys. Lett., 90 (2007) 083114\n[19] I. N. Krivorotov, D. V. Berkov, N. L. Gorn, N. C. Emley, J. C. Sankey, D. C. Ralph, R. A. Buhrman,\nsubm. to Phys. Rev. B, see also: cond-mat/0703458\n [20] S.M. Rezende, F.M. de Aguiar, A. Azevedo, Phys. Rev., B73 (2006) 094402\n \n \n 12 aJ\n0 4 5 6 7Power as \n0.00.10.20.30.40.5aJ\n0 4 5 6 7Frequency f(mz), GHz\n0481216\nW L1\n(a)\n(b)L2\n2D Graph 1\naJ\n4.5 5.0 5.5 6.0f, GHz\n7.27.37.47.5\nFig.1 Dependencies of the frequency f (a) and oscillation power P (b) on the spin torque magnitude aJ for\nthe linear damping case. Frequency and power jumps accompanying the transitions from the extended\nwave mode W to the localized mode L1 and from L1 to the second type of the localized mode L2 are\nclearly seen. Snapshots of the my-components (perpendicular to the film plane) for all three mode types\nare shown at the upper panel as grey-scale maps. The inset on the panel (a) demonstrates the perfectly\nlinear frequency dependence on the current strength for the L1 -mode.13aJ = 3.7\nt, ns\n0 2 4 6 8 10mx(t)\n0.80.91.0\naJ = 4.6\n0 2 4 6 8 10mx(t)\n-1.0-0.50.00.51.0\nt, ns\n0 2 6 8 10 12 14mx(t)\n-1.0-0.50.00.51.0t, ns\n8.0 8.2 8.4-0.8-0.40.0\nt, ns\n1.6 1.8 2.00.00.20.40.6\nt, ns\n2.0 2.2 2.4-0.8-0.4\nt, ns\n12.0 12.2 12.4-0.8-0.4(c)(b)(a)\nt, ns\naJ = 6.4\nFig. 2 Time dependencies of the x-magnetization component (along the external field) averaged across\nthe point contact area for three different values of aJ: (a) - magnetization oscillations for the stable exten-\nded wave mode W for aJ = 3.7, (b) - the dynamical switching process leading to the formation of the loca-\nlized mode L1, (c) - development of the 2nd type of the localized mode L2 from the L1-mode. Insets\ndemonstrate abrupt frequency jumps by the transitions between modes.14t, ns\n0 2 40 42 44mx(t)\n-1.0-0.50.00.51.0\n02 40 42 44DDDDE, 10-12 erg\n04080120(a)\n(b)\nt, ns\n02 40 42 4404080120\nt, ns\n02 40 42 44DDDDE, 10-12 erg\n04080120(c)\n(d)DDDDEexch\nDDDDEextDDDDEtot\nDDDDEdem\nW L1\nFig 3. Energy changes of the magnetization configuration of the total simulation area by the transition\n(see panel (a))from the extended wave mode W to the 1st localized mode L1: (b) - change of the energy\ndue to the external field, (c) - exchange energy and (d) - demagnetizing energy. Although both the oscil-\nlation amplitude and the magnetization gradient within the point contact area for the L1-mode are much\nlarger than for W-mode, all partial contribution to the total magnetic free energy for the L1-mode decrease\nby the transition W \u0003 L1, because magnetization oscillations in the L1-mode are strongly localized.15026 8 10 12DDDDE, 10-12 erg\n04080120t, ns\n0 1 6 7 8 9 10 11mx(t)\n-1.0-0.50.00.51.0(a)\n(b)\nt, ns\n026 8 10 12DDDDE, 10-12 erg\n04080120t, ns\n026 8 10 1204080120(c)\n(d)\nL1\nL2\nDDDDEextDDDDEtot DDDDEtot\nDDDDEtotDDDDEexch\nDDDDEdem\nFig. 4. The same as in Fig. 3 for the transition L1 \u0003 L2. Here both modes are localized, so that the exter-\nnal field and the demagnetizing energies decrease due to a more inhomogeneous magnetization configura-\ntion of the L2-mode in the mode localization area (resulting in the decrease of the total energy), whereby\nthe exchange energy increases for the same reason.16f, GHz0 5 10 15 20 25P(my), a.u.\n0.0000.0020.0040.0060.008\n150 x 150 nm2 400 x 400 nm2\nf, GHz0 5 10 15 20 25 30P(my), a.u.\n0.0000.0010.0020.0030.004\n150 x 150 nm2400 x 400 nm2L1\nL2(a)\n(b)\nFig. 5. Oscillation power spectra of the perpendicular magnetization component (my) averaged over the\nwhole simulation area for localized modes L1 (a) and L2 (b). Spatial maps of the oscillation power for\ncorresponding peaks give mode profiles (left maps) and power emission patterns out of the point contact\narea. Note that spatial power maps correspond to different physical sizes as written on the figure to ensure\nthe representation of oscillations with different localization degree with an adequate resolution.\nr, nm\n0 50 100 150 200 250P(r)\n0.00.20.40.60.81.01.2\npropagating mode\nlocalized mode L10204060801000.010.11\nr, nm 10 100P(r)\n0.11\nRc Rc\nFig. 6. Mode profiles (as the normalized magnetization oscillation power P(r), r being the distance to the\ncontact center) for the propagating wave mode W (triangles) and the first localized mode L1. Note the\nnon-monotonous profile of the localized mode. The left inset shows that for c(20 nm) rR=\u0002 the W-\nmode decays as 1/r (note double-log coordinates), the right inset - that the L1-mode decays exponentially\n(note the logarithmic ordinate scale) with the decay radius (1)\ndec12(1)Lr » ±nm of the same order of magni-\ntude as Rc. See text for details.17aJ\n0 4 5 6 7Power as \n0.00.10.20.30.40.5aJ\n0 4 5 6 7Frequency f(mz), GHz\n0481216\nq1 = 0\nq1 = 1\nW\nL2L1(a)\n(b)\nFig.7. Dependencies of the frequency (a) and oscillation power (b) on the spin torque magnitude aJ for the\nsmall non-linear damping q1 = 1.0 (triangles) compared with the linear damping case (circles). It can be\nseen that for the non-linear damping the aJ-region where the mode L1 exists is surrounded both from\nbelow and above by the existence regions of the L2-mode.\naJ\n5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2t t t t (L1), ns\n051015202530\nfit tttt = A/(aJ - a0)bbbb\nsimulated t t t t (L1)\nq1 = 1.0\nFig. 8. Life times of the first localized mode L1 for aJ-values where this mode decays to the L2- mode. The\nfit of the functional form shown in the figure (with a0 \u0002 5.6 and b \u0002 1.5) strongly indicates that this life\ntimes tends to infinity when aJ decreases, so that the region where the L1 -mode is stable really exists.18aJ\n0 4 5 6 7Frequency f(mz), GHz\n0481216\naJ\n0 4 5 6 7Power as \n0.00.10.20.30.40.5q1 = 0\nq1 = 10\nq1 = 20(a)\n(b)\nFig. 9. The same as in Fig. 7 for the large non-linear damping q1 = 10.0 (triangles) and q1 = 20.0 (squares)\ncompared also to the linear damping case (circles). For such large values of the non-linear parameter q1\nthe first type of the localized mode L1 does not exist anymore. Note also several frequency jumps within\nthe existence region of the W-mode (shown with arrows)." }, { "title": "1805.01815v2.Effective_damping_enhancement_in_noncollinear_spin_structures.pdf", "content": "Effective damping enhancement in noncollinear spin structures\nLevente Rózsa,1,∗Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1\n1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany\n(Dated: August 29, 2021)\nDamping mechanisms in magnetic systems determine the lifetime, diffusion and transport prop-\nerties of magnons, domain walls, magnetic vortices, and skyrmions. Based on the phenomenological\nLandau–Lifshitz–Gilbert equation, here the effective damping parameter in noncollinear magnetic\nsystems is determined describing the linewidth in resonance experiments or the decay parameter\nin time-resolved measurements. It is shown how the effective damping can be calculated from the\nelliptic polarization of magnons, arising due to the noncollinear spin arrangement. It is concluded\nthat the effective damping is larger than the Gilbert damping, and it may significantly differ be-\ntween excitation modes. Numerical results for the effective damping are presented for the localized\nmagnons in isolated skyrmions, with parameters based on the Pd/Fe/Ir(111) model-type system.\nSpinwaves(SW)ormagnonsaselementaryexcitations\nof magnetically ordered materials have attracted signifi-\ncant research attention lately. The field of magnonics[1]\nconcerns the creation, propagation and dissipation of\nSWs in nanostructured magnetic materials, where the\ndispersion relations can be adjusted by the system ge-\nometry. A possible alternative for engineering the prop-\nerties of magnons is offered by noncollinear (NC) spin\nstructures[2] instead of collinear ferro- (FM) or antifer-\nromagnets (AFM). SWs are envisaged to act as informa-\ntion carriers, where one can take advantage of their low\nwavelengths compared to electromagnetic waves possess-\ning similar frequencies[3]. Increasing the lifetime and the\nstability of magnons, primarily determined by the relax-\nation processes, is of crucial importance in such applica-\ntions.\nThe Landau–Lifshitz–Gilbert (LLG) equation[4, 5] is\ncommonly applied for the quasiclassical description of\nSWs, where relaxation is encapsulated in the dimen-\nsionless Gilbert damping (GD) parameter α. The life-\ntime of excitations can be identified with the resonance\nlinewidth in frequency-domain measurements such as fer-\nromagnetic resonance (FMR)[6], Brillouin light scatter-\ning (BLS)[7] or broadband microwave response[8], and\nwith the decay speed of the oscillations in time-resolved\n(TR) experiments including magneto-optical Kerr effect\nmicroscopy (TR-MOKE)[9] and scanning transmission x-\nray microscopy (TR-STXM)[10]. Since the linewidth is\nknowntobeproportionaltothefrequencyofthemagnon,\nmeasuring the ratio of these quantities is a widely ap-\nplied method for determining the GD in FMs[3, 6]. An\nadvantage of AFMs in magnonics applications[11, 12] is\ntheir significantly enhanced SW frequencies due to the\nexchange interactions, typically in the THz regime, com-\npared to FMs with GHz frequency excitations. However,\nit is known that the linewidth in AFM resonance is typ-\nically very wide because it scales with a larger effective\ndamping parameter αeffthan the GD α[13].\nThe tuning of the GD can be achieved in magnonic\ncrystals by combining materials with different values of\nα. It was demonstrated in Refs. [14–16] that this leadsto a strongly frequency- and band-dependent αeff, based\non the relative weights of the magnon wave functions in\nthe different materials.\nMagnetic vortices are two-dimensional NC spin config-\nurations in easy-plane FMs with an out-of-plane magne-\ntized core, constrained by nanostructuring them in dot-\norpillar-shapedmagneticsamples. Theexcitationmodes\nofvortices, particularlytheirtranslationalandgyrotropic\nmodes, havebeeninvestigatedusingcollective-coordinate\nmodels[17] based on the Thiele equation[18], linearized\nSW dynamics[19, 20], numerical simulations[21] and ex-\nperimental techniques[22–24]. It was demonstrated theo-\nretically in Ref. [21] that the rotational motion of a rigid\nvortex excited by spin-polarized current displays a larger\nαeffthan the GD; a similar result was obtained based on\ncalculating the energy dissipation[25]. However, due to\nthe unbounded size of vortices, the frequencies as well\nas the relaxation rates sensitively depend on the sample\npreparation, particularly because they are governed by\nthe magnetostatic dipolar interaction.\nIn magnetic skyrmions[26], the magnetic moment di-\nrections wrap the whole unit sphere. In contrast to vor-\ntices, isolated skyrmions need not be confined for stabi-\nlization, and are generally less susceptible to demagneti-\nzation effects[3, 27]. The SW excitations of the skyrmion\nlattice phase have been investigated theoretically[28–30]\nand subsequently measured in bulk systems[3, 8, 31]. It\nwas calculated recently[32] that the magnon resonances\nmeasured via electron scattering in the skyrmion lattice\nphase should broaden due to the NC structure. Calcula-\ntions predicted the presence of different localized modes\nconcentrated on the skyrmion for isolated skyrmions\non a collinear background magnetization[33–35] and for\nskyrmions in confined geometries[20, 36, 37]. From the\nexperimental side, the motion of magnetic bubbles in a\nnanodisk was investigated in Ref. [38], and it was pro-\nposed recently that the gyration frequencies measured in\nIr/Fe/Co/Pt multilayer films is characteristic of a dilute\narray of isolated skyrmions rather than a well-ordered\nskyrmion lattice[6]. However, the lifetime of magnons in\nskyrmionic systems based on the LLG equation is appar-arXiv:1805.01815v2 [cond-mat.mtrl-sci] 15 Oct 20182\nently less explored.\nIt is known that NC spin structures may influence the\nGD via emergent electromagnetic fields[29, 39, 40] or via\nthe modified electronic structure[41, 42]. Besides deter-\nmining the SW relaxation process, the GD also plays\na crucial role in the motion of domain walls[43–45] and\nskyrmions[46–48] driven by electric or thermal gradients,\nboth in the Thiele equation where the skyrmions are\nassumed to be rigid and when internal deformations of\nthe structure are considered. Finally, damping and de-\nformations are also closely connected to the switching\nmechanisms of superparamagnetic particles[49, 50] and\nvortices[51], as well as the lifetime of skyrmions[52–54].\nTheαeffin FMs depends on the sample geometry due\nto the shape anisotropy[13, 55, 56]. It was demonstrated\nin Ref. [56] that αeffis determined by a factor describing\nthe ellipticity of the magnon polarization caused by the\nshape anisotropy. Elliptic precession and GD were also\ninvestigated by considering the excitations of magnetic\nadatomsonanonmagneticsubstrate[57]. Thecalculation\nof the eigenmodes in NC systems, e.g. in Refs. [6, 20, 35],\nalso enables the evaluation of the ellipticity of magnons,\nbut this property apparently has not been connected to\nthe damping so far.\nAlthough different theoretical methods for calculating\nαeffhave been applied to various systems, a general de-\nscriptionapplicabletoallNCstructuresseemstobelack-\ning. Here it is demonstrated within a phenomenological\ndescription of the linearized LLG equation how magnons\nin NC spin structures relax with a higher effective damp-\ning parameter αeffthan the GD. A connection between\nαeffand the ellipticity of magnon polarization forced by\nthe NC spin arrangement is established. The method\nis illustrated by calculating the excitation frequencies\nof isolated skyrmions, considering experimentally deter-\nmined material parameters for the Pd/Fe/Ir(111) model\nsystem[58]. It is demonstrated that the different local-\nized modes display different effective damping parame-\nters, with the breathing mode possessing the highest one.\nThe LLG equation reads\n∂tS=−γ/primeS×Beff−αγ/primeS×/parenleftBig\nS×Beff/parenrightBig\n,(1)\nwithS=S(r)the unit-length vector field describing\nthe spin directions in the system, αthe GD and γ/prime=\n1\n1+α2ge\n2mthe modified gyromagnetic ratio (with gbeing\ntheg-factor of the electrons, ethe elementary charge and\nmthe electron mass). Equation (1) describes the time\nevolution of the spins governed by the effective magnetic\nfieldBeff=−1\nMδH\nδS, withHthe Hamiltonian or free\nenergy of the system in the continuum description and\nMthe saturation magnetization.\nThe spins will follow a damped precession relaxing\nto a local minimum S0ofH, given by the condition\nS0×Beff=0. Note that generally the Hamiltonian rep-\nresents a rugged landscape with several local energy min-\nima, corresponding to e.g. FM, spin spiral and skyrmionlattice phases, or single objects such as vortices or iso-\nlated skyrmions. The excitations can be determined by\nswitching to a local coordinate system[20, 34, 47] with\nthe spins along the zdirection in the local minimum,\n˜S0= (0,0,1), and expanding the Hamiltonian in the\nvariablesβ±=˜Sx±i˜Sy, introduced analogously to spin\nraising and lowering or bosonic creation and annihila-\ntion operators in the quantum mechanical description of\nmagnons[59–61]. The lowest-order approximation is the\nlinearized form of the LLG Eq. (1),\n∂tβ+=γ/prime\nM(i−α)/bracketleftbig\n(D0+Dnr)β++Daβ−/bracketrightbig\n,(2)\n∂tβ−=γ/prime\nM(−i−α)/bracketleftbig\nD†\naβ++ (D0−Dnr)β−/bracketrightbig\n.(3)\nFor details of the derivation see the Supplemental\nMaterial[62]. The term Dnrin Eqs. (2)-(3) is respon-\nsible for the nonreciprocity of the SW spectrum[2]. It\naccounts for the energy difference between magnons\npropagating in opposite directions in in-plane oriented\nultrathin FM films[63, 64] with Dzyaloshinsky–Moriya\ninteraction[65, 66] and the splitting between clockwise\nand counterclockwise modes of a single skyrmion[20].\nHere we will focus on the effects of the anomalous\nterm[34]Da, which couples Eqs. (2)-(3) together. Equa-\ntions (2)-(3) may be rewritten as eigenvalue equations by\nassuming the time dependence\nβ±(r,t) =e−iωktβ±\nk(r). (4)\nForα= 0, the spins will precess around their equilib-\nriumdirection ˜S0. Iftheequationsareuncoupled, the ˜Sx\nand ˜Syvariables describe circular polarization, similarly\nto the Larmor precession of a single spin in an exter-\nnal magnetic field. However, the spins are forced on an\nelliptic path due to the presence of the anomalous terms.\nThe effective damping parameter of mode kis defined\nas\nαk,eff=/vextendsingle/vextendsingle/vextendsingle/vextendsingleImωk\nReωk/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (5)\nwhich is the inverse of the figure of merit introduced in\nRef. [15]. Equation (5) expresses the fact that Im ωk,\nthe linewidth in resonance experiments or decay coeffi-\ncient in time-resolved measurements, is proportional to\nthe excitation frequency Re ωk.\nInterestingly, there is a simple analytic expression con-\nnectingαk,effto the elliptic polarization of the modes at\nα= 0. Forα/lessmuch1, the effective damping may be ex-\npressed as\nαk,eff\nα≈/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\ndr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\ndr=/integraltext\na2\nk(r) +b2\nk(r)dr/integraltext\n2ak(r)bk(r)dr,\n(6)3\n0.0 0.2 0.4 0.6 0.8 1.00246810\nFIG. 1. Effective damping parameter αk,effas a function of\ninverseaspectratio bk/akofthepolarizationellipse, assuming\nconstantakandbkfunctions in Eq. (6). Insets illustrate the\nprecession for different values of bk/ak.\nwhere the (0)superscript denotes that the eigenvectors\nβ±\nk(r)defined in Eq. (4) were calculated for α= 0, while\nak(r)andbk(r)denote the semimajor and semiminor\naxes of the ellipse the spin variables ˜Sx(r)and ˜Sy(r)\nare precessing on in mode k. Details of the derivation\nare given in the Supplemental Material[62]. Note that\nan analogous expression for the uniform precession mode\nin FMs was derived in Ref. [56]. The main conclusion\nfrom Eq. (6) is that αk,effwill depend on the considered\nSW mode and it is always at least as high as the GD\nα. Although Eq. (6) was obtained in the limit of low\nα, numerical calculations indicate that the αk,eff/αratio\ntends to increase for increasing values of α; see the Sup-\nplementalMaterial[62]foranexample. Theenhancement\nof the damping from Eq. (6) is shown in Fig. 1, with the\nspace-dependent ak(r)andbk(r)replaced by constants\nfor simplicity. It can be seen that for more distorted po-\nlarization ellipses the spins get closer to the equilibrium\ndirectionafterthesamenumberofprecessions, indicating\na faster relaxation.\nSince the appearance of the anomalous terms Dain\nEqs. (2)-(3) forces the spins to precess on an elliptic\npath, it expresses that the system is not axially sym-\nmetric around the local spin directions in the equilib-\nrium state denoted by S0. Such a symmetry breaking\nnaturally occurs in any NC spin structure, implying a\nmode-dependent enhancement of the effective damping\nparameter in NC systems even within the phenomeno-\nlogical description of the LLG equation. Note that the\nNC structure also influences the electronic properties of\nthe system, which can lead to a modification of the GD\nitself, see e.g. Ref. [42].\nIn order to illustrate the enhanced and mode-\ndependent αk,eff, we calculate the magnons in isolated\nchiralskyrmionsinatwo-dimensionalultrathinfilm. Thedensity of the Hamiltonian Hreads[67]\nh=/summationdisplay\nα=x,y,z/bracketleftBig\nA(∇Sα)2/bracketrightBig\n+K(Sz)2−MBSz\n+D(Sz∂xSx−Sx∂xSz+Sz∂ySy−Sy∂ySz),(7)\nwithAthe exchange stiffness, Dthe Dzyaloshinsky–\nMoriya interaction, Kthe anisotropy coefficient, and B\nthe external field.\nIn the following we will assume D>0andB≥0\nwithout the loss of generality, see the Supplemental\nMaterial[62] for discussion. Using cylindrical coordi-\nnates (r,ϕ)in real space and spherical coordinates S=\n(sin Θ cos Φ ,sin Θ sin Φ,cos Θ)in spin space, the equi-\nlibrium profile of the isolated skyrmion will correspond\nto the cylindrically symmetric configuration Θ0(r,ϕ) =\nΘ0(r)andΦ0(r,ϕ) =ϕ, the former satisfying\nA/parenleftbigg\n∂2\nrΘ0+1\nr∂rΘ0−1\nr2sin Θ 0cos Θ 0/parenrightbigg\n+D1\nrsin2Θ0\n+Ksin Θ 0cos Θ 0−1\n2MBsin Θ 0= 0 (8)\nwith the boundary conditions Θ0(0) =π,Θ0(∞) = 0.\nThe operators in Eqs. (2)-(3) take the form (cf.\nRefs. [34, 35, 47] and the Supplemental Material[62])\nD0=−2A/braceleftBigg\n∇2+1\n2/bracketleftbigg\n(∂rΘ0)2−1\nr2/parenleftbig\n3 cos2Θ0−1/parenrightbig\n(∂ϕΦ0)2/bracketrightbigg/bracerightBigg\n−D/parenleftbigg\n∂rΘ0+1\nr3 sin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n−K/parenleftbig\n3 cos2Θ0−1/parenrightbig\n+MBcos Θ 0, (9)\nDnr=/parenleftbigg\n4A1\nr2cos Θ 0∂ϕΦ0−2D1\nrsin Θ 0/parenrightbigg\n(−i∂ϕ), (10)\nDa=A/bracketleftbigg\n(∂rΘ0)2−1\nr2sin2Θ0(∂ϕΦ0)2/bracketrightbigg\n+D/parenleftbigg\n∂rΘ0−1\nrsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n+Ksin2Θ0.(11)\nEquation (11) demonstrates that the anomalous terms\nDaresponsible for the enhancement of the effective\ndamping can be attributed primarily to the NC arrange-\nment (∂rΘ0and∂ϕΦ0≡1) and secondarily to the\nspins becoming canted with respect to the global out-\nof-plane symmetry axis ( Θ0∈ {0,π}) of the system.\nTheDnrtermintroducesanonreciprocitybetweenmodes\nwith positive and negative values of the azimuthal quan-\ntum number (−i∂ϕ)→m, preferring clockwise rotat-\ning modes ( m < 0) over counterclockwise rotating ones\n(m > 0) following the sign convention of Refs. [20, 34].\nBecauseD0andDnrdepend onmbutDadoes not, it is\nexpected that the distortion of the SW polarization el-\nlipse and consequently the effective damping will be more\nenhanced for smaller values of |m|.\nThe different modes as a function of external field\nare shown in Fig. 2(a), for the material parameters de-\nscribing the Pd/Fe/Ir(111) system. The FMR mode at4\n0.7 0.8 0.9 1.0 1.1 1.20255075100125150175\n(a)\n0.7 0.8 0.9 1.0 1.1 1.21.01.52.02.53.0\n(b)\nFIG. 2. Localized magnons in the isolated skyrmion, with the\ninteraction parameters corresponding to the Pd/Fe/Ir(111)\nsystem[58]:A = 2.0pJ/m,D =−3.9mJ/m2,K =\n−2.5MJ/m3,M= 1.1MA/m. (a) Magnon frequencies f=\nω/2πforα= 0. Illustrations display the shapes of the excita-\ntion modes visualized on the triangular lattice of Fe magnetic\nmoments, with red and blue colors corresponding to positive\nand negative out-of-plane spin components, respectively. (b)\nEffective damping coefficients αm,eff, calculated from Eq. (6).\nωFMR =γ\nM(MB−2K), describing a collective in-phase\nprecession of the magnetization of the whole sample, sep-\narates the continuum and discrete parts of the spectrum,\nwith the localized excitations of the isolated skyrmion\nlocated below the FMR frequency[34, 35]. We found a\nsingle localized mode for each m∈{0,1,−2,−3,−4,−5}\nvalue, so in the following we will denote the excita-\ntion modes with the azimuthal quantum number. The\nm=−1mode corresponds to the translation of the\nskyrmion on the field-polarized background, which is a\nzero-frequency Goldstone mode of the system and not\nshown in the figure. The m=−2mode tends to zero\naroundB= 0.65T, indicating that isolated skyrmions\nbecome susceptible to elliptic deformations and subse-\nquently cannot be stabilized at lower field values[68].\nThe values of αm,effcalculated from Eq. (6) for the\ndifferent modes are summarized in Fig. 2(b). It is impor-\ntant to note that for a skyrmion stabilized at a selected\n0 20 40 60 80 100-0.04-0.020.000.020.040.06\n-0.03 0.00 0.03-0.030.000.03FIG. 3. Precession of a single spin in the skyrmion in the\nPd/Fe/Ir(111) system in the m= 0andm=−3modes at\nB= 0.75T, from numerical simulations performed at α=\n0.1. Inset shows the elliptic precession paths. From fitting\nthe oscillations with Eq. (4), we obtained |Reωm=0|/2π=\n39.22GHz,|Imωm=0|= 0.0608ps−1,αm=0,eff= 0.25and\n|Reωm=−3|/2π= 40.31GHz,|Imωm=−3|= 0.0276ps−1,\nαm=−3,eff= 0.11.\nfield value, the modes display widely different αm,effval-\nues, with the breathing mode m= 0being typically\ndamped twice as strongly as the FMR mode. The ef-\nfective damping tends to increase for lower field values,\nand decrease for increasing values of |m|, the latter prop-\nerty expected from the m-dependence of Eqs. (9)-(11)\nas discussed above. It is worth noting that the αm,eff\nparameters are not directly related to the skyrmion size.\nWealsoperformedthecalculationsfortheparametersde-\nscribing Ir|Co|Pt multilayers[69], and for the significantly\nlargerskyrmionsinthatsystemweobtainedconsiderably\nsmaller excitation frequencies, but quantitatively similar\neffective damping parameters; details are given in the\nSupplemental Material[62].\nThe different effective damping parameters could pos-\nsibly be determined experimentally by comparing the\nlinewidths of the different excitation modes at a selected\nfield value, or investigating the magnon decay over time.\nAn example for the latter case is shown in Fig. 3, dis-\nplaying the precession of a single spin in the skyrmion,\nobtained from the numerical solution of the LLG Eq. (1)\nwithα= 0.1. AtB= 0.75T, the frequencies of the\nm= 0breathing and m=−3triangular modes are close\nto each other (cf. Fig. 2), but the former decays much\nfaster. Because in the breathing mode the spin is follow-\ning a significantly more distorted elliptic path (inset of\nFig. 3) than in the triangular mode, the different effective\ndamping is also indicated by Eq. (6).\nIn summary, it was demonstrated within the phe-\nnomenological description of the LLG equation that the\neffective damping parameter αeffdepends on the consid-\nered magnon mode. The αeffassumes larger values if5\nthe polarization ellipse is strongly distorted as expressed\nby Eq. (6). Since NC magnetic structures provide an\nanisotropic environment for the spins, leading to a dis-\ntortion of the precession path, they provide a natural\nchoice for realizing different αeffvalues within a single\nsystem. The results of the theory were demonstrated for\nisolated skyrmions with material parameters describing\nthe Pd/Fe/Ir(111) system. The results presented here\nmay stimulate further experimental or theoretical work\non the effective damping in skyrmions, vortices, domain\nwalls or spin spirals.\nThe authors would like to thank U. Atxitia and G.\nMeier for fruitful discussions. Financial support by the\nAlexander von Humboldt Foundation, by the Deutsche\nForschungsgemeinschaft via SFB 668, by the European\nUnion via the Horizon 2020 research and innovation pro-\ngram under Grant Agreement No. 665095 (MAGicSky),\nand by the National Research, Development and Inno-\nvation Office of Hungary under Project No. K115575 is\ngratefully acknowledged.\n∗rozsa.levente@physnet.uni-hamburg.de\n[1] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. Phys. 43, 264001 (2010).\n[2] M.Garst, J.Waizner, andD.Grundler, J.Phys.D:Appl.\nPhys. 50, 293002 (2017).\n[3] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I.\nStasinopoulos, H. Berger, C. Pfleiderer, and D. Grundler,\nNat. Mater. 14, 478 (2015).\n[4] L. 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Mater.\n138, 255 (1994).\n[68] A. Bogdanov and A. Hubert, Phys. Stat. Sol. B 186, 527\n(1994).\n[69] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sam-\npaio, C. A. F. Vaz, N. Van Horne, K. Bouzehouane, K.\nGarcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J.-M.\nGeorge, M. Weigand, J. Raabe, V. Cros, and A. Fert,\nNat. Nanotechnol. 11, 444 (2016).\n[70] F. Romá, L. F. Cugliandolo, and G. S. Lozano, Phys.\nRev. E 90, 023203 (2014).\n[71] A. O. Leonov, T. L. Monchesky, N. Romming, A. Ku-betzka, A. N. Bogdanov, and R. Wiesendanger, New J.\nPhys. 18, 065003 (2016).\n[72] J. Hagemeister, A. Siemens, L. Rózsa, E. Y. Vedme-\ndenko, and R. Wiesendanger, Phys. Rev. B 97, 174436\n(2018).Supplemental Material to\nEffective damping enhancement in noncollinear spin structures\nLevente Rózsa,1,∗Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1\n1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany\n(Dated: August 29, 2021)\nIn the Supplemental Material the derivation of the linearized equations of motion and the effective\ndamping parameter are discussed. Details of the numerical determination of the magnon modes in\nthe continuum model and in atomistic spin dynamics simulations are also given.\nS.I. LINEARIZED\nLANDAU–LIFSHITZ–GILBERT EQUATION\nHere we will derive the linearized form of the Landau–\nLifshitz–Gilbert equation given in Eqs. (2)-(3) of the\nmaintextanddiscussthepropertiesofthesolutions. The\ncalculation is similar to the undamped case, discussed in\ndetail in e.g. Refs. [1–3]. Given a spin configuration sat-\nisfying the equilibrium condition\nS0×Beff=0, (S.1)\nthe local coordinate system with ˜S0= (0,0,1)may be\nintroduced, andtheHamiltonianbeexpandedinthevari-\nables ˜Sxand˜Sy. Thelineartermmustdisappearbecause\nthe expansion is carried out around an equilibrium state.\nThe lowest-order nontrivial term is quadratic in the vari-\nables and will be designated as the spin wave Hamilto-\nnian,\nHSW=/integraldisplay\nhSWdr, (S.2)\nhSW=1\n2/bracketleftbig˜Sx˜Sy/bracketrightbig/bracketleftbiggA1A2\nA†\n2A3/bracketrightbigg/bracketleftbigg˜Sx\n˜Sy/bracketrightbigg\n=1\n2/parenleftBig\n˜S⊥/parenrightBigT\nHSW˜S⊥. (S.3)\nThe operator HSWis self-adjoint for arbitrary equi-\nlibrium states. Here we will only consider cases where\nthe equilibrium state is a local energy minimum, mean-\ning thatHSW≥0; the magnon spectrum will only be\nwell-defined in this case. Since hSWis obtained as an\nexpansion of a real-valued energy density around the\nequilibrium state, and the spin variables are also real-\nvalued, fromtheconjugateofEq.(S.3)onegets A1=A∗\n1,\nA2=A∗\n2, andA3=A∗\n3.\nThe form of the Landau–Lifshitz–Gilbert Eq. (1) in\nthe main text may be rewritten in the local coordinates\nby simply replacing Sby˜S0everywhere, including the\ndefinitionoftheeffectivefield Beff. TheharmonicHamil-\ntonianHSWin Eq. (S.2) leads to the linearized equation\nof motion\n∂t˜S⊥=γ/prime\nM(−iσy−α)HSW˜S⊥,(S.4)\n∗rozsa.levente@physnet.uni-hamburg.dewithσy=/bracketleftbigg\n0−i\ni0/bracketrightbigg\nthe Pauli matrix.\nBy replacing ˜S⊥(r,t)→˜S⊥\nk(r)e−iωktas usual, for\nα= 0the eigenvalue equation\nωk˜S⊥\nk=γ\nMσyHSW˜S⊥\nk (S.5)\nis obtained. If HSWhas a strictly positive spectrum,\nthenH−1\n2\nSWexists, and σyHSWhas the same eigenvalues\nasH1\n2\nSWσyH1\n2\nSW. Since the latter is a self-adjoint ma-\ntrix with respect to the standard scalar product on the\nHilbert space, it has a real spectrum, consequently all ωk\neigenvalues are real. Note that the zero modes of HSW,\nwhich commonly occur in the form of Goldstone modes\ndue to the ground state breaking a continuous symme-\ntry of the Hamiltonian, have to be treated separately.\nFinally, we mention that if the spin wave expansion is\nperformed around an equilibrium state which is not a\nlocal energy minimum, the ωkeigenvalues may become\nimaginary, meaning that the linearized Landau–Lifshitz–\nGilbert equation will describe a divergence from the un-\nstable equilibrium state instead of a precession around\nit.\nEquations (2)-(3) in the main text may be obtained\nby introducing the variables β±=˜Sx±i˜Syas described\nthere. The connection between HSWand the operators\nD0,Dnr, andDais given by\nD0=1\n2(A1+A3), (S.6)\nDnr=1\n2i/parenleftBig\nA†\n2−A2/parenrightBig\n, (S.7)\nDa=1\n2/bracketleftBig\nA1−A3+i/parenleftBig\nA†\n2+A2/parenrightBig/bracketrightBig\n.(S.8)\nAn important symmetry property of Eqs. (2)-(3) in\nthe main text is that if (β+,β−) =/parenleftbig\nβ+\nke−iωkt,β−\nke−iωkt/parenrightbig\nis an eigenmode of the equations, then (β+,β−) =/parenleftBig/parenleftbig\nβ−\nk/parenrightbig∗eiω∗\nkt,/parenleftbig\nβ+\nk/parenrightbig∗eiω∗\nkt/parenrightBig\nis another solution. Following\nRefs. [1, 3], this can be attributed to the particle-hole\nsymmetry of the Hamiltonian, which also holds in the\npresence of the damping term. From these two solutions\nmentioned above, the real-valued time evolution of the\nvariables ˜Sx,˜Symay be expressed as\n˜Sx\nk=eImωktcos (ϕ+,k−Reωkt)/vextendsingle/vextendsingleβ+\nk+β−\nk/vextendsingle/vextendsingle,(S.9)\n˜Sy\nk=eImωktsin (ϕ−,k−Reωkt)/vextendsingle/vextendsingleβ+\nk−β−\nk/vextendsingle/vextendsingle,(S.10)arXiv:1805.01815v2 [cond-mat.mtrl-sci] 15 Oct 20182\nwithϕ±,k= arg/parenleftbig\nβ+\nk±β−\nk/parenrightbig\n. As mentioned above, the\nImωkterms are zero in the absence of damping close to\na local energy minimum, and Im ωk<0is implied by\nthe fact that the Landau–Lifshitz–Gilbert equation de-\nscribes energy dissipation, which in the linearized case\ncorresponds to relaxation towards the local energy min-\nimum. In the absence of damping, the spins will precess\non an ellipse defined by the equation\n/parenleftBig\n˜Sx\nk/parenrightBig2\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk+β−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ncos2(ϕ+,k−ϕ−,k)\n+2˜Sx\nk˜Sy\nksin (ϕ+,k−ϕ−,k)/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk−β−(0)\nk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk+β−(0)\nk/vextendsingle/vextendsingle/vextendsinglecos2(ϕ+,k−ϕ−,k)\n+/parenleftBig\n˜Sy\nk/parenrightBig2\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk−β−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ncos2(ϕ+,k−ϕ−,k)= 1,(S.11)\nwhere the superscript (0)indicatesα= 0. The semima-\njor and semiminor axes of the ellipse akandbkmay be\nexpressed from Eq. (S.11) as\nakbk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (S.12)\na2\nk+b2\nk= 2/parenleftbigg/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\n.(S.13)\nNote thatβ+\nkandβ−\nk, consequently the parameters of\nthe precessional ellipse akandbk, are functions of the\nspatial position r.\nS.II. CALCULATION OF THE EFFECTIVE\nDAMPING PARAMETER FROM\nPERTURBATION THEORY\nHere we derive the expression for the effective damping\nparameter αeffgiven in Eq. (6) of the main text. By\nintroducingβk=/parenleftbig\nβ+\nk,−β−\nk/parenrightbig\n,\nD=/bracketleftbiggD0+Dnr−Da\n−D†\naD0−Dnr/bracketrightbigg\n,(S.14)\nand using the Pauli matrix σz=/bracketleftbigg\n1 0\n0−1/bracketrightbigg\n, Eqs. (2)-(3)\nin the main text may be rewritten as\n−ωkσzβk=γ/prime\nM(D+iασzD)βk(S.15)\nin the frequency domain. Following standard perturba-\ntion theory, we expand the eigenvalues ωkand the eigen-\nvectorsβkin the parameter α/lessmuch1. For the zeroth-order\nterms one gets\n−ω(0)\nkσzβ(0)\nk=γ\nMDβ(0)\nk, (S.16)\n0.0 0.1 0.2 0.3 0.4 0.50.00.51.01.52.02.5FIG. S1. Effective damping coefficients αm,effof the isolated\nskyrmion in the Pd/Fe/Ir(111) system at B= 1T, calcu-\nlated from the numerical solution of the linearized Landau–\nLifshitz–Gilbert equation (S.15), as a function of the Gilbert\ndamping parameter α.\nwith realω(0)\nkeigenvalues as discussed in Sec. S.I. The\nfirst-order terms read\n−ω(0)\nk/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(1)\nk/angbracketrightBig\n−ω(1)\nk/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(0)\nk/angbracketrightBig\n=γ\nM/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleD/vextendsingle/vextendsingle/vextendsingleβ(1)\nk/angbracketrightBig\n+iαγ\nM/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσzD/vextendsingle/vextendsingle/vextendsingleβ(0)\nk/angbracketrightBig\n,\n(S.17)\nafter taking the scalar product with β(0)\nk. The first terms\non both sides cancel by letting Dact to the left, then\nusing Eq. (S.16) and the fact that the ω(0)\nkare real. By\napplying Eq. (S.16) to the remaining term on the right-\nhand side one obtains\nω(1)\nk=−iαω(0)\nk/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndr,(S.18)\nby writing in the definition of the scalar product. By\nusing the definition αk,eff=|Imωk/Reωk|≈/vextendsingle/vextendsingle/vextendsingleω(1)\nk/ω(0)\nk/vextendsingle/vextendsingle/vextendsingle\nand substituting Eqs. (S.12)-(S.13) into Eq. (S.18), one\narrives at Eq. (6) in the main text as long as/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndoes not change sign under the integral.\nIt is worthwhile to investigate for which values of α\ndoes first-order perturbation theory give a good estimate\nforαk,effcalculated from the exact solution of the lin-\nearized equations of motion, Eq. (S.15). In the materials\nwhere the excitations of isolated skyrmions or skyrmion\nlattices were investigated, significantly different values of\nαhave been found. For example, intrinsic Gilbert damp-\ning parameters of α= 0.02-0.04were determined experi-\nmentallyforbulkchiralmagnetsMnSiandCu 2OSeO 3[4],\nα= 0.28was deduced for FeGe[5], and a total damp-\ning ofαtot= 0.105was obtained for Ir/Fe/Co/Pt mag-\nnetic multilayers[6], where the latter value also includes3\nvarious effects beyond the Landau–Lifshitz–Gilbert de-\nscription. Figure S1 displays the dependence of αm,eff\nonαfor the eigenmodes of the isolated skyrmion in the\nPd/Fe/Ir(111) system, shown in Fig. 2 of the main text.\nMost of the modes show a linear correspondence between\nthe two quantities with different slopes in the displayed\nparameter range, in agreement with Eq. (6) in the main\ntext. For the breathing mode m= 0the convex shape\nof the curve indicates that the effective damping param-\neter becomes relatively even larger than the perturbative\nexpression Eq. (6) as αis increased.\nS.III. EIGENMODES OF THE ISOLATED\nSKYRMION\nHere we discuss the derivation of the skyrmion profile\nEq. (8) and the operators in Eqs. (9)-(11) of the main\ntext. The energy density Eq. (7) in polar coordinates\nreads\nh=A/bracketleftbigg\n(∂rΘ)2+ sin2Θ (∂rΦ)2+1\nr2(∂ϕΘ)2\n+1\nr2sin2Θ (∂ϕΦ)2/bracketrightbigg\n+D/bracketleftbigg\ncos (ϕ−Φ)∂rΘ\n−1\nrsin (ϕ−Φ)∂ϕΘ + sin Θ cos Θ sin ( ϕ−Φ)∂rΦ\n+1\nrsin Θ cos Θ cos ( ϕ−Φ)∂ϕΦ/bracketrightbigg\n+Kcos2Θ−MBcos Θ.\n(S.19)\nThe Landau–Lifshitz–Gilbert Eq. (1) may be rewritten\nas\nsin Θ∂tΘ =γ/primeBΦ+αγ/primesin ΘBΘ,(S.20)\nsin Θ∂tΦ =−γ/primeBΘ+αγ/prime1\nsin ΘBΦ,(S.21)\nwith\nBχ=−1\nMδH\nδχ\n=−1\nM/bracketleftbigg\n−1\nr∂r/parenleftbigg\nr∂h\n∂(∂rχ)/parenrightbigg\n−∂ϕ∂h\n∂(∂ϕχ)+∂h\n∂χ/bracketrightbigg\n,\n(S.22)\nwhereχstands for ΘorΦ. Note that in this form it is\ncommon to redefine BΦto include the 1/sin Θfactor in\nEq. (S.21)[7]. The first variations of Hfrom Eq. (S.19)may be expressed as\nδH\nδΘ=−2A/braceleftbigg\n∇2Θ−sin Θ cos Θ/bracketleftbigg\n(∂rΦ)2+1\nr2(∂ϕΦ)2/bracketrightbigg/bracerightbigg\n−2Ksin Θ cos Θ +MBsin Θ\n−2Dsin2Θ/bracketleftbigg\nsin (ϕ−Φ)∂rΦ + cos (ϕ−Φ)1\nr∂ϕΦ/bracketrightbigg\n,\n(S.23)\nδH\nδΦ=−2A/braceleftbigg\nsin2Θ∇2Φ + sin 2Θ/bracketleftbigg\n∂rΘ∂rΦ +1\nr2∂ϕΘ∂ϕΦ/bracketrightbigg/bracerightbigg\n+ 2Dsin2Θ/bracketleftbigg\nsin (ϕ−Φ)∂rΘ + cos (ϕ−Φ)1\nr∂ϕΘ/bracketrightbigg\n,\n(S.24)\nTheequilibriumconditionEq.(8)inthemaintextmay\nbe obtained by setting ∂tΘ =∂tΦ = 0in Eqs. (S.20)-\n(S.21) and assuming cylindrical symmetry, Θ0(r,ϕ) =\nΘ0(r)and Φ0(r,ϕ) =ϕ. In the main text D>0\nandB≥0were assumed. Choosing D<0switches\nthe helicity of the structure to Φ0=ϕ+π, in which\ncaseDshould be replaced by |D|in Eq. (8). For the\nbackground magnetization pointing in the opposite di-\nrectionB≤0, one obtains the time-reversed solutions\nwith Θ0→π−Θ0,Φ0→Φ0+π,B→−B. Time rever-\nsal also reverses clockwise and counterclockwise rotating\neigenmodes; however, the above transformations do not\ninfluence the magnitudes of the excitation frequencies.\nFinally, we note that the frequencies remain unchanged\neven if the form of the Dzyaloshinsky–Moriya interaction\nin Eq. (S.19), describing Néel-type skyrmions common in\nultrathin films and multilayers, is replaced by an expres-\nsion that prefers Bloch-type skyrmions occurring in bulk\nhelimagnets – see Ref. [3] for details.\nFordeterminingthelinearizedequationsofmotion,one\ncan proceed by switching to the local coordinate system\nas discussed in Sec. S.I and Refs. [1, 3]. Alternatively,\nthey can also directly be derived from Eqs. (S.20)-(S.21)\nby introducing Θ = Θ 0+˜Sx,Φ = Φ 0+1\nsin Θ 0˜Syand\nexpanding around the skyrmion profile from Eq. (8) up\nto first order in ˜Sx,˜Sy– see also Ref. [2]. The operators\nin Eq. (S.3) read\nA1=−2A/parenleftbigg\n∇2−1\nr2cos 2Θ 0(∂ϕΦ0)2/parenrightbigg\n−2D1\nrsin 2Θ 0∂ϕΦ0−2Kcos 2Θ 0+MBcos Θ 0,\n(S.25)\nA2=4A1\nr2cos Θ 0∂ϕΦ0∂ϕ−2D1\nrsin Θ 0∂ϕ,(S.26)\nA3=−2A/braceleftbigg\n∇2+/bracketleftbigg\n(∂rΘ0)2−1\nr2cos2Θ0(∂ϕΦ0)2/bracketrightbigg/bracerightbigg\n−2D/parenleftbigg\n∂rΘ0+1\nrsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n−2Kcos2Θ0+MBcos Θ 0, (S.27)4\nwhich leads directly to Eqs. (9)-(11) in the main text via\nEqs. (S.6)-(S.8).\nThe excitation frequencies of the ferromagnetic state\nmay be determined by setting Θ0≡0in Eqs. (9)-(11) in\nthe main text. In this case, the eigenvalues and eigenvec-\ntors can be calculated analytically[1],\nωk,m=γ/prime\nM(1−iα)/bracketleftbig\n2Ak2−2K+MB/bracketrightbig\n,(S.28)\n/parenleftBig\nβ+\nk,m(r),β−\nk,m(r)/parenrightBig\n= (0,Jm−1(kr)),(S.29)\nwithJm−1theBesselfunction ofthefirstkind, appearing\ndue to the solutions being regular at the origin. Equa-\ntion (S.28) demonstrates that the lowest-frequency exci-\ntation of the background is the ferromagnetic resonance\nfrequencyωFMR =γ\nM(MB−2K)atα= 0. Since the\nanomalous term Dadisappears in the out-of-plane mag-\nnetized ferromagnetic state, all spin waves will be circu-\nlarly polarized, see Eq. (S.29), and the effective damping\nparameterwillalwayscoincidewiththeGilbertdamping.\nRegarding the excitations of the isolated skyrmion, for\nα= 0the linearized equations of motion in Eq. (S.15)\nare real-valued; consequently, β±\nk,m(r)can be chosen to\nbe real-valued. In this case Eqs. (S.9)-(S.10) take the\nform\n˜Sx\nk,m= cos (mϕ−ωk,mt)/parenleftBig\nβ+\nk,m(r) +β−\nk,m(r)/parenrightBig\n,(S.30)\n˜Sy\nk,m= sin (mϕ−ωk,mt)/parenleftBig\nβ+\nk,m(r)−β−\nk,m(r)/parenrightBig\n.(S.31)\nThis means that modes with ωk,m>0form> 0will\nrotate counterclockwise, that is, the contours with con-\nstant ˜Sx\nk,mand ˜Sy\nk,mwill move towards higher values of\nϕastis increased, while the modes with ωk,m>0for\nm < 0will rotate clockwise. Modes with m= 0corre-\nspond to breathing excitations. This sign convention for\nmwas used when designating the localized modes of the\nisolated skyrmion in the main text, and the kindex was\ndropped since only a single mode could be observed be-\nlow the ferromagnetic resonance frequency for each value\nofm.\nS.IV. NUMERICAL SOLUTION OF THE\nEIGENVALUE EQUATIONS\nThe linearized Landau–Lifshitz–Gilbert equation for\nthe isolated skyrmion, Eqs. (2)-(3) with the operators\nEqs.(9)-(11)inthemaintext, weresolvednumericallyby\na finite-difference method. First the equilibrium profile\nwas determined from Eq. (8) using the shooting method\nfor an initial approximation, then obtaining the solution\non a finer grid via finite differences. For the calculationswe used dimensionless parameters (cf. Ref. [8]),\nAdl= 1, (S.32)\nDdl= 1, (S.33)\nKdl=KA\nD2, (S.34)\n(MB)dl=MBA\nD2, (S.35)\nrdl=|D|\nAr, (S.36)\nωdl=MA\nγD2ω. (S.37)\nThe equations were solved in a finite interval for\nrdl∈[0,R], with the boundary conditions Θ0(0) =\nπ,Θ0(R) = 0. For the results presented in Fig. 2 in the\nmain text the value of R= 30was used. It was confirmed\nbymodifying Rthattheskyrmionshapeandthefrequen-\ncies of the localized modes were not significantly affected\nby the boundary conditions. However, the frequencies of\nthe modes above the ferromagnetic resonance frequency\nωFMR =γ\nM(MB−2K)did change as a function of\nR, since these modes are extended over the ferromag-\nnetic background – see Eqs. (S.28)-(S.29). Furthermore,\nin the infinitely extended system the equations of mo-\ntion include a Goldstone mode with/parenleftbig\nβ+\nm=−1,β−\nm=−1/parenrightbig\n=/parenleftbig\n−1\nrsin Θ 0−∂rΘ0,1\nrsin Θ 0−∂rΘ0/parenrightbig\n, corresponding to\nthe translation of the skyrmion on the collinear\nbackground[1]. This mode obtains a finite frequency in\nthe numerical calculations due to the finite value of R\nand describes a slow clockwise gyration of the skyrmion.\nHowever, this frequency is not shown in Fig. 3 of the\nmain text because it is only created by boundary effects.\nIn order to investigate the dependence of the effective\ndamping on the dimensionless parameters, we also per-\nformed the calculations for the parameters describing the\nIr|Co|Pt multilayer system[9]. The results are summa-\nrized in Fig. S2. The Ir|Co|Pt system has a larger di-\nmensionless anisotropy value ( −KIr|Co|Pt\ndl = 0.40) than\nthe Pd/Fe/Ir(111) system ( −KPd/Fe/Ir(111)\ndl = 0.33). Al-\nthough the same localized modes are found in both cases,\nthe frequencies belonging to the m= 0,1,−3,−4,−5\nmodes in Fig. S2 are relatively smaller than in Fig. 2\ncompared to the ferromagnetic resonance frequency at\nthe elliptic instability field where ωm=−2= 0. This\nagrees with the two limiting cases discussed in the lit-\nerature: it was shown in Ref. [1] that for Kdl= 0the\nm= 1,−4,−5modes are still above the ferromagnetic\nresonance frequency at the elliptic instability field, while\nin Ref. [2] it was investigated that all modes become soft\nwithfrequenciesgoingtozeroat (MB)dl= 0inthepoint\n−Kdl=π2\n16≈0.62,belowwhichaspinspiralgroundstate\nis formed in the system. Figure S2(b) demonstrates that\nthe effective damping parameters αm,effare higher at the\nellipticinstabilityfieldinIr|Co|PtthaninPd/Fe/Ir(111),\nshowing an opposite trend compared to the frequencies.\nRegarding the physical units, the stronger exchange\nstiffness combined with the weaker Dzyaloshinsky–5\n0.03 0.04 0.05 0.06 0.07 0.080246810\n(a)\n0.03 0.04 0.05 0.06 0.07 0.081.01.52.02.53.03.5\n(b)\nFIG. S2. Localized magnons in the isolated skyrmion, with\nthe interaction parameters corresponding to the Ir|Co|Pt\nmultilayer system from Ref. [9]: A= 10.0pJ/m,D=\n1.9mJ/m2,K=−0.143MJ/m3,M = 0.96MA/m. The\nanisotropy reflects an effective value including the dipolar in-\nteractions as a demagnetizing term, −K =−K 0−1\n2µ0M2\nwithK0=−0.717MJ/m3. (a) Magnon frequencies f=ω/2π\nforα= 0. Illustrations display the shapes of the excitation\nmodes visualized as the contour plot of the out-of-plane spin\ncomponentsona 1×1nm2grid,withredandbluecolorscorre-\nsponding to positive and negative Szvalues, respectively. (b)\nEffective damping coefficients αm,eff, calculated from Eq. (6)\nin the main text.Moriya interaction and anisotropy in the multilayer sys-\ntem leads to larger skyrmions stabilized at lower field val-\nues and displaying lower excitation frequencies. We note\nthat demagnetization effects were only considered here\nas a shape anisotropy term included in K; it is expected\nthat this should be a relatively good approximation for\nthe Pd/Fe/Ir(111) system with only a monolayer of mag-\nnetic material, but it was suggested recently[6] that the\ndipolar interaction can significantly influence the excita-\ntion frequencies of isolated skyrmions in magnetic multi-\nlayers.\nS.V. SPIN DYNAMICS SIMULATIONS\nFor the spin dynamics simulations displayed in Fig. 3\nin the main text we used an atomistic model Hamiltonian\non a single-layer triangular lattice,\nH=−1\n2/summationdisplay\n/angbracketlefti,j/angbracketrightJSiSj−1\n2/summationdisplay\n/angbracketlefti,j/angbracketrightDij(Si×Sj)−/summationdisplay\niK(Sz\ni)2\n−/summationdisplay\niµBSz\ni, (S.38)\nwith the parameters J= 5.72meV for the Heisenberg\nexchange,D=|Dij|= 1.52meV for the Dzyaloshinsky–\nMoriya interaction, K= 0.4meV for the anisotropy,\nµ= 3µBfor the magnetic moment, and a= 0.271nm\nfor the lattice constant. For the transformation be-\ntween the lattice and continuum parameters in the\nPd/Fe/Ir(111) system see, e.g., Ref. [10]. The simula-\ntionswereperformedbynumericallysolvingtheLandau–\nLifshitz–Gilbert equation on an 128×128lattice with\nperiodic boundary conditions, which was considerably\nlarger than the equilibrium skyrmion size to minimize\nboundary effects. The initial configuration was deter-\nmined by calculating the eigenvectors in the continuum\nmodel and discretizing it on the lattice, as shown in the\ninsets of Fig. 2 in the main text. It was found that such\na configuration was very close to the corresponding exci-\ntation mode of the lattice Hamiltonian Eq. (S.38), simi-\nlarly to the agreement between the continuum and lattice\nequilibrium skyrmion profiles[10].\n[1] C. Schütte and M. Garst, Phys. Rev. B 90, 094423\n(2014).\n[2] V. P. Kravchuk, D. D. Sheka, U. K. Rössler, J. van den\nBrink, andYu.Gaididei, Phys.Rev.B 97, 064403(2018).\n[3] S.-Z. Lin, Phys. Rev. B 96, 014407 (2017).\n[4] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I.\nStasinopoulos, H. Berger, C. Pfleiderer, and D. Grundler,\nNat. Mater. 14, 478 (2015).\n[5] M. Beg, M. Albert, M.-A. Bisotti, D. Cortés-Ortuño, W.\nWang, R. Carey, M. Vousden, O. Hovorka, C. Ciccarelli,\nC. S. Spencer, C. H. Marrows, and H. Fangohr, Phys.\nRev. B95, 014433 (2017).\n[6] B. Satywali, F. Ma, S. He, M. Raju, V. P. Kravchuk,\nM. Garst, A. Soumyanarayanan, and C. Panagopoulos,arXiv:1802.03979 (2018).\n[7] F. Romá, L. F. Cugliandolo, and G. S. Lozano, Phys.\nRev. E90, 023203 (2014).\n[8] A. O. Leonov, T. L. Monchesky, N. Romming, A. Ku-\nbetzka, A. N. Bogdanov, and R. Wiesendanger, New J.\nPhys.18, 065003 (2016).\n[9] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sam-\npaio, C. A. F. Vaz, N. Van Horne, K. Bouzehouane, K.\nGarcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J.-M.\nGeorge, M. Weigand, J. Raabe, V. Cros, and A. Fert,\nNat. Nanotechnol. 11, 444 (2016).\n[10] J. Hagemeister, A. Siemens, L. Rózsa, E. Y. Vedme-\ndenko, and R. Wiesendanger, Phys. Rev. B 97, 174436\n(2018)." }, { "title": "1810.07384v2.Perpendicularly_magnetized_YIG_films_with_small_Gilbert_damping_constant_and_anomalous_spin_transport_properties.pdf", "content": "Perpendicularly magnetized YIG films with small Gilbert \ndamping constant and anomalous spin transport properties \nQianbiao Liu1, 2, Kangkang Meng1*, Zedong Xu3, Tao Zhu4, Xiao guang Xu1, Jun \nMiao1 & Yong Jiang1* \n \n1. Beijing Advanced Innovation Center for Materials Genome Engineering, University of Science and \nTechnology Beijing, Beijing 100083, China \n2. Applied and Engineering physics, Cornell University, Ithaca, NY 14853, USA \n3. Department of Physics, South University of Science and Technology of China , Shenzhen 518055, \nChina \n4. Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China \nEmail: kkmeng@ustb.edu.cn ; yjiang@ustb.edu.cn \n \nAbstract: The Y 3Fe5O12 (YIG) films with perpendicular magnetic anisotropy (PMA) \nhave recently attracted a great deal of attention for spintronics applications. Here, w e \nreport the induced PMA in the ultrathin YIG films grown on \n(Gd 2.6Ca0.4)(Ga 4.1Mg 0.25Zr0.65)O12 (SGGG) substrate s by epitaxial strain without \npreprocessing. Reciprocal space mapping shows that the film s are lattice -matched to \nthe substrate s without strain relaxation. Through ferromagnetic resonance and \npolarized neutron reflectometry measurements, we find that these YIG films have \nultra-low Gilbert damping constant (α < 1×10-5) with a magnetic dead layer as thin as \nabout 0.3 nm at the YIG/SGGG interfaces. Moreover, the transport behavior of the \nPt/YIG/SGGG films reveals an enhancement of spin mixing conductance and a large \nnon-monotonic magnetic field dependence of anomalous Hall effect as compared with \nthe Pt/YIG/Gd 3Ga5O12 (GGG) films. The non- monotonic anomalous Hall signal is \nextracted 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 \nstrain. \n \nThe spin transport in ferrim agnetic insulator (FMI) based devices has received \nconsiderable interest due to its free of current -induced Joule heating and beneficial for \nlow-power spintronic s applications [1, 2]. Especially, the high-quality Y3Fe5O12 (YIG) \nfilm as a widely studied FMI has low damping constant, low magnetostriction and \nsmall magnetocrystalline anisotropy, making it a key material for magnonics and spin \ncaloritronics . Though the magnon s can carr y information over distances as long as \nmillimeters in YIG film , there remain s a challenge to control its magnetic anisotropy \nwhile maintaining the low damping constant [3] , especially for the thin film with \nperpendicular magnetic anisotropy (PMA) , which is very useful for spin polarizers, \nspin-torque oscillators, magneto -optical d evices and m agnon valve s [4-7]. In addition, \nthe spin- orbit torque (SOT) induced magnetization switching with low current \ndensities has been realized in non -magnetic heavy metal (HM)/FMI heterostructures , \npaving the road towards ultralow -dissipation SOT de vices based on FMI s [8-10]. \nFurthermore, p revious theoretical studies have pointed that the current density will \nbecome much smaller if the domain structures were topologically protected (chiral) [11]. However, most FMI films favor in-plane easy axis dominated by shape \nanisotropy , and the investigation is eclipsed as compared with ferromagnetic materials \nwhich 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, \nand the domain walls can be propelled by spin current from an adjacent platinum \nlayer [18]. Similar with the TmIG films, the possible chiral magnetic structures are \nalso expected in the YIG films with lower damping constan t, which would further \nimprove the chiral domain walls’ motion speed. \nRecently, several ways have been reported to attain the perpendicular ly \nmagnetized YIG films , one of which is utiliz ing the lattice distortion and \nmagnetoelastic effect induced by epitaxial strain [1 9-22]. It is noted that the strain \ncontrol can not only enable the field -free magnetization switching but also assist the \nstabilization of the non- collinear magnetic textures in a broad range of magnetic field \nand temperature. Therefore, abundant and interesting physical phenomena would \nemerge in epitaxial grown YIG films with PMA. However, either varying the buffer \nlayer or doping would increase the Gilbert damping constant of YIG, which will \naffect the efficiency of the SOT induced magnetization switching [20, 21]. On the \nother hand, these preprocessing would lead to a more complicate magnetic structures \nand impede the further discussion of spin transport properties such as possible \ntopological Hall effect (THE). \nIn this work, we realized the PMA of ultrathin YIG films deposited on SGGG \nsubstrates due to epitaxial strain . Through ferromagnetic resonance (FMR) and \npolarized neutron reflectometry (PNR) measurements, we have found that the YIG \nfilms had small Gilbert damping constant with a magnetic dead layer as thin as about \n0.3 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 \nmagnet ic field dependence of the anomalous Hall resistivity, which did not exis t in \nthe compared Pt/YIG/GGG films. The non -monotonic anomalous Hall signal was \nextracted in the temperature range from 150 to 350 K, and we ascribed it to the \npossible non -collinear magnetic order at the Pt/YIG interfaces induced by uniaxial \nstrain. \n \nResults \nStructural and magnetic characterization. The epitaxial YIG films with varying \nthickness from 3 to 90 nm were grown on the [111] -oriented GGG substrate s (lattice \nparameter a = 1.237 nm) and SGGG substrates (lattice parameter a = 1.248 nm) \nrespectively by pulsed laser deposition technique (see methods). After the deposition, \nwe have investigated the surface morphology of the two kinds of films using atomic \nforce microscopy (AFM) as shown in Fig. 1 ( a), and the two films have a similar and \nsmall surface roughness ~0.1 nm. Fig. 1 ( b) shows the enlarged XRD ω-2θ scan \nspectra of the YIG (40 nm) thin film s grow n on the two different substrates (more \ndetails are shown in the Supplementary Note 1 ), and they all show predominant (444) \ndiffraction peaks without any other diffraction peaks, excluding impurity phases or other crystallographic orientation s and indicat ing the single -phase nature. According \nto the (444) diffraction pe ak position and the reciprocal space map of the (642) \nreflection of a 40 -nm-thick YIG film grown on SGGG as shown in Fig. 1(c), we have \nfound 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 \nand aIP represent the pseudo cubic lattice constant calculated from the ou t-of-plane \nlattice constant d(4 4 4) OP and in-plane lattice constant d(1 1 0) IP, respectively, \nfollowing the equation of \n2 2 2lkhad\n++= , with h, k, and l standing for the Miller \nindices of the crystal planes . It indicates that the SGGG substrate provides a tensile \nstress ( ξ ~ 0.84%) [21]. At the same time, the magnetic properties of the YIG films \ngrown on the two different substrates were measured via VSM magnetometry at room \ntemperature. According to the magnetic field ( H) dependence of the magnetization (M) \nas shown in Fig. 1 (d), the magnetic anisotropy of the YIG film grown on SGGG \nsubstrate has been modulated by strain, while the two films have similar in -plane \nM-H curves. \nTo further investigate the quality of the YIG films grown on SGGG substrates \nand exclude the possibility of the strain induced large stoichiometry and lattice \nmismatch, compositional analyse s were carried out using x -ray photoelectron \nspectroscopy (XPS) and PNR. As shown in Fig. 2 (a), the difference of binding \nenergy between the 2p 3/2 peak and the satellite peak is about 8.0 eV, and the Fe ions \nare determined to be in the 3+ valence state. It is found that there is no obvious \ndifference for Fe elements in the YIG films grown on GGG and SGGG substrates. \nThe Y 3 d spectrums show a small energy shift as shown in Fig. 2 (b) and the binding \nenergy shift may be related to the lattice strain and the variation of bond length [21]. \nTherefore, the stoichiometry of the YIG surface has not been dramatically modified \nwith 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 \nSGGG substrates. The PNR signals and scattering length density (SLD) profiles for \nYIG (12.8 nm)/SGGG films by applying an in- plane magnetic field of 900 mT at \nroom temperature are shown in Fig. 2 ( c) and ( d), respectively. In Fig. 2(c), R++ and \nR-- are the nonspin -flip reflectivities, where the spin polarizations are the same for the \nincoming and reflected neutrons. The inset of Fig. 2(c) shows the experimental and \nsimulated spin -asymmetry (SA), defined as SA = ( R++ – R--)/(R++ + R--), as a function \nof scattering vector Q. A reasonable fitting was obtained with a three- layer model for \nthe single YIG film, containing the interface layer , main YIG layer and surface layer. \nThe nuclear SLD and magnetic SLD are directly proportional to the nuclear scattering \npotential and the magnetization , respectively . Then, the depth- resolved structural and \nmagnetic SLD profiles delivered by fitting are s hown in Fig. 2(d) . The Z -axis \nrepresents the distance for the vertical direction of the film, where Z = 0 indicates the \nposition at the YIG/SGGG interface. It is obvious that there is few Gd diffusion into \nthe YIG film, and the dead layer (0.3 nm ) is much thinner than the reported values \n(5-10 nm) between YIG (or T mIG) and substrates [23 -25]. The net magnetization of \nYIG is 3.36 μB (～140 emu/cm3), which is similar with that of bulk YIG [2 6]. The \nPNR results also showed that besides the YIG/ SGGG interface region, there is also \n1.51- nm-thick nonmagnetic surface layer, which may be Y2O3 and is likely to be \nextremely important in magnetic proximity effect [ 23]. \n Dynamical characterization and spin transport properties. To quantitatively \ndetermine the magnetic anisotropy and dynamic properties of the YIG films, the FMR \nspectra were measured at room temperature using an electron paramagnetic resonance \nspectrometer 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 \nextract the resonance field (H\nres) and peak -to-peak linewidth ( ΔHpp) at different θ for \nthe 40 -nm-thick YIG fil ms grown on GGG and SGGG substrates, respectively. The \ndetails for 3 -nm-thick YIG film are show n in the Supp lementary Note 2 . According to \nthe angle dependence of H res as shown in Fig. 3(b), one can find that as compared \nwith the YIG films grown on GGG substrate s, the minimum Hres of the 40- nm-thick \nYIG film grown on SGGG substrate increases with varying θ from 0° to 90° .On the \nother hand, according to the frequency dependence of Hres for the YIG (40 nm) films \nwith applying H in the XY plane as shown in Fig. 3(c), in contrast to the YIG/GGG \nfilms, the H res in YIG/SGGG films could not be fitted by the in-plane magnetic \nanisotropy Kittel formula 21)] 4 ( )[2(/\neff res res πM H Hπγ/ f + = . All these results \nindicate that the easy axis of YIG (40 nm) /SGGG films lies out -of-plane. The angle \ndependent ΔHpp for the two films are also compared as shown in Fig. 3(d) , the \n40-nm-thick YIG film grown on SGGG substrate has an optimal value of Δ Hpp as low \nas 0.4 mT at θ =64°, and the corresponding FMR absorption line and Lorentz fitting \ncurve are shown in Fig. 3(e). Generally , the ΔHpp is expected to be minimum \n(maximum) along magnetic easy (hard) axis, which is basically coincident with the \nangle 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. \nThe lowest ΔHpp at θ=64° could be ascribed to the high YIG film quality and ultrathin \nmagnetic dead layer at the YIG/SGGG interface. It should be noted that , as compared \nwith YIG/GGG films , the Δ Hpp is independent on the frequency from 5 GHz to 14 \nGHz as shown in Fig. 3(f). Then, w e have calculate d the Gilbert damping constant α \nof the YIG (40 nm)/SGGG films by extracting the Δ Hpp at each frequency as shown in \nFig. 3(f). The obtained α is smaller tha n 1 × 10−5, which is one order of magnitude \nlower than t he report in Ref. [20] and would open new perspectives for the \nmagnetization dynamics. According to the theor etical theme, the ΔHpp consists of \nthree parts: Gilbert damping, two magnons scattering relaxation process and \ninhomogeneities, in which both the Gilbert damping and the two magnons scattering \nrelaxation process depend on frequency. Therefore, the large Δ Hpp in the YIG/SGGG \nfilms mainly stems from the inhomogeneities, w hich will be discussed next with the \nhelp of the transport measurements. All of the above results have proven that the \nultrathin YIG films grown on SGGG substrate s have not only evident PMA but also \nultra-low Gilbert damping constant. \nFurthermore, we have also investigated the spin transport properties for the high \nquality YIG film s grown on SGGG substrate s, which are basically sensitive to the \nmagnetic details of YIG. The magnetoresistance (MR) has been proved as a powerful \ntool to effectively explore magnetic information originating from the interfaces [ 27]. \nThe temperature dependent spin Hall magnetoresistance (SMR) of the Pt (5 nm)/YIG \n(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 \nis shown in Fig. 4 (a). The β scan of the longitudinal MR, which is defined as \nMR=ΔρXX/ρXX(0)=[ρXX(β) -ρXX(0)]/ρXX(0) in the YZ plane for the two films under a 3 T \nfield (enough to saturate the magnetization of YIG ), shows cos2β behavior s with \nvarying temperature for the Pt/YIG/GGG and Pt/YIG/SGGG films as shown in Fig. 4 \n(b) and (c), respectively. T he SMR of the Pt/YIG /SGGG films is larger than that of \nthe Pt/YIG /GGG films with the same thickness of YIG at room temperature, \nindica ting an enhanced spin mixing conductance ( G↑↓) in the Pt/YIG /SGGG films. \nHere, it should be noted that the spin transport properties for the Pt layers ar e \nexpected to be the same because of the similar resistivity and film s quality . Therefore, \nthe SGGG substrate not only induces the PMA but also enhances G ↑↓ at the Pt/YIG \ninterface. Then, we have also investigated the field dependent Hall resistivities in the \nPt/YIG/SGGG films at the temperature range from 260 to 350 K as shown in Fig. 4(d). \nThough the conduction electrons cannot penetrate into the FMI layer, the possible \nanomalous Hall effect (AHE) at the HM/FMI interface is proposed to emerge, and the \ntotal Hall resistivity can usually be expressed as the sum of various contributions [28, \n29]: \nS-A S H ρ ρ H R ρ + + =0 , (1) \nwhere R0 is the normal Hall coefficient, ρ S the transverse manifestation of SMR, and \nρS-A the spin Hall anomalous Hall effect (SAHE) resistivity. Notably, the external field \nis applied out -of-plane, and ρs (~Δρ1mxmy) can be neglected [ 29]. Interestingly, the \nfilm 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 \ndiscussion, we term the part of extra anomalous signals as the anomalous SAHE resistivity ( ρ\nA-S-A). The ρ A-S-A signals clearly coexist with the large background of \nnormal Hall effect. Notably, the broken (space) inversion symmetry with strong \nspin-orbit coupling (SOC) will induce the Dzyaloshinskii -Moriya interaction (DMI) . \nIf the DMI could be compared with the Heisenberg exchange interaction and the \nmagnetic anisotropy that were controlled by st rain, it c ould stabilize non-collinear \nmagnetic textures such as skyrmions, producing a fictitious magnetic field and the \nTHE . The ρA-S-A signals indicate that a chiral spin texture may exist, which is similar \nwith B20-type compounds Mn 3Si and Mn 3Ge [ 30,31]. To more clearly demonstrate \nthe origin of the anomalous signals, we have subtracted the normal Hall term , and the \ntemperature dependence of ( ρS-A + ρ A-S-A) has been shown in Fig. 4 (e). Then, we can \nfurther discern the peak and hump structure s in the temperature range from 260 to 350 \nK. The SAHE contribution ρS-A can be expressed as 𝜌𝑆−𝐴=𝛥𝜌2𝑚𝑍 [32, 33],\n where \n𝛥𝜌2 is the coefficient depending on the imaginary part of G ↑↓, and mz is the unit \nvector of the magnetization orientation along the Z direction . The extracted Hall \nresist ivity ρA-S-A has been shown in Fig. 4 (f), and the temperature dependence of the \nlargest ρA-S-A (𝜌𝐴−𝑆−𝐴Max) in all the films have been shown in Fig. 4 (g). Finite values of \n𝜌𝐴−𝑆−𝐴Max exist in the temperature range from 150 to 350 K , which is much d ifferen t \nfrom that in B20 -type bulk chiral magnets which are subjected to low temperature and \nlarge 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 \ndiscussion was shown in the Supplementary Note 3. \nTo further discuss the origin of the anomalous transport signals, we have \ninvestigated the small field dependence of the Hall resistances for Pt (5 nm) /YIG (40 \nnm)/SGGG films as shown in Fig. 5(a). The out-of-plane hysteresis loop of \nPt/YIG/SGGG is not central symmetry, which indicates the existence of an internal \nfield leading to opposite velocities of up to down and down to domain walls in the \npresence of current along the +X direction. The large field dependences of the Hall \nresistances are shown in Fig. 5(b), which could not be described by Equation (1). \nThere are large variations for the Hall signals when the external magnetic field is \nlower than the saturation field ( Bs) of YIG film (~50 mT at 300 K and ~150 mT at 50 \nK). More interestingly, we have firstly applied a large out -of-plane external magnetic \nfield of +0.8 T ( -0.8 T) above Bs to saturate the out -of-plane magnetization \ncomp onent MZ > 0 ( MZ < 0), then decreased the field to zero, finally the Hall \nresistances were measured in the small field range ( ± 400 Oe), from which we could \nfind 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 \nsimple linear magnetic order. Theoretically , an additional chirality -driven Hall effect \nmight be present in the ferromagnetic regime due to spin canting [3 5-38]. It has been \nfound that the str ain from an insulating substrate could produce a tetragonal distortion, \nwhich would drive an orbital selection, modifying the electronic properties and the \nmagnetic ordering of manganites. For A\n1-xBxMnO 3 perovskites, a compressive strain makes the ferromagnetic configuration relatively more stable than the \nantiferromagnetic state [3 9]. On the other hand, the strain would induce the spin \ncanting [ 40]. A variety of experiments and theories have reported that the ion \nsubstitute, defect and magnetoelast ic interaction would cant the magnetization of YIG \n[41-43]. Therefore, if we could modify the magnetic order by epitaxial strain, the \nnon-collinear magnetic structure is expected to emerge in the YIG film. For YIG \ncrystalline structure, the two Fe sites ar e located on the octahedrally coordinated 16(a) \nsite and the tetrahedrally coordinated 24(d) site, align ing antiparallel with each other \n[44]. According to the XRD and RSM results, the tensile strain due to SGGG \nsubstrate would result in the distortion ang le of the facets of the YIG unit cell smaller \nthan 90 ° [45]. Therefore, the magneti zations of Fe at two sublattice s should be \ndiscussed separately rather than as a whole. Then, t he anomalous signals of \nPt/YIG/SGGG films could be ascribed to the emergence o f four different Fe3+ \nmagnetic orientation s in strained Pt/YIG films, which are shown in Fig. 5(d). For \nbetter to understand our results, w e assume that, in analogy with ρ S, the ρA-S-A is larger \nthan ρA-S and scales linearly with m ymz and mxmz. With applying a large external field \nH along Z axis, the uncompensated magnetic moment at the tetrahedrally coordinated \n24(d) is along with the external fields H direction for |H | > Bs, and the magnetic \nmoment tends to be along A (-A) axis when the external fields is swept from 0.8 T \n(-0.8 T) to 0 T. Then, if the Hall resistance was measured at small out -of-plane field , \nthe uncompensated magnetic moment would switch from A (-A) axis to B (-B) axis. In \nthis 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 \nstill some problem that needs to be further clarified. There are no anomalous signals \nin Pt/YIG/GGG films that could be ascribed to the weak strength of Δρ3 or the strong \nmagnetic anisotropy . It is still valued for further discussion of the origin of Δ ρ3 that \nwhether it could stem from the skrymions et al ., but until now we have not observed \nany chiral domain structures in Pt/YIG/SGGG films through the Lorentz transmission \nelectron microscopy. Therefore, we hope that future work would involve more \ndetailed magnetic microscopy imaging and microstructure analysis, which can further elucidate the real microscopic origin of the large non -monotonic magnetic field \ndependence of anomalous Hall resistivity. \n \nConclusion \nIn conclusion, the YIG film with PMA could be realized using both epitaxial strain \nand growth -induced anisotropies. These YIG films grown on SGGG substrates had \nlow G ilbert damping constants (<1 ×10\n-5) with a magnetic dead layer as thin as about \n0.3 nm at the YIG/SGGG interface. Moreover, we observe d a large non -monotonic \nmagnetic field dependence of anomalous Hall resistivity in Pt/YIG/SGGG films, \nwhich did not exist in Pt/YIG/GGG films. The non -monotonic anomalous portion of \nthe Hall signal was extracted in the temperature range from 150 to 350 K and w e \nascribed it to the possible non -collinear magnetic order at the Pt/YIG interface \ninduced by uniaxial strain. The present work not only demonstrate that the strain \ncontrol 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 \nstorage technologies based on FMI. \n \nMethods \nSample preparation. The epitaxial YIG films with varying thickness from 3 to 90 \nnm were grown on the [111] -oriented GGG substrate s (lattice parameter a =1.237 nm) \nand SGGG substrates (lattice parameter a =1.248 nm) respectively by pulsed laser \ndeposition technique . The growth temperature was TS =780 ℃ and the oxyg \npressure was varied from 10 to 50 Pa . Then, the films were annealed at 780℃ for 30 \nmin at the oxygen pressure of 200 Pa . The Pt (5nm) layer was deposited on the top of \nYIG films at room temperature by magnetron sputtering. After the deposition, the \nelectron 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 \nμm×120 μm. \nStructural and magnetic characterization. The s urface morphology was measured \nby AFM (Bruke Dimension Icon). Magnetization measurements were carried out \nusing a Physical Property Measurement System (PPMS) VSM. A detailed \ninvestigation of the magnetic information of Y IG was investigated by PNR at the \nSpallation Neutron Source of China. \nFerromagnetic resonance measurements. The measurement setup is depicted in Fig. \n3(a). For FMR measurements, the DC magnetic field was modulated with an AC field. \nThe 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 \nwere then fitted to a sum of symmetric and antisymmetric Lorentzian functions to \nextract the linewidth. \nSpin transport measurements . The measurements were carried out using PPMS \nDynaCool. \n \nAcknowledgments \nThe authors thanks Prof. L. Q. Yan and Y. Sun for the technical assistant in \nferromagnetic resonance measurement . This work was partially supported by the \nNational Science Foundation of China (Grant Nos. 51971027, 51927802, 51971023 , \n51731003, 51671019, 51602022, 61674013, 51602025), and the Fundamental Research Funds for the Central Universities (FRF- TP-19-001A3). \n References \n[1] Wu, M.-Z. & Hoffmann , A. Recent advances in magnetic insulators from \nspintronics to microwave applications. Academic Press , New York, 64 , 408 \n(2013) . \n[2] Maekawa, S. Concepts in spin electronics. Oxford Univ., ( 2006) . \n[3] Neusser, S. & Grundler, D. Magnonics: spin waves on the nanoscale. Adv. Mater., \n21, 2927- 2932 ( 2009) . \n[4] Kajiwara , Y. et al. Transmission of electrical signals by spin -wave \ninterconversion in a magnetic insulator. Nature 464, 262- 266 (2010). [5] Wu, H. et al. Magnon valve effect between two magnetic insulators. Phys. Rev. \nLett. 120, 097205 ( 2018). \n[6] Dai, Y. et al. 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Multiply periodic s tates and isolated skyrmions in \nan anisotropic frustrated magnet. Nat. Commun. 6, 1-8 (2015) . \n[38] Nakatsuji S., Kiyohara N. & Higo T. Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature. Nature 527, 212-215 (2015). \n[39] Quindeau A. et al. Tm\n3Fe5O12/Pt heterostructures with perpendicular magnetic \nanisotropy for spintronic applications. Adv. Electron. Mater. 3, 1600376 (2017). \n[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). \n[41] Parker G. N. & Saslow W. M. Defect interactions and canting in ferromagnets. Phys. Rev. B 38, 11718 (1988). \n[42] Rosencwaig A. Localized canting model for substituted ferrimagnets. I. Singly \nsubstituted YIG systems. Can. J. Phys. 48, 2857- 2867(1970). \n[43] AULD B. A. Nonlinear magnetoelastic interactions. Proceedings of the IEEE, 53, \n1517- 1533 (1965). \n[44] Ching W. Y., Gu Z. & Xu Y N. Th eoretical calculation of the optical properties \nof Y\n3Fe5O12. J. Appl. Phys. 89, 6883- 6885 (2001). [45] Baena A., Brey L. & Calder ón M. J. Effect of strain on the orbital and magnetic \nordering of manganite thin films and their interface with an insulator. Phys. Rev. \nB 83, 064424 (2011). \n \nFigure Captions \n \nFig. 1 Structural and magnetic properties of YIG films. (a) AFM images of the \nYIG films grown on the two substrates (scale bar, 1 μ m). (b) XRD ω-2θ scans of the \ntwo different YIG films grown on the two substrates . (c) High -resolution XRD \nreciprocal space map of t he YIG film deposited on the SGGG substrate. (d) Field \ndependence of the normalized magnetization of the YIG films grown on the two \ndifferent substrates . \n \n \nFig. 2 Structural and magnetic properties of YIG films. Room temperature XPS \nspectra of (a) Fe 2p and (b) Y 3d for YIG films grown on the two substrates . (c) P NR \nsignals (with a 900 mT in -plane field) for the spin -polarized R++ and R-- channels. \nInset: The experimental and simulated SA as a function of scattering vector Q. (d) \nSLD profiles of the YIG/SGGG films. The nuclear SLD and magnetic SLD is directly \nproportional to the nuclear scattering potential and the magnetization , respectively. \n \n \n \n \nFig. 3 Dynamical properties of YIG films . (a) The geometric configuration of the \nangle dependent FMR measurement. (b) The angle dependence of the H res for the YIG \nfilms on GGG and SGGG substrates. (c) The frequency dependence of the H res for \nYIG films grown on GGG and S GGG substrates. (d) The ang le dependence of Δ Hpp \nfor the YIG films on GGG and SGGG substrates. (e) FMR spectrum of the \n40-nm-thick YIG film grown on SGGG substrate with 9.46 GHz at θ =64°. (f) The \nfrequency dependence of Δ Hpp for the 40 -nm-thick YIG films grown on GGG and \nSGGG substr ates. \n \nFig. 4 Spin transport properties of Pt/YIG (3nm) films . (a) The definition of the \nangle, the axes and the measurement configurations. ( b) and ( c) Longitudinal MR at \ndifferent temperatures in Pt/YIG/GGG and Pt/YIG/SGGG films respectively (The \napplied magnetic field is 3 T). (d) Total Hall resistivities vs H for Pt/YIG/SGGG films \nin the temperature range from 260 to 300 K. (e) (ρS-A+ρA-S-A) vs H for two films in the \ntemperature range from 260 to 300 K. (f) ρ A-S-A vs H for Pt/YIG/SGGG films at 300K. \nInset: ρS-A and ρS-A + ρ A-S-A vs H for Pt/YIG/SGGG films at 300K. (g) Temperature \ndependence of the 𝜌𝐴−𝑆−𝐴𝑀𝑎𝑥. \n \n \n \nFigure 5 S pin transport properties of Pt/YIG ( 40 nm) films . (a) and (b) The Hall \nresistances vs H for the Pt/YIG/SGGG films in the temperature range from 50 to 300 \nK in small and large magnetic field range, respectively. (c) The Hall resistances vs H \nat small magnetic field range after sweeping a large out -of-plane magnetic field +0.8 \nT (black line) and - 0.8 T (red line) to zero. (d) An illustration of the orientations of the \nmagnetizations Fe ( a) and Fe ( d) in YIG films with the normal in -plane magnetic \nanisotropy (IMA), the ideal strain induced PMA and the actual magnetic anisotropy \ngrown on SGGG in our work. \n" }, { "title": "1909.09787v1.Resonant_absorption_of_kink_oscillations_in_coronal_flux_tubes_with_continuous_magnetic_twist.pdf", "content": "arXiv:1909.09787v1 [astro-ph.SR] 21 Sep 2019Resonant absorption of kink oscillations in coronal\nflux tubes with continuous magnetic twist\nZanyar Ebrahimi1∗and Karam Bahari2†\n1Research Institute for Astronomy & Astrophysics of Maragha, U niversity of Maragheh, Maragheh, Iran\n2Physics Department, Faculty of Science, Razi University, Kerman shah, Iran\nSeptember 24, 2019\nAbstract\nThere are observational evidences for the existence of twisted m agnetic field in the solar\ncorona. Here, we have investigated resonant damping of the magn etohydrodynamic (MHD)\nkink waves in magnetic flux tubes. A realistic model of the tube with co ntinuous magnetic\ntwist and radially inhomogeneous density profile has been considered . We have obtained the\ndispersion relation of the kink wave using the solution to the linear MHD equations outside\nthe density inhomogeneityand the appropriateconnectionformula to the solutionsacrossthe\nthin transitional boundary layer. The dependence of the oscillation frequency and damping\nrate ofthe waveson the twist parameterand longitudinal wavenum berhas been investigated.\nFor the flux tube parameters considered in this paper, we obtain ra pid damping of the kink\nwaves comparable to the observations. In order to justify this ra pid damping, depending on\nthe sign of the azimuthal kink mode number, m= +1 or m=−1, the background magnetic\nfield must have left handed or right handed twisted profile, respect ively. For the model\nconsidered here, the resonant absorption occurs only when the t wist parameter is in a range\nspecified by the density contrast.\nKey words: Sun: corona – Sun: magnetic fields – Sun: oscillations.\n1 Introduction\nUchida (1970) and Roberts et al. (1984) were among the first re searchers who suggested the\ntechnique of coronal seismology to determine the equilibri um quantities of the solar corona.\nUchida (1970) pointed out that on the basis of knowing the pla sma density of the solar corona\nthe magnetic field can be measured using the seismological di agnosis. The first model of the\ncoronal flux tubes proposed by Edwin and Roberts (1983). They considered the magnetic flux\ntubeas a density enhancement in a straight magnetic field and studied the dispersion diagram of\nthe MHD waves for coronal and photospheric conditions and di scussed the nature of the waves\nsupported by the flux tube.\nUntil now various oscillation modes in the solar coronal loo ps have been observed and inter-\npreted as the trapped MHD waves in magnetic flux tubes. Transv erse oscillations in the coronal\nloops were first reported by Nakariakov et al. (1999) and Asch wanden et al. (1999) using the\nTransition Region and Coronal Explorer (TRACE) telescope o n 1998 July 14 in the 171- ˚A Fe\n∗E-mail: zebrahimi@maragheh.ac.ir\n†E-mail: karam.bahari@gmail.com\n1IX emission lines. Nakariakov et al. (1999) interpreted the se oscillations as MHD kink waves\nand indicated that the oscillations are strongly damped, su ch that the ratio of the damping time\nto the period of the oscillation is about 3-5. Among the mecha nisms believed to be responsible\nfor the strong damping of the coronal loop oscillations, res onant absorption of the MHD waves\nis the most efficient mechanism. Resonant absorption of MHD wa ves established first by Ion-\nson (1978), and is studied by some authors such as Hollweg and Yang (1988), Ruderman and\nRoberts (2002), Ofman (2009) and Morton and Erd´ elyi (2009) to explain the strong damping\nof MHD kink waves. In this mechanism, the frequency of the glo bal mode oscillation equals\nthe background Alfv´ en frequency at some radios inside the t ube called resonance point which\nresults to the cascading of the global mode energy to the loca l Alfv´ en perturbations within a\nlayer around the resonance point called resonance layer. Fo r a good review on this topic, see\ne.g. Goossens et al. (2011).\nHollweg and Yang (1988) studied resonant absorption to expl ain damping of MHD waves\nin planar geometry. They were first to obtain approximate ana lytical expressions for the decay\ntimes of quasi-modes. They also applied their analytical re sults to kink modes m= 1 in\ncylindrical geometry and established the fast damping of ki nk waves. Goossens et al. (1992)\ndetermined the conservation laws and the jump conditions ac ross the slow and Alfv´ en resonance\npoints for a one dimensional cylindrical magnetic flux tube i n the presence of plasma flow and\nmagnetic twist. They used the jump conditions to study surfa ce waves in solar magnetic flux\ntubes. Goossens et al. (2002) mentioned that the quasi-mode kink oscillations in magnetic flux\ntubes can explain the observed rapid damping of the oscillat ions of coronal loops. Goossens\net al. (2009) investigated the nature of MHD kink waves in flux tubes. For this purpose they\ncalculated the eigenfunctions, the frequency and the dampi ng rate of MHD kink waves for three\ndifferent MHD waves cases. They concluded that the kink waves d o not care about the detailed\nqualities of the MHD wave environment, and if an adjective is to be used for the kink waves\nit should be Alfv´ enic. They showed that in the resonance lay er, pressure gradient force can be\nneglected only when the frequency of the kink wave does not di ffers much from the local Alfv´ en\nfrequency. Giagkiozis et al. (2016) for the firsttime studie dresonant absorption of axisymmetric\nMHD waves in twisted tubes. They implemented the conservati on laws and derived a dispersion\nrelation. They showed that the magnetic field and the density have a significant effect on the\ndamping time of axisymmetric MHD waves.\nAs more properties of the waves in the solar corona becomes re vealed, more structured\nmodels of the loops are needed to explain the physics of the wa ves. Verwichte et al. (2004)\nwere first to detect the coexistence of the fundamental and fir st overtone of MHD kink waves\nin closed coronal loops and showed that the ratio of the perio d of the first overtone P2to the\nperiod of the fundamental mode P1is smaller than 2. Andries et al. 2005 used longitudinal\ndensity stratification of the loop to explain the deviation o f the period ratio from 2 (see also\nErd´ elyi and Verth 2007). This deviation can also be explain ed with a twisted magnetic field in\nthe loop (see Erd´ elyi and Carter 2006; Erd´ elyi & Fedun 2006 and Ebrahimi & Karami 2016)\nAn interesting feature of the coronal flux tubes is that they m ay have a twisted magnetic\nfield around the tube axis. Chae et al. (2000) suggested that a twisted magnetic field is needed\nin order to justify the rotational motions observed in coron al flux tubes. The magnetic twist of\n14 coronal loops has been measured by Kwon& Chae (2008). They found that the amount of\nthe twist of the magnetic field, φtwist, was in the range [0 .22π,1.73π]. Here,φtwistis the angel of\nrotation of the background magnetic field around the tube axi s per characteristic length along\nthe flux tube. It has been shown theoretically that if the amou nt of the magnetic twist in a\nflux tube exceeds a critical value which is about φc= 2π, the flux tube would be kink unstable\n(Shafranov 1957; Kruskal et al. 1958). For more details abou t the stability of kink waves in\n2twisted flux tubes see e.g. Hood & Priest (1979); Baty & Heyvae rts (1996); Furno et al. (2006).\nThe effect of twisted magnetic field on the kink waves has been in vestigated for various\nbackground magnetic field and plasma conditions. In some mod els the azimuthal magnetic field\nvaries discontinuously in the tube boundaries. Ruderman (2 007) studied kink oscillations of\na coronal loop in zero beta approximation in the existence of a twisted magnetic field in the\ninternal region of the flux tube. For his model he showed that t he magnetic twist does not affect\nthe kink waves. Karami & Bahari (2010) studied the effect of a tw isted magnetic field on the\nresonant absorption of MHD waves by the assumption of incomp ressibility for the waves in a\nnon-zero beta plasma. In order to have a twisted and continuo us magnetic field in both the\ninterior and exterior regions of the tube, for the sake of sim plicity they considered an unphysical\nprofile for the azimuthal component of the background magnet ic field that goes to ∞asr→ ∞.\nKarami & Bahari (2012) introduced an annulus to the model stu died by Ruderman (2007)\nand showed that the magnetic twist of the annulus region can a lter the oscillation frequency\nof the kink waves substantially. Based on this result they st udied the ratio of the frequency\nof the first overtone and fundamental mode of kink wave. The az imuthal component of the\nbackground magnetic field considered by Karami & Bahari (201 2) was discontinuous at two\ndifferent radiuses. Ebrahimi & Karami (2016) investigated re sonant absorption of MHD kink\nwaves due to the existence of a twisted magnetic field in coron al flux tubes. They considered\na discontinuous magnetic field in their work for the sake of si mplicity. They showed that in\nthe absence of a transition region for the plasma density in t he radial direction the azimuthal\ncomponent of the background magnetic field in flux tubes can in troduce a strong damping for\nkink waves even in the limit of small twist values. Bahari (20 17) investigated the properties\nof standing MHD kink waves in a model coronal loop with an annu lus in the presence of both\nmagnetic twist and plasma flow. He showed that like plasma flow , the twisted magnetic field\nmodifies the symmetry and phase difference of standing kink wav es. He showed that depending\non the direction of the magnetic twist and plasma flow, the pre sence of magnetic twist, like\nplasma flow, can cause underestimation or overestimation in obtaining the flow speed in coronal\nloops. Like Karami & Bahari (2010) but in the zero beta approx imation, Bahari & Khalvandi\n(2017) considered the same unphysical profile for the azimut hal component of the background\nmagnetic field in order to investigate the effect of twisted mag netic field on the nature of MHD\nkink waves. Bahari (2018) has extended the work done by Soler et al. (2011) to the twisted\nloops. Since the kink waves in the long wavelength limit can b e considered as incompressible\nwaves (see Goossens et al. 2009), for simplicity Bahari (201 8) assumed incompressible plasma.\nHe discussed the symmetries in the phase diagram of the kink w aves and showed that the\ndamping length of the waves depends on the propagation direc tion, the amount of magnetic\ntwist, the direction of magnetic twist and the azimuthal mod e number of the waves. In some\nmodels of coronal loops the azimuthal magnetic field varies c ontinuously and is nonzero only\nin a region between internal and external regions of the loop . Terradas & Goossens (2012)\ncalculated the eigenmodes of standing and propagating kink waves numerically for a coronal\nloop with twisted magnetic field. They concluded that in the p resence of a twisted magnetic\nfield, the polarisation of the transverse displacement of th e plasma changes along the loop, and\nthey suggested seismological application of this characte ristic of the kink waves. Ebrahimi et al.\n(2017) showed that depending on the direction of propagatio n of the wave and the twist profile\nconsidered in the tube, a twisted magnetic field can enhance o r reduce the rate of phase-mixing\nof the perturbations of the MHD kink waves. In some other mode ls the magnetic twist is present\nin the whole loop and decreases away from the tube, which is ph ysically more acceptable than\nthe models with discontinuous magnetic field. Ruderman & Ter radas (2015) studied standing\nkink waves in the coronal loops with continuous equilibrium magnetic field. They showed that\n3the effect of the magnetic twist on the period ratio of the kink w aves depends on the density\ncontrast of the tube. Ruderman (2015) studied propagating M HD kink waves in twisted coronal\nloops and showed that in the presence of a twisted magnetic fie ld the symmetry of the phase\nspeed with respect to the propagation direction is broken. H e called these waves accelerated and\ndecelerated kink waves with phase speeds larger and smaller than the kink speed, respectively.\nHe also stated that in the case of simultaneous identificatio n of kink waves propagating with\ndifferent phase speeds, this can be useful for coronal seismol ogy because the amount of the\nmagnetic twist can be estimated from the ratio of the frequen cies.\nTo the authors it is interesting to study the properties of th e MHD kink waves in a model\nflux tube which is more realistic than the models they studied before, a model with continu-\nous magnetic twist and radially varying background density . For this purpose we add a thin\ninhomogeneous boundary layer to the model studied by Ruderm an (2015) and investigate the\noscillation frequency and damping rate of propagating comp onents of standing kink waves. The\npaper is organized as follows: In section 2 we introduce the m odel and the governing equations\nof MHD waves. In section 3, we obtain the dispersion relation and in section 4 we present the\nnumerical results. Section 5 is devoted to our conclusions.\n2 Model and Equations of Motion\nHere, we approximate a real coronal flux tube by a straight cyl inder with circular cross section\nof radius R. We use circular cylindrical coordinates ( r,ϕ,z) such that the zcoordinate is aligned\nwith the tube’s axis. For simplicity we ignore the effect of gra vity on the equilibrium density\nprofile, hence we ignore the density stratification in the axi al direction, also assume no equilib-\nrium plasma flow in the flux tube. The profile of the equilibrium plasma density is assumed to\nbe as follows\nρ(r) =\n\nρi, r≤a,\n1\n2/bracketleftBig\nρi+ρe−(ρi−ρe)sin/parenleftBig\nπ\n2/parenleftBig\n2r−R−a\nR−a/parenrightBig/parenrightBig/bracketrightBig\n, a < r < R,\nρe, r≥R.(1)\nHereρiandρeare the constant densities in the interior and exterior regi ons of the flux tube.\nThe plasma density of the corona is about 10−15gr.cm−3. The background magnetic field is only\na function of rwithBr= 0 and\nBϕ(r) =/braceleftbiggAr, r ≤R,\nR2A\nr, r > R.(2)\nThe constant Ais a parameter that controls the amount of twist in the flux tub e. The profile of\nthe azimuthal component of the background magnetic field is d ue to the existence of a constant\ncurrent density inside the flux tube ( r < R) along the tube axis. Here, we use the zero beta\n(ratio of the gas pressure to the magnetic pressure) approxi mation in our analysis. Hence, the\nbackground magnetic field is force free and satisfies the foll owing equilibrium state condition\ndB2\ndr=−2B2\nϕ\nr(3)\nwhereB2=B2\nϕ+B2\nz. From Eq. (3) and continuity of the magnetic pressure we obta in thez\ncomponent of the background magnetic field as\nB2\nz(r) =/braceleftbiggB2\n0+2A2(R2−r2), r≤R,\nB2\n0, r > R,(4)\n4whereB0is a constant. The Magnetic field in the location of the corona l loops is of the order of\n10G. Itisclear fromEqs. (2)and(4)that boththecomponents oft hebackgroundmagnetic field\nare continuous at boundary of the flux tube. The azimuthal com ponent of the magnetic field is\nzero at the tube axis and increases linearly in the internal p art of the tube and in the external\nregion of the tube decreases like 1 /r. The longitudinal component of the magnetic field takes its\nmaximum value at the tube axis and decreases in the internal r egion of the tube. In the external\nregion the longitudinal component of the magnetic field is co nstant. Since the background\nmagnetic field is continuous everywhere, there is no surface current in our model. Models with\na discontinuous magnetic field introduce a surface current a t the location of discontinuity. The\ntearing mode instability can occur in a thin current sheet wh ere the diffusive effects become\nimportant (e.g. Furth et al. 1963; Goldstone & Rutherford 19 95; Magara & Shibata 1999;\nEbrahimi & Karami 2016). Even a small but non-zero resistivi ty makes the magnetic field lines\ntear and reconnect in the current sheet. In the present model this instability has been avoided\nby considering a continuous magnetic field.\nThe perturbations in the magnetic flux tube are governed by th e linearized ideal MHD\nequations in zero beta approximation as follows\nρ(r)∂2ξ\n∂t2=1\nµ0{(∇×B′)×B+(∇×B)×B′}, (5)\nB′=∇×(ξ×B), (6)\nwhereξis the lagrangian displacement of the plasma and B′is the Eulerian perturbation of the\nmagnetic field. Here, µ0is the magnetic permeability of the free space.\nA standing wave is a superposition of forward and backward pr opagating waves that have\nthe same frequencies. As showed by Terradas and Goossens (20 12), in the presence of a twisted\nmagnetic field, the forward and backward propagating kink wa ves of the same frequency have\ndifferent wave numbers. So it is not possible to Fourier analyz e a standing kink wave in the\nlongitudinal direction in the presence of a twisted magneti c field. In order to overcome this\ndifficulty we study the individual forward and backward trave lling components of a damped\nstanding oscillation, separately. Since the background qu antities are not functions of ϕand\nz, we study the individual Fourier components of the perturba tions and consider the ϕand\nzdependence of the perturbations as exp[ i(mϕ+kzz)], where mandkzare azimuthal mode\nnumber and longitudinal wavenumber, respectively. Here, w e are looking for the global quasi\nmode solution of the waves that are damped in time due to reson ant absorption and assume\nthat the time dependence of the perturbations are of the form exp(−i˜ωt) where the frequency\n˜ωis a complex quantity with dominant real part. Note that a pro pagating wave that starts\nfrom a source point at z= 0 and decays in space as travels in the zdirection have a complex\nwavenumber and a real frequency. But the propagating compon ents of a standing wave decay in\ntime and have a real wavenumber and a complex frequency. Fig. 1 shows these two kinds of the\ndamped propagation. Top panel of the figure exhibit a compone nt of a propagating wave that\ndecays in space at three different times. Bottom panel of the fig ure shows temporal damping\nof a travelling component of a standing wave. As illustrated in the figure, a component of a\npropagating wave with complex wavenumber and real frequenc y decays in the zdirection but\na travelling component of a standing wave needs to have a real wavenumber and a complex\nfrequency in order to decay in time.\nFollowing Ruderman (2015), in thin tube approximation ( kzR≪1) solutions of the Eqs. (5)\nand (6) for radial component of the Lagrangian displacement ,ξr, and Eulerian perturbation of\nthe total (magnetic) pressure, P′, in the internal ( r < a) and external ( r > R) regions of the\nflux tube are as follows\n5Figure 1: top: A propagating wave that decays in space. botto m: A propagating wave that\ndecays in time.\nξr(r) =\n\nξr,i=η, r ≤a,\nξr,e=χ√µ0ρe\n4R2A˜ωmln/vextendsingle/vextendsingle/vextendsingle/vextendsingle(ωA,e−˜ω)(B0kz+˜ω√µ0ρe)\n(ωA,e+˜ω)(B0kz−˜ω√µ0ρe)/vextendsingle/vextendsingle/vextendsingle/vextendsingle, r > R,(7)\nP′(r) =\n\nP′\ni=rηρi/parenleftBig\n˜ω2−B2\n0k2\nz−A2\nµ0ρi/parenrightBig\n, r ≤a,\nP′\ne=ρeχ\nr+\nrρeξr,e/bracketleftBig\n˜ω2−1\nµ0ρe/parenleftBig\nB2\n0k2\nz−R4A4\nr4/parenrightBig/bracketrightBig\n, r > R,(8)\nwhere\nωA,e=1√µ0ρe/parenleftbiggmA2R\nr2+kzB0/parenrightbigg\n. (9)\nHere,ηandχare constant coefficients that are determined by the appropri ate boundary con-\nditions. Notice that in the absence of the magnetic twist, Eq s. (7) and (8) reduce to those\nobtained by Goossens et al. (2009) for pressureless flux tube s with uniform density in thin tube\napproximation.\n3 Connection Formula and Dispersion Relation\nInthissectionweaimtoobtainthedispersionrelationgove rningtheMHDwavesinthefluxtube.\nFor simplicity we assume thin boundary (TB) approximation w hich means that the thickness\nof the inhomogeneous boundary layer is much smaller than the radios of the tube. The TB\napproximation was first used by Hollweg and Yang (1988). Henc e instead of solving the resistive\n6equations of motion in the inhomogeneous boundary layer ( a < r < R ), the ideal solutions\ninside and outside the tube can be related to each other by the connection formula introduced\nby Sakurai et al. (1991). They showed that the jumps in the per turbations across the resonance\nlayer are given by\n[ξr] =−iπ\n|∆|g(rA)\nρiB2(rA)CA(rA), (10)\n/bracketleftbig\nP′/bracketrightbig\n=−iπ\n|∆|2Bϕ(rA)Bz(rA)f(rA)\nµ0ρirAB2(rA)CA(rA), (11)\nwhere\nCA=gBP′(r)−2fBBϕBz\nµ0rAξr(r),\nf=m\nrBϕ+kzBz,\ng=m\nrBz−kzBϕ,\n∆ =−d\ndrω2\nA(r)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=rA,(12)\nandrAis the location of the resonance point. Here\nωA(r) =1/radicalbig\nµ0ρ(r)/parenleftbiggmBϕ(r)\nr+kzBz(r)/parenrightbigg\n(13)\nis defined as the background Alfv´ en frequency.\nNote that the expression for the jump in P′(Eq. 11) has been derived for an equilibrium\nwith continuously varying equilibrium quantities and the j ump ofP′is due to the resonance in\nrAonly. As explained by Goossens et al. (1992), in the absence o f any resonance the radial\ncomponent of displacement ξrand the Lagrangian perturbation of total pressure\nδP=P′+dP0\ndrξr=P′−B2\nϕ\nrξr, (14)\nhave to be continuous. So, in the models with a discontinuous magnetic field at the tube\nboundary where dP0/dris not continuous, an additional jump in P′must be included in the\nright hand side of Eq. 11. As a particular case, in subsection 3.2 of Goossens et al. (1992) a\nmodel with both jumps at the boundary is considered. In the mo del studied by Bahari (2018)\ntwo boundaries exist in different radii, in one of them the reso nance absorption takes place and\ndP0/dris continuous hence only the jump due to resonance absorptio n is included, in the other\none the resonance absorption does not take place but dP0/dris discontinuous which again there\nis a jump in P′. In the present model in addition to the components of the mag netic field Bϕ\nandBzand the equilibrium magnetic pressure P0(r) also the derivative of the magnetic pressure,\ndP0/dr, is continuous at the boundary, hence the jump is due to the Al fv´ en resonance only and\nthe connection formulae introduced by Sakurai et al. (1991) can be used to join the solutions\ninside and outside the tube.\nSubstituting the ideal solutions (7) and (8) in the jump cond itions (10) and (11) and elimi-\nnatingηandχ, we obtain the dispersion relation as\nd0(˜ω)+d1(˜ω)|r=rA= 0, (15)\n7where\nd0(˜ω) =−1+√µ0ρe\n4Am˜ω/bracketleftbigg/parenleftbigga\nRρi\nρe−1/parenrightbigg\n˜ω2+\n/parenleftBig\n1−a\nR/parenrightBig/parenleftbiggB2\n0k2\nz−A2\nµ0ρe/parenrightbigg/bracketrightbigg\n×ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(ωA,e−˜ω)/parenleftbig\nB0kz+ ˜ω√µ0ρe/parenrightbig\n(ωA,e+ ˜ω)/parenleftbig\nB0kz−˜ω√µ0ρe/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,(16)\nand\nd1(˜ω) =−iπ\n|∆|g\nρiB2/bracketleftbigg\ngrρi/parenleftbigg\n˜ω2−B2\n0k2\nz−A2\nµ0ρi/parenrightbigg\n−2fBϕBz\nµ0r/bracketrightbigg\n×/bracketleftbigg√µ0ρe\n4Am˜ω/parenleftbigg2fBϕBz\nµ0ρegrR−/parenleftbigg\n˜ω2−B2\n0k2\nz−A2\nµ0ρe/parenrightbigg/parenrightbigg\n×ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(ωA,e−˜ω)/parenleftbig\nB0kz+ ˜ω√µ0ρe/parenrightbig\n(ωA,e+ ˜ω)/parenleftbig\nB0kz−˜ω√µ0ρe/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1/bracketrightBigg\n.(17)\nHere ˜ω=ω+iγ, in which ωandγare the oscillation frequency and the corresponding dampin g\nrate, respectively. Note that in the limit of a=Rthe resonance layer disappears, ∆ → ∞and\nthe second term in the dispersion relation Eq. (15) disappea rs and Eq. (15) with real ωreduces\nto Eq. (48) of Ruderman (2015) for non-resonant MHD modes.\n4 Numerical results\nIn order to solve Eq. (15) we introduce the following dimensi onless variables\nζ≡ρi\nρe, α≡AR\nB0, ǫ≡kzR. (18)\nAlso we rewrite all lengthes, magnetic fields and frequencie s in new units R,B0andB0\nR√µ0ρi,\nrespectively. Hence, we obtain d0(˜ω) andd1(˜ω) in terms of the dimensionless variables as\nd0(˜ω) =−1+√ζ\n4αm˜ω/bracketleftbigg/parenleftbigg\na−1\nζ/parenrightbigg\n˜ω2+(1−a)/parenleftbig\nǫ2−α2/parenrightbig/bracketrightbigg\n×ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbig√ζ(mα+ǫ)−˜ω/parenrightbig/parenleftbig\nǫ+ ˜ω/√ζ/parenrightbig\n/parenleftbig√ζ(mα+ǫ)+ ˜ω/parenrightbig/parenleftbig\nǫ−˜ω/√ζ/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,(19)\nd1(˜ω) =−iπ\n|∆|g2\nB2/bracketleftbigg/parenleftbig\n˜ω2−ǫ2+α2/parenrightbig\nr−2fBϕBz\ngr/bracketrightbigg\n×/bracketleftbigg√ζ\n4αm˜ω/parenleftbigg2fBϕBz\ngr−/parenleftbigg˜ω2\nζ−/parenleftbig\nǫ2−α2/parenrightbig/parenrightbigg/parenrightbigg\n×ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbig√ζ(mα+ǫ)−˜ω/parenrightbig/parenleftbig\nǫ+ ˜ω/√ζ/parenrightbig\n/parenleftbig√ζ(mα+ǫ)+ ˜ω/parenrightbig/parenleftbig\nǫ−˜ω/√ζ/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1/bracketrightBigg\n.(20)\nSubstituting Eqs. (2) and (4) into Eq. (13) we obtain the dime nsionless form of the background\nAlfv´ en frequency as\nωA(r) =/braceleftBigg1√\nρ(r)/parenleftbig\nmα+ǫ+O(ǫα2)/parenrightbig\n, r≤1,\n√ζ/parenleftbigm\nr2α+ǫ/parenrightbig\n, r > 1.(21)\n8Note that here ρ(r) is in unit of ρi. In weak twist ( α≪1) and thin tube( ǫ≪1) approximations,\nin Eq. (21) we can neglect terms of order ǫα2respect to the other terms. Hence, the background\nAlfv´ en frequency of the interior ( r < R) of the flux tube becomes\nωAi(r)≃1/radicalbig\n��(r)(mα+ǫ). (22)\nTerradas and Goossens (2012) revealed that even if Bϕ≪Bzi.e.α≪1, the competition\nbetween αandǫdetermines the net effect of a twisted magnetic field on the MHD w aves. They\nconsidered a piecewise step function profile for the plasma d ensity in the radial direction and\nlook for the non-resonant solutions in the presence of a cont inuous and twisted magnetic field.\nSince the profile of Bϕchosen by them was parabolic it was possible to have resonanc e inside\nthe flux tube even for a constant density profile. They restric ted their calculations to small twist\nvalues in order to avoid resonance and kink instability in th e flux tube. They also obtained the\ncritical limit of the amount of twist above which the kink osc illation become resonant. Here the\nsituations is different since there are a radial density strat ification in a thin layer at the tube\nboundary that introduces the resonance for the kink waves ev en for zero twist. Since Bϕhas\na linear profile and Bzis almost constant (to first order in α) inside the tube, the occurrence\nof resonance is not related to the magnetic field. However, in what follows the results show\nthat the existence of a twisted magnetic field may suppress th e resonance caused by the radial\ndensity stratification for magnetic twist values higher tha n a critical limit.\nItisclear fromEq. (22) that for m= +1and α=−ǫorm=−1andα= +ǫ, thebackground\nAlfv´ en frequency inside the tube is ωAi= 0. For example, if we insert m= 1 and α=−ǫin the\nnon-resonant dispersion relation d0= 0 obtained by Ruderman (2015), the obtained frequency\nisω= 0. The same result is obtained for m=−1 andα=ǫ. So, in these cases the kink waves\nare not allowed to propagate in the flux tube. In fact, in these cases the wave vector which is\ndefined as k≡m\nrˆϕ+kzˆzis perpendicular to the background magnetic field i.e. k·B= 0. For\nan incompressible plasma ∇·ξ= 0 when k·B= 0, the right hand side of Eq. (6) vanishes\nand the resistive effects become important. In this case, the r esistive diffusion results to the so\ncalled resistive kink instability, that is a reconnecting p rocess (e.g. Biskamp 2000; Wesson 2004;\nPriest 2014).\nHere we take the parameters of a typical oscillating coronal flux tube by the following con-\nsiderations:\n•As stated by Aschwanden et al. (2003), the typical values of t he density ratio ζfor coronal\nflux tubes are in the range [2 ,10]. Here, we take the extreme values ζ= 2 and ζ= 10 in\nour calculations.\n•We investigate both forward ( kz>0) and backward ( kz<0) propagating components of\na standing MHD kink wave for both the right-hand ( α >0) and the left-hand ( α <0)\ntwisted magnetic fields.\n•Since we used the thin tube approximation ( ǫ=kzR≪1), the results are obtained for ǫ\nin the range [ −π\n50,π\n50].\n•If the magnetic twist value in the loop, which here is defined a s\nφ≡L\nRBϕ\nBz/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=R= 2πNtwist (23)\nexceeds a critical value φc, then the loop becomes kink unstable (e.g. Shafranov 1957;\nKruskal et al. 1958). Here, Lis the length scale of the wave propagation in the zdirection\n9which we assume to be equal to the wavelength ( λ) andNtwistis the number of twist turns\nper length L. Substituting L= 2π/kz=λin Eq. (23) we get\nφ= 2πα\nǫ. (24)\nSo, there is a constraint on the maximum value of the twist par ameterαin order to the\nkink oscillations of the flux tube be stable. Note that Ruderm an (2015) derived solutions\n(7) and (8) in the weak twist approximations. He assumed α≤ǫ(i.e.φ≤φc= 2π) in\norder to avoid the kink instability in the flux tube. Hence, th e dispersion relation obtained\nin this paperis valid in this limit. Terradas andGoossens (2 012) obtained the critical value\nof the magnetic twist from equalizing the contribution of th e azimuthal and longitudinal\ncomponents of the magnetic field in the background Alfv´ en co ntinuum. They found that\nin order to have a stable kink oscillation the amount of twist must be smaller than π.\nSince they considered the longitudinal length scale as L=π/kz=λ/2, the critical twist\nobtained by them is equivalent to φc= 2πthat we use in this paper.\nWith these considerations, for a given ǫ∈[−π/50,π/50], from Eq. (24) we deduce that in\norder to have valid results, the twist parameter must be in th e range\n−ǫ≤α≤ǫ. (25)\nFrom Eq. (24) we have\nǫ=2π\nφα. (26)\nThis equation shows that any linear relation between ǫandαcorresponds to a constant φ.\nIn other words, if we adopt a constant φin the flux tube, there exists a set of infinite pairs\nofǫandαsatisfying Eq. (26) that make a straight line in the α−ǫplane. For example, for\nm= +1 ifφ= 2π(right handed twist) then we have α=ǫline (dashed lines in the figures)\nand ifφ=−2π(left handed twist) it yields α=−ǫline (dashed-dotted line in the figures).\nThese lines are the borders between the kink stable and kink u nstable regions. In the con-\ntour plots a point with |α|>|ǫ|that is equivalent to |φ|>2πis in thekink unstableregion.\n•In thin boundary approximation we can estimate the location of the resonance as rA≃\n(a+R)/2.\nWith these considerations we solve Eq. (15) and find its compl ex roots numerically in the\nkink stable regions. We have obtained the oscillation frequ encyωand the damping rate γof\nthe kink waves as a function of αandǫform= +1. Figs. 2 and 3 show the contour plots of\nthe oscillation frequency, the damping rate and the ratio of the frequency to the damping rate\nin theα−ǫplane. Fig. 2 is for ζ= 2 and Fig. 3 is for ζ= 10. In both figures the dashed and\ndashed-dotted lines in the plots correspond to φ= 2π, andφ=−2πrespectively. As illustrated\nin the figures, we do not plot the results in the kink-unstable regions where the magnitude of\nthe magnetic twist, |φ|, is larger than its critical value φc= 2π. Hence, we have plotted the\nresults in the regions which are valid according the stabili ty criterion. It is clear from the top\npanel of Figs. 2 and 3 that in the both cases of ζ= 2 and ζ= 10 when ǫ >0 (ǫ <0) with\nincreasing (decreasing) the twist parameter, α, the frequency of the kink wave increases. This\nresults are in agreement with the results shown in the top pan el of figure 1 in Bahari (2018).\nAlso it is clear from the top panel of Figs. 2 and 3 that in the bo th cases of ζ= 2 and ζ= 10,\nfor a given value of ǫ, the frequency of the kink waves is a monotonic function of αthat starts\n10fromω=ωAi= 0 at a point on the line φ=−2π(i.eǫ=−α) (the dashed-dotted line in the\nfigures). This means that for the model considered in this pap er, when m= +1 and φ=−2π\nthe kink wave do not propagate in the flux tube.\nOur results show that if the amount of the magnetic twist φexceeds a critical value φ∗, the\nreal part of the obtained frequency falls below the backgrou nd Alfv´ en frequency continuum (i.e\nω < ωAi). In this case the oscillation is non-resonant. Note that th e critical point that distinct\nthe resonant and non-resonant regions, is determined by φnotǫneitherαalone. The obtained\nvalue ofφ∗forζ= 2 andζ= 10 isφ∗≃3π/5 andφ∗≃7π/4, respectively. As mentioned before,\nfrom Eq. (26) we see that each of these constant values of φ∗corresponds to a straight line in\ntheα−ǫplane. The red dotted line in Figs. 2 and 3 correspond to φ= 3π/5 andφ= 7π/4\nrespectively, that exhibit the location where ω=ωAi. This result is interesting because it\nshows that if the twist parameter φbecomes larger than a certain value specified by the density\ncontrast, there exists an undamped kink wave in the presence of inhomogeneous boundary layer,\nand resonant absorption cannot be considered as the damping mechanism for this wave. Note\nthat since our model is different form the model studied by Terr adas and Goossens (2012) there\nis differences between the conditions in which resonant absor ption takes place. In Terradas and\nGoossens (2012) the twist has been assumed to be small enough to the waves be undamped,\nbut in our model in order to the resonant absorption does not t ake place and the waves be\nundamped, the twist must be assumed to be large enough.\nThe damping rate of the kink waves in the tubes with density co ntrastsζ= 2 andζ= 10 are\nshown in the middle panels of Figs. 2 and 3 respectively. As il lustrated in the figures, for ζ= 2\nandζ= 10 the damping rate is plotted in the ranges ( −2π,3π/5) and (−2π,7π/4), respectively.\nIt is clear that for any value of ǫthe damping rate is not a monotonic function of α. As the\nmiddle panels of Figs. 2 and 3 show, for both cases of ζ= 2 and ζ= 10, the damping rate\nof the resonant kink wave near the φ=−2πline isγ= 0, and with increasing φfrom−2π\nthe damping rate increases and reaches a maximum value and th en decreases again to γ= 0 at\nφ= 3π/5 and 7π/4, respectively.\nThe absolute value of the ratio of the frequency to the dampin g rate has been illustrated in\nthe bottom panels of Figs. 2 and 3. Note that we only show the re sults for −ω/γ <100. In\norder to justify the rapid damping of MHD kink waves ( ω/(2πγ) =τD/P≃3−5) reported\nby the observations, the ratio −ω/γmust be in the range (20 ,30). As Fig. 3 shows, in the\ncase ofm= +1 and ζ= 10, this happens in a narrow triangular shape region where αand\nǫhave opposite signs. As Fig. 2 shows, the minimum value of −ω/γ= 51 that is equivalent\ntoτD/P≃8 is comparable with observational values. So, the model con sidered in this paper\ncan justify the rapid damping of the MHD kink waves for specifi c pairs of αandǫthat have\nopposite signs if m= +1. The bottom panels of Figs. 2 and 3 also reveal that the con tour lines\nof the plots of ω/γcoincide with φ=constlines. Previous results obtained by e.g. Ruderman\n& Roberts (2002) show that in the absence of the magnetic twis t, the ratio ω/γis not a function\nofǫthat is a special case of the results obtained here for φ= 0. Our results show that in\ngeneral, the ratio ω/γremain unchanged when the twist value is constant in the flux t ube. In\nother words, for a given α/ne}ationslash= 0 the ratio ω/γis a function of ǫ, but ifαandǫvary together so\nthat the twist value remain unchanged, it does not affect the ra tioω/γ. This is an important\nresults which should be considered in the seismology of coro nal flux tubes, since for a given ζby\nmeasuring the ratio of the damping time to the period of a deca ying kink wave, we can estimate\nthe amount of the magnetic twist φin the tube.\nTerradas and Goossens (2012) showed that it is possible to co nstruct a standing kink wave in\nthe presence of a twisted magnetic field from two forward and b ackward propagating kink waves\nof different longitudinal wavenumbers that have the same freq uencies and propagate in opposite\n11directions. They studied undamped kink waves and both of the wavenumber and frequency of\nthe waves were real. But here the damped kink waves have compl ex frequencies. Our results\nshow that even if we can find two propagating waves that have eq ual frequencies their damping\nrate are not the same and as a results it is not possible to make a standing kink oscillation from\ntwo propagating damping kink waves in the presence of a twist ed magnetic field. This result is\nan interesting subject to study that is beyond the goals of th is paper.\n5 Conclusions\nHere, weinvestigated dampingofstandingMHDkinkwaves ina coronal fluxtubeinthepresence\nof a twisted and continuous magnetic field. To this aim we obta ined a dispersion relation for the\nindividual propagating components of the standing kink wav e. In the considered model resonant\nabsorption occurs as a result of the existence of a thin layer of density inhomogeneity at the\nsurface of the tube that connects smoothly the constant dens ities of the interior and exterior\nregions. The kink stability of the flux tube depends on both of the twist parameter and the\nlongitudinal wave number. So, in order to investigate the effe ct of the twisted magnetic field on\nthe MHD kink waves, we used the contour plots of the results ve rsus parameters αandǫ. The\nresults show that:\n•Foragiven ǫ, thefrequencyofthekinkwaves isamonotonicfunctionofth etwistparameter\neven when the range of the variation of the twist parameter in clude both positive and\nnegative values. But for a given value of ǫthe damping rate is not a monotonic function\nof the twist parameter.\n•Resonant absorption occurs only when the value of the magnet ic twist φis in a range\nspecified by the density contrast of the tube.\n•The magnitude of the ratio of the frequency to the damping rat e (ω/γ) is not a monotonic\nfunction of αorǫin general. The results show that for the parameters conside red in this\nstudy (m= 1,ζ= 2,10),αandǫmust have opposite signs in order to justify the rapid\ndamping of the kink oscillations in coronal flux tubes.\n•Forα= 0, the ratio ω/γis not affected by ǫthat is in good agreement with the previous\nresults obtained by e.g. Ruderman & Roberts (2002). Our resu lts show that in general,\nω/γis a function of φand remain unchanged when φ=constfor different values of ǫand\nα.\n•Our results show that for the model considered in this paper, in contrast to the undamped\nkink waves (see Terradas and Goossens 2012), it is not possib le to construct a resonantly\ndamped standing kink wave from two forward and backward prop agating kink waves with\nthe same frequencies, since the corresponding damping rate s are always different.\nReferences\n[1] Andries J., Goossens M., Hollweg J.V., Arregui I., Van Do orsselaere T., 2005, A&A, 430,\n1109\n[2] Aschwanden M. J. et al., 1999, ApJ, 520, 880\n[3] Aschwanden M. J., Nightingale R. W., Andries J., et al., 2 003, ApJ, 598, 1375\n12-0.06 -0.04 -0.02 0 0.02 0.04 0.06-0.063-0.03100.0310.063=2 =-2 =3/5\n00.010.020.030.040.050.060.070.08\nkink-unstable kink-unstableresonant\nresonantnon-resonant\nnon-resonant\n-0.06 -0.04 -0.02 0 0.02 0.04 0.06-0.063-0.03100.0310.063=2 =-2 =3/5\n-1-0.8-0.6-0.4-0.20\n10-3kink-unstable kink-unstableresonant\nresonantnon-resonant\nnon-resonant\n-0.06 -0.04 -0.02 0 0.02 0.04 0.06-0.063-0.03100.0310.063-/=2 =-2 =3/5\n556065707580859095\nkink-unstable kink-unstable55\n5155 non-resonant\nnon-resonantresonantresonant\nFigure 2: Contour plots of the frequency (top), the damping r ate (middle) and the ratio of the\nfrequency to the damping rate (bottom) versus αandǫform= 1 and ζ= 2. The dashed and\ndashed-dotted lines illustrate φ= 2πandφ=−2π, respectively. The red dotted line separate\nthe resonant and non-resonant regions.\n13-0.06 -0.04 -0.02 0 0.02 0.04 0.06-0.063-0.03100.0310.063=2 =-2 =7/4\n00.020.040.060.080.10.12\nkink-unstable kink-unstablenon-resonant\nnon-resonant\nresonantresonant\n-0.06 -0.04 -0.02 0 0.02 0.04 0.06-0.063-0.03100.0310.063=2 =-2 =7/4\n-4-3.5-3-2.5-2-1.5-1-0.50\n10-3kink-unstable kink-unstable\nresonantresonant\nnon-resonantnon-resonant\n-0.06 -0.04 -0.02 0 0.02 0.04 0.06-0.063-0.03100.0310.063-/=2 =-2 =7/4\n102030405060708090\nkink-unstable kink-unstable1040 3020\nresonant\nresonantnon-resonant\nnon-resonant\nFigure 3: Same as Figure 3 but for ζ= 10.\n14[4] Bahari K., 2017, Sol. Phys., 292, 110\n[5] Bahari K., 2018, ApJ, 864, 2\n[6] Bahari K., Khalvandi M.R., 2017, Sol. Phys., 292, 192\n[7] Baty H., Heyvaerts J., 1996, A&A, 308, 935\n[8] Biskamp D., 2000, Magnetic Reconnection in Plasmas. Cam bridge Univ. Press, Cambridge\n[9] Chae J., Wang H., Qiu J., Goode P. 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S., Roberts B., 2002, ApJ, 577, 475\n[40] Ruderman M. S., Terradas J., 2015, A&A, 580, A57\n[41] Sakurai T., Goossens M., Hollweg J. V., 1991, Sol. Phys. , 133, 227\n[42] Shafranov V. D., 1957, J. Nucl. Energy II, 5, 86\n[43] Soler R., Terradas J., Goossens M., 2011, ApJ, 734, 80\n[44] Terradas J., Goossens M., 2012, A&A, 548, A112\n[45] Terradas J., Goossens M., Ballai I., 2010, A&A, 515, A46\n[46] Uchida Y., 1970, Publ. Astron. Soc. Jpn., 22, 341\n[47] Verwichte E., Nakariakov V. M., Ofman L., Deluca E. E., 2 004, Sol. Phys., 223, 77\n[48] Wesson J., 2004, Tokamaks. Oxford Univ. Press, Oxford\n16" }, { "title": "1012.3901v1.Optimal_switching_of_a_nanomagnet_assisted_by_microwaves.pdf", "content": "arXiv:1012.3901v1 [cond-mat.mtrl-sci] 17 Dec 2010Optimal switching of a nanomagnet assisted by microwaves\nN. Barros1, M. Rassam1, H. Jirari2, and H. Kachkachi1\n1LAMPS, Universit de Perpignan Via Domitia, 52 Avenue Paul Al duy, 66860 Perpignan Cedex, France.\n2LPMMC, C.N.R.S-UMR 5493, Universit´ e Joseph Fourier, BP 16 6, 38042 Grenoble-Cedex 9, France\nWe develop an efficient and general method for optimizing the m icrowave field that achieves\nmagnetization switching with a smaller static field. This me thod is based on optimal control and\nrenders an exact solution for the 3 Dmicrowave field that triggers the switching of a nanomagnet\nwith a given anisotropy and in an oblique static field. Applyi ng this technique to the particular\ncase of uniaxial anisotropy, we show that the optimal microw ave field, that achieves switching with\nminimal absorbed energy, is modulated both in frequency and in magnitude. Its role is to drive\nthe magnetization from the metastable equilibrium positio n towards the saddle point and then\ndamping induces the relaxation to the stable equilibrium po sition. For the pumping to be efficient,\nthe microwave field frequency must match at the early stage of the switching process the proper\nprecession frequency of the magnetization, which depends o n the magnitude and direction of the\nstatic field.\nWe investigate the effect of the static field (in amplitude and direction) and of damping on the\ncharacteristics of the microwave field. We have computed the switching curves in the presence of\nthe optimal microwave field. The results are in qualitative a greement with µ-SQUID experiments\non isolated nanoclusters. The strong dependence of the micr owave field and that of the switching\ncurve on the damping parameter may be useful in probing dampi ng in various nanoclusters.\nPACS numbers: 75.50.Tt,75.75.-n,75.10.Hk\nI. INTRODUCTION\nFine magnetic clusters offer tremendous challenges\nboth in the area of fundamental science and practical\napplications. The main reasons for this impetus are the\nnovel features related with their small size, such as the\npossibility of high density storage, short-time switching1\nand fast read-write processes. On the other hand, the\nsmall size is a drawback in many regards. The energy\nbarrier in these systems is too small to ensure a reason-\nable stability, in a given energy minimum, that is neces-\nsary for practical applications at room temperature, e.g.,\nmagnetic recording. This is the problem of superparam-\nagnetism. A possible way out would be to use materials\nwith high anisotropy and thus ensuring a high energy\nbarrier. A consequence of this is that high values of the\nwriting (or switching) fields are required. However, it\nis still unclear how to devise such high fields operating\non the scale of nanoclusters while avoiding the ensuing\nnoise. In order to keep the size small, the energy barrier\nhigh, and the switching field small, other routes are ex-\nplored and a promising one among them is provided by\nmicrowaves. Microwave-assisted magnetization switch-\ning in various magnetic systems, such as thin films, has\nbeen investigated by many groups2. In fact, we have\nat hand a more general and fundamental issue, namely\nthe problem of getting a system out of an energy mini-\nmum by nonlinear resonance. This has previously been\naddressed in many areas of physics and chemistry, espe-\ncially in the context of atomic physics. For example, Liu\netal.4havestudiedthedissociationofdiatomicmolecules\nby a chirped infrared laser pulse and showed that this\nprocess requires a much lower threshold laser intensity\nto achieve dissociation. The quantum regime5has beenstudied in terms of energy-ladderclimbing and givesvery\nsimilarresults. Experimentalevidenceofthis processhas\nbeen provided by the dissociation of HF molecules using\na sub-nanosecond frequency modulated laser pulse6.\nAccording to the classical theory of auto-resonance or\nthe quantum theory of ladder-climbing6–8, exciting an\noscillatory nonlinear system to high energies is possible\nby a weak chirped frequency excitation. Moreover, trap-\nping into resonance followed by a (continuing and stable)\nphase-locking with the drive is possible if the driving fre-\nquencychirp rateissmall enough. It hasalsobeen shown\nthat a slow passage through and capture into resonance\nyields efficient control of the energy of the driven sys-\ntem. Incidentally, an important theoretical result is that\nthe precise form of the time dependence of the oscillating\nfield is not essential for the process to succeed.\nFor nanoclusters, it has been shown in previous works\nhow a monochromatic microwave (MW) pulse can, by\nmeans of a non-linear resonance, substantially reduce the\nrequired static field needed to reverse the magnetization\nofanindividualnanoparticle. Indeed, ithasbeen demon-\nstrated using the µ-SQUID technique on a 20-nm cobalt\nparticle9that adding an MW field with given rising time,\nduration, and frequency, on top of the static (DC) mag-\nnetic field, the switching of the nanocluster is possible\nat a static field lower than the Stoner-Wohlfarth (SW)\nswitching field and within a time interval of the order\nof a nanosecond. The switching curves, or the so-called\nSW astroids, obtained in these measurements present\nsomeirregularfeaturesimplying thatthereductionofthe\nswitching field is not uniform, as is the case with thermal\neffects. The global features depend on several physical\nparameters, such as the MW field pulse duration, its ris-\ning time, and its frequency, the DC field amplitude, and2\nthe damping parameter9. The SW astroid obtained with\nits peculiar features seems to bear the fingerprints of the\nnanocluster and its underlying characteristics such as its\npotential energy. Moreover, the strong dependence of\nthese features on the damping parameter might be used\nto estimate the latter in such clusters.\nOn the theory side, severalworkshavebeen devoted to\nthe understanding of the magnetization dynamics, and\nin particular its reversal, under the effect of a time-\ndependent magnetic field. The theoretical work may be\ndivided into two kinds. The first deals with the effect of\na given MW field with a given polarization10,11while the\nsecond seeks optimal strategies for achieving the magne-\ntization switching12. In particular, a few works, e.g.13,\nassume a given dynamics for the magnetization and at-\ntempt to determine the MW field that realizes it.\nIn the presentwork, weuse ageneralmethod borrowed\nfrom the optimal control theory14–16and apply it to the\nswitchingofananocluster. Thismethod rendersanexact\nsolution for the MW field vectornecessaryfor the switch-\ning of a nanomagnet with a given potential energy (com-\nprising anisotropy and an oblique static field). The stan-\ndard formulation of this method consists in minimizing\na cost functional using the conjugate gradient technique.\nThe latter is known to be a local-convergence method\nand thus renders a solution that is rather sensitive to\nthe initial guess. In order to acquire global convergence\nand thus have solutions for the MW field that are sta-\nble with respect to a change in the initial conditions, we\nhave supplemented the conjugate gradient routine by a\nglobal search using the Metropolis algorithm and simu-\nlated annealing. Then, we have applied our algorithm to\na nanomagnet in the macrospin approximation with uni-\naxial anisotropyand oblique DC magnetic field. We have\ninvestigated the effect of the latter (both in direction and\nmagnitude) and of damping on the characteristics of the\nMW field. Then, we computed the limit-of-metastability\n(orswitching) curvesfor different (small) valuesofdamp-\ning.\nII. METHOD OF OPTIMAL CONTROL\nAPPLIED TO NANOMAGNETS\nOne of our objectives here is to develop a general\nmethod that allowsus tosolvethe followinginverseprob-\nlem: what is the optimal time-dependent magnetic field\nunder which the magnetization of a nanoparticle, with\ngiven potential energy, switches from a given initial state\nto a given final prescribed target state? After formulat-\ning this method we apply it to the case of a macrospin,\nor a nanoparticle in the SW approximation, in an energy\npotential composed of uniaxial anisotropy and a Zeeman\ncontribution from an oblique DC field.\nThe method we propose is borrowed from the opti-\nmal control theory. The main idea is to start with an\narbitrary MW magnetic field hAC(t) with its three com-\nponentshα\nAC,α=x,y,z, that we call the control field , inaddition to the static magnetic field and anisotropy field.\nWe then determine hAC(t) that triggers the switching\nof the cluster’s magnetization between two given states\nwithin a prescribed interval of time.\nA. Model\nConsider a nanomagnet in the macrospin approxima-\ntion where its magnetic state is represented by a macro-\nscopic magnetic moment m=µss,whereµsis its mag-\nnitude and sits direction with |s|= 1. In this ap-\nproximation, the relevant terms entering the energy Eof\nthe nanomagnet are the magneto-crystalline anisotropy\nand the Zeeman energy. The applied DC (static) field\nHDCis assumed to point in an arbitrary direction eh=\nHDC/HDC. Using the convention µ0= 1 so that the\nmagnetic fields are expressed in Tesla, one then defines\nthe effective field\nHeff=−1\nµsδE\nδs(1)\nand writes the damped Landau-Lifshitz equation in the\nGilbert form (LLE) that governs the dynamics of s, as-\nsuming that the module µsremains constant,\n1\nγds\ndt=−s×Heff−αs×(s×Heff) (2)\nwhereγ≃1.76×1011(T.s)−1is the gyromagnetic factor\nandαthe phenomenological damping parameter (taken\nhere in the weak regime).\nWemeasureallappliedfieldsintermsoftheanisotropy\nfield\nHa=2KV\nµs(3)\nand in particular we define the reduced effective field\nheff≡1\nHaHeff=−δE\nδs, (4)\nwithE ≡E/(2KV). In terms of heffLLE becomes\nds\ndτ=−s×heff−αs×(s×heff) (5)\nwhereτ≡t/tsis the dimensionless time and ts=\n1/(γHa) the characteristic scaling time of the system.\nFor instance, for a cobalt particle of 3 nm diameter17\nwithK≃2.2×105J.m−3,µs≃3.8×10−20A.m2we\nhaveHa≃0.3 T and ts≃1.9×10−11s.\nB. Formulation of the optimal control problem\na. General procedure The idea here is to introduce\na control field hAC(τ)≡HAC/Haand then seek its op-\ntimal form that allows for driving the magnetic moment3\ndirection sfrom the given initial state s(i)at timeτi= 0\ninto the desired final state s(f)at the given observation\ntimeτf. Accordingly, we replace in the LLE (5) the (de-\nterministic) field heffby the total (time-dependent) field\nζ(τ) =heff+hAC(τ). (6)\nThis results in the following equation of motion, which\nwill be henceforth referred to as the driven LLE (DLLE)\n˙ s=−s×ζ(τ)−αs×(s×ζ(τ)). (7)\nThe field hAC(τ) is then determined through the mini-\nmization of a cost functional which, in the present case,\nmay be written as\nF[s(τ),hAC(τ)] =1\n2/vextenddouble/vextenddouble/vextenddoubles(τf)−s(f)/vextenddouble/vextenddouble/vextenddouble2\n+η\n2τf/integraldisplay\n0dτh2\nAC(τ)\n(8)\nThefirstterm measuresthedegreeatwhichthe magnetic\nmoment switching is achieved and vanishes in the case\nof full switching. The second term is quadratic in the\ndriving field and is thus proportional to the absorbed\nenergy. The parameter η, called the control parameter ,\nallows us to balance the second condition with respect to\nthe first.\nTherefore, the problem of optimal control boils down\nto minimizing the costfunctional (8) alongthe trajectory\ngiven by DLLE (7). More explicitly, this amounts to\nsolving the following problem\n\n\nmin/braceleftbigg\nF[s,hAC] =1\n2/vextenddouble/vextenddouble/vextenddoubles(τf)−s(f)/vextenddouble/vextenddouble/vextenddouble2\n+η\n2τf/integraltext\n0dτh2\nAC(τ)/bracerightbigg\n˙ s=−s×ζ−αs×(s×ζ), τ∈[0,τf]\ns(0) =s(i).\n(9)\nAn optimal solution of this problem is characterized\nby the first order optimality condition in the form of the\nPontryagin minimum principle (PMP)18. These condi-\ntions are more conveniently formulated with the help of\na Hamilton function which may be in the present case\nwritten in the following form\nH[s(τ),λ(τ),hAC(τ)] =η\n2h2\nAC(τ) (10)\n+λ(τ)·{−s×ζ−αs×(s×ζ)},\nwhereλ(τ), called the adjoint state variable [see be-\nlow], is a Lagrange parameter introduced to implement\nthe constraint and thereby render s(τ) independent of\nhAC(τ). The PMP then states that solving the problem\n(9) is equivalent to solving the following boundary prob-\nlem (i.e., the Hamilton-Jacobi equations with boundaryconditions)\n\n\n˙s=δH\nδλ,s(0) =s(i), τ∈[0,τf],\n˙λ=−δH\nδs,λ(τf) =s(τf)−s(f),\nδH\nδhAC= 0.(11)\nThe last condition is also equivalent to the vanishing of\nthe gradient of the cost functional Fin Eq. (8). It yields\nthe equation\nδH\nδhAC=ηhAC+s×λ−αs×(s×λ),(12)\nwhich is used to compute the variation in the cost func-\ntional, that is\nδF=τf/integraldisplay\n0dτδH\nδhAC·δhAC. (13)\nIn general, this problem is highly nonlinear and con-\nsidering, on top of that, the non-linearity of the Landau-\nLifshitz equation, it isnotpossibletofind analyticalsolu-\ntions. Consequently, we resort to numerical approaches.\nThe advantage of this formulation is manifold: i) the\nMW field hAC(τ) is obtained in 3 D,i.e., one obtains\nthe three functions of time hα\nAC(τ),α=x,y,z; and for\nany potential energy (anisotropy, DC field, etc), ii) the\nfinal time τf, and the absorbed power (second term in\nEq. (8)) can be adjusted; the latter may be achieved by\ntuning the control parameter η, iii) one can generalize\nthis treatment to many-spin problems19and also include\nthermal effects.\nOne of the most efficient techniques for (numerically)\nsolving such a minimization problem is the conjugate-\ngradient method. However, the drawback of this method\nisthatitisalocal-convergencemethod, whichmeansthat\nthesolutionit rendersisstronglydependent onthe initial\nguess. We overcome this inconvenience by supplement-\ning the method by a global search using the Metropolis\nalgorithm with random increments and then proceed by\nthe technique of simulated annealing.\nFor numerical calculations, we have discretized the\nboundary-value problem (11) by subdividing the time in-\nterval [τi= 0,τf] intoNtime slices\nτn=τi+n×∆τ, n= 0,...,N−1, τf=τN−1,\nwhere\n∆τ=τf−τ0\nN−1.\nThen, using the notation vn=v(τn) for a vector v,\nEqs. (7, 8, 13) and the equation for λ, become4\nsn+1=sn+∆τ×[−sn×ζn−αsn×(sn×ζn)],s(τi) =s(i), (14a)\nF=1\n2/vextenddouble/vextenddouble/vextenddoublesN−1−s(f)/vextenddouble/vextenddouble/vextenddouble2\n+η∆τ\n2N−1/summationdisplay\nn=0h2\nAC,n, (14b)\nλn−1=λn−∆τ×Λn,λf=sN−1−s(f), (14c)\nVn=δF\nδhAC,n= ∆τ×[ηhAC,n+sn×λn−αsn×(sn×λn)]. (14d)\nThe explicit expression for Λ nin (14c) depends on\nthe energy potential [see below for the case of uniaxial\nanisotropy].\nWemaysummarizethenumericalprocedureasfollows.\ni) for a given initial guess of the control field hAC(t), we\nfirst solve the state equation (14a) forward in time using\nthe initial condition, and then evaluate the cost func-\ntional (14b), ii) the solution obtained for sis then used\nfor the backward (since the condition now is at tf) inte-\ngration of the equation (14c) for λ, iii) with the trajec-\ntories of sandλthus obtained we compute the gradient\n(14d). The numerical subroutines are standard and can\nbe found in Ref. 20. We emphasize that obtaining the\ncontrol field amounts to solving for 3 ×Nvariables.\nb. Uniaxial anisotropy In the case of uniaxial\nanisotropy with oblique static field the energy of the\nnanomagnet reads (in units of the anisotropy energy\n2KV)\nE=−hDC(eh·s)−1\n2(s·n)2, (15)\nwithKandnbeing the anisotropy constant and easy\naxis,Vthe nanomagnet volume and hDC≡HDC/Ha.\nThe effective field explicitly reads [see Eq. 4]\nheff=hDCeh+(s·n)n. (16)\nFrom the second equation in (11) we obtain the explicit\nequation for λ\n˙λ=ζ×λ+α[ζ×(λ×s)+λ×(ζ×s)]\n+ [λ·(s×n+αs×(s×n))]n (17)\nand in Eq. (14c) we now have\nΛn=ζn×λn+α[ζn×(λn×sn)+λn×(ζn×sn)]\n+[λn+α(λn×sn)]·(sn×n)n.\nIII. RESULTS\nIn the present work, we have considered the case of a\nnanomagnet with uniaxial anisotropy and oblique static\nfield. Unless otherwise stated, the latter is applied in the\nyzplanemakingan angleof170◦with respect tothe easy\nFigure 1: Optimized MW field (upper panel) and the corre-\nsponding spin trajectories (lower panel). The inset is a 3 D\nplot of the spin trajectory on the unit sphere.\naxis (zaxis). Its reduced magnitude is hDC= 0.5, corre-\nspondingtoafieldmagnitude HDC≃150mT.Theinitial\nposition and target states s(i)ands(f), which correspond\nrespectively to the metastable equilibrium state and sta-\nble equilibrium state, arecomputed numerically. The ob-\nservationtimeis τf= 600(i.e.,tf≃11.4ns). Thedamp-\ning parameter is α= 0.05 and the control parameter η\nhas been set to 0 .01. The static field, damping parameter\nand observation time have been varied and their effects\nstudied [see later on]. For simplicity, we have taken a lin-\nearly polarized MW field, i.e.,hAC(t) =hAC(t)ex. This\nchoice also suits the experimental setup9.\nIn Fig. 1 we have plotted the optimized MW field\nmagnitude HAC(t)≡HahAC(t) where tis the time in5\n6 7 8 9012345\nInstantaneous frequency of the MW field\nFMR frequency \nTime (ns)Frequency (GHz)\nFigure 2: Instantaneous frequency of the optimized and fil-\ntered MW field of Fig. 1. The other parameters are the same\nas in Fig. 1.\nseconds, together with the components of the magnetic\nmoment, i.e.,sα(t),α=x,y,z. First, we note that the\namplitude of the MW field is rather small as it does not\nexceed 15 mT, which is 10 times smaller than the static\nfield. Moreover, the summed magnitudes of the DC and\nMW field are smaller than the SW switching field for the\nchosen DC field direction (about 200 mT). This shows\nthat, in the presence of a MW field, magnetic switching\nis achieved at a smaller DC field. Second, the striking\nfeature is that the MW field is modulated both in am-\nplitude and frequency. Its frequency is a slowly varying\nfunction of time in the stage that precedes switching, as\ncanbe seeninFig. 2. Third, asishintedtobythedashed\nvertical lines, the extrema in the MW field and the spin\ncomponents sy(t) andsz(t) match at all times before\nswitching. This simply implies that the magnetic mo-\nment is phase-locked to the MW field. All these features\nagree with the predictions of the classical auto-resonance\nor the ladder-climbing quantum theory, as summarized\nin the introduction.\nThe instantaneous frequency has been obtained after\npassingthe optimized MWfield throughthe Butterworth\nfilter and then applying the Hilbert transformation21,22.\nAs can be seen, for short times the instantaneous fre-\nquency oscillates around the approximate value f0≈\n4.1GHz. This initial frequency is simply the FMR fre-\nquency given by\nfFMR=γHa\n2π/radicalbigg\nh(i)\neff,/bardbl/parenleftBig\nh(i)\neff,/bardbl+k/bracketleftBig/parenleftbig\ns(i).n/parenrightbig2−1/bracketrightBig/parenrightBig\nwhereh(i)\neff,/bardbl≡heffi(i)·s(i)=hDC/parenleftbig\neh·s(i)/parenrightbig\n+ (s(i)·n)2\nis the effective field (16) evaluated at and then pro-\njected onto the initial position s(i). As the magnetic mo-\nmentapproachesthesaddlepointthefrequencydecreases\nrapidly and eventually vanishes when the magnetic mo-\nment crossesthe saddle point into the more stable energy\nminimum.In Fig. 1 it is seen that the time span comprises three\nstages (for the set of physical parameters considered):\n1)Nucleation stage (up to 5 .4ns). The MW field re-\nmains almost zero and the magnetic moment remains in\nthe metastable state. 2) Driven precession (from 5.4ns\nto 9.7ns). Here the MW field and the magnetic moment\naresynchronized. At each processioncycle, the MW field\nhooks up the magnetic moment and pushes it upwards\nin the energy potential towards the saddle point. This\nis the phase-locking process mentioned in the introduc-\ntion and observed above. This is indeed possible because\nthe frequency chirp rate is small as can be seen in Fig.\n2 for 5.4ns≤t≤9.7ns. The MW field thus compen-\nsates for the effect of damping that tends to pull the\nmagnetic moment back towards its initial position. At\naround 9 .7ns, the magnetic moment crosses the saddle\npoint. 3) Free relaxation : from 9.7ns onward, the mag-\nnitude of the MW field dwindles and the synchronization\nwith the magnetic moment is lost. We note that at the\nsaddle point the precession reverses from being counter-\nclockwise to clockwise as the magnetic moment switches\nto the lower half sphere.\nNumerical tests show that the MW field can be re-\nplaced by zero during the nucleation and free relaxation\nstages without noticeably affecting the trajectory of the\nmagnetic moment. This implies that the most relevant\npartofthe signalisthat duringthe drivenprecession; the\nrole of the MW field is thus to drive the magnetic mo-\nment towards the saddle point. Next, the damping takes\nup to lead it to the more stable energy minimum. Dur-\ning the driven precession the frequency of the MW field\nandthe precessionfrequencyofthe magnetic momentare\nsimilar. Consequently, the magnetic moment switching\ncan be viewed as a resonant process: the pumping by\nthe MW field is efficient when its frequency matches the\nfrequency of the magnetization (phase-locking).\nThe same calculation has been carried out with the\nsame sampling time but different values for the total ob-\nservation time tf. The results are shown in Fig. 3. If the\ntotal time is larger than an effective time of 6ns, sim-\nilar values are obtained for the cost functional and the\ncurveshAC(t) can be matched after a time shift. As was\ndiscussed earlier, this effective time corresponds to the\nsum of the time of driven precession and that of free re-\nlaxation. This result implies that the nucleation stage\ncan be suppressed without affecting the final optimized\nMWfield. However,ifthetotaltimeistooshort,thefinal\nvaluefound forthe costfunctional ishigher( i.e.not fully\nminimized). Indeed, we see in Fig. 3 (uppermost panel)\nthat the stage of driven precession is shortened and the\nshape of the control field changes so as to achieve a faster\nswitching and thus comply with the switching-time con-\nstraint [first term in Eq. (8)].\nTheeffectofvaryingtheamplitudeofthestaticfieldon\nthe MWfield isshownin Fig. 4. We seethat the shapeof\nthe MW field envelop remains the same, apart from the\nfact that the smaller the static field, the more symmet-\nrical is the MW field. This shows that for a higher field6\n-1001020 tf = 1.9 ns\n-1001020tf = 7.6 ns\n-1001020 tf = 11.3 ns\n0 5 10-1001020tf = 15.1 ns\nTime (ns)Optimized RF field HAC(t) (mT)\nFigure 3: Optimized MW field obtained with four different\ntotal times tf.\nhDC, the energy potential is less symmetrical. Moreover,\nashDCis increased the energy barrier is lowered and the\nMWfield requiredtoachieveswitchingis smaller. Again,\nthe initial frequency of the oscillations matches the FMR\nfrequency of the system; when hDCincreases, the latter\ndecreases. The cost functional was found to be propor-\ntional to the energy barrier between the saddle point and\nthe metastable minimum. Hence, when the energy bar-\nrier is higher, more energy has to be injected in order to\novercome it. The same study has been carried out upon\nvarying the direction of the static field.\nWe have also investigated the effect of varying the\ndamping parameter αon the MW field. The results are\nsummarized in Fig. 5. We see that the intensity of the\nfield increaseswith α, which is compatible with what was\nsuggested earlier, namely that the role of the MW field is\nto compensate for the damping effect. This effect is sim-\nilar to what happens with a rubber band: the more you\nstretch it the harder it becomes to do so. Moreover, the\neffective duration of the MW field, which mainly corre-\nsponds to the driven precession period, decreases when α\nincreases. We note that, on the contrary, the initial fre-\nquency of the oscillations is independent of α[see Fig. 6].\nThis result can be understood qualitatively if we suppose\nthat, at any time, the MW field exactly compensates for\nthe effect of damping. The spin dynamics is then gov-\nerned by the undamped LLE and the magnetic moment-30-1501530HDC = 0 mT\n-30-1501530HDC = 93 mT\n4 6 8 10-30-1501530HDC = 155 mT\nTime (ns)Optimized RF field HAC(t) (mT)\nFigure 4: Optimized MW field obtained for different magni-\ntudes of the static field hDC, in the same direction making an\nangleθ= 170◦with respect to the anisotropy easy axis.\nprecesses with its proper frequency, which is independent\nof the damping parameter. At short times, since the pre-\ncession angle is small, this precession frequency is equal\nto the FMR frequency.\nAs discussed in the introduction, one of the objectives\nof investigating the magnetization switching assisted by\nMWs is to achieve an optimal switching with smaller DC\nmagnetic fields than it wouldbe necessarywithout MWs.\nThis means that applying the DC field in a given di-\nrection and varying its magnitude one determines the\nswitching field (or the field at the limit of metastability)\nat which the magnetization is reversed. This is the SW\nastroid. Due to the energy brought into the system by\nMWs, the field requiredforswitchingis smaller. This has\nbeen nicely demonstrated using the µ-SQUID technique\non a 20-nm cobalt particle9. The most striking feature\nof the SW astroid obtained by these measurement is its\njaggedness. In other words, the reduction of the switch-\ning field is not uniform and presents a kind of “fractal”\ncharacter. The global features depend on several physi-\ncal parameters, such as the MW field pulse duration, its\nrising time, its frequency, the DC field amplitude, and\nthe damping parameter. In the present work, and in the\nparticular case considered here, namely that of uniaxial\nanisotropy, we first wanted to check whether this reduc-\ntion of the switching field is recovered by our optimal-\ncontrol method. Furthermore, we address the question\nas to whether the SW astroid may be used as a finger-7\n-1001020α = 0.02\n-1001020α = 0.05\n4 6 8 10 12-1001020α = 0.10\nTime (ns)Optimized RF field HAC(t) (mT)\nFigure 5: Optimized MW field obtained for different values\nof the damping parameter α.\nprint of a given nanocluster. More precisely, the ques-\ntion is whether a given SW astroid can provide us with\nspecific information about the corresponding cluster, like\nits energy potential and the physical parameters such as\ndamping.\nAccordingly, we check whether an MW field h0\nAC(t),\nwhich is optimized in the presence of a reference ap-\nplied DC field h0\nDC(t) with given direction and magni-\ntude,e.g.hDC= 0.5 and an angle of 170◦with respect\nto the easy axis, can still induce magnetization switching\nin the presence of another DC field, with different di-\nrection and/or magnitude. To answer this question, the\nMW field h0\nAC(t), was used in the driven LLE (7) and the\ncalculation of the switching field was performed for sev-\neral intensities and directions of the static field hleading\nto the switching curves in presence of h0\nAC(t) as shown in\nFig. 7. As can be seen, the magnetization switching oc-\ncurs only inside the golf-club-shaped green area [see Fig.\n7 (left)]. In the black area, the pumping by the MW field\nis inefficient and switching does not occur. This curve is\nin agreement with the experimental data of Ref. 9.\nThe shape of the green pattern can be explained based\non qualitative arguments about the frequency and mag-\nnitude of the MW field. As has been seen previously, in\norder to achieve the switching, the MW field must fulfill\nthe following conditions: i) it must be synchronized with\nthe proper precession frequency of the magnetization; so\nat short times its frequency must match the FMR fre-024α = 0.02\n024α = 0.05\n8 10 12024α = 0.10\nTime (ns)Instantaneous frequency of the MW field (GHz)\nFigure 6: Instantaneous frequency of the MW field optimized\nfor several values of the damping parameter α. Dotted line:\nFMR frequency.\nquency of the system, and ii) the injected energy, must\nbe sufficient to overcome the energy barrier between the\nmetastable minimum and the saddle point.\nFor any magnitude or direction of the field hDC, both\nthe FMR frequency and the energy barrier can be com-\nputed numerically [see Fig. 7 (right)]. In the black area\nthe value of the FMR frequency is the same as for h0\nDC.\nIn the hatched area the energy barrier is lower than for\ntheh0\nDC. Outside the black zone, the MW field is not\nsynchronized with the precession frequency of the sys-\ntem: the switching can not occur. Outside the hatched\narea the injected energy is not sufficient to overcome\nthe energy barrier. Consequently, the switching is only\nachieved in the intersection between both areas. Indeed,\ncomparing with Fig. 7 (left), this intersection matches\nmore or less the green zone, where the switching occcurs.\nNext, we optimize the MW field h0\nAC(t) for the ref-\nerence DC field h0\nDC(t) with magnitude hDC= 0.5 and\nangle of 170◦with respect to the easy axis, and damping\nα0= 0.05; then we compute the SW astroid for other\nvalues of α, in the presence of the same DC and MW\nfields. The results are shown in Fig. 8.\nWe see that the shape of the switching area strongly\ndepends on the damping parameter α. The largest green\narea is found for α=α0. Then, as αincreases the green\narea shrinks and vanishes for α >0.12. Indeed, for high\nvalues of α, the MW field is not strong enough to com-8\nFigure 7: (a) Switching curve computed in the presence of\nthe MW field h0\nAC(t). The red cross indicates the amplitude\nand direction of the DC field for which the MW field was\noptimized. The area in green is where switching has been\nachieved, the black area is where there is no switching, and\nin the white area the static field is higher than the switching\nfield (i.e.beyond the metastability region). (b) In the black\narea the FMR frequency is the same as for the reference DC\nfieldh0\nDC. In the hatched area the energy barrier between the\nmetastable minimum and the saddle point (computed numer-\nically) is smaller than for the reference DC field h0\nDC.\npensateforthe effect ofdamping. The samephenomenon\nis observed for small values of α, in which case the MW\nfield “overcompensates” for the effect of damping and\nthereby the energy can not be pumped into the system\nin an efficient manner.\nIV. CONCLUSIONS AND OUTLOOK\nWe have developed a general and efficient method for\ndetermining the characteristics (pulse shape, duration,\nintensity, and frequency) of the MW field that triggers\nthe switching of a nanomagnet in an oblique static mag-\nnetic field. We have applied the method to the case ofuniaxial anisotropy and investigated the effect of the DC\nfield and damping on the optimized MW field. We have\nshown that our method does recover the switching field\ncurves as observed on cobalt nanoclusters. It remains\nthough to investigate the origin of the “fractal” charac-\nter observed in the measured switching curves.\nWe have shown that the MW field that triggers the\nmagnetization switching, while minimizing the absorbed\nenergy, can be efficiently calculated using the optimal\ncontroltheory. Accordingtoourresults,theoptimalMW\nfield is modulated both in frequency and in magnitude.\nThe role of this MW field is to drive the magnetization\ntowards the saddle point, then damping leads the mag-\nnetic moment to the stable equilibrium position. For the\npumping to be efficient, the MW field frequency must\nmatch the proper precession frequency of the magnetiza-\ntion, which depends on the magnitude and the direction\nofthe staticfield. Moreover,the intensitydepends on the\ndamping parameter. This result could be used to probe\nthe damping parameter in experimental nanoparticles.\nThe present method is quite versatile and can be ex-\ntended to other anisotropies. It could also be used to\nstudy the dynamics of nanoclusters in the many-spin\napproach19. In this case one will probably have to\ndeal with a nonuniform MW field, especially if surface\nanisotropy is taken into account23. One may then study\nswitchingviainternalspin waveexcitationsandthe effect\nof the MW field on the corresponding relaxation rate24.\nThermal effects can also be accounted for by adding a\nLangevin field on top of the DC and MW fields. In this\ncase, it will be interesting to investigate the interplay be-\ntween the MW field and the Langevin field and to figure\nout when these two fields play concomitant roles.\nAcknowledgments\nWe are grateful to our collaborators E. Bonet, R.\nPicquerel, C. Thirion, W. Wernsdorfer (Institut N´ eel,\nGrenoble) and V. Dupuis (LPMCN,Lyon) for instructive\ndiscussion of their experiments on isolated nanoclusters.\nThis work has been funded by the collaborative program\nPNANO ANR-08-P147-36 of the French Ministry.\n1L. He et al., J. Magn. Magn. Mater. 155, 6 (1996).\n2G. Woltersdorf and Ch. H. Back, Phys. Rev. Lett. 99,\n227207 (2007); J. Podbielski, D. Heitmann, and D.\nGrundler, Phys. Rev. Lett. 99, 207202 (2007); Z. Wang\net al., Phys. Rev. B 81, 064402 (2010).\n3B. Meerson and L. Friedland, Phys. Rev. A 41, 5233\n(1990).\n4W.-K. Liu, B. Wu and J.-M. Yuan, Phys. Rev. Lett. 75,\n1292 (1995).\n5J.-M. Yuan and W.-K. Liu, Phys. Rev. A 57, 1992 (1998).\n6G. Marcus, L. Friedland, and A. Zigler, Phys. Rev. A 69,\n013407 (2004); G. Marcus, A. Zigler, and L. Friedland,Europhys. Lett. 74, 43 (2006).\n7S. Chelkowski, A. Bandrauk, and P. B. Corkum, Phys.\nRev. Lett. 65, 2355 (1990).\n8S. G. Schirmer, H. Fu, and A. I. Solomon, Phys. Rev. B\n63, 063410 (2001).\n9C. 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Hoefel, Signal Processing 84, 1423\n(2004).\n23H. Kachkachi and M. Dimian, Phys. Rev. B 66, 174419\n(2002); D. A. Garanin and H. Kachkachi, Phys. Rev. Lett.\n90, 65504 (2003).\n24D. A. Garanin, H. Kachkachi and L. Reynaud, Europhys.\nLett.82, 17007 (2008); D. A. Garanin and H. Kachkachi,\nPhys. Rev. B 80, 014420 (2009)." }, { "title": "1902.04605v1.Ultra_low_damping_in_lift_off_structured_yttrium_iron_garnet_thin_films.pdf", "content": "1 \n This article may be downloaded for personal use only. Any other use requires prior permission of the \nauthor and AIP Publishing. This article appeared in Applied Physics Letters 111 (19), 192404 (2017) \nand may be found at https://aip.scitation.org/doi/abs/10.1063/1.5002004 \n \n \nUltra -low damping in lift-off structured y ttrium iron garnet thin films \nA. Krysztofik ,1 L. E. Coy,2 P. Kuświk ,1,3 K. Załęski,2 H. Głowiński ,1 \nand J. Dubowik1 \n1Institute of Molecular Physics, Polish Academy of Sciences, PL -60-179 Poznań, Poland \n2NanoBioMedical Centre, Adam Mickiewicz University, PL -61-614 Poznań, Poland \n3Centre for Advanced Technology, Adam Mickiewicz University, PL -61-614 Poznań, Poland \nElectronic mail: adam.krysztofik@ifmpan.poznan.pl , hubert .glowinski @ifmpan.poznan.pl \n \nWe show that using maskless photolithography and the lift-off technique patterned \nyttrium iron garnet thin films possessing ultra -low Gilbert damping can be \naccomplished . The films of the 70 nm thickness we re grown on (001)-oriented \ngadolinium gallium garne t by means of pulsed laser deposition and exhibit high \ncrystalline quality, low surface roughness and effective magnetization of 127 \nemu/cm3. The Gilbert damping parameter is as low as 5×10−4. The obtained \nstructures have well-defined sharp edges which along with good structural and \nmagnetic film properties, pave a path in the fabrication of high -quality magnonic \ncircuits as well as oxide -based spintronic devices. \n \n \nYttrium iron garnet (Y 3Fe5O12, YIG) has become an intensively studied material in recent years due \nto exceptionally low damping of magnetization precession and electrical insulation enabling its \napplication in research on spin -wave propagation1–3, spin-wave based logic devices4–6, spin pumping7, \nand thermally -driven spin caloritronics8. These applications inevitably entail film structurization in \norder to construct complex integrated devices . However, the fabrication of high -quality thin YIG films \nrequires deposition temperatures over 500 C6,9–18 leading to top -down lithographical approach that is \nion-beam etching of a previously deposited plain film where as patterned resist layer serves as a mask. \nConsequently, this metho d introdu ces crystallographic defects , imperfections to surface structure and, \nin the case of YIG films, causes significant increase of the damping parameter .19–21 Moreover, it does \nnot ensure well-defined structure edges for insulators , which play a crucial role in devices utilizing 2 \n edge spin waves22, Goos -Hänchen spin wave shifts23,24 or standing spin waves modes25. On the \ncontrary, t he bottom -up structurization deals with th ese issues since it allows for the film grow th in the \nselect ed, patterned areas followed by a removal of the resist layer along with redundant film during \nlift-off process. Additionally, it reduces the patterning procedure by one step , that is ion etching , and \nimposes room -temperature deposition which both are particularly important whenever low fabrication \nbudget is required. \nIn this letter we report on ultra -low damping in the bottom -up structured YIG film by means of \ndirect writing photolithography technique. In our case, t he method allows for structure patterning \nwith 0.6 µm resolution across full writing area . In order to not preclude the lift -off process, the pulsed \nlaser deposition (PLD) was conducted at room temperature and since such as -deposited films are \namorphous19,27 the ex-situ annealing was performed for recrystallization. Note that post -deposition \nannealing of YIG films is commonly carried out regardless the substrate temperature during film \ndeposition6,12,13,28,29. As a reference we investigated a plain film which was grown in the same \ndeposition process and underwent the same fabrication procedure except for patterning. Henceforth, \nwe will refer to the structured and the plain film as Sample 1 an d Sample 2, respectively . We \nanticipate that such a procedure may be of potential for fabrication of other magnetic oxide structures \nuseful in spintronics. \nStructural characteriza tion of both samples was performed by means of X-Ray Diffraction (XRD). \nAtomic force microscopy (AFM) was applied to investigate surface morphology and the quality of \nstructure edges. SQUID magnetometry provided information on the saturation magnetization and \nmagnetocrystalline anisotropy field . Using a coplanar waveguide connected to a vector network \nanalyzer , broadband ferromagnetic resonance (VNA -FMR) was performed to determine Gilbert \ndamping parameter and anisotropy fields . All the experiments were co nducted at the room \ntemperature. \nThe procedure of samples preparation was as follows. The (001) -oriented gadolinium gallium \ngarnet substrates were ultrasonicated in acetone, trichloroethylene and isopropanol to remove surface \nimpurities. After a 1 minute o f hot plate baking for water evaporation, a positive photoresist was spin -\ncoated onto the substrate (Sample 1). Using maskless photolithography an array of 500 μm x 500 μm \nsquares separated over 500 μm was patterned and the exposed areas were developed. Detailed \nparameters of photolithography process can be found in Ref.26. We chose rather large size of the \nsquares to provide a high signal -to-noise ratio in the latter measurements. Thereafter, plasma etching \nwas performed to remove a residual resist. We would like to emphasize the importance of this step in \nthe fabrication p rocedure as the resist residues may locally affect crystalline structure of a YIG film \ncausing an undesirable increase of overall magnetization damping. Both substrates were then placed in \na high vacuum chamber of 9×10-8 mbar base pressure and a film was d eposited from a stoichiometric \nceramic YIG target under 2×10-4 mbar partial pressure of oxygen. We used a Nd:YAG laser (λ = 355 \nnm) for the ablation with pulse rate of 2 Hz which yielded 1 nm/min growth rate. The target -to-3 \n substrate distance was approximat ely 50 mm. After the deposition the l ift-off process for the Sample 1 \nwas performed using sonication in acetone to obtain the expected structures. Subsequently, both \nsamples were annealed in a tube furnace under oxygen atmosphere (p ≈ 1 bar) for 30 minutes at \n850°C. The heating and cooling rates were about 50 C/min and 10 C/min, respectively. \n \n \n \nFIG. 1. (a) XRD θ−2θ plot near the (004) reflection of structured ( Sample 1 ) and plain ( Sample 2 ) YIG film. Blue arrows \nshow clear Laue reflections of the plain film. Insets show schematic illustration of the structured and plain film used in this \nstudy. (b) Height profile (z(x)) taken from the structured sample (left axis), right shows the differential of the p rofile, clearly \nshowing the slope change. Inset shows 3D map of the structure’s edge. \n \n \nThe structure of YIG films was determined by the X -ray diffraction. Although the as-deposited \nfilms were amorphous, with the annealing treatment they inherited the lattice orientation of the GGG \nsubstrate and recrystallized along [ 001] direction. Figure 1 (a) presents diffraction curves taken in the \nvicinity of ( 004) Bragg reflection. The ( 004) reflection position of structured YIG well coincide s with \nthe reflection of the plain film. The 2 θ=28.70 9 corresponds to the cubic lattice constant of 12.428 Å. \nA comparison of this value with lattice parameter of a bulk YIG (12.376 Å) suggest distortion of unit \n4 \n cells due to slight nonstoichiometry.16,30 Both samples exhibit distinct Laue oscillation s depicted by \nthe blue arrows, indicating film uniformity and high crystalline order , although the structured film \nshowed lower intensity due to the lower mass of the film . From the oscillation period we estimated \nfilm thickness of 73 nm in agreement with the nominal thickness and the value determined using AFM \nfor Sample 1 ( Fig. 1 (b)). By measuring the diffraction in the expanded angle range w e also confirmed \nthat no additional phases like Y 2O3 or Fe 2O3 appeared. \nThe surface morphology of the structured film was investigated by means of AFM. In Fig. 1 (b) \nprofile of a square’s edge is shown. It should be highlighted that no edge irregularities has formed \nduring lift -off process. The horizontal distance between GGG substrate and the surface of YIG film is \nequal to 170 nm as marked in Fig. 1 (b) by the shaded area. A fitting with Gaussian function to the \nderivative of height profile yields the full width at half maximum of 61 nm. This points to the well -\ndefined struct ure edges achieved with bottom -up structurization. Both samples have smooth and \nuniform surface s. The comparable values of root mean square (RMS) roughness (0.306 nm for Sample \n1 and 0.310 nm for Sample 2) indicate that bottom -up structurization process did not leave any resist \nresidues. Note that a roughness of a bare GGG substrate before deposition was 0.281 nm, therefore, \nthe surface roughness of YIG is increased merely by 10%. \n \n \nFIG. 2. Hysteresis loops of structured (Sample 1) and plain (Sample 2) YIG films measured by SQUID \nmagnetometry along [100] direction at the room temperature . \n \nFigure 2 shows magnetization reversal curves measured along [ 100] direction. For each hysteresis \nloop a paramagnetic contribution arising for the GGG substrates was subtracted. The saturation \nmagnetization 𝑀𝑠 was equal to 117 emu/cm3 and 118.5 emu/cm3 for Sample 1 and 2, respectively . \nBoth hysteresis loops demonstrate in -plane anisotropy. For the (001) -oriented YIG the [ 100] direction \nis a “hard” in -plane axis and the magnetization saturates at 𝐻𝑎 = 65 Oe. This value we identify as \n-100 -75 -50 -25 0 25 50 75 100-1.0-0.50.00.51.0 Sample 1\n Sample 2M / MS\nMagnetic Field (Oe)5 \n magnetocrystalline anisotropy field. The VNA -FMR measurements shown in Fig. 3 (a) confirm these \nresults. Using Kitte l dispersion relation, i.e. frequency 𝑓 dependence of resonance magnetic field 𝐻: \n 𝑓=𝛾\n2𝜋√(𝐻+𝐻𝑎cos 4𝜑)(𝐻+1\n4𝐻𝑎(3+cos 4𝜑)+4𝜋𝑀𝑒𝑓𝑓), (1) \n 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑢, (2) \nwe derived 𝐻𝑎 and the effective magnetization 𝑀𝑒𝑓𝑓, both comparable to the values determined using \nSQUID and close to the values of a bulk YIG (see Table I .). Here, the azimuthal angle 𝜑 defines the \nin-plane orientation of the magnetization direction with respect to the [100] axis of YIG and 𝛾 is the \ngyromagnetic ratio ( 1.77×107𝐺−1𝑠−1). To better compare the values of 𝐻𝑎 between samples and to \ndetermine if the results are influenced by additional anisotropic contribution arising from the squares’ \nshape in the structured film we performed angular resolved resonance measurements (inset in Fig. \n3(a)) . The fitting according to Eq. (1) gives |𝐻𝑎| equal to 69.5±0.6 for Sample 1 and 69.74±0.28 for \nSample 2 in agreement with the values derived from 𝑓(𝐻) dependence and better accuracy. Hence, we \nconclude that the structurization did not affect the in -plane anisotropy. The deviations of the derived \n𝑀𝑠 and 𝐻𝑎 from bulk values can be explained in the framework of Fe vacancy model developed for \nYIG films as a result of nonstoichiometry.13,30 For the experimentally determined 𝑀𝑠 and 𝐻𝑎 the \nmodel yields the chemical unit Y 3Fe4.6O11.4 which closely approximates to the composition of a \nstoichiometric YIG Y 3Fe5O12. \n \n \nTABLE I. Key parameters reported for PLD and LPE YIG films. \n AFM SQUID VNA -FMR \n Film \nthickness RMS rough -\nness (nm) Ms \n(emu/cm3) Ha \n(Oe) Field \norientation Meff \n(emu/cm3) |Ha| \n(Oe) Hu \n(Oe) α \n(× 10-4) ΔH 0 \n(Oe) \nSample 1 70 nm 0.306 117±1 65±5 (100): \n(110): \n(001): 125±1 \n126±1 \n129±2 64±1 \n63±1 \n− -101±18 \n-113±18 \n-151±28 5.53±0.13 \n5.24±0.12 \n5.19±0.64 1.45±0.09 \n2.86±0.09 \n2.61±0.34 \nSample 2 70 nm 0.310 118.5±2 65±5 (100): \n(110): \n(001): 124±1 \n127±1 \n131±2 62±1 \n65±1 \n− -69±28 \n-107±28 \n-157±36 5.05±0.07 \n5.09±0.09 \n5.02±0.18 0.97±0.05 \n1.28±0.06 \n1.48±0.09 \nLPE-YIG31 106 nm 0.3 143 − (112): − − − 1.2 0.75 \nLPE-YIG30 120 μm − 139±2 − (111): 133±2 85±6 76±1 0.3 − \n \n \nAlthough the saturation magnetization of the films is decreased by 15% with respect to the bulk \nvalue we can expect similar spin wave dynamics since magnon propagation does not solely depend on \n𝑀𝑠 but on the effective magnetization or equivalently, on the uniaxial anisotropy field 𝐻𝑢.12 \nSubstitution of 𝑀𝑠 into Eq. (2) gives average values of 𝐻𝑢 equal to -122 Oe and -111 Oe for Sample 1 \nand 2, respectively (to determine 𝐻𝑢 from the out -of-plane FMR measurements when H || [001] we 6 \n used the 𝑓=𝛾\n2𝜋(𝐻+𝐻𝑎−4𝜋𝑀𝑒𝑓𝑓) dependence13 to fit the data and assumed the value of 𝐻𝑎 from \nangular measurements ). As 𝑀𝑒𝑓𝑓𝑆𝑎𝑚𝑝𝑙𝑒 1,2≈𝑀𝑒𝑓𝑓𝑏𝑢𝑙𝑘, it follows that the low value of 𝑀𝑠 in room -\ntemperature deposited thin films is “compensated ” by uniaxial anisotropy field. Note that for bulk YIG \nsaturation magnetization is diminished by 𝐻𝑢/4𝜋 giving a lower value of 𝑀𝑒𝑓𝑓 while for Sample 1 \nand 2, 𝑀𝑠 is augmented by 𝐻𝑢/4𝜋 giving a higher value of 𝑀𝑒𝑓𝑓 (Table I .). The negative sign of \nuniaxial anisotropy field is typical for PLD -grown YIG films and originates from preferential \ndistribution of Fe vacancies between different si tes of YIG’s octahedral sublattice.30 This point s to the \ngrowth -induced anisotropy mechanism while the stress -induced contribution is of ≈10 Oe29 and, as it \ncan be estimated according to Ref.32, the transition layer at the substrate -film interface due to Gd, Ga, \nY ions diffusion is ca. 1.5 nm thick for the 30 min of annealing treatment. We argue that the growth -\ninduced anisotropy due to ordering of the magnetic ions is related to the growth condition which in our \nstudy is specific. Namely, it is crystallization of an amorphous material. \nGilbert damping parameter 𝛼 was obtained by fitting dependence of linewidth 𝛥𝐻 (full width at \nhalf maximum ) on frequency 𝑓 as shown in Fig. 3 (b): \n 𝛥𝐻 =4𝜋𝛼\n𝛾𝑓+𝛥𝐻0, (3) \nwhere 𝛥𝐻0 is a zero -frequency linewidth broadening . The 𝛼 parameter of both samples is nearly the \nsame , 5.32×10−4 for Sample 1 and 5.05×10−4 for Sample 2 on average (see Table I.) . It proves \nthat bottom -up patterning does not compromise magnetization damping. The value of 𝛥𝐻0 \ncontribution is around 1.5 Oe although small variations of 𝛥𝐻0 on 𝜑 can be noticed. Additional \ncomments on angular dependencies of 𝛥𝐻 can be found in the supplementary material. The derived \nvalues of 𝛼 remain one order of magnitude smaller than for soft ferromagnets like Ni 80Fe2033, CoFeB34 \nor Finemet35, and are comparable to values reported for YIG film s deposited at hi gh temperatures \n(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 \nsignificantly increased in comparison to the bulk YIG made by means of Liquid Phase Epitaxy (LPE) . \nHowever, recently reported LPE-YIG films of nanometer thickness , suffer from the increased damping \nas well (Table I.) due to impurity elements present in the high -temperature solutions used in LPE \ntechnique31. As PLD method allow s for a good contamination control , we attribute the increase as a \nresult of slight nonstoichiometry determined above with Fe vacancy model .30 Optimization of growth \nconditions , which further improve the film composition may resolve this issue and allow to cross the \n𝛼=1×10−4 limit. We also report that additional annealing of the samples (for 2h) did not influence \ndamping nor it improved the value of 𝐻𝑎 or 𝑀𝑒𝑓𝑓 (within 5% accuracy). 7 \n \nFIG. 3. (a) Kittel dispersion relation s of the structured (Sample 1) and plain (Sample 2) YIG film. The i nset \nshows angular dependence of resonance field revealing perfect fourfold anisotropy for both samples . (b) \nLinewidth dependence on frequency fitted with Eq. (3). The inset shows resonance absorptions peaks with very \nsimilar width (5.3 Oe for Sample 1 and 4.7 Oe for Sample 2 at 10 GHz ). Small differences of the resonance field \noriginate from different values of 4𝜋𝑀𝑒𝑓𝑓. \n \nIn conclusion , the lift-off patterned YIG films possessing low damping have been presented. \nAlthough the structurization procedure required deposition at room temperature , the 𝛼 parameter does \nnot diverge from those reported for YIG thin films grown at temperatures above 500 C. Using the \nplain, reference film fabricated along with the structured one, we have shown that structurization does \nnot significantly affect structural nor magnetic properties of the films, i.e. out-of-plane lattice constant, \nsurface roughness, saturation magnetization, anisotropy fields and damping. The structures obtain ed \nwith bottom -up structurization indeed possess sharp , well-defined edges . In particular, o ur findings \nwill help in the development of magnonic and spintronic devices utilizing film boundary effects and \nlow damping of magnetization precession . \n8 \n \nSupplementary Material \nSee supplementary material for the angular dependence of resonance linewidth . \n \nThe research received funding from the European Union Horizon 2020 research and innovation \nprogra mme under the Marie Skłodowska -Curie grant agreement No 644348 (MagIC). We would like \nto thank Andrzej Musiał for the assistance during film annealing. \n \n1 H. Yu, O. d’Allivy Kelly, V. 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Krysztofik, J. Barnaś, M. Cecot, P. Kuświk, and J. Dubowik, in 21st Int. Conf. Microwave, \nRadar Wirel. Commun. MIKON 2016 (2016). \n \n \n " }, { "title": "2401.00714v1.Calculation_of_Gilbert_damping_and_magnetic_moment_of_inertia_using_torque_torque_correlation_model_within_ab_initio_Wannier_framework.pdf", "content": "Calculation of Gilbert damping and magnetic moment of inertia\nusing torque-torque correlation model within ab initio Wannier\nframework\nRobin Bajaj1, Seung-Cheol Lee2, H. R. Krishnamurthy1,\nSatadeep Bhattacharjee2,∗and Manish Jain1†\n1Centre for Condensed Matter Theory,\nDepartment of Physics,\nIndian Institute of Science,\nBangalore 560012, India\n2Indo-Korea Science and Technology Center,\nBangalore 560065, India\n(Dated: January 2, 2024)\n1arXiv:2401.00714v1 [cond-mat.mtrl-sci] 1 Jan 2024Abstract\nMagnetization dynamics in magnetic materials are well described by the modified semiclassical\nLandau-Lifshitz-Gilbert (LLG) equation, which includes the magnetic damping ˆαand the magnetic\nmoment of inertia ˆItensors as key parameters. Both parameters are material-specific and physi-\ncally represent the time scales of damping of precession and nutation in magnetization dynamics.\nˆαandˆIcan be calculated quantum mechanically within the framework of the torque-torque corre-\nlation model. The quantities required for the calculation are torque matrix elements, the real and\nimaginary parts of the Green’s function and its derivatives. Here, we calculate these parameters\nfor the elemental magnets such as Fe, Co and Ni in an ab initio framework using density functional\ntheory and Wannier functions. We also propose a method to calculate the torque matrix elements\nwithin the Wannier framework. We demonstrate the effectiveness of the method by comparing it\nwith the experiments and the previous ab initio and empirical studies and show its potential to\nimprove our understanding of spin dynamics and to facilitate the design of spintronic devices.\nI. INTRODUCTION\nIn recent years, the study of spin dynamics[1–5] in magnetic materials has garnered\nsignificant attention due to its potential applications in spintronic devices and magnetic\nstorage technologies[6–9]. Understanding the behaviour of magnetic moments and their\ninteractions with external perturbations is crucial for the development of efficient and reliable\nspin-based devices. Among the various parameters characterizing this dynamics, Gilbert\ndamping[10] and magnetic moment of inertia play pivotal roles. The fundamental semi-\nclassical equation describing the magnetisation dynamics using these two crucial parameters\nis the Landau-Lifshitz-Gilbert (LLG) equation[11, 12], given by:\n∂M\n∂t=M× \n−γH+ˆα\nM∂M\n∂t+ˆI\nM∂2M\n∂t2!\n(1)\nwhere Mis the magnetisation, His the effective magnetic field including both external\nand internal fields, ˆαandˆIare the Gilbert damping and moment of inertia tensors with\nthe tensor components defined as αµνand Iµν, respectively, and γis the gyromagnetic\n∗s.bhattacharjee@ikst.res.in\n†mjain@iisc.ac.in\n2ratio. Gilbert damping, ˆαis a fundamental parameter that describes the dissipation of\nenergy during the precession of magnetic moments in response to the external magnetic\nfield. Accurate determination of Gilbert damping is essential for predicting the dynamic\nbehaviour of magnetic materials and optimizing their performance in spintronic devices. On\nthe other hand, the magnetic moment of inertia, ˆIrepresents the resistance of a magnetic\nmoment to changes in its orientation. It governs the response time of magnetic moments to\nexternal stimuli and influences their ability to store and transfer information. The moment\nof inertia[13] is the magnetic equivalent of the inertia in classical mechanics[14, 15] and acts\nas the magnetic inertial mass in the LLG equation.\nExperimental investigations of Gilbert damping[16–23] and moment of inertia involve\nvarious techniques, such as ferromagnetic resonance (FMR) spectroscopy[24, 25], spin-torque\nferromagnetic resonance (ST-FMR), and time-resolved magneto-optical Kerr effect (TR-\nMOKE)[26, 27]. Interpreting the results obtained from these techniques in terms of the\nLLG equation provide insights into the dynamical behaviour of magnetic materials and can\nbe used to extract the damping and moment of inertia parameters. In order to explain\nthe experimental observations in terms of more macroscopic theoretical description, various\nstudies[28–34] based on linear response theory and Kambersky theory have been carried out.\nLinear response theory-based studies of Gilbert damping and moment of inertia involve\nperturbing the system and calculating the response of the magnetization to the pertur-\nbation. By analyzing the response, one can extract the damping parameter. Ab initio\ncalculations based on linear response theory[33] can provide valuable insights into the mi-\ncroscopic mechanisms responsible for the damping process. While formal expression for the\nmoment of inertia in terms of Green’s functions have been derived within the Linear response\nframework[11], to the best of our knowledge, there hasn’t been any first principle electronic\nstructure-based calculation for the moment of inertia within this formalism.\nKambersky’s theory[35–37] describes the damping phenomena using a breathing Fermi\nsurface [38] and torque-torque correlation model [39], wherein the spin-orbit coupling acts\nas the perturbation and describes the change in the non-equilibrium population of electronic\nstates with the change in the magnetic moment direction. Gilmore et al.[32, 34] have reported\nthe damping for ferromagnets like Fe, Ni and Co using Kambersky’s theory in the projector\naugmented wave method[40].\nThe damping and magnetic inertia have been derived within the torque-torque correlation\n3model by expanding the effective dissipation field in the first and second-time derivatives\nof magnetisation[29–31]. In this work, the damping and inertia were calculated using the\ncombination of first-principles fully relativistic multiple scattering Korringa–Kohn–Rostoker\n(KKR) method and the tight-binding model for parameterisation[41]. However, there hasn’t\nbeen any full ab intio implementation using density functional theory (DFT) and Wannier\nfunctions to study these magnetic parameters.\nThe expressions for the damping and inertia involves integration over crystal momentum\nkin the first Brillouin zone. Accurate evaluation of the integrals involved required a dense k-\npoint mesh of the order of 106−108points for obtaining converged values. Calculating these\nquantities using full ab initio DFT is hence time-consuming. To overcome this problem,\nhere we propose an alternative. To begin with, the first principles calculations are done on\na coarse kmesh instead of dense kmesh. We then utilize the maximally localised Wannier\nfunctions (MLWFs)[42] for obtaining the interpolated integrands required for the denser k\nmeshes. In this method, the gauge freedom of Bloch wavefunctions is utilised to transform\nthem into a basis of smooth, highly localised Wannier wavefunctions. The required real\nspace quantities like the Hamiltonian and the torque-matrix elements are calculated in the\nWannier basis using Fourier transforms. The integrands of integrals can then be interpolated\non the fine kmesh by an inverse Fourier transform of the maximally localised quantities,\nthereby enabling the accurate calculations of the damping and inertia.\nThe rest of this paper is organized as follows: In Sec. II, we introduce the expressions for\nthe damping and the inertia. We describe the formalism to calculate the two key quantities:\nGreen’s function and torque matrix elements, using the Wannier interpolation. In Sec. III,\nwe describe the computational details and workflow. In Sec. IV, we discuss the results for\nferromagnets like Fe, Co and Ni, and discuss the agreement with the experimental values\nand the previous studies. In Sec. V, we conclude with a short summary and the future\nprospects for the methods we have developed.\nII. THEORETICAL FORMALISM\nFirst, we describe the expressions for Gilbert damping and moment of inertia within the\ntorque-torque correlation model. Then, we provide a brief description of the MLWFs and\nthe corresponding Wannier formalism for the calculation of torque matrix elements and the\n4Green’s function.\nA. Gilbert damping and Moment of inertia within torque-torque correlation\nmodel\nIf we consider the case when there is no external magnetic field, the electronic structure\nof the system can be described by the Hamiltonian,\nH=H0+Hexc+Hso=Hsp+Hso (2)\nThe paramagnetic band structure is described by H0andHexcdescribes the effective lo-\ncal electron-electron interaction, treated within a spin-polarised (sp) local Kohn-Sham\nexchange-correlation (exc) functional approach, which gives rise to the ferromagnetism. Hso\nis the spin-orbit Hamiltonian. As we are dealing with ferromagnetic materials only, we can\nclub the first two terms as Hsp=H0+Hexc. During magnetization dynamics, (when the\nmagnetization precesses), only the spin-orbit energy of a Bloch state |ψnk⟩is affected, where\nnis the band index of the state. The magnetization precesses around an effective field\nHeff=Hint+Hdamp+HI, where Hintis the internal field due to the magnetic anisotropy\nand exchange energies, Hdampis the damping field, and HIis the inertial field, respectively.\nFrom Eqn. (1), we can see that the damping field Hdamp =α\nMγ∂M\n∂t, while HI=I\nMγ∂2M\n∂t2.\nEquating these damping and inertial fields to the effective field corresponding to the change\nin band energies as magnetization processes, we obtain the mathematical description of the\nGilbert damping and inertia. It was proposed by Kambersky [39] that the change of the\nband energies∂εnk\n∂θµ(θ=θˆndefines the vector for the rotation) can be related to torque\noperator or matrix depending on how the Hamiltonian is being viewed Γµ= [σµ,Hso], where\nσµare the Pauli matrices. Eventually, within the so-called torque-torque correlation model,\nthe Gilbert damping tensor can be expressed as follows:\nαµν=g\nmsπZ Z\u0012\n−d f(ϵ)\ndϵ\u0013\nTr[Γµ(IG)(Γν)†(IG)]d3k\n(2π)3dϵ (3)\nHere trace Tris over the band indices, and msis the magnetic moment. Recently, Thonig\net al [30] have extended such an approach to the case of moment of inertia also, where they\n5deduced the moment of inertia tensor components to be,\nIνµ=gℏ\nmsπZ Z\nf(ϵ)Tr[Γν(IG)(Γµ)†∂2\n∂ϵ2(RG)\n+Γν∂2\n∂ϵ2(RG)(Γµ)†(IG)]d3k\n(2π)3dϵ (4)\nHere f(ϵ) is the Fermi function, ( RG) and ( IG) are the real and imaginary parts of Green’s\nfunction G=(ϵ+ιη− H)−1with ηas a broadening parameter, mis the magnetization in\nunits of the Bohr magneton, Γµ= [σµ,Hso] is the µthcomponent of torque matrix element\noperator or matrix, µ=x, y, z .αis a dimensionless parameter, and I has units of time,\nusually of the order of femtoseconds.\nTo obtain the Gilbert damping and moment of inertia tensors from the above two k-\nintegrals calculated as sums of discrete k-meshes, we need a large number of k-points, such\nas around 106, and more than 107, for the converged values of αand I respectively. The\nreason for the large k-point sampling in the first Brillouin zone (BZ) is because Green’s\nfunction becomes more sharply peaked at its poles at the smaller broadening, ηwhich need\nto be used. For I, the number of k-points needed is more than what is needed for αbecause\n∂2RG/∂ϵ2has cubic powers in RGand/or IGas:\n∂2RG\n∂ϵ2= 2\u0002\n(RG)3−RG(IG)2−IGRGIG−(IG)2RG\u0003\n(5)\nWe note that to carry out energy integration in αit is sufficient to consider a limited\nnumber of energy points within a narrow range ( ∼kBT) around the Fermi level. This\nis mainly due to the exponential decay of the derivative of the Fermi function away from\nthe Fermi level. However, the integral for I involves the Fermi function itself and not its\nderivative. Consequently, while the Gilbert damping is associated with the Fermi surface, the\nmoment of inertia is associated with the entire Fermi sea. Therefore, in order to adequately\ncapture both aspects, it is necessary to include energy points between the bottom of the\nvalence band and the Fermi level.\n6B. Wannier Interpolation\n1.Maximally Localised Wannier Functions (MLWFs)\nThe real-space Wannier functions are written as the Fourier transform of Bloch wave-\nfunctions,\n|wnR⟩=v0\n(2π)3Z\nBZdke−ιk.R|ψnk⟩ (6)\nwhere |ψnk⟩are the Bloch wavefunctions obtained by the diagonalisation of the Hamiltonian\nat each kpoint using plane-wave DFT calculations. v0is the volume of the unit cell in the\nreal space.\nIn general, the Wannier functions obtained by Eqn. (6) are not localised. Usually, the\nFourier transforms of smooth functions result in localised functions. But there exists a phase\narbitrariness of eιϕnkin the Bloch functions because of independent diagonalisation at each\nk, which messes up the localisation of the Wannier functions in real space.\nTo mitigate this problem, we use the Marzari-Vanderbilt (MV) localisation procedure[42–\n44] to construct the MLWFs, which are given by,\n|wnR⟩=1\nNX\nqNqX\nm=1e−ιq.RUq\nmn|ψmq⟩ (7)\nwhere Uq\nmnis a (Nq×N) dimensional matrix chosen by disentanglement and Wannierisation\nprocedure. Nare the number of target Wannier functions, and Nqare the original Bloch\nstates at each qon the coarse mesh, from which Nsmooth Bloch states on the fine k-mesh\nare extracted requiring Nq>Nfor all q,Nis the number of uniformly distributed qpoints\nin the BZ. The interpolated wavefunctions on a dense k-mesh, therefore, are given via inverse\nFourier transform as:\n|ψnk⟩=X\nReιk.R|wnR⟩ (8)\nThroughout the manuscript, we use qandkfor coarse and fine meshes in the BZ, respec-\ntively.\n7FIG. 1: The figure shows the schematic of the localisation of the Wannier functions on a\nRgrid. The matrix elements of the quantities like Hamiltonian on the Rgrid are\nexponentially decaying. Therefore, most elements on the Rgrid are zero (shown in blue).\nWe can hence do the summation till a cutoff Rcut(shown in red) to interpolate the\nquantities on a fine kgrid.\n2.Torque Matrix elements\nAs described in the expressions of αµνand Iνµin Eqns. (3) and (4), the µthcomponent of\nthe torque matrix is given by the commutator of µthcomponent of Pauli matrices and spin-\norbit coupling matrix i.e.Γµ= [σµ,Hso]. Physically, we define the spin-orbit coupling (SOC)\nand spin-orbit torque (SOT) as the dot and cross products of orbital angular momentum and\nspin angular momentum operator, respectively, such that Hso=ξℓ.where ξis the coupling\namplitude. Using this definition of Hso, one can show easily that −ι[σ,Hso] = 2ξℓ×which\nrepresents the torque.\nThere have been several studies on how to calculate the spin-orbit coupling using ab\ninitio numerical approach. Shubhayan et al. [45] describe the method to obtain SOC matrix\nelements in the Wannier basis calculated without SO interaction, using an approximation of\nweak SOC in the organic semiconductors considered in their work. Their method involves\nDFT in the atomic orbital basis, wherein the SOC in the Bloch basis can be related to\nthe SOC in the atomic basis. Then, by the basis transformation, they get the SOC in\nthe Wannier basis calculated in the absence of SO interaction. Farzad et al. [46] calculate\nthe SOC by extracting the coupling amplitude from the Hamiltonian in the Wannier basis,\ntreating the Wannier functions as atomic-like orbitals.\n8We present a different approach wherein we can do the DFT calculation in any basis (plane\nwave or atomic orbital). Unlike the previous approaches, we perform two DFT calculations\nand two Wannierisations: one is with spin-orbit interaction and finite magnetisation (SO)\nand the other is spin-polarised without spin-orbit coupling (SP). The spin-orbit Hamiltonian,\nHsocan then be obtained by subtracting the spin-polarized Hamiltonian, Hspfrom the full\nHamiltonian, HasHso=H−H sp. This, however, can only be done if both the Hamiltonians,\nHandHsp, are written in the same basis. We choose to use the corresponding Wannier\nfunctions as a basis. It should however be noted that, when one Wannierises the SO and\nSP wavefunctions, one will get two different Wannier bases. As a result, we can not directly\nsubtract the HandHspin these close but different Wannier bases. In order to do the\nsubtraction, we find the transformation between two Wannier bases , i.e. express one set of\nWannier functions in terms of the other. Subsequently, we can express the matrix elements\nofHandHspin the same basis and hence calculate Hso. In the equations below, the Wannier\nfunctions, the Bloch wavefunctions and the operators defined in the corresponding bases in\nSP and SO calculations are represented with and without the tilde ( ∼) symbol, respectively.\nTheNSO Wannier functions are given by:\n|wnR⟩=1\nNX\nqNqX\nm=1e−ιq.RUq\nmn|ψmq⟩ (9)\nwhere Uq\nmnis a (Nq× N) dimensional matrix. The wavefunctions and Wannier functions\nfrom the SO calculation for a particular qandRare a mixture of up and down spin states\nand are represented as spinors:\n|ψnq⟩=\n|ψ↑\nnq⟩\n|ψ↓\nnq⟩\n |wnR⟩=\n|w↑\nnR⟩\n|w↓\nnR⟩\n (10)\nThe ˜NsSP Wannier functions are given by:\n|˜ws\nnR⟩=1\nNX\nq˜Ns\nqX\nm=1e−ιq.R˜Uqs\nmn|˜ψs\nmq⟩ (11)\nwhere s=↑,↓.˜Uqs\nmnis a ( ˜Ns\nqטNs) dimensional matrix. Since the spinor Hamiltonian\ndoesn’t have off-diagonal terms corresponding to opposite spins in the absence of SOC, the\nwavefunctions will be |˜ψs\nnq⟩=|˜ψnq⟩⊗|s⟩. The combined expression for ˜Uqfor˜N↑+˜N↓=˜N\n9FIG. 2: This figure shows the implementation flow chart of the theoretical formalism\ndescribed in Sec. II\nSP Wannier functions is:\n˜Uq=\n˜Uq↑0\n0˜Uq↓\n (12)\nwhere ˜Uqis˜NqטNdimensional matrix with ˜Nq=˜N↑\nq+˜N↓\nq. Dropping the sindex for\nSP kets results in the following expression for ˜NSP Wannier functions:\n|˜wnR⟩=1\nNX\nq˜NqX\nm=1e−ιq.R˜Uq\nmn|˜ψmq⟩ (13)\n10We now define the matrix of the transformation between SO and SP Wannier bases as:\nTRR′\nmn=⟨˜wmR|wnR′⟩\n=1\nN2X\nqq′˜Nq,Nq′X\np,l=1eι(q.R−q′.R′)˜Uq†\nmp⟨˜ψpq|ψlq′⟩Uq′\nln\n=1\nN2X\nqq′eι(q.R−q′.R′)[˜Uq†Vqq′Uq′]mn (14)\nHereVqq′\npl=⟨˜ψpq|ψlq′⟩. Eqn. (14) is the most general expression to get the transformation\nmatrix. We can reduce this quantity to a much simpler one using the orthogonality of\nwavefunctions of different q. Eqn. (14) hence becomes,\nTRR′\nmn=1\nN2X\nqeιq.(R−R′)[˜Uq†(NVq)Uq]mn\n=1\nNX\nqeιq.(R−R′)[˜Uq†VqUq]mn (15)\nwhere Vq\npl=⟨˜ψpq|ψlq⟩. Using this transformation, we write SP Hamiltonian in SO Wannier\nbases as:\n(Hsp)RR′\nmn=⟨wmR|Hsp|wnR′⟩\n=X\nplR′′R′′′⟨wmR|˜wpR′′⟩\n⟨˜wpR′′|Hsp|˜wlR′′′⟩⟨˜wlR′′′|wnR′⟩\n=X\nplR′′R′′′(T†)RR′′\nmp(˜Hsp)R′′R′′′\npl TR′′′R′\nln (16)\nSince Wannier functions are maximally localised and generally atomic-like, the major con-\ntribution to the overlap TRR′\nmnis for R=R′. Therefore, we can write TRR\nmn=T0\nmn. The\nreason is that it depends on relative R−R′, we can just consider overlaps at R=0. Eqn.\n16 becomes,\n(Hsp)RR′\nmn=X\npl(T†)0\nmp(˜Hsp)RR′\nplT0\nln (17)\nTherefore, we write the Hsoin Wannier basis as,\n(Hso)RR′\nmn=HRR′\nmn−(Hsp)RR′\nmn (18)\n11The torque matrix elements in SO Wannier bases are given by,\n(Γµ)RR′\nmn= (σµHso)RR′\nmn−(Hsoσµ)RR′\nmn (19)\nConsider ( σµHso)RR′\nmnand insert the completeness relation of the Wannier functions, and\nalso neglecting SO matrix elements between the Wannier functions at different sites because\nof their being atomic-like.\n(σµHso)RR′\nmn=P\npR′′(σµ)RR′′\nmp(Hso)R′′R′\npn\n= (σµ)RR′\nmp(Hso)0\npn (20)\n(σµ)RR′\nmpis calculated by the Fourier transform of the spin operator written in the Bloch\nbasis, just like the Hamiltonian.\n(σµ)RR′\nmp=1\nNX\nqe−ιq.(R′−R)\u0002\nUq†(σµ)qUq\u0003\nmp(21)\nWe interpolate the SOT matrix elements on a fine k-mesh as follows:\n(Γµ)k\nmn=X\nR′−Reιk.(R′−R)(σµ)RR′\nmn (22)\nThis yields the torque matrix elements in the Wannier basis. In the subsequent expres-\nsions, WandHsubscripts represent the Wannier and Hamiltonian basis, respectively. In\norder to rotate to the Hamiltonian gauge, which diagonalises the Hamiltonian interpolated\non the fine kmesh using its matrix elements in the Wannier basis.\n(HW)k\nmn=X\nR′−Reιk.(R′−R)HRR′\nmn (23)\n(HH)k\nmn=\u0002\n(Uk)†(HW)kUk\u0003\nmn(24)\nHere Uk(not to be confused with Uq) are the matrices with columns as the eigenvectors of\n(HW)k, and ( HH)k\nmn=ϵmkδmn. We use these matrices to rotate the SOT matrix elements\nin Eqn. (22) to the Hamiltonian basis as:\n(Γµ\nH)k\nmn=\u0002\n(Uk)†(Γµ\nW)kUk\u0003\nmn(25)\n123.Green’s functions\nThe Green’s function at an arbitrary kandϵon a fine k-mesh in the Hamiltonian basis\nis given by:\nGk\nH(ϵ+ιη) = (ϵ+ιη−(HH)k)−1(26)\nwhere ηis a broadening factor and is caused by electron-phonon coupling and is generally\nof the order 5 −10 meV. Gk\nH(ϵ+ιη) is aN × N dimensional matrix.\nTherefore, we can calculate RG,IGand∂2RG/∂ϵ2as defined in Eqn. (5) and hence, α\nand I.\nIII. COMPUTATIONAL DETAILS\nPlane-wave pseudopotential calculations were carried out for the bulk ferromagnetic tran-\nsition metals bcc Fe, hcp Co and fcc Ni using Quantum Espresso package[47, 48]. The\nconventional unit cell lattice constants ( a) used for bcc Fe and fcc Ni were 5.424 and 6.670\nbohrs, respectively and for hcp Co, a=4.738 bohr and c/a=1.623 were used. The non-\ncollinear spin-orbit and spin-polarised calculations were performed using fully relativistic\nnorm-conserving pseudopotentials. The kinetic energy cutoff was set to 80 Ry. Exchange-\ncorrelation effects were treated within the PBE-GGA approximation. The self-consistent\ncalculations were carried out on 16 ×16×16 Monkhorst-Pack Grid using Fermi smearing of\n0.02 Ry. Non-self-consistent calculations were carried out using the calculated charge den-\nsities on Γ-centered 10 ×10×10 coarse k-point grid. For bcc Fe and fcc Ni, 64 bands were\ncalculated and hcp Co 96 bands were calculated (because there are two atoms per unit cell\nfor Co). We define a set of 18 trial orbitals sp3d2,dxy,dxz, and dyzfor Fe, 18 orbitals\nper atom s,panddfor Co and Ni, to generate 18 disentangled spinor maximally-localized\nWannier functions per atom using Wannier90 package [43].\nFrom the Wannier90 calculations, we get the checkpoint file .chk, which contains all the\ninformation about disentanglement and gauge matrices. We use .spnand.eigfiles generated\nbypw2wannier90 to get the spin operator and the Hamiltonian in the Wannier basis. We\nevaluate the SOT matrix elements in the Wannier Basis.\nWe get αby simply summing up on a fine- kgrid with appropriate weights for the k-\nintegration, and we use the trapezoidal rule in the range [-8 δ,8δ] for energy integration\n1310−610−410−2100\n10−610−410−210010−610−410−210010\n10\n10\n10\n10\n101\n0\n1−2\n−2\n−310−410−310−210−1100101\n10−410−310−210−1100101\nFe Ni Co(a) (b) (c)\nFIG. 3: (a), (b) and (c) shows the αvsηfor Fe, Ni and Co, respectively. Damping\nconstants calculated using the Wannier implementation are shown in blue. Damping\ncalculated using the tight binding method based on Lorentzian broadening and Green’s\nfunction by Thonig et al[29] are shown in brown and green, respectively. Comparison with\ndamping constants calculated by Gilmore et al[32] using local spin density approximation\n(LSDA) are shown in red. The dotted lines are guides to the eye.\naround the Fermi level where δis the width of the derivative of Fermi function ∼kBT. We\nconsider 34 energy points in this energy range. We perform the calculation for T= 300K.\nFor the calculation of I, we use a very fine grid of 400 ×400×400k-points. For η >0.1,\nwe use 320 energy points between VBM and Fermi energy. For 0 .01< η < 0.1, we use\n3200-6400 energy points for the energy integration.\nTABLE I\nMaterial η(meV) -I (fs) α(×10−3) τ(ps)\nFe 6 0.210 3.14 0.42\n8 0.114 2.77 0.26\n10 0.069 2.51 0.17\nNi 10 2.655 34.2 0.48\nCo 10 0.061 1.9 0.21\n14FIG. 4: Schematic explaining the dependence of intraband and interband contribution in\nαwith η.\nIV. RESULTS AND DISCUSSION\nA. Damping constant\nIn this section, we report the damping constants calculated for the bulk iron, cobalt and\nnickel. The magnetic moments are oriented in the z-direction. For reference direction z, the\ndamping tensor is diagonal resulting in the effective damping constant α=αxx+αyy.\nIn Fig. 3, we report the damping constants calculated by the Wannier implementation as\na function of broadening, ηknown to be caused by electron-phonon scattering and scattering\nwith impurities. We consider the values of ηranging from 10−6to 2 eV to understand the\nrole of intraband and interband transitions as reported in the previous studies[29, 32]. We\nnote that the experimental range is for the broadening is expected to be much smaller with\nη∼5−10 meV. The results are found to be in very good agreement with the ones calculated\nusing local spin density approximation (LSDA)[32] and tight binding paramterisation[29].\nThe expression for Gilbert damping[3] is written in terms of the imaginary part of Green’s\nfunctions. Using the spectral representation of Green’s function, Ank(ω), we can rewrite Eqn.\n15(3) as:\nαµν=gπ\nmsX\nnmZ\nTµ\nnm(k)T∗ν\nnm(k)Snmdk (27)\nwhere Snm=R\nη(ϵ)Ank(ϵ)Amk(ϵ)dϵis the spectral overlap. Although we are working in the\nbasis where the Hamiltonian is diagonal, the non-zero off-diagonal elements in the torque\nmatrix lead to both intraband ( m=n) and interband ( m̸=n) contributions. For the\nsake of simple physical understanding, we consider the contribution of the spectral overlaps\nat the Fermi level for both intraband and interband transitions in Fig. 4. But in the\nnumerical calculation temperature broadening has also been considered. For the smaller\nη, the contribution of intraband transitions decreases almost linearly with the increase in\nηbecause the overlaps become less peaked. Above a certain η, the interband transitions\nbecome dominant and the contribution due to the overlap of two spectral functions at\ndifferent band indices m and n becomes more pronounced at the Fermi level. Above the\nminimum, the interband contribution increases till η∼1 eV. Because of the finite Wannier\norbitals basis, we have the accurate description of energy bands only within the approximate\nrange of ( ϵF−10, ϵF+ 5) eV for the ferromagnets in consideration. The decreasing trend\nafter η∼1 eV is, therefore, an artifact.\n10−2\n0.000.050.100.150.20\n10−1100-0.001 0.000 0.001 0.002\n10−1100\nFIG. 5: Plot showing moment of inertia, −I versus broadening, η. The moment of inertia\nin the range 0 .03−3.0 eV is shown as an inset. The values using the Wannier\nimplementation and the tight binding method[30] are shown in blue and green, respectively.\n160.020.040.060.08\nCo(a)\n10−210−11000.0010−1100−0.0075−0.0050−0.0050−0.0025 0.000\n10−210−1100−\n0.00.51.0 1.01.5\nNi(b)\n0.000.010.02\n100101\nFIG. 6: (a) and (b) show negative of the moment of inertia, −I versus broadening, ηfor\nCo and Ni, respectively. The values using the Wannier implementation and the tight\nbinding method are shown in magenta and cyan, respectively. The moment of inertia is\nshown as an inset in the range 0 .03−2.0 eV and 0 .03−3.0 eV for Co and Ni, respectively.\nB. Moment of inertia\nIn Fig. 5, we report the values for the moment of inertia calculated for bulk Fe, Co\nand Ni. Analogous to the damping, the inertia tensor is diagonal, resulting in the effective\nmoment of inertia I = Ixx+ Iyy.\nThe behaviour for I vs ηis similar to that of the damping, with smaller and larger ηtrends\narising because of intraband and interband contributions, respectively. The overlap term\nin the moment of inertia is between the ∂2RG/∂ϵ2andIGunlike just IGin the damping.\nIn Ref. [30], the moment of inertia is defined in terms of torque matrix elements and the\noverlap matrix as:\nIµν=−gℏ\nmsX\nnmZ\nTµ\nnm(k)T∗ν\nnm(k)Vnmdk (28)\nwhere Vnmis an overlap function, given byR\nf(ϵ)(Ank(ϵ)Bmk(ϵ) +Bnk(ϵ)Amk(ϵ))dϵand\nBmk(ϵ) is given by 2( ϵ−ϵmk)((ϵ−ϵmk)2−3η2)/((ϵ−ϵmk)2+η2)3. There are other notable\nfeatures different from the damping. In the limit η→0, the overlap Vmnreduces to 2 /(ϵmk−\nϵnk)3. For intraband transitions ( m=n), this leads to I → −∞ . In the limit η→ ∞ ,\nVmn≈1/η5which leads to I →0. The behaviour at these two limits is evident from Fig. 5.\nThe large τ(small η) behaviour is consistent with the expression I = −α.τ/2πderived by\n17104\n−10−310−2−10−1100\n10−2−10−110−1100101102\nFIG. 7: The damping, magnetic moment of inertia and relaxation rate are shown as a\nfunction of broadening, ηin blue, green and red, respectively. The grey-shaded region\nshows the observed experimental relaxation rate, τ, ranging from 0 .12 to 0 .47 ps. The\ncorresponding range of ηis shown in purple and is 6 −12 meV. This agrees with the\nexperimental broadening in the range of 5 −10 meV, arising from electron-phonon\ncoupling. The numbers are tabulated in Table I\nF¨ahnle et al. [49]. Here τis the Bloch relaxation lifetime. The behaviour of τas a function\nofηusing the above expression in the low ηlimit is shown in Fig. 7. Apart from these\nlimits, the sign change has been observed in a certain range of ηfor Fe and Co. This change\nin sign can be explained by the Eqn. (5). In the regime of intraband contribution, at a\ncertain η, the negative and positive terms integrated over ϵandkbecome the same, leading\nto zero inertia. Above that η, the contribution due to the negative terms decreases until the\ninterband contribution plays a major role leading to maxima in I (minima in −I). Interband\ncontribution leads to the sign change from + to - and eventually zero at larger η.\nThe expression I = −α.τ/2πderived from the Kambersky model is valid for η <10 meV,\nwhich indicates that damping and moment of inertia have opposite signs. By analyzing the\nrate of change of magnetic energy, Ref. [11] shows that Gilbert damping and the moment\nof inertia have opposite signs when magnetization dynamics are sufficiently slow (compared\ntoτ).\nExperimentally, the stiffening of FMR frequency is caused by negative inertia. The\nsoftening caused by positive inertia is not observed experimentally. This is because the\nexperimentally realized broadening, ηcaused by electron-phonon scattering and scattering\nwith impurities, is of the order of 5 −10 meV. The values of Bloch relaxation lifetime, τ\n18measured at the room temperature with the FMR in the high-frequency regime for Ni 79Fe21\nand Co films of different thickness, range from 0 .12−0.47 ps. The theoretically calculated\nvalues for Fe,Ni and Co using the Wannier implementation for the ηranging from 5 −10\nmeV are reported in Table. I and lies roughly in the above-mentioned experimental range\nfor the ferromagnetic films.\nV. CONCLUSIONS\nIn summary, this paper presents a numerical method to obtain the Gilbert damping\nand moment of inertia based on the torque-torque correlation model within an ab initio\nWannier framework. We have also described a technique to calculate the spin-orbit coupling\nmatrix elements via the transformation between the spin-orbit and spin-polarised basis. The\ndamping and inertia calculated using this method for the transition metals like Fe, Co and\nNi are in good agreement with the previous studies based on tight binding method[29, 30]\nand local spin density approximation[32]. We have calculated the Bloch relaxation time\nfor the approximate physical range of broadening caused by electron-phonon coupling and\nlattice defects. The Bloch relaxation time is in good agreement with experimentally reported\nvalues using FMR[27]. The calculated damping and moment of inertia can be used to study\nthe magnetisation dynamics in the sub-ps regime. In future studies, we plan to use the\nWannier implementation to study the contribution of spin pumping terms, arising from\nthe spin currents at the interface of ferromagnetic-normal metal bilayer systems due to\nthe spin-orbit coupling and inversion symmetry breaking to the damping. 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He, Two-dimensional\nferromagnetic materials: From materials to devices, Applied Physics Letters 121(2022).\n23" }, { "title": "1104.1791v1.Spatial_Damping_of_Propagating_Kink_Waves_Due_to_Resonant_Absorption__Effect_of_Background_Flow.pdf", "content": "arXiv:1104.1791v1 [astro-ph.SR] 10 Apr 2011Draftversion June6,2018\nPreprint typesetusingL ATEX styleemulateapjv. 11 /10/09\nSPATIALDAMPINGOFPROPAGATINGKINK WAVESDUE TORESONANTAB SORPTION: EFFECT OF\nBACKGROUND FLOW\nR.Soler1, J.Terradas2,andM.Goossens1\n1Centre for Plasma Astrophysics, Katholieke Universiteit L euven, Celestijnenlaan 200B, 3001 Leuven, Belgium and\n2Departament deF´ ısica, Universitat deles Illes Balears, E -07122, Palma deMallorca, Spain\nDraftversion June 6,2018\nABSTRACT\nObservations show the ubiquitous presence of propagating m agnetohydrodynamic (MHD) kink waves in\nthe solar atmosphere. Waves and flows are often observed simu ltaneously. Due to plasma inhomogeneity in\nthe perpendicular direction to the magnetic field, kink wave s are spatially damped by resonant absorption.\nThe presence of flow may a ffect the wave spatial damping. Here, we investigate the e ffect of longitudinal\nbackground flow on the propagation and spatial damping of res onant kink waves in transversely nonuniform\nmagnetic flux tubes. We combine approximate analytical theo ry with numerical investigation. The analytical\ntheoryusesthethintube(TT)andthinboundary(TB)approxi mationstoobtainexpressionsforthewavelength\nand the damping length. Numerically, we verify the previous ly obtained analytical expressions by means of\nthe full solution of the resistive MHD eigenvalue problem be yond the TT and TB approximations. We find\nthatthebackwardandforwardpropagatingwaveshavedi fferentwavelengthsandaredampedonlengthscales\nthat are inversely proportional to the frequency as in the st atic case. However, the factor of proportionality\ndependsonthecharacteristicsoftheflow,sothatthedampin glengthdiffersfromitsstatic analogue. Forslow,\nsub-Alfv´ enicflows the backward propagatingwave gets damp edon a shorter length scale than in the absence\nof flow, while for the forward propagatingwave the damping le ngth is longer. The di fferent properties of the\nwavesdependingontheirdirectionofpropagationwithresp ecttothebackgroundflowmaybedetectedbythe\nobservationsandmayberelevantforseismologicalapplica tions.\nSubject headings: Sun: oscillations — Sun: corona — Sun: atmosphere — Magnetoh ydrodynamics(MHD)\n— Waves\n1.INTRODUCTION\nThe presence of ubiquitous small-amplitude propagating\nmagnetohydrodynamic (MHD) waves in magnetic waveg-\nuides of the solar corona was first observedwith the Coronal\nMulti-channelPolarimeter(CoMP)(seeTomczyketal.2007;\nTomczyk&McIntosh 2009). The observationshave inspired\na number of recent theoretical works in which the proper-\nties of propagating waves are studied. Based on MHD wave\ntheory (e.g., Erd´ elyi&Fedun 2007; VanDoorsselaereetal.\n2008a,b; Verthet al. 2010) the observations have been in-\nterpreted as propagating kink waves, i.e., transverse MHD\nwaves with mixed fast and Alfv´ enic properties whose\ndominating restoring force is magnetic tension (see, e.g.,\nEdwin& Roberts 1983; Goossenset al. 2009). In particu-\nlar,VanDoorsselaereet al. (2008b) showedthat field-align ed\ndensity enhancements in the corona act as natural waveg-\nuides for MHD waves. Apart from the CoMP observations\nin coronal loops, propagating kink waves have also been\nobserved in chromospheric spicules (e.g., DePontieuetal.\n2007; Heet al. 2009a,b) and in thin threads of solar promi-\nnences (e.g., Linetal. 2007, 2009). In the same man-\nner as standing kink MHD waves are damped in time\nby resonant absorption (see, e.g., Goossenset al. 2002;\nRuderman& Roberts 2002), propagating kink MHD waves\nare damped in space due to naturally occurring plasma in-\nhomogeneity in the direction transverse to the the magnetic\nfield(Terradaset al.2010b). Terradaset al.(2010b),herea fter\nTGV,obtainedtheimportantresultthatthedampinglengthb y\nresonant absorptionis inversely proportionalto the wave f re-\nquency. Thismeansthathigh-frequencywavesaredampedin\nroberto.soler@wis.kuleuven.beshorter length scales than low-frequency waves. Verthet al .\n(2010) showed that the analytical theory of propagating res -\nonant kink waves developed by TGV is fully consistent with\ntheCoMPobservations. Forthetime-dependent,drivenprob -\nlemPascoeet al.(2010)studiednumericallythespatialdam p-\ning of resonant kinkwaves and obtainedequivalentresults t o\nthose analytically predicted by TGV. Importantly,Soleret al.\n(2011a) have shown that the results of TGV still hold when\nthe plasma is partially ionized, so that the theory can be ap-\nplied to kink waves propagating in the chromosphere and in\nprominencesas well. Recently, Soleretal. (2011b) extende d\nthe results of TGV by taking into account for the first time\ndensity variation both transversely and along the magnetic\nfielddirection.\nField-aligned flows are also ubiquitous in magnetic\nstructures in the solar atmosphere (see the observa-\ntional reports by, e.g., Brekkeetal. 1997; Zirkeret al.\n1998; Winebargeret al. 2001, 2002; Okamotoet al. 2007;\nChaeet al. 2008; Ofman& Wang 2008). Typically, the ob-\nserved flow velocities are smaller than 10% of the plasma\nAlfv´ en speed. Faster flows of the order of the Alfv´ en speed\nare much less frequent and are related to energetic events as ,\ne.g., flares and coronal mass ejections (see, e.g, Inneset al .\n2003). The investigation of the e ffect of flow on the proper-\ntiesofthewavesisthereforeofevidentinterest. Forexamp le,\ntheinfluenceofflowandimplicationsforMHDseismologyof\nstanding kink waves in coronal loops have been recently dis-\ncussedbyRuderman(2010)andTerradaset al.(2011). Inthe\ncase of prominences, Soler&Goossens (2011) investigated\nstandingkinkwavesin coronalfluxtubespartiallyfilled wit h\nflowing threads of prominence material. Nevertheless, none\nof these works took damping into account. Also for stand-2\ning waves, Terradaset al. (2010a) studied temporal damping\nbyresonantabsorptioninthepresenceofflowandfoundcor-\nrections to the damping time due to the flow with respect to\nthe static case (Goossenset al. 2002). However, in the case\nof spatial damping of propagating kink waves the e ffect of\nflow has not been investigated. TGV did not include flow in\ntheir study. To our knowledge there is no work in the lit-\nerature that studies in detail the influence of flow on reso-\nnantly damped propagating waves in solar magnetic waveg-\nuides. Existing investigations of surface waves in the sola r\nwind as, e.g., Evansetal. (2009) did not perform a rigorous\ntreatment of the process of resonant absorption and consid-\neredverysimplifiedexpressionsforthedampinglengthofth e\nwaves. Thus, a detailedinvestigationofpropagatingreson ant\nMHDwavesinthepresenceofflowsinneeded.\nHere we investigate the e ffect of flow on the spatial damp-\ning of resonant kink waves in transversely nonuniform solar\nwaveguides. We attack the problembothanalyticallyandnu-\nmerically. The analytical theory uses the thin tube (TT) and\nthin boundary(TB) approximationsto obtain expressionsfo r\nthe wavelength and the damping length. We determine the\ninfluenceofflowandcompareourexpressionswiththoseob-\ntained by TGV in the static case. Later, we use numerical\nmethodstostudythepropagationandspatialdampingofkink\nwaves beyond the TT and TB approximations. The full nu-\nmerical computationsenable us to test the validity of the an -\nalytical expressions. Finally, we discuss the implication s of\nourresultsforMHD seismology.\n2.MODELANDGOVERNINGEQUATIONS\nTheequilibriumconfigurationisastraightcylindricalmag -\nnetic flux tube of radius Rembeddedin a magnetizedplasma\nenvironment. For convenience, we use cylindrical coordi-\nnates,namely r,ϕ,andzfortheradial,azimuthal,andlongitu-\ndinalcoordinates,respectively. The z-directionissetalongthe\naxisofthecylinder. Weusethe β=0approximation,where β\nistheratioofgaspressuretomagneticpressure. The β=0ap-\nproximation enables us to arbitrarily choose the density pr o-\nfile. In what follows, subscripts i and e refer to the internal\nand external plasmas, respectively. For example, we denote\nbyρiandρethe internal and external densities, respectively.\nBoth of these quantitiesare constants. Thereis a nonunifor m\ntransitionallayerin thetransversedirectionthatcontin uously\nconnectstheinternaldensitytotheexternaldensity. Thel ayer\nhasathickness landcoverstheinterval R−l/2≤r≤R+l/2.\nThe equilibrium magnetic field is straight, B=Bˆez, withB\nconstant. We assume a background flow along the magnetic\nfielddirection, U=Uˆez. Wedenoteby UiandUetheinternal\nandexternalflowvelocities,respectively,whichareconst ants.\nWe takeUiandUeas positive quantities. As for the density,\nwe allow the flow velocity to change continuously in the ra-\ndial direction from its internal to its external values with in a\ntransitional layer. The transitional layer for the flow velo city\nextends in the interval R−l⋆/2≤r≤R+l⋆/2, withl⋆the\nthicknessofthetransition.\nLinear, ideal MHD waves propagating in our model are\ngovernedbythefollowingset ofequations,\nρ/parenleftigg∂v\n∂t+U·∇v+v·∇U/parenrightigg\n=1\nµ(∇×b)×B,(1)\n∂b\n∂t−∇×(U×b)=∇×(v×B), (2)\nwhereρis the plasma density, vis the velocity perturbation,bis the magnetic field perturbation, and µis the magnetic\npermittivity. As the equilibrium is uniform in both ϕ- andz-\ndirectionsand we consider waves propagatingalong the tube\nwithafixedfrequency,wewriteallperturbationsproportio nal\ntoexp(imϕ+ikzz−iωt),wheremistheazimuthalwavenum-\nber,kzis the longitudinal wavenumber, and ωis the wave\nangular frequency. Due to the presence of a transverse in-\nhomogeneoustransitional layer, wave modes with m/nequal0 are\nspatially dampeddue to resonant absorption. Here we are in-\nterested in kink waves, which are described by m=1. Kink\nwaves have mixed Alfv´ enic and fast MHD properties. They\naretheonlywavemodesthatcandisplacethemagneticcylin-\nder axis and so producetransverse motionsof the whole flux\ntube(see,e.g.,Edwin&Roberts1983;Goossenset al.2009).\nAs a result of the process of resonant absorption, transvers e\nkink motionsof the flux tube are damped and azimuthal mo-\ntions within the transitional layer are amplified as the wave\npropagatesalongthemagneticcylinder.\nFor realω, resonant damping causes kzto be complex,\nkz=kzR+ikzI, withkzRandkzIthe real and imaginary\nparts ofkz, respectively. For fixed and positive ω, the di-\nrection of wave propagation is determined by the sign of\nkzR. ForkzR>0 the wave propagates towards the posi-\ntivez-direction (forward waves), whereas for kzR<0 the\nwave propagates towards the negative z-direction (backward\nwaves). In the absence of flow both directions of propa-\ngation are equivalent. In the presence of flow forward and\nbackward waves have not the same properties and both di-\nrections of propagation must be taken into account (see,\ne.g., Nakariakov& Roberts 1995; Terra-Homemet al. 2003;\nSoleretal. 2008; VasheghaniFarahaniet al. 2009). Regard-\ning the imaginary part of kz, resonant absorption spatially\ndampsthe wave,sowe expect kzI>0. However,strongflows\nmaytriggertheKelvin-HelmholtzInstability(KHI)(see,e .g.,\nChandrasekhar1961;Drazin&Reid1981),causingmodesto\nbe amplified in z, i.e.,kzI<0. From the real and imaginary\nparts ofkzwe compute the wavelength, λ, and the damping\nlength,LD, as\nλ=2π\nkzR,LD=1\nkzI. (3)\n3.ANALYTICALINVESTIGATION\nTo study analytically the e ffect of flow on the resonantly\ndamped kink waves we use the TT and TB approximations.\nIn the TT approximation we restrict ourselves to waves with\nλ/R≫1. In terms of frequency, the TT approximation is\nequivalent to the low-frequency approximation, i.e., ωτA≪\n1, withτA=R/vAthe Alfv´ en travel time and vA=B/√µρ\ntheAlfv´ envelocity. TheTBapproximationisusedheretoin -\nclude the effect of resonantabsorptionin the inhomogeneous\nlayer, and is valid for l/R≪1 andl⋆/R≪1. In the TB ap-\nproximation,thejumpoftheperturbationsacrosstheinhom o-\ngeneouslayer is assumed to be the same as their jumpacross\nthe resonant layer. The expressions for the jump conditions\ncan be found in, e.g., Sakuraiet al. (1991); Goossenset al.\n(1995); Tirry& Goossens (1996) for the static case, and in\nGoossensetal. (1992); Erd´ elyiet al. (1995) for the statio n-\nary case. Then, the connection formulae at the Alfv´ en res-\nonance are used as jump conditions for the perturbations at\nthe tube boundary (see extensive details about the method in\nGoossensetal. 2006; Goossens 2008; Goossenset al. 2011).\nThe dispersion relation for kink and fluting MHD waves in\nthe TT and TB approximations is (see, e.g., Goossenset al.RESONANTKINK WAVESWITH FLOW 3\n1992)\nρi/parenleftig\nΩ2\ni−ω2\nAi/parenrightig\n+ρe/parenleftig\nΩ2\ne−ω2\nAe/parenrightig\n=\niπm/rA\nρ(rA)|∆A|ρi/parenleftig\nΩ2\ni−ω2\nAi/parenrightig\nρe/parenleftig\nΩ2\ne−ω2\nAe/parenrightig\n,(4)\nwhereΩ=ω−Ukzis the Doppler-shifted frequency, ω2\nA=\nk2\nzv2\nAis the square of the Alfv´ en frequency, rAis the Alfv´ en\nresonanceposition,and\n|∆A|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndr/bracketleftig\nΩ2−ω2\nA/bracketrightig\nrA/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (5)\nThe term on the right-hand side of Equation (4) contains the\neffect of resonant absorption. Note that the dependence on\nmof the dispersion relation is only present in this term. This\nmeansthatintheTTapproximationthee ffectoftheazimuthal\nwavenumber, m, is only felt in the damping of the wave, not\ninitspropagation.\nIn the case of temporal damping of standing waves, i.e.,\nrealkzand complexω, the solutions to Equation (4) have\nbeen investigated in detail by Goossenset al. (1992) and\nTerradaset al.(2010a). Inthecaseofspatialdampingofpro p-\nagating waves, i.e., real ωand complex kz, Equation (4) has\nbeen explored by TGV in the absence of flow. Here our pur-\nposeistoinvestigatethee ffectofflowontheresultsobtained\nbyTGVinthe static case.\n3.1.No resonantdamping\nIn the absence of resonant damping, i.e., for l/R=l⋆/R=\n0,Equation(4)becomes\nρi/parenleftig\nΩ2\ni−ω2\nAi/parenrightig\n+ρe/parenleftig\nΩ2\ne−ω2\nAe/parenrightig\n=0. (6)\nEquation (6) is independent of m. In the static case, Ui=\nUe=0andΩi=Ωe=ω. Thesolutionto Equation(6) is\nkz=±ω\nvk≡±k0, (7)\nwith\nvk=ρiv2\nAi+ρev2\nAe\nρi+ρe1/2\n, (8)\nthe kink velocity. In Equation (7) the +sign stands for the\nforward wave and the −sign for the backward wave. In the\nabsenceofflow bothforwardandbackwardwavesare equiv-\nalentandhavethesame wavelength, λ=2π/k0.\nInthepresenceofflow,werewriteEquation(6)asasecond-\norderpolynomialin kz,namely\n/parenleftig\nv2\nk−v2\nKE/parenrightig\nk2\nz+2vcmωkz−ω2=0, (9)\nwherevcmandvKEaredefinedas\nvcm=ρiUi+ρeUe\nρi+ρe,vKE=ρiU2\ni+ρeU2\ne\nρi+ρe1/2\n.(10)\nvcmis the center-of-mass velocity. vKEis the velocity asso-\nciated to the kinetic energy of the flow. Note that, similarly ,\nthe kink velocity, vk, is the velocity associated to the energy\nof the magnetic field. The two solutions to Equation (9) for\nvKE/nequalvkare\nkz=−vcm\nv2\nk−v2\nKEω±k0v2\nk\nv2\nk−v2\nKE1−ρiρe\n(ρi+ρe)2(Ui−Ue)2\nv2\nk1/2\n,\n(11)wherethe+and−signsin frontof the second term stand for\nforward and backward waves, respectively. If Ui=Ue=0,\nEquation(11) simplyreducesto Equation(7). We clearlysee\ninEquation(11)thattheequivalencebetweenbothdirectio ns\nof propagation is broken by the flow. For vKEvkbothwavespropagateinthesamedirection,i.e.,they\nbothare forwardwaves in practice becausethe flow is strong\nenoughto force the backwardwave to reverseits directionof\npropagation. In the particular case vKE=vk, the solution to\nEquation(9) is\nkz=ω\n2vcm, (12)\nwhich corresponds to the forward wave, while the backward\nwave does not propagate in the static reference frame. For\nUe=0 thecondition vKE=vkisequivalentto\nU2\ni=2v2\nAi. (13)\nWhen the argument of the square root in Equation (11) is\nnegative, kzbecomescomplex. Thisistheclassical KHI(see,\ne.g., Chandrasekhar 1961; Drazin& Reid 1981). Then, the\ntwo solutions correspond to a spatially damped mode and a\nspatiallyamplifiedmode,respectively. TheKHIappearsfor a\ncriticalvelocityshear, ∆U=Ui−Ue, definedas\n(∆U)2>(ρi+ρe)2\nρiρev2\nk≡v2\nKH. (14)\nAgain,inthereferenceframewhere Ue=0Equation(14)can\nberewrittenas\nU2\ni>2/parenleftigg\n1+ρi\nρe/parenrightigg\nv2\nAi. (15)\nFrom Equations(13) and (15) we see that the KHI requiresa\nfaster flow velocity than the one neededto reversethe propa-\ngationofthe backwardwave.\nEquivalentlytothecasewithoutflow(seeEquation(7)),we\ncanrewriteEquation(11)as\nkz=ω\nv±\nkf(16)\nwherewehaveintroducedthee ffectivekinkvelocitymodified\nbythe flow, v±\nkf,definedby\nv±\nkf=−vcm\nv2\nk−v2\nKE±vk\nv2\nk−v2\nKE1−ρiρe\n(ρi+ρe)2(Ui−Ue)2\nv2\nk1/2−1\n.\n(17)\nNotethattheeffectivekinkvelocityisdi fferentforforward(+\nsign)andbackward( −sign)waves. Thismeansthatthephase\nspeedofthewavesdependsontheirdirectionofpropagation .\nAlsonotethat v±\nkfbecomescomplexbeyondthecriticalveloc-\nity shear for the KHI. For slow, sub-Alfv´ enic flows we may\ndrop the quadratic terms in UiandUefrom Equation (17) to\nobtaina first-orderapproximationfor v±\nkf, namely\nv±\nkf≈±vk+vcm. (18)\nIn the absence of flow, Equation (17) reduces to v±\nkf=±vk.\nWhen the effective kink velocity is equal to the external\nAlfv´ en velocity, the wave becomes leaky in the external\nmedium. In terms of kz, waves are leaky for wavenumbers\nlargerthan\nkz=±ω\nvAe≡±kL. (19)4\nThe forward wave becomes leaky for much slower flow ve-\nlocitiesthan the backwardwave. By using Equation(18) and\ntakingUe=0, we find that the forward wave becomes leaky\nfor\nUi/greaterorsimilarρi+ρe√ρiρe−/radicaligg\n2(ρi+ρe)\nρivAi. (20)\nAgain,weusetheapproximationofEquation(18)inEqua-\ntion (16) to obtain a first-order approximation of the wave-\nlengthas\nλ≈λ0/parenleftigg\n±1+vcm\nvk/parenrightigg\n, (21)\nwithλ0=2π/k0. We consider the particular case Ue=0\nand use the dimensionless notation of TGV to rewrite Equa-\ntion(21) as\nλ\nR≈2π/radicaligg\n2ζ\nζ+11\nf±1+/radicaligg\nζ\n2(ζ+1)¯Ui,(22)\nwith\nζ=ρi\nρe,f=ωR\nvAi,¯Ui=Ui\nvAi,(23)\nthe density contrast, the dimensionlessfrequency,and the di-\nmensionlessflow velocity,respectively.\nWe plot in Figure 1(a) |λ|/Rversus¯Uicomputed from the\nfullEquation(11)fortheparticularcase Ue=0. Aspredicted\nby Equation (13), the backward wave reverts its direction of\npropagation for ¯Ui=√\n2. For the set of parameters used in\nFigure1,the forwardwave becomesleakyfora flowvelocity\nslightlysub-Alfv´ enic(Equation(20)). Whenthethreshol dve-\nlocityoftheKHIisreached(Equation(15)),bothforwardan d\nbackward waves merge. We compare the full result with the\napproximationforslowflowsgivenbyEquation(22)(see the\nsymbols in Fig. 1(a)), and obtain a good agreement for sub-\nAlfv´ enic flows, i.e., ¯Ui/lessorsimilar1. On the other hand, Figure 1(b)\ndisplayskzIRversus¯Ui. In the absence of resonant damping,\ntheimaginarypartof kziszeroforflowvelocitiesslowerthan\nthe critical velocity shear of the KHI. For larger velocitie s,\nonedampedsolutionandoneoverstablesolutionarepresent .\n3.2.Effectof resonantdamping\nHere we incorporate the damping due to resonant absorp-\ntion. We take into account the full expression of the disper-\nsionrelation(Equation(4)). We write kz=kzR+ikzIinEqua-\ntion (4) and consider weak damping, so we neglect terms of\nO/parenleftig\nk2\nzI/parenrightig\n. Inaddition,weimplicitlyassumethattheflowveloci-\ntiesareslowerthanthecriticalvelocityoftheKHI.Afterl ong\nbut straightforwardanalytical manipulations, we obtain f rom\nEquation(4)theexpressionforthe ratio kzI/kzR, namely\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglekzI\nkzR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=π\n2m\nrAρ2\ni\nρi+ρe1\nρ(rA)|∆A|/parenleftig\nΩ2\ni−ωAi/parenrightig2\nω(ω−ωcm),(24)\nwithωcm=kzRvcm. In the absence of flows, Equation (24)\nreducestoEquation(8)ofTGV.Duetothedi fferentvaluesof\nΩiandωcmfor forward and backward waves, Equation (24)\npredicts that both waves have di fferent damping ratios. The\nexpression for the ratio kzI/kzRis more complicated in the\npresence of flows compared to the static case of TGV. Let us\nfind a more simple expressionfor kzI/kzRin the limit of slow,\nsub-Alfv´ enicflows. First, weevaluate ρ(rA)|∆|AtoexplicitlyFigure1. (a)|λ|/Rand(b)kzIRversus¯Uicorrespondingtotheforward(solid\nline)andbackward(dashedline)kinkwavesintheabsenceof resonantdamp-\ning. Theverticaldottedlinesinbothpanelscorrespondtot hedifferentcritical\nflow velocities indicated in the text. Thehorizontal dotted line in panel (a) is\nthe wavelength for ¯Ui=0, withλ0=2π/k0. The symbols correspond to the\napproximation for slow flows given in Equation (22). In panel (b) the hori-\nzontal dotted line denotes kzI=0. In this plot, f=0.1,ζ=3, andUe=0.\ntake into account the radial variation of the density and the\nflow velocity at the resonance position. From Equation (5)\nweget\nρ(rA)|∆A|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΩ2(rA)/parenleftiggdρ\ndr/parenrightigg\nrA−2ρ(rA)Ω(rA)kzR/parenleftiggdU\ndr/parenrightigg\nrA/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\n(25)\nwhere we have used the resonant condition Ω2(rA)=\nk2\nzRv2\nA(rA). The quantities present in Equation (25) have to\nbe evaluatedat the resonanceposition, rA. In the TB approx-\nimation it is reasonable to assume rA��R. There are two\nterms in the right-hand side of Equation (25). The first term\nis due to the variation of density and the second term is due\nto the variation of flow velocity. The direction of wave prop-\nagationisalsoimportantinEquation(25)becauseofthesig n\nofkzR. In the absence of flow, Equation (25) simplifies to\nρ(rA)|∆A|=ω2/parenleftigdρ\ndr/parenrightig\nrA.\nFor smooth profiles, the derivatives of the density and the\nflowvelocityprofilesatthe resonancepositioncanbecast as\n/parenleftiggdρ\ndr/parenrightigg\nrA≈Fπ2\n4ρi−ρe\nl,/parenleftiggdU\ndr/parenrightigg\nrA≈Fπ2\n4Ui−Ue\nl⋆(26)\nwithFa factor that depends on the form of the trans-\nverse profile. For example, F=4/π2for a linear profile\n(Goossenset al. 2002) and F=2/πfor a sinusoidal profile\n(Ruderman&Roberts 2002). For simplicity we assume theRESONANTKINK WAVESWITH FLOW 5\nsame profile for both density and flow velocity, but we keep\nl/nequall⋆. Asvaluesofdensityandflowvelocityattheresonance\npositionwe take\nρ(rA)=ρi+ρe\n2,U(rA)=Ui+Ue\n2.(27)\nWe use in Equation (25) the expressions given in Equa-\ntions(26)and(27) toobtain\nρ(rA)|∆A|=Fπ2\n4ρi−ρe\nlΩ2(rA)\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−kzR\nΩ(rA)ρi+ρe\nρi−ρe(Ui−Ue)l\nl⋆/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(28)\nFor our following analysis, we assume that the second term\nwithin the absolute value in Equation (28) is smaller than\none, so that we can drop the absolute value sign. For slow\nflows, this is equivalent to assume l⋆/greaterorsimilarl. In the limit of\nslow, sub-Alfv´ enic flows, we neglect the quadratic terms in\nthe flow velocities. In addition,we write kzR≈ω/v±\nkfanduse\nthe first-order approximation for v±\nkfgiven in Equation (18).\nEquation(28)canthenberewrittenas\nρ(rA)|∆A|≈Fπ2\n4ρi−ρe\nlω2\n×/braceleftigg\n1∓Ui+Ue\nvk/bracketleftigg\n1+ρi+ρe\nρi−ρeUi−Ue\nUi+Uel\nl⋆/bracketrightigg/bracerightigg\n.(29)\nNext, we use Equation (29) in Equation (24) and perform\na first-order expansion in the flow velocities. Equation (24)\nbecomes\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglekzI\nkzR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≈1\n2πm\nFl\nRρi−ρe\nρi+ρe/braceleftigg\n1±vcm\nvk/bracketleftigg\n1−4ρi+ρe\nρi−ρeρiUi−ρeUe\nρiUi+ρeUe\n+Ui−Ue\nvcm/parenleftigg\n1+ρi+ρe\nρi−ρeUi−Ue\nUi+Uel\nl⋆/parenrightigg/bracketrightigg/bracerightigg\n. (30)\nAsinpreviousexpressions,the +and−signsinEquation(30)\nstand for forward and backward waves, respectively. In the\nparticularcase Ue=0,Equation(30)simplifiesto\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglekzI\nkzR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≈1\n2πm\nFl\nRρi−ρe\nρi+ρe\n×/bracketleftigg\n1±Ui\nvk/parenleftigg\n1+ρi\nρi+ρe−4ρi\nρi−ρe+ρi+ρe\nρi−ρel\nl⋆/parenrightigg/bracketrightigg\n.(31)\nWe use again the approximation kzR≈ω/v±\nkfand compute\nfromEquation(31) thedampinglength, LD=1/kzI,as\nLD≈2πF\nmR\nlρi+ρe\nρi−ρevk\nω/bracketleftigg\n1±Ui\nvk/parenleftigg3ρi+ρe\nρi−ρe−ρi+ρe\nρi−ρel\nl⋆/parenrightigg/bracketrightigg\n.\n(32)\nFinally, we express Equation (32) using the dimensionless\nquantitiesdefinedinEquation(23),namely\nLD\nR≈2πξE/radicaligg\n2ζ\nζ+11\nf1±¯Ui/radicaligg\nζ+1\n2ζ/parenleftigg3ζ+1\nζ−1−ζ+1\nζ−1l\nl⋆/parenrightigg,\n(33)\nwith\nξE=F\nmR\nlζ+1\nζ−1. (34)Equation (33) is the key equation of this investigation and\ncontainsbasic propertieson the spatial dampingof propaga t-\ning kink MHD waves. Several important results can be ex-\ntractedfromEquation(33). Equation(33) predictsthat bac k-\nward and forward propagating waves are damped on length\nscalesthatareinverselyproportionaltothefrequency, f. This\nis the same dependence found by TGV in the static case.\nHowever, the factor of proportionality depends on the char-\nacteristics of the flow and the density contrast, so that the\ndamping length di ffers from its static analogue. As for the\nwavelength (see Equation (22)), the damping length for for-\nwardandbackwardwavesisdi fferent.\nTo shed more light on this result, let us consider the case\nl⋆≫l. Equation(33) reducesto\nLD≈2πξE/radicaligg\n2ζ\nζ+11\nf1±¯Ui/radicaligg\nζ+1\n2ζ3ζ+1\nζ−1(35)\nAccording to Equation (35) the backward propagating wave\n(−sign) gets damped on a shorter length scale than in the\nabsence of flow, while for the forward propagating wave ( +\nsign) the damping length is longer. For l≈l⋆the damping\nlength for the forward wave remains longer than that of the\nbackwardwave,butwemustnotethatfor l⋆≪lthesituation\nmay be the opposite. From Equation (33) we can assess the\nrelationbetween landl⋆forwhichthetwotermsmultiplying\ntheflowvelocitycanceleachother,namely\nl⋆=ζ+1\n3ζ+1l. (36)\nThus, the backward wave damping length becomes longer\nthan that of the forward wave for l⋆smaller than the value\ngiven in Equation (36). Strictly, Equation (33) is not valid in\nthe limit l⋆≪l, because we have to properly take into ac-\ncounttheabsolutevalueinEquation(28). Forslowflowsand\nl⋆≪l, Equation(28) canbeapproximatedas\nρ(rA)|∆A|≈Fπ\n4ρi+ρe\nl⋆ω2Ui−Ue\nvk.(37)\nImportantly,wefindthatEquation(37)isindependentof land\nthe same expression holds for forward and backward waves.\nNow, from Equation (37) it is straightforward to obtain that\nLD∼1/l⋆for both forward and backward waves in the limit\nl⋆≪l. As discussed by Terradasetal. (2010a) in the case\nof temporal damping of standing waves, multiple resonances\nmayoccurwithintheinhomogeneoustransitionallayerinth e\nlimitl⋆≪l(see Fig. 5 of Terradasetal. 2010a). In such a\ncase, the total damping rate is the sum of the contributions\nfrom each resonance. We do not study this peculiar situation\nwhichtakesplaceforverysmall,probablynotrealistic l⋆. In-\nstead,wereferthereadertoTerradaset al.(2010a)fordeta ils.\n4.NUMERICALCOMPUTATIONS\nHere, we verify the validity of the analytical expressions\nderived in Section 3 in the TT and TB approximations. To\ndo so, we numerically solve the full eigenvalue problem by\nmeans of the PDE2D code (Sewell 2005). The numerical\nscheme is similar to that used by TGV. The code implements\na method based on finite elements to numerically integrate\nEquations(1)and(2)intheradialdirectionfromthecylind er\naxis,r=0, to the edge of the numerical domain, r=rmax.\nThe boundary conditionsat r=0 are set according the sym-\nmetryarguments,whileweimposeallperturbationstovanis h6\natr=rmax. In order to obtain a good convergence of the\nsolutions, rmaxis located far enough from the magnetic tube\nto avoid numerical errors. We take rmax=100R. We use a\nnonuniformgridwithalargedensityofgridpointswithinth e\ninhomogeneouslayer in order to correctly describe the smal l\nspatial scales of the eigenfunctions due to the Alfv´ en reso -\nnance. To avoid the singularity of the ideal MHD equations\nat the resonance position, we add to the induction equation\n(Equation (2)) a resistive term, i.e., η∇2/vectorb, withηthe mag-\nneticdiffusivity. Theoutputofthecodeisthecomplexeigen-\nfunctions and their corresponding eigenvalues. In the limi t\nof largeRm, withRm=vAR/ηthe magnetic Reynolds num-\nber, the eigenvalues are independent of Rm. In such a case,\nwave damping is due to resonant absorption exclusively. In\nour computations,we have considereda su fficientlylarge Rm\nandhavecheckedthattheeigenvaluesareindeedindependen t\nofRm. We typicallytake Rm≈107in thecomputations.\nThe PDE2D code solves the eigenvalue problem for the\ntemporal damping, i.e., for complex ωprovided a fixed and\nrealkz. As in TGV, we need to convert the results from tem-\nporal damping to spatial damping. We use Equation (40) of\nTGVtoperformtheconversionfromtemporaldamping(com-\nplexωandrealkz)tospatialdamping(real ωandcomplex kz).\nFromEquation(40)ofTGVitisstraightforwardtoobtainthe\nrelationbetweentheimaginarypartsof ωandkz,namely\nkzI=ωI/parenleftigg∂ωR\n∂kzR/parenrightigg−1\n, (38)\nwhereωRandωIare the real and imaginary parts of the fre-\nquency in the temporal damping case. In the spatial damp-\ning caseω=ωR. The factor within the parenthesis is the\ngroup velocity. Note that the flow does not explicitly appear\nin Equation (38), but its e ffect is contained in the value of\ngroupvelocitynumericallycomputed.\nFirst of all, we test the numerical code by considering the\ncase without flow. In this case we fully recover the results\nofTGV. Thus,we areconfidentthat thecodeworksproperly.\nHereonweincorporatethee ffectofflow. Regardingthewave-\nlength, ournumericalresults indicate that the presenceof the\ntransitionallayerhasasmallimpactonthevalueofthewave -\nlength. Thewavelengthinthecasewithoutresonantdamping\n(seeFig.1(a))isaverygoodapproximationtothewavelengt h\nin the resonantcase. The approximationforslow flows given\nin Equation(22) also holds in the case with damping. There-\nfore, our following analysis is focused on the behavior of LD\ninthepresenceofflows. Thenumericalresultsareusedtotes t\ntheapproximationsbehindEquation(33),i.e.,theTTandTB\napproximations,andtheassumptionofslowflows.\nIn Figure 2 we display LD/Ras a function of the dimen-\nsionless frequency, f, for¯Ui=0.1 (the rest of parameters\nare indicated in the caption of the Fig. 2). As in the static\ncase of TGV, the higher the frequency, the shorter the damp-\ning lengthby resonantabsorption. We obtainthe analytical ly\npredictedresultthatforwardandbackwardwaveshavedi ffer-\nent damping lengths. In this example, the damping length of\nthe forward wave is longer than that of the backward wave.\nThe equivalent damping length in the absence of flow is in\nbetween both values (the dotted line in Fig. 2). We compare\nthe numericalresultswith thosein the TT approximationand\nforslowflows(Equation(33)). TheTTapproximationapplies\nwhenf≪1. In Figure 2(a) we plot the results for f≤0.1.\nAn excellent agreement is found between numerical and an-\nalytical solutionsfor both forward and backward waves. TheFigure2. (a) Ratio of the damping length to the radius, LD/R, versus the\ndimensionless frequency, f, corresponding to the forward (solid line) and\nbackward (dashed line) kink waves for ¯Ui=0.1,Ue=0,ζ=3, andl/R=\nl⋆/R=0.1. The symbols correspond to the approximation for slow flows\ngiven in Equation (33). The dotted line is the result in the ab sence of flow\n(seeTGV).(b) Sameas panel (a) butfor larger f.\nagreementbetweenapproximateandnumericalresultsisals o\nremarkably good even when the condition f≪1 of the TT\napproximationisnotstrictlyfulfilled. ThiscanbeseeninF ig-\nure2(b),whereresultsareplottedfor f≤1.\nNext we determine the influence of landl⋆, and test the\nTB approximation used to derive the analytical expressions .\nFirst we consider the case l=l⋆. Figure 3(a) shows the\ndependence of the damping length on l/R. As for the fre-\nquency,LD/Rdecreases as l/Rincreases. The dependence\nof the damping length on l/Rfor both forwardand backward\nwaves is the same. As before, a very good agreement be-\ntween Equation (33) and the numerical result is found even\nwhen the thickness of the transitional layer departs from th e\nlimitl/R≪1. This means that the TB approximationis suf-\nficiently accurate when the condition l/R≪1 is slightly re-\nlaxed. Thisresult enablesustoconfidentlyuse Equation(33 )\nbeyondtheconditionofvalidityoftheTBapproximation. Al -\nternately, in Figure 3(b) we keep l/Rconstant and vary l⋆/R,\nso that the spatial scales for the variation of density and flo w\nvelocity are different. We recover the analytically predicted\nresult that for l⋆≫lthe results are independentof l⋆/R. For\nl⋆≪lEquation(33)doesnotcorrectlydescribethedamping\nlength of the forward wave. As discussed at the end of Sec-\ntion3.2,LD∼1/l⋆in thelimit l⋆≪l, andsoLD/Rincreases\nasl⋆/R→0.\nFinally, we assess the e ffect of the flow velocity. This is\ndone in Figure 4. Again, the linear approximation (Equa-\ntion (33)) is quite accurate and agrees well with the full\nnumerical results for ¯Ui/lessorsimilar0.1. As expected, the di ffer-RESONANTKINK WAVESWITH FLOW 7\nFigure3. (a) Ratio of the damping length to the radius, LD/R, versusl/R\nfor the forward (solid line) and backward (dashed line) kink waves in the\ncasel⋆/R=l/R. The dotted line is the result in the absence of flow. (b)\nDependence on l⋆/Rforl/R=0.15. The vertical dotted line denotes the\nvalue ofl⋆/Rfor which both forward and backward waves have the same\nLD/R(Equation (36)). In both panels the symbols are the approxim ation of\nEquation (33). In these plots, ¯Ui=0.05,Ue=0,f=0.1, andζ=3.\nence between the numerical results and the linear approxi-\nmation increases as the flow velocity gets faster. However,\nfor¯Ui=0.2 the relative difference between the full solu-\ntion and the approximation is only around 10% for the for-\nward wave and 30% for the backward wave. This means\nthat forrealistic flow velocitiesobservedin coronalmagne tic\nloops(e.g.,Brekkeetal.1997;Winebargeretal.2001,2002 ),\nEquation(33)correctlydescribesthebehaviorofthedampi ng\nlength.\nInsummary,inthisSectionwehaveconfirmedthattheana-\nlyticalexpressionsobtainedintheTTandTBapproximation s\nandforslow,sub-Alfv´ enicflowsareveryaccurateevenwhen\nthese expressions are used outside their domain of strict va -\nlidity. This result enables us to use Equation (33), i.e., th e\nkey equation of this investigation, when realistic values o f\nfrequency, flow velocity, and the rest of relevant parameter s\nobtainedfromtheobservationsareused.\n5.DISCUSSION\nNaturally,kinkwavespropagatinginnonuniformmagnetic\nfluxtubesarespatiallydampedbyresonantabsorption. Inth e\nstatic case, TGV showedthat the dampinglengthis inversely\nproportionaltothefrequency. Herewehaveinvestigatedan a-\nlyticallyandnumericallythespatialdampingofresonantk ink\nwaves in a transversely nonuniform magnetic waveguide in\nthe presence of longitudinal background flow. Longitudinal\nflow breaks the equivalence between forward and backward\npropagating waves with respect to the flow direction. TheFigure4. Ratio of the damping length to the radius, LD/R, versus the flow\nvelocity normalized to theinternal Alfv´ en velocity, ¯Ui,for theforward (solid\nline) and backward (dashed line) kink waves. The symbols are the linear\napproximation given in Equation (33). We have used l/R=l⋆/R=0.1,\nUe=0,f=0.1, andζ=3.\nwavelength and the damping length due to resonant absorp-\ntionarebothaffectedbytheflow. Forsub-Alfv´ enicflows,the\nbackwardwavelengthisshorterthanthatoftheforwardwave ,\nandbackwardwavesaredampedinshorterlengthscalesthan\nforward waves. However, as in TGV we have found that the\ndamping length of both forward and backward propagating\nwavesisinverselyproportionaltothe frequency.\nMHD seismology based on propagating waves has at-\ntracted limited attention and definitely less than its count er-\npart based on standing kink waves. Standing kink MHD\nwaves are rare phenomena as they need a violent and en-\nergetic event such as a solar flare for their excitation (see,\ne.g., Aschwandenetal. 1999; Nakariakovet al. 1999). In the\nabsence of flow and for coronal loop standing oscillations\n(see, e.g., Nakariakov&Ofman 2001; Goossensetal. 2002;\nArreguiet al. 2007, 2008; Goossenset al. 2008), MHD seis-\nmology has been used to obtain information of the plasma\nphysical conditions. Particularly, for a given set of param e-\nters providedby the observations, i.e., period, dampingti me,\nand wavelength in the case of standing waves, Arreguiet al.\n(2007) and Goossensetal. (2008) showed that the possible\nvalues of vAi,ζ, andl/Rwhich are consistent with the the-\nory form a one-dimensional curve in the three-dimensional\nparameter space. In principle, any point of this curve can\nequally explain the observations. Soleretal. (2010) and\nArregui& Ballester (2011) showedthat more constrainedes-\ntimations of vAiandl/Rcan be given in the case of promi-\nnence thread oscillations as the limit ζ≫1 can be adopted.\nMore recently, Arregui&Asensio Ramos (2011) found that\nmore accurate estimations of the parameters are possible by\ncombining the analytical theory of Goossensetal. (2008)\nwith statistical Bayesian analysis. In the presence of flows ,\nTerradaset al. (2011) have recently explained also for stan d-\ning waves that the flow velocity can be estimated from the\nwavephasedifferencealongthemagneticloop.\nOn the contrary, propagating MHD waves are ubiq-\nuitous in the solar atmosphere (see, e.g., Tomczyket al.\n2007; Tomczyk&McIntosh 2009) and provide a huge reser-\nvoir of possibilities for seismology. Some examples of\nMHD seismology based on propagating waves are, e.g.,\nVanDoorsselaereetal. (2008b) using numerical simulation s\nof guided MHD waves by density enhancements in the so-\nlarcorona,Linet al.(2009)usingobservationsofkinkwave s\nin prominence threads, Verthet al. (2010) using resonantly8\ndamped kink waves in coronal loops, and Verthet al. (2011)\nexploiting the properties of kink waves in chromospheric\nspicules. However,noneoftheseworksincludedflowintheir\nanalysis. Our theoretical results given in Equations (22) a nd\n(33) have direct implications for MHD seismology based on\npropagatingwavesinaflowingmedium,andcouldbeusedto\ninfer informationabout the plasma properties. Therefore, the\npotentialapplicationofMHD seismologyto the case ofreso-\nnantly damped propagatingkink waves in a flowing medium\nmustbeexplored.\nIn the presence of flow two waves with di fferent wave-\nlengthsand dampinglengthsbut with the same frequency(or\nperiod)are simultaneouslypresent. We denoteas λ+andLD+\nthe wavelength and damping length of the forward wave, re-\nspectively,and asλ−andLD−the equivalentquantitiesof the\nbackwardwave. Observationally,thismeansthatitispossi ble\nto measure five quantities, namely λ+,λ−,LD+,LD−, and the\nperiod,whiletherestofparameters,i.e., vAi,Ui,l/R,andζare\nin principle unknown. In this analysiswe assume l/R=l⋆/R\nforsimplicity. Wealsotake λ−asapositivequantityandsowe\nperform the absolute value of Equation (22) when the −sign\nisused. Ifobservationscanprovideuswithreliablevalues for\nthe wavelengthsand damping lengths of the two waves, then\nwecanuseourtheoreticalresults(Equations(22)and(33)) to\nobtain seismological estimations for the unknown quantiti es\nvAi,Ui,l/R,andζas\nvA=1\nP/radicaligg\nζ+1\n2ζλ++λ−\n2, (39)\nUi=1\nPζ+1\nζλ+−λ−\n2, (40)\nl\nR=Fζ+1\nζ−1λ++λ−\nLD++LD−, (41)\nζ=1+2γ\n1−2γ, (42)\nwithP=2π/ωtheperiodand\nγ=λ+−λ−\nλ++λ−LD++LD−\nLD+−LD−. (43)\nInthe absenceof flow, λ+=λ−andLD+=LD−. Then,Equa-\ntions (39) and (41) are equivalent to the expressions studie d\nby Goossensetal. (2008), and the density contrast (Equa-\ntion (42)) becomes indeterminate. Thanks to the flow, the\ndensitycontrastcanbedeterminedifreliablemeasuresoft he\nwavelengths and damping lengths of both forward and back-\nwardwavesareavailable.\nThe theoretical results of the present paper o ffer new and\nexciting opportunities for MHD seismology in plasma struc-\ntureswithequilibriumflows. Forthefirsttimewehaveshown\nthat an estimation of the density contrast, ζ=ρi/ρe, is possi-\nble. MHD seismology requirestheory and observations. The\nrequired observations might not be available at present tim e.\nHowever, the seismological tool providedin Equations(39) –\n(42) could be used in the future when the required observa-\ntionsbecomeavailable.\nTheinvestigationperformedinthispapermaybeimproved\nin the futureby incorporatingadditional physicsin the MHD\nwave model. Effects that come to mind are the variation of\ntheplasmaparametersalongthemagneticfielddirectionasi n\nSoleret al. (2011b) and magnetic expansion and twist of theflux tube. These and other e ffectsmight be includedin forth-\ncominginvestigationsonpropagatingresonantkinkwaves.\nThis manuscript was finished during a visit of MG to the\nSolar Physics Group of UIB. MG is happy to acknowledge\nthe hospitality of the Solar Physics Group and the financial\nsupport from UIB through grant 40 /2010 under the program\n“Estades breus de professors convidats”. We thank I. Ar-\nregui for useful comments. RS acknowledges support from\na postdoctoral fellowship within the EU Research and Train-\ning Network “SOLAIRE” (MTRN-CT-2006-035484). MG\nacknowledgessupportfromK.U.LeuvenviaGOA /2009-009.\nJTacknowledgessupportfromtheSpanishMinisteriodeEd-\nucaci´ onyCienciathroughaRam´ onyCajalgrantandfunding\nprovidedunderprojectsAYA2006-07637andFEDERfunds.\nREFERENCES\nArregui, I.,Andries, J.,Van Doorsselaere, T.,Goossens,M .,& Poedts, S.\n2007, A&A,463, 333\nArregui, I.,Ballester, J.L.,& Goossens,M. 2008, ApJ,676, L77\nArregui, I.,& Ballester, J.L.2011, Space Sci. 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Keimer1\n1Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany.\n2Laboratoire Léon Brillouin, CEA-CNRS, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France.\n3Institut Laue-Langevin, 156X, 38042 Grenoble cedex 9, France.\n4Institut für Festkörperphysik, Technische Universität Wien, A-1040 Wien, Austria.\nE-mail: d.inosov@fkf.mpg.de\nAbstract. Inelastic neutron scattering (INS) is employed to study damped spin-wave excitations in the\nnoncentrosymmetric heavy-fermion superconductor CePt3Si along the antiferromagnetic Brillouin-zone\nboundary in the low-temperature magnetically ordered state. Measurements along the (1\n21\n2L)and\n(H H1\n2\u0000H)reciprocal-space directions reveal deviations in the spin-wave dispersion from the previously\nreported model. Broad asymmetric shape of the peaks in energy signifies strong spin-wave damping\nby interactions with the particle-hole continuum. Their energy width exhibits no evident anomalies\nas a function of momentum along the (1\n21\n2L)direction, which could be attributed to Fermi-surface\nnesting effects, implying the absence of pronounced commensurate nesting vectors at the magnetic zone\nboundary. In agreement with a previous study, we find no signatures of the superconducting transition\nin the magnetic excitation spectrum, such as a magnetic resonant mode or a superconducting spin\ngap, either at the magnetic ordering wavevector (001\n2)or at the zone boundary. However, the low\nsuperconducting transition temperature in this material still leaves the possibility of such features being\nweak and therefore hidden below the incoherent background at energies ®0.1 meV , precluding their\ndetection by INS.\n1. Introduction\nUnconventional superconductivity in the noncentrosymmetric heavy-fermion superconductor CePt3Si\n[1, 2]emerges below Tc=0.46K[3, 4]out of an antiferromagnetically (AFM) ordered metallic\nphase [5]. It has been suggested that the lack of inversion symmetry mixes the spin-singlet and\nspin-triplet superconducting pairing channels, leading to an exotic ground state with an s+p-wave\nsymmetry of the order parameter [6, 7], supported by recent experimental evidence [8–10]. An\nintense discussion about the details of such pairing is under way, in particular concerning the relative\nmagnitude of the singlet and triplet contributions [7], and the role of the static AFM order and spin\nfluctuations for superconductivity [11]. A detailed knowledge of the spin excitation spectrum is\nindispensable to answer these questions.\nRecently, in an extensive inelastic neutron scattering (INS) study [12], it was shown that\nKondo-type spin fluctuations are found in the vicinity of the AFM ordering wavevector (001\n2)in\nthe paramagnetic state [13]. Below the Néel transition temperature, TN=2.2K, these fluctuations\ngive way to damped spin waves, persisting in a wide range of momenta in the (H0L)plane of the\nreciprocal space. Their dispersion has been successfully parameterized by an effective Heisenberg-type\nmodel that involves five principal exchange integrals between localized Ce moments [12]. However,arXiv:1109.5784v1 [cond-mat.supr-con] 27 Sep 2011Dispersion and damping of zone-boundary magnons in the noncentrosymmetric superconductor CePt3Si\nthe effects of spin-wave damping via coupling to the continuum of particle-hole excitations across the\nFermi level have so far been neglected.\nFigure 1. Mounting of the single-\ncrystalline CePt3Si ingots for low-\ntemperature INS measurements\nin the (H H L )scattering plane.\nThe copper sample holder (bot-\ntom) is shielded with Cd foil.The Fermi surface of CePt3Si, evaluated both from band\nstructure calculations and quantum-oscillation measurements [14],\nsuggests a possible presence of several nesting vectors [15]that\ncould lead to enhanced spin-wave damping or softening of the spin-\nexcitation energies at certain nearly commensurate wavevectors at\nthe Brillouin zone (BZ) boundary. Such anomalies, if found, could\nserve as a direct probe of particle-hole scattering and, hence, would\nshed light on the Fermi surface geometry and electron correlations\nin this material. This argumentation motivated our present study.\n2. Description of the sample and experimental conditions\nIn this paper, we present the results of low-temperature INS\nmeasurements performed along the (1\n21\n2L)direction at the BZ\nboundary — a region not covered in previous experiments. We\nused several single crystals of CePt3Si with a total mass \u00187 g that\nwere coaligned using x-ray Laue diffraction and assembled on a\ncopper sample holder in the (H H L )scattering plane, as shown\nin Fig. 1. The overall mosaicity of the sample, measured at half-\nmaximum of the (110) and (002) Bragg peaks, was better than\n1\u000e. We used a3He dilution insert to cool the sample down to 80 mK. The triple-axis cold-neutron\nspectrometer IN14 was operated in the “W”-geometry in the constant- kfmode, with focusing applied\nboth to the monochromator and analyzer. Second-order neutrons were eliminated by a cold Be-filter\nplaced between the sample and the analyzer. To maximize the intensity, no collimation was used.\n/s45/s48/s46/s54 /s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s49/s49/s48/s49/s48/s48/s49/s48/s48/s48/s107/s102/s32/s61/s32/s49/s46/s53/s53/s32/s197/s45/s49\n/s107/s102/s32/s61/s32/s49/s46/s51/s32/s197/s45/s49\n/s107/s102/s32/s61/s32/s49/s46/s49/s53/s32/s197/s45/s49/s73/s110/s99/s111/s104/s101/s114/s101/s110/s116/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s110/s116/s115/s32/s47/s32/s51/s48/s32/s115/s101/s99/s41\n/s69/s110/s101/s114/s103/s121/s32/s116/s114/s97/s110/s115/s102/s101/s114/s32/s40/s109/s101/s86/s41/s32/s32/s32/s32/s107/s102/s32/s61/s32/s49/s46/s53/s53/s32/s197/s45/s49/s58\n/s84/s32/s61/s32/s48/s46/s56 /s32/s75\n/s84/s32/s61/s32/s50/s48 /s32/s75\nFigure 2. Incoherent scattering intensity from the\nsample for different kf, fitted with Voigt profiles.Energy profiles of the incoherent scattering\nintensity from the sample, measured at Q=\n(0.4 0.4 0)for three values of the final neutron\nmomentum, kf=1.55Å\u00001, 1.3 Å\u00001, and 1.15 Å\u00001\n(Fig. 2), can be well fitted with the Voigt function,\nyielding full widths of the energy resolution\nat half-maximum (FWHM) of 0.20 meV , 0.10\nmeV and 0.077 meV , respectively. The Lorentzian\ncontribution to the peak width, wL=0.015 meV ,\nwas found to be independent of kfwithin the\naccuracy of the fit. Its long “tails”, dominating\nthe incoherent background at low energies, led\nus to the choice of kf=1.55Å\u00001throughout\nthe experiment as a compromise between energy\nresolution and the signal-to-noise ratio.\n–2–Dispersion and damping of zone-boundary magnons in the noncentrosymmetric superconductor CePt3Si\n(a)\n/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 /s52/s46/s53/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48/s52/s48/s48\n/s76/s32/s61/s32/s48/s46/s55/s53\n/s76/s32/s61/s32/s48/s46/s56/s53/s81/s32/s61/s32/s40/s46/s51/s32/s46/s51/s32/s48/s41/s59/s32/s32 /s81/s32/s61/s32/s40/s46/s53/s32/s46/s53/s32 /s76/s41\n/s76/s32/s61/s32/s48\n/s76/s32/s61/s32/s177/s48/s46/s49\n/s76/s32/s61/s32/s48/s46/s49/s53\n/s76/s32/s61/s32/s48/s46/s50\n/s76/s32/s61/s32/s177/s48/s46/s50/s53\n/s76/s32/s61/s32/s48/s46/s51\n/s76/s32/s61/s32/s48/s46/s52\n/s76/s32/s61/s32/s177/s48/s46/s53/s83/s112/s105/s110/s45/s119/s97/s118/s101/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s32/s47/s32/s49/s48/s32/s109/s105/s110/s41\n/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41/s105/s110/s99/s111/s104/s101/s114/s101/s110/s116/s32/s98/s97/s99/s107/s103/s114/s111/s117/s110/s100/s32/s40/s101/s120/s116/s114/s97/s112/s111/s108/s97/s116/s105/s111/s110/s41 (b)\n/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 /s52/s46/s53/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48\n/s76/s32/s61/s32/s48/s46/s55/s53\n/s76/s32/s61/s32/s48/s46/s56/s53/s81/s32/s61/s32/s40/s46/s51/s32/s46/s51/s32/s48/s41/s59/s32/s32 /s81/s32/s61/s32/s40/s46/s53/s32/s46/s53/s32 /s76/s41\n/s76/s32/s61/s32/s48\n/s76/s32/s61/s32/s177/s48/s46/s49\n/s76/s32/s61/s32/s48/s46/s49/s53\n/s76/s32/s61/s32/s48/s46/s50\n/s76/s32/s61/s32/s177/s48/s46/s50/s53\n/s76/s32/s61/s32/s48/s46/s51\n/s76/s32/s61/s32/s48/s46/s52\n/s76/s32/s61/s32/s177/s48/s46/s53/s83/s112/s105/s110/s45/s119/s97/s118/s101/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s32/s47/s32/s49/s48/s32/s109/s105/s110/s41\n/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41 (c)\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s80/s101/s97/s107/s32/s119/s105/s100/s116/s104/s32/s40/s72/s87/s72/s77/s44/s32/s109/s101/s86/s41 /s32/s32/s32/s32/s32/s32/s32/s32ω/s109/s97/s120/s32/s40/s109/s101/s86/s41\n/s76/s32/s105/s110/s32/s40/s48/s46/s53/s32/s48/s46/s53/s32 /s76/s41/s32/s40/s114/s46/s108/s46/s117/s46/s41/s70/s229/s107/s32 /s101/s116/s32/s97/s108/s46\nFigure 3. Evolution of the magnon intensity along the (1\n21\n2L)BZ boundary, measured at T=80mK.\n(a) Raw energy scans measured in the magnetically ordered state at different wave vectors, as indicated\nin the legend. (b) The same after subtraction of the Voigt-shaped incoherent background. The solid\nlines result from a global fit to an empirical model. (c) Dispersion (top) and half-width (bottom) of\nthe peak that resulted from the fit shown in panel (b). The solid line is a guide to the eye. The model\nof B. Fåk et al. [12]is shown by the dashed line for comparison.\n3. Magnons in the magnetically ordered state\nIn Fig. 3, we show a series of energy scans measured at different zone-boundary wave vectors (1\n21\n2L),\nspanning the irreducible part of the BZ between L=0and L=1\n2. Panel (a) shows the raw data,\nwhereas panel (b) results from a subtraction of the incoherent scattering background, fitted to the Voigt\nprofile (Fig. 2), whose extrapolation is given by the solid line in Fig. 3 (a). The strongly asymmetric\nlineshape of the resulting magnetic signal could not be well fitted by the damped harmonic oscillator\nmodel (possibly due to a resolution effect). Therefore, an empirical fit shown by solid lines in Fig. 3 (b),\nwhich includes a constant background offset shared by all curves, was used. The resulting magnon\ndispersion (defined here by the positions of the peak maxima) and the half width at half maximum\n(HWHM) of the peak, related to the magnon lifetime, are shown in Fig. 3 (c). Here, the solid line\nis given by the fit of the experimental dispersion to a sum of sinusoidal functions, whereas the\ndashed black line shows the dispersion given by the spin-wave model of B. Fåk et al. [12]along the\nsame reciprocal-space direction. One can see that the latter is characterized by a somewhat smaller\namplitude as compared to the directly measured one.\nA similar analysis of the magnon dispersion is presented in Fig. 4 for the (H H1\n2\u0000H)direction,\nwhich connects the AFM ordering wavevector (001\n2)with the zone boundary at (1\n21\n20). These data\nwere measured at T=0.8K>Tc, but since the spin-wave spectrum is insensitive to the SC transition,\nthese experimental conditions are practically equivalent to those in Fig. 3. A fit of the experimental\ndispersion, shown in panel (b) by a solid line, exhibits a pronounced nonmonotonic behavior that\ndeviates from the spin-wave model of Ref. 12 (dashed line) by up to 20%.\nThe strong variation in measured inelastic intensity between L=0and L=1=2in both figures\nresults from the momentum-dependent AFM structure factor, which is maximized at the AFM ordering\nwavevector (0 01\n2)and becomes strongly reduced, but not vanishing, at (1\n21\n20). This periodic intensity\n–3–Dispersion and damping of zone-boundary magnons in the noncentrosymmetric superconductor CePt3Si\n/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 /s52/s46/s53/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s81/s32/s61/s32/s40/s72/s32/s72/s32/s49/s47/s50/s8211 /s72/s41\n/s72/s32/s61/s32/s48/s46/s48/s53\n/s72/s32/s61/s32/s48/s46/s49\n/s72/s32/s61/s32/s48/s46/s49/s53\n/s72/s32/s61/s32/s48/s46/s50\n/s72/s32/s61/s32/s48/s46/s50/s53\n/s72/s32/s61/s32/s48/s46/s51\n/s72/s32/s61/s32/s48/s46/s51/s53\n/s72/s32/s61/s32/s48/s46/s52\n/s72/s32/s61/s32/s48/s46/s52/s53\n/s72/s32/s61/s32/s48/s46/s53\n/s32/s83/s112/s105/s110/s45/s119/s97/s118/s101/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s32/s47/s32/s49/s48/s32/s109/s105/s110/s41\n/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41/s40/s98/s41 /s40/s97/s41\nω/s109/s97/s120/s32/s40/s109/s101/s86/s41\n/s72/s32/s105/s110/s32/s40 /s72/s32/s72/s32 /s49/s47/s50/s8211 /s72/s41/s32/s40/s114/s46/s108/s46/s117/s46/s41/s70/s229/s107/s32 /s101/s116/s32/s97/s108/s46\nFigure 4. Magnon dispersion along\nthe(H H1\n2\u0000H)direction, measured\natT=0.8K. (a) Energy scans after\nsubtraction of the Voigt-shaped inco-\nherent background. (b) Dispersion of\nthe peak maximum. The spin-wave\nmodel of B. Fåk et al.[12]is shown by\nthe dashed line for comparison. The\nsolid lines are guides to the eyes.\nmodulation is shown in Fig. 5 after subtraction of the constant background, and can be approximated\nwith a sum of the first- and second-harmonic sinusoidal functions of L(solid line).\nThe same structure-factor modulation can also be seen in the color maps of the INS intensity that\nare shown in Fig. 6 (a) and (b) for the (H H1\n2\u0000H)and(1\n21\n2L)reciprocal-space directions, respectively.\nThe fits of the experimental dispersion (same as in Fig. 3 and 4) are summarized here by solid lines\nand compared to the model of B. Fåk et al. [12]shown in dashed lines. In spite of the apparent\ndifferences between these two fits in both directions, the disparity always remains smaller than the\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s32/s80/s101/s97/s107/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s32/s40/s99/s110/s116/s115/s32/s47/s32/s49/s48/s32/s109/s105/s110/s41/s76/s32/s105/s110/s32/s40/s48/s46/s53/s32/s48/s46/s53/s32 /s76/s41/s32/s40/s114/s46/s108/s46/s117/s46/s41ω /s61/s32/s49/s46/s52/s51/s32/s109/s101/s86\nFigure 5. L-dependence of the background-\nsubtracted intensity at the peak maximum\nalong (1\n21\n2L), modulated due to the AFM\nstructure factor. Empty symbols represent\nthe directly measured intensity at !=\n1.43meV , whereas solid symbols result from\nthe amplitudes of the fits presented in\nFig. 3 (b). The least-squares fit (solid line)\nis given by the following periodic function:\nI(L) =I0[1\u00000.827cos2\u0019L+0.161cos4\u0019L].intrinsic energy width of the overdamped signal.\nThese color maps also demonstrate the absence of\nany pronounced additional branches of paramagnon\nexcitations in the vicinity of (1\n21\n20)that could\noriginate from the nested sections of the normal-state\nFermi surface.\n4. Superconducting state\nIn the unconventional heavy-fermion superconductor\nCeCoIn5(Tc=2.3K), a pronounced magnetic\nresonant mode, centered at \u00180.6 meV , emerges\nbelow Tcout of a weak and featureless normal-state\nspectrum [16]. This resonance is similar to the one\nfound in high- Tccuprates [17]and iron pnictides [18],\nwhere it serves as the hallmark of a sign-changing\nsuperconducting order parameter. However, no such\nresonant enhancement has been reported neither in\nthe spin-triplet p-wave superconductor Sr2RuO4[19]\nnor in the earlier experiments on CePt3Si[12]close to\nthe magnetic ordering wave vector. We have therefore\nperformed a comparison of the inelastic signal in\nCePt3Si above and below Tcto establish the absence of\n–4–Dispersion and damping of zone-boundary magnons in the noncentrosymmetric superconductor CePt3Si\n(a)\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/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\n/s84/s32/s61/s32/s48/s46/s56/s75\n/s72/s32/s105/s110/s32/s40/s72/s32/s72/s32/s48/s46/s53/s45 /s72/s41/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41\n/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48/s57/s48/s49/s48/s48\n/s99/s111/s117/s110/s116/s115/s32/s47/s32/s49/s53/s48/s32/s115/s101/s99/s67/s101/s80/s116/s51/s83/s105 (b)\n/s45/s49/s46/s48 /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 /s49/s46/s48/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\n/s84/s32/s61/s32/s56/s48/s32/s109/s75\n/s76/s32/s105/s110/s32/s40/s48/s46/s53/s32/s48/s46/s53/s32 /s76/s41/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41/s50/s48/s51/s46/s50/s69/s43/s48/s50\n/s99/s111/s117/s110/s116/s115/s32/s47/s32/s49/s48/s32/s109/s105/s110/s67/s101/s80/s116/s51/s83/s105/s32/s51/s50/s48/s32 /s32/s32/s32/s32/s32/s32/s32/s32\nFigure 6. Color maps of the magnon intensity along the (H H1\n2\u0000H)and(1\n21\n2L)reciprocal-space\ndirections. In panel (b), symmetrization with respect to the L=0line has been applied. The solid and\ndashed lines represent fits of the experimental dispersion, defined as the energy of the peak maximum\nat every momentum, and the spin-wave model of B. Fåk et al. [12], respectively.\nresonant effects both near the zone boundary at (1\n21\n20)and at the AFM propagation vector, (0 01\n2). This\ncomparison is presented in Fig. 7. Indeed, within the statistical error, the inelastic intensity remains\nunchanged between 800 mK ( >Tc) and 70 mK (1 mm a good agreement of the derived damping\nis found.\n\u000b. This result was validated by other groups [8], [9]. Whereby\nin reference [9] the line broadening was studied as a function\nof sample thickness with respect to the CPW width and in\nreference [8] as function of the wire aspect ratio p(p=stripe\nthickness / stripe width). Following these arguments we would\nexpect a more homogeneous pulse excitation and hence a\nlower\u000bfor the narrower samples in contradiction to our\nexperimental observations. Furthermore our measured data can\nnot be consistently explained by standing spin wave modes [5].\nOne possible origin of the decrease of \u000bcould be inho-\nmogeneous precession near the edges of the Py stripes. This\nis in good concordance with the theoretical description of\nGuslienko et al. [8] where they consider an effective pinning\nof magnetization at the lateral edges of the stripe. This pinning\nis related to the inhomogeneity of the internal dynamic field\nalong the stripe width and is of dipolar nature.\nWith increasing width of the patterned stripes with respect\nto the center conductor width the edges of the stripes are\nmoved into the gap of the CPW. Thus, their contribution to\nthe measured inductive signal is drastically reduced. This is\nconfirmed by measurements of a further set of test samples\nwith the magnetic stripes situated within the gap. They show\nthat the inductive signal from magnetic material situated in the\ngap can be practically neglected with respect to the magnetic\nmaterial underneath the centre conductor.\nThese results have implications for the reliable determina-\ntion of the intrinsic damping \u000bbased on patterned sample ge-\nometries. Magnetization dynamics in microstructured devices\nare modified with respect to extended unpatterned magnetic\nthin films. Note that in inductive magnetization dynamic\nmeasurements, as PIMM and VNA-FMR, the CPW behaves\nnot only as emitter of the magnetic field pulse perturbation\nbut also as the probe or antenna. It implies that, by these\ntechniques magnetization dynamics are detected locally; i.e.\nonly close to the CPW. Therefore, for microstripe devices withwPy\u0014w, one gets access to the magnetization dynamics of\nthe whole structure including the edges. In this case the total\neffective damping parameter of the whole patterned structure\nwill be measured which includes extrinsic contributions from\nthe edges. However, for wPy\u0015w, the contribution from the\nedges is not directly detected by the inductive experiment.\nTherefore, the influence of the edges on the measured mag-\nnetization dynamics is significantly reduced and the measured\ndamping will approach the effective damping measured on\nextended thin films. Note again that the contribution of two-\nmagnon scattering processes cannot be quantified from our\nexperiments. The role of the sample geomentry on measure-\nments of the intrinsic damping in out-of-plane fields [6], [7]\nwill be the subject of future investigations.\nV. C ONCLUSION\nWe have shown that for inductive magnetization dynam-\nics measurements on microstructured stripes with lateral di-\nmensions comparable to the CPW dimension, the measured\neffective damping parameter \u000bdepends on the lateral size\nof the sample. A decrease of \u000bup to 20 % is observed\nwith increasing width. This effect is ascribed to an effective\n“pinning” of magnetization at the lateral edges of the stripe,\ninducing a dephasing of the magnetization precession and\nhence an enhanced \u000b. Therefore for accurate and reliable\nmagnetization dynamic measurements by inductive techniques\nin microstructured ferromagnetic devices three important pa-\nrameters must be taken into consideration: 1) CPW lateral\ndimension, 2) sample dimension, and 3) relative position and\nsize of the ferromagnetic device with respect to the CPW.\nAll these parameters can modify the resonance frequency and\nespecially the derived effective damping parameter.\nACKNOWLEDGMENT\nThe research within this Euramet joint research project\nreceives funding from the European Community’s Seventh\nFramework Programme, ERA-NET Plus, under iMERA-Plus\nProject-Grant Agreement No. 217257.\nREFERENCES\n[1] T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers, “Inductive\nmeasurement of ultrafast magnetization dynamics in thin-film permal-\nloy,” Journal of Applied Physics , vol. 85, no. 11, pp. 7849–7862, 1999.\n[2] C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P. 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Otani,\n“Spin wave contributions to the high-frequency magnetic response of\nthin films obtained with inductive methods,” Journal of Applied Physics ,\nvol. 95, no. 10, pp. 5646–5652, 2004.[6] S. S. Kalarickal, P. Krivosik, J. Das, K. S. Kim, and C. E. Patton, “Mi-\ncrowave damping in polycrystalline fe-ti-n films: Physical mechanisms\nand correlations with composition and structure,” Phys. Rev. B , vol. 77,\nno. 5, p. 054427, Feb 2008.\n[7] P. Landeros, R. E. Arias, and D. L. Mills, “Two magnon scattering\nin ultrathin ferromagnets: The case where the magnetization is out of\nplane,” Phys. Rev. B , vol. 77, no. 21, p. 214405, Jun 2008.\n[8] K. Y . Guslienko, S. O. Demokritov, B. Hillebrands, and A. N. Slavin,\n“Effective dipolar boundary conditions for dynamic magnetization in\nthin magnetic stripes,” Phys. Rev. B , vol. 66, no. 13, p. 132402, Oct\n2002.\n[9] M. L. Schneider, A. B. Kos, and T. J. Silva, “Finite coplanar waveguide\nwidth effects in pulsed inductive microwave magnetometry,” Applied\nPhysics Letters , vol. 85, no. 2, pp. 254–256, 2004.\n[10] A. Aharoni, “Demagnetizing factors for rectangular ferromagnetic\nprisms,” Journal of Applied Physics , vol. 83, no. 6, pp. 3432–3434,\n1998.\n[11] K. J. Kennewell, M. Kostylev, and R. L. Stamps, “Calculation of spin\nwave mode response induced by a coplanar microwave line,” Journal of\nApplied Physics , vol. 101, no. 9, p. 09D107, 2007." }, { "title": "0908.3821v1.Influence_of_an_external_magnetic_field_on_forced_turbulence_in_a_swirling_flow_of_liquid_metal.pdf", "content": "arXiv:0908.3821v1 [physics.flu-dyn] 26 Aug 2009Influence of an external magnetic field on forced turbulence i n a\nswirling flow of liquid metal\nBasile Gallet, Michael Berhanu, and Nicolas Mordant\nLaboratoire de Physique Statistique,\nEcole Normale Sup´ erieure & CNRS,\n24 Rue Lhomond, 75231 PARIS Cedex 05, France\n(Dated: October 29, 2018)\nAbstract\nWe report an experimental investigation on the influence of a n external magnetic field on forced\n3D turbulence of liquid gallium in a closed vessel. We observ e an exponential damping of the\nturbulent velocity fluctuations as a function of the interac tion parameter N(ratio of Lorentz force\nover inertial terms of the Navier-Stokes equation). The flow structures develop some anisotropy\nbut do not become bidimensional. From a dynamical viewpoint , the damping first occurs homo-\ngeneously over the whole spectrum of frequencies. For large r values of N, a very strong additional\ndamping occurs at the highest frequencies. However, the inj ected mechanical power remains inde-\npendent of the applied magnetic field. The simultaneous meas urement of induced magnetic field\nand electrical potential differences shows a very weak correl ation between magnetic field and ve-\nlocity fluctuations. The observed reduction of the fluctuati ons is in agreement with a previously\nproposed mechanism for the saturation of turbulent dynamos and with the order of magnitude of\nthe Von K´ arm´ an Sodium dynamo magnetic field.\n1Situations were a magnetic field interacts with a turbulent flow are fo und in various\ndomains of physics, including molten metals processing, laboratory fl ows, and astrophysics.\nThe motion of electrically conducting fluid in a magnetic field induces elec trical currents,\nwhich in turn react on the flow through the Lorentz force. The pow er injected in the flow\nis thus shared between two dissipative mechanisms: viscous friction and ohmic dissipation\nof the induced currents. On the one hand, the situation where the flow is laminar is very\nwell understood, and the geometry of the velocity field and of the in duced currents can\nbe computed analytically. On the other hand, several questions re main open in the fully\nturbulent situation: when a statistically steady state is reached, is the mean injected power\nhigher or lower than in the nonmagnetic case? What controls the rat io of ohmic to viscous\ndissipation? How is the turbulent cascade affected by the magnetic fi eld? To adress some\nof these questions, we have designed an experimental device which allows to apply a strong\nmagnetic field on a fully turbulent flow.\nWhen an electrically conducting fluid is set into motion, a magnetic Reyn olds number\nRmcan be defined as the ratio of the ohmic diffusive time to the eddy-tur nover time. This\nnumber reaches huge values in galactic flows, but can hardly exceed one in a laboratory\nexperiment. The common liquids of high electrical conductivity (gallium , mercury, sodium)\nhave a very low kinematic viscosity: their magnetic Prandtl number Pm(ratio of the kine-\nmatic viscosity over the magnetic diffusivity) is less than 10−5. A flow with Rmof order\none is turbulent and thus requires a high power input to be driven. Fo r this reason most\nexperimental studies have been restricted to low Rm. They were conducted mostly in chan-\nnel flows and grid generated turbulence [1, 2]. The most general ob servation is that the\napplication of a strong magnetic field leads to a steeper decay of the power spectra of the\nturbulent velocity fluctuations at large wavenumbers: the decay g oes from a classical k−5/3\nscaling without magnetic field to k−3ork−4for the highest applied fields. In the meantime,\nsome anisotropy is developed leading to larger characteristic scales along the applied mag-\nnetic field. The phenomenology of these transformations is quite we ll understood in terms\nof the anisotropy of the ohmic dissipation [3]. One difficulty arises from the fact that the\nboundary conditions can have a strong effect on the turbulence lev el. In channel flows the\nchoice between conducting and insulating walls strongly impacts the fl ow. A strong external\nmagnetic field perpendicular to the boundaries leads to an increase o f the turbulence level\n2due to modifications of the boundary layers [1]. On the contrary, Ale many et al. observed in\ngrid generated decaying turbulence an enhanced decay of the tur bulent fluctuations. As far\nas forced turbulence is concerned, Sisan et al. studied the influenc e of a magnetic field on a\nflow of liquid sodium inside a sphere [4]. However, the flow again goes thr ough a variety of\ninstabilities which prevents a study solely focussed on the impact of t he magnetic field on\nthe turbulent fluctuations: the geometry of the mean flow and of t he induced magnetic field\nkeeps changing as the magnetic field is increased.\nSeveral numerical simulations of this issue have been conducted [5 , 6, 7, 8]. Despite\nthe rather low spatial resolution available for this problem, these wo rks show the same phe-\nnomenology: development of anisotropy, steepening ofthe spect ra andtrend to bidimension-\nalization of the flow as the magnetic field increases. The direct numer ical simulations (DNS)\nalso allow to compute the angular flux of energy from the energy-co ntaining Fourier modes\n(more or less orthogonal to the applied field) to the modes that are preferentially damped\nby ohmic dissipation. Once again, the magnetic field can affect the larg e scale structure of\nthe velocity field: the interaction between forcing and magnetic field in the DNS of Zikanov\n& Thess [5, 6] leads to an intermittent behavior between phases of r oughly isotropic flow\nand phases of bidimensional columnar vortices that eventually get u nstable. Vorobev et\nal.[7] and Burratini et al.[8] performed DNS in a forced regime which are closely related to\nour experiment. However, the maximum kinetic Reynolds number rea ched in their studies\nremains orders of magnitude below what canbe achieved inthe labora tory. Anexperimental\ninvestigation at very high kinetic Reynolds number thus remains nece ssary to characterize\nthe effect of a strong magnetic field on a fully developped turbulent c ascade.\nThe flow under study in our experiment resembles the Von K´ arm´ an geometry: counter-\nrotating propellers force a flow inside a cylindrical tank. The large sc ale flow is known to\nhave a strong shear layer in the equatorial plane where one observ es the maximum level of\nturbulence [9]. We do not observe any bifurcation of the flow as the m agnetic field increases\nnor any drastic change of the large scale recirculation imposed by th e constant forcing.\nThese ingredients strongly differ from the previously reported exp eriments and allow to\nstudy the influence of an applied magnetic field on the turbulent casc ade in a given flow\ngeometry. This issue arises in the framework of turbulent laborato ry dynamo studies such\nas the Von K´ arm´ an Sodium (VKS) experiment [10] which flow’s geome try is similar to ours.\nIn the turbulent dynamo problem, one issue is to understand the pr ecise mechanism for the\n3FIG. 1: Sketch of the experimental setup. The curved arrows r epresent the average large-scale\nmotion of the fluid.\nsaturation of the magnetic field. Once the dynamo magnetic field get s strong enough, it\nreacts on the flow through the Lorentz force. This usually reduce s the ability of the flow to\nsustain dynamo action. An equilibrium can be reached, so that the ma gnetic field saturates.\nThis backreaction changes the properties of the bulk turbulence. In a situation where an α\neffect takes part in the generation of the dynamo, i.e. if the turbule nt fluctuations have a\nmean field effect, then the changes in the statistics of turbulence s hould be involved in the\nsaturation mechanism of the dynamo.\nI. DESCRIPTION OF THE EXPERIMENT\nA. The flow\nOur turbulent flow resembles the Von K´ arm´ an geometry, which is w idely used in exper-\niments on turbulence and magnetohydrodynamics ([9, 10] for exam ple). 8 liters of liquid\ngallium are contained in a closed vertical cylinder of diameter 20 cm and height 24 cm. The\nthickness of the cylindrical wall is 7.5 mm, and that of the top and bot tom walls is 12 mm\n(Fig. 1). This tank is made of stainless steel. Gallium melts at about 30◦C. Its density is\nρ= 6090 kg/m3and its kinematic viscosity is ν= 3.1110−7m2/s. Its electrical conductivity\nat our operating temperature (45◦C) isσ= 3.9106Ω−1m−1. The main difference with the\nusual Von K´ arm´ an setup is the use of two propellers to drive the fl ow, rather than impellers\nor disks. The two propellers are coaxial with the cylinder. They are m ade of 4 blades\n4inclined 45 degrees to the axis. The two propellers are counter-rot ating and the blades are\nsuch that both propellers are pumping the fluid from the center of t he tank towards its end\nfaces. The large scale flow is similar to the traditional counter-rota ting geometry of the Von\nK´ arm´ an setup. The propellers are 7 cm in diameter and their rotat ion ratefrotranges from\n3 Hz to 30 Hz. They are entrained at a constant frequency by DC mo tors, which drives a\nlarge scale flow consisting in two parts : first the fluid is pumped from t he center of the\nvessel on the axis of the cylinder by the propellers. It loops back on the periphery of the\nvessel and comes back to the center in the vicinity of the equatoria l plane. In addition to\nthis poloidal recirculation, the fluid is entrained in rotation by the pro pellers. Differential\nrotation of the fluid is generated by the counter rotation of the pr opellers, which induces a\nstrong shear layer in the equatorial plane where the variousprobe s arepositioned. The shear\nlayer generates a very high level of turbulence [9]: the velocity fluc tuations are typically of\nthe same order of magnitude as the large scale circulation.\nA stationary magnetic field B0is imposed by a solenoid. The solenoid is coaxial with\nthe propellers and the cylinder. B0is mainly along the axis of the cylinder with small\nperpendicular components due to the finite size of the solenoid (its le ngth is 32 cm and its\ninner diameter is 27 cm). The maximum magnetic field imposed at the cen ter of the tank\nis 1600 G.\nB. Governing equations and dimensionless parameters\nThe fluid is incompressible and its dynamics is governed by the Navier-S tokes equations:\nρ/parenleftbigg∂v\n∂t+(v·∇)v/parenrightbigg\n=−∇p+ρν∆v+j×B (1)\n∇·v= 0 (2)\nwherevis the flow velocity, pis the pressure, Bis the magnetic field and jis the electrical\ncurrent density. The last term in the r.h.s. of (1) is the Lorentz for ce.\nA velocity scale can be defined using the velocity of the tip of the blade s. The radius of\nthe propeller being R= 3.5 cm, the velocity scale is 2 πRfrot. A typical length scale is the\nradius of the cylinder L= 10 cm. The kinetic Reynolds number is then\nRe=2πRLfrot\nν. (3)\n5At the maximum speed reported here it reaches 2106. The flow is then highly turbulent and\nremains so even for a rotation rate ten times smaller.\nIn the approximations of magnetohydrodynamics, the temporal e volution of the magnetic\nfield follows the induction equation:\n∂B\n∂t=∇×(v×B)+η∆B. (4)\nHereη= (µ0σ)−1is the magnetic diffusivity, µ0being the magnetic permeability of vacuum\n(η= 0.20 m2s−1for gallium so that Pm=ν/η= 1.610−6) [11]. The first term in the\nr.h.s describes the advection and the induction processes. The sec ond term is diffusive and\naccounts for ohmic dissipation. A magnetic Reynolds number can be d efined as the ratio of\nthe former over the latter:\nRm=µ0σ2πRLfrot, (5)\nIt is a measure of the strength of the induction processes compar ed to the ohmic dissipation\n[11]. For Rm→0 the magnetic field obeys a diffusion equation. For Rm≫1, it is\ntransported and stretched by the flow. This can lead to dynamo ac tion, i.e. spontaneous\ngeneration of a magnetic field sustained by a transfer of kinetic ene rgy from the flow to\nmagnetic energy. In our study, Rmis about 3 at the maximum speed reported here ( frot=\n30 Hz): Induction processes are present but do not dominate ove r diffusion. As frotgoes\nfrom 3 to 30 Hz, the magnetic Reynolds number is of order 1 in all case s.\nIn our range of magnetic Reynolds number, the induced magnetic fie ldb=B−B0is\nweak relative to B0(of the order of 1%). The Lorentz force is j×B0wherejis the current\ninduced by the motion of the liquid metal (no exterior current is applie d). From Ohm’s\nlaw, this induced current is j∼σv×B0and is thus of order 2 πσRfrotB0. One can define\nan interaction parameter Nthat estimates the strength of the Lorentz force relative to the\nadvection term in the Navier-Stokes equation:\nN=σLB2\n0\n2πρRfrot, (6)\nBecause of the 1 /frotfactor, this quantity could reach very high values for low speeds,\nbut the flow would not be turbulent anymore. In order to remain in a t urbulent regime,\nthe smallest rotation rate reported here is 3 Hz. The maximum intera ction parameter\n(built with the smallest rotation frequency and the strongest applie d magnetic field) is then\napproximately 2 .5. Many studies on magnetohydrodynamics use the Hartmann numb er to\n6quantify the amplitude and influence of the magnetic field. This numbe r measures the ratio\nof the Lorentz force over the viscous one. It is linked to the intera ction parameter by the\nrelationHa=√\nNRewhich gives the typical value Ha≃700 for the present experiment.\nThe influence of the electrical boundary conditions at the end face s is determined by this\nnumber and the conductivity ratio K=σwlw\nσL, whereσwis the electrical conductivity of\nthe walls and lwtheir thickness. A very detailed numerical study on this issue has be en\nperformed in the laminar situation on a flow which geometry is very simila r to the present\none in [12]. As far as turbulent flows are concerned, Eckert et al. st ress the importance of\nthe product KHa, which measures the fraction of the electrical current that leaks from the\nHartmann layers into the electrically conducting walls. With 12 mm thick end faces Kis\nabout 0.05 so that KHa≃35: the walls have to be considered as electrically conducting.\nThese boundary effects are of crucial importance in experiments w here the boundary layers\ncontrol the rate of turbulence of the flow. However, in the prese nt experimental setup the\nfluid is forced inertially and we have checked that the magnetic field ha s only little influence\non the mean flow (see section IIB below). Furthermore, the high va lue of the Hartmann\nnumber implies that the dominant balance in the bulk of the flow is not be tween the Lorentz\nand viscous forces but between the Lorentz force and the inertia l term of the Navier-stokes\nequation: in the following, it is the interaction parameter and not the Hartmann number\nthat leads to a good collapse of the data onto a single curve. For the se reasons the present\nexperimental device allows to study the influence of a strong magne tic field on the forced\nturbulence generated in the central shear layer, avoiding any bou ndary effect.\nC. Probes and measurements\nThe propellers are driven at constant frequency. The current pr ovided to the DC motors\nis directly proportional to the torques they are applying. It is reco rded to access the\nmechanical power injected in the fluid.\nAs liquid metals are opaque, the usual velocimetry techniques such a s Laser Doppler\nVelocimetry or Particle Image velocimetry cannot be used. Hot wire a nemometry is difficult\nto implement in liquid metals even if it has been used in the past [2]. Other v elocimetry\ntechniques have been developed specifically for liquid metals. Among t hose are the potential\n7FIG. 2: Schematics of the potential probes. These probes are vertical and the electrodes are\npositioned in the mid-plane of the tank, where the shear laye r induces strong turbulence.\nprobes. They relyonthemeasurement ofelectric potentialdiffere nces induced bythemotion\nof the conducting fluid in a magnetic field [1, 13, 14]. The latter can be a pplied locally with\na small magnet or at larger scale as in our case. We built such probes w ith 4 electrodes\n(Fig. 2). The electrodes are made of copper wire, 1 mm in diameter, a nd insulated by a\nvarnish layer except at their very tip. The electrodes are distant o fl∼3 mm. The signal\nis first amplified by a factor 1000 with a transformer model 1900 fro m Princeton Applied\nResearch. It is further amplified by a Stanford Research low noise p reamplifier model SR560\nand then recorded by a National Instrument DAQ. Note that beca use of the transformer the\naverage potential cannot be accessed, so that our study focus es on the turbulent fluctuations\nof the velocity field.\nFor a steady flow, and assuming j=0, Ohm’s law gives ∇φ=v×Bso that the electric\npotential difference between the electrodes is directly related to t he local velocity of the\nfluid. One gets δφ=/integraltext\nv×B·dlintegrating between the two electrodes. If Bis uniform\nthenδφ=v⊥Blwherev⊥is the component of the velocity orthogonal to both the magnetic\nfield and the electrodes separation. In the general time-depende nt case, the link is not so\ndirect. Using Coulomb’s gauge and taking the divergence of Ohm’s law, one gets :\n∆φ=ω·B0 (7)\nwhereω=∇×vis the local vorticity of the flow [14]. The measured voltage depends o n\n8(a)00.5 11.5 22.5−30−20−10010203040\ntime [s]δφ [µV]\n(b)10010110210−810−610−410−2\nf [Hz]PSD\nFIG. 3: (a) Time series from a potential probe for frot= 20 Hz and an applied magnetic field\nof 178 G. (b) Corresponding power spectrum density. The dash ed line is a f−5/3scaling and the\nmixed line a f−11/3scaling. These straight lines are drawn as eye guides.\nthe vorticity component parallel to the applied magnetic field, i.e. to g radients of the two\ncomponents ofvelocity perpendicular to B0. The relationbetween the flowand thepotential\nis not straightforward but the potential difference can be seen as a linearly filtered measure-\nment of the velocity fluctuations. For length scales larger than the electrode separation, the\npotential difference can be approximated by the potential gradien t, which has the dimension\nofvB0. The spectral scaling of ∇φ/B0is expected to be the same as that of the velocity,\ni.e. thek−5/3Kolmogorov scaling. For smaller scales, some filtering results from th e finite\nsize of the probe. For scales smaller than the separation l, the values of the potential on the\ntwo electrodes are likely to be uncorrelated: if these scales are also in the inertial range, one\nexpects the spectrum of the potential difference to scale as the p otential itself, i.e. k−11/3\ndue to the extra spatial derivative. Assuming sweeping of the turb ulent fluctuations by the\n9average flow or the energy containing eddies [15] and a k−5/3Kolmogorov scaling for the\nvelocity spectrum, we expect the temporal spectrum of the meas ured potential difference to\ndecay as f−5/3for intermediate frequencies and as f−11/3for high frequencies in the inertial\nrange.\nAn example of a measured time series of the potential difference is sh own in figure 3\ntogether with its power spectrum density. For this dataset, one e xpects a signal of the order\nof 2πRfrotB0l∼20µV which is the right order of magnitude. Because of the quite large\nseparation of the electrodes, the cutoff frequency between the f−5/3and thef−11/3behaviors\nis low so that no real scaling is observed but rather trends. At the h ighest frequencies, the\ndecay is faster than f−11/3.\nBolonov et al. [16] had a rather empirical approach to take into acco unt the filtering\nfrom the probe. Assuming that the velocity spectrum should decay asf−5/3they observed a\nspectral response of the potential probe which displays an expon ential decay exp( −lf/0.6u).\nThe factor 0 .6 is most likely dependent on the geometry. We reproduced their ana lysis in\nfigure 4. In the inset is displayed the spectrum of figure 3 multiplied by the expected\nf−5/3scaling. The decay is seen to be exponential from about 25 Hz to 300 Hz (at higher\nfrequencies the signal does not overcome the noise). The charac teristic frequency of the\ndecay can be extracted and plotted as a function of the rotation f requency of the propeller\n(the local velocity is expected to scale as Rfrot). A clear linear dependence is observed, in\nagreement with the results of Bolonov et al.. The cutoff frequency is about twice frot. It\nis not very high due to the rather large size of the probe. Neverthe less the most energetic\nlength scales are resolved in our measurement. In the following we sh ow only direct spectra,\nand no correction of the filtering is attempted.\nWe can conclude that although some filtering is involved, the measure ment of potential\ndifferences gives an image of the spectral properties of the velocit y fluctuations. Any change\nin the spectral properties of the flow in the vicinity of the probe will t hus be visible on the\nspectrum of the potential difference.\nThe potential probes are quite large in order to fit a gaussmeter Ha ll probe in the vicinity\nof the electrodes (see Fig. 2). These probes are connected to an F.W.Bell Gaussmeter model\n7030 that allows to measure the induced magnetic field down to a few t enths of Gauss. The\nproximity between thevelocity andmagnetic fieldmeasurements allow s tostudy thepossible\ncorrelations between these two fields.\n10(a)10010110210−810−610−410−2\nf [Hz]PSDf−5/3\n010020030040050010−410−310−210−1PSD × f5/3\n(b)051015202530010203040506070\nfrot [Hz]cut−off frequency [Hz]\nFIG. 4: (a) Filtering effect due to the probe geometry. – Inset: semilog plot of the power spectrum\ndensity (PSD) of the potential difference compensated by f5/3. The dashed line is an exponential\nfit. – Main figure: lower curve, PSD; upper solid line: PSD corr ected from the exponential decay\nfitted in the inset. Dashed line: f−5/3decay. Same data set as that of the previous figure. (b)\nCutoff frequency of theproberesponsefor the probeused in (a ) and for various rotation frequencies\nof the propeller. The dashed line is a linear fit. The applied m agnetic field is 178 G.\nII. EFFECT OF THE APPLIED MAGNETIC FIELD ON THE TURBULENCE\nLEVEL\nA. Velocity field\nWe first focus on the fluctuation level of the velocity field accessed through measurements\nofpotentialdifferences. Theevolutionofthe rmsvalueofthepotentialdifference isdisplayed\nin figure 5 asa function of theapplied magnetic field and of the rotatio nrate. Fromequation\n110 500 1000 150002468x 10−5\nB0 [G]δφrms [V]\nFIG. 5: Evolution of the rms value of the potential difference a s a function of the applied vertical\nmagnetic field, for different values of the rotation frequency .•:frot= 5 Hz,△: 10 Hz, /square: 15 Hz,\n⋄: 20 Hz and ⋆: 30 Hz. The solid lines correspond to the azimuthal potentia l difference and the\ndashed line to the radial potential difference for the same fou r electrode potential probe.\n(2), the potential difference should behave as δφrms∝B0v/l. For low values of the applied\nmagnetic field, the interaction parameter Nis low: the magnetic field has almost no effect\non the flow, and the velocity scales as Rfrot. Thus, for high frotand lowB0(i.e. low N),\nδφrmssould be linear in both frotandB0. The upper curve corresponds to the highest\nvelocityfrot= 30 Hz. For low B0there is a linear increase of δφrms. Then it seems to\nsaturate. For smaller rotation rates, the linear part gets smaller a nd the saturation region\ngets wider. Eventually, for frot= 5 Hz, the potential decays for the highest values of the\napplied magnetic field. This demonstrates that there is a strong inte raction between the\nmagnetic field and the flow.\nTo investigate in more details the scaling properties of the potential difference, δφrms/frot\nis displayed versus B0in figure 6. One can clearly see that as frotdecreases, the potential\ndeviates from the linear trend for lower and lower values of the magn etic field.\nFigure 7 shows δφrms/B0as a function of frot. Here a linear trend is observed for large\nfrot. As the external magnetic field is increased, the fluctuations of th e potential are damped\nand the linear trend is recovered for increasingly high values of the r otation rate.\nAll that information can be synthesized by plotting the dimensionless potential\nσ⋆\nv=δφrms\nB0lRfrot. (8)\n120 500 1000 150000.511.522.53x 10−6\nB0 [G]δφrms / frot\nFIG. 6: Evolution of the rms value of the azimuthal potential difference normalized by the rotation\nfrequency, as a function of the applied vertical magnetic fie ld.•:frot= 5 Hz,△: 10 Hz, /square: 15 Hz,\n⋄: 20 Hz and ⋆: 30 Hz. The dashed line is a linear trend fitted on the first four points of the 30\nHz data.\n05101520253000.20.40.60.81x 10−3\nfrot [Hz]δφrms / B0\nFIG. 7: Evolution of the rms value of the azimuthal potential difference normalized by the imposed\nvertical magnetic field, as a function of the rotation freque ncy of the propeller. •:B0= 356 G,\n△: 712 G, /square: 1070 G, ⋄: 1420 G. The parallel dashed lines are used as eye guides. The upper one\ncorresponds to a linear fit of the data at B0= 356 G.\nThis quantity can be understood as the velocity fluctuations of the flow normalized by the\nforcing velocity of the propeller. This quantity is displayed as a funct ion of the interaction\nparameter N=σLB2\n0\n2πρRfrotin figure 8. In this representation, the data collapses fairly well on\na single master curve for high rotation rates. The damping of the tu rbulent fluctuations\n13(a)0 0.5 1 1.510−210−1\nNσv*\n \n5 Hz\n10 Hz\n15 Hz\n20 Hz\n30 Hz\n(b)0 0.5 1 1.510−210−1\nNσv*\n \n3 Hz\n5Hz\n7.5 Hz\n10 Hz\nFIG. 8: Evolution of the dimensionless potential σ⋆\nv(see text) as a function of the interaction\nparameter N. (a) and (b) correspond to the azimuthal potential difference taken from two different\ndatasets. The potential probes are similar but not perfectl y identical. The rotation frequencies\nreach the highest values in (a) and the lowest in (b). The repr esentation is semilogarithmic.\ncan reach one order of magnitude for Nclose to 1. In fig. 8(b) - that corresponds to lower\nrotation rates - there is a slight and systematic drift of the curves with the rotation rate.\nThis indicates a slight dependence in Reynolds number that comes likely from the fact\nthat the flow is not fully similar with frotfor low values of this rotation rate. The main\ndependence is clearly in the interaction parameter. The velocity fluc tuations are seen to\ndecay exponentially with Nfor values of Nup to order 1. For higher Nthe data is affected\nby the noise. The velocity fluctuations are so strongly damped that the signal to noise ratio\ndecreases significantly, ascanbeseen onthespectra inthefollowin gsections. Thedecayrate\nis about 2 .5 in fig. 8(a) and 3 .5 in (b). The difference may come from geometrical factors of\n14the probes which are not exactly similar in both datasets, or froma s light mismatch between\nthe positions of the probes in the two datasets. We observed a simila r collapse of potential\ndata with Nin a different experiment [17]. This older experiment was smaller in size, with\nonly one propeller and smaller magnetic field. Only the beginning of the e xponential decay\ncould be observed in that case.\nB. Induced magnetic field\n(a)0 500 1000 1500051015\nB0 [G]b [G]r rms\n(b)00.20.40.60.8 110−210−1\nNbr rms / ( B0 Rm )\n \n5 Hz\n10 Hz\n15 Hz\n20 Hz\n30 Hz\nFIG. 9: (a) Evolution of the rms value of the induced radial ma gnetic field as a function of the\napplied vertical magnetic field, for different values of the ro tation frequency. •:frot= 5 Hz, △:\n10 Hz,/square: 15 Hz, ⋄: 20 Hz and ⋆: 30 Hz. (b) Evolution of brrms/B0Rmas a function of the\ninteraction parameter N. The representation is semilogarithmic.\nThe induced magnetic field is of the order of one percent of the applie d field. This low\nvalue is due to the low magnetic Reynolds number which is at best of ord er 1. For low Rm\n15the induction equation reduces to\nB0·∇v+η∆b= 0 (9)\n(assumingthat B0isuniform). Theinducedmagneticfieldthusreflectsthevelocitygra dients\nin the direction of the applied magnetic field. We use it as a second tool to investigate the\nstatistical properties of the flow.\nThe fluctuation level of the radial magnetic field brrmsis displayed in figure 9(a). Its\nevolution with B0andfrotis strongly similar to that of the potential differences, as expected\nfrom the previous arguments. The azimuthal component displays t he same behavior (not\nshown).\nFor low values of the applied magnetic field, the flow is not affected ver y much by the\nLorentz force and from equation (9) one expects the induced mag netic field amplitude to\nscale as: b∝B0Rm(see [18] for example). From the previous section, the amplitude of\nthe velocity field decays exponentially with N. We thus expect brms/B0Rmto have the\nsame qualitative behavior. This is what is observed in figure 9(b). The various datasets\nare seen to collapse on a single master curve for Nup to 0.5. For higher N, the sensitivity\nof our gaussmeter is not high enough for the signal to overcome th e electronic noise. As\nfor the velocity, the curve for the lowest frotis below all the others, which confirms the\nslight dependence with Rmobserved on the potential measurements. At a given value of\nN, the collapse of the measurements shows that the fluctuations of the induced magnetic\nfield are indeed linear both in applied magnetic field and in magnetic Reyno lds number:\nbrms≃0.1B0Rm.\nWe have also measured the average value of the induced magnetic fie ld. From equation\n(9) it is linked to the vertical gradients of the time-averaged velocit y field. ∝angb∇acketleftbr���angb∇acket∇ight/B0Rm\nand∝angb∇acketleftbθ∝angb∇acket∇ight/B0Rmare shown in figure 10 as a function of N. For both components, this\nrepresentation leads to a good collapse of the datasets. At a given value ofN, the collapse\nshows again that the average induced magnetic field is linear in RmandB0with∝angb∇acketleftbi∝angb∇acket∇ight ≃\n0.01B0Rmfor the data shown here. In an ideal Von K´ arm´ an experiment, wh en the two\npropellers counter-rotate at the same speed, the time averaged flow is invariant to a rotation\nof angleπaround a radial unit vector taken in the equatorial plane (denoted as/vector eron figure\n1). If the applied field were perfectly symmetric and the probe posit ioned exactly in the\nequatorial plane, this symmetry should lead to ∝angb∇acketleftbr∝angb∇acket∇ight= 0. However, the introduction of the\n16(a)00.20.40.60.81−0.01−0.00500.0050.010.015\nN
/ ( B0 Rm )\n \n5 Hz\n10 Hz\n15 Hz\n20 Hz\n30 Hz\n(b)00.20.40.60.8 100.0020.0040.0060.0080.010.012\nN / ( B0 Rm )\n \n5 Hz\n10 Hz\n15 Hz\n20 Hz\n30 Hz\nFIG. 10: (a) Evolution of∝angb∇acketleftbr∝angb∇acket∇ight\nB0Rmas a function of the interaction parameter N. (b) same for\n∝angb∇acketleftbθ∝angb∇acket∇ight\nB0Rm.\nprobe breaks the symmetry and neither the mecanical device nor t he applied magnetic field\nare perfectly symmetric. It has been observed in our setup and in t he VKS experiment\nthat the measurement of ∝angb∇acketleftbr∝angb∇acket∇ightis extremely sensitive to the position of the probe (private\ncommunicationfromtheVKScollaboration). Forexample, aslightmism atchofthepropeller\nfrequencies or of the relative positions of the probe and mid-plane s hear layer can lead to\nstrong changes in the mean value of the magnetic field. This may be th e reason why ∝angb∇acketleftbr∝angb∇acket∇ight\nis not zero here. Nevertheless there is a systematic change of the average with Nwhich is\nconsistent across the different values of the velocity and of the ap plied magnetic field. It\nindicates a change of the time-averaged flow in the vicinity of the pro be. To the shear layer\nlying in the equatorial plane corresponds strong radial vorticity, o rthogonal to the applied\nmagnetic field. The applied strong magnetic field will impact the shear la yer, as it tends to\n17elongate the flow structures along its axis. Even a small change in th e shear layer geometry\naffects strongly the measured ∝angb∇acketleftbr∝angb∇acket∇ight. Here we see that its sign is changed at high N.\nThe time-averaged induced azimuthal magnetic field ∝angb∇acketleftbθ∝angb∇acket∇ightis due to ω-effect from the\ndifferential rotation of the propellers [11, 18]. It is seen to decay slig htly (about 35%) with\nN. This effect may be due to some magnetic braking that leads to an elon gation of the shear\nlayer and thus to weaker differential rotation in the vicinity of the mid -plane. Nevertheless\nthis effect is limited and we expect the average large scale structure of the flow to remain\nalmost unchanged. The small change of the large scale flow is not the reason for the strong\ndamping of the turbulent fluctuations by an order of magnitude.\nC. Injected mechanical power\n0 5 10 15 20050100150200250300\nfrot [Hz]ε [W]\n \n101100102\nfrot [Hz]ε [W]\n \nFIG. 11: Injected mechanical power as a function of the rotat ion frequency of\nthe propellers. The circles correspond to measurements wit hout magnetic field, and\nthe triangles correspond to measurements at constant rotat ion frequency and B0=\n90,180,530,705,880,1000,1230,1240,1320,1410, and 1500 Gauss. The solid line has equation\ny= 0.032f3\nrotwhich comes from the turbulent scaling law. The inset is a log -log representation.\nAn interesting issue in MHD turbulence is to understand how the injec ted mechanical\npower is shared between ohmic and viscous dissipations. The torque sT1andT2provided\nby the motors can be accessed through measurements of the cur rent delivered to them. The\ninjected mechanical power is then ǫ= (T1+T2)2πfrot. This quantity has been measured as a\n18function of the rotation frequency with and without applied magnet ic field. The results are\ndrawn on figure 11: without magnetic field, the injected power follow s the turbulent scaling\nlawǫ∼f3\nrot. Moresurprisingly, wenoticethatatagivenrotationfrequencyth eexperimental\npointscorresponding todifferent amplitudes ofthemagnetic fieldar eindistinguishable. This\nobservation confirms that no change of the global structure of t he flow is induced by the\nmagnetic field. The fact that the injected power is independent of t he applied magnetic\nfield seems to be in contradiction with results from numerical simulatio ns where an ohmic\ndissipation of the same order of magnitude as the viscous one is obse rved when a strong\nmagnetic field is applied (the ohmic dissipation is around three times the viscous dissipation\nforN= 1inBurattinietal.[8]). Onecouldarguethattheremaybealargeohm icdissipation\ncompensated by a drop in the viscous dissipation: the energy flux in t he turbulent cascade\nwould then be dissipated mainly through ohmic effect without changing the overall injected\npower. However, a rough estimate of the ohmic dissipation Djgives values which are rather\nlow: with the maximum value of the induced magnetic field b≃0.1B0Rm, and assuming\nthat the magnetic field is dissipated mostly at large scale, one gets:\nDj∼j2\nσL3∼1\nσ(0.1RmB0)2\nL2µ2\n0L3≃4W (10)\nwhere we used the values Rm= 1, and B0= 1500G. This estimate is much lower than\nthe injected power and we may expect ohmic dissipation to remain neg ligible compared\nto viscous dissipation, even at order one interaction parameter. H owever, one also needs\nto know the current that leaks through the boundaries to evaluat e the additional ohmic\ndissipation that takes place inside the walls. As these effects are diffic ult to quantify, we are\nnot able to determine the ratio of ohmic to viscous dissipation. Never theless it is interesting\nto notice that the injected mechanical power remains the same whe n a strong magnetic field\nis applied, although velocity fluctuations are decreased by a factor 10 in the mid-plane of\nthe tank.\nD. Development of anisotropy\nThe application of a uniform magnetic field on a turbulent flow is known t o elongate\nthe flow structures in the direction of the applied field [3]. In decaying turbulent flows,\nthis effect eventually leads to the bidimensionalization of the flow [19]. As far as forced\n1900.20.40.60.8 10123456\nNa\n \n5 Hz\n10 Hz\n15 Hz\n20 Hz\n30 Hz\nFIG. 12: Evolution of the anisotropy parameter a=ηbrrms\nδφrmsas a function of the interaction\nparameter.\nturbulence is concerned, only numerical simulations have demonstr ated this effect [7]. The\nmeasurement of both the induced magnetic field and the electric pot ential allows to quantify\nthe elongation of the turbulent structures in the zdirection: on the one hand, equation (7)\nlinks the electric potential to the vertical component of vorticity, i.e. to horizontal gradients\nof velocity. On the other hand the induced magnetic field is related to vertical gradients of\nvelocity through equation (9). We expect then that\n∆b\n∆φ∼1\nη∂||v\n∂⊥v(11)\nwhere∂||and∂⊥denote derivatives in directions parallel and perpendicular to the ap plied\nmagneticfield. Aswecannotaccessexperimentally theorderofmag nitudeoftheLaplacians,\nwe define the quantity a=ηbrrms\nδφrmswhich we expect to give a crude estimate of the ratio of\nthe vertical to the horizontal gradients of velocity. It is somewha t related to the parameter\nG1defined in Vorobev et al. [7]. This anisotropy parameter ais represented as a function\nof the interaction parameter Nin fig. 12 for different values of the rotation frequency: it\ndecreases from about 4 until it saturates around 1.5. This decrea se of the parameter aby a\nfactor about 3 when a strong magnetic field is applied provides eviden ce for the elongation\nof the flow structures in the zdirection: the derivatives of the velocity field in the direction\nofB0are much smaller than its derivatives in directions perpendicular to B0. Although the\nturbulence becomes moreanisotropic, itremains threedimensional even forthehighest value\nof the interaction parameter reached in this experiment. This is due to the fact that the\n20forcing imposed by the propellers is three dimensional and prevents the flow from becoming\npurely two dimensional.\nIII. TEMPORAL DYNAMICS\nWe observe that the turbulent fluctuations are being damped by ma gnetic braking when\na strong magnetic field is applied to homogeneous and nearly isotropic turbulence. An\ninteresting question is to know how this damping is shared among scale s. In this section we\nstudy the evolution of the power spectrum densities of the potent ial difference and induced\nmagnetic field.\nA. Potential\nWe show in figure 13 the power spectrum densities (PSD) of the dimen sionless potential\nv⋆=φ/B0lRfrot. For a rotation frequency of 20 Hz, the spectra are seen to deca y as the\nmagnetic field is increased, the shape of the different spectra rema ining the same (fig. 13(c)).\nThe decay of the PSD is at most of a factor 10. This dataset reache s a maximum value of the\ninteraction parameter about 0.4. For the lowest displayed value of t he rotation frequency\nfrot= 5 Hz (fig. 13(a)), Nreaches 1.5. As the interaction parameter increases, two distinct\nregimes are observed: first an overall decay of the PSD and secon d a change in the shape\nof the PSD. The highest frequencies are overdamped compared to the lowest ones: for the\nhighest value of N, the PSD decays by about 6 orders of magnitude at twice the rotat ion\nfrequency, whereas it decays by only two orders of magnitude at lo w frequency. We already\nobserved the first regime in a previous experiment performed on a d ifferent flow [17]. The\ninteraction parameter was not high enough in this experiment to obs erve the second regime.\nTo quantify more precisely the relative decay, we plot in figure 14 the ratio of the PSD\nofv⋆over the PSD at B0= 178 G (and at the same rotation rate of the propeller). At\nthe smallest values of N, the ratio weakly changes across the frequencies but it decays wit h\nN. WhenNis increased over approximately 0 .1 an overdamping is observed at the highest\nfrequencies. This extra damping can be qualitatively characterized by a cutoff frequency,\nwhich decreases extremely rapidly and seems to reach the rotation frequency for N≃0.3.\nAbove the cutoff frequency, the decay rate seems to behave as a power law of the frequency.\n21(a)10−110010110−2100102104\nf / frotPSD(v*) frot\n(b)10−110010110−2100102104\nf / frotPSD(v*) frot\n(c)10−110010110−2100102104\nf / frotPSD(v*) frot\nFIG. 13: Evolution of the power spectrum density of the dimen sionless potential v⋆=δφ\nB0lRfrot\nfor the radial potential difference. The subfigures correspon d to different rotation rates of the\npropellers: (a) frot= 5 Hz, (b) 10 Hz, (c) 15 Hz. In each subfigure, the different curve s correspond\nto the different values of the applied magnetic field B0= 178, 356, 534, 712, 890, 1070, 1250, 1420\nand 1600 G. They are naturally ordered from top to bottom as th e magnetic field is increased. The\nnoise part of the spectra has been removed to improve the clar ity of the figures.\nThe exponent of this power law gets more and more negative as Nincreases and seems to\nreach−5 at the highest value of Ndisplayed here ( frot= 5 Hz,B0= 1600 G, N= 1.5).\nIn figure 15, we gathered the dimensionless spectra at various rot ation rates that corre-\nspond to the same interval of N. The spectra are collapsing fairly well onto each other. A\nlittle bit of scatter is observed, most likely due to the slight dependen ce onRmdescribed\npreviously. The shape of the spectra is essentially a function of the interaction parameter.\nIn experiments on the influence of a magnetic field on decaying turbu lence, the velocity\nspectrum goes from an f−5/3to anf−3behavior as the interaction parameter is increased.\nThis−3 exponent is attributed either to two-dimensional turbulence or t o a quasi-steady\n22(a)10−110010110−610−410−2100\nf / frotratio of PSDs\n(b)10−110010110−410−310−210−1100\nf / frotratio of PSDs\n(c)10−110010110−310−210−1100\nf / frotratio of PSDs\nFIG. 14: Evolution of the shape of the dimensionless potenti al’s power spectrum. The spectra of\nfigure 13 are divided by the spectrum obtained at the smallest value of the applied magnetic field\nB0= 178 G and at the same frot. The subfigures correspond to different rotation rates of the\npropellers: (a) frot= 5 Hz, (b) 10 Hz, (c) 15 Hz.\nequilibrium between velocity transfer and ohmic dissipation. As far as forced turbulence\nis concerned, we observe a strong steepening of the velocity spec trum, with slopes already\nmuch steeper than f−3forN= 1. We do not observe any signature of this quasi-steady\nequilibrium or of 2D turbulence. Once again, this comes from the thre e-dimensional forcing\nof the propellers which prevents the flow from becoming purely 2D.\nUsing the full ensemble of datasets at the various rotation rates o ne can interpolate the\nevolution of the shape of the dimensionless spectra as a function of bothf/frotandN. The\nresult is shown in figure 16. This representation summarizes all prev ious observations. As\nNincreases, first there is a self similar decay of the spectra. When Nreaches approximately\n0.1, an additional specific damping of the high frequencies is observe d. The high frequency\nspectrum gets extremely steep. This extremely steep regime cove rs the full inertial range\n2310−110010110−21001021041060 < N < 0.13PSD( v* ) frot\n10−110010110−21001021041060.13 < N < 0.26\n10−110010110−21001021041060.26 < N < 0.38PSD( v* ) frot\n10−110010110−21001021041060.38 < N < 0.51\n10−110010110−21001021041060.51 < N < 0.64\nf / frotPSD( v* ) frot\n10−110010110−21001021041060.64 < N < 0.77\nf / frot\nFIG. 15: Evolution of the shape of the normalized potential’ s power spectrum as a function of the\ninteraction parameter. Colors corresponds to the different r otation speeds: black 3 Hz, blue 5 Hz,\ngreen 7.5 Hz and red 10 Hz. Each subfigure corresponds to data r estricted to the specified interval\nofN. The dashed line corresponds to the spectrum at the smallest non zero value of N.\n10−1\n100\n10110−2\n10−1\n100−1012345\nf / frot NPSD(v*) frot\nFIG. 16: Evolution of the shape of the power spectrum of the no rmalized potential as a function\nof the interaction parameter. Data for 5, 10, 15 and 20 Hz have been used for this representation.\n24forNclose to 1.\nB. Induced magnetic field\n(a)10−110010−2100102104\nf / frotPSD(br) frot / ( B0 Rm )\n(b)10−110010−2100102104\nf / frotPSD(br) frot / ( B0 Rm )\n(c)10−110010−2100102104\nf / frotPSD(br) frot / ( B0 Rm )\nFIG. 17: Evolution of the power spectrum of the dimensionles s induced magnetic fieldbr\nB0Rm.\nThe subfigures correspond to different rotation rates of the pr opellers: (a) 10 Hz, (b) 15 Hz and (c)\n20 Hz. In each subfigure, the various curves correspond to diffe rent applied magnetic fields. The\ncurves are naturally ordered from top to bottom as the magnet ic field is increased. The results\nfor the following values of B0are displayed: B0= 178, 356, 534, 712, 890, 1070, 1250, 1420 and\n1600 G. The noise part of the spectra has been removed for the c larity of the figures.\nThe same analysis can be performed on the induced magnetic field. Th e dimensionless\nspectra of brare shown in figure 17. One concern is that the induced magnetic field is low, so\nthat the signal to noise ratio of the gaussmeter is not as goodas th at of the potential probes.\nThe magnetic field spectrum reaches the noise level at a frequency which is approximately\n2frot. This is also related to the fact that the scaling law expected from a K olmogorov-like\nanalysis for the magnetic field is much steeper than the one of the ve locity (f−11/3in the\n25dissipative range of the magnetic field, at frequencies correspond ing to the inertial range of\na velocity scaling as f−5/3). Nevertheless, the same two regimes are observed: at low N\nthe spectra are damped in a self similar way. At large Nthe high frequencies seem to be\noverdamped. However, the picture is not so clear because of the lo w signal to noise ratio.\n(a)10−110010−310−210−1100\nf / frotratio of PSDs\n(b)10−110010−210−1100\nf / frotratio of PSDs\n(c)10−110010−1100\nf / frotratio of PSDs\nFIG. 18: Evolution of the shape of the power spectrum of the in duced magnetic field br. The\nspectra of figure 17 are divided by the spectrum obtained at th e smallest value of the applied\nmagnetic field B0= 178 G and at the same frot. The subfigures correspond to different rotation\nrates of the propellers: (a) 10 Hz, (b) 15 Hz and (c) 20 Hz.\nThis last point is better seen on the ratio of the spectra in figure 18. The ratio corre-\nsponding to the highest applied magnetic field is decaying at large freq uencies.\nC. Coherence between velocity and magnetic field\nThe experiment has been designed to record the potential differen ce and the magnetic\nfield in the vicinity of the same point. Figure 19 displays the spectral c oherence between\nthese two quantities. In this figure we have identified the azimuthal potential difference with\n26(a)10010110200.050.10.150.20.25\nf [Hz]coherence of vr, vθ with br\nvθvr 101102PSDvθ\nbr\n(b)10010110200.050.10.150.2\nf [Hz]coherence of vr, vθ with bθ\nvθ\nvr101102PSD\nbθvr\nFIG. 19: Spectral coherence between the velocity and the ind uced magnetic field. (a) coherence\nbetween brandvrorvθ. The inset recalls the spectra of brandvθ. (b) coherence between bθand\nvrorvθ. The inset recalls the spectra of bθandvr. The data correspond to frot= 15 Hz and\nB0= 356 G.\nthe radial component of velocity vrand the radial potential difference with the azimuthal\ncomponent of velocity vθ, even though this identification may be somewhat abusive. We\nrecall that the coherence is one when both signals are fully correlat ed at a given frequency\nand zero if they are uncorrelated at this frequency. We have plott ed in the insets the\npower spectrum of each signal. The magnetic field spectrum falls belo w the noise level at\nabout 2frot, which explains why all coherence curves go to zero above 2 frot(there is no\nmagnetic signal at these frequencies). For lower frequencies, a s mall but non-zero value of\nthe coherence is observed for the following pairs: ( br,vr) with a coherence level above 0.1,\n(br,vθ) with a coherence close to 0.05 and ( bθ,vθ) with a coherence level barely reaching 0.1.\nThe coherence value changes a little for other values of frot, but the global picture remains\nthe same, with ( br,vr) being the most coherent.\nThe time correlations are shown in figure 20. No clear correlation is ob served except for\nthe pair ( br,vr) which displays a small peak reaching 0.1. The weak coherence obser ved in\nthe previous figure is barely seen here, probably because of an insu fficient convergence of\nthe correlation function.\nAt the low values of magnetic Reynolds number attained in our experim ent, the magnetic\nfield is diffused through Joule effect, so that its structure is mainly lar gescale. The Reynolds\nnumber being large, the velocity fluctuations develop down to much s maller scales. Because\nof the strong ohmic diffusion, the magnetic field is expected to be sen sitive mostly to large-\n27−5 0 5−0.100.1\ntime (s)Cb\nrv\nθ , Cb\nrv\nr vrvθ\n−5 0 5−0.100.1\ntime (s)Cb\nθv\nθ , Cb\nθv\nr\nvθvr\nFIG. 20: Correlation coefficients between the velocity and th e induced magnetic field. (a) Correla-\ntion between brandvr(upper curve) or vθ(lower curve). (b) Correlation between bθandvr(lower\ncurve) or vr(upper curve). The data correspond to frot= 15 Hz and B0= 356 G.\nscale and low frequency fluctuations of the velocity field. For examp le one could expect\nto observe a correlation or coherence between vθandbθ: fluctuations of the differential\nrotation can induce fluctuations in the conversion of the axial B0into azimuthal magnetic\nfield through ωeffect. As far as bris concerned, the poloidal recirculation bends the vertical\nfield lines towards the exterior of the tank in the vicinity of the mid-pla ne, a process which\ninduces radial magnetic field from the vertical applied field B0. Fluctuations of this poloidal\nrecirculation thus directly impacts the radial induced magnetic field, hence the correlation\nbetween vrandbr. Although such low frequency coherence is indeed observed in figur e 19\nits amplitude remains very low, so that there is almost no correlation b etween the velocity\nfield and the magnetic field measured in the vicinity of the same point.\nIV. DISCUSSION AND CONCLUSION\nThe effect of a strong magnetic field on forced turbulence is studied experimentally with\npotential probes and induced magnetic field mesurements. The velo city fluctuations are\nstrongly damped as the applied magnetic field increases: for N≃0.6, the turbulence inten-\nsity in the mid-plane of the tank is decreased by an order of magnitud e. As a consequence,\nthe standard deviation of the induced magnetic field - normalized by B0Rm- also diminishes\nby a factor ten. The spectrum of the non-dimensional potential v∗is affected in two differ-\n28ent ways by the magnetic field: for low values of the interaction para meter the spectrum\ndecays uniformly at all frequencies, its shape remaining the same. F or higher values of N,\nwe observe an overdamping of the high frequencies. The same effec t is seen on the induced\nmagnetic field spectra.\nWe identify several features which highlight the very different beha viors of forced and\ndecaying turbulence when they are subject to a strong magnetic fi eld: decaying turbulence\nis thought to evolve towards a bidimensional structure. Its velocit y spectrum displays a −3\nexponent which can be attributed either to this bidimensionalization o r to a quasi-steady\nequilibrium between velocity transfer and ohmic dissipation. In the pr esent experiment,\nno such−3 exponent is observed, and the spectra are much steeper (expo nent−5 to−6)\nfor values of Nhigher than 0 .5. Moreover, we have introduced a parameter a=ηbrms\nδφrmsto\nquantify the anisotropy of the turbulence in the mid-plane of the ta nk. When Nincreases,\nthe decrease of this parameter by a factor 3 is the signature of th e elongation of the flow\nstructures along the applied magnetic field. However, the flow alway s remains 3D since ais\nnon-zero even for the highest value of the interaction parameter reached in this experiment.\nThese differences between forced anddecaying turbulence come f romthe 3Dforcing imposed\nby the propellers, which rules out the possibility of a 2D statistically st eady state of the flow.\nIt would be interesting to perform the same kind of experimental st udy of the anisotropy\nwith other forcing mechanisms, such as current-driven MHD flows o r turbulent thermal\nconvection in a liquid metal.\nWe have studied the evolution of the injected mechanical power as t he applied magnetic\nfield increases and found almost no influence of the latter: the injec ted mechanical power\nremains the same although velocity fluctuations are decreased by a n order of magnitude in\nthe central shear layer.\nFinally, we stress the poor level of correlation between the velocity and induced magnetic\nfields measured in the vicinity of the same point, and attribute it to th e scale separation\nbetween the two fields.\nThe strong damping of turbulent fluctuations by the magnetic field c an be invoked as\na saturation mechanism for turbulent dynamos: in a dynamo experim ent, one observes\nspontaneous generation of magnetic field when the magnetic Reyno lds number is above a\ncritical value Rmc. If the turbulent fluctuations are involved in the generating proce ss of the\nmagnetic field (through αωorα2mechanisms for instance), there is a critical level of rms\n29turbulent fluctuations σvcabovewhich magneticfield isgenerated( σv=σvcforRm=Rmc).\nForRm > Rm c, the initial level of turbulent fluctuations is above σvc, andthe magnetic field\ngrows exponentially from a small perturbation: the interaction par ameter increases, and the\nturbulent fluctuations are damped according to figure 8. An equilibr ium is reached when the\ndamping is such that the rms turbulent fluctuations are reduced to σvc. For small values of\nNweobserved that σv(N) =σv(N= 0)e−γN≃σv(N= 0)(1−γN), with2.5< γ <3.5. The\nsaturated value NsatofNfollows from the equality σv(Nsat) =σvc=σv(N= 0)(1−γNsat).\nAsσvis proportional to Rm, we getNsat=Rm−Rmc\nγRmc, hence the following scaling law for the\nmagnetic field:\nB2\nsat=2πρRfrot\nσLγRm−Rmc\nRmc(12)\nThis is the turbulent scaling law for the saturation of a dynamo, which was originally\ndescribed by P´ etr´ elis et al. (see [20] for instance). Although the exact geometry of the\nexperimental setup and large scale magnetic fields are different fro m that of the present\nexperimental study, results from the VKS dynamo can be used to t est this relationship:\nusingR= 15.5 cm,L= 21 cm, frot= 16 Hz, ρ= 930 kgm−3,σ= 9.5106Ω−1m−1and\nγ= 3, the computed magnetic field amplitude is Bsat≃290 G forRm−Rmc\nRmc=1\n3. This is the\nright order of magnitude: the amplitude of the VKS dynamo field meas ured in the vicinity\nof the axis of the cylinder is approximately 150 Gfor this value of Rm([21]: figure 3(b)).\nTheauthorswouldlike tothankF.P´ etr´ elisforhiscomments andfo rhishelpinthedesign\nof the experimental setup, and S Fauve for insightful discussions . This work is supported\nby ANR BLAN08-2-337433.\n[1] S. Eckert, G. Gerbeth, W. Witke and H. Langenbrunner, “MH D turbulence measurements in\na sodium channel flow exposed to a transverse magnetic field,” Int. J. Heat Fluid Flow 22,\np. 358-364, (2001).\n[2] A. Alemany, R. Moreau, P.L. Sulem and U. Frisch, “Influenc e of an external magnetic field\non homogeneous MHD turbulence,” J. M´ ecanique 18, 2, (1979).\n[3] B. Knaepen, R. Moreau, Magnetohydrodynamic turbulence at low magnetic Reynolds n umber,\nAnn. Rev. Fluid Mech. 40, p. 25-45 (2008).\n30[4] D. R. Sisan, W. L. Shew and D. P. Lathrop, “Lorentz force effe cts in magneto-turbulence,”\nPhys. Earth Planet. Int. 135, p. 137-159, (2003).\n[5] O. Zikanov and A. Thess, “Direct numerical simulation of forced MHD turbulence at low\nmagnetic Reynolds number,” J. Fluid Mech. 358, p. 299-333, (1998).\n[6] T. Boeck, D. Krasnov, A. Thess and O. Zikanov, “Large-Sca le Intermittency of Liquid-Metal\nChannel Flow in a Magnetic Field,” Phys. Rev. Lett. 101, 244501, (2008).\n[7] A. Vorobev, O. Zikanov, P. A. Davidson and B. Knaepen, “An isotropy of magnetohydrody-\nnamic turbulence at low magnetic Reynolds number,” Phys. Fl uids17, 125105, (2005).\n[8] P. Burattini, M. Kinet, D. Carati and B. Knaepen, “Anisot ropy of velocity spectra in qua-\nsistatic magnetohydrodynamic turbulence,” Phys. Fluids 20, 065110, (2008).\n[9] L. Mari´ e, F. Daviaud, “Experimental measurement of the scale-by-scale momentum transport\nbudget in a turbulent shear flow,” Phys. Fluids 16, p. 457-461, (2004).\n[10] R. Monchaux, M. Berhanu, M. Bourgoin, M. Moulin,P. Odie r, J.-F. Pinton, R. Volk, S. Fauve,\nN. Mordant, F. P´ etr´ elis, A. Chiffaudel, F. Daviaud, B. Dubru lle, C. Gasquet, L. Mari´ e and F.\nRavelet, “Generation of a Magnetic Field by Dynamo Action in a Turbulent Flow of Liquid\nSodium,” Phys. Rev. Lett. 98, 044502, (2007).\n[11] H. K. Moffatt, Magnetic Field Generation in Electrically conducting Flui ds, (Cambridge\nUniversity Press, 1978).\n[12] A. Kharicha, A. Alemany, D. Bornas, “Influence of the mag netic field and the conductance\nratio on the mass transfer rotating lid driven flow,” Int. J. H eat Mass Transfer 47, p. 1997-\n2014, (2004).\n[13] R. Ricou and C. Vives, “Local velocity and mass transfer measurements in molten metals\nusing an incorporated magnet probe,” Int. J. Heat Mass Trans fer25, p. 1579-1588, (1982).\n[14] A. Tsinober, E. Kit, M. Teitel, “On the relevance of the p otential-difference method for\nturbulence measurements,” J. Fluid Mech. 175, p. 447-461, (1987).\n[15] H. Tennekes and J.L. Lumley, A First Course in Turbulence , (MIT Press, Cambridge, 1972).\n[16] N. I. Bolonov, A. M. Kharenko, A. E. ´Eidel’man, “Correction of spectrum of turbulence in the\nmeasurement by a conduction anemometer,” Inzhernerno-Fiz icheskii Zhurnal 31, 243, (1976).\n[17] M. Berhanu, B. Gallet, N. Mordant and S. Fauve, “Reducti on of velocity fluctuations in a\nturbulent flow of liquid gallium by an external magnetic field ,” Phys. Rev. E 78, 015302,\n(2008).\n31[18] P. Odier, J.-F. Pinton and S. Fauve, “Advection of a magn etic field by a turbulent swirling\nflow,” Phys. Rev. E 58, p. 7397-7401, (1998).\n[19] J. Sommeria and R. Moreau, “Why, how, and when, MHD turbu lence becomes two-\ndimensional,” J. Fluid Mech. 118, p. 507-518, (1982).\n[20] F. P´ etr´ elis, N. Mordant, S. Fauve, “On the magnetic fie lds generated by experimental dy-\nnamos,” Geophysical and Astrophysical Fluid Dynamics 101, p. 289-323, (2007).\n[21] R. Monchaux, M. Berhanu, S. Aumaˆ ıtre, A. Chiffaudel, F. D aviaud, B. Dubrulle, F. Ravelet,\nS. Fauve, N. Mordant, F. P´ etr´ elis, M. Bourgoin, P. Odier, J .-F. Pinton, N. Plihon and R.\nVolk, “The Von K´ arm´ an Sodium experiment: turbulent dynam ical dynamos,” Phys. Fluids,\n21, 035108 , (2009).\n32" }, { "title": "2205.06399v1.Precession_dynamics_of_a_small_magnet_with_non_Markovian_damping__Theoretical_proposal_for_an_experiment_to_determine_the_correlation_time.pdf", "content": "arXiv:2205.06399v1 [cond-mat.mes-hall] 13 May 2022Precession dynamics of a small magnet with non-Markovian da mping: Theoretical\nproposal for an experiment to determine the correlation tim e,✩✩\nHiroshi Imamura, Hiroko Arai, Rie Matsumoto, Toshiki Yamaj i, Hiroshi Tsukahara✩\nNational Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan\nAbstract\nRecent advances in experimental techniques have made it pos sible to manipulate and measure the magnetization dynamics on\nthe femtosecond time scale which is the same order as the corr elation time of the bath degrees of freedom. In the equations of\nmotion of magnetization, the correlation of the bath is repr esented by the non-Markovian damping. For development of th e science\nand technologies based on the ultrafast magnetization dyna mics it is important to understand how the magnetization dyn amics\ndepend on the correlation time. It is also important to deter mine the correlation time experimentally. Here we study the precession\ndynamics of a small magnet with the non-Markovian damping. E xtending the theoretical analysis of Miyazaki and Seki [J. C hem.\nPhys. 108, 7052 (1998)] we obtain analytical expressions of the prece ssion angular velocity and the e ffective damping constant for\nany values of the correlation time under assumption of small Gilbert damping constant. We also propose a possible experi ment for\ndetermination of the correlation time.\nKeywords: non-Markovian damping, generalized Langevin equation, LL G equation, ultrafast spin dynamics, correlation time\n1. Introduction\nDynamics of magnetization is the combination of precession\nand damping. The precession is caused by the torque due to\nthe internal and external magnetic fields. Typical time scal e\nof the precession around the external field and the anisotrop y\nfield is nanosecond. The damping is caused by the coupling\nwith the bath degrees of freedom such as conduction electron s\nand phonons. The typical time scale of the relaxation of con-\nduction electrons and phonons is picosecond or sub-picosec ond\nwhich is much faster than precession. In typical experiment al\nsituations such as ferromagnetic resonance and magnetizat ion\nprocess, the time correlation of the bath degrees of freedom\ncan be neglected and the magnetization dynamics is well repr e-\nsented by the Landau-Lifshitz-Gilbert (LLG) equation with the\nMarkovian damping term[1–3].\nRecent advances in experimental techniques such as fem-\ntosecond laser pulse and time-resolved magneto-optical Ke rr\neffect measurement have made it possible to manipulate and\nmeasure magnetization dynamics on the femtosecond time\nscale[4–11]. In 1996, Beaurepaire et al. observed the femto sec-\nond laser pulse induced sub-picosecond demagnetization of a\nNi thin film[4], which opens the field of ultrafast magnetiza-\ntion dynamics. The all-optical switching of magnetization in a\n✩Permanent address: High Energy Accelerator Research Organ ization\n(KEK), Institute of Materials Structure Science (IMSS), Ts ukuba, Ibaraki 305-\n0801, Japan\n✩✩This work is partly supported by JSPS KAKENHI Grant Numbers\nJP19H01108 and JP18H03787.\nEmail addresses: h-imamura@aist.go.jp (Hiroshi Imamura),\narai-h@aist.go.jp (Hiroko Arai)ferrimagnetic GdFeCo alloy was demonstrated by Stanciu et a l.\nusing a 40 femtosecond circularly polarized laser pulse[5] . The\nhelicity-dependent laser-induced domain wall motion in Co /Pt\nmultilayer thin films was reported by Quessab et al.[11].\nTo understand the physics behind the ultrafast magnetizati on\ndynamics it is necessary to take into account the time correl a-\ntion of bath in the equations of motion of magnetization. The\nfirst attempt was done by Kawabata in 1972[12]. He derived the\nBloch equation and the Fokker-Planck equation for a classic al\nspin interacting with the surroundings based on the Nakajim a-\nZwanzig-Mori formalism[13–15]. In 1998, Miyazaki and Seki\nconstructed a theory for the Brownian motion of a classical\nspin and derived the integro-di fferential form of the generalized\nLangevin equation with non-Markovian damping[16]. They\nalso showed that the generalized Langevin equation reduces to\nthe LLG equation with modified parameters in a certain limit.\nAtxitia et al. applied the theory of Miyazaki and Seki to the\natomistic model simulations and showed that materials with\nsmaller correlation time demagnetized faster[17].\nDespite the experimental and theoretical progresses to dat e\nlittle attention has been paid to how to determine the correl a-\ntion time experimentally. For development of the science an d\ntechnologies based on the ultrafast magnetization dynamic s it\nis important to determine the correlation time experimenta lly\nas well as to understand how the magnetization dynamics de-\npend on the correlation time.\nIn this paper the precession dynamics of a small magnet with\nnon-Markovian damping is theoretically studied based on th e\nmacrospin model. The magnet is assumed to have a uniaxial\nanisotropy and to be subjected to an external magnetic field\nparallel to the magnetization easy axis. The non-Markovian ity\nPreprint submitted to Journal of Magnetism and Magnetic Mat erials May 16, 2022enhances the precession angular velocity and reduces the da mp-\ning. Assuming that the Gilbert damping constant is much\nsmaller than unity, the analytical expressions of the prece ssion\nangular velocity and the e ffective damping constant are derived\nfor any values of the correlation time by extending the analy sis\nof Miyazaki and Seki[16]. We also propose a possible exper-\niment for determination of the correlation time. The correl a-\ntion time can be determined by analyzing the external field at\nwhich the enhancement of the precession angular velocity is\nmaximized.\nThe paper is organized as follows. Section 2 explains the\ntheoretical model and the equations of motion. Section 3 giv es\nthe numerical and theoretical analysis of the precession dy nam-\nics in the absence of an anisotropy field. The e ffect of the\nanisotropy field is discussed in Sec. 4. A possible experimen t\nfor determination of the correlation time is proposed in Sec . 5.\nThe results are summarized in Sec. 6.\n2. Theoretical model\nWe calculate the magnetization dynamics in a small mag-\nnet with a uniaxial anisotropy under an external magnetic fie ld\nbased on the macrospin model. The magnetization easy axis\nis taken to be z-axis and the magnetic field is applied in the\npositive z-direction. In terms of the magnetization unit vector,\nm=(mx,my,mz), the energy density is given by\nE=K(1−m2\nz)−µ0MsH m z, (1)\nwhere Kis the effective anisotropy constant including the crys-\ntalline, interfacial, and shape anisotropies. µ0is the vacuum\npermeability, Msis the saturation magnetization, His the exter-\nnal magnetic field. The e ffective field is obtained as\nHeff=(Hkmz+H)ez, (2)\nwhere ezis the unit vector in the positive zdirection and Hk=\n2K/(µ0Ms) is the effective anisotropy field.\nThe magnetization precesses around the e ffective field with\ndamping. The energy and angular momentum are absorbed by\nthe bath degrees of freedom such as conduction electrons and\nphonons until the magnetization becomes parallel to the e ffec-\ntive field. The equations of motion of mcoupled with the bath\nis given by the Langevin equation with the stochastic field re p-\nresenting the bath degrees of freedom. If the time scale of th e\nbath is much smaller than the precession frequency the stoch as-\ntic field can be treated as the Wiener process[18] as shown by\nBrown[3].\nSince we are interested in the ultrafast magnetization dyna m-\nics of which time scale is the same order as the correlation ti me\nof the bath degrees of freedom, the stochastic field should be\ntreated as the Ornstein-Uhlenbeck process[18, 19]. As show n\nby Miyazaki and Seki [16] the equations of motion of mtakes\nthe following integro-di fferential form:\n˙m=−γm×(Heff+r)+αm×/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′,(3)whereγis the gyromagnetic ratio, αis the Gilbert damping con-\nstant, and ris the stochastic field. The first term represents the\nprecession around the sum of the e ffective field and the stochas-\ntic field, and the second term represents the non-Markovian\ndamping. The memory function in the non-Markovian damping\nterm is defined as\nν(t−t′)=1\nτcexp/parenleftigg\n−|t−t′|\nτc/parenrightigg\n, (4)\nwhereτcis the correlation time of the bath degrees of freedom.\nThe stochastic field, r, satisfies/angbracketleftri(t)/angbracketright=0 and\n/angbracketleftrj(t)rk(t′)/angbracketright=µ\n2δj,kν(t−t′), (5)\nwhere/angbracketleft/angbracketrightrepresents the statistical mean, and\nµ=2αkBT\nγMsV. (6)\nThe subscripts jandkstand for x,y, orz,kBis the Boltzmann\nconstant, Tis the temperature, Vis the volume of the mag-\nnet, andδj,kis Kronecker’s delta. The LLG equation with the\nMarkovian damping derived by Brown [3] is reproduced in the\nlimit ofτc→0 because lim τc→0ν(t−t′)=2δ(t−t′), where\nδ(t−t′) is Dirac’s delta function. Equation (3) is equivalent to\nthe following set of the first order di fferential equations,\n˙m=−γm×[Heff+δH] (7)\n˙δH=−1\nτcδH−α\nτ2cm−γ\nτcR, (8)\nwhere Rrepresents the stochastic field due to thermal agita-\ntion. Equations (7), (8) are used for numerical simulations . The\nstochastic field, R, satisfies/angbracketleftRj(t)/angbracketright=0 and\n/angbracketleftRj(t)Rk(t′)/angbracketright=µδj,kδ(t−t′). (9)\n3. Precession dynamics in the absence of an anisotropy field\nIn this section the precession dynamics in the absence of an\nanisotropy field, i.e. Hk=0, is considered. The initial di-\nrection of magnetization is assumed to be m=(1,0,0). The\nnumerical simulation shows that the non-Markovian damping\nenhances the precession angular velocity and reduces the da mp-\ning. The numerical results are theoretically analyzed assu ming\nthatα≪1. The analytical expressions of the precession an-\ngular velocity and the e ffective damping constant are obtained.\nThe case with Hk/nequal0 will be discussed in Sec. 4.\n3.1. numerical simulation results\nWe numerically solve Eqs. (7), (8) for H=10 T,α=0.05,\nandτc=1 ps. The temperature is assumed to be low enough\nto set R=0 in Eq. (8). Figure 1(a) shows the trajectory of m\non a unit sphere. The initial direction is indicated by the fil led\ncircle. The plot of the temporal evolutions of mx,my, and mzare\nshown in Fig. 1(b). The magnetization relaxes to the positiv ez\ndirection with precessing around the external field. The res ults\n2t [ps] 0.00 0.01 0.03 \n0.02 0.04 \n100 200 0\nt [ps] \na) b) \nc) d) z\nx yHφ [rad / ps] \n1.76 1.77 1.79 \n1.78 1.80 100 200 0\n100 200 0-1 1\n0mx, m y, m z\nt [ps] \nφ00.05 \nαmzmymx\nyαeff \nFigure 1: (a) Trajectory of mon a unit sphere. The external field of H=\n10 T is applied in the positive zdirection. The initial direction is assumed to\nbem=(1,0,0) as indicated by the filled circle. The other parameters are\nτc=1 ps, andα=0.05. (b) Temporal evolution of mx,my,mz. (c) Temporal\nevolution of the precession angular velocity, ˙φ. The solid red curve shows the\nsimulation result. The dotted black line indicates the resu lt of the Markovian\nLLG equation, i.e. ˙φ0=γH/(1+α2). (d) Temporal evolution of the e ffective\ndamping constant, αeff. The solid red curve shows the simulation result. The\ndotted black line indicates α=0.05.\nare quite similar to that of the Markovian LLG equation, whic h\nimplies that the non-Markovianity in damping causes renorm al-\nization of the gyromagnetic ratio and the Gilbert damping co n-\nstant in the Markovian LLG equation.\nThe renormalized value of the gyromagnetic ratio can be\nobserved as a variation of the precession angular velocity, ˙φ,\nwhere the polar and azimuthal angles are defined as m=\n(sinθcosφ,sinθsinφ,cosθ). Figure 1(c) shows that temporal\nevolution of ˙φ(solid red) together with the precession angular\nvelocity without non-Makovianity, ˙φ0=γH/(1+α2), (dotted\nblack). The precession angular velocity increases with inc rease\nof time and saturates to a certain value around 1.798. The sha pe\nof the time dependence of ˙φis quite similar to that of mzshown\nin Fig. 1(b), which suggests that the non-Markovian damping\nacts as an effective anisotropy field in the precession dynamics.\nThe renormalization of the Gilbert damping constant can be\nobserved as a variation of the temporal evolution of the pola r\nangle, ˙θ. Rearranging the LLG equation for ˙θ, the effective\ndamping constant can be defined as\nαeff=−˙θ/(γHsinθ). (10)\nIn Fig. 1(d)αeffis shown by the red solid curve as a function of\ntime. The effective damping constant is reduced to about one-\nfifth of the original value of α=0.05 (dotted black). Contrary\nto˙φ,αeffdoes not show clear correlation with the dynamics of\nm. During the precession, αeffis kept almost constant.\nThe enhancement of the precession angular velocity and the\nreduction of the Gilbert damping constant due to the non-\nMarkovian damping will be explained by deriving the e ffectiveLLG equation that is valid up to the first order of αin the next\nsubsection.\n3.2. Theoretical analysis\nSince the Gilbert damping constant, α, of a conventional\nmagnet is of the order of 0 .01∼0.1, it is natural to take the\nfirst order ofαto derive the effective equations of motion for\nm. The other parameters related to the motion of mareγ,H,\nandτc. Multiplying these parameters we can obtain the dimen-\nsionless parameter, ξ=γHτc, which represents the increment\nof the precession angle during the correlation time.\nIn the case ofξ<1 Miyazaki and Seki dereived the e ffective\nLLG equation using time derivative series expansion[16]. W e\nfirst briefly review their analysis. Then we derive the e ffective\nLLG equation forξ>1 using the time-integral series expansion\nand show that the e ffective LLG equation has the same form for\nbothξ< 1 andξ> 1. Therefore, it is natural to assume that\nthe derived effective LLG equation is valid for any values of ξ\nincludingξ=1.\n3.2.1. Brief review of Miyazaki and Seki’s derivation of the ef-\nfective LLG equation for ξ<1\nIn Ref. 16, Miyazaki and Seki derived the e ffective LLG\nequation with renormalized parameters using the time deriv a-\ntive series expansion. Similar analysis of the LLG equation\nwas also done by Shul in the study of the damping due to\nstrain[20, 21]. The following is the brief summary of the der iva-\ntion.\nSuccessive application of the integration by parts using ν(t−\nt′)=τc[dν(t−t′)/dt′] gives the following time derivative se-\nries expansion:\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay\nn=1(−τc)n−1dnm\ndtn. (11)\nThen the non-Markovian damping term in Eq. (3) is expressed\nas\nα∞/summationdisplay\nn=1(−τc)n−1/parenleftigg\nm×dnm\ndtn/parenrightigg\n. (12)\nThe first derivative, n=1, is given by\n˙m=−γHm×ez+O(α), (13)\nwhere Ois the Bachmann–Landau symbol. For n=2, substi-\ntution of Eq. (13) into the time derivative of Eq. (13) gives\n¨m=(−γH)2(m×ez)×ez+O(α). (14)\nThe n-th order time derivative is obtained by using the same\nalgebra as\ndn\ndtnm=(−γH)n/bracketleftbig(m×ez)×ez.../bracketrightbig+O(α), (15)\nwhere ezappears ntimes. Expanding the vector products we\nobtain for even order time derivatives\nd2nm\ndt2n=(−1)n(γH)2n/bracketleftbigm−mzez/bracketrightbig+O(α), (16)\n3and for odd order time derivatives\nd2n+1m\ndt2n+1=(−1)n(γH)2n˙m+O(α). (17)\nSubstituting Eqs. (16) and (17) into Eq. (12) the non-\nMarkovian damping term is expressed as\n−∞/summationdisplay\nn=1γ2nm×ez+∞/summationdisplay\nn=0α2n+1m×˙m, (18)\nwhere\nγ2n=αγH m z(−1)n−1ξ2n−1(19)\nα2n+1=α(−1)nξ2n. (20)\nThe sums in Eq. (18) converge for ξ<1. Introducing\n˜γ=γ/parenleftigg\n1+αmzξ\n1+ξ2/parenrightigg\n(21)\n˜α=α\n1+ξ2, (22)\nEq. (3) can be expressed as the following e ffective LLG equa-\ntion with renormalized gyromagnetic ratio, ˜ γ, and damping\nconstant, ˜α:\n˙m=−˜γm×(H+r)+˜αm×˙m+O(α2). (23)\n3.2.2. Derivation of the e ffective LLG equation for ξ>1\nForξ>1 we expand Eq. (3) in power series of 1 /ξusing the\ntime integral series expansion approach. Using the integra tion\nby parts with dν(t−t′)/dt′=ν(t−t′)/τcthe integral part of the\nnon-Markovian damping can be written as\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=1\nτc/integraldisplayt\n−∞˙m(t′)dt′\n−1\nτc/integraldisplayt\n−∞ν(t−t′)/bracketleftigg/integraldisplayt′\n−∞˙m(t′′)dt′′/bracketrightigg\ndt′. (24)\nSuccessive application of the integration by parts gives\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=−∞/summationdisplay\nn=1/parenleftigg\n−1\nτc/parenrightiggn\nJn, (25)\nwhere Jnis the nth order multiple integral defined as\nJn=/integraldisplayt\n−∞/integraldisplayt1\n−∞···/integraldisplaytn−1\n−∞˙m(tn)dtn···dt2dt1. (26)\nFrom Eq. (17), on the other hand, ˙ mis expressed as\n˙m=1\n(−1)n(γH)2nd2n\ndt2n˙m+O(α). (27)\nSubstituting Eq. (27) into Eq. (26) the multiple integrals a re\ncalculated as\nJ2n=1\n(−1)n(γH)2n˙m (28)\nJ2n−1=1\n(−1)n(γH)2n¨m. (29)Then Eq. (25) becomes\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay\nn=11\n(−1)n−1ξ2n˙m\n+∞/summationdisplay\nn=1τc\n(−1)nξ2n¨m. (30)\nSubstituting Eq. (30) into the second term of Eq. (3) the non-\nMarkovian damping term is expressed as\nα∞/summationdisplay\nn=11\n(−1)n−1ξ2nm×[ ˙m+τc¨m]+O(α2). (31)\nFrom Eq. (16) ¨ mis expressed as\n¨m=(−1)(γH)2/bracketleftbigm−mzez/bracketrightbig. (32)\nSubstituting Eqs. (31) and (32) into Eq. (3) we obtain\n˙m=−γH∞/summationdisplay\nn=1/bracketleftigg\n1+αmz\n(−1)n−1ξ2n−1/bracketrightigg\nm×ez−γm×r\n+α∞/summationdisplay\nn=11\n(−1)n−1ξ2nm×˙m+O(α2). (33)\nThe sums converge for ξ>1, and the effective LLG equation\nforξ>1 has the same form as ξ<1, i.e. Eq. (23). Since the\neffective LLG equation has the same form for both ξ< 1 and\nξ>1, it is natural to Eq. (23) is valid for any values of ξ.\nAs pointed out by Miyazaki and Seki, and independently by\nSuhl the effect of the non-Markovian damping on the precession\ncan be regarded as the renormalization of the e ffective field [16,\n20, 21]. Equation (23) can be expressed as\n˙m=−γm×/parenleftigg\nH+αHξ\n1+ξ2mz/parenrightigg\nez−γm×r\n+˜αm×˙m+O(α2). (34)\nThe second term in the bracket represents the fictitious unia xial\nanisotropy field originated from the non-Markovian damping .\nThe fictitious anisotropy field increases with increase of ξfor\nξ<1 and takes the maximum value of αH m z/2 atξ=1, i.e.\nγHτc=1. Forξ>1 the fictitious anisotropy field decreases\nwith increase ofξand vanishes in the limit of ξ→∞ because\nthe non-Markovian damping term vanishes in the limit of τc→\n∞. The precession angular velocity, ˙φ, is expected to have the\nsameξdependence as the fictitious anisotropy field and to have\nthe same temporal evolution as mzas shown in Figs. 1(b) and\n1(c).\n3.2.3. The Correlation time dependence of the precession an -\ngular velocity, and e ffective damping constant\nEquation (21) tells us that up to the first order of αthe pre-\ncession angular velocity can be approximated as\n˙φ≃˜γH=γH/bracketleftigg\n1+αmzγHτc\n1+(γHτc)2/bracketrightigg\n, (35)\n4τc’ =1/( γH) \n0.1 1 10 0.01 \nτc [ps] τc [ps] a) b) \nφ [rad / ps] \n1.76 1.77 1.79 \n1.78 1.80 \n0.1 1 10 0.01 \nα, αeff ~\nαeffα0.04 \n0.00 0.02 0.05 \n0.01 0.03 \nααα\neffeffeffαeffαeff~sim. approx. \nFigure 2: (a) The correlation time, τc, dependence of the precession angular\nvelocity,δ˙φ, atθ=5◦forH=10 T. The solid yellow curve shows the ap-\nproximation result, ˜ γH. The dotted black curve shows the simulation results\nobtained by numerically solving Eqs. (7) and (8). The thin ve rtical dotted line\nindicates the critical value of the correlation time, τ′\nc=1/(γH). (b)τcdepen-\ndence of ˜α(solid yellow) andαeff(dotted black). The parameters and the other\nsymbols are the same as panel (a).\nwhere the second term in the square bracket represents the en -\nhancement due to the fictitious anisotropy field.\nIn Fig. 2(a) the approximation result of Eq. (35) at θ=5◦\nwhere ˙φis almost saturated is plotted as a function of τcby the\nsolid yellow curve. The external field and the Gilbert damp-\ning constant are assumed to be H=10 T andα=0.05, re-\nspectively. The corresponding simulation results obtaine d by\nnumerically solving Eqs. (7) and (8) are shown by the dotted\nblack curve. Both curves agree well with each other because\nαis as small as 0.05. The precession angular velocity is maxi-\nmized at the critical value of the correlation time τ′\nc=1/(γH).\nFigure 2(b) shows the τcdependence of ˜α(solid yellow) and\nαeff(dotted black) for the same parameters as panel (a). Both\ncurves agree well with each other and are monotonic decreasi ng\nfunctions ofτc. They vanish in the limit of τc→∞ similar to\nthe non-Markovian damping term.\n4. Effect of an anisotropy field on precession dynamics\nThe theoretical analysis given in the previous section can\nbe applied to the case with Hk/nequal0 by replacingξwithξk=\nγ(H+Hkmz)τc. Following the same procedure as for Hk=0\nEq. (3) can be expressed as\n˙m=−γm×/parenleftig\nH+αHξk\n1+ξ2\nkmz+αHkξk\n1+ξ2\nkm2\nz/parenrightig\nez\n−γm×r+α\n1+ξ2\nkm×˙m+O(α2). (36)\nThe second and the third terms in the bracket can be regarded\nas the fictitious uniaxial and unidirectional anisotropy fie lds\ncaused by the non-Markovian damping. Similar to the re-\nsults for Hk=0 the precession angular velocity is maximized\natξk=1. The renormalized damping constant is given by\nα/(1+ξ2\nk) which is a monotonic decreasing function of ξkand\nvanishes in the limit of ξk→∞ .c) d) a) b) \n24 6 14 12 10 8 0\nH [T] δφ /φ0 [%] 2\n013\nδφ /φ0 [%] 2\n01\n24 6 14 12 10 8 0\nH [T] Hk = 0 H’=1/( γτ c) \nτc = 1 ps \nθ = 5 oθ = 5 oHk = 1 T \nτc = 1 ps H’=1/( γτ c) − HkmzH = 6, 7, 8, 9 T \nH = 2, 3, 4, 5 T \n = 1 ps \n = 5 = 0 \n = 1 ps τ\nθH\nτ = 1 ps = 1 ps = 1 ps τc = 1 ps τ = 1 ps τ = 1 ps \nθτH\nτ\n = 5 = 1 ps = 1 T \n = 1 ps = 1 ps = 1 T \n = 1 ps = 1 ps τc = 1 ps ττ = 1 ps = 1 ps 3δφ /φ0 [%] 2\n013\nδφ /φ0 [%] 2\n013\nt [ps] 100 200 0\nt [ps] 100 200 0Hk = 0 \nτc = 1 ps Hk = 0 \nτc = 1 ps \nFigure 3: (a)τcdependence ofδ˙φ/˙φ0atθ=5◦. From top to bottom the\nexternal field is H=2,3,4,5 T. The parameters are Hk=0, andτc=1 ps. (b)\nThe same plot as panel (a) for H≥5 T. From top to bottom the external field is\nH=6,7,8,9 T. (c) Hdependence ofδ˙φ/˙φ0atθ=5◦obtained by solving Eqs.\n(7) and (8). The parameters are Hk=0, andτc=1 ps. The critical value of\nthe external field, H′=1/(γτc), is indicated by the thin vertical dotted line. (d)\nThe same plot as panel (c) for Hk=1 T. The thin vertical dotted line indicates\nthe critical value of the external field, H′=1/(γτc)−Hkmz.\n5. A possible experiment to determine the correlation time\nBased on the results shown in Secs. 3 and 4 we propose a\npossible experiment to determine the correlation time, τc. Sim-\nilar to the previous sections we first discuss the case withou t\nanisotropy field, i.e. Hk=0, and then extend the discussion to\nthe case with Hk/nequal0.\nIn Figs. 3(a) and 3(b) we show the temporal evolution of the\nenhancement of angular velocity, δ˙φ/˙φ0, obtained by the solv-\ning Eqs. (7) and (8) for various values of H. The increment\nof the precession angular velocity is defined as δ˙φ=˙φ−˙φ0.\nThe initial state and the correlation time are assumed to be\nm=(1,0,0) andτc=1 ps, respectively. As shown in Fig.\n3(a),δ˙φ/˙φ0increases with increase of HforH≤5T. Once\nthe external field exceeds the critical value of 1 /(γτc)=5.7 T,\nδ˙φ/˙φ0decreases with increase of Has shown in Fig. 3 (b). The\nresults suggest that correlation time can be determined by a na-\nlyzing the external field that maximizes the enhancement of t he\nprecession angular velocity.\nFigure 3(c) shows the Hdependence ofδ˙φ/˙φ0atθ=5◦\nwhereδ˙φ/˙φ0is almost saturated. The enhancement is maxi-\nmized at the critical value of the external field, H′=5.7 T. The\ncorrelation time is calculated as τc=1/(γH′)=1 ps.\nIf the system has a uniaxial anisotropy field, Hk, the en-\nhancement of the precession angular velocity is maximized a t\nH′=1/(γτc)−Hkmzas shown in Fig. 3(d). The correlation\ntime is obtained as τc=1/γ(H′+Hkmz).\nThe above analysis is expected to be performed experimen-\n5tally using the time resolved magneto optical Kerr e ffect mea-\nsurement technique. In the practical experiments the analy sis\ncan be simplified as follows. The polar angle of the initial st ate\nis not necessarily large. It can be small as far as the preces-\nsion angular velocity can be measured. Instead of analyzing\nδ˙φ/˙φ0, one can analyze ˙φ/Hor˙φ/(H+Hkmz) because they are\nmaximized at the same value of Hasδ˙φ/˙φ0. Since the required\nmagnetic field is as high as 10 T, a superconducting magnet [22 ]\nis required.\n6. Summary\nIn summary we theoretically analyze the ultrafast precessi on\ndynamics of a small magnet with non-Markovian damping. As-\nsumingα≪1, we derive the effective LLG equation valid for\nany values ofτc, which is a direct extension of Miyazaki and\nSeki’s work[16]. The derived e ffective LLG equation reveals\nthe condition for maximizing ˙φin terms of Handτc. Based on\nthe results we propose a possible experiment for determinat ion\nofτc, whereτccan be determined from the external field that\nmaximizesδ˙φ/˙φ0.\nReferences\n[1] L. Landau, E. Lifshits, ON THE THEORY OF THE DISPER-\nSION OF MAGNETIC PERMEABILITY IN FERROMAGNETIC\nBODIES, Physikalische Zeitschrift der Sowjetunion 8 (1935 ) 153.\ndoi:10.1016/B978-0-08-010586-4.50023-7 .\n[2] T. Gilbert, Classics in Magnetics A Phenomenological Th eory of Damp-\ning in Ferromagnetic Materials, IEEE Transactions on Magne tics 40 (6)\n(2004) 3443–3449. doi:10.1109/TMAG.2004.836740 .\n[3] W. F. Brown, Thermal Fluctuations of a Single-Domain Par ticle, Physical\nReview 130 (5) (1963) 1677–1686. doi:10.1103/PhysRev.130.1677 .\n[4] E. Beaurepaire, J.-C. Merle, A. Daunois, J.-Y . Bigot, Ul trafast Spin Dy-\nnamics in Ferromagnetic Nickel, Physical Review Letters 76 (22) (1996)\n4250–4253. doi:10.1103/PhysRevLett.76.4250 .\n[5] C. D. Stanciu, F. Hansteen, A. V . Kimel, A. Kirilyuk, A. Ts ukamoto,\nA. Itoh, T. Rasing, All-Optical Magnetic Recording with Cir cu-\nlarly Polarized Light, Physical Review Letters 99 (4) (2007 ) 047601.\ndoi:10.1103/PhysRevLett.99.047601 .\n[6] G. P. Zhang, W. H¨ ubner, G. Lefkidis, Y . Bai, T. F. George, Paradigm of the\ntime-resolved magneto-optical Kerr e ffect for femtosecond magnetism,\nNature Physics 5 (7) (2009) 499–502. doi:10.1038/nphys1315 .\n[7] J.-Y . Bigot, M. V omir, E. Beaurepaire, Coherent ultrafa st magnetism in-\nduced by femtosecond laser pulses, Nature Physics 5 (7) (200 9) 515–520.\ndoi:10.1038/nphys1285 .\n[8] A. Kirilyuk, A. V . Kimel, T. Rasing, Ultrafast optical ma nipulation of\nmagnetic order, Reviews of Modern Physics 82 (3) (2010) 2731 –2784.\ndoi:10.1103/RevModPhys.82.2731 .\n[9] J.-Y . Bigot, M. V omir, Ultrafast magnetization dynamic s of nanostruc-\ntures: Ultrafast magnetization dynamics of nanostructure s, Annalen der\nPhysik 525 (1-2) (2013) 2–30. doi:10.1002/andp.201200199 .\n[10] J. Walowski, M. M¨ unzenberg, Perspective: Ultrafast m agnetism and\nTHz spintronics, Journal of Applied Physics 120 (14) (2016) 140901.\ndoi:10.1063/1.4958846 .\n[11] Y . Quessab, R. Medapalli, M. S. El Hadri, M. Hehn, G. Mali nowski, E. E.\nFullerton, S. Mangin, Helicity-dependent all-optical dom ain wall motion\nin ferromagnetic thin films, Physical Review B 97 (5) (2018) 0 54419.\ndoi:10.1103/PhysRevB.97.054419 .\n[12] A. Kawabata, Brownian Motion of a Classical Spin, Progr ess of Theoret-\nical Physics 48 (6) (1972) 2237–2251. doi:10.1143/PTP.48.2237 .\n[13] S. Nakajima, On Quantum Theory of Transport Phenomena: Steady\nDiffusion, Progress of Theoretical Physics 20 (6) (1958) 948–95 9.\ndoi:10.1143/PTP.20.948 .[14] R. Zwanzig, Ensemble Method in the Theory of Irreversib il-\nity, The Journal of Chemical Physics 33 (5) (1960) 1338–1341 .\ndoi:10.1063/1.1731409 .\n[15] H. Mori, Transport, Collective Motion, and Brownian Mo -\ntion, Progress of Theoretical Physics 33 (3) (1965) 423–455 .\ndoi:10.1143/PTP.33.423 .\n[16] K. Miyazaki, K. Seki, Brownian motion of spins revisite d,\nThe Journal of Chemical Physics 108 (17) (1998) 7052–7059.\ndoi:10.1063/1.476123 .\n[17] U. Atxitia, O. Chubykalo-Fesenko, R. W. Chantrell, U. N owak,\nA. Rebei, Ultrafast Spin Dynamics: The E ffect of Col-\nored Noise, Physical Review Letters 102 (5) (2009) 057203.\ndoi:10.1103/PhysRevLett.102.057203 .\n[18] C. W. Gardiner, Stochastic Methods: A Handbook for the N atural and\nSocial Sciences, 4th Edition, no. 13 in Springer Series in Sy nergetics,\nSpringer, Berlin Heidelberg, 2009.\n[19] G. E. Uhlenbeck, L. S. Ornstein, On the Theory of the\nBrownian Motion, Physical Review 36 (5) (1930) 823–841.\ndoi:10.1103/PhysRev.36.823 .\n[20] H. Suhl, Theory of the magnetic damping constant, IEEE T ransactions on\nMagnetics 34 (4) (1998) 1834–1838. doi:10.1109/20.706720 .\n[21] H. Suhl, Relaxation Processes in Micromagnetics, Oxfo rd University\nPress, 2007. doi:10.1093/acprof:oso/9780198528029.001.0001 .\n[22] H. W. Weijers, U. P. Trociewitz, W. D. Markiewicz, J. Jia ng, D. My-\ners, E. E. Hellstrom, A. Xu, J. Jaroszynski, P. Noyes, Y . Viou chkov,\nD. C. Larbalestier, High field magnets with HTS conductors, I EEE\nTransactions on Applied Superconductivity 20 (3) (2010) 57 6–582.\ndoi:10.1109/TASC.2010.2043080 .\n6" }, { "title": "1808.09110v1.Enhancement_of_zonal_flow_damping_due_to_resonant_magnetic_perturbations_in_the_background_of_an_equilibrium__E__times_B__sheared_flow.pdf", "content": "Enhancement of zonal \row damping due to\nresonant magnetic perturbations in the\nbackground of an equilibrium E\u0002Bsheared\n\row\nM. Leconte and R. Singh\nNational Fusion Research Institute, Daejeon 34133, South Korea\nEmail: mleconte@nfri.re.kr\nAugust 29, 2018\nAbstract\nUsing a parametric interaction formalism, we show that the equi-\nlibrium sheared rotation can enhance the zonal \row damping e\u000bect\nfound in Ref. [M. Leconte and P.H. Diamond, Phys. Plasmas 19,\n055903 (2012)]. This additional damping contribution is proportional\nto (Ls=LV)2\u0002\u000eB2\nr=B2, whereLs=LVis the ratio of magnetic shear\nlength to the scale-length of equilibrium E\u0002B\row shear, and \u000eBr=B\nis the amplitude of the external magnetic perturbation normalized to\nthe background magnetic \feld.\n1 Introduction\nThe high con\fnement mode (H-mode) regime is a reference operation sce-\nnario for future tokamak experiments like ITER. In this regime, boundary\ndisturbances known as Edge Localized Modes need to be either avoided or\ncontrolled. One candidate control method uses external magnetic pertur-\nbations known as Resonant Magnetic Perturbations i.e. RMP [1]. RMPs\nwere shown to damp GAM zonal \rows [2, 3] and to enhance turbulent den-\nsity \ructuations [4, 5, 6]. The modi\fcation of turbulence and \rows was also\n1arXiv:1808.09110v1 [physics.plasm-ph] 28 Aug 2018observed for the case of a large-scale static magnetic island m:n= 2 : 1\n[7, 8, 9]. Proposed mechanisms to explain the enhanced zonal \row damping\ninclude modi\fcations to the Rosenbluth-Hinton residual zonal \rows due to\n3D magnetic geometry [10, 11, 12]. An alternative mechanism was described\nby Leconte & Diamond [13]. In this article, we recast the latter theory in\nthe language of parametric interaction, and consider important additional\ne\u000bects on zonal \rows due to the synergy between RMPs and the equilibrium\n\row shear. Our proposed physical mechanism for RMP-induced zonal \row\ndamping by a large-scale static magnetic perturbation can be understood as\nthecoupling of the zonal \row branch to a damped Alfven wave branch. In\nthe parametric interaction analysis, the sideband response to zonal poten-\ntial\u001eqscales as s\u0018 \u0003\n0\u001eq=B\u0002(\n +i\u0011q2\nr)\u00001with 0the external magnetic\nperturbation, qrthe radial wavenumber of zonal \rows, \n = \n r+i\rqthe com-\nplex frequency of the modulational instability, and \u0011the Spitzer resistivity.\nReplacing sin the expression for the Maxwell stress , the resulting disper-\nsion relation is: (\n \u0000i\r0\nq)(\n +i\u0011q2\nr) =c2\nAj~Brj2=B2, where\r0\nqdenotes the\nunperturbed zonal \row growth-rate (balance between turbulence drive and\nneoclassical damping), and cA=B=p4\u0019n0miis the Alfven speed. Without\nmagnetic perturbation j~Brj= 0, the two branches decouple, one becomes the\nusual turbulence-driven zonal \row branch \n \u0018i\r0\nq, while the other is purely\ndamped \n\u0018\u0000i\u0011q2\nr. This shows that the external magnetic perturbation\ne\u000bectively couples the two branches of the dispersion relation, resulting in a\nmodi\fcation of zonal \row growth.\n2 Model\nIn order to calculate the e\u000bect of the nonlinear Reynolds stress due to 3D\n\felds on zonal \row dynamics, we \frst present the linear relation between\nan externally-imposed \u000e 0perturbation and a helical stream function \u000e\u001ehin\nweak shear rotating plasmas. We consider a slab geometry ( x;y;z ), where\nxdenotes the local radial coordinate, and ydenotes the local poloidal coor-\ndinate, and zis the local toroidal coordinate, in a fusion device. We intro-\nduce a \rux function , viaB?=\u0000^z\u0002r and a stream-function \u001e, with\nvE=^z\u0002r\u001e. For small perturbations, the \rux and stream functions are\nexpressed as = eq+\u000e hand\u001e=\u001eeq+\u000e\u001eh, with the mean magnetic\n\rux eq(x) =B0x2=2Ls. Here, we impose the radial pro\fle of the mean\nstream function as ( c=B)\u001eeq(x) =VE0\nLV[x2\n2+x3\n6LV+:::] (E\u0002Bsheared \row)\n2near the rational surface [14, 15]. For weak \row shear \u0001 x=LV<1, with\n\u0001xthe characteristic radial scale, and in the limit of low-resistivity, like\nthe frozen-in dynamics, i.e. Ek= 0, the linearized Ohm's law is given by:\n^z\u0002r\u001eeq\u0001r\u000e h+^z\u0002r\u000e\u001eh\u0001r eq= 0. TheE\u0002B\row is related to \u001eeq\nviaVE(x) = (c=B)\u001e0\neq(x)'V0\nExwithV0\nE=V0x=LV. Integrating between\nparallel and poloidal coordinates, this equation then gives the linear relation:\n\u000e\u001eh=LsVE0\nLV\u000e h.\nUsing normalizations \u001eh= (e\u000e\u001eh=Te)(Ln=\u001as) and h= (e\u000e h=Te)(Ln=\u001as)(2cs=c\f),\nwithLn=n0=jrn0jthe density-gradient length, we obtain the desired rela-\ntion - in normalized form - which is later used in the zonal \row dynamics.\n\u001eh=\f\n2jV0j h: (1)\nwith the normalized \row-shear V0= (Ls=LV)VE0=cs,Ls=q0R. Note that\nEq. (1) is valid for \u0001 x\u0018\u0001w < LV(small island width \u0001 w), withLn=\nn0=jrn0jthe density-gradient length.\nWriting the magnetic \feld B=B0\u0000^z\u0002r , and the current jk=r2\n? ,\nwhere is the magnetic \rux and B0=B0^zis the toroidal magnetic \feld, the\nmodel equations for coupled zonal \rows and external magnetic perturbation\nare the vorticity equation (charge balance) and Ohm's law:\n@r2\n?~\u001ek\n@t\u0000rk0r2\n?~ k=\f\n2f~ ;r2\n?~ g\u0000f ~\u001e;r2\n?~\u001eg+\u0017r4\n?\u001ek; (2)\n@ k\n@t\u00002\n\frk0\u001ek=\u0000f~\u001e;~ g+\u0011jkk: (3)\nHere, we have written the perturbed parallel gradient as rk=rjj0+\u000eB\nB\u0001r,\nwithrjj0the equilibrium part, and\u000eB\nB\u0001rthe contribution due to external\nmagnetic perturbations. Time is normalized ascs\nLnt!t, and the perpen-\ndicular and parallel scales are normalized respectively as: \u001asr?!r?and\nLnrk0!rk0. Other normalizations are:Ln\n\u001ase\nTe\u001e!\u001e,2csLn\nc\u001as\fee\nTe ! , and\nLn\n\u001ascsVE!VE, with\f=8\u0019n0Te\nB2the ratio of kinetic to magnetic energy,\n^\u0011=\u0011c2Ln\n4\u0019\u001a2\nscs=\u00152\ns\n\u001a2\ns\u0017eiLn\ncsthe normalized resistivity, , \u0015s=c\n!pethe electron\nskin-depth and ^ \u0017=Ln\n\u001a2scs\u0017the normalized - turbulent - viscosity. In the fol-\nlowing, we drop the^on normalized quantities for clarity.\n3To describe the perturbed \rux surface geometry due to RMPs, we use\nthe following ansatz for the total magnetic \rux:\n = eq+ hcosk0y (4)\nwhere eqis the unperturbed poloidal \rux, and k0is the poloidal wavenumber\nof the RMP. Physically, this represents a long-wavelength modulation of the\nmagnetic \feld [Fig. 1].\n2.1 Parametric interaction analysis\nWe vizualize the interaction between zonal \rows and external magnetic per-\nturbation as a four-wave parametric interaction [16, 17, 18]. A schematic\ndiagram of the interaction is shown [Fig. 2]. Parametric interaction is as-\nsociated with a phase-instability, as described e.g. in Ref. [20]. Here, the\nmagnetic perturbation acts like a stationary long-wavelength modulation of\nthe background magnetic \feld ( !0= 0;k0=k0^y), and zonal \rows act like\na long-scale wave (\n ;q=qx^x). Here,k0\u001dqy, sinceqy= 0 for zonal \rows.\nIn practical experiments, we expect the magnetic perturbation to evolve in\ntime, but since this evolution is very slow, we can treat it as stationary. The\nMP can be called the pump, although it provides a damping rather than\na drive, as far as zonal \rows are concerned. In this parametric process, a\nlong-scale wave (ZF) at (\n ;q) interacts with the magnetic perturbation at\n(!0= 0;k0) and generates two side-bands at ( !1;k1) and (!2;k2), where\nk1;2=q\u0006k0and!1;2=!(k1;2). The resonant interaction condition for the\nwaves!0\u0006Ref\ng= Ref!1;2gis approximately satis\fed since, to \frst approx-\nimation, zonal \rows have zero frequency. The two side-bands couple with\nthe magnetic perturbation to produce electrostatic and magnetostatic pon-\nderomotive forces (poloidal torques) on the plasma, which can excite and/or\ndamp the low-frequency mode (ZF).\nLet us now obtain the equations for the two sideband amplitudes, follow-\ning the notations of Lashmore-Davies et al. [16]. To calculate the parametric\ninteraction between short-scale MP and long-scale zonal \rows, we take the\nmagnetic perturbation as:\n~ h= h(x)[exp(ik0y\u0000i!0t) +c:c:] (5)\n~\u001eh=\u001eh(x)[exp(ik0y\u0000i!0t) +c:c:] (6)\n4Figure 1: The radial perturbation \u000eBrdue to 3D \felds can be viewed as a\nlong-wavelength modulation of the background magnetic \feld B.\nFigure 2: Schematic diagram of the parametric interaction.\n5Here, the pump frequency is !0\u00180, for static MPs. This mode can couple\nto long scale zonal wave, which is represented by:\nVZF(x;t) =iqx\u001eq(t) exp(iqxx\u0000i\nt) +c:c: (7)\nwhereqxis the dimensionless wavenumber along the radial direction, and \u001eq\nis the potential amplitude.\nThe resonant coupling between the pump mode ( !0;k0) and zonal \row\n(\n;q) can generate two sideband waves: ( !1;2);k1;2,!1;2=!0\u0006\n, with\nk1;2=q\u0006k0. The desired sideband \feld ( 1;2;\u001e1;2) can be represented as:\n~ 1;2(r;t) = 1;2[exp(ik1;2\u0001r\u0000i!1;2t) +c:c:] (8)\n~\u001e1;2(r;t) =\u001e1;2[exp(ik1;2\u0001r\u0000i!1;2t) +c:c:] (9)\nUsing equations (5,6,7,8,9), the vorticity equation (2) for zonal \row can\nbe written:\n@\u001eq\n@t=\u0000\f\n4(^z\u0002q)\u0001k0( \u0003\nh 1\u0000 h 2) +1\n2(^z\u0002q)\u0001k0(\u001e\u0003\nh\u001e1\u0000\u001eh\u001e2) (10)\nNote that the subscript \"h\" corresponds to the externally applied helical\nperturbation. The perturbation with subscript \"q\" represents the zonal \row,\nand the \feld with subscript \"1,2\" represents the driven sideband perturba-\ntion.\nFrom Ohm's law Eq. (3), the equations of the two sideband waves 1;\u001e1\nand 2;\u001e2are:\n@ 1\n@t+\u0011k2\n1 1= (^z\u0002q)\u0001k0 h\u001eq (11)\n@ 2\n@t+\u0011k2\n2 2=\u0000(^z\u0002q)\u0001k0 \u0003\nh\u001eq (12)\n@\u001e1\n@t+\u0017k2\n1\u001e1= (^z\u0002q)\u0001k0k2\n0\u0000q2\nk2\n1\u001eh\u001eq (13)\n@\u001e2\n@t+\u0017k2\n2\u001e2=\u0000(^z\u0002q)\u0001k0k2\n0\u0000q2\nk2\n2\u001e\u0003\nh\u001eq; (14)\nwhere the sideband complex amplitudes 1;2can be further decomposed\nas: \u0014 1;2(t)\n\u001e1;2(t)\u0015\n=\u0014\t1;2(t)\n\b1;2(t)\u0015\nei\u000e1;2t; (15)\n6with\u000e1;2=!1;2\u0000!0, and the unperturbed frequencies !1;2are given by:\n!1;2=!(k1;2) (16)\nwithjk1;2j2=jk0^y\u0006qx^xj2=k2\n0+q2\nx. Note that, in the present case:\n!1=!2= Re(\n)'0 (17)\nIn the following, we derive the parametric interaction equations. We\nobtain, after some algebra, the following system of coupled equations:\n@\u001eq\n@t\u0000(\u000bDW\u000f\u0000\u0017q2\nx\u0000\u0016)\u001eq=\f\n4\u0003h\n \u0003\nh\t1\u0000 h\t2i\n\u0000\u0003\n2h\n\u001e\u0003\nh\b1\u0000\u001eh\b2i\n; (18)\n\u0014@\n@t+\u0011(q2\nx+k2\n0)\u0015\n\t1+i\u000e1\t1= \u0003 h\u001eq; (19)\n\u0014@\n@t+\u0011(q2\nx+k2\n0)\u0015\n\t2+i\u000e2\t2=\u0000\u0003 h\u001e\u0003\nq; (20)\n\u0014@\n@t+\u0017(q2\nx+k2\n0)\u0015\n\b1+i\u000e1\b1= \u0003k2\n0\u0000q2\nx\nk2\n0+q2\nx\u001eh\u001eq; (21)\n\u0014@\n@t+\u0017(q2\nx+k2\n0)\u0015\n\b2+i\u000e2\b2=\u0000\u0003k2\n0\u0000q2\nx\nk2\n0+q2\nx\u001e\u0003\nh\u001eq; (22)\nwith the coe\u000ecients: \u0003 = ( ^z\u0002q)\u0001k0=qxk0, and\u000e1;2= 0. We also included\nthe turbulence drive and neoclassical damping via the term \u000bDW\u000f\u0000\u0016, where\n\u000f=P\nkj\u001eDW\nkj2denotes the turbulence energy, \u000bDWis the DW-ZF coupling\nparameter and \u0016=\u0016neois the neoclassical friction. In the following, we\nneglect viscous dissipation for zonal \rows since \u0017q2\nx\u001c\u0016.\nNote that here, since \u000e1=\u000e2= 0, the two sidebands are directly related\nvia:\n h\t2=\u0000 \u0003\nh\t1and\u001eh\b2=\u0000\u001e\u0003\nh\b1 (23)\nHence, only one sideband (\t 1;\b1) appears, and the system reduces to:\n@\u001eq\n@t\u0000(\u000bDW\u000f\u0000\u0016)\u001eq=\u0000\f\n2\u0003 \u0003\nh\t1+ \u0003\u001e\u0003\nh\b1; (24)\n\u0014@\n@t+\u0011(q2\nx+k2\n0)\u0015\n\t1= \u0003 h\u001eq; (25)\n\u0014@\n@t+\u0017(q2\nx+k2\n0)\u0015\n\b1= \u0003k2\n0\u0000q2\nx\nk2\n0+q2\nx\u001eh\u001eq (26)\n7The \frst term on the r.h.s. of the ZF evolution (24) is the direct contribution\nfrom the MP-induced nonlinearity f ;r2\n? g, in the Maxwell stress-like form\nwhereas the second term on the r.h.s. comes from the indirect f\u001e;r2\n?\u001eg\nnonlinearity due to the helical potential \u001ehassociated to the MP, i.e. Eq.\n(1).\nAfter some algebra, one can obtain the following 'nonlinear' dispersion\nrelation for zonal \rows:\n\u0000i\n\u0000(\u000bDW\u000f\u0000\u0016) =\u0000\fq2\nx\n2k2\n0 2\nh\n\u0000i\n +\u0011(q2\nx+k2\n0);\n\u0000q2\nxq2\nx\u0000k2\n0\nq2\nx+k2\n0\u0001k2\n0\u001e2\nh\n\u0000i\n +\u0017(q2\nx+k2\n0)= 0; (27)\nwhere we replaced \u0003 with its expression.\nIn the case without mean sheared \row ( V0= 0, i.e.\u001eh= 0, cf. Eq. 1),\nthe associated 'nonlinear' dispersion relation is approximately:\n\rq\u0000(\u000bDW\u000f\u0000\u0016)'\u0000\fq2\nx=2\n\u0011(q2\nx+k2\n0)k2\n0 2\nh: (28)\nwith\rq= Im \n the zonal \row growth-rate. The enhancement of zonal \row\ndamping is shown v.s. qxschematically [Fig. 3a]. To guide the reader, we can\nevaluate a typical normalized ZF radial wavenumber as qx\u001as\u0018(\u001asLn)\u00001=2\u0018\n0:2 orjqx=k0j\u0018(a=nq )=p\u001asLn'20 forn= 1 RMPs.\nIn the limitjqxj\u001dk0, we recover the results of Leconte & Diamond [13]\nfor the enhancement of ZF damping, in dimensional form:\n\u0001\rd\n\rd'C1\u0014Bvac\nr\nB\u00152\n(29)\nwith the coe\u000ecient C1=c2\nA=(\u0017ii\u0017ei\u00152\nskin) in our notation. Here, \u0001 \rd=\n\rd\u0000\r0\nd, with\r0\ndthe reference zonal \row damping without external magnetic\nperturbation Bvac\nr=B= 0. This reference zonal \row damping is of the order of\nthe ion-ion collision frequency \u0017ii. The enhancement over this value due to the\nexternal perturbation (Eq. 29) is of the order of c2\nA=(\u0017ii\u0017ei\u00152\nskin)\u0002(Bvac\nr=B)2.\nFor typical parameters \u0017ei'5:105s\u00001,\u0017ii'\u0017ei=40,\u0015skin'10\u00003m, and\ncA'106m/s, this yields: \u0001 \rd=\rd'1:6 for typical external perturbation\namplitudeBvac\nr=B\u001810\u00004.\n800.20.40.60.811.21.41.6\n-10 -5 0 5 10∆ γd / γd\nqx / k0a)\n012345\n-10 -5 0 5 10∆ γd / γd\nqx / k0b)Figure 3: Relative change in zonal \row damping \u0001 \rd=\rdv.s. ZF radial\nwavenumber qx, forBvac\nr=B= 10\u00004, given by Eq. (30). a) case without\nmean \row shear V0= 0 and b) case with mean \row shear V0= 0:1 (solid),\nV0= 0:15 (dash) and V0= 0:2 (dash-dotted).\nIn the case with mean sheared \row ( V06= 0, i.e.\u001eh6= 0), we obtain the\nfollowing modi\fed 'nonlinear' dispersion relation:\n\rq\u0000(\u000bDW\u000f\u0000\u0016)'\u0000\fq2\nx=2\n\u0011(q2\nx+k2\n0)k2\n0 2\nh\u0000q2\nx\u0000k2\n0\nq2\nx+k2\n0\u0001\f2q2\nx=4\n\u0017(q2\nx+k2\n0)V02k2\n0 2\nh;(30)\nwhere we expressed \u001ehin terms of handV0, using the relation (1). Eq.\n(30) is the main result of this Letter.\nThe e\u000bect of mean \row shear on zonal \row damping is shown v.s. qx\nschematically [Fig. 3b]. Parameters are the same as in [Fig. 3a].\nIn the limit k0\u001cjqxj, the enhancement of zonal \row damping becomes:\n\u0001\rd\n\rd'C1\u0014Bvac\nr\nB\u00152\n+C2V02\u0014Bvac\nr\nB\u00152\n(31)\nwithC1given below Eq. (29), and the new coe\u000ecient C2=L2\nn=(\u001a2\ns\u0017\u0003\ni2)\u0001\n(qR=Ln)4, with\u0017\u0003\ni=\u0017iiqR=vth;ithe ion collisionality. For typical parame-\ntersR= 2m,q= 3,Ln= 5:10\u00002m,\u0017\u0003\ni'0:4,V0'0:1 and eddy viscosity\n\u0017\u001810m2:s\u00001, this yields \u0001 \rd=\rd'2:3, which represents a signi\fcant en-\nhancement of zonal \row damping. Morevover, for a \row shear V0>1:5, the\nrelative zonal \row damping becomes negative for zonal \row wavenumbers\nqx< k 0. Physically, this suggests that the synergy between RMPs and the\n9relative zonal \row damping \u0001 \rd=\rd\nw/o mean \row shear Ref. [13] C1h\nBvac\nr\nBi2\nwith mean \row shear [this work] C1h\nBvac\nr\nBi2\n+C2V02h\nBvac\nr\nBi2\nTable 1: Main scalings of the enhancement of zonal \row damping-rate \u0001 \rd=\rd\nby 3D \felds, in the limit jqxj\u001dk0. The coe\u000ecients C1,C2are given in the\ntext.\nmean \row shear can excite relativively large-scale zonal \rows at wavenumber\nqxqx, the zonal \row damp-\ning becomes negative, i.e. RMPs are predicted to enhance the drive of zonal\n\rows for large mean \row shear, via this mechanism. Collision-free gyroki-\nnetic simulations presented in Ref. [19] did not observe any e\u000bect on zonal\n\rows from external magnetic perturbations. Electron-ion collisions treated\nin our model may play a role and partially explain this discrepancy. If future\nimproved simulations show a damping, it would be interesting to see if this\ndamping depends on the ZF radial wavenumber. Due to energy conservation\namong turbulence/zonal \row system, this additional damping of zonal \rows\nimplies a simultaneous increase of turbulence intensity, which can enhance\nthe turbulent transport.\nThere are limitations to our model (i) We use the vacuum \feld approxi-\nmation and thus neglect the plasma response (ii) We do not explicitely treat\n10the spatial resonance aspect of the problem.\nIn conclusion, we found a new contribution to the zonal \row damping\ne\u000bect due to non-axisymmetric \feld. This contribution is proportional to\nthe square of the equilibrium E\u0002B\row shear, and may be important in the\npedestal region where E0\nris large. This additional damping of zonal \rows\nimplies a simultaneous increase of turbulence intensity, which can enhance\nthe turbulent transport.\nAcknowledgements\nThe authors would like to thank Z.X. Wang, M.J. Choi, W.H. Ko and J.M.\nKwon for usefull discussions. 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Safeer1,2,3, Mohamed-Ali Nsibi1, Jayshankar Nath1, Mihai Sebastian Gabor4, Haozhe Yang1, Isabelle Joumard1, Stephane Auffret1, Gilles Gaudin1, & Ioan-Mihai Miron1* 1Univ. Grenoble Alpes CNRS, CEA, Grenoble INP, SPINTEC, Grenoble, France 2CIC nanoGUNE BRTA, 20018 Donostia-San Sebastian, Basque Country, Spain. 3Department of Physics, Clarendon Laboratory, University of Oxford, Oxford, United Kingdom 4C4S, Physics and Chemistry Department, Technical University of Cluj-Napoca, Cluj-Napoca, Romania * To whom correspondence should be addressed. E-mail: mihai.miron@cea.fr Abstract Friction plays an essential role in most physical processes that we experience in our everyday life. Examples range from our ability to walk or swim, to setting boundaries of speed and fuel efficiency of moving vehicles. In magnetic systems, the displacement of chiral domain walls (DW) and skyrmions (SK) by Spin Orbit Torques (SOT), is also prone to friction. Chiral damping (ac), the dissipative counterpart of the Dzyaloshinskii Moriya Interaction (DMI), plays a central role in these dynamics. Despite experimental observation, and numerous theoretical studies confirming its existence, the influence of chiral damping on DW and SK dynamics has remained elusive due to the difficulty of discriminating from DMI. Here we unveil the effect that ac has on the flow motion of DWs and SKs driven by current and magnetic field. We use a static in-plane field to lift the chiral degeneracy. As the in-plane field is increased, the chiral asymmetry changes sign. When considered separately, neither DMI nor ac can explain the sign reversal of the asymmetry, which we prove to be the result of their competing effects. Finally, numerical modelling unveils the non-linear nature of chiral dissipation and its critical role for the stabilization of moving SKs. Introduction The observation of DMI1,2 in metallic multilayers has allowed significant advances in understanding the mechanism of the current induced DW motion3–6 as well as the development of the current induced SK dynamics7–14. These phenomena, potentially useful for applications are closely related. Their efficiency relies on two pillars: efficient SOT and strong DMI. In materials with broken inversion symmetry and with large spin orbit interaction, the electric current produces a damping like torque (DL)-SOT15,16, and the DMI effective field, deriving from a chiral energy contribution, imposes a Néel DW structure4 . In this configuration, the magnetization of the DWs is parallel to the current (Figure 1a), which makes the SOT most efficient in producing DW displacements. By analogy with DMI, which is a chirality dependent energy, the chiral damping is a chirality dependent dissipation. Several theoretical approaches have successfully predicted different possible origins, as well as diverse manifestations17–22. The main feature though, is that in practice, as the chirality is modified, the damping coefficient changes. A simple way to express this for the case of a DW in a perpendicularly magnetized material of interest here, is α!=α\"+α#$m&&&⃗$%·∇m&&&⃗&*. Here m&&&⃗$%·∇m&&&⃗& is the projection of the in-plane DW magnetization (m&&&⃗$%) on the normal to the DW position (∇m&&&⃗&) (Figure 1a, 1c). Because this projection can be negative, a1 must be larger than a2 to ensure that the damping always remains positive. Having the same symmetry and possibly driven by the same microscopic interactions, the effects of ac cannot be easily separated from those of DMI. The ability to isolate them experimentally is crucial for the fundamental understanding of chiral phenomena in magnetism23, as well as from the purely pragmatic perspective of applications. Dissipative processes are fundamental to enable high performance, energy efficient, computing and storage devices based on magnetic SKs and chiral DWs24–26. In this work, we separate the respective contributions of DMI and ac on the dynamics of DWs and skyrmionic bubbles and prove that the two phenomena compete (Figure 1a). In particular, ac modulates the DW and SK velocity and modifies the shape of the SK bubbles during their motion (Figure 1b), affecting their dynamical stability13,27. Results and discussion As a material platform for our study, we employ a Pt3 nm/Co0.6 nm/Pt1.5 nm tri-layer. The first reason is that in this material we have already observed evidence of chiral damping17. The second reason is that DMI is relatively weak, and allows controlling the chirality by moderate in-plane magnetic fields (Hip). We study the chiral dynamics in the flow regime, by measuring the displacements of DWs and SKs induced by nanosecond current or out-of-plane field pulses (see methods) using wide field Kerr microscopy. Our study covers all possible types of chiral dynamics: field induced DW motion (FIDM), current induced DW motion (CIDM) and current induced SK motion (CISKM). These dynamics are very different: FIDM can be either turbulent or steady motion, the CIDM has a single highly non-linear regime, while the CISKM includes 2D effects, not present in the case of CIDM. The chiral effects are extracted by monitoring the asymmetric motion of DWs with opposite polarity (up/down and down/up). When applying a sufficiently strong Hip, as the magnetization of both DWs aligns with the field, their chirality becomes opposite (Figure 1a). In this situation, the two DWs will have different energy (due to DMI) and different damping (due to ac). This reveals the competing effects of DMI and ac on the DW motion asymmetry in all experiments. Our analysis is supported by a numerical model, based on a q-f approach, (see supplementary information S1). The physical parameters used in the simulations are either measured independently or obtained from fitting the experimental data. Because the experimental features outnumber the free variables in the model, the parameters are uniquely determined (S1). The simulations reproduce accurately the ensemble of our experimental results, by using the same set of values for all cases. Micromagnetic simulations based on MuMax34 (Supplementary information S10) are further used to assess the influence of ac on the SK stability. For the sake of clarity, in sections I, II, and III we present separately the three different experiments. In section IV, we present theoretical predictions of the model, using parameter values required for applications. I) Current Induced DW motion The samples are patterned in the form of 10 µm long and 1 µm wide wires. The DWs are prepared using a perpendicular magnetic field. Under the effect of ns current pulses, the DWs shift in the direction of the current, indicating a left-handed DW chirality4. The DW velocity increases rapidly above 1·108A/cm2 (Figure 2c). This rapid increase characteristic of the thermally activated DW motion stops above 2·108A/cm2, signaling the limit of the flow regime. The saturation of the velocity observed above 2·108A/cm2 is characteristic of the SOT-DMI mechanism in the flow regime, typically observed for materials with weak DMI4. To evidence directly the chiral effects, we compare the displacements of up/down and down/up DWs as we apply Hip (See Figure S7 for the detailed Kerr images). The current density is fixed at 1· 108 A/cm2, sufficient to induce DW motion in the absence of Hip, but not to provoke nucleations at the largest Hip. Without Hip, both DWs move in the same direction, shifting the domain that they enclose (Figure 2a) along the current flow. As Hip is increased, the two DWs in the wire no longer move with the same velocity, leading to a contraction of the domain (Figure 2a; 2b). This is because Hip magnetizes them in the same direction. According to the DMI-SOT model it is expected that, as Hip becomes much larger than HDMI, their motion should be symmetric5,6, without any net displacement of the central domain (Figure 2b). Experimentally, the DWs indeed move symmetrically, but only at fields of approximately 80 mT. Above this value, the slow moving DW (blue arrow) becomes faster (Figure 2a; 2b), shifting the enclosed domain against the electric current. This is to say the unidirectional component of the motion has been reversed. Though the DW motion at Hip =0 is hindered by pinning, the reversed unidirectional motion is observed at large velocity (> 20 m/s) in the flow regime. Therefore, we can safely test the reversal of the unidirectional component of motion within our model. This result can only be reproduced by including both DMI and ac (light blue and red arrows in Figure 2b). In the absence of DMI there is no DW motion at Hip=0 (Figure S5), and without ac, the DW motion at large Hip is fully symmetric (orange and green arrows in Figure 2b). The reason for this dependence is that at large Hip, the influence of HDMI becomes negligible and the magnetization of the two DWs has the same orientation. Therefore, the effective field produced by SOT on the two DWs is identical. The reversal of the velocity asymmetry proves that DWs with opposite chirality have different damping. Thus, ac is required to explain quantitatively and qualitatively the DW dynamics. II) Field induced DW motion We start by measuring the DW velocity vs. the out-of-plane magnetic field (HZ) without applying Hip (Figure 3a). We prepare bubble domains in an un-patterned film by applying a small HZ field. To reach the flow DW motion and avoid the influence of pinning, we apply strong ( up to -217 mT) but short (40 ns) HZ pulses. The DW velocity (Figure 3a) increases linearly from 40 m/s to 70 m/s when HZ is larger than -110 mT, confirming the observation of the flow regime. For weaker HZ values, the velocity drops fast, indicating a significant DW pinning below 40 m/s. In agreement with the low DW mobility, the numerical fitting of this curve reveals that the motion of the DW is turbulent. This is consistent with the low Walker breakdown field characteristic for materials with weak DMI4. We then measure the asymmetry induced by Hip 17,28–30 on the velocity of up/down and down/up DWs (See Figure S8 for the detailed Kerr images). The effective in-plane field will be Hip+HDMI for down/up DWs, and Hip-HDMI for up/down DWs. Therefore, the presence of DMI is recognized by a lateral shift of the velocity vs Hip curves. To a first approximation, HDMI can be extracted directly from the magnitude of the lateral shift29,30. The measured DW velocity (Figure 3b) exhibits an almost parabolic dependence on Hip. The small lateral shifting of the curves indicates a small HDMI. Both the sign and magnitude of HDMI are consistent with the CIDM experiment presented above. However, at large Hip (> 150 mT) there is a deviation from this behavior: as the velocity curves for the up/down (blue) and down/up (red) DWs cross, the DW motion asymmetry reverses. The fact that the asymmetry reversal occurs at large velocity ( > 100 m/s), where the DW motion is in the flow regime, allows us to apply our model to reproduce this experimental feature. The simulations show that at low Hip, where DW motion is turbulent, because of the periodic changes of the chirality, ac has little effect on the DW velocity. As Hip exceeds a critical value required to stabilize the DW magnetization the DW periodic transformations cease. In this second regime of steady motion, the effect of chiral damping on the asymmetry is stronger than the opposing effect of DMI, leading to the change in sign. Indeed, if a1=0.4 and a1=0, in the steady regime the curves approach asymptotically but the crossing never occurs. A simple way to verify this scenario experimentally is to repeat the measurements at weaker Hz. In this case, a smaller Hip should be sufficient to restore the steady DW motion and achieve the asymmetry reversal. Indeed, the velocity curves measured at HZ= -55 mT exhibit a reversed asymmetry over an extended interval (Figure 3c), starting at Hip= 40 mT, while the positive asymmetry region is nearly suppressed. Once again, the model reproduces this tendency (Figure 3f). Because the DW motion at HZ= -55 mT starts to be affected by pinning, which is not included in the model, a better match between model and experiment is obtained using HZ= -10 mT in the calculation (inset of Figure 3f). This confirms that the reversal of the velocity asymmetry is caused by ac and occurs as the DW motion becomes steady at large Hip. III) Current induced bubble deformation Our next step is to evidence the effect of chiral damping on the current induced motion of the SK bubbles. For these experiments, the layers are patterned into 10 µm wide wires, large enough to contain the displacements of a single bubble (Figure 4a). The bubbles can undergo plastic deformations, as they are not stabilized by the dipolar energy or DMI, but by local pinning. In this sense, they differ from stable SKs, but they have an important advantage: the shape resulting from the deformation is stable over time. Therefore, it is possible to measure the current induced distortion of the bubbles. Such effects cannot be evidenced in stable SKs, which would return to their equilibrium position after the current is removed. The experiments involving SKs also exhibit the features that we have observed in the CIDM experiments present above: i) bubbles are initially shifted in the direction of the current; ii) as Hip increases, the bubble growth (shrinking) becomes predominant over the shifting; iii) at the largest Hip ( > 80 mT) the shift is reversed, and the bubbles displace in the opposite direction. The 2D geometry employed here allows to evidence two additional features. First, at zero Hip the motion is asymmetric with respect to the angle between the current and the DW, leading to a distortion of the initially circular bubble towards an elliptical shape. Second, when Hip is applied, the bubble not only shifts parallel to the current, it also shifts in the perpendicular direction. These distorted bubble displacements stem from the specific angular dependence of the DW velocity with respect to the electric current31–33. To verify whether these observations are consistent with our understanding of the FIDM and CIDM, we use the 1D model to calculate the velocity as a function of this angle. The shifting of the bubbles (white arrows in Figure 4a) is consistent with the asymmetry of the DW velocity predicted by the model (black arrows in Figure 4b), only if ac is included. This shows that ac is fully responsible for the bubble distortion observed at large Hip. IV) Model predictions Up to now, we have shown that all our experiments can be understood by including simultaneously the effects of both ac and DMI. Moreover, the successful fitting of the entire dataset has allowed to establish numerical values for the different parameters (see Methods). However, one should be cautious when further using these values. They are not real values, but rather “effective values” which are model dependent in the sense that they incorporate all the approximations made in the model. For example, we neglect the second order uniaxial anisotropy, the even components of chiral damping21 as well as the corrections to the gyromagnetic ratio19,22, we use an effective DW width; the temperature variation of the parameters with Joule heating is not considered (saturation magnetization, uniaxial anisotropy, DMI, SOT etc.). Therefore, these values should not be used to assess the strength of the microscopic interactions causing ac and DMI. Nevertheless, since the model is explaining well all our experiments, we can use it to distinguish the roles played in the dynamics by each individual parameter (Figure S5). Most importantly, we can also predict how these dynamics will evolve as these values change. In particular, we can focus on the range of values required for practical applications. From this perspective, there are two main requirements: i) In order to stabilize the rigid Néel DW structure or small size SKs, the DMI has to be much stronger than in the case of our experiments. The ceiling value for HDMI is imposed by the spontaneous formation of helical phases, which obstructs the stabilization of single DWs or SKs4. ii) Efficient current induced motion requires larger DL-SOT compared to the Pt/Co/Pt layers used here, but not too large compared to HDMI. In this sense, Figure 5a shows a calculation of DW velocity as a function of current density expressed as HDL/HDMI. The dependence can be approximately divided into two parts: in the low current regime, DW velocity increases with current, while in the second part, it reaches a plateau. The best efficiency of the CIDM is achieved at intermediate current (HDL/HDMI < 0.5) where the velocity is large, but not saturated. The numerical calculations indicate that the influence of the chiral damping will become stronger, as the DL-SOT and DMI approach the optimal values for applications (Figure 5b, c). The angular dependence of the DW velocity (black curve in Figure 5b, c), will tend to distort the SK circular structure affecting its dynamical stability. The chiral damping will reduce the distortion and improve the stability (blue curve in Figure 5b, c). On the contrary, if a2 changes sign, it will accentuate the SK deformation (red curve in Figure 5b, c). To understand the effect of the deformation on the SK stability, we perform micromagnetic simulations (S10), including chiral damping. We observe (Figure 5e) that as long as the effect of current is insufficient to destabilize the SK structure, ac modifies the SK velocity, but its influence on the anisotropic velocity is concealed by the rigidity of the magnetic structure. However, at larger driving force, close to the SK instability, the anisotropic velocity becomes relevant: ac can either limit or enhance the SK distortion, influencing its stability. NB. The values of the damping coefficient used in the simulations are intentionally smaller than those of the 1D model. This is imposed by our simplified numerical integration of the chiral damping in the MuMax, which becomes approximate for larger damping. Moreover, the definition of the micromagnetic chiral component of the damping (aµc) differs by a factor of 2 from its definition in the q-f model (see Supplementary information S10). Furthermore, the simulations of the SK trajectories (Figure 5d), allow to evidence the effects of ac by order of importance: 1st order: The effects of damping can be observed in the linear response regime, where the driving force does not modify the chirality of the magnetic texture significantly. In this case, ac, can be approximated by a constant value that depends on the chirality (Figure 5d). Note that it is very different compared to the Gilbert damping, which does not depend on the chirality. In a given material, the Gilbert damping coefficient is the same, regardless of the type of dynamics (uniform mode precession, SK motion etc…), while in presence of ac the effective damping depends on the chirality of the magnetic texture involved in the dynamics. 2nd order: As the driving force increases, the chiral texture distorts, and ac creates non-linear damping (Figure 5e). The SK trajectories deviate significantly from those obtained using a constant value. In this case, the damping varies in time and space (along the SK perimeter), and the evolution of the micromagnetic structure cannot be reproduced using a constant damping. In conclusion, by performing a full set of experiments of DW and SK dynamics in Pt/Co/Pt, we were able to observe how chiral dissipation influences the DW motion in the flow regime, and also to understand how it affects the distortion of skyrmionic bubbles. In all experiments, we observe a reversal of the chiral asymmetry when an in-plane field is applied. The numerical model reproduces this behavior only if both ac and DMI are included. In this case, all experimental features are comprehensively reproduced using a unique set of physical parameters. By extrapolating our numerical model to the range of large SOT and strong DMI, which is more suitable for applications, we evidence a strong influence of ac on the SK dynamics and stability. Understanding and controlling ac is thus essential for engineering devices employing DWs or SKs. Methods FIDM: We prepare DW bubbles in the un-patterned film by applying a small perpendicular field (5 mT for 1 s). Then we apply nanosecond field pulses (up to 217 mT for 40 ns) and extract the velocity from the DW displacements. Because of its small size and of the short pulses, the magnetic field of the micro-coil cannot be measured directly. It is calibrated by comparing low field DW velocity produced by the micro-coil to low field DW velocity produced by a macroscopic, calibrated coil. CIDM: In order to achieve large current densities, we pattern the samples into micrometer sized wires. The DWs are prepared using a perpendicular magnetic field. Nanosecond current pulses with current densities ranging from 0.1·108 A/cm2 to 2·108 A/cm2 are used to propagate the DWs in the micro-wires (Figure 2a). The velocity is determined at every current density from the linear dependence of the DW displacement vs pulse length. When measuring the CIDM in presence of Hip, we cannot use the same procedure as described above. In these experiments, the Joule heating together with Hip cause unwanted nucleations limiting the range of pulse amplitude and duration that we can use. Since the range of current pulses is reduced, we cannot quantify the DW velocity by the linear regression of the displacement vs. pulse length. Therefore, we have evidenced the chiral effects using the following procedure: 1. We set the current density to a value strong enough to produced DW motion for Hip= 0, but not so strong to produce nucleations for the largest Hip (145 mT). 2. The exact pulse width and pulse sequence (number and frequency) are optimized for every Hip value in order to maximize the displacements, while avoiding nucleations. 3. As the current density remains constant, when Hip is decreased, the number and frequency of pulses have to be increased in order to maintain the size of the measured displacements. This is a consequence of the weaker driving force provoking more frequent DW pinning to defects. 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Angular dependence of current-driven chiral walls. Appl. Phys. Express 9, 063008 (2016). 34. Vansteenkiste et al. “The design and verification of MuMax3”, AIP Adv. 4, 107133 (2014) Acknowledgements We thank J. Vogel and S. Pizzini for their help with the MOKE imaging and the micro-coil experiments, as well as for their critical reading and commenting on the manuscript. We acknowledge funding for this work from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 638653 – Smart Design). MSG acknowledges funding from MRI-CNCS/UEFISCDI through grant PN-III-P4-ID-PCE-2020-1853. Figure 1. Effect of chiral damping on the DW and SK dynamics a. Schematic representation of current induced motion of chiral DWs by SOT. The DWs with the same chirality (set by DMI) are displaced by the same amount (D represented by the dotted arrows) and in the same direction by the current induced SOT. When an external in-plane field sets the magnetization of the two DWs in the same direction, it imposes opposite chiralities. In this case, the two DWs still move by the same amount, but in opposite directions (D and -D). In the presence of chiral damping, the two DWs with different chirality experience different damping (red and blue). This engenders different mobility, which causes the different magnitude of the opposite displacements (D′ and -Dʺ). b. Kerr images of magnetic bubble motion driven by electric current pulses (5000 · 20 ns at 1·1012A/cm2) in Pt/Co/Pt tri-layers. The two initial states correspond to bubbles magnetized either up (top-left) or down (top-right). The black dotted lines are guides for the eye indicating the initial bubble position. The orange dotted lines indicate the lateral edge of the magnetic pad. As seen in the differential Kerr images (below), the bubbles are displaced along the direction of the electric current (white arrow), but also undergo a chiral distortion (from circular to elliptical; the orientation of the ellipse depends on the chirality). The white dotted lines indicate the final state of the bubble. c. DW magnetization profile (black arrows) along the bubble perimeter under the effect of DMI and horizontally applied electric current (HDMI =200 mT; the damping-like and field-like components of SOT are HFL=33 mT and HDL=66 mT). The dark/bright grey indicates the up/down magnetization. This image is to illustrate that as the chirality changes along the perimeter of the bubble, the value of the chiral damping (color scale) will also vary, leading to an angular dependent DW mobility. \n Figure 2. Current induced DW motion a. Differential Kerr images showing CIDM for different values of Hip, shown in white on each image. Dark and white contrast indicates that up/down and down/up DWs are displaced in the same direction. Red and blue lines mark the starting positions for the DWs with opposite up/down polarities. The current density is fixed at 1·108A/cm2. The pulse sequences used for each image are, from top to bottom: 10000 · 2 ns; 1000 · 1.5 ns; 100 · 1.5 ns; 200 · 1.2 ns; 300 · 1 ns b. Comparison of relative DW displacements extracted from the MOKE images (bold red and blue arrows) with numerical simulations. Since for every value of Hip in the experiment we had to use a different pulse sequence, it is impossible to extract accurately the DW velocity. Therefore, to evidence chiral effects, we compare the respective displacement of the two DWs by normalizing to the value of the positive displacement (in red). Orange and green arrows correspond to the scenario of a1=0.4 and a2 = 0. Red and blue arrows correspond to a1=0.4 and a2 = 0.2. In the absence of chiral damping, the displacements at large Hip become symmetric. The DWs move by the same amount in opposite directions. The reversal of the asymmetry observed experimentally, with the backward moving DW displacing a larger distance (blue arrows), is only reproduced in our model if we include the chiral damping. c. DW velocity as a function of current density (black symbols) for Hip=0. The numerical model reproduces well the data in the range of high current density, above 2·108A/cm2 and 20 m/s, within the boundaries of the flow regime. The dotted line is an exponential fit of the velocity in the intermediate range of current from 1·108A/cm2 to 2·108A/cm2. It is a guide for the eye to visualize the separation of the flow regime from the thermally activated DW motion. \n Figure 3 Field induced DW motion a. The experimentally measured DW velocity as a function of HZ (black symbols). The lines are the results of the model with DMI only (black) or including ac (blue and red). The model reproduces accurately the velocity data in the flow regime (the region with constant DW mobility). The inset shows a typical differential Kerr image. The bright ring represents the DW displacement produced by the field pulse. b. Measured DW velocity vs. Hip (red for down/up DW, blue for up/down DW, see Figure S8 for detailed Kerr images) at constant HZ= -82 mT and c. HZ=-55 mT. The inset is a zoom of the low field region where the crossing of two curves occurs. d, e, f, Calculated DW velocity with (d) HZ=-82 mT and ac=0.4; (e) HZ=-82 mT and ac=0.4+0.2·mx. (f) HZ=-55 mT and inset HZ=-10 mT \n Figure 4. Chiral distortion of skyrmionic bubbles. a. Kerr images of the bubble displacement under the effect of current pulses (20 ns at 1·108A/cm2) and Hip. On the first row we show images of a bubble magnetized up, on the second row, a bubble magnetized down. The first column shows the direct image of the initial bubble prepared by applying HZ. The rest of the columns show differential images of displacements induced by the current for different values of Hip (from left to right: 0, 10, 20, 37, 65, 120, 145 mT). White dotted lines indicate the contours of the initial and final bubble. The white arrows point towards the direction of bubble displacement. The number of pulses is different for each image (top: 5000; 30000; 10000; 2000; 1500; 360; 120; bottom: 20000; 50000; 2500; 5000; 400; 100; 100;). b. Polar plot of computed DW velocity as a function of the angle between the DW orientation and electric current, corresponding to the same current density and Hip values as the experimental data. In the top row, we show in red results from a scenario with DMI but without ac. The bottom row shows in blue results including both ac and DMI. From the velocity dependence, we extract the direction of bubble displacement indicated by the black arrows. They are in agreement with the white arrows on the MOKE images that indicate the experimental bubble displacement. \n Figure 5. Modelling of the SK distortion in presence of the chiral damping a. CIDM plotted as a function of the ratio of HDL/HDMI. The dotted line approximately separates the linear variation regime from the plateau. Here, HDMI= 0.2 T and HDL=2·HFL. The values chosen for these simulations are close to the optimal values and very importantly, they are realistic (typical for Pt/Co/AlOx systems). b. angular dependence of the velocity in the linear regime c. angular dependence of the velocity in the saturation regime. (black: a1=0.4; blue: a1 =0.4, a2 =0.2; red: a1 =0.4, a2 = -0.2) d. Skyrmion position as a function of time for different current and damping values. e. Superposed images of the out-of-plane component of the micromagnetic structure at different moments of time during the simulation. On each image, the SK on the left corresponds to the initial state. At low current density (top row) the SK structure remains intact during the displacement. Chiral damping only influences the trajectory. At larger currents, at first the SK undergoes a slight distortion, which in the case of a1 = 0.2 and a µ2 = – 0.05, leads to its instability and the SK explodes. Here the electric current is applied at 45° with respect to the stripe major axis. \nSupplementary information S1. Numerical model S2. Influence of HDMI S3. Influence of ac S4. Dipolar field of the DW S5. Temperature S6. Effect of Hip on the DW motion asymmetry S7. Limit of the 1D model for 2D dynamics: the local approximation S8. MOKE images of CIDM in nanowires in presence of Hip S9. MOKE images of FIDM of a bubble in presence of Hip S10. Micromagnetic simulations S1. Numerical model The DW motion at large driving forces is independent of material imperfections. For this reason, since the complexity of dealing with pinning and de-pinning effects is removed, this motion regime can be modelled more easily. Analytical models have been used successfully for materials with in-plane and out-of-plane anisotropy. Two motion regimes have been identified: steady DW motion, occurring at relatively low DW velocity, and turbulent motion at larger fields. In the steady regime, the DW structure is increasingly distorted as the velocity increases. Beyond a critical velocity, the DW internal structure is no longer stationary and transforms periodically. In materials with perpendicular anisotropy, such as the ones used in the present study, the DW dynamics can be modeled by a few equations1–3. The field driven DW velocity in the steady regime: in the turbulent regime: The Walker critical field separating the two regimes: 𝐻!\t\t≈\t𝛼\t%𝐻\"#$+12⁄\t𝐻\"%&*\t\t\t (3)\tThe current driven DW velocity in the SOT-DMI model4: 𝑣'\t=(\t∆\t+!\"#,\"-./$∙\t%\t'\t(!\"#\t)∙\t+!,-./0\t11\t (4) With: Here, g is the gyromagnetic ratio; a is the damping; D is the DW width ; HDMI is the DMI field; HDip is the dipolar field associated to the Néel DW structure; C0 is a constant depending on physical constants and sample parameters; j is the current density; xDL-SOT is the efficiency of the damping like component of the SOT; Hip is the external in-plane field, HK is the anisotropy field of the sample. We note that the dynamics are determined by a relatively small number of parameters. For the modelling of the DW motion, due to the complexity of our experiments, which include both magnetic fields and electric current, applied simultaneously at different angles, the numerical modelling is more convenient than analytical calculations. For this, we have chosen to model the DW motion using a collective coordinates q-f approach7. This has the advantage of providing accurate results for cases where analytical solutions may be too complex. To ensure the validity of our numerical model, we have tested it for the simpler cases with well-known analytical results described by the above equations. v234567\t\t=\t8\t∆9𝐻& (1) \n𝑣:;<=;>?@:\t\t=\tA\t(\tB\"-A1𝐻& (5) 𝑣:;<=;>?@:\t\t=\tA\t(\t∆\"-A1𝐻& (2) Such numerical models have been widely used to study the DW dynamics. Their advantage over micromagnetic simulations is that the computation time is much faster. At the same time, they yield the same physical results in the vast majority of situations7. The reason for this is that the approximation that they rely on, that of a rigid DW as a solitonic quasiparticle, is sufficient in most instances. Moreover, since they rely on effective parameters, the results can be interpreted easily, and the influence of each physical parameter can be identified directly. Our numerical approach includes both the chiral energy and chiral damping. Chiral energy appears as an effective magnetic field acting on the core magnetization of the DW. It has been shown theoretically that the chiral damping can take different forms: it can have a high-order dependence8 on mx and it can even affect the gyromagnetic ratio5. Our model only uses the simplest form (ac=a1+a2·mx), which is sufficient to account for all the chiral features seen in our experiments. We note that while higher order contributions may exist and can affect the dynamics, disentangling the second order effects, or the different forms of chiral damping is beyond the purpose of our present work. Non-chiral parameters such as the current induced torques, the DW dipolar field, domain wall width are included. The effect of disorder is included in the form of temperature fluctuations. The current induced torques as well as the anisotropy field were determined experimentally using independent (second harmonic) measurements. xDL-SOT = 0.017 (T/1012 A/m2), xFL-SOT = 0.004 (T/1012 A/m2). The values that fit best our results (D = 6 nm; HDMI = 30 mT; HDip = 30 mT; ac = 0.4+0.2·mx) are uniquely determined and allow to model simultaneously the entire range of different experiments. The exact values of the non-chiral parameters affect little the general shape of the curves and have a weak influence on the emergence of chiral features (S6, S7). The fitting procedure The model is based on 5 independent fitting parameters, 2 fixed measured parameters: Independent: a1, a2, HDMI, D, HDip. Measured: xDL-SOT, xFL-SOT. Equations (1) – (5) show that these are all the parameters that can influence the DW motion. We fit the following experimental features: - The value of the velocity saturation of the CIDM in the flow regime (Figure 2c) - The value of the DW mobility in the flow regime of FIDM (Figure 3a) - Reversal of the unidirectional component of the CIDM (Figure 2a, b) - Reversal of the chiral asymmetry of the FIDM (Figure 3b, c, d, e, f) - Asymmetry of the current-induced 2D bubble distortion produced by Hip (Figure 4) From previous measurements in the creep regime we can apply more constraints on the chiral damping and DMI. From the magnitude and the saturation field of the anti-symmetric component of the DW motion in the creep regime6, we had previously determined that a2/a1 ≈ 0.5 and HDMI + HDip ≈ 50 mT. When including these additional constraints, the total number of independent experimental features (7) exceeds the number of free parameters (5). This means that the values that we extract for these parameters are uniquely determined. To determine their numerical values, we varied the values manually to improve the fit. After a few iterations, we converged toward a set of values that reproduce reasonably well all the different experiments simultaneously. In order to illustrate the influence of each parameter on the dynamics, in the following we will show how the independent variations of each parameter affect the results. S2. Influence of HDMI HDMI field models the effect of the DMI interaction on the DW magnetization. In a q-f model, this field either points in the direction of DW motion, or is opposed to it, thereby favoring the Néel type DW configuration. It has a drastic effect on current induced DW motion, as it tends to align the magnetization with the electric current, thus making the DL-SOT more efficient in producing DW displacements (Figure S1a). Moreover, by stabilizing the DW structure, it also increases the HW value (Figure S1b) for the field driven DW motion. \n Figure S1. a. Current induced DW velocity, b. field induced DW velocity, calculated for three different HDMI values. S3. Influence of ac For the field induced DW motion in the turbulent regime, there is a periodic change of the chirality, thus canceling the effects of the chiral dependent damping. In the case of current induced motion, the insensitivity to damping is due to the fact that the ratio HDL-SOT/HDMI >1. This is seen directly in equation (4): when the HDL-SOT is significantly larger than HDMI, as it is in Pt/Co/Pt tri-layers, the velocity becomes less sensitive to the damping value4. \n Figure S2. a. Current induced DW velocity, b. Field induced DW velocity, for different ac values S4. Dipolar field of the DW This effective field models the dipolar energy difference between the Bloch and Néel configurations. It will increase the stability of the Bloch DW structure and enhance the DW stability. Consequently, the increase of HDip will tend to lower the efficiency of the CIDM and to increase the Walker breakdown field. \n Figure S3. a. Current induced DW velocity, b. Field induced DW motion, for different HDip S5. Temperature By increasing the disorder, the temperature lowers the mx component of the magnetization, thus reducing the efficiency of the CIDM. For the FIDM, the main effect of temperature is to smoothen the sharp features of the curves, such as Walker breakdown. Outside the range where the sharp features occur, the temperature value has little influence. The temperature is fixed at 300 K, while the “macrospin” volume subjected to temperature fluctuations is set to 0.5×6×10 nm for all the calculations. These dimensions correspond to the 0.5 nm = layer thickness, 6 nm= the DW width, and 10 nm = 2.5×lex (the exchange length) an estimate of \nthe length over which the thermal fluctuations do not disrupt significantly the parallel orientations of the spins9. \n Figure S4. a. Current induced DW velocity, b. Field induced DW velocity for different temperatures S6. Effect of Hip on the DW motion asymmetry a) FIDM First we analyse the field induced DW motion in absence of any chiral effects (Figure S6). Hip increases the DW velocity as the motion passes from turbulent to steady. The velociy has an even dependence on Hip; the effects for the two DW polarities are identical. Moreover, Hip has no effect on the steady motion regime. The main effect of DMI is to compete with Hip, shifting the curves for the two polarities in opposite directions. It is only when we include the chiral damping that the velocity asymmetry changes sign in the steady regime. However, ac alone can not produce a significant asymmetry in the turbulent regime. In order to reproduce the reversal of the asymmetry observed experimentally, it is required to include both ac and DMI. b) CIDM In the absence of chiral effects, the velocity curves for the two DWs reflect the symmetry of the SOT: the DWs do not move at Hip = 0; when Hip ≠ 0 the two DWs move in opposite directions with the same velocity. The main effect of DMI is to shift the curves in opposite directions. Consequently, the DWs move even without Hip. However, when Hip becomes sufficiently strong, the DWs still end up moving with the same velocity in opposite directions. In order to obtain different DW velocities at large Hip, we need to introduce ac. On the other hand, ac alone cannot be responsible for the DW motion at zero field. Both ac and DMI are required to reproduce the experimental observations. \n Figure S5. (left column) Effect of Hip on field driven motion (HZ= 82 mT). We consider all the possible scenarios for the presence of chiral effects: no chiral damping and no DMI; only DMI; only chiral damping; both chiral damping and DMI. Temperature is also included. (right column) Effect of Hip on current driven motion (j=108A/cm2). We consider the same scenarios as for the field driven motion. \n S7. Limit of the 1D model for 2D dynamics: the local approximation We have extended the use of the 1D model to study the motion and distortion of the 2D bubbles under magnetic field and current. For this we have performed 1D calculations for all the angles of the DW canting with respect to the current and the applied field, and from here we have reconstructed the angular dependence of the velocity vs. DW angle. This approach is not exact, because the DW behaves similarly to an elastic membrane so that the behavior at a given angle is influenced by the DW behavior at adjacent positions. For this reason, the measured data must be considered with care when comparing with models. Nevertheless, experimentally we observe that the bubbles can have complex shapes (Figure S6); the shape depends on the magnetic history and is weakly influenced by the DW elastic energy. This can be understood when considering that the bubbles are relatively large and the pinning relatively strong. When the pinning energy overcomes the elasticity, the bubbles can undergo plastic deformation, and the different sections of the bubble evolve almost independently of the rest. In this case, the 1D approximation is sufficient to describe their deformation. \n Figure S6. Differential Kerr images of bubble growth. The shape of the initial bubble is extracted from each image and depicted on the right column. The bubble can have vastly different shapes, depending on \nthe precise sequence of magnetic field and current that was used to create it. This indicates an important plasticity of the bubble perimeter. S8. MOKE images of CIDM in nanowires in presence of Hip \n Figure S7. Differential Kerr images showing CIDM in the presence of Hip= 40 mT, 80 mT, and 120 mT for 4 different combinations of current and Hip directions, depicted above the images. The motion of down/up and up/down DWs are schematically shown with red and blue arrows respectively. For each case, we clearly observe that the asymmetric DW motion at Hip= 40 mT along one direction becomes almost symmetric at Hip= 80 mT. At Hip= 120 mT, the DW motion becomes asymmetric again, but the asymmetry direction is opposite to that observed at Hip= 40 mT. \n S9. MOKE images of FIDM of a bubble in presence of Hip \n Figure S8. a) Differential Kerr images showing bubble expansion for Hz= -82 mT and -55 mT for different values of Hip. The red and blue arrows schematically represent down/up and up/down DW displacements. The velocities extracted from these images were used in the plots shown in Figure 3 of the main text. Asymmetry= C2$34/67DC67/2$34(C2$34/67-C67/2$34)/#, calculated from the images b) at Hz= -82 mT c) at Hz= -55 mT. For both cases, the asymmetry direction reverses as Hip increases as explained in Figure 3 of the main text. \nS10. Micromagnetic simulations The micromagnetic simulations were performed using the MuMax3 micromagnetic simulation software10,11. The chiral damping was implemented using the custom fields feature of MuMax3. As such, it is not included directly in the dissipative term, but rather as an effective field. The Landau–Lifshitz–Gilbert (LLG) governing the magnetization dynamics: 𝑑𝒎𝑑𝑡=−𝛾𝒎×𝑯?HH+𝛼𝒎×𝑑𝒎𝑑𝑡 can be transformed to: 𝑑𝒎𝑑𝑡=−𝛾1+𝛼#𝒎×8𝑯?HH+𝛼$𝒎×𝑯?HH*9 where γ is the gyromagnetic ratio and 𝑯?HH=−\"I8J9KLK𝒎 is the effective field. Here, the energy includes the exchange, dipolar, anisotropy, DMI and SOT terms. Adding the chiral damping as an effective field is correct up to the renormalization of the gyromagnetic factor (1+a2)-1. Using this approximate implementation, we cannot model exactly the experimental results in our samples, because the experimental damping is too large for this approximation to be applicable. Nevertheless, this correction becomes negligible when the damping is significantly smaller than one (so a2 becomes negligible relative to unity). For this reason, in the micromagnetic simulations we use small values of damping. The comparison to experiments is thus not quantitative, but qualitatively correct. Including the chiral damping the LLG reads: 𝑑𝒎𝑑𝑡=−𝛾1+𝛼\"#𝒎×8𝑯?HH+𝛼\"$𝒎×𝑯?HH*+𝜙/$𝒎×𝑯?HH*9 where the last term in the brackets represents the chiral damping effective field and 𝜙/ is given by4: 𝜙/=𝛼N#𝛥<𝑚O𝜕𝑚&𝜕𝑥−𝑚&𝜕𝑚O𝜕𝑥+𝑚P𝜕𝑚&𝜕𝑦−𝑚&𝜕𝑚P𝜕𝑦A where we introduce a characteristic exchange length Δ=CQR:;;\t with 𝐴 the exchange constant and 𝐾?HH the effective anisotropy, and 𝛼N# the chiral component of the damping coefficient. Note that the micromagnetic definition of chiral damping is different from the q - f model definition. In the q - f model the chirality is expressed only with regard to the in-plane magnetization, while in the micromagnetic simulations the chirality is expressed both using the in-plane as well as the out-of-plane components. For this reason, equivalent results are obtained when the numerical value of chiral damping in the q - f model is approximately a factor of 2 larger than the value used in the micromagnetic simulations. For the simulations we used cells with a size of 1 nm. We chose realistic material parameters that were shown to stabilize skyrmions at room temperature in Pt/CoFeB/MgO layers12. Explicitly, the saturation magnetization 𝑀S=1.12×10T\t𝐴𝑚D\", perpendicular uniaxial anisotropy 𝐾;\"=10T\t𝐽𝑚DU, exchange constant 𝐴=10×10D\"#\t𝐽𝑚D\", the DMI constant 𝐷=1.5\t𝑚𝐽𝑚D#, the spin Hall angle 𝜃V+=0.07 and the ratio between the field-like and the damping like SOTs 𝜉=0.5. During the SK simulations, an out-of-plane field of -10.4mT was applied. Before studying the effect of chiral damping on the SK motion and their stability, we used the simulations to reproduce the main features of our experimental results. The goal is to validate the micromagnetic model that we will use further for studying the effect of chiral damping on the SK dynamics. It is to be noted that in this case we have used a lower DMI energy equivalent to a DMI field of 𝐻WJX=10.4\t𝑚𝑇. The simulations reproduce well all the characteristic features observed experimentally and exhibit a very good qualitative agreement with our q-f model (Figures S9 and S10 compared to Figure S5). This is an important point because the micromagnetic simulations require fewer assumptions compared to the q-f model. For example, they automatically account for the changes of the DW structure (such as the variation of the DW width) during the application of the electric current and the magnetic fields. Furthermore, to test the robustness of our conclusions, we also used different values of DMI and chiral damping (Figure S9, Figure S10). They confirm that the competition between DMI and chiral damping asymmetries is a general effect, occurring throughout the entire parameter space; the asymmetry reversal only occurs when the chiral damping is sufficiently strong to overcome the asymmetry produced by DMI. Figure S9. Reversal of the asymmetry of the Field Induced DW motion (HZ = 30 mT). a. HDMI = 0, a1 = 0.2, a µ2 = 0. b. HDMI= 10.4 mT, a1 = 0.2, a µ2 = 0 c. HDMI = 0, a1 = 0.2, a µ2 = 0.05. d. HDMI = 10.4 mT, a1 = 0.2, a µ2 = 0.05, e. HDMI = 30 mT, a1 = 0.2, a µ2 = 0.05, f. HDMI = 30 mT, a1 = 0.2, a µ2 = 0.1 \n Figure S10. Reversal of the asymmetry of the Current Induced DW motion (j=2×1011A/m2) a. HDMI = 0, a1 = 0.2, a µ2 = 0. b. HDMI= 10.4 mT, a1 = 0.2, a µ2 = 0 c. HDMI = 0, a1 = 0.2, a µ2 = 0.05. d. HDMI = 10.4 mT, a1 = 0.2, a µ2 = 0.05, e. HDMI = 30 mT, a1 = 0.2, a µ2 = 0.05, f. HDMI = 30 mT, a1 = 0.2, a µ2 = 0.1 \n Figure S11. Effect of the chiral damping on the Skyrmion dynamic stability. a. Evolution of the area of a dipolar stabilized skyrmion (seen in figure 5 of the main text) as a function of time for different chiral damping scenarios at 1.0×1012 A/m2 . b. Area of a smaller, skyrmion stabilized by DMI (seen in figure 5 of the main text) as a function of time at 1.7×1012 A/m2. c. Micromagnetic simulation of the deformation of the small topological skyrmion for different chiral damping scenarios. In order to evidence the effect of the chiral damping on the SK stability we stabilize skyrmions with an equilibrium diameter of 40 nm and we simulate the effect of the SOT as a function of the current density for different values of damping. The first effect of chiral damping is to provoke a variation of the SK velocity. Second, the damping influences the distortion of the skyrmion’s circular shape. This is in agreement with the 2D velocity charts (Figure 5) calculated using the q-f model. The magnitude of the distortion is established by the competing effect of the asymmetric DW velocity and SK energy. If the SK is not sufficiently stable or the velocity asymmetry is too large, the SK size can diverge. In order to quantify this effect we use the total SK size as an indicator of the stability. In Figure S11 we plot the time evolution of SK size for a “large” dipolar stabilized SK as well as for a “small” SK stabilized by DMI. The majority of experimental studies realized at room temperature until now use the dipolar stabilized SKs. However, smaller skyrmions can be stabilized by using their topological protection. In our simulations, we were able to produce such “small” skyrmions, by slightly increasing the exchange coupling (𝐴=12×10D\"#\t𝐽𝑚D\"). We confirm that in this case too, the chiral damping has a strong influence on the stability (Figure S11). References 1. Thiaville, A., Garcia, J. M. & Miltat, J. Domain wall dynamics in nanowires. 245, 1061–1063 (2002). 2. Slonczewski, J. C. DYNAMICS OF MAGNETIC DOMAIN WALLS. 5, 170 (1972). 3. Schryer, N. L. & Walker, L. R. The motion of 180 ° domain walls in uniform dc magnetic fields. 45, 5406 (1974). 4. Thiaville, A., Rohart, S., Jué, É., Cros, V. & Fert, A. Dynamics of Dzyaloshinskii domain walls in ultrathin magnetic films. Europhys. Lett. 100, 57002 (2012). \n5. Akosa, C. A., Takeuchi, A., Yuan, Z. & Tatara, G. Theory of chiral effects in magnetic textures with spin-orbit coupling. Phys. Rev. B 98, 184424 (2018). 6. Jué, E. et al. Chiral damping of magnetic domain walls. Nat. Mater. 15, 272–277 (2016). 7. Boulle, O., et al. \"Current induced domain wall dynamics in the presence of spin orbit torques.\" Journal of Applied Physics 115.17, 17D502 (2014). 8. Kim, J. Von. Role of nonlinear anisotropic damping in the magnetization dynamics of topological solitons. Phys. Rev. B 92, 014418 (2015). 9. Abo, G. S., et al. \"Definition of magnetic exchange length.\" IEEE Transactions on Magnetics 49.8 4937-4939 (2013) 10. Vansteenkiste et al. “The design and verification of MuMax3”, AIP Adv. 4, 107133 (2014) 11. Mulkers et al., “Effects of spatially engineered Dzyaloshinskii-Moriya interaction in ferromagnetic films”, Phys. Rev. B 95, 144401 (2017) 12. Felix Büttner. et al.\" Field-free deterministic ultrafast creation of magnetic skyrmions by spin–orbit torques\" Nat. Nano. 12, 1040 (2017) " }, { "title": "1009.4871v1.Spatial_Damping_of_Propagating_Kink_Waves_in_Prominence_Threads.pdf", "content": "arXiv:1009.4871v1 [astro-ph.SR] 24 Sep 2010SPATIAL DAMPING OF PROPAGATING KINK WAVESIN\nPROMINENCE THREADS\nR. Soler, R. Oliver,and J. L.Ballester\nDepartamentdeF´ ısica,Universitatdeles IllesBalears,E -07122,PalmadeMallorca,Spain\nroberto.soler@uib.es\nABSTRACT\nTransverse oscillations and propagating waves are frequen tly observed in threads\nof solar prominences /filaments and have been interpreted as kink magnetohydrody-\nnamic (MHD) modes. We investigate the spatial damping of pro pagating kink MHD\nwavesintransverselynonuniformandpartiallyionizedpro minencethreads. Resonant\nabsorption and ion-neutral collisions (Cowling’s di ffusion) are the damping mecha-\nnismstakenintoaccount. Thedispersionrelationofresona ntkinkwavesinapartially\nionized magnetic flux tube is numerically solved by consider ing prominence condi-\ntions. Analytical expressions of the wavelength and dampin g length as functions of\nthe kink mode frequency are obtained in the Thin Tube and Thin Boundary approxi-\nmations. For typically reported periods of thread oscillat ions, resonant absorption is\nanefficientmechanismforthekinkmodespatialdamping,whileion -neutralcollisions\nhave a minor role. Cowling’s di ffusion dominates both the propagation and damping\nfor periods much shorter than those observed. Resonant abso rption may explain the\nobserved spatial damping of kink waves in prominence thread s. The transverse in-\nhomogeneity length scale of the threads can be estimated by c omparing the observed\nwavelengths and damping lengths with the theoretically pre dicted values. However,\nthe ignorance of the form of the density profile in the transve rsely nonuniform layer\nintroducesinaccuracies in thedeterminationoftheinhomo geneitylengthscale.\nSubject headings: Sun: oscillations – Sun: filaments, prominences – Sun: coron a –\nMagnetohydrodynamics(MHD)– Waves\n1. INTRODUCTION\nWaves and oscillatory motions are frequently reported in th e observations of solar promi-\nnences and filaments (see reviews by Oliver& Ballester 2002; Ballester 2006; Engvold 2008;– 2 –\nMackay et al. 2010). In high-resolution observations, the p rominence fine structures (threads) of-\ntendisplaytransverseoscillationsofsmallamplitude(e. g.,Linet al.2005,2007,2009;Okamotoet al.\n2007;Ninget al.2009),whichhavebeeninterpretedaskinkm agnetohydrodynamic(MHD)waves\n(e.g., D´ ıazet al. 2002; Dymova& Ruderman 2005; Terradas et al. 2008; Lin etal. 2009). The ob-\nserved threads in H αimages are between 3,000 km and 28,000 km long, and between 10 0 km\nand 600 km wide (Lin 2004; Linet al. 2008). The threads outlin e part of much larger magnetic\nflux tubes which are probably rooted in the solar photosphere . The majority of observed periods\nof transverse thread oscillations roughly range between 1 m in and 10 min, but a few detections\nof longer periods of about 20 min have been also informed (e.g ., Yiet al. 1991; Lin 2004). The\nwavelengths are usually between 700 km and 8,000 km, althoug h values up to 250,000 km have\nbeen reported (Okamotoet al. 2007). Recently, Soleret al. ( 2010a) pointed out that the short pe-\nriods and wavelengths are consistent with an interpretatio n in terms of propagating waves, while\nperiods larger than 10 min and wavelengths longer than 100,0 00 km could correspond to stand-\ning oscillations of the whole magnetic tube. In the case of st anding oscillations, the value of the\nwavelengthisnotstronglyinfluencedbythethreadproperti esbutismainlydeterminedbythetotal\nlength of the magnetic tube, since the fundamental kink mode wavelength is twice the length of\nthe tube, approximately (see details in Soler et al. 2010a). Although there are no direct measure-\nments of the length of prominence magnetic tubes, this param eter is estimated around 105km.\nThisroughestimationis inagreement withthewavelengthsr eported byOkamotoet al. (2007). In\naddition, a common feature of the observations is that the os cillations are strongly damped (e.g.,\nTerradas et al. 2002; Ninget al. 2009;Lin et al.2009).\nMotivatedbytheobservationalevidence,greate fforthasbeenrecentlydevotedtothetheoret-\nicalstudyofbothtemporalandspatialdampingofMHDwavesi nprominenceplasmas. Temporal\ndamping is investigated for waves with fixed wavelength, whi le spatial damping is studied for\npropagating waves with fixed frequency. Both phenomena have been extensively investigated in\nunbounded and homogeneous prominence plasmas by assuming d ifferent damping mechanisms\n(e.g.,Carbonell et al.2004,2006,2010;Forteza etal.2007 ,2008). ThereaderisreferredtoOliver\n(2009),Arregui &Ballester(2010),andreferencestherein foracompleteaccountofthetheoretical\nworks.\nInthecaseofprominencethreadoscillations,workssofarh avefocusedontemporaldamping\nbymechanismsas,e.g.,non-adiabatice ffects(Soleret al.2008),ion-neutralcollisions(Soleret a l.\n2009b), and resonantabsorption(Arreguiet al. 2008, 2010; Soleret al.2009a,c, 2010a). Thecon-\nclusions of these works indicate that resonant absorption i s efficient enough to provide realistic\nkink mode damping times consistent with the reported strong damping, whereas non-adiabatic\neffects are negligible and ion-neutral collisions are only imp ortant for shorter wavelengths than\nthoseobserved. Inthecaseofspatialdamping,P´ ecseli & En gvold(2000)studiedthee ffectofion-\nneutralcollisionsbutrestrictedthemselvestoAlfv´ enwa vesandkinkmodeswerenotinvestigated.– 3 –\nAlthough spatial damping of kink waves has been studied in th e context of coronal loops (e.g.,\nPascoeet al. 2010; Terradas etal. 2010a), to our knowledge n o detailed investigation taking into\naccount the peculiar properties of prominence threads can b e found in the existing literature. The\npurposeofthispaperistofillthisgapintheliterature,ast herecentobservationsofwavedamping\ninsolarprominencesneed tobeunderstood.\nHere, we study the spatial damping of propagating kink waves in prominence threads. Our\nmodel is composed of a cylindrical magnetic flux tube with par tially ionized prominence plasma,\nrepresenting a thread, surrounded by a fully ionized corona l environment. The thread is non-\nuniforminthetransversedirection. Resonantabsorptiona ndion-neutralcollisionsareassumedas\nthe damping mechanisms. We use the β=0 approximation, with βthe ratio of the gas pressure\nto the magnetic pressure, and the Thin Boundary approach to d escribe the effect of resonant ab-\nsorption in the Alfv´ en continuum using the connection form ulas for the perturbations across the\nresonantlayer(e.g.,Sakurai et al.1991;Goossenset al.19 92). Wedeterminethedominantdamp-\ningmechanismandobtainanalyticalexpressionsforthewav elength,thedampinglength,andtheir\nratio asfunctionsofthekinkmodefrequency.\nThis paper is organized as follows. Section 2 contains the de scription of the model configu-\nrationand thebasicEquations. First, theproblemisattack edanalyticallyinSection 3byadopting\nthethintubeapproximation. Lateron,thefulldispersionr elationisnumericallysolvedandapara-\nmetric study of the wavelength and damping length of thekink modeas functions of the period is\nperformed inSection 4. Finally,theconclusionsofthiswor k aregiveninSection 5.\n2. MODEL ANDDISPERSION RELATION\nFig. 1.—Sketch oftheprominencethread modeladoptedinthi swork.– 4 –\nThe equilibrium configuration is composed of a straight magn etic cylinder of radius aem-\nbedded in a homogeneous environment representing the coron al medium (see Figure 1). We use\ncylindrical coordinates, namely r,ϕ, andzfor the radial, azimuthal, and longitudinal coordinates,\nrespectively. The magnetic field is uniform and along the axi s of the cylinder, B=Bˆez, withB\nconstant everywhere. Hereafter, subscripts p and c denote p rominence and coronal quantities, re-\nspectively. The density within the prominence thread is den oted byρp, while the coronal density\nisρc. Bothρpandρcare homogeneous. A transverse transitional layer is includ ed in the radial\ndirection, where the density varies continuously between t he internal and external densities. We\ndo not specify the form of the density profile at this stage. Th e transverse inhomogeneous length\nscale in the transitional layer is given by the radio l/a, withlthe thickness of the layer. This ratio\nranges from l/a=0 if no transitional layer is present, to l/a=2 if the whole tube is radially\ninhomogeneous. Due to the presence of the transverse transi tional layer, the kink mode is reso-\nnantlycoupledtoAlfv´ encontinuummodes. Theresonancele adstothekinkmodedampingasthe\nenergyistransferredtoAlfv´ enmodesattheAlfv´ enresona nceposition. Thismechanismisknown\nas resonantabsorption.\nThe prominence plasma is partially ionized and we adopt the s ingle-fluid formalism (e.g.,\nBraginskii1965). Theionizationdegreeisarbitraryandis denotedherebythemeanatomicweight\nof the prominence material, ˜ µp. This parameter takes values in the range 0 .5≤˜µp≤1, where\n˜µp=0.5 corresponds to a fully ionized plasma and ˜ µp=1 to a fully neutral gas (see details\nin, e.g., Forteza etal. 2007; Soleret al. 2009b). The extern al coronal medium is assumed fully\nionized. The basic MHD Equations governing a partially ioni zed plasma can be found in, e.g.,\nFortezaet al. (2007); Pinto et al. (2008); Soler (2010). By a ssuming theβ=0 approximation and\nlinear perturbations from the equilibrium state, the basic MHD Equations discussed in this work\nare\nρ∂v\n∂t=1\nµ(∇×b)×B, (1)\n∂b\n∂t=∇×(v×B)+∇×/braceleftbiggηC\nB2[(∇×b)×B]×B/bracerightbigg\n, (2)\nwhereρisthelocaldensity,and v=/parenleftig\nvr,vϕ,vz/parenrightig\nandb=/parenleftig\nbr,bϕ,bz/parenrightig\narethevelocityandthemagnetic\nfieldperturbations,respectively. Notethat vz=0intheβ=0approximation. Inapartiallyionized\nplasma,theinductionequationcontainsatermaccountingf orCowling’sdiffusion,i.e.,thesecond\nterm on the right-hand side of Equation (2). Cowling’s di ffusion represents enhanced magnetic\ndiffusioncausedbyion-neutralcollisions,whichisseveralor dersofmagnitudemoree fficientthan\nclassical Ohm’s diffusionand is thedominante ffect in partially ionized plasmas (Cowling 1956).\nForthisreason,hereweneglectOhm’sdi ffusionandothertermsofminorimportancepresentinthe\ngeneralizedinductionequation(see,e.g.,Equation(14)o fFortezaet al.2007). Cowling’sdi ffusion\ncoefficient,ηC, depends on the ionization degree through ˜ µpas well as on the plasma physical\nconditions. The expression of ηCfor a hydrogen plasma can be found in, e.g., Pinto et al. (2008 )– 5 –\nand Soleret al. (2009b), whereas for a plasma composed of hyd rogen and helium see Soleret al.\n(2010b). Astheeffectofheliumisnegligibleforrealisticheliumabundances inprominences,here\nweconsiderapurehydrogenplasma. Thee ffectofCowling’sdi ffusionisneglectedintheexternal\nmediumbecause thecoronais assumedfullyionized.\nWe follow an approach based on normal modes. Since ϕandzare an ignorable coordinates,\nthe perturbations are expressed proportional to exp (imϕ+ikzz−iωt), whereωis the oscillatory\nfrequency, kzis the longitudinal wavenumber, and mis the azimuthal wavenumber ( m=1 for the\nkinkmode). Alternatively,theproblemcouldbeinvestigat edby meansoftime-dependentsimula-\ntionsof drivenwaves as in Pascoeet al. (2010). However, in t helinear regimethe di fferent values\nofmandkzare decoupled from each other, and a normalmodeanalysisis a simplerprocedure for\nlinear waves. If Cowling’s di ffusionis neglected, our configuration corresponds to that st udiedby\nTerradas et al. (2010a) for propagatingkinkwaves in corona l loops. Weextend theirinvestigation\nby incorporatingthee ffect ofCowling’sdi ffusionduetoion-neutralcollisions\nByusingtheThinBoundary(TB)approach(seedetailsin,e.g .,Goossenset al.2006;Goossens\n2008), theanalyticaldispersionrelationfor resonantlyd ampedkink wavespropagatingin atrans-\nversely nonuniform and partially ionized prominence threa d was obtained by Soleret al. (2009c,\nEquation (25)). If partial ionization is not considered and the effect of Cowling’s di ffusion is ab-\nsent, the dispersion relation of Soleret al. (2009c) reduce s to that investigated by Terradas et al.\n(2010a, Equation (28)) for kink waves in coronal loops. Sole ret al. (2009c) checked that the so-\nlutionsof theirdispersionrelation are in excellent agree ment with the solutionsobtained from the\nfull numerical integration of the MHD equations beyond the T B approximation. Therefore, the\ndispersion relation derived by Soleret al. (2009c) correct ly describes the kink mode behavior in\nourmodelandcomplicatednumericalintegrationsarenotne eded. Thedispersionrelationobtained\nby Soleret al.(2009c)inthecase ofa straightand homogeneo usmagneticfield is\nnc\nρc/parenleftig\nω2−k2zv2\nAc/parenrightigK′\nm(nca)\nKm(nca)−mp\nρp/parenleftig\nω2−k2zΓ2\nAp/parenrightigJ′\nm/parenleftig\nmpa/parenrightig\nJm/parenleftig\nmpa/parenrightig=−iπm2/r2\nA\nω2|∂rρ|rA, (3)\nwhereJmandKmaretheBesselfunctionandthemodifiedBesselfunctionofth efirstkindoforder\nm(Abramowitz& Stegun1972), respectively,and thequantiti esmpandncare defined as\nm2\np=/parenleftig\nω2−k2\nzΓ2\nAp/parenrightig\nΓ2\nAp,n2\nc=/parenleftig\nk2\nzv2\nAc−ω2/parenrightig\nv2\nAc, (4)\nwhereΓ2\nAp=v2\nAp−iωηCis the modified prominence Alfv´ en speed squared (Forteza et al. 2008),\nwithvAp=B√µρpandvAc=B√µρcthe prominence and coronal Alfv´ en speeds, respectively, a nd\nµ=4π×10−7N A−2the magnetic permeability. In addition, rAis the Alfv´ en resonance position,\nand|∂rρ|rAistheradialderivativeofthetransversedensityprofileat theAlfv´ enresonanceposition.– 6 –\nSoleret al.(2009c)studiedthetemporaldampingofkinkwav es,hencetheyassumedafixed,\nrealkzand solved Equation (3) to obtain the complex frequency. Her e, we investigate the spatial\ndampingandproceedtheotherwayround,i.e.,wefixareal ωandsolveEquation(3)toobtainthe\ncomplexwavenumber. Then, theperiod, P, wavelength,λ, and dampinglength, LD, are computed\nas follows,\nP=2π\nω, λ=2π\nkzR,LD=1\nkzI, (5)\nwithkzRandkzIthereal and imaginaryparts of kz, respectively.\n3. ANALYTICALAPPROXIMATIONS\nSomeanalyticalprogresscanbeperformedbeforesolvingEq uation(3)bymeansofnumerical\nmethods. To do so, we adopt the Thin Tube (TT) limit, i.e., λ/a≫1. A fist-order expansion of\nEquation(3)givesthedispersionrelationin boththeTT and TBapproximations,namely\nρp/parenleftig\nω2−k2\nzΓ2\nAp/parenrightig\n+ρc/parenleftig\nω2−k2\nzv2\nAc/parenrightig\n−iπm\nrAρpρc\n|∂rρ|rA/parenleftig\nω2−k2\nzΓ2\nAp/parenrightig/parenleftig\nω2−k2\nzv2\nAc/parenrightig\nω2=0.(6)\nIfbothCowling’sdi ffusionand resonant absorptionare omitted,thesolutiontoE quation(6)is\nk2\nz=ω2\nc2\nk, (7)\nwithc2\nk=2B2\nµ(ρp+ρc)the kink speed squared. Equation (7) corresponds to the idea l, undamped kink\nmode. The solutionsto Equation(6) considering thedi fferent dampingmechanisms are discussed\nnext.\n3.1. Damping by Cowling'sdi ffusion\nIn theabsence of transversetransitionallayer, i.e., l/a=0, resonant absorptiondoes not take\nplace and the damping is due to Cowling’sdi ffusion exclusively. In such a case, the third term on\nthe left-hand side of Equation (6) is absent. We write the wav enumber as kz=kzR+ikzIand use\nEquation(6)toobtaintheexactexpressionsfor k2\nzRandk2\nzI, namely\nk2\nzR=1\n2ω2c2\nk\nc4\nk+ω2¯η2\nC/radicaligg\n1+ω2¯η2\nC\nc4\nk+1, (8)\nk2\nzI=1\n2ω2c2\nk\nc4\nk+ω2¯η2\nC/radicaligg\n1+ω2¯η2\nC\nc4\nk−1, (9)– 7 –\nwith ¯ηC=ρp\nρp+ρcηC. By combining Equations (8) and (9), we compute the ratio of t he damping\nlengthtothewavelengthas\nLD\nλ=kzR\n2πkzI=1\n2πc2\nk+/radicalig\nc4\nk+ω2¯η2\nC\nω¯ηC. (10)\nEquations(8)–(10)areexactexpressionsthatcanbefurthe rsimplifieddependingonthevalue\noftheratioω2¯η2\nC\nc4\nk. Forω2¯η2\nC\nc4\nk≪1, i.e., in thelimitoflowfrequency ( ωsmall)and/orlargeionization\ndegree(¯ηCsmall),Equations(8)–(10)simplifyto\nk2\nzR≈ω2\nc2\nk/parenleftbigg\n1+ω2¯η2\nC\nc4\nk/parenrightbigg≈ω2\nc2\nk, (11)\nk2\nzI≈1\n4ω4¯η2\nC\nc6\nk/parenleftbigg\n1+ω2¯η2\nC\nc4\nk/parenrightbigg≈1\n4ω4¯η2\nC\nc6\nk, (12)\nLD\nλ≈1\nπc2\nk\nω¯ηC. (13)\nOn the contrary, ifω2¯η2\nC\nc4\nk≫1, i.e., high frequency and /or small ionization degree, the equivalent\nexpressionsare\nk2\nzR≈k2\nzI≈1\n2ω\nc2\nk¯ηC, (14)\nLD\nλ≈1\n2π. (15)\nThus, forω2¯η2\nC\nc4\nk≪1 the ratio of the damping length to the wavelength is inverse ly proportional to\nbothωand ¯ηC, andk2\nzRcoincides with the ideal value (Equation (7)). This case cor responds to a\nweaklydampedkinkmode. Ontheotherhand,forω2¯η2\nC\nc4\nk≫1,LD/λisindependentofωand ¯ηCand\nthe wave behavior is governed by di ffusion. By assuming typical values for the parameters in the\ncontext of oscillating prominence threads, e.g., P=3 min,B=5 G, andρp/ρc=200, we obtain\nω2¯η2\nC\nc4\nk≈6×10−17for ˜µp=0.5andω2¯η2\nC\nc4\nk≈1.6×10−4for ˜µp=0.99,meaning thatthecaseω2¯η2\nC\nc4\nk≪1\nismorerealisticin thecontextofoscillatingthreads even foran almostneutral plasma.\n3.2. Damping by resonant absorption andCowling's di ffusion\nNext,wetakethecase l/a/nequal0intoaccountandstudythecombinede ffectofresonantabsorp-\ntion and Cowling’s di ffusion. The third term on the left-hand side of Equation (6) is now present.– 8 –\nAsbefore, wewrite kz=kzR+ikzIand putthisexpressioninEquation(6). Sinceitisvery di fficult\ntogiveexactexpressionsfor kzRandkzIinthegeneralcase,wefocuson LD/λandrestrictourselves\ntoω2¯η2\nC\nc4\nk≪1. Following the procedure of Terradas et al. (2010a), we ass ume weak damping, i.e.,\nkzI≪kzR, and neglect terms with k2\nzI. The following process is long but straightforward, and we\nrefer the reader to Terradas et al. (2010a) for details. Fina lly, we arrive at the expression for the\nratio ofthedampinglengthtothewavelengthas\nLD\nλ≈/parenleftigg\nπω¯ηC\nc2\nk+m\nFl\naρp−ρc\nρp+ρc/parenrightigg−1\n, (16)\nwhere the first term within the parentheses accounts for Cowl ing’s diffusion and the second term\nfor resonant absorption. The factor Fin the second term takes di fferent values depending on\nthe density profile within the inhomogeneous layer. For exam ple,F=4/π2for a linear profile\n(Goossenset al. 2002), while F=2/πfor a sinusoidal profile with rA≈a(Ruderman &Roberts\n2002). If the term related to resonant absorption is absent, Equation (16) reverts to Equation (13).\nOn the other hand, if the term related to Cowling’s di ffusion is dropped, Equation (16) coincides\nwithEquation(13)ofTerradas et al.(2010a).\nTherelativeimportanceofthetwotermsinEquation(16)can beassessedbyperformingtheir\nratio as\nǫ≡(LD/λ)RA\n(LD/λ)C≈πFa\nlω¯ηC\nc2\nkmρp+ρc\nρp−ρc=πFa\nlωηC\nc2\nkmρp\nρp−ρc, (17)\nwhere(LD/λ)RAand(LD/λ)Cstand for the damping ratio by resonant absorption and Cowli ng’s\ndiffusion, respectively. By considering as before P=3 min,B=5 G, andρp/ρc=200, and\nadopting a linear profile with l/a=0.2, we obtainǫ≈8×10−8for ˜µp=0.5 andǫ≈0.12\nfor ˜µp=0.99, meaning that in the TT limit resonant absorption dominat es the kink mode spatial\ndamping for typical parameters of thread oscillations. Thi s result is equivalentto that obtained by\nSoleret al. (2009c) inthecaseoftemporaldamping.\n4. NUMERICALRESULTS\nNow, we solve the dispersion relation (Equation (3)) by mean s of standard numerical proce-\ndures. In the following figures, both the wavelength, λ, and the damping length, LD, are plotted\nin dimensionless form with respect to the thread mean radius ,a. The period, P, is computed in\nunits of the internal Alfv´ en travel time, τAp=a/vAp. Unless otherwise stated, the results have\nbeen computed with ρp=5×10−11kg m−3,ρp/ρc=200, and B=5 G. With these parameters,\nvAp≈63 kms−1and, fora=100km,τAp≈1.59 s.\nFigure 2 displaysλ/a,LD/a, andLD/λversusP/τApfor the case l/a=0, i.e., the damping– 9 –\nis due to Cowling’s di ffusion exclusively. We compute the results for di fferent values of ˜µp. The\nshaded areas in Figure 2 and in the other figures represent the range of observed periods of trans-\nversethread oscillations,i.e., 1min–10min,correspondi ngto40/lessorsimilarP/τAp/lessorsimilar400,approximately.\nRegardingthewavelength,weseethatthee ffectofCowling’sdi ffusionisonlyrelevantforperiods\nmuchshortedthanthoseobserved. ThisisinagreementwithE quations(11)and(14). Ontheother\nhand, an almost neutral plasma, i.e., ˜ µp→1, has to be considered to obtain an e fficient damping\nand to achieve small values of LD/λwithin the relevant range of periods. Although we do not\nknowtheexactionizationdegreein prominencethreads,suc hvery largevaluesof ˜ µpare probably\nunrealistic (see, e.g., Gouttebroze& Labrosse 2009). The a nalytical expressions for λ,LD, and\nLD/λintheTTcasegivenbyEquations(8),(9),and(10),respecti vely,areingoodagreementwith\nthefullresultsinthewholerange ofperiods(seesymbolsin Figure 2).\nNext, we study the general case l/a/nequal0. We adopt a sinusoidal density profile within the\ninhomogeneoustransitionallayer(Ruderman &Roberts2002 ). AstheAlfv´ enresonanceposition,\nrA,isneededforthecomputationsoftheresonantdamping,wef ollowatwo-stepprocedure. First,\nwe solve the dispersion relation for a fixed ωin the case l/a=0 and determine kzR. Then, we\nassume that the value of kzRis approximately the same in the case l/a/nequal0, meaning that the\nresonant condition is ω=kzRvA(rA). In the case of a sinusoidal profile, the expression of the\nresonantpositioncan beanalyticallyobtainedfrom theres onantconditionas\nrA=a+l\nπarcsinρp+ρc\nρp−ρc−2v2\nApk2\nzR\nω2ρp\nρp−ρc. (18)\nFinally, we compute |∂rρ|rAusing the previously determined rAby means of Equation (18) and\nsolvethedispersionrelationwith theseparameters to obta intheactual kzRandkzI. Figure3 shows\nthe results of these computations for di fferent values of l/awhen the ionization degree has been\nfixed to ˜µp=0.8, whereas Figure 4 displaystheequivalentcomputationsfo r different valuesof ˜µp\nwhen thetransverseinhomogeneitylengthscale hasbeen fixe d tol/a=0.2. Since thewavelength\nis not affected by the value of l/aand has the same behavior as in Figure 2(a), both Figures 3 and\n4 focus on LD/aandLD/λ. We obtain two di fferent behaviors of the solutions depending on the\nperiod. For small P/τAp, the damping length is independent of l/aand is governed by the value\nof ˜µp. On the contrary, for large P/τApthe damping length depends on l/abut is independent of\n˜µp. This result indicates that resonant absorptiondominates thedamping for large P/τAp, whereas\nCowling’sdiffusionismorerelevantforsmall P/τAp. Theapproximatetransitionalperiod,namely\nPtr,inwhichthedampinglengthbyCowling’sdi ffusionbecomessmallerthanthatduetoresonant\nabsorption can be estimated by setting ǫ≈1 in Equation (17) and writing Ptr=2π/ω. Then, one\nobtains\nPtr≈2π2Fa\nl¯ηC\nc2\nkmρp+ρc\nρp−ρc=2π2Fa\nlηC\nc2\nkmρp\nρp−ρc. (19)\nThistransitionalperiodisingoodagreementwiththenumer icalresults(seetheverticaldottedline– 10 –\nFig. 2.—Resultsforthekinkmodespatialdampinginthecase l/a=0: (a)λ/a,(b)LD/a, and(c)\nLD/λversusP/τApfor ˜µp=0.5,0.6,0.8,and0.95. Symbolsinpanels(a),(b),and(c)co rrespondto\ntheanalyticalsolutionintheTTapproximationgivenbyEqu ations(8),(9), and(10),respectively,\nwhile the horizontal dotted line in panel (c) corresponds to the limit of LD/λfor high frequencies\n(Equation(15)). Theshadedarea denotestherangeofobserv edperiodsofthread oscillations.– 11 –\nFig. 3.— Results for the kink mode spatial damping in the case l/a/nequal0: (a)LD/aand (b)LD/λ\nversusP/τApforl/a=0.05, 0.1, 0.2, and 0.4, with ˜ µp=0.8. Symbols in panel (b) correspond to\nthe analytical solution in the TT approximation given by Equ ation (16), while the vertical dotted\nline is the approximate transitional period given by Equati on (19) for l/a=0.1. The shaded area\ndenotestherangeofobservedperiodsofthread oscillation s.\nFig. 4.— Results for the kink mode spatial damping in the case l/a/nequal0: (a)LD/aand (b)LD/λ\nversusP/τApfor ˜µp=0.5, 0.6, 0.8, and 0.95, with l/a=0.2. Symbols in panel (b) correspond to\nthe analytical solution in the TT approximation given by Equ ation (16). The shaded area denotes\ntherangeofobservedperiodsofthread oscillations.– 12 –\nin Figure 3(b)). In addition, we see that Ptris much smaller than the typically observed periods,\nindicatingthat resonantabsorptionisthedominantdampin gmechanismin therelevantrange.\nFinally, we check that the analytical approximationof LD/λgivenby Equation (16) provides\nanaccuratedescriptionofthekinkmodespatialdampingint herelevantrangeofperiods(compare\nthesymbolsandthesolidlinesinFigures 3(b)and 4(b)).\n5. DISCUSSION AND CONCLUSION\nIn this paper, we have studied the spatial damping of kink wav es in prominence threads.\nResonant absorption and Cowling’s di ffusion are the damping mechanisms taken into account.\nBoth analytical expressions and numerical results indicat e that, in the range of typically observed\nperiods of prominence thread oscillations, the e ffect of Cowling’s di ffusion (and so the ionization\ndegree) is negligible. On the other hand, resonant absorpti on provides an efficient damping in\nagreement with the study of Terradas et al. (2010a) in the con text of coronal loop oscillations.\nThese conclusions are equivalent to those obtained by Soler etal. (2009c) in the case of temporal\ndamping.\nWe point out that small values of LD/λare obtained by resonant absorption in the observa-\ntionally relevant range of periods, which is consistent wit h the reported strong damping of the\noscillations. The analytical estimation of LD/λgiven by Equation (16) is very accurate in the ob-\nservationallyrelevantrangeofperiods,andthecontribut ionofCowling’sdi ffusioncanbedropped\nfromEquation(16)becausetheplasmaionizationdegreetur nsouttobeirrelevantforthedamping.\nTherefore, forkinkmodes( m=1)theradio LD/λsimplifiesto\nLD\nλ≈Fa\nlρp+ρc\nρp−ρc, (20)\nwhich coincides with the expression providedby Terradas et al. (2010a). Asρp+ρc\nρp−ρc→1 for typical\nprominenceandcoronaldensities,thisfactorcanbedroppe dfromEquation(20),meaningthatthe\nratioLD/λdepends almost exclusively on the transverse inhomogeneit y length scale, l/a, and the\nform ofthedensityprofilethrough Fas\nLD\nλ≈Fa\nl. (21)\nInthecaseofcoronallooposcillationsstudiedbyTerradas et al.(2010a),thefactorρp+ρc\nρp−ρccannotbe\ndroppedfromtheirexpressions,meaningthatincoronalloo pstheratio LD/λsignificantlydepends\non the density contrast. Therefore, information about the p arameters l/aandFin prominence\nthreads could be determined by using Equation (21) along wit h accurate measurements of the– 13 –\ndamping length and the wavelength provided from the observa tions. However, since the precise\nform of the transverse density profile in prominence threads is unknown, we have to assume an\nad hoc profile, i.e., a value of F, to infer the transverse inhomogeneity length scale from th e\nobservations,whichcan introducesomeuncertaintiesinth eestimationof l/a.\nFor example, let us assume that the ratio LD/λhas been determined from an observation of\ndampedkinkwavesinaprominencethreadandwewanttocomput ethetransverseinhomogeneity\nlength scale of the thread. For simplicity, we consider that the transverse density profile in the\ninhomogeneous layer is either linear or sinusoidal. Denoti ng as(l/a)linthe value of l/acomputed\nassuming a linear profile, and (l/a)sinthe corresponding value for a sinusoidalprofile, the relati on\nbetween bothofthemis(l/a)lin\n(l/a)sin=π\n2≈1.57, (22)\npointingoutthattherelativeuncertaintyof l/aislargerthan50%,andtheinaccuracycouldbeeven\nlargerifotherprofilesareconsidered. Thisfactshouldbet akenintoaccountinfutureseismological\ndeterminationsofthisparameter.\nThe present investigation is a first step for the study of the s patial damping of kink waves\nin prominence fine structures. Here, we have adopted a simple model of a prominence thread.\nSome effects that might influence the kink mode propagation and dampi ng are not included in\nthe present paper. Among them, plasma inhomogeneity along t he thread may affect somehow\nthe amplitude of a propagating kink mode, whereas the presen ce of flows affects the damping by\nresonantabsorption(seeTerradas et al.2010b). Theinfluen ceoftheseandothere ffectswillbethe\nsubjectofforthcomingworks.\nTheauthorsacknowledgethefinancialsupportreceivedfrom theSpanishMICINNandFEDER\nfunds (AYA2006-07637). The authors also acknowledge discu ssion within ISSI Team on Solar\nProminence Formation and Equilibrium: New data, new models . 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Phys.,132,63\nThispreprintwaspreparedwiththeAAS L ATEXmacrosv5.2." }, { "title": "1610.06661v1.Spin_transport_and_dynamics_in_all_oxide_perovskite_La___2_3__Sr___1_3__MnO__3__SrRuO__3__bilayers_probed_by_ferromagnetic_resonance.pdf", "content": "Spin transport and dynamics in all-oxide perovskite La 2=3Sr1=3MnO 3/SrRuO 3bilayers\nprobed by ferromagnetic resonance\nSatoru Emori,1,\u0003Urusa S. Alaan,1, 2Matthew T. Gray,1, 2Volker\nSluka,3Yizhang Chen,3Andrew D. Kent,3and Yuri Suzuki1, 4\n1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305 USA\n2Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305 USA\n3Department of Physics, New York University, New York, NY 10003, USA\n4Department of Applied Physics, Stanford University, Stanford, CA 94305 USA\n(Dated: November 11, 2021)\nThin \flms of perovskite oxides o\u000ber the possibility of combining emerging concepts of strongly\ncorrelated electron phenomena and spin current in magnetic devices. However, spin transport and\nmagnetization dynamics in these complex oxide materials are not well understood. Here, we ex-\nperimentally quantify spin transport parameters and magnetization damping in epitaxial perovskite\nferromagnet/paramagnet bilayers of La 2=3Sr1=3MnO 3/SrRuO 3(LSMO/SRO) by broadband ferro-\nmagnetic resonance spectroscopy. From the SRO thickness dependence of Gilbert damping, we\nestimate a short spin di\u000busion length of <\u00181 nm in SRO and an interfacial spin-mixing conductance\ncomparable to other ferromagnet/paramagnetic-metal bilayers. Moreover, we \fnd that anisotropic\nnon-Gilbert damping due to two-magnon scattering also increases with the addition of SRO. Our\nresults demonstrate LSMO/SRO as a spin-source/spin-sink system that may be a foundation for\nexamining spin-current transport in various perovskite heterostructures.\nI. INTRODUCTION\nManipulation and transmission of information by spin\ncurrent is a promising route toward energy-e\u000ecient mem-\nory and computation devices1. Such spintronic devices\nmay consist of ferromagnets interfaced with nonmagnetic\nconductors that exhibit spin-Hall and related spin-orbit\ne\u000bects2{4. The direct spin-Hall e\u000bect in the conductor\ncan convert a charge current to a spin current, which ex-\nerts torques on the adjacent magnetization and modi\fes\nthe state of the device5,6. Conversely, the inverse spin-\nHall e\u000bect in the conductor can convert a propagating\nspin current in the magnetic medium to an electric signal\nto read spin-based information packets7. For these device\nschemes, it is essential to understand the transmission of\nspin current between the ferromagnet and the conductor,\nwhich is parameterized by the spin-mixing conductance\nand spin di\u000busion length. These spin transport parame-\nters can be estimated by spin pumping at ferromagnetic\nresonance (FMR), in which a spin current is resonantly\ngenerated in the ferromagnet and absorbed in the adja-\ncent conductor8,9. Spin pumping has been demonstrated\nin various combinations of materials, where the magnetic\nlayer may be an alloy (e.g., permalloy) or insulator (e.g.,\nyttrium iron garnet) and the nonmagnetic conductor may\nbe a transition metal, semiconductor, conductive poly-\nmer, or topological insulator10{16.\nTransition metal oxides, particularly those with the\nperovskite structure, o\u000ber the intriguing prospect of\nintegrating a wide variety of strongly correlated elec-\ntron phenomena17,18with spintronic functionalities19,20.\nAmong these complex oxides, La 2=3Sr1=3MnO 3(LSMO)\nand SrRuO 3(SRO) are attractive materials for epitaxial,\nlattice-matched spin-source/spin-sink heterostructures.\nLSMO, a metallic ferromagnet known for its colossalmagnetoresistance and Curie temperature of >300 K,\ncan be an excellent resonantly-excited spin source be-\ncause of its low magnetization damping21{26. SRO, a\nroom-temperature metallic paramagnet with relatively\nhigh conductivity27, exhibits strong spin-orbit coupling28\nthat may be useful for emerging spintronic applications\nthat leverage spin-orbit e\u000bects2{4.\nA few recent studies have reported dc voltages at FMR\nin LSMO/SRO bilayers that are attributed to the in-\nverse spin-Hall e\u000bect in SRO generated by spin pump-\ning24{26. However, it is generally a challenge to separate\nthe inverse spin-Hall signal from the spin recti\fcation sig-\nnal, which is caused by an oscillating magnetoresistance\nmixing with a microwave current in the conductive mag-\nnetic layer29{31. Moreover, while the spin-mixing con-\nductance is typically estimated from the enhancement in\nthe Gilbert damping parameter \u000b, the quanti\fcation of \u000b\nis not necessarily straightforward in epitaxial thin \flms\nthat exhibit pronounced anisotropic non-Gilbert damp-\ning23,32{37. It has also been unclear how the Gilbert and\nnon-Gilbert components of damping in LSMO are each\nmodi\fed by an adjacent SRO layer. These points above\nhighlight the need for an alternative experimental ap-\nproach for characterizing spin transport and magnetiza-\ntion dynamics in LSMO/SRO.\nIn this work, we quantify spin transport parameters\nand magnetization damping in epitaxial LSMO/SRO bi-\nlayers by broadband FMR spectroscopy with out-of-plane\nandin-plane external magnetic \felds. Out-of-plane FMR\nenables straightforward extraction of Gilbert damping as\na function of SRO overlayer thickness, which is repro-\nduced by a simple \\spin circuit\" model based on di\u000busive\nspin transport38,39. We \fnd that the spin-mixing conduc-\ntance at the LSMO/SRO interface is comparable to other\nferromagnet/conductor interfaces and that spin current\nis absorbed within a short length scale of <\u00181 nm in thearXiv:1610.06661v1 [cond-mat.mtrl-sci] 21 Oct 20162\n42 44 46 48 50LSMO(10)\n /SRO(18)LSMO(10)\n log(intensity) (a.u.)\n2 (deg.)LSAT(002) \nLSMO(002) \nSRO(002) \nFigure 1. 2 \u0012-!x-ray di\u000braction scans of a single-layer\nLSMO(10 nm) \flm and LSMO(10 nm)/SRO(18 nm) bilayer.\nconductive SRO layer. From in-plane FMR, we observe\npronounced non-Gilbert damping that is anisotropic and\nscales nonlinearly with excitation frequency, which is ac-\ncounted for by an existing model of two-magnon scat-\ntering40. This two-magnon scattering is also enhanced\nwith the addition of the SRO overlayer possibly due to\nspin pumping. Our \fndings reveal key features of spin\ndynamics and transport in the prototypical perovskite\nferromagnet/conductor bilayer of LSMO/SRO and pro-\nvide a foundation for future all-oxide spintronic devices.\nII. SAMPLE AND EXPERIMENTAL DETAILS\nEpitaxial \flms of LSMO(/SRO) were grown on\nas-received (001)-oriented single-crystal (LaAlO 3)0:3\n(Sr2AlTaO 6)0:7(LSAT) substrates using pulsed laser de-\nposition. LSAT exhibits a lower dielectric constant than\nthe commonly used SrTiO 3substrate and is therefore\nbetter suited for high-frequency FMR measurements.\nThe lattice parameter of LSAT (3.87 \u0017A) is also closely\nmatched to the pseudocubic lattice parameter of LSMO\n(\u00193.88 \u0017A). By using deposition parameters similar to\nthose in previous studies from our group41,42, all \flms\nwere deposited at a substrate temperature of 750\u000eC with\na target-to-substrate separation of 75 mm, laser \ruence\nof\u00191 J/cm2, and repetition rate of 1 Hz. LSMO was de-\nposited in 320 mTorr O 2, followed by SRO in 100 mTorr\nO2. After deposition, the samples were held at 600\u000eC\nfor 15 minutes in \u0019150 Torr O 2and then the substrate\nheater was switched o\u000b to cool to room temperature. The\ndeposition rates were calibrated by x-ray re\rectivity mea-\nsurements. The thickness of LSMO, tLSMO , in this study\nis \fxed at 10 nm, which is close to the minimum thickness\nat which the near-bulk saturation magnetization can be\nattained.\nX-ray di\u000braction results indicate that both the LSMO\n\flms and LSMO/SRO bilayers are highly crystalline\nand epitaxial with the LSAT(001) substrate, with high-\nresolution 2 \u0012-!scans showing distinct Laue fringes\naround the (002) Bragg re\rection (Fig. 1). In this study,\nthe maximum thickness of the LSMO and SRO layers\n770 780 790 800 810 main mode\n sec. mode\n dIFMR/dH (a.u.)\n0H (mT) data\n fit(a) (b) \n770 780 790 800 810 data\n fit\n dIFMR/dH (a.u.)\n0H (mT)Figure 2. Exemplary FMR spectra and \ftting curves: (a)\none mode of Lorentzian derivative; (b) superposition of a main\nmode and a small secondary mode due to slight sample inho-\nmogeneity.\ncombined is less than 30 nm and below the threshold\nthickness for the onset of structural relaxation by mis\ft\ndislocation formation41,42. The typical surface roughness\nof LSMO and SRO measured by atomic force microscopy\nis<\u00184\u0017A, comparable to the roughness of the LSAT sub-\nstrate surface.\nSQUID magnetometry con\frms that the Curie tem-\nperature of the LSMO layer is \u0019350 K and the room-\ntemperature saturation magnetization is Ms\u0019300 kA/m\nfor 10-nm thick LSMO \flms. The small LSMO thickness\nis desirable for maximizing the spin-pumping-induced en-\nhancement in damping, since spin pumping scales in-\nversely with the ferromagnetic layer thickness8,9. More-\nover, the thickness of 10 nm is within a factor of \u00192\nof the characteristic exchange lengthp\n2Aex=\u00160M2s\u00195\nnm, assuming an exchange constant of Aex\u00192 pJ/m in\nLSMO (Ref. 43), so standing spin-wave modes are not\nexpected.\nBroadband FMR measurements were performed at\nroom temperature. The \flm sample was placed face-\ndown on a coplanar waveguide with a center conductor\nwidth of 250 \u0016m. Each FMR spectrum was acquired at a\nconstant excitation frequency while sweeping the exter-\nnal magnetic \feld H. The \feld derivative of the FMR\nabsorption intensity (e.g., Fig. 2) was acquired using an\nrf diode combined with an ac (700 Hz) modulation \feld.\nEach FMR spectrum was \ftted with the derivative of the\nsum of the symmetric and antisymmetric Lorentzians, as\nshown in Fig. 2, from which the resonance \feld HFMR\nand half-width-at-half-maximum linewidth \u0001 Hwere ex-\ntracted. In some spectra (e.g., Fig. 2(b)), a small sec-\nondary mode in addition to the main FMR mode was\nobserved. We \ft such a spectrum to a superposition of\ntwo modes, each represented by a generalized Lorentzian\nderivative, and analyze only the HFMR and \u0001Hof the\nlarger-amplitude main FMR mode. The secondary mode\nis not a standing spin-wave mode because it appears\nabove or below the resonance \feld of the main mode\nHFMR with no systematic trend in \feld spacing. We\nattribute the secondary mode to regions in the \flm with3\n0.360.400.440.48\n 0Meff (T)\n0 5 10 15 201.952.002.05\ntSRO (nm)\n gop\n0 5 10 15 200.00.20.40.60.81.01.2\nLSMO/SRO\nLSMO\n 0HFMR (T)\nf (GHz)(a) (b) \n(c) \nFigure 3. (a) Out-of-plane resonance \feld\nHFMR versus excitation frequency ffor\na single-layer LSMO(10 nm) \flm and a\nLSMO(10 nm)/SRO(3 nm) bilayer. The\nsolid lines indicate \fts to the data using\nEq. 1. (b,c) SRO-thickness dependence of\nthe out-of-plane Land\u0013 e g-factor (b) and ef-\nfective saturation magnetization Me\u000b(c).\nThe dashed lines indicate the values aver-\naged over all the data shown.\nslightly di\u000berent Msor magnetic anisotropy. More pro-\nnounced inhomogeneity-induced secondary FMR modes\nhave been observed in epitaxial magnetic \flms in prior\nreports22,44.\nIII. OUT-OF-PLANE FMR AND ESTIMATION\nOF SPIN TRANSPORT PARAMETERS\nOut-of-plane FMR allows for conceptually simpler ex-\ntraction of the static and dynamic magnetic properties\nof a thin-\flm sample. For \ftting the frequency depen-\ndence ofHFMR, the Land\u0013 e g-factor gopand e\u000bective sat-\nuration magnetization Me\u000bare the only adjustable pa-\nrameters in the out-of-plane Kittel equation. The fre-\nquency dependence of \u0001 Hfor out-of-plane FMR arises\nsolely from Gilbert damping, so that the conventional\nmodel of spin pumping8,9,38,39can be used to analyze the\ndata without complications from non-Gilbert damping.\nThis consideration is particularly important because the\nlinewidths of our LSMO(/SRO) \flms in in-plane FMR\nmeasurements are dominated by highly anisotropic non-\nGilbert damping (as shown in Sec. IV). Furthermore, a\nsimple one-dimensional, time-independent model of spin\npumping outlined by Boone et al.38is applicable in the\nout-of-plane con\fguration, since the precessional orbit of\nthe magnetization is circular to a good approximation.\nThis is in contrast with the in-plane con\fguration with\na highly elliptical orbit from a large shape anisotropy\n\feld. By taking advantage of the simplicity in out-of-\nplane FMR, we \fnd that the Gilbert damping parame-\nter in LSMO is approximately doubled with the addition\nof a su\u000eciently thick SRO overlayer due to spin pump-\ning. Our results indicate that spin-current transmission\nat the LSMO/SRO interface is comparable to previously\nreported ferromagnet/conductor bilayers and that spin\ndi\u000busion length in SRO is <\u00181 nm.\nWe \frst quantify the static magnetic properties of\nLSMO(/SRO) from the frequency dependence of HFMR.\nThe Kittel equation for FMR in the out-of-plane con\fg-\nuration takes a simple linear form,\nf=gop\u0016B\nh\u00160(HFMR\u0000Me\u000b); (1)\nwhere\u00160is the permeability of free space, \u0016Bis the Bohrmagneton, and his the Planck constant. As shown in\nFig. 3(a), we only \ft data points where \u00160HFMR is at\nleast 0.2 T above \u00160Me\u000bto ensure that the \flm is sat-\nurated out-of-plane. Figures 3(b) and (c) plot the ex-\ntractedMe\u000bandgop, respectively, each exhibiting no\nsigni\fcant dependence on SRO thickness tSROto within\nexperimental uncertainty. The SRO overlayer therefore\nevidently does not modify the bulk magnetic proper-\nties of LSMO, and signi\fcant interdi\u000busion across the\nSRO/LSMO interface can be ruled out. The averaged\nMe\u000bof 330\u000610 kA/m (\u00160Me\u000b= 0:42\u00060:01 T) is close\ntoMsobtained from static magnetometery and implies\nnegligible out-of-plane magnetic anisotropy; we thus as-\nsumeMs=Me\u000bin all subsequent analyses. The SRO-\nthickness independence of gop, averaging to 2 :01\u00060:01,\nimplies that the SRO overlayer does not generate a signif-\nicant orbital contribution to magnetism in LSMO. More-\nover, the absence of detectable change in gopwith in-\ncreasingtSRO may indicate that the imaginary compo-\nnent of the spin-mixing conductance8,9is negligible at\nthe LSMO/SRO interface.\nThe Gilbert damping parameter \u000bis extracted from\nthe frequency dependence of \u0001 H(e.g., Figure 4(a)) by\n\ftting the data with the standard linear relation,\n\u0001H= \u0001H0+h\ngop\u0016B\u000bf: (2)\nThe zero-frequency linewidth \u0001 H0is typically attributed\nto sample inhomogeneity. We observe sample-to-sample\nvariation of \u00160\u0001H0in the range\u00191\u00004 mT with no\nsystematic correlation with tSRO or the slope in Eq. 2.\nMoreover, similar to the analysis of HFMR, we only \ft\ndata obtained at \u00150.2 T above \u00160Me\u000bto minimize spu-\nrious broadening of \u0001 Hat low \felds. The linear slope of\n\u0001Hplotted against frequency up to 20 GHz is therefore\na reliable measure of \u000bdecoupled from \u0001 H0in Eq. 2.\nFigure 4(a) shows an LSMO single-layer \flm and an\nLSMO/SRO bilayer with similar \u0001 H0. The slope, which\nis proportional to \u000b, is approximately a factor of 2 greater\nfor LSMO/SRO. Figure 4(b) summarizes the dependence\nof\u000bon SRO-thickness, tSRO. For LSMO single-layer\n\flms we \fnd \u000b= (0:9\u00060:2)\u000210\u00003, which is on the same\norder as previous reports of LSMO thin \flms21{23,26.\nThis low damping is also comparable to the values re-4\n0 5 10 15 200123\n (10-3)\ntSRO (nm)\n1/Gext 1/G↑↓m LSMO SRO\n0 5 10 15 201.01.52.02.53.0\nLSMO\n 0H (mT)\nf (GHz)LSMO/SRO(a) (b) (c) \nFigure 4. (a) Out-of-plane FMR linewidth \u0001 Hversus excitation frequency for LSMO(10 nm) and LSMO(10 nm)/SRO(3 nm).\nThe solid lines indicate \fts to the data using Eq. 2. (b) Gilbert damping parameter \u000bversus SRO thickness tSRO. The solid\ncurve shows a \ft to the di\u000busive spin pumping model (Eq. 5). (c) Schematic of out-of-plane spin pumping and the equivalent\n\\spin circuit.\"\nported in Heusler alloy thin \flms45,46and may arise from\nthe half-metal-like band structure of LSMO (Ref. 47).\nLSMO can thus be an e\u000ecient source of spin current\ngenerated resonantly by microwave excitation.\nWith a few-nanometer thick overlayer of SRO, \u000bin-\ncreases to\u00192\u000210\u00003(Fig. 4(b)). This enhanced damping\nwith the addition of SRO overlayer may arise from (1)\nspin scattering48,49at the LSMO/SRO interface or (2)\nspin pumping8,9where nonequilibrium spins from LSMO\nare absorbed in the bulk of the SRO layer. Here, we\nassume that interfacial spin scattering is negligible, since\n<\u00181 nm of SRO overlayer does not enhance \u000bsigni\fcantly\n(Fig. 4(b)). This is in contrast with the pronounced in-\nterfacial e\u000bect in ferromagnet/Pt bilayers48,49, in which\neven<1 nm of Pt can increase \u000bby as much as a fac-\ntor of\u00192 (Refs. 50{52). In the following analysis and\ndiscussion, we show that spin pumping alone is su\u000ecient\nfor explaining the enhanced damping in LSMO with an\nSRO overlayer.\nWe now analyze the data in Fig. 4(b) using a one-\ndimensional model of spin pumping based on di\u000busive\nspin transport38,39. The resonantly-excited magnetiza-\ntion precession in LSMO generates non-equilibrium spins\npolarized along ^ m\u0002d ^m=dt, which is transverse to the\nmagnetization unit vector ^ m. This non-equilibrium spin\naccumulation di\u000buses out to the adjacent SRO layer\nand depolarizes exponentially on the characteristic length\nscale\u0015s. The spin current density ~jsat the LSMO/SRO\ninterface can be written as38,53\n~jsjinterface =~2\n2e2^m\u0002d^m\ndt\u0010\n1\nG\"#+1\nGext\u0011; (3)\nwhere ~is the reduced Planck constant, G\"#is the inter-\nfacial spin-mixing conductance per unit area, and Gextis\nthe spin conductance per unit area in the bulk of SRO.\nIn Eq. 3, 1/ G\"#and 1/Gextconstitute spin resistors in\nseries such that the spin transport from LSMO to SRO\ncan be regarded analogously as a \\spin circuit,\" as il-\nlustrated in Fig. 4(c). In literature, these interfacialand bulk spin conductances are sometimes lumped to-\ngether as an \\e\u000bective spin-mixing conductance\" Ge\u000b\n\"#=\n(1=G\"#+ 1=Gext)\u00001(Refs. 10{13, 16, 20, 23, 26, 44). We\nalso note that the alternative form of the (e\u000bective) spin-\nmixing conductance g(e\u000b)\ne\u000b, with units of m\u00002, is related to\nG(e\u000b)\n\"#, with units of \n\u00001m\u00002, byg(e\u000b)\ne\u000b= (h=e2)G(e\u000b)\n\"#\u0019\n26 k\n\u0002G(e\u000b)\n\"#.\nThe functional form of Gextis obtained by solving the\nspin di\u000busion equation with appropriate boundary condi-\ntions38,39,53. In the case of a ferromagnet/nonmagnetic-\nmetal bilayer, we obtain\nGext=1\n2\u001aSRO\u0015stanh\u0012tSRO\n\u0015s\u0013\n; (4)\nwhere\u001aSROis the resistivity of SRO, tSROis the thick-\nness of the SRO layer, and \u0015sis the di\u000busion length of\npumped spins in SRO. Finally, the out\row of spin cur-\nrent (Eq. 3) is equivalent to an enhancement of Gilbert\ndamping9with respect to \u000b0of LSMO with tSRO = 0\nsuch that\n\u000b=\u000b0+gop\u0016B~\n2e2MstLSMO\u00141\nG\"#+ 2\u001aSRO\u0015scoth\u0012tSRO\n\u0015s\u0013\u0015\u00001\n:\n(5)\nThus, two essential parameters governing spin transport\nG\"#and\u0015scan be estimated by \ftting the SRO-thickness\ndependence of \u000b(Fig. 4(b)) with Eq. 5.\nIn carrying out the \ft, we \fx \u000b0= 0:9\u000210\u00003. We note\nthat\u001aSROincreases by an order of magnitude compared\nto the bulk value of \u00192\u000210\u00006\nm astSROis reduced to\na few nm; also, at thicknesses of 3 monolayers ( \u00191.2 nm)\nor below, SRO is known to be insulating54. We there-\nfore use the tSRO-dependent \u001aSRO shown in Appendix\nA while assuming \u0015sis constant. An alternative \ftting\nmodel that assumes a constant \u001aSRO, which is a common\napproach in literature, is discussed in Appendix A.\nThe curve in Fig. 4(b) is generated by Eq. 5 with G\"#=\n1:6\u00021014\n\u00001m\u00002and\u0015s= 0:5 nm. Given the scatter of5\n170175180185\n170\n175\n180\n185[010]\n[110]\n[100] \n \n0HFMR (mT)\nLSMO\nLSMO/SRO\n0 5 10 15 200100200300400500\n 0HFMR (mT)\nf (GHz)H||[100]\nH||[110](a) (b) (c) \n0 5 10 15 201.952.002.05\ntSRO (nm)\n gip\n-6-4-20\n 0H||,4 (mT)\n(d) \n14 15330360 \n \n \nFigure 5. (a) Angular dependence of HFMR at 9 GHz for LSMO(10 nm) and LSMO(10 nm)/SRO(7 nm). The solid curves\nindicate \fts to the data using Eq. 6. (b) Frequency dependence of HFMR for LSMO(10 nm)/SRO(7 nm) with \feld applied in the\n\flm plane along the [100] and [110] directions. Inset: close-up of HFMR versus frequency around 14-15 GHz. In (a) and (b), the\nsolid curves show \fts to the Kittel equation (Eq. 6). (c,d) SRO-thickness dependence of the in-plane cubic magnetocrystalline\nanisotropy \feld (c) and in-plane Land\u0013 e g-factor (d). The dashed lines indicate the values averaged over all the data shown.\nthe experimental data, acceptable \fts are obtained with\nG\"#\u0019(1:2\u00002:5)\u00021014\n\u00001m\u00002and\u0015s\u00190:3\u00000:9\nnm. The estimated ranges of G\"#and\u0015salso depend\nstrongly on the assumptions behind the \ftting model.\nFor example, as shown in Appendix A, the constant- \u001aSRO\nmodel yields G\"#>\u00183\u00021014\n\u00001m\u00002and\u0015s\u00192:5 nm.\nNevertheless, we \fnd that the estimated G\"#\nis on the same order of magnitude as those\nof various ferromagnet/transition-metal heterostruc-\ntures39,55,56, signifying that the LSMO/SRO interface\nis reasonably transparent to spin current. More impor-\ntantly, the short \u0015simplies the presence of strong spin-\norbit coupling that causes rapid spin scattering within\nSRO. This \fnding is consistent with a previous study on\nSRO at low temperature in the ferromagnetic state show-\ning extremely fast spin relaxation with Gilbert damping\n\u000b\u00181 (Ref. 28). The short \u0015sindicates that SRO may be\nsuitable as a spin sink or detector in all-oxide spintronic\ndevices.\nIV. IN-PLANE FMR AND ANISOTROPIC\nTWO-MAGNON SCATTERING\nIn epitaxial thin \flms, the analysis of in-plane FMR\nis generally more complicated than that of out-of-plane\nFMR. High crystallinity of the \flm gives rise to a non-\nnegligible in-plane magnetocrystalline anisotropy \feld,\nwhich manifests in an in-plane angular dependence of\nHFMR and introduces another adjustable parameter in\nthe nonlinear Kittel equation for in-plane FMR. More-\nover, \u0001Hin in-plane FMR of epitaxial thin \flms often\ndepends strongly on the magnetization orientation and\nexhibits nonlinear scaling with respect to frequency due\nto two-magnon scattering, a non-Gilbert mechanism for\ndamping23,32{37. We indeed \fnd that damping of LSMO\nin the in-plane con\fguration is anisotropic and domi-\nnated by two-magnon scattering. We also observe ev-idence of enhanced two-magnon scattering with added\nSRO layers, which may be due to spin pumping from\nnonuniform magnetization precession.\nFigure 5(a) plots HFMR of a single-layer LSMO \flm\nand an LSMO/SRO bilayer as a function of applied \feld\nangle within the \flm plane. For both samples, we observe\nclear four-fold symmetry, which is as expected based on\nthe epitaxial growth of LSMO on the cubic LSAT(001)\nsubstrate. Similar to previous FMR studies of LSMO on\nSrTiO 3(001)57,58, the magnetic hard axes (corresponding\nto the axes of higher HFMR) are alongh100i. The in-\nplane Kittel equation for thin \flms with in-plane cubic\nmagnetic anisotropy is59,\nf=gip\u0016B\nh\u00160\u0002\nHFMR +Hjj;4cos(4\u001e)\u00031\n2\u0002\n\u0014\nHFMR +Me\u000b+1\n4Hjj;4(3 + cos(4\u001e))\u00151\n2\n;\n(6)\nwheregipis the Land\u0013 e g-factor that is obtained from in-\nplane FMR data, Hjj;4is the e\u000bective cubic anisotropy\n\feld, and\u001eis the in-plane \feld angle with respect to\nthe [100] direction. Given that LSMO is magnetically\nvery soft (coercivity on the order of 0.1 mT) at room\ntemperature, we assume that the magnetization is par-\nallel to the \feld direction, particularly with \u00160H\u001d10\nmT. In \ftting the angular dependence (e.g., Fig. 5(a))\nand frequency dependence (e.g., Fig. 5(b)) of HFMR to\nEq. 6, we \fx Me\u000bat the values obtained from out-of-\nplane FMR (Fig. 3(b)) so that Hjj;4andgipare the\nonly \ftting parameters. For the two samples shown in\nFig. 5(a), the \fts to the angular dependence and fre-\nquency dependence data yield consistent values of Hjj;4\nandgip. For the rest of the LSMO(/SRO) samples, we\nuse the frequency dependence data with Hjj[100] and\nHjj[110] to extract these parameters. Figures 5(c) and\n(d) show that Hjj;4andgip, respectively, exhibit no sys-\ntematic dependence on tSRO, similar to the \fndings from6\nout-of-plane FMR (Figs. 3(b),(c)). The in-plane cubic\nmagnetocrystalline anisotropy in LSMO(/SRO) is rela-\ntively small, with \u00160Hjj;4averaging to\u00192.5 mT.gipav-\nerages out to 1 :99\u00060:02, which is consistent with gop\nfound from out-of-plane FMR.\nWhile the magnetocrytalline anisotropy in\nLSMO(/SRO) is found to be modest and indepen-\ndent oftSRO, we observe much more pronounced\nin-plane anisotropy and tSRO dependence in linewidth\n\u0001H, as shown in Figs. 6(a) and (b). Figure 6(a)\nindicates that the in-plane dependence of \u0001 His four-\nfold symmetric for both LSMO(10 nm) and LSMO(10\nnm)/SRO(7 nm). \u0001 His approximately a factor of 2\nlarger when the sample is magnetized along h100icom-\npared to when it is magnetized along h110i. One might\nattribute this pronounced anisotropy to anisotropic\nGilbert damping60, such that the sample magnetized\nalong the hard axes h100imay lead to stronger damp-\ning. However, we \fnd no general correlation between\nmagnetocrystalline anisotropy and anisotropic \u0001 H: As\nwe show in Appendix B, LSMO grown on NdGaO 3(110)\nwith pronounced uniaxial magnetocrystalline anisotropy\nexhibits identical \u0001 Hwhen magnetized along the easy\nand hard axes. Moreover, whereas Gilbert damping\nshould lead to a linear frequency dependence of \u0001 H,\nfor LSMO(/SRO) the observed frequency dependence\nof \u0001His clearly nonlinear as evidenced in Fig. 6(b).\nThe pronounced anisotropy and nonlinear frequency\ndependence of \u0001 Htogether suggest the presence of a\ndi\u000berent damping mechanism.\nA well-known non-Gilbert damping mechanism in\nhighly crystalline ultrathin magnetic \flms is two-magnon\nscattering23,32{37,40,61,62, in which uniformly precessing\nmagnetic moments (a spin wave, or magnon mode, with\nwavevector k= 0) dephase to a k6= 0 magnon mode with\nadjacent moments precessing with a \fnite phase di\u000ber-\nence. By considering both exchange coupling (which re-\nsults in magnon energy proportional to k2) and dipolar\ncoupling (magnon energy proportional to \u0000jkj) among\nprecessing magnetic moments, the k= 0 andk6= 0\nmodes become degenerate in the magnon dispersion re-\nlation61as illustrated in Fig. 6(c).\nThe transition from k= 0 tok6= 0 is activated by\ndefects that break the translational symmetry of the\nmagnetic system by localized dipolar \felds40,61,62. In\nLSMO(/SRO), the activating defects may be faceted such\nthat two-magnon scattering is more pronounced when\nthe magnetization is oriented along h100i. One possibil-\nity is that LSMO thin \flms naturally form pits or islands\nfaceted alongh100iduring growth. However, we are un-\nable to consistently observe signs of such faceted defects\nin LSMO(/SRO) samples with an atomic force micro-\nscope (AFM). It is possible that these crystalline defects\nare smaller than the lateral resolution of our AFM setup\n(<\u001810 nm) or that these defects are not manifested in sur-\nface topography. Such defects may be point defects or\nnanoscale clusters of distinct phases that are known to\nexist intrinsically even in high-quality crystals of LSMO(Ref. 63).\nAlthough the de\fnitive identi\fcation of defects that\ndrive two-magnon scattering would require further in-\nvestigation, we can rule out (1) atomic step terraces\nand (2) mis\ft dislocations as sources of anisotropic two-\nmagnon scattering. (1) AFM shows that the orienta-\ntion and density of atomic step terraces di\u000ber randomly\nfrom sample to sample, whereas the anisotropy in \u0001 H\nis consistently cubic with larger \u0001 HforHjjh100ithan\nHjjh110i. This is in agreement with the recent study\nby Lee et al. , which shows anisotropic two-magnon scat-\ntering in LSMO to be independent of regularly-spaced\nparallel step terraces on a bu\u000bered-oxide etched SrTiO 3\nsubstrate23. (2) Although Woltersdorf and Heinrich have\nfound that mis\ft dislocations in Fe/Pd grown on GaAs\nare responsible for two-magnon scattering33, such dis-\nlocations are expected to be virtually nonexistent in\nfully strained LSMO(/SRO) \flms on the closely-latticed\nmatched LSAT substrates41,42.\nWe assume that the in-plane four-fold anisotropy and\nnonlinear frequency dependence of \u0001 Hare entirely due\nto two-magnon scattering. For a sample magnetized\nalong a given in-plane crystallographic axis hhk0i=h100i\norh110i, the two-magnon scattering contribution to \u0001 H\nis given by40\n\u0001Hhhk0i\n2m = \u0000hhk0i\n2m sin\u00001sp\nf2+ (fM=2)2\u0000fM=2p\nf2+ (fM=2)2+fM=2;(7)\nwherefM= (gip\u0016B=h)\u00160Msand \u0000hhk0i\n2m is the two-\nmagnon scattering parameter. The angular dependence\nof \u0001His \ftted with33\n\u0001H= \u0001H0+h\ngip\u0016B\u000bf\n+ \u0001Hh100i\n2m cos2(2\u001e) + \u0001Hh110i\n2m cos2(2[\u001e\u0000\u0019\n4]):(8)\nSimilarly, the frequency dependence of \u0001 Hwith the sam-\nple magnetized along [100] or [110], i.e., \u001e= 0 or\u0019=4,\nis well described by Eqs. 7 and 8. In principle, it should\nbe possible to \ft the linewidth data with \u0001 H0,\u000b, and\n\u00002mas adjustable parameters. In practice, the \ft car-\nried out this way is overspeci\fed such that wide ranges\nof these parameters appear to \ft the data. We there-\nfore impose a constraint on \u000bby assuming that Gilbert\ndamping for LSMO(/SRO) is isotropic: For each SRO\nthicknesstSRO,\u000bis \fxed to the value estimated from\nthe \ft curve in Fig. 4(c) showing out-of-plane FMR data.\n(This assumption is likely justi\fed, since the damping\nfor LSMO(10 nm) on NdGdO 3(110) with strong uniaxial\nmagnetic anisotropy is identical for the easy and hard\ndirections, as shown in Appendix B.) To account for the\nuncertainty in the Gilbert damping in Fig. 4(c), we vary \u000b\nby\u000625% for \ftting the frequency dependence of in-plane\n\u0001H. Examples of \fts using Eqs. 7 and 8 are shown in\nFig. 6(a),(b).\nFigure 6(d) shows that the SRO overlayer enhances\nthe two-magnon scattering parameter \u0000 2mby up to a7\n0 5 10 15 2004812\nLSMOLSMO/SRO\n 0H (mT)\nf (GHz)(a) (b) \n(c) (d) \n \nFMR \nfreq. \nk f \nk=0 k≠0 \n0 5 10 15 200102030\n 02m (mT)\ntSRO (nm)H||[100]\nH||[110]\n0510\n0\n5\n10\nLSMO\nLSMO/SRO[110][010]\n[100]\n 0H (mT)\nFigure 6. (a) In-plane angular dependence of\nlinewidth \u0001H at 9 GHz for LSMO(10 nm) and\nLSMO(10 nm)/SRO(7 nm). The solid curves\nindicate \fts to Eq. 8. (b) Frequency depen-\ndence of \u0001H for LSMO(10 nm) and LSMO(10\nnm)/SRO(7 nm) with Happlied along the [100]\ndirection. The solid curves indicate \fts to\nEq. 7. The dashed and dotted curves indicate\nestimated two-magnon and Gilbert damping\ncontributions, respectively. (c) Schematic of a\nspin wave dispersion curve (when the magne-\ntization is in-plane and has a \fnite component\nparallel to the spin wave wavevector k) and two-\nmagnon scattering. (d) Two-magnon scattering\ncoe\u000ecient \u0000 2m, estimated for the cases with H\napplied along the [100] and [110] axes, plotted\nagainst SRO thickness tSRO. The dashed curve\nis the same as that in Fig. 4(c) scaled to serve\nas a guide for the eye for \u0000 2mwith H along\n[100].\nfactor of\u00192 forHjj[100]. By contrast, for Hjj[110], al-\nthough LSMO/SRO exhibits enhanced \u0001 Hcompared to\nLSMO, the enhancement in \u0000 2mis obscured by the un-\ncertainty in Gilbert damping. In Table I, we summa-\nrize the Gilbert and two-magnon contributions to \u0001 H\nfor LSMO single layers and LSMO/SRO (averaged val-\nues for samples with tSRO>4 nm) with Hjj[100] and\nHjj[110]. Comparing the e\u000bective spin relaxation rates,\n(gip\u0016B=h)\u00160Ms\u000band (gip\u0016B=h)\u00160\u00002m, reveals that two-\nmagnon scattering dominates over Gilbert damping.\nWe now speculate on the mechanisms behind the\nenhancement in \u0000 2min LSMO/SRO, particularly for\nHjj[100]. One possibility is that SRO interfaced with\nLSMO directly increases the rate of two-magnon scat-\ntering, perhaps due to formation of additional defects at\nthe surface of LSMO. If this were the case we might ex-\npect a signi\fcant increase and saturation of \u0000 2mat small\ntSRO. However, in reality, \u0000 2mincreases for tSRO>1 nm\n(Fig. 6(d)), which suggests spin scattering in the bulk\nof SRO. We thus speculate another mechanism, where\nk6= 0 magnons in LSMO are scattered by spin pump-\nTable I. Spin relaxation rates extracted from in-plane FMR\n(106s\u00001)\nLSMO LSMO/SRO*\nGilbert:gip\u0016B\nh\u00160Ms\u000b 11\u00062 23\u00064\ntwo-magnon:gip\u0016B\nh\u00160\u00002m(Hjj[100]) 290\u000650 550\u0006100\ntwo-magnon:gip\u0016B\nh\u00160\u00002m(Hjj[110]) 140\u000660 250\u000660\n* Averaged over samples with tSRO>4 nm.ing into SRO. As shown by the guide-for-the-eye curve\nin Fig. 6(d), the tSROdependence of \u0000 2m(forHjj[100])\nmay be qualitatively similar to the tSROdependence of\n\u000bmeasured from out-of-plane FMR (Fig. 4(c)); this cor-\nrespondence would imply that the same spin pumping\nmechanism, which is conventionally modeled to act on\nthek= 0 mode, is also operative in the degenerate k6= 0\nmagnon mode in epitaxial LSMO. Indeed, previous stud-\nies have electrically detected the presence of spin pump-\ning fromk6= 0 magnons by the inverse spin-Hall e\u000bect in\nY3Fe5O12/Pt bilayers64{66. However, we cannot conclu-\nsively attribute the observed FMR linewidth broadening\nin LSMO/SRO to such k6= 0 spin pumping, since it\nis unclear whether faster relaxation of k6= 0 magnons\nshould necessarily cause faster relaxation of the k= 0\nFMR mode. Regardless of its origin, the pronounced\nanisotropic two-magnon scattering introduces additional\ncomplexity to the analysis of damping in LSMO/SRO\nand possibly in other similar ultrathin epitaxial magnetic\nheterostructures.\nV. SUMMARY\nWe have demonstrated all-oxide perovskite bilayers\nof LSMO/SRO that form spin-source/spin-sink systems.\nFrom out-of-plane FMR, we deduce a low Gilbert damp-\ning parameter of \u00191\u000210\u00003for LSMO. The two-fold en-\nhancement in Gilbert damping with an SRO overlayer\nis adequately described by the standard model of spin\npumping based on di\u000busive spin transport. We ar-\nrive at an estimated spin-mixing conductance G\"#\u0019\n(1\u00002)\u00021014\n\u00001m\u00002and spin di\u000busion length \u0015s<\u00181\nnm, which indicate reasonable spin-current transparency\nat the LSMO/SRO interface and strong spin scattering8\nwithin SRO. From in-plane FMR, we reveal pronounced\nnon-Gilbert damping, attributed to two-magnon scatter-\ning, which results in a nonlinear frequency dependence\nand anisotropy in linewidth. The magnitude of two-\nmagnon scattering increases with the addition of an SRO\noverlayer, pointing to the presence of spin pumping from\nnonuniform spin wave modes. Our \fndings lay the foun-\ndation for understanding spin transport and magneti-\nzation dynamics in epitaxial complex oxide heterostruc-\ntures.\nACKNOWLEDGEMENTS\nWe thank Di Yi, Sam Crossley, Adrian Schwartz, Han-\nkyu Lee, and Igor Barsukov for helpful discussions, and\nTianxiang Nan and Nian Sun for the design of the copla-\nnar waveguide. This work was funded by the National\nSecurity Science and Engineering Faculty Fellowship of\nthe Department of Defense under Contract No. N00014-\n15-1-0045.\nAPPENDIX A: SPIN PUMPING AND SRO\nRESISTIVITY\nWhen \ftting the dependence of the Gilbert damping\nparameter\u000bon spin-sink thickness, a constant bulk re-\nsistivity for the spin sink layer is often assumed in lit-\nerature. By setting the resistivity of SRO to the bulk\nvalue\u001aSRO= 2\u000210\u00006\nm and \ftting the \u000b-versus-tSRO\ndata (Fig. 4(c) and reproduced in Fig. 7(a)) to Eq. 5,\nwe arrive at G\"#>\u00183\u00021014\n\u00001m\u00002and\u0015s\u00192:5 nm.\nThe \ft curve is insensitive to larger values of G\"#because\nthe bulk spin resistance 1/ Gext, with the relatively large\nresistivity of SRO, dominates over the interfacial spin re-\nsistance 1/G\"#(see Eqs. 4 and 5). As shown by the dot-\nted curve in Fig. 7, this simple constant- \u001aSROmodel ap-\npears to mostly capture the tSRO-dependence of \u000b. This\nmodel of course indicates \fnite spin pumping at even\nvery small SRO thickness <\u00181 nm, which is likely non-\nphysical since SRO should be insulating in this thickness\nregime54. Indeed,\u0015sestimated with this model should\nprobably be considered a phenomenological parameter:\nAs pointed out by recent studies, strictly speaking, a\nphysically meaningful estimation of \u0015sshould take into\naccount the thickness dependence of the resistivity of the\nspin sink layer39,56,67, especially for SRO whose thickness\ndependence of resistivity is quite pronounced.\nFigure 7(b) plots the SRO-thickness dependence of the\nresistivity of SRO \flms deposited on LSAT(001) mea-\nsured in the four-point van der Pauw geometry. The\ntrend can be described empirically by\n\u001aSRO=\u001ab+\u001as\ntSRO\u0000tth; (9)\nwhere\u001ab= 2\u000210\u00006\nm is the resistivity of SRO in the\nbulk limit,\u001as= 1:4\u000210\u000014\nm2is the surface resistivity\n0 5 10 15 200123\n (10-3)\ntSRO (nm)\n0 10 20 3010-61x10-51x10-4\n SRO (m)\ntSRO (nm)(a) (b) Figure 7. (a) Gilbert damping parameter \u000bversus SRO\nthicknesstSRO. The solid curve is a \ft taking into account\nthetSROdependence of SRO resistivity, whereas the dotted\ncurve is a \ft assuming a constant bulk-like SRO resistivity.\n(b) Resistivity of SrRuO 3\flms on LSAT(001) as a function\nof thickness.\n0 5 10 15 20012345\nhard\neasy\n 0H (mT)\nf (GHz)\n0 5 10 15 200123450H (mT)\n \nf (GHz)hard\neasy(b) (a) \nFigure 8. Frequency dependence of in-plane FMR\nlinewidth \u0001 Hof LSMO(10 nm) on (a) LSAT(001) and (b)\nNdGaO 3(110), with the magnetization along the magnetic\neasy and hard axes. The solid curves are \fts to Eq. 7 with\nthe Gilbert damping parameter \u000b\fxed to 0:9\u000210\u00003.\ncoe\u000ecient, and tth= 1 nm is the threshold thickness\nbelow which the SRO layer is essentially insulating. The\nvalue oftthagrees with literature reporting that SRO\nis insulating at thickness of 3 monolayers ( \u00191.2 nm) or\nbelow54. Given the large deviation of \u001aSROfrom the bulk\nvalue, especially at small tSRO, the trend in Fig. 7(b)\nsuggests that taking into account the tSRO dependence\nof\u001aSROis a sensible approach.\nAPPENDIX B: IN-PLANE DAMPING OF LSMO\nON DIFFERENT SUBSTRATES\nIn Fig. 8, we compare the frequency dependence of \u0001 H\nfor 10-nm thick LSMO \flms deposited on di\u000berent sub-\nstrates: LSAT(001) and NdGaO 3(110). (NdGaO 3is an\northorhombic crystal and has ap\n2-pseudocubic param-\neter of\u00193.86 \u0017A, such that (001)-oriented LSMO grows\non the (110)-oriented surface of NdGaO 3.) As shown\nin Sec. 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Bose National Centre for Basic \nSciences, Block JD, Sector III, Salt Lake, Kolkata 700 098 \n2Advanced Science Institute, RIKEN, 2-1 Hirosawa, Wa ko, Saitama 351-0198, Japan \n3Institute for Solid State Physics, University of To kyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, \nJapan \n \n \n*CORRESPONDING AUTHOR, EMAIL ADDRESS: yotani@issp.u -tokyo.ac.jp \n†abarman@bose.res.in 2 ABSTRACT: We report an all-optical time-domain dete ction of picosecond magnetization \ndynamics of arrays of 50 nm Ni 80Fe 20 (permalloy) dots down to the single nanodot regime . In the \nsingle nanodot regime the dynamics reveals one domi nant resonant mode corresponding to the \nedge mode of the 50 nm dot with slightly higher dam ping than that of the unpatterned thin film. \nWith the increase in areal density of the array bot h the precession frequency and damping \nincreases significantly due to the increase in magn etostatic interactions between the nanodots and \na mode splitting and sudden jump in apparent dampin g are observed at an edge-to-edge \nseparation of 50 nm. \n \nKEYWORDS: Single Nanomagnets Dynamics, Time-resolve d Magneto-optical Kerr Effect, \nMagnetization Precession, Damping, Magnetostatic In teraction. 3 The quest to measure the ultrafast magnetization d ynamics of nanomagnets continues to be an \nimportant problem in nanoscience and nanotechnology .1-9 Picosecond magnetization dynamics of \nnanoscale magnetic structures is important for many present and future technologies including magnetic \ndata storage,10-11 , logic devices,12-14 spintronics,15 and magnetic resonance imaging.16 Emerging \ntechnologies such as spin torque nano-oscillators 17 and magnonic crystals 18-19 rely heavily upon the fast \nand coherent dynamics of nanomagnets and the genera tion and manipulation of spin waves in spatially \nmodulated magnetic nanostructures. Novel techniques for fabrication of nanomagnets arrays 20 and \napplications towards biomedicine 21 show exciting new promises. Overall, the detection and \nunderstanding of nanomagnet dynamics down to the si ngle nanomagnet regime have become \nincreasingly important. Investigation of picosecond dynamics of arrays of nano-scale magnetic dots has \ninferred that, for dot sizes less than 200 nm, the response of the magnetization to a pulsed magnetic field \nis spatially non-uniform and is dominated by locali zed spin wave modes.22 This non-uniformity may \nresult in a degradation of the signal to noise rati o in future nanomagnetic devices. However, the \nmeasurements were done in densely packed arrays whe re the intrinsic dynamics of the individual dots \nare strongly influenced by the magnetostatic stray fields of the neighbouring dots. Magnetostatically \ncoupled nanomagnets in a dense array may show colle ctive behaviors both in the quasistatic \nmagnetization reversal 23 and in the precessional dynamics.22, 24-28 In the quasistatic regime the strong \ninter-dot magnetostatic interactions result in coll ective rotation of magnetic spins and formation of flux \nclosure through a number of dots during the reversa l. On the other hand, in the collective precessiona l \ndynamics the constituent nanomagnets maintain defin ite amplitude and phase relationships. \nMagnetization dynamics in dense arrays of nanomagne ts have been studied both experimentally by time-\ndomain,22,24 frequency-domain 25-26 and wave-vector-domain 27-28 techniques; and theoretically by \nanalytical 29-30 and micromagnetic 31 methods. To this end the frequency, damping and sp atial patterns of \nspin waves and dispersion relations of frequency wi th wave-vector of spin wave propagation have been \nstudied. \n 4 On the other hand, magnetization dynamics of isolat ed nanomagnets with lateral dimensions \ndown to 125 nm have been reported by time-resolved magneto-optical techniques.3,5,6,8,9 However, \npicosecond magnetization dynamics including the dam ping behavior of isolated nanomagnets down to \n50 nm size has never been reported. Here, we presen t an all-optical far field measurement of the \npicosecond magnetization dynamics of arrays of squa re Ni 80 Fe 20 (permalloy) dots with 50 nm width and \nwith varying edge-to-edge separation ( S) between 200 nm and 50 nm. When the dots are separ ated by \nlarge distance ( S ≥ 150 nm) they reveal the dynamics of the isolated n anomagnet. The isolated \nnanomagnets revealed a single resonant mode, whose damping is slightly higher than the unpatterned \nthin film value. With the decrease in inter-dot sep aration the effects of dipolar and quadrupolar \ninteractions become important, and we observe an in crease in precession frequency and damping. At the \nhighest areal density a sudden jump in the apparent damping is observed due to the mutual dephasing of \ntwo closely spaced eigenmodes of the array. \n \nRESULS AND DISCUSSION \n10 µm × 10 µm square arrays of permalloy dots with nominal dim ensions as 50 nm width, 20 \nnm thickness and separation S varying from 50 nm to 200 nm were prepared by a co mbination of \nelectron beam evaporation and electron-beam lithogr aphy. Figure 1(a) presents the scanning electron \nmicrographs of three of these dot arrays, which sho w that there are some deviations in the shape and \ndimensions of the samples from the nominal shape an d dimensions, although the general features are \nmaintained. A square permalloy dot with 10 µm width and 20 nm thickness was also prepared to ob tain \nthe magnetic parameters of the unpatterned sample. The ultrafast magnetization dynamics was measured \nby using a home-built time-resolved magneto-optical Kerr effect microscope based upon a two-color \ncollinear pump-probe set-up.32 The two-color collinear arrangement enabled us to achieve a very good \nspatial resolution and sensitivity even in an all-o ptical excitation and detection scheme of the \nprecessional dynamics. A schematic of the measureme nt geometry is shown in Fig. 1(b). The time- 5 resolved data was recorded for a maximum duration o f 1 ns and this was found to be sufficient to recor d \nall important features of the dynamics including th e spectral resolution of the double peaks for the \nsample with S = 50 nm and measurement of the damping coefficient . Figure 1(c) shows the time-\nresolved reflectivity and Kerr rotation data from t he array with separation S = 50 nm at a bias field H = \n2.5 kOe. The reflectivity shows a sharp rise follow ed by a bi-exponential decay. On the other hand the \ntime-resolved Kerr rotation shows a fast demagnetiz ation within 500 fs and a bi-exponential decay with \ndecay constants of about 8 ps and 116 ps. The demag netization and decay times are found to be \nindependent of the areal density of the arrays. The precessional dynamics appears as an oscillatory \nsignal 2 above the decaying part of the time-resolved Kerr rotation data. The bi-exponential background \nis subtracted from the time-resolved Kerr signal be fore performing the fast Fourier transform (FFT) to \nfind out the corresponding power spectra. \n \nFigure 2 shows the time-resolved Kerr rotation from the permalloy dot arrays with S varying \nbetween 50 nm and 200 nm at H = 2.5 kOe. Clear precession is observed down to S = 200 nm, where the \ndots are expected to be magnetostatically isolated and hence exhibit single dot like behavior. The \ncorresponding FFT spectrum (Fig. 2(b)) shows a domi nant single peak at 9.04 GHz. As S decreases the \nprecession continues to have a single resonant mode but the peak frequency generally increases with th e \ndecrease in S. For S = 150 nm, the peak frequency decreases slightly al though the errors bars are large \nenough to maintain the general trend of increase in the frequency with the decrease in S as stated above. \nAt S = 50 nm the single resonant mode splits into two c losely spaced modes with the appearance of a \nlower frequency peak. In Fig. 2(c), we show the FFT spectra of the time-resolved magnetization \nobtained from micromagnetic simulations of arrays o f 7 × 7 dots using the OOMMF software.33 In \ngeneral, the deviation in the shape and dimensions as observed in the experimental samples are include d \nin the simulated samples but the precise edge rough ness profiles and deformations are not always \npossible to include in the finite difference method based micromagnetic simulations used here, where \nsamples are divided into rectangular prism like cel ls. In the simulation the arrays were divided into cells 6 of 2.5 × 2.5 × 20 nm 3 dimensions and material’s parameters for permalloy were used as γ = 18.5 \nMHz/Oe, HK = 0, MS = 860 emu/cc and A = 1.3 × 10 -6 erg/cm. The material's parameters for permalloy \nwere obtained by measuring the precession frequency of the unpatterned thin film as a function of the in-\nplane bias field and by fitting the bias field vari ation of frequency with Kittel's formula. The excha nge \nstiffness constant A was obtained from literature.34 The lateral cell size is well below the exchange \nlength 2\n02\nSex MAlµ= of permalloy (5.3 nm) and further reduction of cell size does not change the \nmagnetic energies appreciably. Test simulations wit h discretization along the thickness of the samples \ndo not show any variation in the resonant modes, wh ich is expected as this will only affect the \nperpendicular standing spin waves, whereas in the p resent study we have concentrated on the spin-\nwaves with in-plane component of wave-vector. The equilibrium states are obtained by allowing the \nsystem to relax under the bias field for sufficient time so that the maximum torque ( m × H, m = M/MS) \ngoes well below 10 -6 A/m. The dynamic simulations were obtained for a total duration of 4 ns at time \nsteps of 5 ps. Consequently, the simulated linewidt hs of the resonant modes are narrower, which enable d \nus to clearly resolve the mode splitting in the sim ulation. The simulation reproduces the important \nfeatures as observed in the experiment, namely the observation of a single resonant mode for the array s \nwith S varying between 200 nm and 75 nm, a systematic inc rease in the resonant mode frequency with \nthe decrease in S, and finally a mode splitting at S = 50 nm. However, the increase in the resonant \nfrequency with decrease in S is less steep as compared to the experimental resu lt. The deviation is larger \nfor smaller values of S possibly due to the increased non-idealities in th e physical structures of the \nsamples in this range, as discussed earlier. Furthe rmore the relative intensities of the two modes \nobserved for the array with S = 50 nm are not reproduced by the simulation. This is possibly because the \nlower frequency mode is a propagating mode and the finite boundary of the simulated array of 7 × 7 \nelements may cause much faster decay of the propaga ting mode as opposed to that in the much larger \narray of 100 × 100 elements studied experimentally. In Fig. 3(a), we plot the precession frequency as a \nfunction of the ratio of width ( w) to centre to centre separation ( a), where a = w + S. For w/a ≤ 0.25 ( S ≥ 7 150 nm) the frequency is almost constant but for w/a > 0.25 ( S < 150 nm) the frequency increases \nsharply both for the experimental and simulated dat a. We fit both data with Eq. 1 including both dipol ar \nand quadrupolar interaction terms.35 \n 5 3\n0 \n\n+\n\n− =awBawAff [1], \nwhere A and B are the strengths of the dipolar and quadrupolar i nteractions. The fitted data are shown by \nsolid lines in Fig. 3(a). The simulated data fits w ell with Eq. 1, while the fit is reasonable for the \nexperimental data, primarily due to the large devia tion in data points for the arrays with S = 75 and 150 \nnm. However, the theoretical curve passes through t he error bars for those data points. The quadrupola r \ncontribution is dominant over the dipolar contribut ion as is also evident from the sharp increase in t he \nfrequency for w/a > 0.25. The dipolar contributions extracted from t he curve fitting are almost identical \nfor both experimental and simulated results, wherea s for the experimental data the quadrupolar \ncontribution is about 30% greater than that for the simulated data.. \n \nVARIATION OF DAMPING OF PRECESSION WITH AREAL DENSI TY OF THE ARRAYS \nWe have further investigated the damping behavior o f the nanomagnets in the array. The time \ndomain data was fitted with a damped sine curve \n( ) φ πτ− =−\nft e M tMt\n2 sin ) 0 ( )( [2], \nwhere the relaxation time τ is related to the Gilbert damping coefficient α by the relation \nαπτf21= , f is the experimentally obtained precession frequenc y and φ is the initial phase of the \noscillation. The fitted data is shown by solid line s in Fig. 2(a). The damping coefficient α, as extracted \nfrom the above fitting, is plotted as a function of the inter-dot separation S along with the error bars in 8 Fig. 3(b). The sample with S = 200 nm shows the lowest α of about 0.023. This value of α is slightly \nhigher than the damping coefficient (0.017) measure d for a permalloy film of 20 nm thickness grown \nunder identical conditions to those for the arrays of permalloy dots. Since the dots are magnetostatic ally \nisolated, the increase in damping due to the mutual dynamic dephasing of the permalloy dots is unlikel y \nfor S = 200 nm. Another possibility is the dephasing of more than one mode within the individual dots,36 \nwhich is also ruled out due to the appearance of a dominant single mode in the individual dots. Hence, \nwe believe this increase in damping is possibly due to the defects 37 produced in these dots during \nnanofabrication, which is quite likely due to the s mall size of these dots. As S decreases, the \nmagnetostatic interaction between the dots becomes more prominent and hence the mutual dephasing of \nslightly out-of-phase magnetization precession of t he dots in the array becomes more prominent,31 and \nconsequently α increases systematically with the decrease in S down to 75 nm. At S = 50 nm a different \nsituation arises, where the single resonant mode sp lits into two closely spaced modes and the apparent \ndamping (square symbol in Fig. 3(b)) of the time-do main oscillatory signal jumps suddenly from 0.032 \nto 0.066. Clearly, this is due to the out-of-phase superposition of two closely spaced modes within th e \narray, as shown later in this article. In order to understand the correct damping behavior of the unif orm \nresonant mode we have isolated the time-domain sign al for the mode 1 from the lower lying mode \n(mode 2) by using fast Fourier filtering. The extra cted damping of the filtered time-domain signal for the \nsample with S = 50 nm is about 0.033, which is consistent with t he systematic increase in the damping \ncoefficient of the arrays with decreasing S, as shown by the circular symbols in Fig. 3(b). \n \nMICROMAGNETIC ANALYSIS OF THE OBSERVED PRECESSIONAL DYNAMICS \nIn order to gain more insight into the dynamics, we have calculated the magnetostatic field \ndistribution of the simulated arrays and the contou r plot of the magnetostatic fields from the 3 × 3 d ots \nat the centre of the array is shown in Fig. 4. At l arger separations the stray fields from the dots re main \nconfined close to their boundaries and the interact ions between the dots is negligible. As the inter-d ot \nseparation decreases the stray fields of the neighb oring dots start to overlap causing an increase in the 9 effective field acting on the dots and consequently the corresponding precession frequency. At S = 50 \nnm the stray field is large enough to cause a stron g magnetostatic coupling between the dots and hence \nthe collective precession modes of the dots in the array.31 The spatial natures of the modes were \ninvestigated by numerically calculating the spatial distributions of amplitudes and phases correspondi ng \nto the resonant modes of the samples. The amplitude and phase maps of resonant modes for the arrays \nwith S = 50 nm and 200 nm are shown in Figs. 5 (a) - (b). For S = 50 nm, the main resonant mode \n(mode 1) corresponds to the in-phase precession of majority of the dots in the array apart from the do ts \nnear the edges. The intensities of the dots increas e from the edge to the centre of the array. The lo wer \nfrequency peak (mode 2), on the other hand, shows t hat the dots in the consecutive columns precess out -\nof-phase, while the dots in the alternative columns precess in-phase. The intensity again shows small \nvariation from the edge to the centre of the array. The spatial variation of the phase of precession o f the \ndots is similar to the magnetostatic backward volum e modes with the wave-vector parallel to the bias \nmagnetic field ( H) and both lie within the plane of the sample. For S = 200 nm, the single resonant mode \n(mode 1) corresponds to the precession of the indiv idual dots and hence all of them have identical \namplitude and phase. For comparison we have calcula ted the amplitude and phase maps of the only \nresonant mode of a single 50 nm wide dot (Figs. 5(c ) – (e)) with different cell size ((c): 2.5 × 2.5 × 20 \nnm 3, (d): 1 × 1 × 20 nm 3, and (e): 2.5 × 2.5 × 5 nm 3), which is found to be the edge mode 22, 35, 38 that \noccupies the major fraction of the volume of the do t. Important to note that the mode structure remai ns \nindependent on the chosen cell size. A closer view to the central dot of the array with S = 200 nm shows \n(Fig. 5(f)) an identical mode structure to that of the single dot, ensuring that in this array the dyn amics is \ndominated by that of the single dot. \n \nCONCLUSIONS \nIn summary, we have detected the picosecond precess ional dynamics in arrays of 50 nm \npermalloy dots down to the single nanodot regime by an all-optical time-resolved magneto-optical Kerr \neffect microscope. The inter-dot separation ( S) varies from 200 nm down to 50 nm and numerical 10 calculation of magnetostatic fields shows a transit ion from magetostatically isolated regime to strong ly \ncoupled regime as S decreases. Consequently, we observe a single prece ssional mode for S down to 75 \nnm, whose frequency increases with the decrease in S. This has been analytically modeled by \nintroducing the dipolar and quadrupolar contributio ns to the precession frequency. At the smallest \nseparation S = 50 nm, we observe a splitting of the resonant mo de and a lower frequency mode appears \nin addition to the existing mode. Micromagnetic sim ulations reproduce the above observations \nqualitatively. Analyses of amplitude and phase maps of the resonant modes reveal that the dynamics of a \nsingle dot with 50 nm width is dominated by the edg e mode. In sparsely packed arrays ( S ≥ 150 nm) we \nprimarily observe the isolated dynamics of the cons tituent dots, all in phase. For S = 50 nm, the observed \nmodes correspond to the uniform collective precessi on of the array (higher frequency mode) and an out-\nof-phase precession of the alternative columns of t he array parallel to the bias field (lower frequenc y \nmode). The damping also shows significant variation with the areal density. For S = 200 nm i.e. , in the \nsingle nanodot regime, the damping is minimum at ab out 0.023, which is slightly higher than the \ndamping coefficient (0.017) of a permalloy thin fil m of same thickness. We understand this slight \nincrease in damping is a result of the defects intr oduced in the dots during nanofabrication. However, \nthe damping increases further with the decrease in S as a result of the dynamic dephasing of the \nprecession of the weakly interacting dots. At S = 50 nm, the dephasing due to the superposition of two \nresonant modes results in a sudden increase in the apparent damping of the precession. The ability of all-\noptical detection of the picosecond dynamics of 50 nm dots down to the single nanomagnet regime and \nunderstanding of the effects of magnetostatic inter action on those dots when placed in a dense array w ill \nbe important from fundamental scientific viewpoint as well as for their future applications in various \nnanomagnetic devices. \n \nMETHODS \nSquare arrays of permalloy dots were prepared by a combination of electron beam evaporation \nand electron-beam lithography. A bilayer PMMA (poly methyl methacrylate) resist pattern was first 11 prepared on thermally oxidized Si(100) substrate by using electron-beam lithography and permalloy was \ndeposited on the resist pattern by electron-beam ev aporation at a base pressure of about 2 × 10 -8 Torr. A \n10 nm thick SiO 2 capping layer was deposited on top of permalloy to protect the dots from degradation \nwhen exposed to the optical pump-probe experiments in air. This is followed by the lifting off of the \nsacrificial material and oxygen plasma cleaning of the residual resists that remained even after the l ift-\noff process. \nThe ultrafast magnetization dynamics was measured b y using a home-built time-resolved \nmagneto-optical Kerr effect microscope based upon a two-color collinear pump-probe set-up. The \nsecond harmonic ( λ = 400 nm, pulse width ~ 100 fs) of a Ti-sapphire la ser (Tsunami, SpectraPhysics, \npulse-width ~ 80 fs) was used to pump the samples, w hile the time-delayed fundamental ( λ = 800 nm) \nlaser beam was used to probe the dynamics by measur ing the polar Kerr rotation by means of a balanced \nphoto-diode detector, which completely isolates the Kerr rotation and the total reflectivity signals. The \npump power used in these measurements is about 8 mW , while the probe power is much weaker and is \nabout 1.5 mW. The probe beam is focused to a spot s ize of 800 nm and placed at the centre of each arra y \nby a microscope objective with numerical aperture N . A. = 0.65 and a closed loop piezoelectric scannin g \nx-y-z stage. The pump beam is spatially overlapped with the probe beam after passing through the same \nmicroscope objective in a collinear geometry. Conse quently, the pump spot is slightly defocused (spot \nsize ~ 1 µm) on the sample plane, which is also the focal pla ne of the probe spot. The probe spot is \nplaced at the centre of the pump spot as shown in F ig. 1(b). A large magnetic field is first applied a t a \nsmall angle (~ 15°) to the sample plane to saturate its magnetization. The magnetic field strength is t hen \nreduced to the bias field value ( H = component of bias field along x-direction), which ensures that the \nmagnetization remains saturated along the bias fiel d direction. The bias field was tilted 15° out of t he \nplane of the sample to have a finite demagnetizing field along the direction of the pump pulse, which is \neventually modified by the pump pulse to induce pre cessional magnetization dynamics within the dots. \nThe pump beam was chopped at 2 kHz frequency and a phase sensitive detection of the Kerr rotation \nwas used. 12 Acknowledgement The authors gratefully acknowledge the financial s upports from Department of \nScience and Technology, Government of India under t he grant numbers SR/NM/NS-09/2007, \nSR/FTP/PS-71/2007, INT/EC/CMS (24/233552) and INT/J P/JST/P-23/09 and Japan Science and \nTechnology Agency Strategic International Cooperati ve Program under the grant numbers 09158876. \n \n \nREFERENCES AND NOTES \n \n1. Krivorotov, I. N.; Emley, N. C.; Sankey, J. C.; Kiselev, S. I.; Ralph, D. C.; Buhrman, R. A. Time-\nDomain Measurements of Nanomagnet Dynamics Driven b y Spin-transfer Torques. Science 2005 , 307 , \n228-231. \n2. 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B 2007 , 75 , 024416. \n30. Bondarenko, P. V.; Galkin, A. Yu.; Ivanov, B. A .; Zaspel, C. E. Collective Modes for an Array of \nMagnetic Dots with Perpendicular Magnetization. Phys. Rev. B 2010 , 81 , 224415. \n31. Barman, A.; Barman, S. Dynamic Dephasing of Mag netization Precession in Arrays of Thin \nMagnetic Elements. Phys. Rev. B 2009 , 79 , 144415. \n32. Pal, S.; Rana, B.; Hellwig, O.; Thomson, T.; Ba rman, A. Tunable Magnonic Frequency and \nDamping in [Co/Pd] 8 Multilayers. Appl. Phys. Lett. 2011 , 98 , 082501. \n33. Donahue, M.; Porter, D. G. OOMMF User’s guide, Version 1.0, Interagency Report NIST IR 6376, \nNational Institute of Standard and Technology, Gait hersburg, MD (1999): URL: \nhttp://math.nist.gov/oommf. \n34. Buschow, K. H. J.: Handbook of Magnetic Materials (North Holland, Amsterdam, 2009) vol. 18, p. \n168. \n35. Awad, A. A.; Aranda, G. R.; Dieleman, D.; Gusli enko, K. Y.; Kakazei, G. N.; Ivanov, B. A.; Aliev, \nF. G. Spin Excitation Frequencies in Magnetostatica lly Coupled Arrays of Vortex State Circular \nPremalloy Dots. Appl. Phys. Lett. 2010 , 97 , 132501. 16 36. Barman, A.; Kruglyak, V. V.; Hicken, R. J.; Row e, J. M.; Kundrotaite, A.; Scott, J.; Rahman, M. \nImaging the Dephasing of Spin Wave Modes in a Squar e Thin Film Magnetic Element. Phys. Rev. B \n2004 , 69 , 174426. \n37. Sparks, M. Ferromagnetic Relaxation Theory, McGraw-Hill, New York, 1966 . \n38. Jorzick, J.; Demokritov, S. O.; Hillebrands, B. ; Bailleul, M.; Fermon, C.; Guslienko, K. Y.; Slavi n, \nA. N.; Berkov, D. V.; Gorn, N. L. Spin Wave Wells i n Nonellipsoidal Micrometer Size Magnetic \nElements. Phys. Rev. Lett. 2002 , 88 , 047204. 17 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1. (a) Scanning electron micrographs of arrays of perm alloy dots with width = 50 nm, thickness \n= 20 nm and with varying separation S = 50 nm, 100 nm, and 200 nm. (b) A schematic of th e two color \npump-probe measurement of the time-resolved magneti zation dynamics of the nanomagnets. (c) Typical \ntime-resolved reflectivity and Kerr rotation data a re shown for the array with S = 50 nm at H = 2.5 kOe. \n \n \n \n B. Rana et al. \nS = 50nm S = 100nm S = 200nm \n(b) xyz\nHbias \n~ 15 °80 fs \n100 fs Pump Probe \n∆t(a) \n0.0 0.5 1.0 \n0 200 400 600 800 1000 -1.0 -0.5 0.0 Width = 50 nm \nSeparation = 50 nm \nTime (ps) Kerr rotation (arb. unit) Reflectivity (arb. unit) \n \n \n(c) \n \n \nS = 50nm S = 100nm S = 200nm \n S = 50nm S = 100nm S = 200nm \n(b) xyz\nHbias \n~ 15 °80 fs \n100 fs Pump Probe \n∆t\n(b) xyz\nHbias \n~ 15 °80 fs \n100 fs Pump Probe \n∆t80 fs 80 fs \n100 fs Pump Probe \n∆t(a) \n0.0 0.5 1.0 \n0 200 400 600 800 1000 -1.0 -0.5 0.0 Width = 50 nm \nSeparation = 50 nm \nTime (ps) Kerr rotation (arb. unit) Reflectivity (arb. unit) \n \n \n(c) \n \n \n 18 \n \n \n \n \n \n \n \n \n \n \nFigure 2. (a) Experimental time-resolved Kerr rotations and ( b) the corresponding FFT spectra are \nshown for arrays of permalloy dots with width = 50 nm, thickness = 20 nm and with varying inter-dot \nseparation S at H = 2.5 kOe. (c) The FFT spectra of the simulated ti me-resolved magnetization are \nshown. The peak numbers are assigned to the FFT spe ctra. The dotted line in (c) shows the simulated \nprecession frequency of a single permalloy dot with width = 50 nm, thickness = 20 nm. \n \n \n \n B. Rana et al. 0 250 500 750 1000 0 5 10 15 20 5 10 15 \n S = 100 nm \n \n S = 75 nm \n \n S = 50 nm \n \n S = 150 nm \n S = 200 nm \nTime (ps) Kerr rotation (arb. unit) \n 1 \n12\n1\n(b) \nFrequency (GHz) Power (arb. unit) \n \n \n 1\n \n 1\n \n 1\n2 1\n(a) \n \n 1\n 1\n(c) \n \n 1\n 19 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3. (a) The precession frequency is plotted as a functi on of w/a . The circular and square symbols \ncorrespond to the experimental and simulated result s, respectively, while the solid curves correspond to \nthe fit to Eq. 1. (b) The damping coefficient α is plotted as a function of S. The symbols correspond to \nthe experimental data, while the solid line corresp onds to a linear fit. The dashed line corresponds t o the \nmeasured value of α for a continuous permalloy film grown under identi cal conditions. \n \n \n \n B. Rana et al. 50 100 150 200 0.01 0.02 0.03 0.04 0.05 0.06 0.07 \n \nαααα\nS (nm) Mode 1 \n Apparent \n Linear Fit \n(b) 0.2 0.3 0.4 0.5 8.8 9.2 9.6 10.0 10.4 \n(a) \n Frequency (GHz) \nw/a Experimental \n Simulation \n50 100 150 200 0.01 0.02 0.03 0.04 0.05 0.06 0.07 \n \nαααα\nS (nm) Mode 1 \n Apparent \n Linear Fit \n(b) 0.2 0.3 0.4 0.5 8.8 9.2 9.6 10.0 10.4 \n(a) \n Frequency (GHz) \nw/a Experimental \n Simulation \n 20 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 4. Simulated magnetostatic field distributions (x-comp onent) are shown for arrays of permalloy \ndots with S = 50 nm, 75 nm, 100 nm, 150 nm and 200 nm at H = 2.5 kOe. The arrows inside the dots \nrepresent the magnetization states of the dots, whi le the strengths of the stray magnetic fields are \nrepresented by the color bar at the top right corne r of the figure. \n \nB. Rana et al. \nS= 75 nm \n S= 50 nm \nS= 100 nm \n+4.5kOe \n-4.5kOe \nS= 150 nm \n S= 200 nm \nS= 75 nm \n S= 50 nm \nS= 100 nm \n+4.5kOe \n-4.5kOe \n+4.5kOe \n-4.5kOe \nS= 150 nm \n S= 200 nm 21 \n \n \n \n \n \n \n \n \n \n \nFigure 5. The amplitude and phase maps corresponding to diffe rent resonant frequencies are shown for \nthe arrays with (a) S = 50 nm and (b) S = 200 nm. We have also simulated the amplitude map s for a \nsingle 50 nm dot with 20 nm thickness with differen t cell size as (c) 2.5 × 2.5 × 20 nm 3, (d) 1 × 1 × 20 \nnm 3, and (e) 2.5 × 2.5 × 5 nm 3 and compared it with (f) the central dot from the 7 × 7 array with S = 200 \nnm. The color bars at the top of the images represe nt the amplitude and phase values within the images . \n \nB. Rana et al. \n(a) \nS= 200 nm, mode 1 (b) \n(c) (d) \n(e) (f) S= 50 nm, mode 1 \n S= 50 nm, mode 2 Amplitude (arb. unit) Phase (radian) \n-3-2-10123\n60 80 100 120 140 160 180 200 220 \n(a) \nS= 200 nm, mode 1 (b) \n S= 200 nm, mode 1 (b) \n(c) (d) \n(e) (f) \n(c) (d) \n(e) (f) S= 50 nm, mode 1 \n S= 50 nm, mode 2 \nS= 50 nm, mode 2 Amplitude (arb. unit) Phase (radian) \n-3-2-10123\n-3-2-10123\n60 80 100 120 140 160 180 200 220 \n " }, { "title": "1711.05142v1.Spin_Noise_and_Damping_in_Individual_Metallic_Ferromagnetic_Nanoparticles.pdf", "content": "arXiv:1711.05142v1 [cond-mat.mes-hall] 14 Nov 2017Spin-Noise and Damping in Individual Metallic Ferromagnet ic Nanoparticles\nW. C. Jiang, G. Nunn, P. Gartland, and D. Davidovi´ c\nSchool of Physics, Georgia Institute of Technology, Atlant a, GA 30332\n(Dated: September 10, 2018)\nWe introduce a highly sensitive and relatively simple techn ique to observe magnetization motion\nin single Ni nanoparticles, based on charge sensing by elect ron tunneling at millikelvin temperature.\nSequential electron tunneling via the nanoparticle drives nonequilibrium magnetization dynamics,\nwhich induces an effective charge noise that we measure in rea l time. In the free spin diffusion\nregime, where the electrons and magnetization are in detail ed balance, we observe that magnetic\ndamping time exhibits a peak with the magnetic field, with a re cord long damping time of ≃10 ms.\nMeasuring magnetization motion in single magnetic\nnanoparticles in real time has been a longstanding goal\nin solid state physics. Magnetic nanoparticles are a\nbridge between bulk ferromagnets and single electron\nspins, and can have extraordinary magnetic character-\nistics. Metallic ferromagnetic nanoparticles, for exam-\nple, exhibit competitions between superconductivity and\nferromagnetism, [1] entanglement between charge and\nspin degrees of freedom, [2, 3] and geometric quantum\nnoises of spin. [4] We may also suggest that the damp-\ning characteristics of nanoparticles are extraordinary. It\nhas been widely believed, until recently, that the damp-\ning time in ferromagnets cannot be arbitrarily long. [5–\n7] However, spins in semiconducting quantum dots prove\notherwise: exceptionally long relaxation times of single\nelectron spins have been observed, of up to ∼170 ms\nin GaAs, [8] and ∼6 s on P-donors in Si. [9] Being\nthe bridge between bulk and single electron spins, mag-\nnetic nanoparticles may also have unusually long damp-\ning time. However, the damping time in single metallic\nferromagnetic nanoparticles has not yet been measured.\nAmong various techniques that determine the mag-\nnetization motion of individual ferromagnetic nanopar-\nticles, SQUIDs (superconducting-quantum-interference-\ndevices) have the highest sensitivity, of up to ∼\n1µB/√\nHz, [10] where µBis the Bohr magneton.\nSQUIDs allow measurements of the magnetization rever-\nsal process with unprecedented detail. [11, 12] Notable\nexamples include studies of FePt nanobeads [13], Co\nnanoparticles, [14] and ferritin, [15] with magnetic mo-\nments of approximately 106µB, 2200µB, and 300 µB, re-\nspectively. Due to the relatively large size of the SQUID\npickup loop, however, measuring nanoparticles becomes\nprogressively more difficult as the magnetic moments of\nthe nanoparticles are reduced. Consequently, SQUIDs\narenotusedfordetecting spin-1/2statesin semiconduct-\ning quantum dots or P-donors in Si. Instead, these spin-\n1/2 states are measured using single-shot spin-readout\ntechnique, [16–18] where a spin signal is converted into\na charge signal. The latter signal can be measured with\nrelative ease using single-electronics or quantum point\ncontacts. In that vein, here we adapt spin-to-charge\nconversion to observe magnetization motion in individ-\nual metallic ferromagnetic nanoparticles, by measuring\nFIG. 1. (a) Transmission electron microscope image of Ni\nnanoparticles on amorphous alumina substrate. The square\nin the upper right has area of 10 ×10 nm2. (b) Zoomed-in\nnanoparticles within the square demonstrate that they are\nsingle crystal. (c) Schematic of our tunneling device used f or\nmagnetic sensing of a single nanoparticle. The silver-pain ted\nregions indicate the Al leads. The white marbleized region\nindicates the alumina tunneling barrier.\nan internal charge displacement induced by the magne-\ntization displacement. Since spin-to-charge conversion is\neffective in detecting motion of single spins, [16, 18] our\ntechnique does not suffer from the difficulty due the re-\nduced magnetic moment of the nanoparticles. Although\nwe measure the magnetization indirectly, the technique\nis self-calibrating, because the chemical potential versus\nmagnetization orientation is measured independently us-\ning tunneling spectroscopy of discrete levels in magnetic\nfield. Our main results are the observation of record long\nmagnetic damping time of approximately 10 ms in a fer-\nromagnet, finding a peak in damping time versus mag-\nnetic field, and a physical interpretation of the effect.\nSingle crystal Ni nanoparticles that we measure are\nshownin Fig. 1. In the ferromagneticstate, the minority-\nspin electron-box level spacing can be estimated using\nthe nanoparticle volume and band structure calculation\nofthe density ofstates, [19] δ= 0.45±0.18meV, with the\nuncertaintydue tovolumefluctuations amongnanoparti-\ncles. Thespinmagnitude S= 420isestimatedasthevol-\nume times bulk magnetization divided by µB. Fig. 1(c)\nshows a schematic of the tunneling device we use for\nmeasuringthe nanoparticle. All measurements presented2\nhere are at 70 mK ambient temperature, using a current\namplifier with time constant T= 0.3 s. More details\naboutsample fabricationand measurementsareprovided\nin the supplementary document S1.\nFig. 2(a) displays the IV-curves of sample 1 measured\nat fixedB. NearB= 0, the Coulomb blockade voltage\nthreshold is 15 .2 mV. The current exhibits discrete steps\nwithVdue to the electron-in-a-box quantization. [20]\nThe electron temperature in the leads 150 mK is ob-\ntained from the width of the steps at B= 0. When we\nsweepBat fixed V, current does not display magnetic\nhysteresis. Fig. 2(c,d) displays dI/dVversus voltage at\nfixedB. AsBvaries, the peak voltages [or step voltages\ninI(V)] shift non-monotonically and differently among\nthe levels, resembling prior work on magnetic quantum\ndots. [21–25] These type of shifts are the consequence of\nspin-orbit coupling between the magnetization orienta-\ntion and the electronic states. [26, 27] To minimize the\nZeemanplusspin-orbitenergy, the groundstatemagneti-\nzation unity vector m(B) changes orientation with mag-\nnetic field, thereby inducing the energy level spin-orbit\nshiftǫn[m(B)]. It is striking that the levels shift nonlin-\nearly even in the high magnetic field range of 6 −11.5 T.\nThe magnetization is mostly collinear with the magnetic\nfield at high magnetic fields, [28] implying that the effect\nof spin-orbit coupling on electronic states is so strong\nthat the small magnetization displacements that remain\nabove 6 T lead to significant changes of the level ener-\ngies. That is, the level energy is in a sense a sensitive\ndetector of magnetization displacement. The levels ap-\nproach negative slope in voltage versus field at 11 .5 T,\ncomparable to the expected slope −µBfrom the Zeeman\nshift [red line in Fig. 2(d)]. At 11 .5 T, the difference ∆ n\nbetween the measured (e.g., spin-orbit plus Zeeman) and\nthe expected (e.g., Zeeman) shifts fluctuates among the\nlowest three levels, i.e., ∆ n= 0.50, 0.90, and 1 .05 meV\nforn= 1,2,3, with the corresponding root-mean-square\nvaluerms(∆) = 0 .23 meV. The rmsvalue is also an\nestimate of the rmsenergy level for the isotropic mag-\nnetization distribution, within a factor of 2. (See supple-\nmentary document S2 for the derivation.)\nA shift of the electronic energy induced by a magneti-\nzation displacement implies that the magnetization will\nalso shift if the electronic subsystem is displaced. A sig-\nnature of this magnetic reaction is the magnetic field de-\npendence of current noise, which is the first key result\nof this paper. As shown by the IV curves and conduc-\ntance maps in Fig. 2, the field intervals of ±(0.7,1.5) T\nexhibit enhanced current and conductance noise at high\nvoltage, with the typical tunneling current in the noisy\nregionIn= 2 pA. This magnetic field induced enhance-\nment of the noise is particularly striking if we plot the\ndifferential conductance traces [Fig. 2(b)]. Fig. 3(a) dis-\nplaysrmscurrent noise versus magnetic field, obtained\nfrom those conductance traces. (We calculate rmscon-\nductance in voltage interval (17 ,22) mV and multiply by\nFIG. 2. Sample 1 at 70 mK: (a) Current versus voltage at\nfixed magnetic fields indicated by the numbers on panel a\nand between pairs of panels on left and right. (b) Differentia l\nconductance versus voltage, at three fixed fields, showing pr o-\nnouncednoise at1 .2Tandhighvoltage. The spacingbetween\ntick marks is 10−4e2/h. In (a,b) the curves are offset verti-\ncally for clarity. (c,d) Differential conductance maps, sho wing\ndiscrete level shifts with magnetic field. The enhanced cur-\nrent noise in the narrow magnetic field range is indicated by\nyellow boxed arrows to the right. Color bar indicates the con -\nductance interval ( −10−4,5×10−4)e2/h. The yellow straight\nlines are the expected Zeeman shifts of the lowest level.\nthe voltage increment.) By fitting rms(I) versus Bto a\nGaussian, we obtain peak noise field of BP≈1.4 T and\nthe excess noise of rmsP(I)≈30 fA at B=BP.\nBearing in mind that the tunneling current in a single\nelectron transistor is sensitive to charge fluctuations in\nthe surrounding dielectric, we may calculate the chem-\nical potential fluctuations of the nanoparticle [ rms(µ)]\nmeasured at the amplifier output, that would correspond\nto the observed current noise. First we find the aver-\nage slope of the IV-curve at voltages where we measure\nthe noise and multiply rms(I) with that slope to find\nrms(V). Then we convert from voltage to energy and\nfindrms(µ)≈48µeV atB=BP.\nThe strong magnetic field dependence of the current\nnoise, along with the symmetry about B= 0, implies\nmagnetic rather than electric origin of the excess noise.\nThis conclusion is further supported by the observation\nthat the enhanced current noise is suppressed in the\nvoltage region that includes well resolved step-voltages\n(<17 mV). The sensitivity of the current to a chemical\npotential fluctuation is generallyhighest at step voltages,\nwhereI(V) is the steepest. [29] So if the excess noise\nnearB=BPwere induced by the fluctuating charges, it\nwould be more pronounced about the step voltages com-\npared to higher voltages where I(V) curve is less steep.\nFig. 3(b) sketches the effect of electron transport on\nmagnetic damping. For simplicity, we may assume that\nonly minority levels are involved in transport and that\nthe electronic system of the nanoparticle is fully relaxed.3\n(We revisit the assumption later on.) A tunneling tran-\nsition into a discrete level of the magnetic nanoparticle\ncan either be direct or assisted by a spin-flip transition\n∆Sz=±1 in the magnetic subsystem. If the Fermi\nlevel (EF) is above a discrete level energy [level 1 in\nFig. 3(a)], but below that energy plus magnetic quantum\n¯hΩ≈gµBB, then the lead can absorb but not emit mag-\nnetic quanta by tunneling into that level. This will cause\nstrong damping by electron transportconsistent with the\nsuppression of chemical potential noise at the step volt-\nages. [30, 31] If, however, EFis much higher than the\ntransition energy [level 2 in Fig. 3(a)], the Fermi distri-\nbutionwill notfavorabsorptiontoemissionandtherefore\ndamping by tunneling via level 2 will be suppressed. At\nhigh voltage, only one level (level 1) near EFmay con-\ntribute to damping, while the remaining energetically ac-\ncessible levels still contribute to symmetric emission and\nabsorption of magnetic quanta. Since the relative damp-\ning rate in that case is lower, the magnetic displacement\nwill be higher, consistent with the enhanced chemical po-\ntential noise we measure at high voltages.\nIf all energetically accessible levels are well below EF\nand no other damping mechanisms are present, the mag-\nnetization will be freely diffusing, with an approximately\nisotropic distribution in solid angle. [30, 31] We note a\nsubtle point that the magnetic damping time is finite in\nthe free diffusion regime, since the principle of detailed\nbalance demands symmetry between time averagedemis-\nsion and absorption powers. This free-diffusion damping\ntime is the one that we measure here.\nIt may be apparent that the measured rms(µ)\nis related to the intrinsic rms-fluctuation /angbracketleft|δǫ|/angbracketrightof\nthe nanoparticle chemical potential as rms(µ) =\n/angbracketleft|δǫ|/angbracketright/radicalbig\nT1,d/T. (The derivation is provided in the sup-\nplementary document S3.) Hence,\nT1,d=T/bracketleftbiggrms(µ)\n/angbracketleft|δǫ|/angbracketright/bracketrightbigg2\n. (1)\nLet us assume that the magnetization is freely diffus-\ning. In that case, /angbracketleft|δǫ|/angbracketrightis the isotropic rms-shift and we\nmay substitute /angbracketleft|δǫ|/angbracketright=rms(∆) and obtain T1,d= 13 ms\natB≈BP. Now we make our central hypothesis, which\nis also the second main result of this paper, that the ex-\nperimenter can identify free spin diffusion by observing\nthatrms(µ) increases with magnetic field at fixed bias\nvoltage. On the other hand, if they observe that rms(µ)\ndecreases with field, the magnetization will be in the lin-\near, strongly damped, and harmonic oscillator regime,\nwhile/angbracketleft|δǫ|/angbracketrightand by extension T1,dcannot be determined\nfrom the data.\nBefore we justify the hypothesis, we evaluate the sta-\ntus of the electronic subsystem in the nanoparticle in\nour measurement. The status is not critical for the\nmechanistics of spin diffusion and damping, however,\nit is related to prior experiments in the field which we\nFIG. 3. (a) Root-mean-square current versus magnetic field\nin sample 1. The line between the points is a fittoaGaussian.\n(b) Schematic of magnetic damping by electron transport,\nshowing enhanced damping of level 1, and suppressed damp-\ning of level 2. (c) Same as in (a) but in sample 2. (d) Differen-\ntial conductance in sample 2 at 70 mK, showing pockets of en-\nhanced conductance noise indicated by yellow boxed arrows.\nColor bar indicates conductance range ( −2×10−5,10−4)e2/h.\nuse as references. Fig. 4 sketches the ground state and\nvarious electronic and spin excitations in the simplest\nand exactly solvable model of metallic ferromagnetic\nnanoparticles. [32, 33] In the ground state displayed in\nFig. 4(a), the minority and majority quasiparticle states\nare shifted in energy by the exchange splitting, which\nbreaks the time reversal symmetry and the associated\nKramers degeneracy. Stoner excitations are spin-flip\nparticle-holeexcitationsinvolvingdifferent Kramersdou-\nblets, as sketched in Fig. 4(b,c). Prior measurements of\nthe relaxation time T1,sof Stoner excitations in metal-\nlic ferromagnetic nanoparticles yield unusually large val-\nues, ofT1,s∼0.1µsusing nonequilibriumtunneling spec-\ntroscopy at mK-temperature, [22] and T1,s∼0.1−10µs\nusing spin accumulation. [34–36] Since in our measure-\nment the electron tunneling time e/Inis shorter than\nT1,s, the electronic subsystem is out of equilibrium, fluc-\ntuating between Stoner excitations within the energy\nrangeeV/kB∼100 K. [33, 37]\nNext we examine how electron tunneling and the\nnonequilibrium electronic distribution impart dynamics\nin the magnetic subsystem. Tunneling and internal re-\nlaxation transitions in the electronic subsystem produce\nspin-orbit energy fluctuations, which can induce transi-\ntions between the states of the magnetic subsystem. A\nuseful intuitive picture of this effect is that the fluctu-4\nFIG. 4. Spin excitations in an idealized metallic feromagne tic\nnanoparticle. The spin direction is indicated by the color g ra-\ndient, e.g. the left levels are spin down, and the right level s\nare spin up. (a) The ground state. (b),(c) Stoner excita-\ntions with ∆ Sz=−1 and−2, respectively. The electronic\ndensities in (a,b,c) vary. (d,e,f) State representations a fter\nsubtracting the exchange splitting. (e,f) Components of th e\ncollective magnetic excitations, for ∆ Sz=−1 and ∆Sz=−3,\nrespectively. The electronic densities in (d,e,f) are the s ame.\nating anisotropy energy induces noncollinear magneti-\nzation orientations among the magnetic ground states\ncorresponding to different electronic configurations. In\ngeneral, electron transfer combined with noncollinear-\nity of the initial and final magnetizations implies spin-\ntransfer. [38] We may suppose that the ground state\nmagnetizations of different electronic configurations be-\ncome more collinear in the strong magnetic field, as the\nZeeman energy overtakes the anisotropy energy, thereby\nsuppressing the spin-transfer rates between the magneti-\nzation and electrons. Hence, free spin diffusion embodies\na magnetic damping time that increases with magnetic\nfield. (We further illuminate this argument in the sup-\nplementary document S4, where we also estimate the de-\npendence quantitatively.)\nHowever, the damping time due to phonons decreases\nversusBwith a power law, analogous to the spin relax-\nation time in semiconducting quantum dots. [8, 9] There-\nfore, the magnetic field is a lever that changes the dom-\ninant environment for magnetic damping. In the strong\nmagnetic field, spin fluctuations decrease rapidly with B,\nas the magnetization localizes about the ground state di-\nrection due to the strong damping by phonons. Thus,\nthe measured peak in rms(µ) versusBis consistent with\nthe crossoverfrom free spin diffusion to strongly damped\nmagnetic dynamics. The key effects described here are\nobserved in two additional samples. Fig. 3(c,d) shows\nthoseeffectsinsample2, wherewealsofind T1,d≃10ms.\nIs there a physical justification for such long T1,d? The\nrelation between Stoner and collective spin excitations\nin a metallic ferromagnetic nanoparticles is analogous to\nthat between the triplet-singlet and intra-Kramers (e.g.,\nsublevel-to-sublevel) transitions in semiconducting quan-tum dots. The relaxation time between the triplet and\nsinglet states is much shorter than that between the\nKramers sublevels, [16, 39] since triplet to singlet transi-\ntions involve states with different electronic densities at\nB= 0, while the transitions between Kramers sublevels\ninvolve states with equal electronic density at B= 0.\nConsider Figs. 4(d,e,f) that display the ground state\nand the collective magnetic excitations in the simplest\ntheoretical model of metallic ferromagnetic nanoparti-\ncle. [32, 33] These excitations are admixtures of particle-\nhole excitations, with example components illustrated in\nFigs. 4(e,f). Within the model, they all have the same\nelectronicdensity, incontrasttoStonerexcitations. Since\nthe measured values of T1,sin metallic ferromagnetic\nnanoparticles are up to 10 µs long, [22, 34–36] we find\nthat the observed value T1,d≃10 ms is plausible.\nIn conclusion, we present real-time detection of mag-\nnetic motion in single metallic ferromagnetic nanopar-\nticles, using the conversion of magnetic dynamics into\nan effective charge dynamics. We observe nonmonotonic\nmagnetic field dependence of magnetic damping time,\nwhich we attribute to the crossover in damping between\nthe electronic and ambient environments. The magnetic\ndampingtimeofapproximately10msinNi nanoparticles\nestablishes a benchmark for magnetic damping in mag-\nnetic nanoparticles, and provides the relevant time scale\nwhere the magnetic dynamics can be studied. 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Rev.\nLett.74, 3241 (1995)." }, { "title": "1807.07897v2.Another_view_on_Gilbert_damping_in_two_dimensional_ferromagnets.pdf", "content": "Another view on Gilbert damping in two-dimensional\nferromagnets\nAnastasiia A. Pervishko1, Mikhail I. Baglai1,2, Olle Eriksson2,3, and Dmitry Yudin1\n1ITMO University, Saint Petersburg 197101, Russia\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75 121 Uppsala, Sweden\n3School of Science and Technology, ¨Orebro University, SE-701 82 ¨Orebro, Sweden\nABSTRACT\nA keen interest towards technological implications of spin-orbit driven magnetization dynamics requests a proper theoretical\ndescription, especially in the context of a microscopic framework, to be developed. Indeed, magnetization dynamics is so far\napproached within Landau-Lifshitz-Gilbert equation which characterizes torques on magnetization on purely phenomenological\ngrounds. Particularly, spin-orbit coupling does not respect spin conservation, leading thus to angular momentum transfer to\nlattice and damping as a result. This mechanism is accounted by the Gilbert damping torque which describes relaxation of the\nmagnetization to equilibrium. In this study we work out a microscopic Kubo-St ˇreda formula for the components of the Gilbert\ndamping tensor and apply the elaborated formalism to a two-dimensional Rashba ferromagnet in the weak disorder limit. We\nshow that an exact analytical expression corresponding to the Gilbert damping parameter manifests linear dependence on the\nscattering rate and retains the constant value up to room temperature when no vibrational degrees of freedom are present\nin the system. We argue that the methodology developed in this paper can be safely applied to bilayers made of non- and\nferromagnetic metals, e.g., CoPt.\nIntroduction\nIn spite of being a mature field of research, studying magnetism and spin-dependent phenomena in solids still remains one of\nthe most exciting area in modern condensed matter physics. In fact, enormous progress in technological development over the\nlast few decades is mainly held by the achievements in spintronics and related fields1–11. However the theoretical description of\nmagnetization dynamics is at best accomplished on the level of Landau-Lifshitz-Gilbert (LLG) equation that characterizes\ntorques on the magnetization. In essence, this equation describes the precession of the magnetization, mmm(rrr;t), about the effective\nmagnetic field, HHHeff(rrr;t), created by the localized moments in magnetic materials, and its relaxation to equilibrium. The latter,\nknown as the Gilbert damping torque12, was originally captured in the form ammm\u0002¶tmmm, where the parameter adetermines the\nrelaxation strength, and it was recently shown to originate from a systematic non-relativistic expansion of the Dirac equation13.\nThus, a proper microscopic determination of the damping parameter a(or, the damping tensor in a broad sense) is pivotal to\ncorrectly simulate dynamics of magnetic structures for the use in magnetic storage devices14.\nFrom an experimental viewpoint, the Gilbert damping parameter can be extracted from ferromagnetic resonance linewidth\nmeasurements15–17or established via time-resolved magneto-optical Kerr effect18, 19. In addition, it was clearly demonstrated\nthat in bilayer systems made of a nonmagnetic metal (NM) and a ferromagnet material (FM) the Gilbert damping is drastically\nenhanced as compared to bulk FMs20–24. A strong magnetocrystalline anisotropy, present in CoNi, CoPd, or CoPt, hints\nunambiguously for spin-orbit origin of the intrinsic damping. A first theoretical attempt to explain the Gilbert damping\nenhancement was made in terms of sdexchange model in Ref.25. Within this simple model, magnetic moments associated with\nFM layer transfer angular momentum via interface and finally dissipate. Linear response theory has been further developed\nwithin free electrons model26, 27, while the approach based on scattering matrix analysis has been presented in Refs.28, 29. In\nthe latter scenario spin pumping from FM to NM results in either backscattering of magnetic moments to the FM layer or\ntheir further relaxation in the NM. Furthermore, the alternative method to the evaluation of the damping torque, especially\nin regard of first-principles calculations, employs torque-correlation technique within the breathing Fermi surface model30.\nWhile a direct estimation of spin-relaxation torque from microscopic theory31, or from spin-wave spectrum, obtained on the\nbasis of transverse magnetic field susceptibility32, 33, are also possible. It is worth mentioning that the results of first-principles\ncalculations within torque-correlation model34–38and linear response formalism39, 40reveal good agreement with experimental\ndata for itinerant FMs such as Fe, Co, or Ni and binary alloys.\nLast but not least, an intensified interest towards microscopic foundations of the Gilbert parameter ais mainly attributed\nto the role the damping torque is known to play in magnetization reversal41. In particular, according to the breathing Fermi\nsurface model the damping stems from variations of single-particle energies and consequently a change of the Fermi surfacearXiv:1807.07897v2 [cond-mat.mes-hall] 21 Nov 2018z\nyx\nFM\nNMFigure 1. Schematic representation of the model system: the electrons at the interface of a bilayer, composed of a\nferromagnetic (FM) and a nonmagnetic metal (NM) material, are well described by the Hamiltonian (1). We assume the\nmagnetization of FM layer depicted by the red arrow is aligned along the zaxis.\nshape depending on spin orientation. Without granting any deep insight into the microscopic picture, this model suggests that\nthe damping rate depends linearly on the electron-hole pairs lifetime which are created near the Fermi surface by magnetization\nprecession. In this paper we propose an alternative derivation of the Gilbert damping tensor within a mean-field approach\naccording to which we consider itinerant subsystem in the presence of nonequilibrium classical field mmm(rrr;t). Subject to the\nfunction mmm(rrr;t)is sufficiently smooth and slow on the scales determined by conduction electrons mean free path and scattering\nrate, the induced nonlocal spin polarization can be approached within a linear response, thus providing the damping parameter\ndue to the itinerant subsystem. In the following, we provide the derivation of a Kubo-St ˇreda formula for the components of\nthe Gilbert damping tensor and illustrate our approach for a two-dimensional Rashba ferromagnet, that can be modeled by\nthe interface between NM and FM layers. We argue that our theory can be further applied to identify properly the tensorial\nstructure of the Gilbert damping for more complicated model systems and real materials.\nMicroscopic framework\nConsider a heterostructure made of NM with strong spin-orbit interaction covered by FM layer as shown in Fig. 1, e.g., CoPt.\nIn general FMs belong to the class of strongly correlated systems with partially filled dorforbitals which are responsible\nfor the formation of localized magnetic moments. The latter can be described in terms of a vector field mmm(rrr;t)referred to as\nmagnetization, that in comparison to electronic time and length scales slowly varies and interacts with an itinerant subsystem.\nAt the interface (see Fig. 1) the conduction electrons of NM interact with the localized magnetic moments of FM via a certain\ntype of exchange coupling, sdexchange interaction, so that the Hamiltonian can be written as\nh=p2\n2m+a(sss\u0002ppp)z+sss\u0001MMM(rrr;t)+U(rrr); (1)\nwhere first two terms correspond to the Hamiltonian of conduction electrons, on condition that the two-dimensional momentum\nppp= (px;py) =p(cosj;sinj)specifies electronic states, mis the free electron mass, astands for spin-orbit coupling strength,\nwhile sss= (sx;sy;sz)is the vector of Pauli matrices. The third term in (1) is responsible for sdexchange interaction with the\nexchange field MMM(rrr;t) =Dmmm(rrr;t)aligned in the direction of magnetization and Ddenoting sdexchange coupling strength. We\nhave also included the Gaussian disorder, the last term in Eq. (1), which represents a series of point-like defects, or scatterers,\nhU(rrr)U(rrr0)i= (mt)\u00001d(rrr\u0000rrr0)with the scattering rate t(we set ¯h=1throughout the calculations and recover it for the final\nresults).\nSubject to the norm of the vector jmmm(rrr;t)j=1remains fixed, the magnetization, in broad terms, evolves according to (see,\ne.g., Ref.42),\n¶tmmm=fff\u0002mmm=gHHHeff\u0002mmm+csss\u0002mmm; (2)\nwhere fffcorresponds to so-called spin torques. The first term in fffdescribes precession around the effective magnetic field\nHHHeffcreated by the localized moments of FM, whereas the second term in (2) is determined by nonequilibrium spin density of\nconduction electrons of NM at the interface, sss(rrr;t). It is worth mentioning that in Eq. (2) the parameter gis the gyromagnetic\nratio, while c= (gmB=¯h)2m0=dis related to the electron g\u0000factor ( g=2), the thickness of a nonmagnetic layer d, with mBand\nm0standing for Bohr magneton and vacuum permeability respectively. Knowing the lesser Green’s function, G<(rrrt;rrrt), one\ncan easily evaluate nonequilibrium spin density of conduction electrons induced by slow variation of magnetization orientation,\nsm(rrr;t) =\u0000i\n2Tr\u0002\nsmG<(rrrt;rrrt)\u0003\n=Qmn¶tmn+:::; (3)\n2/8where summation over repeated indexes is assumed ( m;n=x;y;z). The lesser Green’s function of conduction electrons\ncan be represented as G<=\u0000\nGK\u0000GR+GA\u0001\n=2, where GK,GR,GAare Keldysh, retarded, and advanced Green’s functions\nrespectively.\nKubo-St ˇreda formula\nWe further proceed with evaluating Qmnin Eq. (3) that describes the contribution to the Gilbert damping due to conduction\nelectrons. In the Hamiltonian (1) we assume slow dynamics of the magnetization, such that approximation MMM(rrr;t)\u0019\nMMM+(t\u0000t0)¶tMMMwith MMM=MMM(rrr;t0)is supposed to be hold with high accuracy,\nH=p2\n2m+a(sss\u0002ppp)z+sss\u0001MMM+U(rrr)+(t\u0000t0)sss\u0001¶tMMM; (4)\nwhere first four terms in the right hand side of Eq. (4) can be grouped into the Hamiltonian of a bare system, H0, which\ncoincides with that of Eq. (1), provided by the static magnetization configuration MMM. In addition, the expression (4) includes the\ntime-dependent term V(t)explicitly, as the last term. In the following analysis we deal with this in a perturbative manner. In\nparticular, the first order correction to the Green’s function of a bare system induced by V(t)is,\ndG(t1;t2) =Z\nCKdtZd2p\n(2p)2gppp(t1;t)V(t)gppp(t;t2); (5)\nwhere the integral in time domain is taken along a Keldysh contour, while gppp(t1;t2) =gppp(t1\u0000t2)[the latter accounts for the\nfact that in equilibrium correlation functions are determined by the relative time t1\u0000t2] stands for the Green’s function of the\nbare system with the Hamiltonian H0in momentum representation. In particular, for the lesser Green’s function at coinciding\ntime arguments t1=t2\u0011t0, which is needed to evaluate (3), one can write down,\ndG<(t0;t0) =i\n2¥Z\n\u0000¥de\n2pZd2p\n(2p)2n\ngR\npppsm¶g<\nppp\n¶e\u0000¶gR\nppp\n¶esmg<\nppp+g<\npppsm¶gA\nppp\n¶e\u0000¶g<\nppp\n¶esmgA\npppo\n¶tMm; (6)\nwhere m=x;y;z, while gR,gA, and gjDjwe can establish that d=1=(2t)\nandh=0 in the weak disorder regime to the leading order.\nWithout loss of generality, in the following we restrict the discussion to the regime m>jDj, which is typically satisfied with\nhigh accuracy in experiments. As previously discussed, the contribution owing to the Fermi sea, Eq. (7), can in some cases be\nignored, while doing the momentum integral in Eq. (8) results in,\n1\nmtZd2p\n(2p)2gR\nppp(e)sssgA\nppp(e) =D2\nD2+2ersss+Dd\nD2+2er(sss\u0002zzz)+D2\u0000er\nD2+2er(sss\u0002zzz)\u0002zzz; (10)\nwhere r=ma2. Thus, thanks to the factor of delta function d(e\u0000m) =\u0000¶f(e)=¶e, to estimate Q(2)\nmnat zero temperature one\nshould put e=min Eq. (10). As a result, we obtain,\nQ(2)\nmn=\u00001\n4pm\nD2+2mr0\n@2tmr D 0\n\u0000D 2tmr 0\n0 0 2 tD21\nA: (11)\nMeanwhile, to properly account the correlation functions which appear when averaging over disorder configuration one has\nto evaluate the so-called vertex corrections, which from a physical viewpoint makes a distinction between disorder averaged\nproduct of two Green’s function, hgRsngAidis, and the product of two disorder averaged Green’s functions, hgRidissnhgAidis, in\nEq. (8). Thus, we further proceed with identifying the vertex part by collecting the terms linear in dexclusively,\nGGGs=Asss+B(sss\u0002zzz)+C(sss\u0002zzz)\u0002zzz; (12)\nprovided A=1+D2=(2er),B= (D2+2er)Dd=(D2+er)2, and C=D2=(2er)\u0000er=(D2+er). To complete our derivation\nwe should replace snin Eq. (8) by Gs\nnand with the aid of Eq. (10) we finally derive at e=m,\nQ(2)\nmn=0\n@Qxx Qxy 0\n\u0000QxyQxx\n0 0\u0000mtD2=(4pmr)1\nA: (13)\nWe defined Qxx=\u0000mtmr=[2p(D2+mr)]andQxy=\u0000mD(D2+2mr)=[4p(D2+mr)2], which unambiguously reveals that\naccount of vertex correction substantially modifies the results of the calculations. With the help of Eqs. (3), (11), and (13) we\ncan write down LLG equation. Slight deviation from collinear configurations are determined by xandycomponents ( mxand\nmyrespectively, so that jmxj;jmyj\u001c1). The expressions (11) and (13) immediately suggest that the Gilbert damping at the\ninterface is a scalar, aG,\n¶tmmm=˜gHHHeff\u0002mmm+aGmmm\u0002¶tmmm; (14)\nwhere the renormalized gyromagnetic ratio and the damping parameter are,\n˜g=g\n1+cDQxy;aG=\u0000cDQxx\n1+cDQxy\u0019\u0000cDQxx: (15)\nIn the latter case we make use of the fact that mc\u001c1for the NM thickness d\u0018100mm — 100 nm. In Eq. (14) we have\nredefined the gyromagnetic ratio g, but we might have renormalized the magnetization instead. From physical perspective,\nthis implies the fraction of conduction electrons which become associated with the localized moment owing to sdexchange\ninteraction. With no vertex correction included one obtains\naG=mc\n2p¯htmrD\nD2+2mr; (16)\n4/8t=1ns\nt=10ns\nD=0.2meV\nD=0.3meV\nD=1meV\n501001502002503000.0000.0010.0020.0030.004\nT,KaGFigure 2. Gilbert damping, obtained from numerical integration of Eq. (8), shows almost no temperature dependence\nassociated with thermal redistribution of conduction electrons. Dashed lines are plotted for D=1meV for t=1andt=10ns,\nwhereas solid lines stand for D=0:2, 0:3, and 1 meV for t=100 ns.\nwhile taking account of vertex correction gives rise to a different result,\naG=mc\n2p¯htmrD\nD2+mr: (17)\nTo provide a quantitative estimate of how large the St ˇreda contribution in the weak disorder limit is, on condition that m>jDj,\nwe work out Q(1)\nmn. Using ¶gR=A(e)=¶e=\u0000[gR=A(e)]2and the fact that trace is invariant under cyclic permuattaions we conclude\nthat only off-diagonal components m6=ncontribute. While the direct evaluation results in Q(1)\nxy=3mD=[2(D2+2mr)]in the\nclean limit. It has been demonstrated that including scattering rates dandhdoes not qualitatively change the results, leading to\nsome smearing only52.\nInterestingly, within the range of applicability of theory developed in this paper, the results of both Eqs. (16) and (17)\ndepend linearly on scattering rate, being thus in qualitative agreement with the breathing Fermi surface model. Meanwhile, the\nlatter does not yield any connection to the microscopic parameters (see, e.g., Ref.53for more details). To provide with some\nquantitative estimations in our simulations we utilize the following set of parameters. Typically, experimental studies based on\nhyperfine field measurements equipped with DFT calculations54reveal the sdStoner interaction to be of the order of 0.2 eV ,\nwhile the induced magnetization of s-derived states equals 0.002–0.05 (measured in the units of Bohr magneton, mB). Thus,\nthe parameter of sdexchange splitting, appropriate for our model, is D\u00180.2–1 meV . In addition, according to first-principles\nsimulations we choose the Fermi energy m\u00183 eV . The results of numerical integration of (8) are presented in Fig. 2 for several\nchoices of sdexchange and scattering rates, t. The calculations reveal almost no temperature dependence in the region up to\nroom temperature for any choice of parameters, which is associated with the fact that the dominant contribution comes from the\nintegration in a tiny region of the Fermi energy. Fig. 2 also reveal a non-negligible dependence on the damping parameter with\nrespect to both Dandt, which illustrates that a tailored search for materials with specific damping parameter needs to address\nboth the sdexchange interaction as well as the scattering rate. From the theoretical perspective, the results shown in Fig. 2\ncorrespond to the case of non-interacting electrons with no electron-phonon coupling included. Thus, the thermal effects are\naccounted only via temperature-induced broadening which does not show up for m>jDj.\nConclusions\nIn this paper we proposed an alternative derivation of the Gilbert damping tensor within a generalized Kubo-St ˇreda formula.\nWe established the contribution stemming from Eq. (7) which was missing in the previous analysis within the linear response\ntheory. In spite of being of the order of (mt)\u00001and, thus, negligible in the weak disorder limit developed in the paper, it should\nbe properly worked out when dealing with more complicated systems, e.g., gapped materials such as iron garnets (certain half\nmetallic Heusler compounds). For a model system, represented by a Rashba ferromagnet, we directly evaluated the Gilbert\ndamping parameter and explored its behaviour associated with the temperature-dependent Fermi-Dirac distribution. In essence,\nthe obtained results extend the previous studies within linear response theory and can be further utilized in first-principles\ncalculations. We believe our results will be of interest in the rapidly growing fields of spintronics and magnonics.\n5/8References\n1.Žuti´c, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323 (2004).\n2.Bader, S. D. & Parkin, S. S. P. 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Rev.\nLett.117, 046601 (2016).\n52.Nunner, T. S., et al. Anomalous Hall effect in a two-dimensional electron gas. Phys. Rev. B 76, 235312 (2007).\n53.Eriksson, O., Bergman, A., Bergqvist, L. & Hellsvik, J. Atomistic spin dynamics: Foundations and applications (Oxford\nUniversity Press, Oxford, 2017).\n54.Brooks, M. S. S. & Johansson, B. Exchange integral matrices and cohesive energies of transition metal atoms. J. Phys. F:\nMet. Phys. 13, L197 (1983).\n7/8Acknowledgements\nA.A.P. acknowledges the support from the Russian Science Foundation Project No. 18-72-00058. O.E. acknowledges support\nfrom eSSENCE, the Swedish Research Council (VR), the foundation for strategic research (SSF) and the Knut and Alice\nWallenberg foundation (KAW). D.Y . acknowledges the support from the Russian Science Foundation Project No. 17-12-01359.\nAuthor contributions statement\nD.Y . conceived the idea of the paper and contributed to the theory. A.A.P. wrote the main manuscript text, performed numerical\nanalysis and prepared figures 1-2. M.I.B. and O.E. contributed to the theory. All authors reviewed the manuscript.\nAdditional information\nCompeting interests The authors declare no competing interests.\n8/8" }, { "title": "1310.7657v1.Observational_Study_of_Large_Amplitude_Longitudinal_Oscillations_in_a_Solar_Filament.pdf", "content": "arXiv:1310.7657v1 [astro-ph.SR] 29 Oct 2013Nature of Prominences and their role in Space Weather\nProceedings IAU Symposium No. 300, 2014\nB. Schmieder, JM. Malherbe & S. Wu, eds.c/circlecopyrt2014 International Astronomical Union\nDOI: 00.0000/X000000000000000X\nObservational Study of Large Amplitude\nLongitudinal Oscillations in a Solar Filament\nKalman Knizhnik1,2,Manuel Luna3, Karin Muglach2,4\nHolly Gilbert2, Therese Kucera2, Judith Karpen2\n1Department of Physics and Astronomy\nThe Johns Hopkins University, Baltimore, MD 21218\nemail: kalman.knizhnik@nasa.gov\n2NASA/GSFC, Greenbelt, MD 20771, USA\n3Instituto de Astrof´ ısica de Canarias, E-38200 La Laguna, T enerife, Spain\n4ARTEP, Inc., Maryland, USA\nAbstract. On 20 August 2010 an energetic disturbance triggered damped large-amplitude lon-\ngitudinal (LAL) oscillations in almost an entire filament. I n the present work we analyze this\nperiodic motion in the filament to characterize the damping a nd restoring mechanism of the\noscillation. Our method involves placing slits along the ax is of the filament at different angles\nwith respect to the spine of the filament, finding the angle at w hich the oscillation is clearest,\nand fitting the resulting oscillation pattern to decaying si nusoidal and Bessel functions. These\nfunctions represent the equations of motion of a pendulum da mped by mass accretion. With\nthis method we determine the period and the decaying time of t he oscillation. Our preliminary\nresults support the theory presented by Luna and Karpen (201 2) that the restoring force of LAL\noscillations is solar gravity in the tubes where the threads oscillate, and the damping mechanism\nis the ongoing accumulation of mass onto the oscillating thr eads. Following an earlier paper, we\nhave determined the magnitude and radius of curvature of the dipped magnetic flux tubes host-\ning a thread along the filament, as well as the mass accretion r ate of the filament threads, via\nthe fitted parameters.\nKeywords. solar prominences, oscillations, magnetic structures\n1. Procedure\nLAL oscillations consist of periodic motions of the prominence thread s along the mag-\nnetic field that are disturbed by a small energetic event close to the filament (see Luna\net al. paper in this volume). Luna and Karpen (2012) argue that pro minence oscillations\ncan be modeled as a damped oscillating pendulum, whose equation of mo tion satis-\nfies a zeroth-order Bessel function. In their model, a nearby trig ger event causes quasi-\nstationary preexisting prominence threads sitting in the dips of the magnetic structure\nto oscillate back and forth, with the restoring force being the proj ected gravity in the\ntubes where the threads oscillate (e.g. Luna et al. (2012)). In this paper, we report pre-\nliminary results of comparisonsof observations of prominence oscilla tions with the model\npresented by Luna and Karpen (2012). More details will be available in the forthcoming\npaper by Luna et al. (2013).\nIn this analysis, we place slits along the filament spine and measure the intensity along\neach slit as a function of time. Fig. 1 (left) shows the filament in the AI A 171˚A filter\nwith the slits overlaid. Each slit is then rotated in increments of 0.5◦from 0◦to 60◦\nwith respect to the filament spine. We select the best slit according t o the following\ncriteria: (a) continuity of oscillations, (b) amplitude of the oscillation is maximized, (c)\nclear transition from dark to bright regions, (d) maximum number of cycles.\nThe oscillation for a representative slit is shown in Figure 1 (right), wh ich corresponds\nto the grey slit in Figure 1 (left). We identify the position of the cente r of mass of the\n12 K. Knizhnik, M. Luna, K. Muglach, H. Gilbert, T. Kucera, J. Karpen\nFigure 1. Left: Filament seen in AIA 171 with best slits overlaid. Right: An intensity distance–\ntime slit, showing an oscillation with the Bessel fit (white c urve) to equation (2.1) in Luna et\nal. (this volume). The sinusoidal fit was not as good as the Bes sel fit and is not shown.\nthread by finding the intensity minimum along the slit, indicated by black crosses in\nFigure 1 (right). These points are then fit to equation (2.1) of Luna et al. (this volume),\nand the resulting fit is shown in white.\n2. Results\nFitting our data to equation (2.1) of Luna et al. (this volume) yields va lues ofχ2\nranging between 1-13. Using equation (2.2) of Luna et al. (this volum e), we find the\naverageradiusofcurvatureofthe magneticfield dips that suppor t the oscillatingthreads.\nWe find it to be approximately 60 Mm. We also calculate a threshold value for the field\nitself that would allow it to support the observedthreads. Using equ ation (3.1) of Luna et\nal.(this volume), wefind anaveragemagneticfield of ∼20G, assumingatypicalfilament\nnumber density of 1011cm−3, in good agreement with measurements (e.g. Mackay et al.\n2010). On average, the oscillations form an angle of ∼25owith respect to the filament\nspine, and have a period of ∼0.8 hours. To explain the very strong damping mass must\naccrete onto the threads at a rate of about 60 ×106kg/hr.\n3. Conclusions\nWe conclude that the observedoscillationsarealongthe magneticfie ld, which formsan\nangle of∼25owith respect to the filament spine (Tandberg-Hanssen & Anzer, 19 70). We\nfind that both the curvature and the magnitude of the magnetic fie ld are approximately\nuniform on different threads. Both the Bessel and sinusoidal func tions are well fitted,\nindicating that mass accretion is a likely damping mechanism of LAL oscilla tions, and\nthat the restoring force is the projected gravity in the dips where the threads oscillate.\nThe mass accretion rate agrees with the theoretical value (Karpe n et al., 2006, Luna,\nKarpen, & DeVore, 2012).\nReferences\nKarpen, J. T., Antiochos, S. K., Klimchuk, J. A. 2006, ApJ, 63 7, 531\nLuna, M., Karpen, J. T., & Devore, C. R. 2012a, ApJ, 746, 30\nLuna, M., & Karpen, J. 2012, ApJ, 750, L1\nLuna, M., Knizhnik, K., Muglach, K., Gilbert, H, Kucera, T. & Karpen, J., this volume , 2014\nLuna, M., Knizhnik, K., Muglach, K., Gilbert, H, Kucera, T. & Karpen, J., ApJ, 2013,in prep.\nMackay, D., Karpen, J., Ballester, J., Schmieder, B., Aulan ier, G. 2010, Sp. Sci. Rev. , 151, 333\nTandberg-Hanssen, E. and Anzer, U. 1970, Solar Physics 15, 158T" }, { "title": "1710.07690v2.Tidal_dissipation_in_rotating_fluid_bodies__the_presence_of_a_magnetic_field.pdf", "content": "MNRAS 000, 1{14 (2017) Preprint 23 November 2021 Compiled using MNRAS L ATEX style \fle v3.0\nTidal dissipation in rotating \ruid bodies: the presence of a\nmagnetic \feld\nYufeng Lin?and Gordon I. Ogilvie\nDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences,\nWilberforce Road, Cambridge CB3 0WA, UK\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nWe investigate e\u000bects of the presence of a magnetic \feld on tidal dissipation in rotating\n\ruid bodies. We consider a simpli\fed model consisting of a rigid core and a \ruid en-\nvelope, permeated by a background magnetic \feld (either a dipolar \feld or a uniform\naxial \feld). The wavelike tidal responses in the \ruid layer are in the form of magnetic-\nCoriolis waves, which are restored by both the Coriolis force and the Lorentz force. En-\nergy dissipation occurs through viscous damping and Ohmic damping of these waves.\nOur numerical results show that the tidal dissipation can be dominated by Ohmic\ndamping even with a weak magnetic \feld. The presence of a magnetic \feld smooths\nout the complicated frequency-dependence of the dissipation rate, and broadens the\nfrequency spectrum of the dissipation rate, depending on the strength of the back-\nground magnetic \feld. However, the frequency-averaged dissipation is independent\nof the strength and structure of the magnetic \feld, and of the dissipative parame-\nters, in the approximation that the wave-like response is driven only by the Coriolis\nforce acting on the non-wavelike tidal \row. Indeed, the frequency-averaged dissipation\nquantity is in good agreement with previous analytical results in the absence of mag-\nnetic \felds. Our results suggest that the frequency-averaged tidal dissipation of the\nwavelike perturbations is insensitive to detailed damping mechanisms and dissipative\nproperties.\nKey words:\n1 INTRODUCTION\nTidal interactions may have played an important role in the\nevolution of short-period exoplanetary systems, binary stars\nand planet-satellite systems. The e\u000eciency of tidal dissipa-\ntion, which is usually parameterized as the tidal quality fac-\ntorQ(Goldreich & Soter 1966), will determine the fate of\nthese systems. With the accumulation of observations over\nseveral decades, the observational constraint of the tidal\nquality factor is now possible, especially in hot Jupiter sys-\ntems (e.g. Wilkins et al. 2017; Patra et al. 2017). However,\nit is still very challenging to theoretically predict the tidal\nquality factor owing to intrinsic di\u000eculties and uncertainties\nof the problem.\nTidal responses can be generally separated into two\nparts: the equilibrium tide and the dynamic tide (e.g. Ogilvie\n2014). The equilibrium tide is a quasi-hydrostatic deforma-\ntion to the gravitational pulling of the orbital companion.\nThe dynamic tide is usually associated with di\u000berent kinds\nof internal waves such as internal gravity waves (Zahn 1975;\n?E-mail: yl552@cam.ac.ukSavonije & Papaloizou 1983; Goldreich & Nicholson 1989;\nGoodman & Dickson 1998; Barker & Ogilvie 2010; Essick\n& Weinberg 2016) and inertial waves (Ogilvie & Lin 2004,\n2007; Wu 2005; Ogilvie 2009, 2013; Goodman & Lackner\n2009; Rieutord & Valdettaro 2010; Papaloizou & Ivanov\n2010). The tidal dissipation associated with these hydro-\ndynamic waves exhibits very complicated dependences on\nthe tidal frequency (Ogilvie 2014). The non-linear e\u000bect and\nthe interactions with convection and magnetic \felds may\nwash out some of the frequency-dependence (Ogilvie 2013),\nyet these e\u000bects on tides remain to be elucidated. Magnetic\n\felds are ubiquitous in astrophysical bodies, where they can\ninteract with tidal \rows in electrically conducting \ruid lay-\ners and lead to additional Ohmic dissipation. The magnetic\n\feld will modify the propagation and dissipation of hydrody-\nnamic waves through the coupling with Alfv\u0013 en waves. The\npresent study aims to investigate the magnetic e\u000bects on\ntidally forced inertial waves. In the presence of a magnetic\n\feld and rotation, wavelike motions are hybrid inertial waves\nand Alfv\u0013 en waves, which we collectively refer to as magnetic-\nCoriolis waves (Finlay 2008).\nMagnetic-Coriolis (MC) waves were \frst investigated\nc\r2017 The AuthorsarXiv:1710.07690v2 [astro-ph.EP] 21 Nov 20172 Y. Lin and G. I. Ogilvie\nby Lehnert (1954). He showed the rotation can split the\nAlfv\u0013 en waves into two groups: fast waves and slow waves, al-\nthough this separation is vague in some parameter regimes.\nMC waves have been studied in di\u000berent geophysical and\nastrophysical contexts since the pioneering study of Lehnert\n(1954). In geophysics, slow MC waves (also called magne-\ntostrophic waves) are of particular interest, as these waves\nare thought to be important in the generation and secular\nvariations of the Earth's magnetic \feld through the dynamo\nprocess in the liquid outer core (Hide 1966; Malkus 1967;\nJault 2008; Finlay 2008; Bardsley & Davidson 2017). The\nslow MC waves usually have frequencies much lower than\nthe rotation frequency and are quasi-geostrophic, i.e. nearly\ninvariant along the rotation axis. In astrophysics, studies of\nstellar oscillations have shown that MC waves can be excited\nin rotating magnetized stars (Lander et al. 2010; Abbassi\net al. 2012). MC waves have also been observed recently\nin liquid metal experiments (Nornberg et al. 2010; Schmitt\n2010; Schmitt et al. 2013). However, tidally forced MC waves\nand the energy dissipation associated with MC waves have\nnot been well studied, except for a recent study using a pe-\nriodic box (Wei 2016). Bu\u000bett (2010) calculated the Ohmic\ndissipation of tidally driven (free inner-core nutation) iner-\ntial waves in the Earth's outer core, but the Lorentz force is\ndropped in his calculations.\nThe present paper studies the propagation and the\nenergy dissipation of tidally forced MC waves in spher-\nical shells. We use a simpli\fed model consisting of a\nrigid core and a homogeneous \ruid envelope, of which the\nhydrodynamic responses have previously been considered\n(Ogilvie 2009, 2013; Rieutord & Valdettaro 2010). Tidal\nresponses are decomposed into non-wavelike and wavelike\nparts (Ogilvie 2013), and we focus on the latter in this study.\nThe wavelike perturbations are MC waves in the presence\nof a magnetic \feld, where both the Lorentz force and the\nCoriolis force act as the restoring force. The linearized equa-\ntions describing the wavelike perturbations are numerically\nsolved using a pseudo-spectral method. The total energy dis-\nsipation can be contributed to by both viscous damping and\nOhmic damping, but is dominated by the latter in most\ncases. We investigate the dependence of the total dissipa-\ntion rate on the magnetic \feld strength, the \feld structure,\nthe dissipative parameters and the tidal frequency. We also\nexamine the frequency-averaged dissipation, which has been\nused to study tidal dissipation and evolution of stars re-\ncently (Guenel et al. 2014; Mathis 2015; Bolmont & Mathis\n2016; Gallet et al. 2017; Bolmont et al. 2017). Remark-\nably, the frequency-averaged dissipation quantity is in good\nagreement with previous analytical results in the absence of\nmagnetic \felds (Ogilvie 2013). Our results suggest that the\nfrequency-averaged tidal dissipation is insensitive to the de-\ntailed damping mechanisms, at least for the wavelike tides.\nThis may have important implications for studying the long-\nterm tidal evolution, as the detailed damping mechanism\nand dissipative properties are not well constrained in stars\nand planets.\nThe paper is organized as follows. Section 2 introduces\nthe simpli\fed model, the basic equations and the numeri-\ncal method. Section 3 presents numerical results. Section 4\nsummarizes the key \fndings of this study.\nΩ\nB0\nRigid insulator\nConducting fluid\nInsulatingFigure 1. Illustration of the model. The dashed lines are a\nschematic representation of the background magnetic \feld B0.\n2 THE SIMPLIFIED MODEL\nOur model essentially builds upon a hydrodynamic model\nconsidered by Ogilvie (2009, 2013). We consider a uniformly\nrotating spherical body consisting of a rigid inner core and\na homogeneous incompressible \ruid envelope. In order to\ntake into account magnetic e\u000bects, we assume that the whole\nbody is permeated by a steady axisymmetric magnetic \feld\nand the \ruid is electrically conducting (see Fig. 1). We fo-\ncus on the wavelike tidal perturbations in the \ruid envelope,\nwhich may be regarded as an idealized model of the convec-\ntive zones of stars and giant planets as it lacks stable strat-\ni\fcation. This section introduces basic equations describing\nthe wavelike perturbations and the numerical method we\nused to solve these equations.\n2.1 Basic equations\nA perfectly rigid inner core of radius r=\u000bRis enclosed by\na homogeneous, incompressible and electrically conducting\n\ruid shell with a free surface at r=R. We assume that\nthe rigid core and the \ruid envelope have the same density\n\u001a. The whole body uniformly rotates at \n= \n^zand is\npermeated by a steady magnetic \feld B0. For simplicity, we\nassume that the background magnetic \feld B0is either a\ndipolar \feld\nB0=B0\"\n^r\u0012R\nr\u00133\ncos\u0012+^\u0012\u0012R\nr\u00133sin\u0012\n2#\n; (1)\nor a uniform axial \feld\nB0=B0^z=B0(^rcos\u0012\u0000^\u0012sin\u0012); (2)\nwhere we have used spherical coordinates ( r;\u0012;\u001e ). Note that\nfor both cases, B0is a potential \feld, i.e. r\u0002B0= 0.\nWe consider the linear responses of the \ruid enve-\nlope to a tidal potential \t = A(r=R)lYm\nl(\u0012;\u001e)e\u0000i!t, where\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 3\nYm\nl(\u0012;\u001e) is a spherical harmonic and !is the tidal frequency\nin the rotating frame. The tidal responses can be decom-\nposed into non-wavelike and wavelike parts as introduced\nby Ogilvie (2013). In the absence of a magnetic \feld, the\nnon-wavelike part represents the instantaneous response to\nthe tidal potential, while the wavelike part is in the form of\ninertial waves excited by the e\u000bective force (Ogilvie 2013)\nf= 2\n\u0002r[Xl(r)Ym\nl(\u0012;\u001e)]e\u0000i!t; (3)\nwhereXlis associated with the non-wavelike motions. This\ne\u000bective force is not generally irrotational and results from\nthe failure of the non-wavelike tide to satisfy the equation\nof motion when the Coriolis force is included. For a homo-\ngeneous incompressible \ruid of the same density as that of\nthe rigid core, Xl(r) is given as (Ogilvie 2013)\nXl(r) =Cl\"\u0010r\nR\u0011l\n+\u000b2l+1l\nl+ 1\u0012R\nr\u0013l+1#\n: (4)\nThe constant Clis given by\nCl=i!(2l+ 1)R3A\n2l(l\u00001)(1\u0000\u000b2l+1)GM; (5)\nwhereGis the gravitational constant and Mis the total\nmass of the body. In this paper, we consider only the domi-\nnant tidal component of l= 2 andm= 2, unless otherwise\nspeci\fed.\nIn the presence of a magnetic \feld, we assume that the\nlarge-scale non-wave-like part is unchanged, and that the\nwave-like perturbations are still driven by the e\u000bective force\ngiven in equation (3), but we allow the magnetic \feld to\na\u000bect the small-scale wave-like motions through the induc-\ntion equation and the Lorentz force. The linearized equations\ngoverning the wavelike velocity perturbation uand magnetic\n\feld perturbation bin the rotating frame can be written as\n\u0000i!u+2\n\u0002u=\u00001\n\u001arp+1\n\u001a\u00160(r\u0002b)\u0002B0+\u0017r2u+f;(6)\n\u0000i!b=r\u0002(u\u0002B0) +\u0011r2b; (7)\nr\u0001u= 0; (8)\nr\u0001b= 0; (9)\nwhere\u0017is the \ruid viscosity, \u00160is the magnetic permeability\nand\u0011is the magnetic di\u000busivity. Equation (6) is the Navier-\nStokes equation including the Coriolis force and the Lorentz\nforce. Equation (7) is the magnetic induction equation.\nUsingR, \n\u00001,B0as units of length, time and magnetic\n\feld strength, we get the non-dimensional equations:\n\u0000i!u+2^z\u0002u=\u0000rp+Le2(r\u0002b)\u0002B0+Ekr2u+f;(10)\n\u0000i!b=r\u0002(u\u0002B0) +Emr2b; (11)\nwhere (and whereafter) !,u,p,b,B0andfdenote the\ncorresponding non-dimensional quantities. The dimension-\nless parameters in equations (10-11) are the Lehnert num-\nberLe, the Ekman number Ekand the magnetic Ekman\nnumberEm:\nLe=B0p\u001a\u00160\nR; Ek=\u0017\n\nR2; Em=\u0011\n\nR2: (12)The Lehnert number Lemeasures the strength of the back-\nground magnetic \feld with respect to rotation, i.e. the ratio\nbetween the Alf\u0013 ven velocity B0=p\u001a\u00160and the rotation ve-\nlocity \nRat the equator. The Ekman number Ekis the\nratio between the rotation time scale \n\u00001and the viscous\ntime scaleR2=\u0017, while the magnetic Ekman number Emis\nthe ratio between the rotation time scale \n\u00001and the mag-\nnetic di\u000busion time scale R2=\u0011. The last two parameters are\nrelated by the magnetic Prandtl number\nPm=Ek\nEm=\u0017\n\u0011: (13)\nWe set the dimensionless forcing term as\nf=^z\u0002r[(r2+\u000b5=r3)Y2\n2(\u0012;\u001e)](1\u0000\u000b5)\u00001e\u0000i!t; (14)\nforl= 2 andm= 2.\nIn order to minimize the viscous and electromagnetic\ncouplings between the \ruid layer and the rigid core, we use\nthe stress-free boundary condition for the velocity uand\nthe insulating boundary condition for the magnetic \feld b\nat both inner and outer boundaries. The stress-free bound-\nary condition implies that the tangential component of the\nviscous stress should vanish:\nur=@\n@r\u0010u\u0012\nr\u0011\n=@\n@r\u0010u\u001e\nr\u0011\n= 0 (15)\nThe insulating boundary condition indicates that no electric\ncurrent can go through the boundaries:\n(r\u0002b)r= 0; (16)\nand the magnetic \feld in the conducting \ruid should also\nmatch a potential \feld Be=\u0000rPin the exterior insulating\nregions, where Pis a scalar potential. This condition can\nbe readily expressed in terms of spherical harmonics (see\nAppendix A).\nThe viscous dissipation rate in dimensionless form is\nDvis=1\n2EkZ\nVRe[(r2u)\u0001u\u0003]dV; (17)\nand the dimensionless Ohmic dissipation rate is\nDohm=1\n2Le2EmZ\nVjr\u0002bj2dV; (18)\nwhere integrals are evaluated over the \ruid domain. It can\nbe shown from the integrated energy equation that the total\nenergy dissipation rate equals the power input by the tidal\nforcingf\nDvis+Dohm\u00111\n2Z\nVRe[f\u0001u\u0003]dV; (19)\nwhereu\u0003is the complex conjugate of u.\n2.2 Numerical method\nEquations (10-11) are solved using a pseudo-spectral\nmethod. We use a spheroidal-toroidal decomposition and\nthen project the equations on to spherical harmonics in a\nsimilar way as in Rincon & Rieutord (2003). The velocity\nand magnetic \feld perturbations are expanded as:\nu=X\num\nl(r)Rm\nl+X\nvm\nl(r)Sm\nl+X\nwm\nl(r)Tm\nl; (20)\nb=X\nam\nl(r)Rm\nl+X\nbm\nl(r)Sm\nl+X\ncm\nl(r)Tm\nl; (21)\nMNRAS 000, 1{14 (2017)4 Y. Lin and G. I. Ogilvie\nwith the summation is carried out over integers l\u0015m\u00150.\nHereRm\nl,Sm\nl,Tm\nlare vector spherical harmonics:\nRm\nl=Ym\nl(\u0012;\u001e)^r;Sm\nl=rrYm\nl(\u0012;\u001e);Tm\nl=rr\u0002Rm\nl:\n(22)\nFor the divergence-free \felds, the \frst two terms in equations\n(20-21) are also referred to as the poloidal part and the last\nterm is the toroidal part. The divergence-free conditions of\nuandb(equations 8-9) are satis\fed by\nvm\nl=1\nl(l+ 1)rd(r2um\nl)\ndr; (23)\nbm\nl=1\nl(l+ 1)rd(r2am\nl)\ndr: (24)\nThe equations projected on to spherical harmonics are given\nin Appendix A. We can see that equations (A1-A4) are de-\ncoupled for each mowing to the axisymmetric rotation and\nmagnetic \feld B0. The Coriolis force and the magnetic \feld\nonly couple the neighbouring spherical harmonics l\u00001 and\nl+ 1. Numerically, the system is truncated at a spherical\nharmonical degree L. In radial direction, we use Chebyshev\ncollocation on N+ 1 Gauss-Lobatto nodes. We use a typ-\nical truncation of L=N= 400 in most of calculations,\nbut higher resolutions up to L=N= 600 are also used\nfor a few more demanding calculations (when Em\u001410\u00005\nandLe\u001410\u00004). The numerical discretization leads to linear\nequations involving a large block-tridiagonal matrix, which\nis solved using the standard direct method based on LU fac-\ntorization.\nThe stress-free boundary condition at the inner and\nouter boundaries becomes\num\nl=d\ndr\u0012vm\nl\nr\u0013\n=d\ndr\u0012wm\nl\nr\u0013\n= 0: (25)\nThe insulating boundary condition requires vanishing\ntoroidal \feld, i.e. cm\nl= 0, at the inner and outer bound-\naries. The poloidal \feld needs to match a potential \feld,\nleading to (see Appendix A)\ndam\nl\ndr\u0000l\u00001\nram\nl= 0; (26)\nat the inner boundary and\ndam\nl\ndr+l+ 2\nram\nl= 0; (27)\nat the outer boundary.\nThe dissipation rates in equations (17-18) can be re-\nduced to integrals in radius only using the orthogonality of\nspherical harmonics:\nDvis=1\n2EkZ1\n\u000bLX\nl=ml(l+ 1)\f\f\f\fum\nl(r) +r2d[vm\nl(r)=r]\ndr\f\f\f\f2\n+l(l+ 1)\f\f\f\fr2d[wm\nl(r)=r]\ndr\f\f\f\f2\n+ 3\f\f\f\frdum\nl(r)\ndr\f\f\f\f2\n+ (l\u00001)l(l+ 1)(l+ 2)\u0000\njvm\nl(r)j2+jwm\nl(r)j2\u0001\ndr;(28)Dohm=1\n2Le2EmZ1\n\u000bLX\nl=ml2(l+ 1)2jcm\nl(r)j2\n+l(l+ 1)\f\f\f\fd[rcm\nl(r)]\ndr\f\f\f\f2\n+l(l+ 1)\f\f\f\fam\nl(r)\u0000d[rbm\nl(r)]\ndr\f\f\f\f2\ndr:\n(29)\nThe integrals are evaluated by a Chebyshev quadrature\nformula which uses the function values at the collocation\npoints. We also calculate the total dissipation rate using\nequation (19), which involves only spectral coe\u000ecients of\nl= 1 andl= 3. This is because the e\u000bective forcing term\nprojected onto spherical harmonics has only l= 1 andl= 3\ncomponents for the l= 2 tidal forcing (see equations (A33-\nA35) in Appendix A), and because of the orthogonality\nof spherical harmonics. The identity (19) can be used to\ncheck the numerical accuracy and convergence. Our numer-\nical code is also validated by comparing with some of the\nresults in Rincon & Rieutord (2003) and Ogilvie (2009).\n2.3 Dispersion relation of magnetic-Coriolis waves\nBefore presenting our numerical results, let us brie\ry recall\nthe dispersion relation of magnetic-Coriolis waves, which is\nuseful for the discussion of some results. Substituting the\nplane wave ansatz u;b/ei(k\u0001r\u0000!t)into equations (6-7),\nand neglecting the di\u000busive terms and the forcing term, we\ncan obtain the dispersion relation of magnetic-Coriolis waves\nin a uniform \feld B0(e.g. Finlay 2008):\n!=\u0006\n\u0001k\njkj\u0006\u0012(\n\u0001k)2\njkj2+(B0\u0001k)2\n\u001a\u00160\u00131=2\n: (30)\nIn the absence of a magnetic \feld, i.e. B0= 0, equation (30)\nrecovers the dispersion relation of inertial waves\n!=\u00062\n\u0001k\njkj; (31)\nwhich exist only when j!j<2\n. The group velocity of iner-\ntial waves is\nVg=\u00062k\u0002(\n\u0002k)\njkj3: (32)\nIn the absence of rotation, i.e. \n= 0, the dispersion relation\nof Alfv\u0013 en waves is obtained:\n!=Va\u0001k; (33)\nwhereVa=B0=p\u001a\u00160is the group velocity.\nThe propagation of MC waves is more complicated, de-\npending on the Lehnert number which measures the impor-\ntance of the magnetic \feld with respect to the rotation. The\ndispersion relation (30) in dimensionless form can be written\nas\n!=\u0006^z\u0001k\njkj\u0006\u0012(^z\u0001k)2\njkj2+Le2k2\nB\u00131=2\n; (34)\nwherekBis the wavenumber along the magnetic \feld B0.\n3 RESULTS\n3.1 Overview\nIn this section, we show a general overview of the spatial\nstructure and the dissipation rate of tidally forced MC waves\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 5\n(a)\n(b)\n(c)\n(d)\n(e)\n(f)\nFigure 2. Structure of the velocity perturbation jujand the mag-\nnetic \feld perturbation jbjin the meridional plane with an axial\n\feldB0at di\u000berent Le.Ek= 1:0\u000210\u00009,Em= 1:0\u000210\u00005,\n!= 1:1,\u000b= 0:5. Note that the colour scales may be di\u000berent for\ndi\u000berentLe.by varying the Lehnert number Le. We consider a case of\nthe radius ratio \u000b= 0:5 and the tidal frequency != 1:1,\nof which the hydrodynamic response has been studied in\ndetail (Ogilvie 2009). In the absence of a magnetic \feld, in-\nertial waves propagate along the characteristics (at a \fxed\nangle with respect to the rotation axis) and form two sim-\nple wave attractors after multiple re\rections (see Fig. 9 in\nOgilvie (2009)). The dissipation rate associated with iner-\ntial wave attractors is independent of the viscosity (the Ek-\nman number) provided that the Ekman number is asymp-\ntotically small (Ogilvie 2005, 2009). We use this case as a\nreference, mainly because of its relatively simple hydrody-\nnamic responses, to examine the e\u000bects of magnetic \felds.\nFig. 2 shows the velocity perturbation jujand the mag-\nnetic \feld perturbation jbjin the meridional plane in the\npresence of an axial magnetic \feld with various values of the\nLehnert number Le. The dissipative parameters are \fxed at\nEk= 10\u00009andEm= 10\u00005, meaning that the magnetic\nPrandtl number Pm= 10\u00004. When the Lehnert number is\nsu\u000eciently small ( Le\u0014O(E2=3\nm) as we shall show later),\nMC waves retain the rays of inertial waves, leading to the\nwave attractors as for purely inertial waves (Fig 2 a). Weak\nmagnetic \feld perturbations are induced along the attrac-\ntors by the velocity perturbations, but the Lorentz force\nhas negligible in\ruence on the propagation of waves. As the\nLehnert number is gradually increased, the perturbations do\nnot concentrate on the wave attractors any more, because\nthe Lorentz force starts to play a part. In Fig 2 (b), however,\nwe can still see the predominant e\u000bect of the rotation as the\nperturbations are mainly organized along the characteristics\nof inertial waves, but slightly modi\fed.\nAs we increase the Lehnert number further, the e\u000bect of\nrotation becomes less visible and the perturbations spread\nout to the whole \ruid domain (Fig 2 c). At certain values of\nthe Lehnert number, e.g. Le= 0:1 for this case, we observe\nsome large-scale structures in the polar region in Fig 2 (d).\nThese structures may be associated with eigen-modes of the\nsystem. We shall show more examples of such structures at\ndi\u000berent frequencies in section 3.2.\nAt relatively large values of Le, i.e.Le> 0:1, the mag-\nnetic e\u000bect become predominant as we can see from Fig. 2(e-\nf) that the perturbations concentrate along certain magnetic\n\feld lines. In this regime, the perturbations are essentially\nin the form of Alfv\u0013 en waves, where the magnetic tension acts\nas the restoring force. Each magnetic \feld line can be anal-\nogous to a string, which has a natural frequency depending\non the length and strength of the \feld line. Perturbations\nmainly concentrate along certain \feld lines, where a reso-\nnance may occur if the tidal frequency matches the natural\nfrequency of the \feld line.\nFig. 3 shows the structure of the perturbations as in\nFig. 2 but for a dipolar \feld B0. The spatial structures vary\nin a similar way as in the case of an axial \feld, as we grad-\nually increase Le. However, there are some local di\u000berences\nbecause the dipolar \feld B0is spatially non-uniform, being\nstronger at the polar regions than at the equator and de-\ncaying as a function of the radius. For instance, in Fig. 3(c),\nthe perturbations are less in\ruenced by the magnetic \feld\nnear the equator compared to other regions. Nevertheless,\nthe general picture is qualitatively similar to Fig. 2, from\ninertial wave attractors at small Leto nearly Alfv\u0013 en wave\nperturbations at relatively large Le.\nMNRAS 000, 1{14 (2017)6 Y. Lin and G. I. Ogilvie\n(a)\n(b)\n(c)\n(d)\n(e)\n(f)\nFigure 3. Same as Fig. 2 but for a dipolar \feld B0. Dashed lines\nshow some \feld lines of B0.\n10-510-410-310-210-1100\nL e10-710-610-510-410-310-210-1Dissipation rate\nViscous\nOhmic\nTotal(a)\n10-510-410-310-210-1100\nL e10-810-710-610-510-410-310-210-1100Dissipation rate\nViscous\nOhmic\nTotal\n(b)\nFigure 4. Dissipation rate versus the Lehnert number Lefor (a)\nan axial \feld, (b) a dipolar \feld. Vertical dash lines represent\nLe=E2=3\nm.Ek= 1:0\u000210\u00009,Em= 1:0\u000210\u00005,!= 1:1,\u000b= 0:5.\nBlack triangles correspond to cases shown in Figs. 2-3.\nWe have mentioned that the perturbations retain the\nray dynamics of inertial waves when Le\u0014O(E2=3\nm). We can\nderive this scaling by simply comparing several typical time\nscales in the system when Le\u001c1,Ek\u001c1,Em\u001c1 and\nPm\u001c1. The inertial wave propagation time in the \ruid\ndomain is\n\u001ci=L\njVgj\u0018\u0012l\nL\u0013\u00001\n\n\u00001: (35)\nThe time scale for Alfv\u0013 en waves to transversely cross the\ninertial wave beams is\n\u001ca=l\njVaj\u0018l\nLLe\u00001\n\u00001: (36)\nThe magnetic di\u000busion time across the beams is\n\u001c\u0011=l2\n\u0011\u0018\u0012l\nL\u00132\nE\u00001\nm\n\u00001; (37)\nand the viscous di\u000busion time is\n\u001c\u0017=l2\n\u0017\u0018\u0012l\nL\u00132\nE\u00001\nk\n\u00001: (38)\nHerelis the typical width of the wave beams, whereas L\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 7\nis the domain size, i.e. L=R. The viscous time scale is\nirrelevant here because of the assumption Pm\u001c1. In order\nto keep the perturbations within the inertial wave beams,\nthe crossing time of Alfv\u0013 en waves \u001cashould be longer than\nthe inertial wave propagation time \u001ci:\n\u001ca\u0015\u001ci: (39)\nMeanwhile, the width of the wave beams is set by the di\u000bu-\nsion time across the beams:\n\u001ci=\u001c\u0011: (40)\nCombining equations (39-40) and using equations (35-37),\nwe obtain\nLe\u0014O(E2=3\nm): (41)\nNote that this scaling analysis is merely heuristic. The\nprefactor of the scaling (41) varies depending on the fre-\nquency and the structure of background \felds. Neverthe-\nless, the above scaling provides an approximate threshold,\nabove which the presence of a magnetic \feld would modify\nthe propagation of inertial waves. This scaling is also clearly\nevidenced from the dissipation rate in Fig. 4.\nFig. 4 shows the viscous dissipation rate, the Ohmic\ndissipation rate and the total dissipation rate as a function\nofLefor an axial \feld (a) and a dipolar \feld (b). When\nLe\u0014O(E2=3\nm), the dissipation rate due to the Ohmic damp-\ning grows and then saturates as Leincreases, while the vis-\ncous dissipation rate drops and eventually becomes negligi-\nble. However, the total dissipation rate remains unchanged\nin the range of Le\u0014O(E2=3\nm). This observation is remi-\nniscent of the analytical result by Ogilvie (2005), who has\nshown that the total rate of energy dissipation of a wave\nattractor is independent of the dissipative properties and\nthe detailed damping mechanisms. The theoretical analysis\nhas been con\frmed by hydrodynamic calculations in spher-\nical shells (Ogilvie 2009; Rieutord & Valdettaro 2010). Here\nwe show that the theory is still valid in the presence of a\nmagnetic \feld as long as the wave attractors are retained.\nWhenLe > O (E2=3\nm), the total dissipation is almost\ntotally contributed by the Ohmic damping, whereas the vis-\ncous dissipation is negligible. The dissipation rate \ructuates\nas a function of Le, exhibiting several peaks and troughs. In\nthis range of parameters, the magnetic \feld modi\fes the\npropagation of waves depending on the Lehnert number.\nResonance may occur at certain values of Lefor a given fre-\nquency, leading to enhanced dissipation. Indeed, these peaks\nin the dissipation rate usually correspond to either large-\nscale perturbations, e.g. Fig 2(d) and Fig 3 (d), or pertur-\nbations concentrating on certain \feld lines, e.g. Fig 2 (f) and\nFig 3 (e).\nNote that we have restricted our investigations in the\nrange ofLe\u00141. We found that strong magnetic bound-\nary layers arise when Le\u001d1, and thus the energy dissi-\npation is mainly contributed by the boundary layers, which\nmay be not realistic because of our idealized boundary con-\nditions. For instance, a rigid core of \fnite electric conduc-\ntivity (rather than insulating) would relax the accumula-\ntion of the electrical current near the inner boundary. Any-\nway, the Lehnert number should be smaller than unity in\nmost stars and planets, although the speci\fc value is di\u000e-\ncult to estimate owing to the uncertainties of the magnetic\n\feld strength and \ruid properties. The Lehnert number forthe Sun is estimated to be around 10\u00005, if we assume the\ntypical magnetic \feld strength is a few 10\u00003T (Charbon-\nneau 2014), and use the mean density of the Sun. Mean-\nwhile, the magnetic Ekman number is also very small for\nthe Sun, i.e. Em<10\u000010, as the magnetic di\u000busion time is\naround 1010year (Charbonneau 2014). The magnetic \feld\nis strong enough to modify the propagation of inertial waves\nasLe > O (E2=3\nm) for the Sun, despite the small Lehnert\nnumber.\n3.2 Frequency-dependence\nWe now investigate the frequency-dependence of the dissi-\npation rate. Fig 5 shows a frequency scan of the total dissi-\npation rate in a frequency range of \u00003\u0014!\u00143 at various\nvalues ofLewith an axial \feld, and \u000b= 0:5,Ek= 10\u00008\nandEm= 10\u00004. For comparison, we show also the dissipa-\ntion rate in the absence of a magnetic \feld, i.e. Le= 0, in\nthe frequency range of inertial waves. In the presence of the\nmagnetic \feld, we show only the cases when Le>O (E2=3\nm),\nbecause the total dissipation rate shows similar behaviour\nto that of inertial waves when Le\u0014O(E2=3\nm).\nFig 5 (a) shows the dissipation rate at Le= 3:2\u000210\u00003\n(the blue curve) and Le= 2:4\u000210\u00002( the red curve). We\ncan see that these curves are very bumpy, likewise the curve\nin the absence of a magnetic \feld (the black dashed line).\nThe peaks and troughs are closely related to those the black\ndashed line. These cases can be regarded as weakly modi\fed\ninertial waves. In particular, we can see from the red curve\nthat the peaks shift to higher frequencies, which is in line\nwith the dispersion relation of magnetic-Coriolis waves.\nFig 5 (b) shows the dissipation rate at Le= 0:1 (the\nblue curve) and Le= 0:42 ( the red curve). These two curves\nare signi\fcantly smoothed out by the presence of a mag-\nnetic \feld compared to the black dashed line. In addition,\nthe spectra of the dissipation rate are broadened beyond\nthe frequency range of inertial waves, as expected from the\ndispersion relation (34).\nAlthough the dissipation rate curves become smooth at\nrelatively large Le, they still exhibit a few peaks in the fre-\nquency range we have shown. Fig. 6 shows structures of the\nperturbations at some peak frequencies of the blue curve in\nFig. 5 (b). We can see that all these cases feature smooth\nlarge-scale structures, in particular in the polar region. We\nalso note that the number of nodes in the vertical direc-\ntion increases as the frequency increases, which is reminis-\ncent of the dispersion relation of magnetic-Coriolis waves.\nWe have mentioned that such smooth structures may be\neigen-modes of the system, which are resonantly excited at\nthe eigen-frequencies, leading to the enhanced energy dissi-\npation. However, the theoretical analysis of the eigen value\nproblem is beyond the scope of this paper.\nFor the case of a dipolar \feld, the frequency-dependence\nof the total dissipation rate is qualitatively similar to that\nof an axial \feld, although details are di\u000berent, such as the\npeak frequencies.\n3.3 Frequency-averaged dissipation rate\nThe tidal dissipation leads to a long-term evolution of the\nspin and orbital parameters through the exchange of angu-\nlar momentum. As the system evolves, the tidal frequency\nMNRAS 000, 1{14 (2017)8 Y. Lin and G. I. Ogilvie\n-3 -2 -1 0 1 2 3\nFrequency10-610-510-410-310-210-1100D i s s i p a t i o n r a t e\n(a)\n-5-4-3-2-1012345\nFrequency10-610-510-410-310-210-1100D i s s i p a t i o n r a t e\n(b)\nFigure 5. Total dissipation rate versus the tidal forcing frequency. The background magnetic \feld is set to be an axial \feld. \u000b= 0:5,\nEk= 10\u00008andEm= 10\u00004. (a)Le= 3:2\u000210\u00003(blue) and Le= 2:4\u000210\u00002(red); (b)Le= 0:1 (blue) and Le= 0:42 (red). Black\ndashed lines represent the dissipation rate in the absence of a magnetic \feld. Blue triangles correspond to cases shown in Fig 6.\nvaries over time. It is very di\u000ecult to estimate the instanta-\nneous tidal dissipation owing to the complicated frequency-\ndependence. However, the frequency-averaged dissipation\nrate can be useful to study the long-term evolution of the\nsystem (Mathis 2015; Bolmont & Mathis 2016; Gallet et al.\n2017; Bolmont et al. 2017). Ogilvie (2013) has shown that\nthe frequency-averaged dissipation rate of inertial waves is\nindependent of the dissipative properties, but strongly de-\npends on the size of the rigid core. Here we examine the\nfrequency-averaged dissipation in the presence of a magnetic\n\feld.\nThe frequency-averaged dissipation can be measured by\nthe dimensionless quantity (Ogilvie 2013)\n\u0003 =Z1\n\u00001Im[Km\nl(!)]d!\n!; (42)\nwhere Im[Km\nl] is the imaginary part of the potential Lovenumber, and is related to the dissipation rate by\n^D=(2l+ 1)R\n8\u0019GA2\n!Im[Km\nl(!)]: (43)\nwhereAis the tidal amplitude with the unit of gravitational\npotential and ^Dis the dimensional dissipation rate with the\nunit of power. With our normalization of the forcing in equa-\ntion (14) for l= 2 andm= 2, the frequency-averaged quan-\ntity becomes\n\u0003 =Z1\n\u00001Im[K2\n2(!)]d!\n!=15\n2\u000f2Z1\n\u00001D(!)d!; (44)\nwhere\u000f2= \n2R3=GM andD(!) =Dvis+Dohmis the di-\nmensionless total dissipation rate as shown in Fig. 5. Note\nthat\u000fis a small parameter for astrophysical bodies, but we\nsimply set \u000f= 1 in our linear calculations of the wavelike\nperturbations. The above integral can be carried out only in\na \fnite frequency range numerically. For the hydrodynamic\ncase (Le= 0), the integral is evaluated over the frequency\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 9\n(a)\n(b)\n(c)\n(d)\n(e)\n(f)\nFigure 6. Structure of the velocity perturbation jujand the mag-\nnetic \feld perturbation jbjin the meridional plane with an axial\n\feldB0at di\u000berent tidal frequencies. Le= 0:1,Ek= 10\u00008,\nEm= 10\u00004,\u000b= 0:5.\n10-310-210-1100\nL e10-310-210-1100$ H y d r o\nU n i f o r m B 0 , E k = 1 0! 8, E m = 1 0! 4\nU n i f o r m B 0 , E k = 1 0! 1 0, E m = 1 0! 5\nD i p o l a r B 0 , E k = 1 0! 8, E m = 1 0! 4Figure 7. Frequency-averaged quantity \u0003 versus the Lehnert\nnumberLefor various di\u000berent parameters but for \fxed inner\ncore radius \u000b= 0:5.\nrange of inertial waves, i.e. \u00002 0:1. We also doubled the frequency range at large Le\nand found that the results of the integral are converged. Fig.\n7 shows the frequency-averaged quantity \u0003 at several di\u000ber-\nent parameters but with the \fxed inner core size \u000b= 0:5.\nWe can see that the frequency-averaged dissipation rate is\nindependent of the strength and the structure of the mag-\nnetic \feld, and the dissipative parameters EkandEm. Sim-\nilar results have been observed in a previous study using a\nperiodic box (Wei 2016). Also, the frequency-averaged dissi-\npation rate in the presence of a magnetic is nearly the same\nas that in the absence of a magnetic \feld. The small discrep-\nancies are likely due to the errors of numerical integrals.\nHowever, the frequency-averaged dissipation rate\nstrongly depends on the size of the inner core. Ogilvie\n(2013) derived an analytical expression by considering low-\nfrequency hydrodynamic responses to an impulsive forcing,\nwhich is given as\n\u0003 =Z1\n\u00001Im[K2\n2(!)]d!\n!=100\u0019\n63\u000f2\u0012\u000b5\n1\u0000\u000b5\u0013\n; (45)\nfor a homogeneous \ruid of the same density as that of the\nrigid core, and for the tidal component of l=m= 2. Equa-\ntion (45) has been veri\fed numerically (Ogilvie 2013), and\nresembles the scaling of \u000b5for the small inner core size found\nby previous hydrodynamic studies (Goodman & Lackner\n2009; Ogilvie 2009; Rieutord & Valdettaro 2010).\nFig. 8 show the frequency-averaged quantity \u0003 as a\nfunction of the radius ratio \u000bat di\u000berent values of Le. Re-\nmarkably, this frequency-averaged quantity in the presence\nof a magnetic \feld is still in very good agreement with equa-\ntion (45), which is derived in the absence of magnetic \felds.\nIn the presence of a magnetic \feld, the energy is mainly\ndissipated through the Ohmic damping of magnetic-Coriolis\nMNRAS 000, 1{14 (2017)10 Y. Lin and G. I. Ogilvie\n00.2 0.4 0.6 0.8 1\n,10-510-410-310-210-1100101102$\nL e = 0 ( a n a l y t i c a l )\nL e = 0 : 0 1\nL e = 0 : 1\nFigure 8. Frequency-averaged quantity \u0003 versus the radius ratio\n\u000bat di\u000berent values of Lewith an axial \feld and Ek= 10\u00008,\nEm= 10\u00004. The solid line represents the analytical expression\nfrom Ogilvie (2013).\nwaves, which occur over a wider frequency range, but the\nfrequency-averaged dissipation rate is the same as that of\nviscous dissipation of inertial waves. This suggests that the\nfrequency-averaged dissipation rate is independent of the de-\ntailed damping mechanisms. Indeed, we show in Appendix\nB that the analytical results on the frequency-averaged dis-\nsipation in Ogilvie (2013) are not altered by the presence\nof a magnetic \feld, in the approximation that the wave-like\nresponse is driven only by the Coriolis force acting on the\nnon-wavelike tidal \row, if we assume that the frequencies\nof MC waves are small compared to those of acoustic and\nsurface gravity waves. This is the case in our numerical cal-\nculations of the Lehnert number Le\u00141 (see Fig. 5). In real\nastrophysical \ruid bodies, this assumption may be justi\fed\nby the fact the tidal frequency is usually small compared to\nthe frequencies of acoustic and surface gravity waves.\n3.4 Obliquity tide\nSo far, we considered only the tidal component of l= 2\nandm= 2, which mainly determines the orbital evolution\nand synchronization. In this section, we brie\ry consider an-\nother important tidal component of l= 2 andm= 1, the\nso-called obliquity tide, which exists only in spin-orbit mis-\naligned systems and mainly determines the evolution of the\nspin-orbit angle. The obliquity tide is peculiar because the\ntidal frequency in the rotating frame is always equal to \u0000\nregardless of the orbital frequency, i.e. !=\u00001 in dimension-\nless form. In addition, the obliquity tide is responsible for\nthe precessional motion of the spin axis and the orbital nor-\nmal around the total angular momentum vector. We have\nshown that dissipative inertial waves can be excited by the\nobliquity tide on top of precession in a hydrodynamic study\n(Lin & Ogilvie 2017). To examine the magnetic e\u000bect on the\nwavelike responses of the obliquity tide, we need to replace\nthe forcing in equation (3) by\nf= 2\n\u0002r[Xl(r)Y1\n2(\u0012;\u001e)]ei\nt+ (\n\u0002\np)\u0002r; (46)\n0.2 0.4 0.6 0.8\n,10-610-410-2100102Dissipation rateL e = 0\nL e = 0 : 0 1\nL e = 0 : 1Figure 9. Total dissipation rate of the obliquity tide as a function\nof radius ratio \u000bat di\u000berent values of Lewith an axial \feld.\nEk= 10\u00007,Em= 10\u00004.\nwhere the last term arises from the precessional motion\naround the total angular momentum vector. The precession\nfrequency is determined by the tidal amplitude and given as\n(Lin & Ogilvie 2017)\n\np=\u000015\n8r\n5\n6\u0019R3\nA\n(1\u0000\u000b5)GMsini; (47)\nwhereiis the angle between the spin angular momentum\nand the total angular momentum.\nFig. 9 shows the total dissipation as a function of the\nradius ratio \u000bat di\u000berent value of Lewith an axial magnetic\n\feld. For comparison, we show also the dissipation rate in\nthe absence of a magnetic \feld, which exhibits complicated\ndependence of the core size owing to varied ray dynamics of\ninertial waves, especially when \u000b>0:5 (Lin & Ogilvie 2017).\nThe major e\u000bect of a magnetic \feld is, again, to smooth out\nthe dissipation rate curves, but the overall level is similar to\nthat of the hydrodynamic case. Note that the frequency of\nthe obliquity tide is always !=\u00001 in the rotating frame,\nso we did not explore the frequency-dependence and the\nfrequency-averaged quantity for the obliquity tide.\n4 CONCLUSIONS\nWe have investigated the magnetic e\u000bects on the tidal dis-\nsipation in rotating \ruid bodies using a simpli\fed model.\nThe tidal responses are decomposed into the non-wavelike\nand wavelike parts, but we have focused on the latter in\nthis study, which is in the form of magnetic-Coriolis waves.\nThe linearized wave equations are numerically solved using\na pseudo-spectral method. The major e\u000bects of the presence\nof a magnetic \feld can be summarized as follows.\n(i) When the magnetic \feld is very weak, namely\nLe\u0014O(E2=3\nm), the wavelike perturbations retain the ray\ndynamics of inertial waves, while the energy can be dissi-\npated through the viscous damping and the Ohmic damping.\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 11\nWhenLe>O (E2=3\nm), the magnetic \feld start to modify the\npropagation of waves, and the dissipation of energy occurs\nmainly through the Ohmic damping.\n(ii) The magnetic \feld smooths out the complicated de-\npendence of the total dissipation rate on the tidal frequency,\nand broadens the frequency spectrum of the dissipation rate,\ndepending on the Lehnert number Le.\n(iii) However, the frequency-averaged dissipation quan-\ntity is independent of the magnetic \feld strength, the \feld\nstructure and the dissipative parameters, but increases as\nthe relative size of the rigid core in our simpli\fed model. In\nmore realistic models, this frequency-averaged quantity may\ndepend on other properties of the internal structure such as\nthe density pro\fle. Indeed, the frequency-averaged quantity\nis in very good agreement with previous analytical results in\nthe absence of magnetic \felds (Ogilvie 2013). In Appendix\nB, we show that the magnetic \feld has no e\u000bect on the\nfrequency-averaged dissipation, in the approximation that\nthe wave-like response is driven only by the Coriolis force\nacting on the non-wavelike tidal \row, if we assume that the\nfrequencies of magnetic-Coriolis waves are small compared\nto those of acoustic and surface gravity waves.\nOwing to the complicated frequency-dependence of the\ndissipation rate of inertial waves, it is very di\u000ecult to di-\nrectly apply the instantaneous tidal dissipation to astro-\nphysical bodies. Alternatively, the frequency-averaged dis-\nsipation rate has been used to study tidal dissipation and\nevolution of stars (Mathis 2015; Bolmont & Mathis 2016;\nGallet et al. 2017; Bolmont et al. 2017). It has been con-\njectured that other e\u000bects such as non-linear interactions,\nconvection, di\u000berential rotations and magnetic \felds may\nwash out some of the complicated frequency-dependence of\npurely inertial waves (Ogilvie 2013). Indeed, our numeri-\ncal results show a smoothing e\u000bect of the magnetic \feld\non the frequency-dependence. More complicated processes\ncould lead to smoother dissipation curves. Therefore, the\nfrequency-averaged dissipation quantity is probably a useful\nindicator of the e\u000eciency of tidal dissipation. In addition,\nour results suggest that the frequency-averaged quantity is\ninsensitive to the detailed damping mechanisms and dissi-\npative properties. If this is still the case in more realistic\nmodels, it would be very useful in applications as these de-\ntails are not well understood in stars and planets. Note that\nthe frequency-averaged dissipation does depend on the in-\nternal structure of the bodies, which is merely determined\nby the size of the rigid core in our simpli\fed model.\nIt is worthwhile to mention that we considered only the\nmagnetic e\u000bects on the wave-like tidal perturbations due\nto the Coriolis force in this study. The magnetic \feld can\nalso interact with large-scale non-wave-like motions to pro-\nduce a further wave-like response, which remains to be stud-\nied. Our tentative investigations based on a radially forced\nmodel (Ogilvie 2009) suggest that the interactions between\na magnetic \feld and the non-wavelike motions may be not\nnegligible when the Lehnert number Le > 0:1. However,\nthe boundary conditions probably need to be treated more\ncarefully with a magnetic \feld, as the non-wavelike part is\nassociated with the instantaneous tidal deformation.ACKNOWLEDGEMENTS\nWe would like to thank the anonymous referee for a set of de-\ntailed and constructive comments, which helped improve the\npaper. YL acknowledges the support of the Swiss National\nScience Foundation through an advanced Postdoc.Mobility\nfellowship.\nREFERENCES\nAbbassi S., Rieutord M., Rezania V., 2012, MNRAS, 419, 2893\nBardsley O., Davidson P., 2017, Geophysical Journal Interna-\ntional, 210, 18\nBarker A. J., Ogilvie G. I., 2010, MNRAS, 404, 1849\nBolmont E., Mathis S., 2016, Celestial Mechanics and Dynamical\nAstronomy, 126, 275\nBolmont E., Gallet F., Mathis S., Charbonnel C., Amard L., Al-\nibert Y., 2017, A&A, 604, A113\nBu\u000bett B. A., 2010, Nature, 468, 952\nCharbonneau P., 2014, ARA&A, 52, 251\nEssick R., Weinberg N. N., 2016, ApJ, 816, 18\nFinlay C. 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J., Deming D., Hamilton D.,\nGillon M., Jehin E., 2017, ApJ, 836, L24\nWu Y., 2005, ApJ, 635, 674\nZahn J.-P., 1975, A&A, 41, 329\nMNRAS 000, 1{14 (2017)12 Y. Lin and G. I. Ogilvie\nAPPENDIX A: EQUATIONS PROJECTED ON\nTO SPHERICAL HARMONICS\nTaking the curl and the curl of the curl of equation (10),\ntaking the curl of equation (11) and using the orthogonality\nof the vector spherical harmonics (Rieutord 1987), we obtain\nthe following projected equations. Similar equations were\ngiven in the appendix of Rincon & Rieutord (2003), but\nwithout the Coriolis term. The projection of the Coriolis\nforce can be found in, e.g. Rieutord (1987). The projected\nequations are given as\n\u0000i!lDl(um\nl) =\u00002\fm\nlDw\nl\u00001(wm\nl\u00001)\u00002\fm\nl+1Dw\nl+1(wm\nl+1)\n+Le2[imLc\nl(cm\nl) +\u000bm\nlLa\nl\u00001(am\nl\u00001) +\u000bm\nl+1La\nl+1(am\nl+1)]\n+EDu\nl(um\nl) +fl;(A1)\n\u0000i!lwm\nl=\u00002\u000bm\nlDu\nl\u00001(um\nl\u00001)\u00002\u000bm\nl+1Du\nl+1(um\nl+1)\n+Le2[imLa\nl(am\nl) +\u000bm\nlLc\nl\u00001(cm\nl\u00001) +\u000bm\nl+1Lc\nl+1(cm\nl+1)]\n+EDw\nl(wm\nl) +fl\u00001+fl+1;(A2)\n\u0000i!am\nl= imLw\nl(wm\nl) +\u000bm\nlLu\nl\u00001(um\nl\u00001) +\u000bm\nl+1Lu\nl+1(um\nl+1)\n+EmDa\nl(am\nl);(A3)\n\u0000i!cm\nl= imLu\nl(um\nl) +\u000bm\nlLw\nl\u00001(wm\nl\u00001) +\u000bm\nl+1Lw\nl+1(wm\nl+1)\n+EmDc\nl(cm\nl);(A4)\nwhereDandLdenote linear di\u000berential operators:\nDl=rd2\ndr2+ 4d\ndr\u0000(l2+l\u00002)1\nr; (A5)\nDw\nl\u00001=d\ndr\u0000(l\u00001)1\nr; (A6)\nDw\nl+1=d\ndr+ (l+ 2)1\nr(A7)\nDu\nl=rd4\ndr4+ 8d3\ndr3\u00002(lp\u00006)1\nrd2\ndr2\n\u00004lp1\nr2d\ndr+lp(lp\u00002)1\nr3;(A8)\nLc\nl=Brd2\ndr2+\u00122Br\nr+dBr\ndr\u0013d\ndr\n+dBr\ndr\u0000lpd\ndr\u0012B\u0012\nr\u0013\n;(A9)\nLa\nl\u00001=lp\u0014\nrBrd3\ndr3+\u0012\nlB\u0012+rdBr\ndr+ 6Br\u0013d2\ndr2\u0015\n\u0000lp\u0014\n(l+ 2)(l\u00003)Br\nr\u00004dBr\ndr\u00004lB\u0012\nr\u0015d\ndr\n\u0000lp(l+ 1)(l\u00002)\u0012\nlB\u0012\nr2+1\nrdBr\ndr\u0013\n;(A10)\nLa\nl+1=lp\u0014\nrBrd3\ndr3\u0000\u0012\n(l+ 1)B\u0012\u0000rdBr\ndr\u00006Br\u0013d2\ndr2\u0015\n\u0000lp\u0014\n(l+ 4)(l\u00001)Br\nr\u00004dBr\ndr\u00004(l+ 1)B\u0012\nr\u0015d\ndr\n+lpl(l+ 3)\u0012\n(l+ 1)B\u0012\nr2\u00001\nrdBr\ndr\u0013\n;(A11)Du\nl\u00001=rd\ndr\u0000(l\u00002); (A12)\nDu\nl+1=rd\ndr+ (l+ 3); (A13)\nDw\nl=d2\ndr2+2\nrd\ndr\u0000lp1\nr2; (A14)\nLa\nl=\u0000Br\nl2p\u0012\nrd2\ndr2+ 4d\ndr\u0000(lp\u00002)1\nr\u0013\n; (A15)\nLc\nl\u00001=l(l\u00001)\u0012\nBrd\ndr+Br\nr+lB\u0012\nr\u0013\n; (A16)\nLc\nl+1= (l+ 1)(l+ 2)\u0012\nBrd\ndr+Br\nr\u0000(l+ 1)B\u0012\nr\u0013\n;(A17)\nDa\nl=d2\ndr2+4\nrd\ndr+ (2\u0000lp)1\nr2; (A18)\nLw\nl=Br\nr; (A19)\nLu\nl\u00001=lp\u0012\nBrd\ndr+ 2Br\nr+lB\u0012\nr\u0013\n; (A20)\nLu\nl+1=lp\u0012\nBrd\ndr+ 2Br\nr\u0000(l+ 1)B\u0012\nr\u0013\n; (A21)\nlp=l(l+ 1) (A22)\nDc\nl=d2\ndr2+2\nrd\ndr+lp1\nr2; (A23)\nLu\nl=\u00001\nl2p\u0014\nrBrd2\ndr2+\u0012\n4Br+rdBr\ndr\u0013d\ndr\u0015\n1\nl2p\u0012\nlpdB\u0012\ndr\u0000lpB\u0012\nr\u0000dBr\ndr\u0000Br\nr\u0013\n;(A24)\nLw\nl\u00001=l(l\u00001)\u0012\nBrd\ndr+Br\nr+dBr\ndr+lB\u0012\nr\u0013\n; (A25)\nLw\nl+1= (l+ 1)(l+ 2)\u0012\nBrd\ndr+Br\nr+dBr\ndr\u0000(l+ 1)B\u0012\nr\u0013\n:\n(A26)\nIn above operators, we have used the following notations:\n!l=!+2m\nl(l+ 1); (A27)\nlp=l(l+ 1); (A28)\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 13\nqm\nl=\u0012l2\u0000m2\n4l2\u00001\u00131=2\n; (A29)\n\u000bm\nl=1\nl2qm\nl; \fm\nl= (l2\u00001)qm\nl; (A30)\nandBrandB\u0012represent the radial dependence of the back-\nground magnetic \feld B0. For the dipolar \feld, we have\nBr=1\nr3; B\u0012=1\n2r3; (A31)\nwhile for the uniform vertical \feld\nBr= 1; B\u0012=\u00001: (A32)\nThe vortical tidal forcing fcan be also project on to spher-\nical harmonics\nfl=\u00002im\nl(l+ 1)\u0012\nl(l+ 1)Xl(r)\nr\u0000rd2Xl(r)\ndr2\u00002dXl(r)\ndr\u0013\n;\n(A33)\nfl\u00001=\u00002qm\nl\nl\u0012dXl(r)\ndr+ (l+ 1)Xl(r)\nr\u0013\n; (A34)\nfl+1= 2qm\nl+1\nl+ 1\u0012dXl(r)\ndr\u0000lXl(r)\nr\u0013\n: (A35)\nFor a homogeneous \ruid, fl\u00110 because Xl(r) satis\fes\n(equation (86) in Ogilvie 2013)\n1\nr2d\nr\u0012\nr2dXl(r)\ndr\u0013\n\u0000l(l+ 1)\nr2Xl(r) = 0: (A36)\nThe boundary conditions can be also projected onto\nspherical harmonics. The stress-free boundary condition (15)\nleads to\num\nl=d\ndr\u0012vm\nl\nr\u0013\n=d\ndr\u0012wm\nl\nr\u0013\n= 0: (A37)\nThe insulating boundary condition ( r\u0002b)rleads tocm\nl=\n0. The poloidal magnetic \feld continuously extends to the\ninsulating regions as a potential \feld Be=\u0000rP, where the\nscalar potential Psatis\fes Laplace's equation\nr2P= 0: (A38)\nThe solution of the above equation is\nP=XX\ngm\nlrlYm\nl(\u0012;\u001e); (A39)\nin the rigid inner core and\nP=XX\nhm\nlr\u0000(l+1)Ym\nl(\u0012;\u001e); (A40)\noutside the body. Substituting those solutions into Be=\n\u0000rPand comparing with the spherical harmonics expan-\nsion ofbin the \ruid region, we obtain the boundary condi-\ntions\nam\nl(r)\u0000lbm\nl(r) = 0; (A41)\nat the inner boundary and\nam\nl(r) + (l+ 1)bm\nl(r) = 0; (A42)\nat the outer boundary. Using the divergence-free conditionofbin equation (24), we can write the boundary condi-\ntions (A41-A42) as\ndam\nl\ndr\u0000l\u00001\nram\nl= 0; (A43)\nat the inner boundary and\ndam\nl\ndr+l+ 2\nram\nl= 0; (A44)\nat the outer boundary.\nAPPENDIX B: FREQUENCY-AVERAGED\nDISSIPATION IN THE PRESENCE OF A\nMAGNETIC FIELD\nOgilvie (2013) found a way of calculating a certain\nfrequency-average of the tidal response of a slowly and uni-\nformly rotating barotropic \ruid body to harmonic forcing.\nIn this Appendix we consider how the argument and re-\nsults presented in Section 4 of that paper are a\u000bected by the\npresence of a magnetic \feld. All the section numbers and\nequation numbers used below refer to Ogilvie (2013).\nThe equilibrium condition (21) is modi\fed to\n0 =\u0000r(h+ \b g+ \b c) +1\n\u00160\u001a(r\u0002B)\u0002B (B1)\nand the linearized equations are modi\fed to\n\u0018+ 2\n\u0002_\u0018=\u0000rW+F\u0018; (B2)\nW=h0+ \b0+ \t; (B3)\n\u001a0=\u0000r\u0001(\u001a\u0018); (B4)\nr2\b0= 4\u0019G\u001a0; (B5)\nwhereFis the linearized Lorentz force operator, which is\nself-adjoint with respect to a mass-weighted inner product\nand is given by\nF\u0018=1\n\u00160\u001a\u0002\n(r\u0002B0)\u0002B+ (r\u0002B)\u0002B0\u0003\n; (B6)\nwhere\nB0=r\u0002(\u0018\u0002B): (B7)\nIn making the low-frequency asymptotic analysis in Sec-\ntion 4.4, we wish to assume that the frequencies of Alfv\u0013 en\nand slow magnetoacoustic waves are, like those of the in-\nertial waves, small compared to those of acoustic (or fast\nmagnetoacoustic) and surface gravity waves. Unlike the in-\nertial waves, however, the Alfv\u0013 en and slow magnetoacoustic\nwaves are not bounded in frequency if we allow ourselves to\nconsider arbitrarily short wavelengths. We therefore need to\napply a high-wavenumber cuto\u000b to the response in order to\ncontain the spectrum of low-frequency oscillations. This may\nbe justi\fed by assuming that the tidal response is smooth or\nby appealing to resistivity to eliminate disturbances of small\nscale.\nThe relevant scaling assumptions are then that the\nMNRAS 000, 1{14 (2017)14 Y. Lin and G. I. Ogilvie\nLehnert number is O(1) (or smaller) and that the pertur-\nbations are of large scale. Formally this can be achieved by\nsaying that both \n and BareO(\u000f). With the arbitrary nor-\nmalization \t = O(1), we then have (as before) \u0018=O(1),\nW=O(\u000f2),h0=O(1), \b0=O(1) and nowB0=O(\u000f). Our\nreduced system of linearized equations at leading order is\nthen\n\u0018+ 2\n\u0002_\u0018=\u0000rW+F\u0018; (B8)\nh0+ \b0+ \t = 0; (B9)\n\u001a0=\u0000r\u0001(\u001a\u0018); (B10)\nr2\b0= 4\u0019G\u001a0; (B11)\nand is to be solved on a spherically symmetric, hydrostatic\nbasic state una\u000bected by rotation or magnetic \felds.\nWe again decompose the perturbations into non-\nwavelike and wavelike parts, satisfying respectively\n\u0018nw=\u0000rWnw; (B12)\nh0\nnw+ \b0\nnw+ \t = 0; (B13)\n\u001a0\nnw=\u0000r\u0001(\u001a\u0018nw); (B14)\nr2\b0\nnw= 4\u0019G\u001a0\nnw; (B15)\nand\n\u0018w+ 2\n\u0002_\u0018w=\u0000rWw+F\u0018w+f; (B16)\nr\u0001(\u001a\u0018w) = 0; (B17)\nwhere\u001a0\nw=h0\nw= \b0\nw= 0 and\nf=\u00002\n\u0002_\u0018nw+F\u0018nw (B18)\nis the e\u000bective force per unit mass driving the wavelike part\nof the solution. As before, the non-wavelike tide may be\nassumed to be instantaneously related to the tidal potential\nthrough\n\u0018nw=\u0000rX; (B19)\nwhereXis the solution of the elliptic equation (61). The\nenergy equation for the wavelike part is\nd\ndt\u00141\n2Z\n\u001a\u0000\njuwj2\u0000\u0018w\u0001F\u0018w\u0001\ndV\u0015\n=Z\n\u001auw\u0001fdV; (B20)\nwhereuw=_\u0018wis the wavelike velocity and the second term\nin the integral on the left-hand side is the magnetic energy\nassociated with the wavelike displacement.\nTurning now to the impulsive forcing analysed in Sec-\ntion 4.6, we again consider a tidal potential of the form\n\t = ^\t(r)H(t); (B21)whereH(t) is the Heaviside step function. This implies that\n\u0018nw=^\u0018nw(r)H(t); (B22)\nleading to an e\u000bective force\nf=^f(r)\u000e(t) +~f(r)H(t); (B23)\nwhere\u000e(t) is the Dirac delta function,\n^f=\u00002\n\u0002^\u0018nw (B24)\nderives solely from the Coriolis force and\n~f=F^\u0018nw (B25)\nderives solely from the Lorentz force. Therefore the impul-\nsive contibution to the e\u000bective force comes only from the\nCoriolis force and not from the Lorentz force. The solution\nof equations (B16) and (B17) in this case involves a wave-\nlike displacement \u0018wthat is continuous in tbut has a dis-\ncontinuous \frst derivative at t= 0. The wavelike velocity\nimmediately after the impulse is again\n^uw=^f\u0000r^Ww; (B26)\nwhere ^Wwis chosen to satisfy the anelastic constraint\nr\u0001(\u001a^uw) = 0 and the boundary conditions ^ uw;r= 0. The\nenergy transferred in the impulse is equal to the kinetic en-\nergy immediately after the event,\n^E=1\n2Z\n\u001aj^uwj2dV: (B27)\nThere is no change in the wavelike magnetic energy at t= 0\nbecause\u0018wis continuous there. Since ^uwderives solely from\nthe Coriolis force, we conclude that the magnetic \feld has\nno e\u000bect either on the impulsive energy transfer associated\nwith this term, or on the frequency-averaged dissipation re-\nlated to it through equation 100. We note, however, that\nthe Lorentz part of the e\u000bective force in equation (B23),\nwhich is neglected in the main part of this paper, could al-\nter the energy of the wave-like disturbance after the impulse\natt= 0 and therefore make an additional contribution to\nthe frequency-averaged dissipation, which requires further\ninvestigation.\nThis paper has been typeset from a T EX/LATEX \fle prepared by\nthe author.\nMNRAS 000, 1{14 (2017)" }, { "title": "1812.01237v1.Optical_excitation_of_single__and_multi_mode_magnetization_precession_in_Galfenol_nanolayers.pdf", "content": "arXiv:1812.01237v1 [cond-mat.mes-hall] 4 Dec 2018Optical excitation of single- and multi-mode magnetizatio n precession in Galfenol\nnanolayers\nA. V. Scherbakov,1,2A. P. Danilov,1F. Godejohann,1T. L. Linnik,3B. A. Glavin,3L. A. Shelukhin,2\nD. P. Pattnaik,4M. Wang,4A. W. Rushforth,4D. R. Yakovlev,1,2A. V. Akimov,4and M. Bayer1,2\n1Experimentelle Physik 2, Technische Universit¨ at Dortmun d, D-44227 Dortmund, Germany\n2Ioffe Institute, Russian Academy of Science, 194021 St.Peter sburg, Russia\n3Department of Theoretical Physics, V.E. Lashkaryov Instit ute of Semiconductor Physics, 03028 Kyiv, Ukraine\n4School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK\nWe demonstrate a variety of precessional responses of the ma gnetization to ultrafast optical\nexcitation in nanolayers of Galfenol (Fe,Ga), which is a fer romagnetic material with large saturation\nmagnetization and enhanced magnetostriction. The particu lar properties of Galfenol, including\ncubic magnetic anisotropy and weak damping, allow us to dete ct up to 6 magnon modes in a 120nm\nlayer, and a single mode with effective damping αeff= 0.005 and frequency up to 100 GHz in a 4-\nnm layer. This is the highest frequency observed to date in ti me-resolved experiments with metallic\nferromagnets. We predict that detection of magnetisation p recession approaching THz frequencies\nshould be possible with Galfenol nanolayers.\nWithin the last decade magnetization precession has\nbecome an actively exploited tool in nanoscale mag-\nnetism. The precessing magnetization of a ferromag-\nnet is an effective, tunable and nanoscopic source of\nmicrowave signals of various types. Generation of mi-\ncrowavemagneticfieldsbyprecessingmagnetizationisal-\nready implemented in magnetic storage technology such\nasmicrowaveassistedmagneticrecording(MAMR)[1]by\nmeans of spin-torque nano-oscillators [2]. Spin waves or\nmagnons, i.e. the waves of precessing magnetization, are\ninformation carriers and encoders in magnon spintronics\n[3] aimed to substitute conventional CMOS technology.\nThe precessing magnetization is also an effective tool to\ngenerate a pure spin current in a nonmagnetic material\nby means of spin pumping [4].\nThe common way to excite magnetization precession\nin a ferromagnet is the technique of ferromagnetic res-\nonance (FMR). A monochromatic microwave magnetic\nfield drives the magnetization precession, the frequency\nof which is tuned into resonance with the microwaves\nby an external magnetic field. This technique, which\ncan provide comprehensive information about the main\nprecession parameters, is not adaptable for practical use\nwith nanostructures due to the need to use bulky elec-\ntromagnetic resonators and waveguides. An alternative\napproach is broad-band excitation induced by dc-current\n[5], picosecond magnetic field pulses [6] and ultrashort\nlaser [7] and strain [8] pulses. In those cases the param-\neters of the excited magnetization precession, i.e. the\nspectral content, lifetime, spatial distribution and their\ndependences on external magnetic field, are determined\nby the properties of the ferromagnetic material and the\ndesign of the nanostructure [9]. The ability to con-\ntrol these dynamical parameters is of crucial importance\nfor nanoscale magnetic applications. For practical use,\nan ideal combination of dynamical parameters includes\na tunable and narrow spectral band in the GHz and\nTHz frequency ranges; large saturation magnetizationand high precessionamplitude for high microwavepower;\nand ultrafast triggering for high-frequency modulation.\nAchieving this combination has been an unmet challenge\nuntil now. High precession frequency, f≫10 GHz, can\nbe reached by using ferrimagnetic materials [10, 11], but\nthe weak net magnetization limits their functionality. In\nthe case of metallic ferromagnets with large net magne-\ntization, the direct way to achieve high frequency pre-\ncession is to apply a strong external magnetic field, B,\nwhich, however, drastically decreases the precession am-\nplitude. Earlier experiments on the excitation of magne-\ntization precession in metallic ferromagnets by femtosec-\nond optical pulses [7, 12–18], i.e. the fastest method of\nlaunching precession, report also high values of the ef-\nfective damping coefficient αeff= (2πτf)−1>0.01 (τ\nis the precession decay time). Thus, the excitation and\ndetection of sub-THz narrow band precession in metallic\nferromagnets remains extremely challenging.\nIn the present letter, we report the results of ultrafast\nmagneto-optical experiments with nanolayersof (Fe,Ga),\ni.e. Galfenol. This metallic ferromagnet with large net\nmagnetization is considered as a prospective material for\nmicrowave spintronics due to the narrow ferromagnetic\nresonance [19, 20] and enhanced magnetostriction [21],\nwhichallowsmanipulationofthe magnetizationdirection\nand precession frequency by applying stress, i.e. with-\nout changing the external magnetic field [19, 22]. Our\nstudy extends significantly the application potential of\nGalfenol. We show that in a Galfenol layer with a thick-\nness of several nanometers, the femtosecond optical ex-\ncitation leads to the generation of single-mode magneti-\nzation precession with frequency f >100 GHz and large\namplitude. Despite the strong interaction between the\nmagnetization and the lattice, we observe a weak damp-\ning of precession with αeff≈0.005. Thus, we demon-\nstratethepossibilitytoachievethedesirablecombination\nof sub-THz magnetization precession with large ampli-\ntude and tunable narrow spectral band. Moreover, we2\nshow that, depending on the nanolayer thickness, we can\nexcite multi- or single-mode magnetization precession: in\na thick 120-nm Galfenol layer we observe multimode pre-\ncession and resolve up to 6 precessional localized magnon\nmodes. Thisallowscontrolofthe precessionspectralcon-\ntent and spatial profile by adjusting the film thickness\nand excitation regime.\nThe samples studied are four Fe 0.81Ga0.19nanolayers\nwith thicknesses d= 4, 5, 20 and 120 nm grown by mag-\nnetron sputtering on (001) semi-insulating GaAs sub-\nstrates and covered by a 3-nm Al or Cr cap layer to\nprevent oxidation. A 150-nm thick SiO 2cap was de-\nposited on the Galfenol layers with a thickness ≤20 nm\nfor amplification of the magnetooptical Kerr effect [23].\nRoom temperature experiments were carried out with an\nexternal magnetic field Bapplied in the layer plane. The\nin-plane direction of Bis defined by the azimuthal an-\ngleϕB[see the inset in Fig. 1(a)]. In all studied layers\nthe easy axes of magnetization are in the layer plane and\nclose to the [100]/[010] crystallographic directions, while\nthe hard axes lie along the [110] and [1 ¯10] diagonals. All\nnanolayers possess a weak uniaxial in-plane anisotropy,\nwhich is typical for thin Galfenol films on GaAs sub-\nstrates [22]. We have checked that the SiO 2cap does not\naffect the anisotropy parameters of the layers.\nThe magnetization precession was excited by 150-fs\npumppulsesfromamode-lockedErbium-dopedringfiber\nlaser(80 MHz repetition rate, 1050nm wavelength). The\npump beam, focused to a spot of 20 µm diameter with\nan energy density of ≈1 mJ/cm2, launched the magne-\ntization precession by ultrafast changes of the magnetic\nanisotropy altered by the optically-induced heating [24].\nThe magnetization response was monitored using 150-\nfs linearly polarized probe pulses of 780-nm wavelength\nfrom another ring-fiber laser oscillator focused to a 5 µm\nspot in the center of the pump beam. For monitoring the\ntime evolution of the magnetization precession, we uti-\nlizedthetransientmagneto-opticalKerreffect(TMOKE)\nand detected the rotation of polarization of the probe\nbeam reflected from the (Fe,Ga) layer. In this detec-\ntion scheme the signal is proportional to the changes of\nthe magnetization projection ∆ Mz, wherezis the nor-\nmal to the (Fe,Ga) layer. The temporal resolution was\nachieved by means of an Asynchronous Optical Sampling\nSystem (ASOPS) [25]. The pump and probe oscillators\nwere locked with a frequency offset of 800 Hz. In com-\nbination with the 80-MHz repetition rate, it allows mea-\nsurement of the time-resolved signal in a time window of\n12.5 ns with time resolution limited by the probe pulse\nduration.\nFor the measurements at magnetic fields B >1 T,\nthe samples were mounted in an optical cryostat with a\nsuperconducting solenoid. In this case, the temperature\nof the sample was 150 K. The source of the laser pulses\nwas a regenerative amplifier RegA (wavelength 800 nm,\nrepetition rate 100 kHz) and a standard scanning delay0 1 2 3\n Time (ns) ∆Mz (arb. units) Fe 0.81 Ga 0.19 d = 120 nm \nT = 290 K \nB = 200 mT; ϕB = - π/8 (a) \n0 20 40 60 (b) \n 0 - 4 \n 0.6 - 4.6 \n Frequency (GHz) 52\n31 Amplitude (arb.units) n = 0 \n4Time window \n (ns) \n0 1 2 3 4 510 20 30 40 50 \nFrequency (GHz) 400 mT 600 mT \n100 mT \nMode number 300 mT (c) [100] [010] xy\nϕB\nB\nFIG. 1. Multimode magnon excitation; T= 290 K. (a)\nTemporal evolution of the magnetization precession in a 120 -\nnm thick Fe 0.81Ga0.19layer. (b) Fast Fourier transform of\nthe signal shown in (a) performed in a time window of 4 ns,\nwith the start point at t= 0 (blue curve) and t= 0.6 ns\n(red curve); vertical bars point at the frequency of resonan ce\nmodes with n= 0,1,2,3...(c) Measured (symbols) and cal-\nculated by Eq.(1) (lines) dependences of resonance frequen cy\non the mode number for several in-plane magnetic fields. The\ninset in (a) shows the in-plane magnetic field configuration.\nline was used to monitor the temporal evolution of the\nmagnetization.\nFigure1showstheexperimentalresultsforthethickest\nd= 120 nm Fe 0.81Ga0.19layer obtained at ϕB=−π/8,\nwhen the precession amplitude is maximal. The magne-\ntization precession shown in Fig. 1(a) decays in a time\nmuch less than 1 ns, which is consistent with the result\nfor (Fe,Ga) films reported earlier [24, 26]. However, in\ncontrast with the previous experiments, temporal beat-\nings with a long-living tail are clearly observed. The fast\nFouriertransform(FFT) ofthe measuredsignalobtained\nin a time window of 4 ns is shown in Fig. 1(b). The blue\nline possesses a band spectrum where overlapping peaks\nare marked by integer numbers. Six spectral bands with\nfrequencies fn(n= 0...5) are recognized in the spec-\ntrum. We attribute these bands to standing spin wave\n(magnon) modes. This conclusion is based on a compar-\nison of the experimental dependence of fnonn,shown\nin Fig. 1(c) by symbols, with the well-known dispersion\nrelation for magnon modes:\nfn=f0+1\n2πγ0βDq2\nn, (1)3 \n d = 20 nm (a) \nαeff = 0.014 \n Amplitude (arb. units) (b) ∆Mz (arb. units) d = 5 nm \nαeff = 0.01 \n0 1 2 3(c) \n Time (ns) d = 4 nm \n10 20 30 40 \n Frequency (GHz) αeff = 0.008 \nFIG. 2. Single-mode magnon excitation; T= 290 K. Tempo-\nral evolutions (left panels) and corresponding spectra (ri ght\npanels) of the magnetization precession for Fe 0.81Ga0.19lay-\ners with different thickness measured at B= 200 mT and\nϕB=−π/8.\nwhereqnis the wavevector of the mode n= 0,1,2,3...,\nDis the exchange spin stiffness, γ0is the gyromagnetic\nratio andβis a field dependent coefficient determined by\nthe anisotropy parameters of the ferromagnet [27]. With\nthe assumption of free boundary conditions, qn=πn/d,\nwe get an excellent agreement of the measured magnon\nfrequencies with the curves calculated using Eq.(1) for\nD= 2.6×10−17Tm2, shown in Fig. 1(c) by lines [29].\nThis allows us to attribute unambiguously the bands in\nthe measured spectra in this (Fe,Ga) film to magnon\nmodes [30].\nIt is interesting that the FFT obtained in a temporal\nwindowwhich starts600ps afterthe pump pulse [redline\nin Fig. 1(b)] shows only two spectral lines with frequen-\ncies corresponding to n= 0 and 2. We may conclude\nthat different magnon modes have different decay times\nand that modes with uneven ndecay more quickly than\nmodes with even n. The explanation of such behavior\nis related to the magnon decay mechanisms which are\nwidely discussed in the literature [9] but still not fully\nunderstood. Two-magnon scattering [31] and the related\nselection rules could be the explanation, but this requires\na comprehensive theoretical study which is beyond the\nscope of the present work.\nThe precession kinetics change drastically in thin\nnanolayers with d= 4, 5 and 20 nm. Figure 2 shows\nthe temporal evolutions (left panels) and their FFTs\n(right panels) of magnetization precession measured forϕB = - π/8 B = 3 T; \n TMOKE d = 4 nm (a) \n40 60 80 100 120 \n Frequency (GHz) Precession Amplitude Brillouin \n0.0 0.1 0.2 0.3 0.4 ∆Mz(arb. units) (b) \nTime (ns) After filtering \n40 60 80 100 120 Amplitude Frequency (GHz) αeff =0.005 \n0 1 2 30123\n ASOPS \n RegA \n Theory \n ∆Mz/M 0(x10 3)\nMagnetic field (T) d=4 nm \n ϕ B=-π/8 (c) \nFIG. 3. (a) TMOKE signal and its FFT spectrum (inset)\nmeasured in 4-nm thick Fe 0.81Ga0.19nanolayer at B= 3 T\nandT= 150 K; (b) Temporal evolution of the magnetization\nprecession obtained by high-pass filtering of the signal in ( a);\ndots - experimental data; red line - fit with a single-frequen cy\ndecaying sine function; inset is the corresponding FFT spec -\ntrum; (c) Normalized experimental (symbols) and calculate d\n(solid line) dependences of the precession amplitude on ext er-\nnal magnetic field; squares and stars correspond to the data\nmeasured by the ASOPS system and by scanning delay line\nwith excitation by the RegA, respectively.\nB= 200 mT applied at ϕB=−π/8. Only one spec-\ntral line is observed in the magnon spectrum, which cor-\nresponds to the fundamental mode with n= 0. The\nprecession damping is well described with a single expo-\nnential decay with constants τ= 1.05, 0.85, and 0.6 ns,\nwhich correspond to αeff= 0.008, 0.01 and 0.014 for the\n4, 5, and 20-nm layers respectively.\nFigure 3(a) shows the temporal evolution measured in\nthe thinnest 4-nm nanolayer for B= 3 T. The preces-\nsion frequency is f= 108 GHz, which corresponds to the\nmaximum precession frequency in the present work. The\nFFT spectrum shown in the inset of Fig. 3(a) consists\nalso of a Brillouin line at 44 GHz due to dynamical in-\nterference of the probe pulse on the strain pulse injected\ninto the GaAs substrate [32], which is not related to the\nmagneticpropertiesofthe(Fe,Ga) layer. Singlemodeex-\ncitation is observedfor the filtered signal (high-pass filter\nwith 50 GHz cutoff frequency) shown in Fig. 3(b). The\ndecay time of the magnetization precession in the 4 nm\nnanolayer at f= 108 GHz is τ= 0.29 ns, which corre-\nsponds to an effective damping parameter αeff= 0.005.\nThe line in Fig. 3(b) is a fit to the experimental data\nby an exponentially decaying sine function:\n∆Mz=Aexp(−t/τ)sin(ωt+ψ), (2)\nwhereω= 2πf(fis obtained from the FFT spec-\ntrum). The fitting parameters A,τ, andψare the ampli-\ntude, decay time and the initial precession phase, respec-\ntively. The dependence of the amplitude, AonBfor the4\nthinnest (Fe,Ga) nanolayer is shown by symbols in Fig.\n3(c). It is seen that Adecreaseswith increasing B, but in\nour experiment it is still possible to detect the precession\nwith frequency higher than 100 GHz at B= 3 T.\nThe main experimental results of the present work are\nthe demonstration of excitation of a multimode quan-\ntized precession spectrum in a thick, 120-nm, (Fe,Ga)\nlayer, and a long-living single mode magnetization pre-\ncession with a frequency >100 GHz in a thin, 4-nm,\n(Fe,Ga) nanolayer. Our qualitative explanation for these\nexperimentalfactsisbasedonacomparisonoftheoptical\npenetration depth in (Fe,Ga) with the layer thickness, d.\nThe penetration depth for the pump light is η≈20 nm,\nwhich is larger than the thickness of the films where only\none magnon mode is excited. In this case, the optical\nexcitation, which kicks the magnetization precession, is\nalmosthomogeneousalongthethicknessofthenanolayer.\nAssuming free boundary conditions at the nanolayer in-\nterfaces, only the excitation of the ground uniform mode\nis efficient, while the higher order magnon modes are not\nexcited due to their sign-changing spatial profile [7]. In\ncontrast, in thick films η < d, and the excitation is in-\nhomogeneous, being stronger near the surface, resulting\nin the efficient excitation of high-energy magnon modes.\nThe efficiency ofsuch excitation should decreasewith the\nincrease of n, which is clearly observed in Fig. 1(b): the\nspectral amplitude of the magnon spectral line decreases\nby more than one order of magnitude with nincreasing\nfrom 0 to 5. It is important to note that due to the\nshallow penetration depth of the probe pulse, both even\nand odd magnonmodescontribute tothe TMOKEsignal\nand we observe monotonic decrease of the magnon mode\namplitude with increase of its number.\nWe now consider the observation of precession with\nfrequency ≈100 GHz. Fitting the measured temporal\nsignal shown in Fig. 3(b) with a single harmonic func-\ntion gives a decay τ=0.29 ns and a respective value for\nαeff= 0.005. This value is close to the smallest damping\nparameters measured in pure Fe on semiconductor sub-\nstrates by the FMR technique [33–36], but has not been\nreported in experiments using ultrafast optical excitation\nof the magnetization precessionin metallic ferromagnetic\nmaterials so far.\nWe have performed a theoretical analysis of the pre-\ncessional response of the magnetization and its depen-\ndence on magnetic field strength and direction using the\napproach presented in earlier work [24], which consid-\ners launching of the magnetization precession by ultra-\nfast modification of the magnetic anisotropy. The com-\nprehensive study of the angular dependences f(ϕB) and\nA(ϕB,B), which can be found in the Supplemental Ma-\nterial [37], allows us to obtain the main film param-\neters: saturation magnetization µ0M0= 1.72 T, cu-\nbic anisotropy coefficient K1= 15 mT and uniaxial\nanisotropy coeficient Ku= 5 mT. We also confirmed\nexperimentally that for the used pump excitation den-sity, thedemagnetizationisnegligible[37]. Theoptically-\ninduced changes of the anisotropy coefficients were esti-\nmated by using the data from Ref. [24]: ∆ K1=−3.7\nmT and ∆Ku=−0.8 mT. The respective dependence\nof the precession amplitude on magnetic field calcu-\nlated atϕB=−π/8 is shown by the solid line in Fig.\n3(c). A good agreement between the experimental de-\npendence, which is normalized accordingly, and the the-\noretical curve is clearly observed. Moreover, for rela-\ntively small changes of the anisotropy coefficients, and\nneglecting demagnetization, we can simplify the depen-\ndenceA(ϕB,B) to:\nA≈(∆K1/2)sin4ϕB−∆Kucos2ϕB/radicalbig\nB(B+µ0M0).(3)\nThis expression is valid with high accuracy at B≥\n200 mT. As one can see from Eq.(3), the precession am-\nplitude is maximal at ϕB=−π/8 (−3π/8, 5π/8, and\n7π/8), and remains nonzero with increase of magnetic\nfield due to the field-independent ∆ K1and ∆Ku. At\nB= 9 T, when the precession frequency approaches the\nterahertz range ( f= 300 GHz), the estimated precession\namplitude ∆ Mz/M0= 3×10−4is expected to be easily\ndetectable.\nIt is worth noting that in the 4-nm layer αeffdemon-\nstrates a pronounced anisotropy and is 1.5 times smaller\natϕB=π/4 than atϕB=−π/8 [37]. Unfortunately,\nthe small precession amplitude at B/bardbl[110] does not\nallow us to detect the magnetization precession at high\nmagnetic fields applied along this direction. Anisotropic\ndamping has been previously observed in Fe nanolayers\nand is actively studied nowadays [34–36].\nIn conclusion, we have demonstrated multimode exci-\ntation of magnetization precession in Fe 0.81Ga0.19layers\nwith a thickness of 120 nm and single-mode precession in\nthin Fe 0.81Ga0.19nanolayers. We show that the param-\neters of (Fe,Ga) provide the possibility to detect magne-\ntization precession with frequency higher than 100 GHz,\nand small effective damping parameter αeff≈0.005.\nThese are record values for experiments using optical ex-\ncitationofmagnetizationprecessioninmetallicferromag-\nnets. Due to the large saturation magnetization, the pre-\ncession amplitude of 10−3M0observed at high magnetic\nfields generates an ac-induction of 1 mT, which may be\nexploitedfornanoscalegeneratorsofmicrowavemagnetic\nfield [38] and pure spin currents [39]. Our analysis shows\nthat 100 GHz is not the limit for the detectable magneti-\nzation precession and the THz range can be achieved by\napplying an appropriate external magnetic field.\nACKNOWLEDGEMENTS\nWe are grateful to Serhii Kukhtaruk and Alexandra\nKalashnikovafor fruitful discussions. This work was sup-5\nported by the Deutsche Forschungsgemeinschaft and the\nRussian Foundation for Basic Research in the frame of\ntheInternationalCollaborativeResearchCenterTRR160\n[project B6] and by the Bundesministerium f¨ ur Bildung\nund Forschung through the project VIP+ ”Nanomag-\nnetron”. The experimental studies in the Laboratory of\nPhysics of Ferroics (Ioffe Institute) were performed un-\nder support of the Russian Science Foundation [grant no.\n16-12-10485]. The Volkswagen Foundation supported\nthe cooperation with the Lashkarev Institute [grant no.\n90418].\n[1] J.-G. Zhu, X. Zhu, Y. Tang, IEEE Trans. Magn. 44, 125\n(2008).\n[2] T. Chen, R. K. Dumas, A. Eklund, P. K. 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Phys. 101, 09D120 (2007).\n[36] L. Chen, S. Mankovsky, S. Wimmer, M. A. W. Schoen,\nH. S. K¨ orner, M. Kronseder, D. Schuh, D. Bougeard, H.\nEbert, D. Weiss and C. H. Back, Nat. Phys. 14, 490\n(2018).\n[37] See Supplemental Material at http://...\n[38] A. S. Salasyuk, A. V. Rudkovskaya, A. P. Danilov, B.\nA. Glavin, S. M. Kukhtaruk, M. Wang, A. W. Rush-6\nforth, P. A. Nekludova, S. V. Sokolov, A. A. Elistratov,\nD. R. Yakovlev, M. Bayer, A. V. Akimov, and A. V.\nScherbakov, Phys. Rev. B 97, 060404(R) (2018).[39] A. P. Danilov, A. V. Scherbakov, B. A. Glavin, T. L.\nLinnik, A. M. Kalashnikova, L. A. Shelukhin, D. P. Pat-\ntnaik, A. W. Rushforth, C. J. Love, S. A. Cavill, D. R.\nYakovlev, and M. Bayer, Phys. Rev. B 98, 060406(R)\n(2018)." }, { "title": "1508.07517v1.Spin_transfer_torque_based_damping_control_of_parametrically_excited_spin_waves_in_a_magnetic_insulator.pdf", "content": "arXiv:1508.07517v1 [cond-mat.mes-hall] 30 Aug 2015Spin-transfer torque based damping control of parametrica lly excited spin\nwaves in a magnetic insulator\nV. Lauer,1D. A. Bozhko,1,2T. Brächer,1P. Pirro,1,a)V. I. Vasyuchka,1A. A. Serga,1M. B. Jungfleisch,1,b)M.\nAgrawal,1Yu. V. Kobljanskyj,3G. A. Melkov,3C. Dubs,4B. Hillebrands,1and A. V. Chumak1\n1)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern,\n67663 Kaiserslautern, Germany\n2)Graduate School Materials Science in Mainz, Gottlieb-Daim ler-Strasse 47, 67663 Kaiserslautern,\nGermany\n3)Faculty of Radiophysics, Electronics and Computer Systems , Taras Shevchenko National University of Kyiv,\n01601 Kyiv, Ukraine\n4)Innovent e.V., Prüssingstraße 27B, 07745 Jena, Germany\n(Dated: 26 September 2018)\nThe damping of spin waves parametrically excited in the magn etic insulator Yttrium Iron Garnet (YIG)\nis controlled by a dc current passed through an adjacent norm al-metal film. The experiment is performed\non a macroscopically sized YIG( 100nm )/Pt(10nm ) bilayer of 4×2mm2lateral dimensions. The spin-wave\nrelaxation frequency is determined via the threshold of the parametric instability measured by Brillouin light\nscattering (BLS) spectroscopy. The application of a dc curr ent to the Pt film leads to the formation of a\nspin-polarized electron current normal to the film plane due to the spin Hall effect (SHE). This spin current\nexerts a spin transfer torque (STT) in the YIG film and, thus, c hanges the spin-wave damping. Depending\non the polarity of the applied dc current with respect to the m agnetization direction, the damping can be\nincreased or decreased. The magnitude of its variation is pr oportional to the applied current. A variation in\nthe relaxation frequency of ±7.5%is achieved for an applied dc current density of 5·1010A/m2.\nThe injection of a spin current into a magnetic film\ncan generate a spin-transfer torque (STT) that acts on\nthe magnetization collinearly to the damping torque.1–3\nThus, it can be used for tuning of the damping of a mag-\nnetic film4–7as well as for the excitation of magnetiza-\ntion precession in the film.8–12In the first experimental\nrealizations, a dc charge current was sent through an ad-\nditional magnetic layer with a fixed magnetization direc-\ntion in order to generate a spin-polarized current.4,8–10\nA different way to generate a spin current is based on\nthe spin Hall effect (SHE)13,14caused by spin-dependent\nscattering of electrons in a non-magnetic metal with\nlarge spin-orbit interaction.5,6,11One of the advantages\nof the SHE is that it does not require a dc current in\nthe magnetic layer and, thus, allows for the applica-\ntion of a STT to a magnetic dielectric such as yttrium\niron garnet (YIG),15which is of particular interest for\nmagnon spintronics16–18due to its extremely small damp-\ning parameter.19–23Moreover, a great advantage of the\nSHE is that a STT can be applied not only locally but\nto a large area of a magnetic film5and, thus, can be\npotentially used to compensate the damping in a whole\ncomplex magnonic circuit.18\nUp to now, the auto-oscillatory regime and the mag-\nnetization precession generation has only been reached\nin patterned structures.11,12,24Nonlinear multi-magnon\nscattering phenomena are assumed to be the reason\na)Present address: Institut Jean Lamour, Université Lorrain e,\nCNRS, 54506 Vandoeuvre-lès-Nancy, France\nb)Present address: Materials Science Devision, Argonne Nati onal\nLaboratory, Argonne, Illinois 60439, USAthat disturbs the generation process in un-patterned\nstuctures. At the same time, nonlinear scattering\nshould not affect the SHE-STT-based damping compen-\nsation since spin-wave amplitudes in this case are much\nsmaller. However, most experimental studies concern-\ning this phenomena using metallic samples4,6as well\nas YIG structures7,12were performed with laterally-\nconfined nano- or micro-structures.\nHere, we use a YIG/Pt bilayer of macroscopic size to\ninvestigate SHE-STT damping variation. The measured\nvariation of the damping is proportional to the applied\ndc current and no influence of multi-scattering magnon\nprocesses is observed.\nThe investigated sample consists of a YIG/Pt bilayer\nwith macroscopic lateral dimensions of 4mm×2mm,\nand film thicknesses of tYIG= 100nm andtPt= 10nm .\nThe YIG film is grown by liquid phase epitaxy on a\ngadolinium gallium garnet (GGG) substrate and the Pt\nfilm is deposited afterwards using plasma cleaning and\nRF sputtering, as described in Ref. 20. Measurements of\nthe ferromagnetic resonance (FMR) linewidth and of the\ninverse SHE induced by spin pumping in a wide frequency\nrange yield the Gilbert damping parameters αYIG=\n(1.3±0.1)·10−4for the bare YIG film and αYIG/Pt=\n(5.3±0.1)·10−4for the YIG/Pt bilayer. In addition, a\nsaturation magnetization µ0Ms= (173±1)mT , an in-\nhomogeneous broadening µ0∆H0= 0.26mT , a resistiv-\nity of the Pt film ρPt= 1.475·10−7Ωm, and an effective\nspin mixing conductance g↑↓\neff= 3.68·1018m−2are deter-\nmined at room temperature.\nThe investigation of the SHE-STT damping variation\nis based on the analysis of the threshold of the parametric\ninstability. Figure 1 shows a sketch of the experimental2\nMicrowave\npumpingBiasing\nfield, Happl Probing\nlaser beamdc\ncurrent\ncontactsx y\nz\nPt (10 nm)\nYIG (100 nm)\nGGG (500 µm)4 mm2 mm\nCu stripline\nFIG. 1. (color online) Sketch of the sample and the setup in\nthe parallel pumping geometry used for the parametric exci-\ntation of spin waves and their detection by BLS spectroscopy .\nThe externally applied biasing field and the pumping field\nin the examined area of the sample are both along the x-\ndirection, the charge current is applied in-plane along the y-\ndirection.\narrangement, in which the parametric instability thresh-\nold is measured in the parallel pumping geometry.17,25,26\nAn external biasing field Happlmagnetizes the YIG film\nin-plane along the x-axis. The parametric excitation of\nspin waves is achieved by an alternating pumping mag-\nnetic field hporiented parallel to Happl. For this purpose,\na microwave signal at a fixed frequency of fp= 14GHz\nis applied to a 50µm wide Cu microstrip antenna placed\non top of the sample. A 10µm thick polyethylene inter-\nlayer separates the antenna from the sample electrically.\nIn our experiment, spin waves at half of the pumping\nfrequency ( fsw=fp/2 = 7GHz ) are excited as soon\nas the pumping field amplitude hpovercomes a critical\nthreshold value hth. The detection of these spin waves\nis realized by means of BLS spectroscopy.27The incident\nprobing laser beam in our experiment, which accesses\nthe YIG film through the GGG substrate (see Fig. 1),\nis always perpendicular to the film plane and only spin\nwaves with wavenumbers in a range of k/lessorsimilar104rad/cm\nare detected.28Since the threshold hthdepends on the bi-\nasing field, both parameters, Happlandhp, are varied in\neach measurement. Furthermore, Au wires are mounted\nto the edges of the sample by silver conductive adhesive\nto apply an in-plane dc current to the Pt film along the\ny-axis. Since the dc current is perpendicular to Happl,\nthe SHE generates an out-of-plane spin current (along\nthez-direction) in the Pt film that exerts a STT on the\nYIG magnetization at the YIG/Pt interface. A maximal\ncurrent of 1Ais used, corresponding to a current density\nofjc= 5·1010A/m2. In order to reduce the influence of\nJoule heating in the Pt film, the experiment is performed\nin the pulsed regime with a pulse duration of 10µs and\na repetition time of 1ms. All measurements presented\nhere are performed at room temperature.\nFigures 2(a)-(c) exemplary show BLS intensities of\nspin waves at fsw= 7GHz when the pumping field\nhpand the biasing field Happl(in+x-direction) are\nswept, measured for three different current densities jcApplied magnetic field µ0 applH(mT)177 179 181 183 18520222426(a)\n(b)\n(c)Pumping field (arb .units)µ h0 p\nPumping field (arb .units)µ h0 p\nPumping field (arb .units)µ h0 p20222426\n20222426\nµ h0 th,min = 22.48 arb. units\nµ0 FMRH= 179.8 mT\nµ h0 th,min = 19.32 arb. units\nµ0 FMRH= 180.7 mTthreshold value ( )µ h H0 th appl\nµ h0 th,min = 20.39 arb. units\nµ0 FMRH= 177.9 mTjc= +5·10 A/m²10\njc= −5·10 A/m²10jc= 0\nmax\nNormalized BLS intensity: min\nFIG. 2. (color online) BLS intensity of parametrically\npumped spin waves at 7GHz while scanning the biasing field\nHappland the pumping field hp, for different applied cur-\nrent densities (a) jc= +5·1010A/m2, (b)jc= 0, (c)\njc=−5·1010A/m2. The dashed white line in (b) represents\nthe butterfly curve for spin waves with k/lessorsimilar104rad/cm.\n(in+y-direction). The applied current densities are (a)\njc= +5·1010A/m2, (b) reference measurement jc= 0,\n(c)jc=−5·1010A/m2. Blue colored areas in the inten-\nsity graphs correspond to the case hp< hthwhen no spin\nwaves are parametrically excited.\nThe density n(/vectork,t)of parametrically excited magnons\nof a certain /vectork-vektor at the time tin the vicinity of the\nthreshold is given by29,30\nn(/vectork,t) =n0(/vectork)·exp/bracketleftBig\n−2/parenleftBig\nΓ(/vectork)−|V(/vectork)|µ0hp/parenrightBig\nt/bracketrightBig\n,(1)\nwheren0is the initial magnon density (thermal level),\nΓis the relaxation frequency, V(/vectork)is the coupling pa-\nrameter between magnons and the pumping field, and µ0\nis the vacuum permeability. The magnon density grows3\n- 1 18- 0 18-179- 7 1 81 8 7179180181\nCu ren dr t ensity jc( 10 × A/m )10 2-5.0 -2.5 0 5.0 2.5Variation of (mT)µ H0 FMR\n-0.4-0.200 2 .0 4 .fit to Eq. 4 with fitting parameters:\n= 8.0 10 Vs/Aa ×-15\n= 1.6 ×b 10 m /A-23 4 2\nµ0 ind cH a j= ·\nµ µ0 s 0 s,0 cM M b j= ·(1 · −2)\nOersted field (mT)µ H0 ind\n165167169171173\nVariation of (mT)µ M0 sµ H0 FMR > 0(a)\n(b)\nFIG. 3. (color online) (a) Obtained HFMRvalues (in +x-\ndirection) at hth,minas function of jc. The solid lines are fits\nto Eq. 4. (b) The induced Oersted field (in +x-direction) and\nthe variation of the saturation magnetization as functions of\njc, whenHFMR>0(in+x-direction).\nexponentially, as soon as the pumping field overcomes a\ncritical threshold value of hp≥hth= min/braceleftBigg\nΓ(/vectork)\nµ0|V(/vectork)|/bracerightBigg\nfor one mode with the wavevector /vectork. For a fixed pumping\nfrequency, the hthvalue strongly depends on the biasing\nfieldHappl, which determines the wavevector of the avail-\nable spin waves. The diagram of hthvs.Happl, called\nbutterfly curve,29exhibits a minimum hth,minat the res-\nonant field HFMRwhich corresponds to the excitation\nof spin waves with k→0. The white dashed line in\nFig. 2(b) represents only the part of the butterfly curve\n(for the BLS accessible wavenumbers) in a small range\nof the biasing field around its minimum. For the further\nstudy, only the variations of hth,minandHFMR, related\ntok→0spin waves, are analyzed as functions of the\napplied current density jc.\nFirst, the variation of the resonant biasing field HFMR\ncorresponding to the minimal threshold pumping field\nhth,minis investigated. Figure 3(a) presents all HFMR\nvalues obtained from BLS measurements (as shown, e.g.,\nin Figs. 2(a)-(c)) for both directions of the biasing field\nand both directions of the current. HFMRevidently shifts\nalways to values of higher magnitude with increasing jc,\nregardless of the current direction, but the magnitudes\nof the shift are different for opposite current directions.\nThe observed behavior can be understood by taking into\naccount the contributions of two effects: (i) The dc cur-\nrent in the Pt induces an Oersted field Hindcollinear to\nHappl, which is proportional to jc. Thus, the total mag-netic field in the YIG film reads\nHtotal=Happl+Hind=Happl+a\nµ0·jc, (2)\nwhere the proportionality constant ais introduced as a\nfitting parameter accounting for the sample geometry.\n(ii) Joule heating in the Pt film leads to a temperature\nincrease in the bilayer, and decreases the saturation mag-\nnetization of the YIG film following\nMs(jc) =Ms(0)·(1−β·∆T) =Ms(0)·(1−b·jc2).(3)\nHere,µ0Ms(0) = (173 ±1)mT is the initial saturation\nmagnetization at room temperature ( 300K) atjc= 0.\nThe factor β= (2.2±0.1)·10−3K−1accounts for the\nchange of Mswith a temperature change of ∆T,31which\nin turn is assumed to depend on the applied current as\n∆T=b/β·jc2by introducing the fitting parameter b.\nUsing Eq. 2, Eq. 3 and the Kittel equation ( k→0)\nfsw=γµ0/radicalbig\n(HFMR+Hind)(HFMR+Hind+Ms), the\nHFMRvalues in Fig. 3(a) can be fitted by\n±µ0|HFMR|=−a·jc∓1\n2µ0Ms(0)·(1−b·jc2)\n±/radicalBigg/parenleftbiggfsw\nγ/parenrightbigg2\n+/parenleftbigg1\n2µ0Ms(0)·(1−b·jc2)/parenrightbigg2\n,(4)\nwhereγ= 28GHz /Tdenotes the gyromagnetic ra-\ntio. The sign of ±µ0|HFMR|indicates the field polar-\nity (\"+\" corresponds to the +x-direction). The solid\nlines in Fig. 3(a) represent the fit according to Eq. 4\nyielding the fitting parameters a= 8.0·10−15Vs/Aand\nb= 1.6·10−23m4/A2. Based on the values of aandb\nthe variation of the Oersted field µ0Hindand the magne-\ntizationµ0Mswithjcare shown in Fig. 3(b) for the case\nofµ0HFMR>0. The highest applied current density of\njc=±5·1010A/mdecreases µ0Msby≈7.0mT, yield-\ning a corresponding temperature increase of ∆T≈18K\nfor the YIG film. The change of µ0Hindfor the maximal\njcis≈ ±0.4mT. This magnitude agrees with the the-\noretically expected Oersted field given by (µ0/2)jctPt≈\n±0.3mT, if the sample width is considered to be much\nlarger than the film thickness tYIG. This suggests that\nno STT-based variation of the saturation magnetization\nof YIG is observed in our experiment.6\nFigure 4(a) presents the minimal threshold pumping\nfieldhth,minas function of jc, extracted from data such\nas shown in Figs. 2(a)-(c). The shift of hth,minobviously\ndepends on the orientation of jcwith respect to Happl\nand can be understood in terms of the SHE-STT effect.\nAccording to our experimental geometry, the direction of\nthe SHE-generated spin current in the Pt film is given by\nthe unity vector (/vectorjc×/vectorHappl)/|/vectorjc×/vectorHappl|.32If the spin\ncurrent, for example, is oriented in +z-direction (from Pt\nto YIG), the damping in the YIG film and consequently\nhth,mindecreases (compare Fig. 2(b) with (c)). In the\ncase of a reversed spin current direction (from YIG to\nPt),hth,minincreases (compare Fig. 2(b) with (a)). The4\n> µ0HFMR 0\n< µ0HFMR 023\n22\n21\n20\n19\n- . 0 15- . 0 10- . 0 0500 05 .0 10 .0 15 .\n-5.0 -2.5 0 5.0 2.5\nCurrent density (×10 A/m ) jc10 2> µ0HFMR 0\n< µ0HFMR 0\nlinear fitVariation of (arb. units)µ h0 th,min(a)\n(b)\nΔΓ( )jc\nΓ(0)\nRelative variation relaxationof\nFIG. 4. (color online) (a) Measured minimum threshold val-\nueshth,minextracted from BLS as a function of jcfor positive\nand negative HFMR. (b) Relative change of the relaxation\nfrequency according to Eq. 6. The solid lines are linear fits.\nrelaxation frequency Γof spin waves with k→0as a\nfunction of jcreads29,30\nΓ(jc) =π\n2γ2\nfsw·µ0Ms(jc)·µ0hth,min(jc), (5)\nand the relative variation of Γis given by\n∆Γ(jc)\nΓ(0)=Γ(jc)−Γ(0)\nΓ(0)=hth,min(jc)Ms(jc)\nhth,min(0)Ms(0)−1.(6)\nFigure 4(b) shows the dependence of the relative vari-\nation of Γas a function of jc. These values are ob-\ntained according to Eq. 6 using the hth,min(jc)values\nfrom Fig. 4(a), as well as Eq. 3 and the fitting param-\neterbforMs(jc). The relative variation of the relax-\nation is proportional to the current within the error bars,\nwhich is illustrated by the linear fit in Fig 4(b). Γis\nchanged by approximately 7.5%for the highest applied\ncurrent densities of jc=±5·1010A/m2in our experi-\nment. In order to estimate the required critical current\ndensity for the complete compensation of damping, and\nfor the triggering of auto-oscillations in our experiments ,\nthe extrapolation of the linear fit to Γ(jc) = 0 is per-\nformed, yielding a value of jexp\nc,crit≈6.7·1011A/m2. This\nvalue agrees well with the results obtained with patternedstructures,7,12and with the theoretically estimated value\nofjtheo\nc,crit≈8.3·1011A/m2calculated on the basis of an-\nalytical expressions provided in Ref. 7. For this calcula-\ntion, the parameters of our system indicated above, the\ntransparency of the YIG/Pt interface33T= 0.12(esti-\nmated for a spin diffusion length of λ= 3.4nm) and a\nspin Hall angle of θSH= 0.056are used.34Heating ef-\nfects are neglected. In addition, reference measurements\nare performed in a geometry with jcparallel to Happl, so\nthat the SHE is eliminated. As a result, no variation of\nΓwithjcis observed within the error bars (not shown).\nIn summary, the threshold of the parametric excita-\ntion of spin waves in a macroscopic YIG/Pt bilayer is\ndetected by BLS spectroscopy. A change of the damp-\ning in the YIG due to SHE-STT is observed when a dc\ncurrent is passed through the adjacent Pt film. Based on\nour calculations, including current induced Joule heat-\ning and Oersted fields, the damping is found to change\nlinearly with the current. 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George, L. Vila, and H.\nJaffrès, Phys. Rev. Lett. 112, 106602 (2014)." }, { "title": "2103.03885v1.Universal_spin_wave_damping_in_magnetic_Weyl_semimetals.pdf", "content": "arXiv:2103.03885v1 [cond-mat.str-el] 5 Mar 2021Universal spin wave damping in magnetic Weyl semimetals\nPredrag Nikoli´ c1,2\n1Department of Physics and Astronomy, George Mason Universi ty, Fairfax, VA 22030, USA and\n2Institute for Quantum Matter at Johns Hopkins University, B altimore, MD 21218, USA\n(Dated: March 9, 2021)\nWe analyze the decay of spin waves into Stoner excitations in magnetic Weyl semimetals. The\nlifetime of a mode is found to have a universal dependence on i ts frequency and momentum, and\non a few parameters that characterize the relativistic Weyl spectrum. At the same time, Gilbert\ndamping by Weyl electrons is absent. The decay rate of spin wa ves is calculated perturbatively\nusing the s-d model of itinerant Weyl or Dirac electrons coup led to local moments. We show that\nmany details of the Weyl spectrum, such as the momentum-spac e locations, dispersions and sizes\nof the Weyl Fermi pockets, can be deduced indirectly by probi ng the spin waves of local moments\nusing inelastic neutron scattering.\nI. INTRODUCTION\nWeyl semimetals are condensed matter realizations\nof massless fermions with a chiral relativistic three-\ndimensional spectrum1–3. Topologically protected gap-\nless Fermi “arc” states on the system boundaries, and\nunconventional transport properties such as the intrinsic\nanomalous Hall effect, set Weyl semimetals apart from\nother weakly interacting conductors. One way to ob-\ntain a Weyl spectrum involves breaking the time-reversal\nsymmetry in a material that has Dirac quasiparticles.\nThe presence of magnetization, for example, will remove\nthe spin degeneracy of a Dirac node by splitting it into\na dipole of opposite-chirality Weyl nodes in momentum\nspace. Magnetism then becomes intimately related to\nthe presence of Weyl electrons. Alternatively, Weyl spec-\ntrum of itinerant electrons can be created by a broken\ninversion symmetry, e.g. due to the crystal structure,\nand then coupled to magnetism if the material possesses\nadditional local moments or undergoes a spin density\nwave instability. Some of these theoretical scenarios are\nslowly finding their actualization in experimentally stud-\nied magnetic Weyl semimetals4–13.\nHere we analyze an important imprint of Weyl elec-\ntrons on the magnetic dynamics – the damping of spin\nwaves via particle-hole (Stoner) excitations. This basic\ninteractioneffect revealsthe definingfeatures ofthe Weyl\nspectrum, relativity and chirality. We will show that the\nlifetime of spin waves exhibits a universal dependence on\nthe modefrequency andmomentum whichcanbe used to\nextract detailed properties of the underlying Weyl elec-\ntrons. By measuring the mode lifetime throughout the\nfirst Brillouin zone, it is possible to discern the locations\nof the Weyl nodes in momentum space, their relative chi-\nralities, slope of the energy versusmomentum dispersion,\nand the size of the Fermi pockets on the Weyl nodes.\nThe spin wave lifetime is obtained from the width of the\nscattering intensity peaks in inelastic neutron scattering\nexperiments, provided that a sufficient energy resolution\nis available and other sources of decoherence (thermal\nbroadening, disorder, phonons) do not mask the elec-\ntronic source.\nEven though neutron scattering is a powerful Green’sfunction probe, its ability to detect fermionic quasipar-\nticles is normally ruined by the incoherent continuum\nof excitations that can absorb an angular momentum\nquantum. Interestingly, this problem is reduced in Weyl\nsemimetals14, and fortunately it is also possible to in-\ndirectly characterize the quasiparticles via collective ex-\ncitations. The latter has been achieved in the neutron\nstudies of samarium hexaboride (SmB 6)15,16, where the\nmeasured dispersion of a “spin exciton” has revealed a\nnon-trivial topology of the underlying electronic quasi-\nparticles. An energygap protects the exciton’s coherence\nin SmB 6, but the gaplessquasiparticlesin Weyl semimet-\nals will generally induce ubiquitous damping of collective\nmodes. Such a damping can in fact reveal the existence\nand properties of chiral fermionic quasiparticles. The\nWeyl electron characterization through damping could\npotentially overcome various issues that plague other ap-\nproaches, such as correlation effects in the case of band-\nstructure calculations, limited resolution in the case of\nARPES, sensitivity to conventional bands (that coexist\nwith Weyl nodes) in transport measurements, etc.\nClosely related to the physics we pursue here is the ex-\ntensively studied damping in metallic ferromagnets17–29.\nStoner excitations provide a mechanism for the decay\nof spin waves, and also typically give rise to Gilbert\ndamping30– the dissipated precession of uniform mag-\nnetization in an external magnetic field. Many works\nhave been devoted to the calculation of Gilbert damp-\ning since it is possible to measure it by ferromagnetic\nresonance31,32and time-resolved magneto-optical Kerr\neffect33,34. A careful consideration of the relativistic\nelectron dynamics has revealed that Gilbert damping\noriginates in the spin-orbit coupling and depends on\nthe electrons’ mass25. In the case of massless Weyl\nelectrons, we show here that Gilbert damping is ab-\nsent. However, spin waves unavoidably decay via Stoner\nexcitations35–39,41,42, and their damping features “non-\nreciprocity” – different polarization modes that carry\nthe same momentum have different damping rates. This\naccompanies the non-dissipative aspects of chiral spin-\nmomentumlocking44,45. Spinwave“non-reciprocity”has\nbeen anticipated in spiral magnets46, magnetic interfaces\nwith a Dzyaloshinskii-Moriya interaction derived from2\nthe Rashba spin-orbit coupling43,47–52, and observed in\nseveral experiments53–58. In the context of magnetic\nWeyl semimetals, initial theoretical studies have been fo-\ncused on the domain wall dynamics59,60.\nThe rest of this paper is organized as follows. Section\nII presentsthe approachand the main results ofthe anal-\nysis, focusing on the observable physical characteristics\nof the spin wave damping by Weyl electrons. Section III\nis devoted to the technical development of the damping\ntheory. It contains separate derivations of the dissipative\ntermsintheeffectivespinaction(IIIA),spinwavedamp-\ning (IIIB), and Gilbert damping from the semiclassical\nfield equation (IIIC). The last section IV summarizesthe\nconclusions and discusses the broader applicability and\nlimitations of the damping theory.\nII. SUMMARY OF THE RESULTS\nIn this paper, we work with the s-d model of Weyl\nelectrons coupled to local moments. We perturbatively\ncalculate the dissipative non-Hermitian parts of the mo-\nments’effectiveaction,whichdeterminetherate γofspin\nwavedamping. γalsodependsonthe magneticorderand\nthe wave’s propagation direction relative to the magneti-\nzation, but it is always controlled by the components of\nthe universal damping rate tensor given by\nγab\nmn(q) =a3J2\nKΩ2\n128πSv3fab\nmn/parenleftbigg|Ω|\nvq,|Ω|\n2|µ|;sign(µ,Ω)/parenrightbigg\n(1)\nfor ferromagnetic local moments of spin magnitude S.\nThe upper indices a,b∈ {x,y,z}refer to spin projec-\ntions. The universal scaling functions fab\nmnare dimen-\nsionless, the factor a3is the unit-cell volume of the local\nmoment’s lattice, JKis the Kondo or Hund coupling en-\nergy scale, vandµare the Fermi velocity and Fermi\nenergy of the Weyl electrons respectively, and Ω is the\nreal spin wave frequency (we use the units /planckover2pi1= 1). The\nspin wave momentum qin this expression is measured\nrelative to the difference ∆ Q=Qm−Qnbetween the\nwavevectors Qm,Qnof any two Weyl nodes in the first\nBrillouin zone. Coherent collective excitations that span\nthe entire first Brillouin zone can be used to separately\naddress many pairs of Weyl nodes – by tuning the total\nwavevector ∆ Q+qto the vicinity of ∆ Q. Representa-\ntive functions fab\nmnfor the Weyl nodes with finite Fermi\nsurfaces are plotted in Figures 1 and 2\nWemakeanalyticalprogressandgainvaluablephysical\ninsight through several idealizations: all Weyl nodes are\nassumed to be identical, sphericallysymmetric and living\nat the same node energy. Their chiralities χm=±1 and\nlocations Qmare arbitrary (as long as the total chirality\nin the first Brillouin zone vanishes). Under these condi-\ntions, only three tensor components of γabare finite and\nindependent, γ/bardbl/bardbl,γ⊥⊥andγ⊥⊥′. Here and throughout\nthe paper ∝bardblindicates the spin direction parallel to the\nmode’s wavevector q, and⊥,⊥′are the spin directions(a)\n(b)\nFIG. 1. The plots of functions (a) f⊥⊥and (b) f/bardbl/bardblfor the\ndamping rates of transverse and longitudinal spin waves re-\nspectively, contributed by the Fermi surfaces on a particul ar\npair of Weyl nodes. Solid red lines are for the same-chiralit y\nnodes, and the dashed blue lines are for the opposite-chiral ity\nnodes.|Ω|= 1.4|µ|was assumed in this example.\nFIG. 2. The plots of selected universal functions fabfeatured\nin the damping rate γ∼Ω2f(vq/|Ω|;xµ). The functions\nare parametrized by xµ= 2|µ/Ω|, with finer dashes corre-\nsponding to larger Weyl Fermi pockets (solid lines refer to\nthe Fermi level that crosses the Weyl nodes). Shown func-\ntionsincludetransverse( ⊥⊥)andchiral( ⊥⊥′)dampingchan-\nnels shaped by electron scattering between equal-chiralit y (+)\nandopposite-chirality ( −)Weyl nodes. Longitudinal channels\n(∝bardbl∝bardbl) are similar to the shown transverse channels, compare\nwith Fig.1.3\n(a)\n (b)\nFIG. 3. Examples of the damping rate map in momentum\nspace for (a) µ∝negationslash= 0 and (b) µ= 0 (with and without a Fermi\nsurface of Weylelectrons respectively). Brightness depic ts the\nrateγ(q) of spin wave damping, and the red crosshair shows\nthe reference ∆ Qfor the local wavevector q= 0. These are\nqz= 0 slices through the full 3D map. Observing patterns\nof this kind in the full Brillouin zone scan will indicate the\nWeyl-electron origin of damping and reveal the complete set\nof ∆Q=Qm−Qnwavevectors from which the individual\nnode wavevectors Qmcan be deduced (assuming, for exam-\nple,/summationtext\nmQm= 0). The bright outer ring, which shrinks and\ncloses when 2 |µ|<|Ω|, originates in the inter-band electron\nscattering and gains strength from the rapidly growing Weyl\nelectron density of states. Note that various details in the se\nmaps, such as the anisotropy and ring sizes, will generally\ndepend on the concrete spin-wave dispersion Ω( q+∆Q), po-\nlarization, type and orientation of magnetic order, as well as\nthe chiralities and symmetries of the Weyl nodes.\nwhich areperpendicular to qand eachother. The full ex-\npression for damping rates is presented in Section IIIB;\nin Weyl ferromagnets, it becomes\nγmn=γ⊥⊥\nmn±γ⊥⊥′\nmn (2)\nfor the two polarizations of spin waves propagating along\nthe magnetization direction.\nThe essential utility of the universal damping comes\nfrom its qualitative features that reflect the relativistic\nnature of Weyl electrons. If the Fermi energy µlies away\nfrom the energy of the Weyl nodes, Fermi surfaces will\nform. Then, the spin wave damping rate is expected to\nexhibitasetofminimumsandmaximumsasafunctionof\nthefrequencyΩandmomentum q. Thelocationsofthese\nextremums depend on the parameters that characterize\nthe Weyl nodes: Fermi velocity v, chemical potential µ\nand even their relative chiralities χmχn=±1. Fig.3\ndemonstrates how the locations Qmof Weyl nodes can\nbe extracted from the full Brillouin zone map of the spin\nwave’s damping rate γ(q). Once the wavevectors Qmare\nknown, Fig.4 illustrates how the observation of enough\nextremums enables indirect measurements of the Weyl\nelectron spectra on multiple Weyl nodes. The presence\nof Weyl Fermi pockets also introduces spin-momentum\nlocking into the damping rates ( γ⊥⊥′\nmn∝ne}ationslash= 0), but only on\nthe pairs of Weyl nodes with opposite chiralities. As a\nvqΩ\n02\u00012 \u0000\u0002\n\u0003Ω=vq\nΩ=vq-2\u0004\nFIG. 4. A density plot of the collective mode damping rate\nγ(q,Ω) induced by Weyl electrons. Thin solid green lines in-\ndicateγ= 0, and the thin dashed green line indicates the\nlocal maximum of γ. The thick dashed yellow line represents\nthe dispersion Ω( q+ ∆Q) of a hypothetical spin-wave exci-\ntation (note that the origin of the plot corresponds to the\nmomentum difference ∆ Qof two Weyl nodes in the first Bril-\nlouin zone). The spin-wave’s damping rate will exhibit loca l\nminimums and maximums at the shown red points, which are\ncharacteristic for the relativistic spectrum of Weyl elect rons.\nResolving two of these points is enough for the determinatio n\nof the Weyl Fermi velocity vand the chemical potential µof\nthe Weyl nodes addressed via ∆ Q. Resolving three points al-\nlows an independent verification that Weyl nodes are indeed\nresponsible for the damping. The two-parameter scaling of\nthe damping rate (1) across a range of energies is the most\ngeneral signature of Weyl electrons, and can be used to verif y\nthe Weyl-electron origin of damping even if the visible spin\nwave dispersion does not cross any of the shown characterist ic\npoints.\nconsequence, the two spin wave modes that carry oppo-\nsite spin currents at the same wavevector qhavedifferent\npeak widths in inelastic neutron scattering.\nThe above qualitative features of damping disappear if\nthe Fermi energy sits exactly at the Weyl nodes. How-\never, the damping rate then becomes a universalfunction\nofa single parameter |Ω|/vq. This kind of scalingis a sig-\nnature of the relativistic Weyl electrons – it is caused by\n“inter-band” transitions in which an electron below the\nWeyl nodeisexcited toastate abovethe Weyl node. The\nplots of universal functions fab\nmnthat appear in Eq. 1 at\nµ= 0 are shown in Fig.2.\nThe magnitude of the damping rate depends\non the Kondo/Hund scale JKwhich may not be\nknown. However, the spin wave damping caused\nby Weyl electrons is always related to the effec-\ntive strength Jof the Weyl-electron-induced Ruder-\nman–Kittel–Kasuya–Yosida(RKKY)interactionsamong4\nqqJKJK\nFIG. 5. The Feynman diagram for two-spin interactions.\nThick external lines represent local moment fields and thin\nlines represent Weyl electron propagators. The two-spin co u-\nplings include Heisenberg, Kitaev and Dzyaloshinskii-Mor iya\ninteractions, but the Weyl-electron origin of spin dynamic s\nalso creates a dissipation channel in which spin waves decay\ninto electron-hole pairs.\nthe local moments45:\nγ\nJ∼1\n(aΛ)3/parenleftBigq\nΛ/parenrightBig2\n×/parenleftbiggΩ\nvq/parenrightbigg2\n, J∼vΛ/parenleftbigga3Λ2JK\nv/parenrightbigg2\n.\n(3)\nHere, Λisthemomentumcut-offforthelinearWeylspec-\ntrum,|q|<Λ. SinceaΛ<1 and the characteristic fea-\ntures of the universal damping appear near |Ω| ∼vq, the\ndamping rates are generally comparable to the energy\nscaleJof the induced RKKY interactions. For example,\nthe RKKY energy scale in the magnetic Weyl semimetal\nNdAlSi13can be crudely estimated as J∼1 meV. Even\nif the damping rate is more than an order of magnitude\nbelow this value of J, it should be detectable with high\nresolutionneutron instruments (a spin echospectrometer\ncan achieve energy resolution below 10 µeV).\nIII. DISSIPATION BY WEYL ELECTRONS\nHere we calculate the Gaussian dissipative part of the\neffective action for local moments which arises due to\ntheir coupling to itinerant Weyl electrons. The non-\ndissipative part of this action, computed in Ref.45, cap-\ntures the induced RKKY interactions among the lo-\ncal moments: Heisenberg, Kitaev and Dzyaloshinskii-\nMoriya. All Gaussian terms δnaΓabδnbof the action ob-\ntain from a single two-point Feynman diagram which in-\nvolves momentum integration of a singular function; the\nprincipal part of this integral yields the interactions, and\nthe contribution of its pole singularity amounts to dissi-\npation. We will focus only on the latter, following the\nprocedure from Ref.45.\nThe essential dynamics of local moments ˆnicoupled to\nconduction electrons ψiis given by the Hamiltonian:\nH0=Hn+/summationdisplay\nkǫkψ†\nkψk+JK/summationdisplay\niˆniψ†\niσψi.(4)\nBoth the local moments and electrons live on a lattice\nwhose sites are labeled by i, but we will immediately\ntake the continuum limit. The basic two-spin correla-\ntions∝an}bracketle{tˆna\niˆnb\nj∝an}bracketri}htare contained in the second-order Feynman\ndiagram shown in Fig.5:\nΓab\nmn(q) =iJ2\nK\n2/integraldisplayd4k\n(2π)4tr/bracketleftBig\nGm/parenleftBig\nk−q\n2/parenrightBig\nσaGn/parenleftBig\nk+q\n2/parenrightBig\nσb/bracketrightBig\n(5)The Weyl electron Green’s functions\nGn(ω,k) =/bracketleftBig\nω−Hn(k)+isign(ǫn(k))0+/bracketrightBig−1\n(6)\nare treated as spinor matrices and refer to the low-energy\nelectronic states near any Weyl node nwhose wavevector\nin the first Brillouin zone is Qn; the wavevector kis a\n“small” displacement |k|<Λ fromQn, where Λ is the\nmomentum cut-off for the linear Weyl dispersion. These\nlow-energy electrons are described by the Hamiltonian\nHn(k+Qn) =vχnσk−µ , (7)\nwhereµis the chemical potential that determines the\nWeyl Fermi pocket character and size, vis the Fermi\nvelocity, and χn=±1 is the Weyl node chirality. We\nassume for simplicity that all Weyl nodes are spherically\nsymmetric, share the same node energy, chemical poten-\ntial and Fermi velocity, but have arbitrary wavevectors\nQnand chiralities χn=±1 (as long as the chiralities of\nall nodes in the first Brillouin zone add up to zero). By\nthis construction, the expression (5) is associated with a\npairm,nof Weyl nodes, and qis a “small” wavevector\nmeasured relative to Qm−Qn.\nWe will carry out all calculations with the formal as-\nsumption that no external or effective magnetic field is\nexerted on electrons. Realistically, however, we are inter-\nested in magnetic Weyl semimetals whose local moments\nmay carry a non-zero net magnetization ˆn0that presents\nitself as an effective magnetic field B=−JKˆn0to elec-\ntrons. This is of no concern because the correction of the\nspectrum (7) amounts merely to a shift of the wavevector\nk→k−B/vχn. Hence, an effective magnetic field only\nalters the locations Qnof the Weyl nodes in momentum\nspace, which are arbitrary in our formalism.\nThe full effective action matrix Γ for local moments\ntakes contributions from all Weyl node pairs:\nΓ(Q,Ω) =/summationdisplay\nm,nΓmn(Q−Qm+Qn,Ω).(8)\nIn this sense, it is possible to experimentally address a\nparticular pair of Weyl nodes, or a set of pairs, by prob-\ningthe momentumspaceinthe vicinityof Q∼Qm−Qn.\nThe dissipative part of Γ mnwill contain information\nabout the addressed Weyl nodes.\nA. Calculation of the dissipative terms in the\neffective spin Lagrangian\nThe calculationof(5) is lengthy, sowe will onlyoutline\nits key steps. The trace has been evaluated before45, and\nthe frequency integration yields:5\nΓab\nmn(q) =−J2\nK\n2/integraldisplayd3k\n(2π)3/bracketleftBigg\nXab(Ω,q;vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle+Ω\n2−µ,k)\n2vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ−vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/parenrightbig\nΩ+vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−vsχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle+i0+F(sχn,χm)\n−Xab(Ω,q;−vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle+Ω\n2−µ,k)\n2vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ+vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/parenrightbig\nΩ−vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−vsχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle+i0+F(sχn,−χm)\n+Xab(Ω,q;vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω\n2−µ,k)\n2vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ−vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/parenrightbig\nvχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω−vsχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−i0+F(χn,sχm)\n−Xab(Ω,q;−vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω\n2−µ,k)\n2vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ+vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/parenrightbig\n−vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω−vsχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−i0+F(−χn,sχm)\nHere,θ(x) is the step function, and two more functions,\nXab(Ω,q;ω,k) andF(s+,s−) are introduced to simplify\nnotation. The function Xab(Ω,q;ω,k) obtains from the\nnumerator of the trace in (5). Introducing the Kronecker\nsymbolδaband the Levi-Civita symbol ǫabc, we have:\nXab(Ω,q;ω,k) =/bracketleftbigg\n(ω+µ)2−Ω2\n4/bracketrightbigg\nδab\n+v2χmχn/bracketleftbigg\n2/parenleftbigg\nkakb−qaqb\n4/parenrightbigg\n−δab/parenleftbigg\nkckc−qcqc\n4/parenrightbigg/bracketrightbigg\n+ivǫabc/bracketleftbigg\nχm/parenleftbigg\nω+Ω\n2+µ/parenrightbigg/parenleftbigg\nkc−qc\n2/parenrightbigg\n−χn/parenleftbigg\nω−Ω\n2+µ/parenrightbigg/parenleftbigg\nkc+qc\n2/parenrightbigg/bracketrightbigg\n. (9)\nThe function F(s+,s−) withs+,s−=±1 keeps track of\nthe infinitesimal imaginary terms in the denominators of\nGreen’s functions:\nF(s+,s−) = sign/parenleftBig\nvs+/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle−µ/parenrightBig\n−sign/parenleftBig\nvs−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n=θ/parenleftbigg\n|qk|−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigµ\nv/parenrightBig2\n−k2−q2\n4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n×/bracketleftbigg/parenleftbigg\nsign(µ)+s++s−\n2/parenrightbigg\nsign(qk)+s+−s−\n2/bracketrightbigg\n+(s+−s−)θ/parenleftbigg\nk2+q2\n4−|qk|−/parenleftBigµ\nv/parenrightBig2/parenrightbigg\n.(10)\nAt this point, we use the relationship\n1\nx±i0+=P1\nx∓iπδ(x) (11)\nto isolate the dissipative processes that curb the x→0\nresonances. Dropping all terms that involve the principal\npartP, we get:\n/tildewideΓab\nmn(q) =iπJ2\nK\n8v2/summationdisplay\nsm,snsmsn/integraldisplayd3k\n(2π)3F′(snχn,smχm)\nχmχn/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle\n×Xab/parenleftbigg\nΩ,q;vsmχm/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle+Ω\n2−µ,k/parenrightbigg\n(12)\n×δ/parenleftBig\nΩ+vsmχm/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle−vsnχn/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n×/bracketleftBig\nθ/parenleftBig\nµ−vsmχm/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n−θ/parenleftBig\nµ−vsnχn/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig/bracketrightBigWe introduced F′= sign(F)(1−δF,0), and the sum goes\noversm,sn=±1. All chirality factors χm,χn=±1\nthat appear outside of Xabare clearly eliminated by the\nsummation over sm,sn, so it will be convenient do define\ns−=smχm=±1 ands+=snχn=±1. The Dirac\nδ-function in (12) imposes:\ns+/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle−s−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle=Ω\nv. (13)\nThis pins the magnitude of the wavevector kto\nk=|Ω|\n2v/radicalBigg\nΩ2−v2q2\nΩ2−v2q2cos2θ, (14)\nassuming qk=qkcosθ, and further requires satisfying\none of these two conditions:\n|Ω|>vq ∧s±=±sign(Ω)\n|Ω|2|µ|. Defining\nλ=vq\n|Ω|, x=2|µ|\n|Ω|, κ=/radicalBigg\n1−λ2\n1−λ2ξ2(17)\nwith|ξ|=|cosθ|, we have:\nα⊥⊥\nmn=1/integraldisplay\n0dξθ/parenleftbig\n2κλξ−|x2−κ2−λ2|/parenrightbig\nκ2(18)\n×/bracketleftBigg/parenleftBigg\n1−χmχn−κ2ξ2+λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(1−λ)+/parenleftBigg\n1+χmχn−κ2ξ2+λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(λξ−1)/bracketrightBigg\nα/bardbl/bardbl\nmn=1/integraldisplay\n0dξθ/parenleftbig\n2κλξ−|x2−κ2−λ2|/parenrightbig\nκ2\n×/bracketleftBigg/parenleftBigg\n1−χmχnκ2(2ξ2−1)−λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(1−λ)+/parenleftBigg\n1+χmχnκ2(2ξ2−1)−λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(λξ−1)/bracketrightBigg\nα⊥⊥′\nmn=−iǫ/bardbl⊥⊥′1/integraldisplay\n0dξθ/parenleftbig\n2κλξ−|x2−κ2−λ2|/parenrightbig\nκ2\n×/summationdisplay\ns=±1(χm+χn)sign(µ)+s(χm−χn)sign(Ω)\n2/radicalbig\nκ2+λ2−2sκλξ(κξ−sλ)/bracketleftBig\nθ(1−λ)−sθ(λξ−1)/bracketrightBig\n,\nand\nβ⊥⊥\nmn=1/integraldisplay\n−1dξθ/parenleftbig\nκ2+λ2−2κλ|ξ|−x2/parenrightbig\nκ2\n1−χmχn−κ2ξ2+λ2\n/radicalBig\n(κ2+λ2)2−(2κλξ)2\nθ(1−λ) (19)\nβ/bardbl/bardbl\nmn=1/integraldisplay\n−1dξθ/parenleftbig\nκ2+λ2−2κλ|ξ|−x2/parenrightbig\nκ2\n1−χmχnκ2(2ξ2−1)−λ2\n/radicalBig\n(κ2+λ2)2−(2κλξ)2\nθ(1−λ)\nβ⊥⊥′\nmn=−iǫ/bardbl⊥⊥′1/integraldisplay\n−1dξθ/parenleftbig\nκ2+λ2−2κλ|ξ|−x2/parenrightbig\nκ2/summationdisplay\ns=±1s(χm−χn)sign(Ω)\n2/radicalbig\nκ2+λ2−2sκλ|ξ|(κ|ξ|−sλ)θ(1−λ).\nThe functions fabhave the same characteristics in all\nspin channels a,b∈ {⊥⊥,∝bardbl∝bardbl,⊥⊥′}. Their plots in Fig-\nures 1, 2, 4 illustrate that fabvanish for |Ω||Ω|>v|q|and|Ω|=v|q|. Thedissipationat|Ω|>max(2|µ|,v|q) is dominated by the collective mode\ndecay into “high energy” particle-hole pairs which are\nexcited across the Weyl node. Outside of this frequency-\nmomentum region, the decay occurs by generating “low7\nenergy” particle-hole pairs across the Fermi surface on\nthe Weyl node. This “low energy” channel is weaker,\nbut has several features that clearly reveal the relativis-\ntic properties of the Weyl spectrum. Fig. 4 shows how\ntheminimumsandmaximumsofacollectivemodedamp-\ning rate can be used to characterize the Fermi surface of\nWeyl electrons.\nB. Spin wave damping\nThe actual damping rate of collective excitations gen-\nerally obtains from a mixture of spin channels. Consider\nthe spin waves with wavevectors ∆ Q+qin the vicinity\nof the momentum-space separation ∆ Q=Qm−Qnbe-\ntween two particular Weyl nodes. Let −SΩab\n0(q) be the\nintrinsic part of the effective Lagrangian density δLeff\nfor the local moment fluctuations δn, excluding the spin\nBerry phase SΩδab(Sis the spin magnitude of local\nmoments). This can contain any exchange interactions\nof the localized electrons and crystal field anisotropies.\nThe Lagrangian density terms induced by the itinerant\nWeyl electrons are all contained in the Γabtensor (5).\nThe principal part of (5) yields a variety of induced\nRKKY interactions45, while its dissipative components\n/tildewideΓabare collected in (16). The presence of magnetic or-\nder in the ground state further affects the dynamics of\nspin waves because the small spin fluctuations δnof low-\nenergy modes must be orthogonal to the local spins ˆn.\nThis can be incorporated into the general analysis44, but\nwe will simplify the discussion here by considering only a\nferromagnetic ground state ˆn(r) =ˆn0. The spectrum of\ndamped spin waves is extracted from the Gaussian part\nof the Lagrangian density in momentum space\nδLeff= (δna)∗/bracketleftBig\nSΩδab−SΩab\n0(q)+a3Γab(Ω,q)/bracketrightBig\nδnb(20)\nThe factor of a unit-cell volume a3converts the energy\ndensity Γabto the energy per lattice unit-cell, and the\nfactor of1\n2in the Berry phase term Ω is appropriate for\nthe local moments with spin S=1\n2. Introducing\ngab= Ωab\n0−a3\nSΓab(21)\nto simplify notation, the spin wave modes obtain by di-\nagonalizing PMP, wherePab=δab−ˆna\n0ˆnb\n0projects-out\nthe high-energy amplitude fluctuations (keeps δn⊥ˆn0)\nand\nMab= Ωδab−g⊥⊥(δab−ˆqaˆqb)−g/bardbl/bardblˆqaˆqb−g⊥⊥′ǫabcˆqc\nis the matrix embedded in (20). An arbitrary choice of\nthe background magnetization ˆn0=ˆzreveals two polar-\nization modes δn= (δnx,δny) atq=qˆq\nδn±∝/parenleftBigg\ng/bardbl/bardbl−g⊥⊥\n2(ˆq2\nx−ˆq2\ny)±δǫ\n(g/bardbl/bardbl−g⊥⊥)ˆqxˆqy−g⊥⊥′ˆqz/parenrightBigg\n(22)with energies\nΩ±=g⊥⊥\n0+g/bardbl/bardbl−g⊥⊥\n2(1−ˆq2\nz)±δǫ(23)\nwhereδǫ=1\n2/radicalbig\n(g/bardbl/bardbl−g⊥⊥)2(1−ˆq2z)2−(2g⊥⊥′ˆqz)2.\nThese polarizations are generally elliptical, but become\ncircularδn∝(±i,1) with Ω ±=g⊥⊥∓ig⊥⊥′for the\nmodes that propagate along the magnetization direction\n(q∝bardblˆn0), and linear δn+∝ˆq,δn−∝ˆn0׈qwith\nΩ+=g/bardbl/bardbl, Ω−=g⊥⊥respectively for the modes that\npropagate in the plane perpendicular to the magneti-\nzation (q⊥ˆn0). The character and non-degeneracy\nof the two polarization modes is the hallmark of the\nRKKY interactions induced through the spin-orbit cou-\npling: Dzyaloshinskii-Moriya(DM) in the caseof circular\npolarizations, and Kitaev in the case of linear polariza-\ntions.\nThe equation (23) has to be solved self-consistently\nsince the components of the gabtensor on its right-hand\nside depend on frequency, but the revealed form of its\nsolutions ensures all of the spin wave properties that\nwe discuss. The two circular polarizations at the same\nwavevector q∝bardblˆn0carry opposite spin currents\nja\ni=−iqiǫabc(δnb)∗δnc∝ ∓|g⊥⊥′|2qiδaz,(24)\nso their energy difference Ω ±=g⊥⊥∓ig⊥⊥′due to the\nDM interaction implies spin-momentum locking. Note\nthat the DM interactions appears as gab\nDM∝ǫabc(iqc), so\nit does shift the spin wave energy. The dissipative com-\nponents/tildewidegab∝/tildewideΓabofgabimpart an imaginary part on\nthe pole frequency Ω, which corresponds to the damping\nrate. The signs of both /tildewideΓ⊥⊥,/tildewideΓ/bardbl/bardbl(f⊥⊥,f/bardbl/bardbl>0) in-\ndeed correspond to damping and not an instability, and\nthe chiral contributions are not large enough to overturn\nthis at any Ω. The chiral dissipative part extracted from\n(16) is real,/tildewidegab\nDM∝ǫabcqc, and hence introduces differ-\nent damping rates for the two circular spin waves. These\nqualitative conclusions hold for the elliptical modes as\nwell.\nC. The absence of uniform precession damping\nThe universal dependence of (16) on |Ω|/vqintroduces\nanon-analyticbehavioratΩ ,q→0inthedampingterms\n/tildewideLof the spin Lagrangian density. Therefore, one cannot\nstrictly expand /tildewideLin powers of Ω ,qto represent the dis-\nsipation as a result of local processes. /tildewideLcan be approx-\nimated by an expansion only in special limits. Suppose\nthe spin waves have dispersion |Ω|=uqat low ener-\ngies (in the vicinity of ∆ Q=Qm−Qn→0 for intra-\nnode scattering m=n). If the spin wave velocity uis\nsmaller than the Weyl electrons’ velocity v, then a suf-\nficiently large qpushes the spin waves into the regime\n|Ω|< vq−2|µ|where/tildewideΓab= 0 in (16) and the damping\nis absent (see Fig.2). Alternatively, if u≫v, then the8\nspin waves are in the regime |Ω| ≫vqand their damp-\ning at energies |Ω|>2|µ|is approximately characterized\nby the dominant local terms /tildewideΓ/bardbl/bardbl,/tildewideΓ⊥⊥∼i(AΩ2+Bq2)\nand a smaller chiral term /tildewideΓ⊥⊥′∼DqΩ. Together with\nthe non-dissipative Hermitian terms χ−1\n0, the electron-\ninduced part of the local moments’ effective Lagrangian\ndensity (20) contains\nΓab|Ω|≫vq− −−−− →1\n2/bracketleftBig\n(χ−1\n0)ab+i(AabΩ2+Babq2)+DǫabcqcΩ/bracketrightBig\n(25)\nwithAab=A⊥⊥(δab−qaqb/q2)+A/bardbl/bardblqaqb/q2andlikewise\nforBab. By construction (5), Γ ≡1\n2χ−1is the inverse\ntime-ordered correlation function\n∝an}bracketle{tδsa(q,Ω)δsb(q′,Ω′)∝an}bracketri}ht=iχab(q,Ω)δ(q+q′)δ(Ω+Ω′)\nfor the small fluctuations δsof the Weyl electron spins\naway from their equilibrium magnetization. We will con-\nsider only the simplest case of a collinear ferromagnet\nin the following analysis. The equilibrium state will be\ngiven by the uniform magnetization of local moments ˆn0\nand electrons ∝an}bracketle{ts0∝an}bracketri}ht ∝bardblˆn0.\nA semiclassical representation of the local moment dy-\nnamics is given by the field equation for ˆn. The pres-\nence of non-Hermitian damping terms in the effective ac-\ntion for local moments prevents us from deriving the field\nequation by considering the stationary action condition.\nInstead, we can use linear response theory to learn about\nthe semiclassical dynamics. The retarded electrons’ spin\ncorrelation function\nχR(q,Ω) =/braceleftBigg\nχ(q,Ω),Ω>0\nχ†(q,Ω),Ω<0(26)\nis readily obtained from (25)\n(χ−1\nR)ab|Ω|≫vq− −−−− →(χ−1\n0)ab(27)\n+sign(Ω)/bracketleftBig\ni(AabΩ2+Babq2)+DǫabcqcΩ/bracketrightBig\n,\nand then the response of electron spins to the local mo-\nment field is\n∝an}bracketle{tδsa(r,t)∝an}bracketri}ht=JK\na3/integraldisplay\ndt′d3r′χab\nR(r−r′,t−t′)δnb(r′,t′).\n(28)\nThis follows from the Kondo interaction JKin (4) be-\ntween the “perturbation” field nand the responding\nelectrons spin s=ψ†σψon a lattice site (the unit-cell\nvolumea3effectively converts the integration over coor-\ndinates to a summation over lattice sites). Note that\nχab\nR(q,Ω) = (χab\nR)∗(−q,−Ω) is established globally in\nmomentumspace(notnecessarilyintheimmediatevicin-\nity of the Weyl node wavevector ∆ Q)61, so that its in-\nverse Fourier transform χab\nR(r,t) is real. The thermody-\nnamic potential for local moments is simply\nF[ˆn] =JK∝an}bracketle{ts∝an}bracketri}htˆn. (29)The local moment dynamics is driven by an effective\n“magnetic” field in units of energy\nHeff(r,t) =−δF[ˆn]\nδˆn(r,t)=−JK∝an}bracketle{ts(r,t)∝an}bracketri}ht(30)\nTaking into account the Berry phase of local moments\nyields the usual semiclassical field equation\n∂ˆn\n∂t=ˆn×Heff. (31)\nwith\nHa\neff(r,t)≈ −JKˆna\n0−J2\nK\na3/integraldisplay\ndδtd3δrχab\nR(δr,δt) (32)\n×δˆnb(r+δr,t+δt)\n≈ −JKˆna\n0−J2\nK\na3/integraldisplay\ndδtd3δrχab\nR(δr,δt)\n×/bracketleftbigg\n1+δr∇+δt∂\n∂t+···/bracketrightbigg\nδˆnb(r,t)\nThis is seen to generate Gilbert damping which dissi-\npates the precession of uniform magnetization in typical\nferromagnets\n∂ˆn\n∂t=ˆn×Heff=···+ˆn×αG∂ˆn\n∂t(33)\nwith the damping tensor\nαab\nG=−J2\nK\na3/integraldisplay\ndδtd3δrχab\nR(δr,δt)δt (34)\n=iJ2\nK\na3/integraldisplaydΩ\n2π/integraldisplay\ndδte−iΩδt∂χab\nR(0,Ω)\n∂Ω\n=−J2\nK\na3∂Imχab\nR\n∂Ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n(q,Ω)=0.\nThe real part of χR(q,Ω) generally does not contribute\nbecause it is an even function of Ω at q= 0 (even though\nit diverges for gapless spin waves when Ω →0). In the\ncase of damping induced by Weyl electrons, the imagi-\nnary part of χRbecomes zero when 2 |µ|−vq>|Ω| ≥vq,\nfollowing the behavior of the time-ordered χ−1≡Γab\nthat was discussed earlier (see Fig.4). Therefore, χRis\nreal in the limit Ω ,q→0 and the decay of spin waves\ninto Stoner excitations of the Weyl electrons does not\ngenerate Gilbert damping.\nThe complete equation of motion for local moments\ncanbe extractedfrom(31)and(32), but thenon-analytic\nfrequency dependence of the dissipative terms in (27) in-\ntroduces (via its Fourier transform) non-local relation-\nships between the fields ˆn(t) at different times t. If one\nwere to ignore this issue, or approximate the non-local\neffect by couplings over small time intervals, then a local\nfield equation would be obtained from the expansion in-\ndicated in (32). We will not pursue this here any further.9\nIV. CONCLUSIONS AND DISCUSSION\nWe analyzed the dynamics of local magnetic moments\ncoupled to itinerant Weyl electrons, and focused on the\ndissipation of spin waves via the continuum of Stoner\nparticle-holeexcitations. Wedescribedthisdissipationat\nthe level of the effective Lagrangian of local moments, or\nequivalently the spin-spin correlation function (dynamic\nsusceptibility). For the spin waves at wavevector∆ Q+q\nand frequency Ω in the vicinity of the momentum differ-\nence∆Q=Qm−QnbetweentwoWeylnodes,thedamp-\ning rate is proportional to Ω2and a universal function of\n|Ω|/v|q|wherevis the Weyl electron (Fermi) velocity.\nThe presence of Fermi pockets with chemical potential µ\nintroduces additional dependence of the damping rate on\n|Ω/µ|. If the Weyl nodes are well-separated in momen-\ntum space, then there is no cross-talk between them in\nthe damping rates and the momentum-space locations of\nthe Weyl nodes can be discerned from the wavevectors\nat which the spin wave dissipation is locally maximized.\nThe Weyl-electron origin of dissipation can be experi-\nmentally verified by the universal relativistic properties\nof damping over a range of mode frequencies and mo-\nmenta, while various parameters of the Weyl spectrum\ncan be extracted from the momentum space locations\nof the characteristic damping features (e.g. local maxi-\nmums and points where damping vanishes). The damp-\ning rates involving Weyl electrons also generally exhibit\n“non-reciprocity”or chirality– the modes ofdifferent po-\nlarizationsthat propagateatthe samemomentum qhave\ndifferent lifetimes. We presented a procedure to obtain\nthe field equation for the semi-classical dynamics of the\nlocalmomentmagnetizationfield, andfoundthatthedis-\nsipation on Weyl electrons does not give rise to Gilbert\ndamping.\nOne important conclusion of this study is that the\nspin wave damping rate reveals the relativistic nature\nof Weyl electrons – both through its universal depen-\ndence on |Ω|/v|q|and the places in momentum space\nwhere it vanishes. We computed the damping rate asso-\nciated with Stoner excitations, but similar results should\nhold for zero-spin particle-hole excitations as well. Then,\notherkindsofcollectivemodescoupledtoWeylelectrons,\ne.g. the phonons of the crystal or a charge density wave,\nshouldexhibitsimilaruniversalityintheirdampingrates.\nThiswouldbeinterestingtoexploreinthefuturesincein-\nelastic neutron scattering is sensitive to phonons as well.\nThe developed theory is very general within its limi-\ntations. It makes no assumptions about the Weyl node\nlocations, so it applies to Diracsemimetals aswell (where\nthe opposite-chirality Weyl nodes coexist at the samewavevectors). It also makes no assumptions about the\nmagnetic order, so it holds for ferromagnets, antiferro-\nmagnets and paramagnets, with or without local spin\nanisotropy. In this regard, however, the damping rates\nof spin waves are affected by the nature of magnetic or-\nder; we demonstrated the calculations only in the fer-\nromagnetic (and implicitly also the paramagnetic) case.\nAnalytical progress was made by simplifying the model\nto spherically symmetric Weyl nodes that all live at the\nsame energy. This is the main limitation of the current\ntheory, although many implications of realistic model ex-\ntensions can be readily anticipated. Energy differences\nbetween the nodes are easily included by associating dif-\nferent chemical potentials to the nodes, while a small\nWeyl node anisotropy is expected to introduce a simi-\nlar anisotropy in the induced dynamics and dissipation\nof local moments. It is possible that type-II Weyl nodes\nfalloutsideofthistheory’sdomain, sotheirexplorationis\nleft for future study. We also did not consider corrections\ndue to finite temperature and disorder.\nTheusefulnessofthistheoryfortheexperimentalchar-\nacterization of magnetic Weyl semimetals is guarantied\nin principle, but depends on several factors in reality.\nThe needed level of detail is not easy to achieve in the\nmeasurements of spin wave spectra. It requires at least\nvery clean samples, low temperatures, as well as a suffi-\nciently high energy resolution and adequate statistics to\nresolvewithlowerrorbarsthe energy/momentumdepen-\ndence of the inelastic neutron scattering. These aspects\nof measurements can always be improved, but there are\nalso material-related constraints: phonons, for example,\nmust not coexist with spin waves at the same momenta\nand frequencies. Still, some regions of the first Brillouin\nzone should expose the electronic damping mechanism\nand enable the proposed experimental characterization\nof magnetic Weyl semimetals. On the purely theoretical\nfront, the present study was concerned with a basic but\nintricate and important aspect of interaction physics in\na topological system. It plays a role in piecing together\na broader picture of magnetic correlated topological ma-\nterials, which can host non-trivial anisotropic magnetic\norders13, chiral magnetic states and excitations44, and\npossibly even exotic spin liquids62.\nV. ACKNOWLEDGMENTS\nI am very grateful for insightfull discussions with\nJonathanGaudet and Collin Broholm. 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Nikoli´ c, Physical Review B 101, 115144 (2020)." }, { "title": "1504.02940v3.Evolution_of_Kinetic_and_Magnetic_Energy_in_Intra_Cluster_Media.pdf", "content": "arXiv:1504.02940v3 [physics.plasm-ph] 13 Jul 2015Evolution of Kinetic and Magnetic Energy in Intra Cluster Me dia\nKiwan Park\nDepartment of Physics, UNIST, Ulsan, 689798, Korea; pkiwan @unist.ac.kr\nDongho Park\nAPCPT, Pohang, 790784, Korea; dongho@apctp.org\nABSTRACT\nIntra Cluster Media (ICMs) located at galaxy clusters is in the state of hot,\ntenuous, magnetized, and highly ionized X-ray emitting plasmas. This overall\ncollisionless, viscous, and conductive magnetohydrodynamic (MHD) turbulence\nin ICM is simulated using hyper and physical magnetic diffusivity. The re sults\nshow that fluctuating random plasma motion amplifies the magnetic fie ld, which\ncascades toward the diffusivity scale passing through the viscous s cale. The ki-\nnetic eddies in the subviscous scale are driven and constrained by th e magnetic\ntension which finally gets balanced with the highly damping effect of the kinetic\neddies. However, the saturated kinetic energy spectrum is deepe r than that of\nthe incompressible or compressible hydrodynamics fluid. To explain th is unusual\nfield profile we set up two simultaneous differential equations for the kinetic and\nmagnetic energy spectrum using an Eddy Damped Quasi Normal Mar kovian-\nized (EDQNM) approximation. The analytic solution tells us that the ma gnetic\nenergy in addition to the viscous damping effect constrains the plasm a motion\nleading to the power spectra: kinetic energy spectrum Ek\nV∼k−3and correspond-\ning representative magnetic energy spectrum Ek\nM∼k−1/2. Also the comparison\nof simulation results with different resolutions and magnetic diffusivitie s implies\nthe role of small scale magnetic energy in dynamo.\nSubject headings: galaxies: clusters: intracluster medium, magnetic fields\n1. Introduction\nICM located at the center of galaxy cluster is composed of fully ionize d hot plasmas\n(T∼108K). The gas includes most of the cluster baryons ( >85%) and heavy elements, but\noverall density ‘ n’ is very low ( n <103cm−3). As a result ICM has very small diffusivity– 2 –\n‘η’ (∼T−3/2) while viscosity ‘ ν’ (∼1/n) is very large (Schekochihin et al. 2005). Dynamo\neffect from the turbulent flow motions in ICM exceeds the dissipation of such high viscous\nplasmas causing the growth of seed magnetic fields, which react bac k the plasma motion\nthrough magnetic tension ( B·∇B).\nThe amplified magnetic field constrains some intrinsic properties in ICM . Non-magnetized\nthermal Spitzer conductivity ‘ κSP’ is like (Narayan & Medvedev 2001):\nκSP∼λ2\ne\ntcoul∼4×1032T5/2\n1n−1\n−3cm2s−1, (1)\nwhereT1=kT/10 Kev,λeis the mean free path of an electron and tcoulis the time between\ncoulomb collisions. If magnetic field is injected, κ⊥, conductivity perpendicular to the mag-\nnetic field is reduced to ∼(ρe/λe)2κSP(≪κSP) so that the conductivity becomes anisotropic\nwith one third of the thermal Spitzer conductivity: κ=κ⊥+κ/bardbl→κ/bardbl. This anisotropic con-\nductivity brings about temperature distribution that depends on t he direction and strength\nof magnetic field. Also as the conservation of magnetic moment µB=u2\n⊥/Band kinetic\nenergy (∼u2\n/bardbl+u2\n⊥) implies, magnetic field makes pressure tensor Pij=mnuiujanisotropic\nalong the field. The ratio of pressure anisotropy ‘ △’ is implicitly related to the viscosity and\nstability of ICM (Schekochihin et al. 2010):\n△ ≡p⊥−p/bardbl\np∼1\nνii1\nBdB\ndt∈/bracketleftbigg\n−2\nβ,1\nβ/bracketrightbigg\n,(β= 8πp/B2). (2)\nHere, kinematic viscosity ν∼u2\nth/νii, ion-ion collision frequency νii= 4πne4lnΛm−1/2\niT3/2.\nIf△is smaller than −2/βor larger than 1 /β, firehose or mirror instability occurs in the\nrange between ion Larmor radius ( ρi∼104−106km) and the mean free path ( λmfp∼1015\nkm) (Schekochihin et al. 2008). Since ICM has high ‘ β’, stability range is very narrow, i.e.,\npractically unstable. Instability is thought to redistribute the plasm a motions faster than\ncoulomb collision does, but its role is not fully understood yet (Jones 2 008).\nBy now we may begin to wonder if a typical MHD theory is applicable to th e weakly colli-\nsionalICMplasma. Sincecollisionbetweenparticlestransfersmomen tumtomakethesystem\nisotropic, most phenomenological or analytic methods assume the c ollisional MHD system\nsystem. However, since the tenuous plasmas cannot expect suffic ient collisions, Chew, Gold-\nberger, andLow(Chew et al.1956)developed anadiabaticMHDequa tionwiththeconsider-\nation of anisotropic pressure tensor (CGL-MHD model). Recently S antos-Liman insisted the\nvalidness of the typical MHD theory with these additional constrain ts (Santos-Lima et al.\n2011, 2014). Their model, a kinetic MHD based on CGL-closure with th e limit of anisotropy,– 3 –\nshows the growth rate of magnetic energy is similar to that of a typic al MHD theory in the\nearly time regime, but the saturation is much smaller. In fact, the eff ect of anisotropy due\nto the magnetic field in the plasma does not appear in the kinematic reg ime while magnetic\nback reaction is negligibly small.\nHowever, regardless of the anisotropic limit, the fluids driven by a ra ndom and isotropic\nexternal force eventually become isotropic followed by the strong ly frozen magnetic fields if\nPrM→ ∞(η→0). Even for a unit magnetic prandtl number ( PrM=ν/η= 1), where\nmagneticfieldsarenotsostronglyfrozen, thelargescalefieldsdriv en bytherandomisotropic\nforce tend to remain independent of direction if a background magn etic field ‘ bext’ is not very\nstrong (Cho et al. 2009). Moreover thegrowing anisotropy insmalle r scales cannot decisively\naffect the energy spectrum due to the small eddy turnover time. A s long as the isotropic\nforce continuously drives the system, especially large scales, the t heoretical fluid model that\nassumes an isotropic system is valid except the case of microscale ins tability or very strong\n‘bext’. In addition there is an interesting report that the magnetic energ y spectrum around\nthe core of cluster is possibly Kolmogorov’s spectrum ∼k−5/3(Kolmogorov 1941) which\nassumes an isotropic system (Vogt & Enßlin 2003). Now we can focus our interest on the\nisotropic properties in ICM of large magnetic prandtl number: ener gy spectra.\nDynamo in magnetized plasma with large PrM, which is not rarely observed in space, occurs\neasilyandhasitspeculiarproperties. Astheexcitedmagneticfields( energy)areinstilledinto\nthedampedkinetic eddies, theviscous damping scale kν∼1/lνisextended towardresistivity\nscalekη∼1/lη(kν≪kη). Then, these two coupled velocity and magnetic field generate spe -\ncificpowerspectrawhicharedifferentfromthetypicalspectrumo fKolmogorov’sincompress-\nible fluid or Burger’s compressible fluid. There have been works on the small scale dynamo\nfor large PrMplasma (Schekochihin et al. 2002; Cho et al. 2003; Schekochihin et al. 2004;\nYousef et al. 2007) in addition to the several analytic methods (Bat chelor 1950; Kazantsev\n1968; Kulsrud and Anderson 1992; Schober et al. 2012; Bovino et a l 2013). To solve the\nmagnetic induction equation MHD theories based on Kazantsev’s wor k (Kazantsev 1968)\nassume the second order velocity field correlation /angb∇acketleftv·v/angb∇acket∇ight ∼(δij−rirj\nr2)TN(r) +rirj\nr2TL(r).\nHowever to explain the formation of this second order velocity field c orrelation, i.e., ki-\nnetic energy spectrum, we need to solve the coupled momentum and magnetic induction\nequation simultaneously instead of making an assumption of the kinet ic energy spectrum in\nadvance. Here we use an Eddy Damped Quasi Normal Markovianizat ion method EDQNM,\n(Kraichnan & Nagarajan 1967; McComb 1990; Park 2013) and dimen sional approach in a\nlimited way. We simplified the resultant simultaneous differential equat ions and found out\nthe solutions for kinetic and magnetic energy spectrum.– 4 –\nIn chapter 2, we briefly introduce simulation tool and analytic metho d used in this paper.\nSimulational and analytic results are introduced in chapter 3. And in t he final chapter\nwe discuss about the results, their physical meanings, and future topics. Detailed analytic\ncalculations are discussed in the appendix.\n2. Numerical and analytic method\nWe have solved the incompressible MHD equations using a pseudo-spe ctral code with a\nperiodic box of size ‘(2 π)3’:\n∂v\n∂t=−v·∇v−∇P+(∇×B)×B+ν∇2v+f, (3)\n∂B\n∂t=∇×(v×B)+bext, (4)\nwhere ‘f’ is a random mechanical force driving a system at k∼2−3 in fourier space,\nand ‘bext’ is a weak background (guide) magnetic field1(bext= 0.0001) covering the whole\nmagnetic eddy scales. Here, ‘ B’, magnetic field divided by ‘(4 πρ)1/2’, has the unit of Alfv´ en\nvelocity, and ‘ v’ is ‘rms velocity’, and ‘t’ has the unit of large scale eddy turnover time ‘L/v’.\nFor example, if ‘ L’ of a cluster is ∼400 kpc and ‘ v’ is∼400 km/s, then ‘ L/v’ is∼109year.\nThe time ‘ t’ has the unit of ‘109’ year. The system realizes the state of ICM of the high\nviscous plasma state with ideally frozen magnetic fields.\nTo compare the effect of hyper diffusivity (incompressible fluid) and p hysical diffusivity\n(compressible fluid), we also used PENCIL CODE (Brandenburg 2001 ) with message pass-\ning interface(MPI) in a periodic box of spatial volume (2 π)3with mesh size 2883. The basic\nequations solved in the code are,\nDρ\nDt=−ρ∇·u (5)\nDu\nDt=−c2\ns∇lnρ+J×B\nρ+ν/parenleftbig\n∇2u+1\n3∇∇·u/parenrightbig\n+f (6)\n∂A\n∂t=u×B−η∇×B. (7)\n1‘bext’ is not indispensable for the growth of magnetic fields if there is a sub stituting seed magnetic field.\nThe turbulence by cosmological shocks can amplify the weak seed fie ld of any origin (Ryu et al. 2008). It\nwas pointed out that subsonic turbulence could develop with a very w eak seed magnetic field (Ryu et al.\n2012). Moreover turbulence can amplify a localized seed magnetic fie ld (Brandenburg 2001; Cho & Yoo\n2012). Cho (Cho 2014) investigated the origin of seed magnetic field s and insisted that the origin of the seed\nfield should be more like the localized seed magnetic fields ejected from the astrophysical bodies– 5 –\nρ: density; u: velocity; B: magnetic field; A: vector potential; J: current density; D/Dt(=\n∂/∂t+u· ∇): advective derivative; η: magnetic diffusivity(= c2/4πσ,σ: conductivity); ν:\nkinematic viscosity(= µ/ρ,µ: viscosity); cs: sound speed. Velocity is expressed in units of\ncs, and magnetic fields in units of ( ρ0µ0)1/2cs([B] =/radicalbig\nρ0[µ0][v] fromEM∼B2/µ0and\nEkin∼ρ0v2).µ0is magnetic permeability and ρ0is the initial density. Note that ρ0∼ρin\ntheweakly compressible simulations. Theseconstants cs,µ0, andρ0areset tobe‘1’. f(x,t)is\nrepresented by Nf0(t)exp[ikf(t)·x+iφ(t)](N: normalization factor, f0: forcing magnitude,\nkf(t): forcing wave number2. The variables are also independent of a unit system. However,\ninstead of bext, PENCIL CODE gives a system initial seed magnetic field in small scales.\nThis seedfield disappearsinafewsimulation timesteps. PENCIL CODEa ndtheother code\n(Cho et al. 2003) are not the same in various ways, but we will see the y produce practically\nthe same energy spectra.\n3. Results\n3.1. Simulation results\nFig.1(a) shows the normalized kinetic energy EV(t) (black dashed line) and magnetic\nenergyEM(t) (red solid line) of incompressible MHD fluid ( ν= 0.015,η→0,PrM→ ∞,\nβ= 8πP/B2→ ∞) with resolution of 2563(thinner line) and 5123(thicker line). Reynolds\nnumberReof both cases are ∼42, and their magnetic Reynolds number ReMare actually\ninfinity. When the random isotropic forcing begins to drive the syste m,EV(t) quickly grows,\nbecomes saturated, and keeps the status quo until EM(t) begins to arise at t∼15−20. As\nEM(t) grows, the energy transfer from EV(t) toEM(t) gets accelerated. For this nonlocal en-\nergy transfer, the gap between kinetic and magnetic energy is an im portant factor. However,\nasB·∇vin Eq.(4) implies, the geometrical constraint between vandBalso plays a role of\ndeterminant. If magnetic field is parallel to the gradient of velocity fi eld, kinetic energy is\ntransferred to magnetic eddies. If not, magnetic field frozen to t he plasma fluid moves freely\nalong the fluid or gets annihilated. Around the onset position at t∼15−20, structural\nchange between EVandEMseems to get accelerated and completed at the saturation. Also\nthe plot shows the onset position of EM(t) is proportional to the resolution. Higher resolu-\ntion without resistivity makes more space where the forward casca dedEM(t) can stay not\nbeing dissipated much. This leads to the imbalanced energy distributio n like the relatively\nsmaller amount of magnetic energy in large scales and more amount of magnetic energy in\n2Pencil code selects one of 350 vectors in k fvector set at each time step. f0is 0.08 and injection scale\n|kf|is∼1−2– 6 –\nsmall scales. This makes the gap between EVandEMin large scales, especially around\nthe injection scale, increase so that the nonlocal energy transfe r become accelerated. The\nsaturated magnetic energy EM,satgrows with the increase of resolution, but the saturated\nkinetic energy EV,satdecreases. This isthe special featureof η∼0with a fixed high viscosity.\nFig.1(b) includes the normalized EV(t) andEM(t) of compressible fluids. Their properties\nare asfollows; for thicker line, PrM=75,ν= 0.015,η= 2×10−4,Re∼170,ReM∼1.3×104,\nβ∼83; for thethinner line, PrM=7500,ν= 0.015,η= 2×10−6,Re∼120,ReM∼8.8×105,\nβ∼430. The resolutions in both cases are 2883. Since the average of Mach number is at\nmost 0.18, the effect of compressibility is not much. We can infer more EVcan be trans-\nferred to EMwith higher PrM(lowerη). However, the plot shows the actual EV,sat&EM,sat\nare inversely proportional to PrM. Moreover EV,satis slightly larger than EM,sat, and this\ntendency is opposite to that of hyper diffusivity (Fig.1(a)). If ηis small, magnetic energy\ncan migrate into the smaller scales. But since the effect of dissipation grows with the wave\nnumberk2, moreEMis dissipated to increase energy gap between EVandEM. This boosts\nthe nonlocal energy transfer bringing about the accelerated diss ipation of energy ( ∼k2E(k))\nin small scales. So the saturated energy EV,sat,EM,satare smaller than those of lower PrM.\nHowever, it is not easy to conclude whether they converge onsome lower limit asthe physical\ndiffusivity η→0 with the limited resolution and PrMat this moment.\nFig.2(a) shows the evolving energy spectrum EV(k) andEM(k) of the incompressible fluid\nin the early time regime t∼0−10.5 (from bottom to top). And the plot in Fig.2(b) is a de\nfacto the same one in the range of t= 24.9−32.6. When the kinetic energy begins to drive\nthe system at k= 2.5,EV(black dashed line, t∼0) grows prior to EM(red solid line).\nAs the advection term v·∇vindicates, EMis not necessary for the local energy transfer in\nkinetic eddies. However, without EV(k) magnetic eddies cannot receive energy from kinetic\neddies nor transfer its energy to the neighboring eddies. Only when magnetic fields run into\nthe plasma fluids with the nontrivial EVof which gradient is in the direction of magnetic\nfields, dynamo process occurs leading to the growth of EM. Thus, in the very early time\nregime the evolution of magnetic field looks subsidiary. But around t∼1.4−2.1EM(k) in\nsmall scales begins to surpass EV(k), and gets past EV(k) which suffers from the viscous\ndissipation. So most of EVis located near the injection scale where the viscous damping\neffect is not so much, but EMcascades forward and stays in the smaller scales. Kinetic\neddies in k <∼15 lose energy (or magnetic eddies receive energy through B·∇v) whereas\nsmaller scale kinetic eddies ( k >∼15) receive energy through magnetic tension B·∇B. As\nthe fluid motion can amplify the magnetic field through dynamo proces s, also magnetic fields\ncan increase the kinetic energy through Lorentz force ( J×B). The magnetic fields press the– 7 –\nfluid through magnetic pressure ( −∇B2/2) and stretch the fluid through magnetic tension.\nWe will see that the unusual EV(k) spectrum k−3is due to not only the viscous damping\nbut also the interaction with EM.\nFig.3(a), 3(b) include the saturated energy spectra of incompres sible fluids with hyper diffu-\nsivity. Also the saturated energy spectra of compressible fluids wit h physical diffusivity are\nshown in Fig.3(c), 3(d). These plots show EV,sat(k) eventually converges to ∼k−3ifPrM\nis not too small regardless of the different evolving profiles of EV(t) andEM(t) in Fig.1(a),\n1(b). So the comparison of these plots will give us some clues to the f ormation of EV(k) in\nsmall scale range. A quick look shows EV(k) of 5123has clear spectrum of k−3compared\nwith the kinetic energy spectrum of 2563. However, these two simulation sets have the same\nconditions: weak bext= 0.0001, isotropic random driving force f, injection scale kf∼2.5,\nviscosity ν= 0.015, and negligible magnetic diffusivity η. Also the saturated energy levels\nshown in Fig.1(a) are not much different. Just the distribution of EMof 5123in small scale\nis flatter than that of 2563. This implies EMin smaller scales may be a determinant of EV\nspectrum.\nOn the other hand, for the compressible system in Fig.3(c) and Fig.3( d),EV(k) ofPrM=\n7500 has clearer and longer spectrum of k−3despite its smaller saturated energy level:\nE(t)sat.,Pr M=75> E(t)sat.,Pr M=7500(Fig.1(b)). This means EV,sat.(k) is not so much influ-\nenced by the magnitude of EM(k). However, careful look shows the peak of EMin Fig.3(d)\nis located at smaller scale regime than that of Fig.3(c). Then kinetic e ddies in wider range\ncan interact with magnetic energy EM(k) whose power spectrum is less steep. In other\nwords, Fig.3(b) and Fig.3(d) commonly show the slope of EM(k) is not so slanted as that of\nFig.3(a) and Fig.3(c).\nSo if we assume a critical representative spectrum EM(k)∼kmin the subviscous regime,\nwe can infer that steeper spectrum than kmcannot provide the kinetic eddies with enough\nenergy for EV∼k−3. However the exact value of ‘ m’ is not known yet. Just we can guess\nthis index should be negative. The scaling factor ‘ k−1’, so called an invariant scaling factor\n(Ruzmaikin et al. 1982; Kleeorin et al. 1996), is drawn together in nex t figure for reference.\nBut we will not discuss this concept further in this paper.\nFig.4(a), 4(b) include the compensated k3EVandk1/2EM.EV(k) of high PrMhas clear\nspectrum of k−3in the subviscous scale. But for the spectrum of EMin Fig.4(b), it is\nambiguous to pinpoint any scaling factor. Instead, we choose ‘ k−1/2’, i.e.,m=−1/2, an– 8 –\napproximate median value of the previous results (Cho et al. 2003; L azarian et al. 2004;\nSchekochihin et al. 2004). We will see analytic method derives quite ex actEV(k) with this\nscaling factor k−1/2. A reference line ‘ k−1’ for the scaling invariant factor is drawn together.\nFig.5(a) shows EM/EVfor fig.3(a)-3(d). The ratio of hyper diffusivity is larger than that\nof physical diffusivity, which is consistent. As mentioned, with the ne gligible diffusivity the\nratio proportionally depends on the resolution because of the large rEM. In contrast, for\nthe case of physical diffusivity (black line), the dissipation effect gro wing with wave number\n(∼k2) dissipates EMmore efficiently until the energy states are balanced to be in the sta te\nof equipartition.\n3.2. Analytic methods & results\n3.2.1. Eddy Damped Quasi Normal Markovianization\nWe start from the momentum (Eq.3) and magnetic induction equation (Eq.4). Taking di-\nvergence of the momentum equation we can replace pressure by co nvection and magnetic\ntension (Leslie and Leith 1975, Yoshizawa 2011)3:\n∂vi(k,t)\n∂t=/summationdisplay\np+r=kMiqm(k)[vq(p,t)vm(r,t)−Bq(p,t)Bm(r,t)]+ν∇2vi(k,t),\n(8)\n∂Bi(k,t)\n∂t=/summationdisplay\np+r=kMB\niqm(k)vq(p,t)Bm(r,t), (9)\nwith the definition of algebraic multipliers\nMiqm(k) =−i\n2/parenleftbig\nkmδiq+kqδim−2kikqkm\nk2/parenrightbig\n,\nMB\niqm(k) =i(kmδiq−kqδim). (10)\n3‘v(k,t)’ and ‘B(k,t)’ depend on time ‘ t’ and wavenumber ‘ k’, but ‘t’ will be omitted for simplicity.– 9 –\nThenwecangettheevolvingsecondordercorrelationequationsof /angb∇acketleftvi(k)vi(−k)/angb∇acket∇ightand/angb∇acketleftBi(k)Bi(−k)/angb∇acket∇ight:\n/parenleftbig∂\n∂t+2νk2/parenrightbig\n/angb∇acketleftvi(k)vi(−k)/angb∇acket∇ight=/summationdisplay\np+r=kMiqm(k)[A1/bracehtipdownleft/bracehtipupright/bracehtipupleft /bracehtipdownright\n/angb∇acketleftvq(p)vm(r)vi(−k)/angb∇acket∇ight−A2/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright\n/angb∇acketleftvq(−p)vm(−r)vi(k)/angb∇acket∇ight\n−A3/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright\n/angb∇acketleftBq(p)Bm(r)vi(−k)/angb∇acket∇ight+A4/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright\n/angb∇acketleftBq(−p)Bm(−r)vi(+k)/angb∇acket∇ight],(11)\n∂\n∂t/angb∇acketleftBi(k)Bi(−k)/angb∇acket∇ight=/summationdisplay\np+r=kMB\niqm(k)[B1/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright\n/angb∇acketleftvq(p)Bm(r)Bi(−k)/angb∇acket∇ight−B2/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright\n/angb∇acketleftvq(−p)Bm(−r)Bi(k)/angb∇acket∇ight].(12)\nThe third order correlation, ‘ A1,A2,...,B2’, is called a transport function. These triple\ncorrelations play a role of transfer and dissipation of energy to dec ide the field profiles of\nthe system. If the field is helical, ‘ α’ coefficient ( ∼ /angb∇acketleftj·b/angb∇acket∇ight−/angb∇acketleftv·ω/angb∇acket∇ight) for the inverse cascade\nof magnetic energy can be derived. Also other terms for the forwa rd cascade of energies\nare derived (Krause & R¨ adler 1980; Moffatt 1978; Park & Blackman 2012a,b; Pouquet et al.\n1976). However the helical field is not assumed in our system, which m eans ‘α’ coefficient or\nthe inverse cascade of magnetic energy is excluded. We use a quasi normalization approxi-\nmation, sort of an iterative method, to find the transport functio n (appendix).\nThe formal representations of EV(k) andEM(k) are as follows:\n∂EV(k)\n∂t= +1\n2/integraldisplay\ndpdrΘννν\nkpr(t)k3\npr(1−2y2z2−xyz)EV(p)EV(r)\n−/integraldisplay\ndpdrΘννν\nkpr(t)p2\nr(xy+z3)EV(r)EV(k)−2νk2EV(k)\n+1\n2/integraldisplay\ndpdrΘνηη\nkpr(t)k3\npr(1−2y2z2−xyz)EM(p)EM(r)\n+/integraldisplay\ndpdrΘνηη\nkpr(t)p2\nr(y2z−z)EM(r)EV(k),\n(13)\nand\n∂EM(k)\n∂t=−/integraldisplay\ndpdrΘηην\nkrp(t)p2\nrz(1−x2)EM(r)EM(k)\n−/integraldisplay\ndpdrΘηην\nkrp(t)r2\np(y+xz)EV(p)EM(k)\n+/integraldisplay\ndpdrΘηην\nkrp(t)k3\npr(1+xyz)EV(p)EM(r).\n(14)– 10 –\nThe integral variables ‘p’, ‘r’, i.e., wavenumbers, are constrained by t he relation of p+r=k.\n‘x’, ‘y’, and ‘z’ are cosines of the angles formed by three vectors ‘ k’, ‘p’, and ‘r’. (Fig.5(a)).\nAlgebraically ‘ k’, ‘p’, and ‘r’ should satisfy a condition like (Leslie and Leith 1975):\n|k−r| ≤p≤k+r. (15)\nTo derive analytically solvable equations from Eq.(13), (14), we need to simplify these two\nequations considering the interaction between ‘ k’ and its close wave number. So, we take\naccount of only two cases: large p(k∼p≫r) and large r(k∼r≫p). In principle\n‘k/2∼p∼r’ should also be included. But since the interaction between the close wave\nvectors, local energy transfer, is dominant in the nonhelical small scale dynamo, the latter\ncase can be ignored.\nWith the assumption of EV(k)∼kvandEM(k)∼km, we simplify the first term in Eq.(14)\nlike below:\n(i) Large p(smallr, i.e.,k∼p≫r,η∼0)\nEddy damping function Θηην\nkrp(k,t) is approximately\nΘηην\nkrp(k,t) =1−e−[νp2+η(k2+r2)+µkpr]t\nνp2+η(k2+r2)+µkpr/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nη=0∼1\nνp2,(t→ ∞). (16)\nAlso as Fig.5(a), 6(a) show, we can use the relations x∼y∼0,z∼1, andr=k−p. Then,\n−/integraldisplayk\ndp1\nνp2p2\nr(1−x2)(k−p)mEM(k)∼ −1\nνkmEM(k). (17)\n(ii) Small p(larger, i.e.,k∼r≫p,x∼z∼0,y∼1)\nIn this case only the triad relaxation time µkpris left. We assume that it is independent of\ntime, so we can write µkprlike\nΘηην\nkrp(k,t)∼1\nνp2+µkpr∼1\nµkpr. (18)\nThen,\n∼ −/integraldisplayk\ndr1\nµkpr(k2rm−1−2krm+rm+1)x(1−x2)EM(k)∼0. (19)\nThe results, Eq.(17), (19) represent the first term in ‘ D’ in Eq.(21). The other terms can be\nfound in a similar way.– 11 –\nThen, the coupled equations of EV(k) andEM(k) are\n∂EV(k)\n∂t=−/parenleftbig\na1kv+2νk2/parenrightbig\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nAEV(k)+/parenleftbig\nb1km−b2kv/parenrightbig\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nBEM(k), (20)\n∂EM(k)\n∂t=/parenleftbig\nc1km+c2km+2/parenrightbig\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nCEV(k)−/parenleftbig\nd1km+d2kv+2/parenrightbig\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nDEM(k). (21)\nThe coefficients ‘ ai’, ‘bi’, ‘ci’, and ‘di’ are independent of ‘ k’, and assumed to be independent\nof time for simplicity. The matrix form of these simultaneous different ial equations is simply\n/bracketleftbiggE′\nV(k)\nE′\nM(k)/bracketrightbigg\n=/bracketleftbigg−A B\nC−D/bracketrightbigg\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nM/bracketleftbiggEV(k)\nEM(k)/bracketrightbigg\n. (22)\nEq.(22) can be solved diagonalizing the matrix ‘ M’. For this, the bases EV(k) andEM(k)\nneed to be transformed using a transition matrix ‘ P’ which is composed of eigenvectors of\n‘M’:\n/bracketleftbiggEV(k)\nEM(k)/bracketrightbigg\n=P/bracketleftbiggV(k)\nM(k)/bracketrightbigg\n,P=/bracketleftbiggB B\nA+λ1A+λ2/bracketrightbigg\n. (23)\nThen,\n/bracketleftbiggV′(k)\nM′(k)/bracketrightbigg\n=P−1MP/bracketleftbiggV(k)\nM(k)/bracketrightbigg\n=/bracketleftbiggλ10\n0λ2/bracketrightbigg/bracketleftbiggV(k)\nM(k)/bracketrightbigg\n⇒/bracketleftbiggV(k)\nM(k)/bracketrightbigg\n=/bracketleftbiggV0(k)eλ1t\nM0(k)eλ2t/bracketrightbigg\n. (24)\nλ1andλ2are eigenvalues:\nλ1=−(A+D)+/radicalbig\n(A+D)2−4AD+4BC\n2, (25)\nλ2=−(A+D)−/radicalbig\n(A+D)2−4AD+4BC\n2. (26)\nWe choose a leading term in each elements with the consideration of th eir coefficients k∼p\nork∼r. Then the coefficients are like\nA∼2νk2, B∼b1km, C∼c2km+2, D∼d2kv+2. (27)– 12 –\nHere4, sinceAD−BC∼kv+4−k2m+2∼0 ask→ ∞, the eigenvalues converge to\nλ1∼0, λ2∼ −2νk2as ‘k’ increases.\nEV(k) andEM(k) are expressed like\n/bracketleftbiggEV(k)\nEM(k)/bracketrightbigg\n=/bracketleftbiggb1kmb1km\n2νk20/bracketrightbigg/bracketleftbiggV0(k)e∼0·t\nM0(k)e−2νk2t/bracketrightbigg\n. (28)\nIf ‘V0’ and ‘M0’ are replaced by EV0(k)∼kvandEM0(k)∼km, we find the saturated\nsolutions:\nEM(k) = 2νk2V0(k)∼2νk2EM0(k)\n2νk2∼km, (29)\nEV(k) =b1kmEM0(k)\n2νk2+b1km/bracketleftbigg\n−EM0(k)\n2νk2+EV0(k)\nb1km/bracketrightbigg\ne−2νk2t∼k2m−2.(30)\nFor complete energy spectra, ‘ m’ is required. But it is difficult to pinpoint a representative\nmagneticpowerspectrumbecause EM(k)isacontinuouslychangingcurve. Schekochihin et al.\n(2004) found ‘ m= 0’, peak of EM(k), but we do not think a peak can be a representative\npower spectrum that drops continuously with the wavenumber ‘ k’. On the other hand,\nLazarian et al. (2004) found ‘ m=−1’ using a simulation with a strong background mag-\nnetic field bext, a filling factor, and balance relation B· ∇B∼ν∇2v. So we infer the\nindex ‘m=−1/2’ for the magnetic scaling factor in a system under the influence of a weak\nbackground magnetic field. Then, from Eq.(30) we get EV(k) ‘k−3’, which matches the sim-\nulation results well. Also this makes ‘ AD−BC∼0’, i.e., ‘λ1∼0’ and ‘λ2<0’ so that\nthe energy spectra become independent of time when they are sat urated. If ‘ m=−1’ is\nchosen, ‘ EV(k)∼k−4’ andEM(k)∼k−1’, coincident with the results of (Cho et al. 2003;\nLazarian et al. 2004). A simple relation E2\nM/EV=k2can be derived in high PrM.\n4. Discussion\nThe analytic and simulation job in this paper are to realize the high PrMplasma state\nin ICM. Magnetic field affects the fluid motion through Lorentz force qv×Bto cause the\nrotational motion of ionized particles around the magnetic field. The effect of magnetic field,\nleading to the anisotropic system, competes with that of collision whic h transfers momen-\ntum to make the system isotropic. In fact the weakly collisional ICM p lasma has few proper\n4The dimensional analysis of Eq.(20) implies 2 m∼v+2 when the system is saturated.– 13 –\nways to constrict the anisotropic tendency. However, if backgro und magnetic field is not\ntoo strong, a system driven by a random isotropic force in large sca les eventually becomes\nisotropic overall although small scale eddies still tend to be anisotro pic under the influence\nof magnetic field.\nSimulation of high PrMtells us some important features. The viscous scale kνis extended\ntoward the much smaller diffusivity scale kη, and nontrivial EVin this extended scale is a\nprerequisite to the growth of EM. For the local and nonlocal energy transfer, besides energy\ngap in kinetic and magnetic eddies, additional specific geometrical re lation between vand\nBis required. For example smaller EM(k) thanEV(k) in larger scales boosts the nonlocal\nkinetic energy transfer from kinetic to magnetic eddies; and, smalle rEV(k) thanEM(k) in\nsmall scale helps the energy transfer from magnetic to kinetic eddie s. However, the energy\ntransfer in magnetic eddies is possible only when B·∇V(nonlocal transfer) or −V·∇B(lo-\ncal transfer) is nontrivial. As a result of all these effects with the v iscous damping, EV∼k−3\nand smoothly changing EMare finally saturated in the subviscous regime. Analytic analysis\nshows the viscous effect ∼νk2coupled with E2\nMinduces this unusual spectrum. (Eq.29, 30).\nFinally, as mentioned the anisotropic features of small scale cannot affect the large scale\ndrivenbytheisotropicforce. However, ifthereisaninstabilityduet otheanisotropicpressure\nin microscale, plasma distribution in the whole system may change. We h ave not discussed\nthe influence of microscale instability due to the anisotropic pressur e including viscosity and\nconductivity on the MHD system in this paper, but we will leave these im portant topics for\nthe future research on ICM.\n5. Acknowledgements\nKP acknowledges support from the National Research Foundation of Korea through\ngrant 2007-0093860. DP acknowledges the Korea Ministry of Educ ation, Science and Tech-\nnology, Gyeongsangbuk-Do and Pohang City for the support of th e Junior Research Group\nat APCTP– 14 –\nA. Appendix\nFor ‘A1’ in Eq.(11), we differentiate this triple correlation term over time to use Eq.(8),\n(9). Then, we have\n/bracketleftbig∂\n∂t+ν/parenleftbig\nk2+p2+r2/parenrightbig/bracketrightbig/angbracketleftbig\nvq(p)vm(r)vi(−k)/angbracketrightbig\nA1=\n/angbracketleftbig/bracketleftbig/parenleftbig∂\n∂t+νk2/parenrightbig\nvi(−k)/bracketrightbig\nvq(p)vm(r)/angbracketrightbig\n+/angbracketleftbig\nvi(−k)/bracketleftbig/parenleftbig∂\n∂t+νp2/parenrightbig\nvq(p)/bracketrightbig\nvm(r)/angbracketrightbig\n+/angbracketleftbig\nvi(−k)vq(p)/bracketleftbig/parenleftbig∂\n∂t+νr2/parenrightbig\nvm(r)/bracketrightbig/angbracketrightbig\n=/angb∇acketleftvvvv/angb∇acket∇ight+/angb∇acketleftvvBB/angb∇acket∇ight... (A1)\nIf we see the first term, for example,\n/angbracketleftbig/bracketleftbig/parenleftbig∂\n∂t+νk2/parenrightbig\nvi(−k)/bracketrightbig\nvj(p)vm(r)/angbracketrightbig\nδp+r,k=/summationdisplay\nj,l/bracketleftbig\nMins(−k)/angb∇acketleftvn(j)vs(l)vq(p)vm(r)/angb∇acket∇ight\n−Mins(−k)/angb∇acketleftBn(j)Bs(l)vq(p)vm(r)/angb∇acket∇ight/bracketrightbig\nδj+l,−k.(A2)\nthedifferentiationgeneratesthefourthordercorrelation. Anot herdifferentiationjustinduces\nthe fifth order correlation. So we need an assumption to close this e quation. It is known\nthat statistically turbulent quantities follow a normal distribution. A nd the fourth-order\nterm/angb∇acketleftuuuu/angb∇acket∇ightcan be decomposed into the combination of second-order correlat ion terms like\nbelow: quasi-normal approximation (Proudman et al. 1954; Tatsum i 1957):\n/angb∇acketleftu1u2u3u4/angb∇acket∇ight=/angb∇acketleftu1u2/angb∇acket∇ight/angb∇acketleftu3u4/angb∇acket∇ight+/angb∇acketleftu1u3/angb∇acket∇ight/angb∇acketleftu2u4/angb∇acket∇ight+/angb∇acketleftu1u4/angb∇acket∇ight/angb∇acketleftu2u3/angb∇acket∇ight. (A3)\nSo if these second order correlation terms are replaced by energy spectrum expressions as\nfollows:\n4πk2/angb∇acketleftvi(k)vq(k′)/angb∇acket∇ight=Piq(k)EV(k)δk+k′,0, (A4)\n4πk2/angb∇acketleftBi(k)Bq(k′)/angb∇acket∇ight=Piq(k)EM(k)δk+k′,0,\n4πk2/angb∇acketleftBi(k)vq(k′)/angb∇acket∇ight=Piq(k)HBV(k)δk+k′,0.\n(Piq=δiq−kikq\nk2)– 15 –\nEq.(A1) can be rewritten like\n/bracketleftbig∂\n∂t+ν/parenleftbig\nk2+p2+r2/parenrightbig/bracketrightbig/angbracketleftbig\nvq(p)vm(r)vi(−k)/angbracketrightbig\nA1=\n2Mins(−k)/summationdisplay\np,r(4πp2)−1(4πr2)−1Pnq(p)Pms(r)[EV(p)EV(r)−HBV(p)HBV(r)]+\n2Mqns(p)/summationdisplay\np,r(4πk2)−1(4πr2)−1Pin(k)Pms(r)[EV(k)EV(r)−HBV(k)HBV(r)]+\n2Mmns(r)/summationdisplay\np,r(4πk2)−1(4πr2)−1Pin(k)Pjs(p)[EV(k)EV(p)−HBV(k)HBV(p)]\n≡Lvv1\niqm(k,p,r;t). (A5)\nHowever, when the fourth-order correlation is decomposed into t he combinations of second\norder terms, the summation of decomposed ones, i.e., right hand sid e of Eq.(A3), can be\nlarger than its actual value. This can cause a negative energy spec trum, which cannot be\nallowed(Ogura1963). So(Orszag1970)introducedaneddydampin gcoefficient µkprofwhich\ndimension is ‘ ∼1/t’. Its more detailed expression (Pouquet et al. 1976) can be contriv ed,\nbut the dimension ‘ ∼1/t’ does not change. We assume it to be sort of a reciprocal of time\nconstant for simplicity in this paper. Then, with a simple integration we can find the third\ncorrelation term:\n/angbracketleftbig\nvq(p)vm(r)vi(−k)/angbracketrightbig\nA1=/integraldisplayt\ne−(ν(k2+p2+r2)+µkpr)(t−τ)Lvv1\niqm(k,p,r;τ)dτ. (A6)\nWe can also calculate the representation of ‘ A3’ in the same way:\n/angbracketleftbig\nBq(p)Bm(r)vi(−k)/angbracketrightbig\nA3=/integraldisplayt\ne−(νk2+µkpr)(t−τ)Lvv3\niqm(k,p,r;τ)dτ. (A7)\nSince ‘A2’ and ‘A4’ are ‘-A1’ and ‘-A3’ respectively, Eq.(11) are\n/parenleftbig∂\n∂t+2νk2/parenrightbig\n/angb∇acketleftvi(k)vi(−k)/angb∇acket∇ight=/summationdisplay\np+r=k2Miqm(k)/bracketleftbigg/integraldisplayt\ne−(ν(k2+p2+r2)+µkpr)(t−τ)Lvv1\niqm(k,p,r;τ)dτ\n−/integraldisplayt\ne−(νk2+µkpr)(t−τ)Lvv3\niqm(k,p,r;τ)dτ/bracketrightbigg\n.\n(A8)\nIfLvv1\niqm(k,p,r;τ) orLvv3\niqm(k,p,r;τ) is larger than ( ν(k2+p2+r2)+µkpr)−1or (νk2+µkpr)−1,\nthis equation can be markovianized. Thus, with the definition of a tria d relaxation time Θ( t)– 16 –\n(Frisch et al. 1975) we get\n/parenleftbig∂\n∂t+2νk2/parenrightbig\nEV(k) =\n/summationdisplay\np+r=k4πk2Miqm(k)/bracketleftbigg/parenleftbigg1−e−(ν(k2+p2+r2)+µkpr)t\nν(k2+p2+r2)+µkpr/parenrightbigg\nLvv1\niqm(k,p,r;t)\n−/parenleftbigg1−e−(νk2+µkpr)t\nνk2+µkpr/parenrightbigg\nLvv3\niqm(k,p,r;t)/bracketrightbigg\n≡/summationdisplay\np+r=k4πk2Miqm(k)/bracketleftbigg\nΘννν\nkpr(t)Lvv1\niqm(k,p,r;t)−Θνηη\nkpr(t)Lvv3\niqm(k,p,r;t)/bracketrightbigg\n.\n(A9)\nUsing a trigonometric relation: k·p=kpz,k·r=kry,p·r=−prx(Fig.5(a)) and\ndpdr=2πpr\nkdpdr, we can simplify this expression:\n1\n2/integraldisplay\ndpdrΘννν\nkpr(t)k3\npr(1−2y2z2−xyz)[EV(p)EV(r)−HBV(p)HBV(r)].(A10)\nEq.(13), (14) can be derived using a similar way.\nREFERENCES\nBatchelor, G. K. 1950, Proc. Roy. Soc. London, Ser. A, 201, 405\nBovino, S., Schleicher, D., R., G., & Schober, J., 2013, New J. 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S., 2010, MNR AS, 405, 291\nSchober, J., Schleicher, D., Federrath, C., Klessen, R., & Banerjee , R., 2012, Phys. Rev. E.,\n85, 026303\nTatsumi, T., 1957, Royal Society of London Proceedings Series A, 2 39, 16\nVogt, C., & Enßlin, T., A., 2003, A&A, 412, 373\nYoshizawa, A., 2011,HydrodynamicandMagnetohydrodynamicTur bulentFlows: Modelling\nand Statistical Theory (Fluid Mechanics and Its Applications\nYousef, T., A., Rincon, F., & Schekochihin, A., A., 2007, JFM, 575, 111\nThis preprint was prepared with the AAS L ATEX macros v5.2.– 19 –\n(a) (b)\nFig. 1.—(a)Normalized EV(t)andEM(t)fortheincompressible fluid( PrM→ ∞,resolution\n2563and 5123). (b) Normalized EV(t) andEM(t) for the compressible fluid ( PrM= 75 and\nPrM= 7500, resolution 2883). Time scale is contracted by 10% ( ×0.1).\n(a) (b)\nFig. 2.— Energy spectra of incompressible fluid ( PrM→ ∞) (a)EV(k) andEM(k) in the\nearly time regime ( t≤10.5) (b)EV(k) andEM(k) in 24.9≤t≤32.6. AsEMgrows, large\nscale kinetic energy is transferred to magnetic eddies, but kinetic e ddies in subviscous scale\nreceive energy from the magnetic eddies.– 20 –\n(a) (b)\n(c) (d)\nFig. 3.— (a), (b) Saturated EV(k) andEM(k) with resolution 2563and 5123for the in-\ncompressible fluid. (c), (d) Saturated EV(k) andEM(k) for the compressible fluids with\nresolution 2883. Saturated energy level of PrM= 75 is higher than that of PrM= 7500.\nBottle neck effect appears.– 21 –\n(a) (b)\nFig. 4.— (a) Compensated energy spectrum k3EV(k). (b)k0.5EM(k). The flat reference line\nk0meansk−1/2inEM(k), and slanted line k−0.5is the scaling invariant line k−1.\n(a)\nFig. 5.— The ratio EM/EVof hyper diffusivity increases with resolution. In case of physical\ndiffusivity the equipartition of EMandEVappears.– 22 –\n(a)p+r+k (b)k∼p≫r,k∼r≫p\nFig. 6.—(a)Thesummation ofthreewave numbers shouldsatisfy the condition p+r+k= 0\n(b) Upper triangle is the case of large p(smallr):k∼p≫r,x∼y,z∼1, and lower one\nis for large r(smallp):k∼r≫p,x∼z,y∼1." }, { "title": "1706.09665v1.Resonant_Absorption_of_Axisymmetric_Modes_in_Twisted_Magnetic_Flux_Tubes.pdf", "content": "arXiv:1706.09665v1 [astro-ph.SR] 29 Jun 2017Resonant Absorption of Axisymmetric Modes\nin Twisted Magnetic Flux Tubes\nI. Giagkiozis1, M. Goossens2, G. Verth1, V. Fedun3, T. Van Doorsselaere2\nReceived ; accepted\n1Solar Plasma Physics Research Centre, School of Mathematics and Statistics, University\nof Sheffield, Hounsfield Road, Hicks Building, Sheffield, S3 7RH, UK\n2Centre for mathematical Plasma Astrophysics, Mathematics Depa rtment, KU Leuven,\nCelestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium\n3Department of Automatic Control and Systems Engineering, Unive rsity of Sheffield,\nMappin Street, Amy Johnson Building, Sheffield, S1 3JD, UK– 2 –\nABSTRACT\nIthasbeenshownrecentlythatmagnetictwistandaxisymmetricMH Dmodes\nare ubiquitous in the solar atmosphere and therefore, the study o f resonant ab-\nsorption for these modes have become a pressing issue as it can hav e important\nconsequences for heating magnetic flux tubes in the solar atmosph ere and the ob-\nserved damping. Inthisinvestigation, forthefirst time, we calculat ethedamping\nrate for axisymmetric MHD waves in weakly twisted magnetic flux tube s. Our\naim is to investigate the impact of resonant damping of these modes f or solar\natmospheric conditions. This analytical study is based on an idealized config-\nuration of a straight magnetic flux tube with a weak magnetic twist ins ide as\nwell as outside the tube. By implementing the conservation laws deriv ed by\nSakurai et al. (1991a) and the analytic solutions for weakly twisted flux tubes\nobtained recently by Giagkiozis et al. (2015), we derive a dispersion relation for\nresonantly damped axisymmetric modes in the spectrum of the Alfv´ en contin-\nuum. We also obtain an insightful analytical expression for the damp ing rate\nin the long wavelength limit. Furthermore, it shown that both the long itudinal\nmagnetic field and the density, which are allowed to vary continuously in the\ninhomogeneous layer, have a significant impact on the damping time. G iven the\nconditions in the solar atmosphere, resonantly damped axisymmetr ic modes are\nhighly likely to be ubiquitous and play an important role in energy dissipat ion.\nWe also suggest that given the character of these waves, it is likely t hat they\nhave already been observed in the guise of Alfv´ en waves.\nSubject headings: axisymmetric modes, mhd, resonant absorption– 3 –\n1. Introduction\nInhomogeneities, such as a density variation across a magnetic flux tube, produce a\ncontinuous spectrum of eigenfrequencies. For instance, conside r a straight magnetic flux\ntube of radius reand constant temperature, where the density varies smoothly fr om its\ncenter to its boundary, such that cylindrical surfaces have cons tant density. This means\nthat also the sound and Alfv´ en speeds within every cylindrical surf ace are constant. These\nconcentric cylindrical sheaths comprise the flux tube. Due to the d ifference in characteristic\nspeeds, every surface will have its own eigenfrequency. This resu lts in an infinite set of\neigenfrequencies, a continuum. One of the consequences of this c ontinuum in driven systems\nis resonant absorption, assuming the driving frequency is within the continuum.\nGiven that inhomogeneities are the rule rather than the exception in the solar\natmosphere, resonant absorption is bound to occur there. This h as long been recognized,\nfrom the first suggestion by Ionson(1978) to subsequent studies motivated by advances\nin solar observations, see for example the following works ( Poedts et al. 1989 ,1990;\nRuderman & Roberts 2002 ;Goossens et al. 2002 ;Andries et al. 2005 ;Goossens et al.\n2009;Van Doorsselaere et al. 2009 ;Verth et al. 2010 ;Terradas et al. 2010 ;Antolin et al.\n2015;Okamoto et al. 2015 ) to name but a few. In general, resonant absorption in\nmagnetohydrodynamic (MHD) modes is important for the solar atmo sphere. Some of\nthe many reasons for this are the following. Resonant damping of Alf v´ en waves is a\nnatural and efficient mechanism for energy dissipation of MHD waves in inhomogeneous\nplasmas ( Ionson 1978 ,1985;Hollweg & Yang 1988 ). It can also provide an explanation\nfor the observed loss of power of acoustic modes in sunspots ( Hollweg 1988 ;Sakurai et al.\n1991a,b;Goossens & Poedts 1992 ;Keppens et al. 1994 ), and, it has been shown that it is\nof importance in transverse oscillations (kink mode), see for examp le (Aschwanden et al.\n1999;Nakariakov et al. 1999 ;Ruderman & Roberts 2002 ;Goossens et al. 2002 ). Resonant– 4 –\nAlfv´ en waves can be an energy conduit between photospheric mot ions at the footpoints\nof coronal loops (see for example De Groof & Goossens 2000 ;De Groof et al. 2002 ;\nDe Groof & Goossens 2002 ), and, resonant dissipation plays an important role in the\nobserved damped oscillations in prominences (see Terradas et al. 2008 ;Arregui et al. 2012 ).\nFor an in depth review of resonant absorption in the solar atmosphe re seeGoossens et al.\n(2011).\nSince 1999, when the first post-flare standing mode transverse o scillations were\ndetected using the Transition Region and Coronal Explorer (TRACE ) (Aschwanden et al.\n1999;Nakariakov et al. 1999 ) there has been a growth in studies of resonant absorption\nfor the kink mode. Ruderman & Roberts (2002) produced relations describing the\nexpected damping for coronal loops using the long wavelength and p ressure-less plasma1\napproximations, a result that was previously obtained by Goossens et al. (1992) using the\nconnection formulae derived by Sakurai et al. (1991a,b) for the driven problem and by\nTirry & Goossens (1996) for the eigenvalue problem. Later Goossens et al. (2002) and\nAschwanden et al. (2003) used these results and calculated the expected damping times for\na sequence of observed parameters for coronal flux tubes. Goossens et al. (2002) concluded\nthat for the parameter sample used, resonant absorption can ex plain the observed damping\ntimes well, provided that the density contrast is allowed to vary from loop to loop. Another\nimportant result in this work is that the observed damping does not r equire modification\nof the order of magnitude estimates of the Reynolds number (1014) as suggested by\nNakariakov et al. (1999).Aschwanden et al. (2003) also arrived at the conclusion that,\non average, the theoretical predictions of the damping rate deriv ed byGoossens et al.\n(1992) andRuderman & Roberts (2002), are consistent with observations and suggested\nthat damping times of coronal loops can be used to infer their densit y contrast with the\n1Also referred to as coldplasma approximation.– 5 –\nsurrounding plasma. Coronal flux tubes tend to deform in their middlesection due to\nbuoyancy, effectively resulting in cross-sections that are approx imately elliptical. Ruderman\n(2003) studied the damping of the kink mode in flux tubes with an elliptical cro ss-section\nand found that for moderate ratios of the minor to major semi-axis the difference of the\ndamping rate for resonant absorption compared with flux tubes wit h circular cross-section\nis not very large. Another deviation from the ideal straight magnet ic flux tube is axial\ncurvature. Van Doorsselaere et al. (2004) studied the effect of this curvature and also found\nthat the longitudinal curvature of flux tubes does not significantly alter the damping time\nof kink modes. Progressively the theoretical models for kink oscillat ions have become more\nelaborate, for example, Andries et al. (2005) considered longitudinal density stratification.\nAlso, methods for kink wave excitation have been studied, see for e xampleTerradas (2009).\nThe increased body of observations of kink waves allowed Verwichte et al. (2013) to perform\na statistical study to constrain the free parameters present in t heoretical models of resonant\nabsorption in kink modes.\nIn contrast to this avalanche of theoretical and observational a dvances related to the\nkink mode, resonant absorption for axisymmetric modes has not re ceived much attention.\nOne reason for this is that it was believed that the sausage mode had a long wavelength\ncuto��� (e.g. Edwin & Roberts 1983 ) which suggested that observation of the sausage mode\nwould be quite challenging. Furthermore, it was correctly believed th at for a straight\nmagnetic field, axisymmetric modes could not be resonantly damped. However, it is\napparent, even in early works in resonant absorption (see for exa mpleSakurai et al. 1991a ,b;\nGoossens et al. 1992 ), that for weakly twisted magnetic field axisymmetric modes can and\nare resonantly damped. What was not known until recently, howev er, was that the long\nwavelength cutoff for these modes is also removed in the presence o f weak magnetic twist\n(Giagkiozis et al. 2015 ). Therefore these modes can freely propagate for all wavelengt hs.\nAnd so, at least in principle, these modes should be observable. Addit ionally, recent works– 6 –\nsuggest that magnetic twist and axisymmetric modes are ubiquitous throughout the solar\natmosphere. Therefore, the study of these modes has become q uite relevant and important.\nSome examples of magnetic twist in the solar atmosphere are, flux tu bes emerging from\nthe convection zone (see for example Hood et al. 2009 ;Luoni et al. 2011 ), sunspot rotation\ncan result in twisted magnetic fields ( Brown et al. 2003 ;Yan & Qu 2007 ;Kazachenko et al.\n2009), spicules are observed to have twist ( De Pontieu et al. 2012 ;Sekse et al. 2013 ) as well\nas solar tornadoes ( Wedemeyer-B¨ ohm et al. 2012 ). Lastly observations of axisymmetric\nmodes have been recently reported in Morton et al. (2012) andGrant et al. (2015).\nIn this work, we focus on the resonant absorption of axisymmetric MHD modes in\nweakly twisted magnetic flux tubes. Axisymmetric modes correspon d to modes with\nazimuthal wavenumber m= 0. We accomplish this using the following sequence. First\nwe recall recent results for axisymmetric modes in magnetic flux tub es with weak twist\n(Giagkiozis et al. 2015 ). In that work the longitudinal component of the magnetic field, an d\nthe density were discontinuous across the flux tube boundary. Th is choice was intentional\nas it avoids the MHD continua and simplifies the analysis. However, this also left out\nrelevant physics. Then having as a starting point the setup in Giagkiozis et al. (2015) we\nintroduce an intermediate layer about the flux tube boundary. With in this layer, we allow\nthe magnetic field and density to vary smoothly, resulting in an overa ll continuous profile\nfor the longitudinal magnetic field and density. This in turn allows for t he existence of the\ntwo MHD continua, the slow and Alfv´ en continuum. Next, we assume that the layer that\nconnects the internal and external quantities, is thin, namely we a ssume that ℓ≪rewhere\nℓis the width of the layer and reis the flux tube radius. Then we use the conservation laws,\nand the resulting jump conditions, for the Alfv´ en continuum by Sakurai et al. (1991a), and\nwe derive the resulting complex dispersion relation. We then solve this dispersion relation\nnumerically. Lastly, to better understand the predicted damping t imes we apply the long\nwavelength limit approximation to the resulting complex dispersion rela tion. These simpler– 7 –\nrelations allow us to compare our results with the expected damping f or the kink mode\npredicted using the results by Goossens et al. (1992) andRuderman & Roberts (2002).\nWe conclude this investigation with a statistical analysis of the result ing approximations\nto further understand the necessary conditions for the observ ation of resonantly damped\naxisymmetric modes. The main contributions of this work can be summ arized as follows.\n•For the first time, we uncover a dispersion relation for axisymmetric modes in\nmagnetic flux tubes with internal and external twist, including the r esonance with the\nAlfv´ en continuum. We produce simplified expressions for the frequ ency and damping\ntime in the long wavelength limit, for which the axisymmetric modes are n o longer\nleaky.\n•Given that there are four parameters required for the evaluation of the aforementioned\nrelation, namely density contrast, magnetic field contrast, thickn ess of the\ninhomogeneous layer and magnetic twist, we present a statistical f ramework to infer\nwhat can be drawn from observations.\n•We use this statistical framework and show that the predictions of our theoretical\nmodel are in agreement with observed damping times that are in agre ement with\nobserved damping times of quasi periodic pulsations (QPPs). QPPs a re interpreted\nas axisymmetric modes (sausage modes) ( Kolotkov et al. 2015 ).\nThe plan of this paper is as follows. In Section 2we present the model, and include\nprior theoretical results required for the derivation of the disper sion relation leading to\nresonant absorption. In Section 3, using the jump relations in Sakurai et al. (1991a) we\nderive a dispersion equation. In Section 4we use the dispersion relation derived in Section 3\nto obtain an expression for the damping rate in the long wavelength lim it and then in\nSection5we elaborate on the significance of the results in this work for the ob servation– 8 –\nof axisymmetric modes in the solar atmosphere. Lastly, in Section 6we summarize and\nconclude this work.\n2. Model\nIn this work we assume an idealized cylindrically symmetric magnetic flux tube in\nstatic equilibrium. We employ cylindrical coordinates r,ϕandz, with the zcoordinate\nalong the axis of symmetry of the flux tube. The linearized ideal MHD e quations are,\n(1a) ρ∂2ξ\n∂t2+∇p′+1\nµ0(B′×(∇×B)+B×(∇×B′)) = 0,\n(1b) p′+ξ·∇p+γp∇·ξ= 0,\n(1c) B′+∇×(B×ξ) = 0,\nwhereρ,pandBare the density, plasma kinetic pressure and magnetic field, respec tively, at\nequilibrium, ξis the Lagrangian displacement, p′andB′are the Eulerian variations of the\npressure and magnetic field, γis the ratio of specific heats (taken to be 5 /3 in this work),\nandµ0is the permeability of free space. In what follows an index, i, indicates quantities\ninside the flux tube ( r < ri) while variables indexed by, e, refer to the environment outside\nthe flux tube ( r > re). The inhomogeneous layer has a width equal to ℓ=re−riand it\nis assumed that ℓ≪re. Note that in Giagkiozis et al. (2015),ra, was used to denote the\ntube radius, this is equivalent to rein this work. The model configuration is illustrated\nin Figure 1whenBϕe∝1/r. The quantities ρ,pandBare assumed to have only an\nr-dependence, therefore, the following balance equation must be s atisfied when ℓ= 0,\n(2)d\ndr/parenleftbigg\np+B2\nϕ+B2\nz\n2µ0/parenrightbigg\n=−B2\nϕ\nµ0r.\nThe equilibrium magnetic field is taken to be B= (0,Bϕ,Bz), withBϕi=Sr,\nBϕe=r1+κ\neS/rκandBzi,Bzeconstant. By substituting BϕiandBϕeinto Eq. ( 2) and– 9 –\nFig. 1.— Illustration of the model used in this paper. Straight magnet ic cylinder with\nvariable twist inside ( r < ri) and outside ( r > re) the tube. The region where ri< r < r e\nis the inhomogeneous layer, where the Bzcomponent of the magnetic field and the density\nare varying continuously across this layer. The parameters ρi,piandTiare respectively the\ndensity, kinetic pressure and temperature at equilibrium inside the t ube, i.e. for r < ri. The\ncorresponding quantities outside the tube ( r > re) are denoted with a subscript e. Also,rA\nis the radius at the resonance. The dark blue surface emanating ra dially outwards inside the\ntube represents the influence of Bϕ∝r. The yellow surface outside the tube corresponds\ntoBϕ∝1/rdependence. The dashed red rectangle depicts a magnetic surfac e which would\ncorrespond to a magnetic field with only a longitudinal ( z) magnetic field component. The\ninhomogeneous layer is bounded between riandreand is of width ℓ. Note that the radius\nof the tube with the inhomogeneous layer is re.– 10 –\ndefiningBϕA=Bϕ(re) =Sre, we obtain:\n(3) p(r) =\n\nB2\nϕA\nµ0/parenleftbigg\n1−r2\nr2e/parenrightbigg\n+pe forr≤re,\nr2κ\neB2\nϕA(1−κ)\n2µ0κ/parenleftbigg1\nr2κ−1\nr2κ\ne/parenrightbigg\n+peforr > re,\nwhere,pe, is the pressure at the boundary of the magnetic flux tube and the parameter\nκ→1 corresponds to external twist proportional to 1 /rwhileκ→0 to constant\nexternal twist. Note that although p(r) is continuous, for solar atmospheric conditions\nand for weak magnetic twist (sup( B2\nϕ/B2\nz)≪1) its variation is much smaller than pe\nand therefore can be assumed to be constant ( Giagkiozis et al. 2015 ). However, in the\nmodel used by Giagkiozis et al. (2015) the equilibrium density and the zcomponent of the\nmagnetic field are discontinuous, therefore the Alfv´ en continuum was avoided. Note that in\nGiagkiozis et al. (2015) the equivalent to Eq. ( 3) had a typographical error, (1 −2κ) should\nread (1−κ).\nIn the present investigation both the density and the magnetic field are continuous,\nsee Figure 2, which introduces the slow and fast continua into our model. Specific ally, the\ndensity is assumed to be a piecewise linear function of the following for m,\n(4) ρ(r) =\n\nρi forr < ri,\nρi+r−ri\nℓ(ρe−ρi) forri≤r≤re,\nρe forr > re,\na similar form for the variation in the longitudinal component of the ma gnetic field is\nassumed, namely,\n(5) Bz(r) =\n\nBzi forr < ri,\nBzi+r−ri\nℓ(Bze−Bzi) forri≤r≤re,\nBze forr > re.– 11 –\nNote that the assumption here is that ℓ≪re, so that pressure balance is maintained (see\nEq. (2)). Also note that allowing both the density and the magnetic field to v ary results in\na non-monotonic variation in the Alfv´ en frequency across the inho mogeneous layer as seen\nin Figure 3.\nThe equilibrium quantities depend only on rand therefore the perturbed quantities\ncan be Fourier analyzed with respect to the ϕandzcoordinates, namely,\n(6) ξ,p′\nT∝ei(mϕ+kzz−ωt).\nHere,ωis the angular frequency, mis the azimuthal wavenumber, kzis the longitudinal\nwavenumber, and p′\nTis the Eulerian total pressure perturbation defined as p′+BB′/µ0. Our\nfocus is on axisymmetric modes (sausage waves) and therefore th e azimuthal wavenumber is\ntaken to be m= 0. The Lagrangiandisplacement vector in flux coordinates is ξ= (ξr,ξ⊥,ξ/bardbl)\nwhere,\nξ⊥=Bzξϕ−Bϕξz\n|B|, ξ /bardbl=Bϕξϕ+Bzξz\n|B|, (7)\nassuming Br= 0. Using Eq. ( 6), Eq. (1) can be transformed to the following two coupled\nfirst order differential equations,\n(8a) Dd(rξr)\ndr=C1(rξr)−rC2p′\nT,\n(8b) Ddp′\nT\ndr=1\nrC3(rξr)−C1p′\nT.\n(8c) ρ(ω2−ω2\nA)ξ⊥=ı\n|B|CA,\n(8d) ρ(ω2−ω2\nc)ξ/bardbl=ıfB\n|B|v2\ns\nv2s+v2\nACS,\n(8e) ∇·ξ=−ω2CS\nρ(v2s+v2\nA)(ω2−ω2c)– 12 –\nFig. 2.— Density profile as a function of rin the inhomogeneous layer of the magnetic flux\ntube. Here, riandrearetheradiusatwhichtheinhomogeneousbeginsandendsrespect ively,\nalso,reis the flux tube radius. Lastly, rA, is the radius at the resonance.– 13 –\nand,\n(9a) D=ρ(ω2−ω2\nA)C4,\n(9b) C1=2Bϕ\nµ0r/parenleftBig\nω4Bϕ−m\nrfBC4/parenrightBig\n,\n(9c) C2=ω4−/parenleftbigg\nk2\nz+m2\nr2/parenrightbigg\nC4,\n(9d)C3=ρD/bracketleftbigg\nω2−ω2\nA+2Bϕ\nµ0ρd\ndr/parenleftbiggBϕ\nr/parenrightbigg/bracketrightbigg\n+4ω4B4\nϕ\nµ2\n0r2−ρC44B2\nϕω2\nA\nµ0r2,\n(9e) C4= (v2\ns+v2\nA)(ω2−ω2\nc),\n(9f) CA=gBp′\nT−2fBBϕBzξr\nµ0r, CS=p′\nT−2B2\nϕξr\nµ0r\nwhere,\nv2\ns=γp\nρ, v2\nA=B2\nµ0ρ,\nω2\nc=v2\ns\nv2\nA+v2sω2\nA, ω2\nA=f2\nB\nµ0ρ,\nfB=k·B=m\nrBϕ+kzBz, g B= (k×B)r=m\nrBz−kzBϕ.\nHere,k= (0,m/r,k z) is the wavevector, CAandCSare the coupling functions, vsis\nthe sound speed, vAis the Alfv´ en speed, ωcis the cusp angular frequency and ωAis the\nAlfv´ en angular frequency. Eq. ( 8) was initially derived by Hain & Lust (1958) and later by\nGoedbloed (1971);Sakurai et al. (1991a). The first order coupled ODEs in Eq. ( 8) can be\nreduced to a single second order ODE for ξr,\n(10)d\ndr/bracketleftbiggD\nrC2d\ndr(rξr)/bracketrightbigg\n+/bracketleftbigg1\nD/parenleftbigg\nC3−C2\n1\nC2/parenrightbigg\n−rd\ndr/parenleftbiggC1\nrC2/parenrightbigg/bracketrightbigg\nξr= 0.– 14 –\nThe assumption of axisymmetry ( m= 0) leads to,\nfB=kzBz, g B=−kzBϕ, C A=−kzBϕ/parenleftbigg\np′\nT+2B2\nz\nµ0rξr/parenrightbigg\n.(11)\nAnd therefore,\n(12a) ρ(ω2−ω2\nA)ξ⊥=−ıkzBϕ\n|B|/parenleftbigg\np′\nT+2B2\nz\nµ0rξr/parenrightbigg\n,\n(12b) ρ(ω2−ω2\nc)ξ/bardbl=ıkzBz\n|B|v2\ns\nv2s+v2\nA/parenleftbigg\np′\nT−2B2\nϕ\nµ0rξr/parenrightbigg\n.\nNote that Eq. ( 12) suggests that the solutions for the components of the Lagrang ian\ndisplacement vector are coupled. Coupled in the sense that eliminatio n of one component,\ne.g. by setting it to be identical to zero, has direct implications to the remaining\ncomponents. To see this, consider a solution for which ξr= 0, then by Eq. ( 10),p′\nTmust\nalso be equal to zero and as a consequence of Eq. ( 12a) and Eq. ( 12b) it follows immediately\nthatξ⊥andξ/bardblmust also be identically equal to zero. Namely setting ξr= 0 leads to the\ntrivial solution. Alternatively, let us assume that ξ⊥= 0. In this case, by Eq. ( 12a) the\nfollowing relation must hold,\n(13) p′\nT=−2B2\nz\nµ0rξr.\nThis in turn implies,\n(14) ρ(ω2−ω2\nc)ξ/bardbl=−2ıkzBz|B|\nµ0rv2\ns\nv2\ns+v2\nAξr,\nwhich in general is non-zero. Now, if we assume ξ/bardbl= 0 then,\n(15) p′\nT= 2B2\nϕ\nµ0rξr\nwhich leads to,\n(16) ρ(ω2−ω2\nA)ξ⊥=−2ıkzBϕ|B|\nµ0rξr.– 15 –\nIn the case where Bϕ= 0 then ξ⊥decouples from ξrandξ/bardbl. At this point it is instructive\nto mention the interpretation of the three components of ξin flux coordinates by\nGoossens et al. (2011).Goossens et al. (2011) suggest that ξ⊥is the dominant component\nfor Alfv´ en waves and for low plasma- βthe slow and fast magnetoacoustic waves ξ/bardblandξris\nthe dominant component, respectively. A quick check, by setting Bϕ= 0 in Eq. ( 7), renders\nξ⊥equivalent to ξϕ. This illuminates the connection of ξ⊥with torsional Alfv´ en waves.\nGiagkiozis et al. (2015) solved Eq. ( 10) for weak internal and external magnetic\ntwist, albeit with the density profile assumed piecewise constant. Wit h the help of the\nconservation relations for the Alfv´ en continuum derived by Sakurai et al. (1991a), these\nsolutions, which are for ideal MHD, can be used to produce a dispers ion relation for MHD\nwaves that undergo damping in the continuum. The solutions by Giagkiozis et al. (2015)\nare as follows,\n(17a) ξri(s) =Ais1/2\nE1/4e−s/2M(a,b;s),\n(17b)p′\nTi(s) =AikaDi\nn2\ni−k2\nze−s/2/bracketleftbiggni+kz\nkzsM(a,b;s)\n−2M(a,b−1;s)/bracketrightbigg\n,\nand,\n(18a) ξre(r) =AeKν(krer),\n(18b)p′\nTe=Ae/parenleftbiggµ0(1−ν)De−2B2\nϕAn2\ne\nµ0r(k2z−n2e)Kν(krer)\n−De\nkreKν−1(krer)/parenrightbigg\n.\nM(·) is the Kummer function and K(·) is the modified Bessel function of the second kind\n(Abramowitz & Stegun 2012 ). The solutions in Eq. ( 17a) and Eq. ( 17b) were initially– 16 –\nderived by Erd´ elyi & Fedun (2007). The parameters in Eq. ( 17) and Eq. ( 18) are,\na= 1+k2\nri\n4k2zE1/2, b = 2, (19)\nka=kz(1−α2)1/2, α2=4B2\nϕAω2\nAi\nµ0r2eρi(ω2−ω2\nAi)2, (20)\ns=k2\naE1/2r2, E =4B4\nϕAn2\ni\nµ2\n0r4\neD2\nik2\nz(1−α2)2, (21)\nk2\nr=k2\nz/parenleftbigg\n1−n2\nk2z/parenrightbigg\n, k2\nr=(k2\nzv2\ns−ω2)(k2\nzv2\nA−ω2)\n(v2\nA+v2s)(k2zv2\nT−ω2),(22)\nn2=k2\nzω4\n(ω2s+ω2\nA)(ω2−ω2c), v2\nT=v2\nAv2\ns\nv2\nA+v2s, (23)\nDi=ρi(ω2−ω2\nAi), D e=ρe(ω2−ω2\nAe). (24)\nandνis,\n(25)ν2(κ;r) = 1+2r2κ\neB2\nϕA\nµ2\n0D2er2κ/braceleftbigg\n2r2κ\neB2\nϕAn2\nek2\nz\nr2κ+µ0ρe/bracketleftbig\nω2\nAe(n2\ne(3+κ)\n−k2\nz(1−κ))−(n2\ne+k2\nz)(1+κ)ω2/bracketrightbig/bracerightbigg\n.\nThis function in Giagkiozis et al. (2015) is evaluated for κ= 0, resulting in an exact\nsolution for constant twist outside the flux tube which is also a zero o rder approximation\nfor the external solution when magnetic twist is proportional to 1 /r:\n(26) ν2(0;r) = 1+2B2\nϕA\nµ2\n0D2\ne/braceleftbig\n2B2\nϕAn2\nek2\nz\n+µ0ρe/bracketleftbig\nω2\nAe(3n2\ne−k2\nz)−ω2(n2\ne+k2\nz)/bracketrightbig/bracerightbig\n.\nUsingν=ν(0;r), i.e. constant external magnetic twist, results in solutions, name ly (18a)\nand (18b), that have approximately 5% root mean squared error when comp ared with the\nexact solution corresponding to ν=ν(1;r), that corresponds to external magnetic twist\n∼1/r. For more details see ( Giagkiozis et al. 2015 ).– 17 –\nImposing continuity for the Lagrangian displacement in the radial dir ection and total\npressure continuity across the flux tube,\n(27a) ξri|r=re=ξre|r=re,\n(27b) p′\nTi−B2\nϕi\nµ0rξri/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=re=p′\nTe−B2\nϕe\nµ0rξre/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=re,\nthe following dispersion relation was derived,\nreDe\nkreKν−1(krere)\nKν(krere)=ρiv2\nAϕi/bracketleftbigg1\nk2\nri(ni+kz)2−1\nk2re(n2\ne+k2\nz)/bracketrightbigg\n+(1−ν)De\nk2re−2Di\nk2\nriM(a,b−1;s)\nM(a,b;s),\n(28)\nwherev2\nAϕi=B2\nϕA/µ0ρiandω2\nAϕi=k2\nzB2\nϕA/µ0ρi.\n2.1. Long Wavelength Limit\nThe long wavelength limit of Eq. ( 28) is needed for the approximation of the location\nof the resonant point used in subsequent sections and is obtained a s follows. From Eq.\n(13.5.5) in Abramowitz & Stegun (2012) we have,\n(29) lim\nǫ→0M(a,b−1;s)\nM(a,b;s)= 1,\nfurthermore, rewriting ν2(0;r) as,\n(30) ν2(0;r) = 1+2ρeB2\nϕA\nµ0D2e/parenleftbigg\nω2\nAe/parenleftbigg\n3n2\ne\nk2z−1/parenrightbigg\n−ω2/parenleftbiggn2\ne\nk2z+1/parenrightbigg/parenrightbigg\nk2\nz+4B4\nϕA\nµ2\n0D2en2\ne\nk2zk4\nz,\nbecomes apparent that ν= 1 +O(ǫ2), where ǫ=rekz. Therefore using Eq. (9.6.8) and\n(9.6.9) in Abramowitz & Stegun (2012) we obtain that,\n(31) lim\nǫ→0K0(krere)\nK1(krere)= 0.– 18 –\nUsing Eq. ( 29) and31in Eq. (28) we have,\n(32) 2(ω2−ω2\nAi) =v2\nAϕi/bracketleftbigg\n(ni+kz)2−k2\nri\nk2re/parenleftbig\nn2\ne−k2\nz/parenrightbig/bracketrightbigg\n.\nExpanding the part in square brackets on the right hand side of this equation about ǫ= 0\nleads to,\n(33) (ni+kz)2−k2\nri\nk2re/parenleftbig\nn2\ne−k2\nz/parenrightbig\n= 2ω\n(v2\nAi+v2\nsi)1/2kz+O/parenleftbig\nǫ2/parenrightbig\n.\nUsing this approximation in Eq. ( 32) the positive solution of the dispersion relation Eq. ( 28)\nin the long wavelength limit to first order is,\n(34) ω=1\n2/bracketleftBigg\nω2\nAϕi\n(ω2\nAi+ω2\nsi)1/2+/parenleftbiggω4\nAϕi\nω2\nAi+ω2\nsi+4ω2\nAi/parenrightbigg1/2/bracketrightBigg\n.\nFor notational convenience Eq. ( 34) is rewritten as follows,\nω=ωAih, (35)\nh=1\n2/bracketleftBigg\nq2\ni\n(1+d2)1/2+/parenleftbigg\n4+q4\ni\n1+d2/parenrightbigg1/2/bracketrightBigg\n(36)\nwhere,qi=BϕA/Bziandd=vsi/vAi. Thisωis used as an approximation to the resonance\nfrequency, ω0, in Section 4. Lastly, we should note that given this value for ω0, although\nthe variation of the Alfv´ en speed across the inhomogeneity in the fl ux tube is quadratic (see\nFigure3), sinceωAi< ω0< ωAe, there will only be a single resonance point.\n3. Alfv´ en Continuum\nFor an equilibrium with magnetic twist, such as the model used in this wo rk, the\ntotal pressure perturbation is no longer a conserved quantity an d therefore Eq. ( 27b) and\nEq. (27a) require modification. Sakurai et al. (1991a) derived new conserved quantities for– 19 –\nFig. 3.— An example of Alfv´ en frequency variation across the reson ant layer when Bz=\nBz(r) andρ=ρ(r), forχ=ρe/ρi= 0.1,ζ=Bze/Bzi= 0.35 andℓ/re= 0.2. Herer= 1 is\nthe tube boundary and ωAis the normalised normalized Alfv´ en frequency, the normalization\nis with respect to the internal Alfv´ en frequency, ωAi.– 20 –\nthe Alfv´ en and slow continua. Specifically for the Alfv´ en continuum the conserved quantity\nis,\n(37) CA=gBp′\nT−2fBBϕBzξr\nµ0r.\nUsing this conserved quantity they derived jump conditions forξrandp′\nT, namely a\nprescription on how the radial component of the Lagrangian displac ement and the total\npressure perturbation can vary across the inhomogeneous layer connecting the internal with\nthe external solutions. This prescription then implies, that the follo wing conditions must\nbe satisfied,\n(38) ξri(r)|r=ri+/llbracketξr(r)/rrbracket=ξre(r)|r=re\nand\n(39) p′\nTi(r)|r=ri+/llbracketp′\nT(r)/rrbracket=p′\nTe(r)|r=re,\nwhere /llbracketξr/rrbracketand/llbracketp′\nT/rrbracketare the jump conditions across the resonant layer in the inhomogen eous\nsection of the flux tube, in the radial displacement and total press ure perturbation\n(Sakurai et al. 1991a ). They are given by\n(40) /llbracketξr/rrbracket=−ıπ\n|∆A|gB\nµ0ρ2v2\nACA,\nand\n(41) /llbracketp′\nT/rrbracket=−ıπ\n|∆A|2TBz\nµ0ρ2v2\nArCA,\nwhereT=fBBϕ/µ0=kzBzBϕ/µ0and,\n(42) ∆A=d\ndr/parenleftbig\nω2−ω2\nA(r)/parenrightbig\n.– 21 –\nTaking into account that m= 0 and Bϕ∝ne}ationslash= 0 and Eq. ( 11) the jump conditions, Eq. ( 40)\nand (41), can be written as,\n(43)/llbracketξr/rrbracket=ıπ\n|∆A|kzBϕ\nµ0ρ2v2\nACA\n=−ıπ\n|∆A|k2\nzB2\nϕ\nµ0ρ2v2\nA/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=rA·/braceleftbigg\np′\nT+2B2\nz\nµ0ξr\nr/bracerightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=ri,\nand,\n(44)/llbracketp′\nT/rrbracket=−ıπ\n|∆A|2kzBϕB2\nz\nµ2\n0ρ2v2\nArCA\n=2ıπ\nr|∆A|/parenleftbiggkzBϕBz\nµ0ρvA/parenrightbigg2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=rA·/braceleftbigg\np′\nT+2B2\nz\nµ0ξr\nr/bracerightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=ri.\nGiven that Bz=Bz(r) andρ=ρ(r) in the inhomogeneous layer, see Eq. ( 4), Eq. (5) and\nFigure2, we have,\n(45)∆A=ωA(r)2∆AF\n=ωA(r)2/bracketleftbigg1\nρdρ\ndr−21\nBzdBz\ndr/bracketrightbigg\n.\nObviously when Bzis constant across the inhomogeneous layer,\n(46) ∆A=ω2\nA(r)1\nρdρ\ndr.\nSubstituting Eq. ( 43),44and45into Eq. ( 38) and Eq. ( 39) we obtain the dispersion relation\nfor axisymmetric MHD waves that undergo resonant absorption in t he Alfv´ en continuum of\nfrequencies due to the twist in the magnetic field:\n(47) DAR(ω,kz)+ıDAI(ω,kz) = 0,\nwhere,\n(48)DAR= 2Di\nk2\nriM(a,b−1;si)\nM(a,b;si)−2ρini(ni+kz)v2\nAϕi\nk2\nri\n+riDe\nkreKν−1(krere)\nKν(krere)−ri\nrek2re/braceleftbig\n(1−ν)De−2ρen2\nev2\nAϕe/bracerightbig\n,– 22 –\nand,\n(49)DAI=π\nρ|∆AF|v2\nAϕ\nv4\nA/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=rA /bracketleftbigg2\nk2\nri/parenleftbigg\nDiM(a,b−1;si)\nM(a,b;si)−ni(ni+kz)ρiv2\nAϕi/parenrightbigg\n+2ρiv2\nAi/bracketrightbigg\n/bracketleftBigg\n2B2\nz\nµ0r/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nrA+1\nrek2\nre/braceleftbig\n(1−ν)De−2ρen2\nev2\nAϕe/bracerightbig\n−De\nkreKν−1(krere)\nKν(krere)/bracketrightBigg\n.\nIn these equations the following definitions were used,\nv2\nAϕA=B2\nϕA\nµ0ρA, v2\nAA(rA) =B2\nzA\nµ0ρA,(50)\n|∆AF(rA)|=1\nℓ/vextendsingle/vextendsingle/vextendsingle/vextendsingleρe−ρi\nρA−2Bze−Bzi\nBz(rA)/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (51)\nwhereρA=ρ(rA),vAA=vA(rA) andBzA=Bz(rA). To find the radius at the resonance\npoint, namely the radius where vA(r) =v0(=ω0/kz)2, we can express rAas a convex\ncombination of the radius riand the width of the inhomogeneous layer ℓsincerAmust be\nwithin the interval ( ri,re). Therefore we can write rA=ri+wℓ, wherew∈(0,1). Now\nwe have transformed the problem of solving for rAto a problem where we have to solve\nforw, the convex combination parameter. Given this formulation for rAand Eq. ( 4) and\nEq. (5) we can write BzA=Bz(rA) =Bzi+w(Bze−Bzi) andρA=ρ(rA) =ρi+w(ρe−ρi).\nEquipped with these definitions, the equation that we need to solve t o findwbecomes,\n(52) v2\nAA=(Bzi+w(Bze−Bzi))2\nµ0(ρe+w(ρe−ρi))=v2\n0\nUsing the definitions χ=ρe/ρi,ζ=Bze/BziEq. (52), simplifies to\n(53) v2\nAi(1+w(ζ−1))2\n1+w(χ−1)=v2\n0.\nThis equation is solved for win the next section.\n2For the definition of ω0see Eq. (35).– 23 –\n4. Long Wavelength Limit - Alfv´ en Continuum\nE1\nTaking the long wavelength limit, ǫ≪1, of Eq. ( 48) and Eq. ( 49) and using Eq. ( 29),\nEq. (31) then, Eq. ( 48) and Eq. ( 49) reduce to,\n(54) DAR=ω2−ω2\nAi\nk2\nri−n2\ni\nk2\nriv2\nAϕi+χn2\ne\nk2rev2\nAϕe,\n(55)DAI=π\nre/bracketleftBigg\nπ\n∆AFv2\nAϕ\nv2\nA/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=rA/bracketrightBigg/bracketleftbigg\n1−ρe\nρAn2\ne\nk2\nrev2\nAϕe\nv2\nAA/bracketrightbigg\n/bracketleftbiggω2−ω2\nAi\nk2\nri−n2\ni\nk2\nriv2\nAϕi+v2\nAi/bracketrightbigg\n.\nThese equations can be solved if we allow a complex frequency ω=ωr+ıγA, and\nwhenγA≪ωrwe can obtain the damping rate, γAin the Alfv´ en continuum frequencies\n(Goossens et al. 1992 ) to second order is given by,\n(56) γA=−DAI(ω0)/parenleftBigg\n∂DAR\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=ω0/parenrightBigg−1\n.\nThis equation results in an expression that is difficult to interpret, an d for this reason, given\nthat we seek an expression for the damping rate in the long waveleng th limit we expand it\nin a series about ε= 0 where ε=rekz. This expansion results in,\n(57) γA=ω0π\nZℓ\nreρA\nρiB2\nϕA\nB2\nzA/parenleftbigg\n1+B2\nϕA\nB2\nzA/parenrightbigg/parenleftbigg\n1+B2\nϕA\nB2\nzi/parenrightbigg\n+O/parenleftbig\nε2/parenrightbig\n,\nwhere\n(58) Z=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(χ−1)−2ρA\nρiBzi\nBzA(ζ−1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nE1NOTE TO EDITOR: Figure 4and Figure 5should appear side-by-side in the text.– 24 –\nFig. 4.— Contour map of the damping time τd(see Eq. ( 66)) as a multiple of the period\nτ, plotted for density contrast in the interval χ∈(0,ζ2/h2), and longitudinal magnetic field\ncontrast in the interval ζ∈(0,1). The remaining parameters in ( 66) are set as follows:\nℓ/re= 0.1 andBϕA/BzA= 0.15. The redline marks ζ2/h2above which the resonance\nfrequency is outside of the continuum. The gray region in this plot de notes damping times\nof 30 and above.– 25 –\nFig. 5.— Contour map of the ratio τd(χ,ζ) versusτd(χ,1), see (69). The density contrast\nis allowed to vary in the interval χ∈(0,ζ2/h2), and longitudinal magnetic field contrast in\nthe interval ζ∈(0,1). Theredline is the same as in Figure 4, whilst values within the gray\nregion correspond to ratios larger than 4.– 26 –\nNow, in this investigation we assume weak magnetic twist ( q=BϕA/BzA≪1) and\ntherefore,\n(59)B2\nϕA\nB2\nzA/parenleftbigg\n1+B2\nϕA\nB2\nzA/parenrightbigg/parenleftbigg\n1+B2\nϕA\nB2\nzi/parenrightbigg\n=B2\nϕA\nB2\nzA+O(q4),\nand so Eq. ( 57) can be simplified to,\n(60) γA=ω0π\nZℓ\nreρA\nρiB2\nϕA\nB2\nzA.\nHereω0is approximated by ( 34), i.e.ω0≈ωAih, and the radius at the resonance point, rA,\nis obtained analogously to Eq. ( 53), by solving,\n(61)(1+w(ζ−1))2\n1+w(χ−1)=h2.\nThere are two cases to be considered. First, when ζ= 1, that is Bzi=Bze, and assuming\nχ∈(0,1/h2), the solution for wis\n(62) w=h2−1\nh2(1−χ),\nwhen 1/h2< χ <1 in this case the resonant point is outside of the continuum and ther e\nis no resonant absorption and in the limit χ→1 the external and internal Alfv´ en speeds\nbecome equal and there are no propagating waves either. The sec ond case is for values of\nζ∈(0,1) andχ∈(0,ζ2/h2) for which the admissible solution is,\n(63) w=2(1−ζ)+h(4(ζ−1)(ζ−χ)+h2(χ−1)2)1/2+h2(χ−1)\n2(1−ζ)2.\nWhenχ > ζ2/h2, similarly to the first case there is no resonant absorption since the\nresonance frequency ( ω0) is outside of the continuum. For ζ2/h2< χ < ζ2there exist\nundamped propagating waves, however, when χ > ζ2the external Alfv´ en speed is smaller– 27 –\nthan the internal and no waves propagate. Lastly note, that in th is investigation we assume\nthatBzi≥Bzeand therefore ρi=ρehas no admissible solution for wwhenBzi=Bze.\nNow, when Bzis assumed to be constant, i.e. Bzi=Bze=Bz, using Eq. ( 62),ρA/ρi,\nBzA/BziandZsimplify to,\nρA\nρi=1\nh2,BzA\nBzi= 1, Z =|1−χ|, (64)\nresulting in γA(Eq. (60))\nγA=ω0\nh2π\n|1−χ|ℓ\nreB2\nϕA\nB2\nzA=ωAi\nhπ\n|1−χ|ℓ\nreB2\nϕA\nB2\nzA. (65)\nTo obtain the damping time normalized by the period of the wave, we us e a typical\nwavelength kz=π/L, whereLis the characteristic length of the tube, the associated period\nisτ= 2L/hvAi(see Eq. ( 35)) the damping time (1 /γA) for modes in the continuum as a\nmultiple of the wave period is,\n(66) τd=Z\n2π2re\nℓρi\nρAB2\nzA\nB2\nϕAτ,\na contour map of this equation for ζ∈(0,1) andχ∈(0,ζ2/h2), can be seen in Figure 4.\nWhenBzi=Bzethe damping time becomes,\n(67) τd=h2|1−χ|\n2π2re\nℓB2\nzA\nB2\nϕAτ.\nThe long wavelength limit approximation of the damping rate γAin Eq. (57), is accurate to\n≈10−6atkzre= 1 when compared with the numerical solution of the dispersion relat ion\nin Eq. (47). This accuracy is better than 10−6forkzre<0.1 and is calculated using the\nmaximum of the root mean squared error (RMSE),\n(68) RMS Error =/parenleftBigg\n1\nN−1N/summationdisplay\ni=1/parenleftbiggγA−ˆγA\nγA/parenrightbigg2/parenrightBigg1/2\n.– 28 –\nIn this equation, γAis the numerically calculated damping rate, ˆ γAis the theoretical\napproximation in Eq. ( 57) andNis the number of samples. For this error estimate we used\n104samples in the parameter space ( χ,ζ,ℓ/r e,Bϕ/Bz), uniformly distributed3.\nWorks investigating resonant absorption in the context of solar at mospheric conditions\ntend to consider solely a radial non-uniformity in either the magnetic field or density.\nHowever, accounting for radial variation in both the magnetic field a nd density can lead to\nsignificant variation in the estimated damping times. The ratio of Eq. ( 66) over Eq. ( 67) is,\n(69)τd(χ,ζ)\nτd(χ,1)=Z\n|1−χ|ρi\nρA1\nh2\nand in Figure 5a contour map is shown for ζ∈(0,1) andχ∈(0,ζ2/h2).\nIt can be seen from Figure 4that the behavior of the damping rate with respect to\nchanges in the density contrast is in some regions exactly the oppos ite to that for the kink\nmode (Goossens & Poedts 1992 , see for example). Namely, in a roughlytriangular region\nin Figure 4the damping rate is proportional to ∼1/χ, in contrast to the kink mode where\nthe damping rate is proportional to ∼χ. Similar behavior has been been shown to exist\nin the leaky regime for sausage modes ( Vasheghani Farahani et al. 2014 ). The factor in\nEq. (66) that determines this behavior is Zρi/ρA. We approximate the local minimum in\ntheχdirection by evaluating the partial derivative of Zwith respect to χ,\n(70)∂Z\n∂χ=−2ζ+h2(χ−1)+2\nh/radicalbig\n4(ζ−1)(ζ−χ)+h2(χ−1)2,\nwhich is subsequently equated to 0. From this we obtain a relation χ= (2/h2)ζ+bandbis\nidentified by noting that at ζ= 1, the maximum value for χis 1/h2, thus the approximation\n3Since the parameter space is not a hypercube, e.g. see Figure 5, we used rejection sam-\npling for invalid parameter combinations until the desired number of s amples was achieved.– 29 –\nis,\n(71) χ=2\nh2ζ−1\nh2.\nAs the remaining terms in Eq. ( 66) do not vary with χandζ(note the ratio Bϕ/Bzis\nheld fixed) this approximation holds for all valid parameters. This app roximation allows us\nto estimate in which regime a specific parameter combination is. Namely , for parameter\ncombinations that are below the line described by Eq. ( 71), for increasing density contrast\n(χ↓), damping will be slower ( τd↑). For parameter combinations that result in points\nabove this line, increasing the density contrast ( χ↓) results in decreasing damping time\n(τd↓), namely waves will decay faster. This is illustrated in Figure 4as a yellow line (( 71))\nand the exact inflection points are marked with a green line.\nGiven the form of Eq. ( 66), and especially that of Eq. ( 67), a comparison with\nprevious results for the kink mode is in order, particularly the expre ssion for the damping\nrate obtained by Goossens et al. (1992) and later by Ruderman & Roberts (2002). In\n(Ruderman & Roberts 2002 ), and Equation (73) in that work, using the notation in this\nwork, reads as follows,\n(72) τd=2\nπre\nℓ1−χ\n1+χτ.\nThe relative magnitude of the damping time shown in Eq. ( 72) and Eq. ( 67) is,\n(73)τd,Axisymmetric\nτd,Kink=h2\n4π(1−χ)2\n1+χB2\nzA\nB2\nϕA.\nIt is evident that there exists a region in the parameter space of ( χ,Bϕ/Bz) for which\nτd,Axisymmetric is smaller when compared with τd,Kink, however, this comparison is given\nhere just as a reference and caution should be exercised in its inter pretation since the\ndamping, τd,KinkinRuderman & Roberts (2002) was calculated for the kink mode without\nmagnetic twist. It is possible that magnetic twist amplifies dissipation in the kink mode,– 30 –\nand therefore still, dissipation for the kink mode may be larger than t hat of axisymmetric\nmodes.\n4.1. Numerical Solution of Dispersion Relation in the Alfv´ en Continuum\nWe have solved Eq. ( 47) numerically, using ω0obtained in Eq. ( 34) as an initial point\nin the solver. Additionally, by means of investigating whether anothe r solution exists, we\nsolved the dispersion relation again with a random ω0in the range ( vAi,vAe). The solutions\nand their associated damping rates can be seen in Figure 6. It is interesting that there\nexists another solution in the long wavelength limit that we could not ob tain from our\nanalysis in Section 4. However, given that for this solution τd≪τ, it is unlikely that this\nmode will be observed.\nNow, it has been shown that the singularity about the resonance po int atrAis\nlogarithmic for ξr(ln(|r−rA|)) and 1/(r−rA) forξ⊥, so the dynamics will be governed by\nξ⊥sinceξr/ξ⊥→0 asr→rAand therefore ξ⊥≫ξrein the neighbourhood of the resonant\npoint (Poedts et al. 1989 ;Sakurai et al. 1991a ). Also the ξrcomponent provides its energy\nto the resonant layer ( Goossens et al. 2011 ) and therefore the characteristic expansion and\ncontraction of axisymmetric modes will be reduced. These facts, a long with the proximity\nof the solution corresponding to the long wavelength limit approximat ion to Section 4to\nthe internal Afv´ en speed suggest that these waves would appea r in observations to have\nproperties similar to Alfv´ en waves. Given that pure Alfv´ en waves r equire∇ ·ξto be\nidentically zero, and, a driving mechanism that is solely torsional, we ar gue that observed\nAlfv´ enwaves are much more likely to be axisymmetric waves, as these do not have these\nstrict requirements. In Figure 6panels(a)through(d)show solutions for different values\nofχwhile panels (e)through (h)solutions are shown when qis allowed to vary. The\ndamping time for the solution for which we have an analytical approxim ation (see panels– 31 –\nFig. 6.— Numerical solutions of the dispersion equation Eq. ( 47) forχ=\n{0.1,0.2,0.3,0.4,0.5},q= 0.15,ζ= 1 and ℓ/re= 0.1 for panels (a)to(d)(χ=ρe/ρi,ζ=\nBze/Bzi,q=BϕA/BzA) andχ= 0.2,q={0.1,0.1375,0.175,0.2125,0.25},ζ= 1 and\nℓ/re= 0.1 for panels (e)to(h). The panels (a),(c),(e)and(g)depict the normalized\nphase velocity and the panels (b),(d),(f)and(h)the corresponding normalized damping\nrates. The bottom panel shows a logarithmic plot of the damping time versus magnetic twist\nfor different values of kzre. All solutions have been obtained numerically by solving Eq. ( 47).– 32 –\n(c)) and(d)) increases ( τd↑) for increasing density contrast ( χ↓) while the other solution\nexhibits the opposite behaviour (see panels (a)and(b)) namely τd↓forχ↓. However,\nthe damping time for both solutions decreases ( τd↓) for increasing magnetic twist q↑. The\nbottom panel of Figure 6shows a different view of the damping times as a function of the\nmagnetic tiwst ( q) shown in panels (f) and (h) at kzre={0.1,0.15,0.2,0.25,0.3}. From\nthis view it can be seen that the solutions in (e) are much more sensitiv e to variations in\nthe mangetic twist when compared with the solutions in panel (g). Th is sensitivity, in\ncombination with the fact that for extremely small twist the sausag e cut-off is reintroduced\n(Giagkiozis et al. 2015 ), means that this mode will be observable for a very small interval\nof magnetic twist. The mode shown in panels (c) and (d) does not pre sent this difficulty,\nand therefore we expect that observation of this mode is more likely . In both cases the\nsolution corresponding to the analytic approximation remains very c lose to the internal\nAlfv´ en speed which is equal to 1 in Figure 6. Sinceω0from Eq. ( 35) depends on q,BzA\nand the internal sound speed, these modes will appear to have a st rong Alfv´ en character\nfor virtually all valid parameter combinations. Lastly, krcan be likened to the wavenumber\nin the radial direction, and, since in the long wavelength limit kris proportional to kz, as\nkzincreases, the wavelength in the radial direction decreases and co uples with the thin\ninhomogeneous layer more closely and therefore more energy per w avelength is absorbed\nand thus the damping time is reduced (see Figure 6).\n5. Connection to Observations\nE2\nReports of observations of axisymmetric modes (sausage modes) are increasing\nE2NOTE TO EDITOR: Figure 7and Figure 8should appear side-by-side in the text.– 33 –\nFig. 7.— Contour map of the estimated probability (see Eq. ( 74)) that an axisymmetric\nmode can be observed to have a normalized damping time ¯ τdin the range (1 ,3), for a given\ncombination of ( ζ,q), i.e. magnetic field contrast and twist respectively. The free para meters\nareζ∈(0.35,1) andq=BϕA/BzA∈(0,0.3) and the integration parameters are χ∈(0.5,1)\nandℓ/re∈(0.1,0.5). The white region represents 0 probability.– 34 –\nFig. 8.—Contour mapoftheestimated probabilityforanaxisymmetric modetobeobserved\nto have a normalized damping time ¯ τdin the range (1 ,3) for a point in ( ζ,χ), i.e. magnetic\nfield and density contrast respectively. The free parameters are ζ∈(0,1) andχ∈(0,1) and\nthe integration parameters are q=BϕA/BzA∈(0,0.3) andℓ/re∈(0.1,0.5). Similarly to\nFigure7the white region in this map represents an estimated probability of 0 o f observing\nresonantly absorbed axisymmetric modes for the particular set of parameter combinations.– 35 –\nin frequency in the recent literature. For example quasi-periodic pu lsations in\nsolar flares are believed to be associated with the kink and sausage m ode (see for\nexample Van Doorsselaere et al. 2011 ;Nakariakov & Zimovets 2011 ;Nakariakov 2012 ;\nDe Moortel & Nakariakov 2012 ;Kolotkov et al. 2015 ). Even more interestingly some of\nthese pulsations appear to have periods in the interval (15 ,100) seconds which could be\nconsistent with the results in the present investigation if the length -scale of these pulsations\nis on the same order as the length-scale of coronal flux tubes ≈100Mm. Furthermore\nthe results by Morton et al. (2012) suggest that axisymmetric modes are ubiquitous and\nthat they appear to be coexistent with kink modes. This coexistenc e further supports the\nargument by Arregui et al. (2015);Arregui & Soler (2015);Arregui(2015) that Bayesian\nanalysis is an essential tool for the identification of the likelywave modes present in\nobservations as well as a more systematic way for the appropriate model selection. The\nuncertainty in determining the parameters for the kink mode led Verwichte et al. (2013) to\nperform a statistical analysis as a way to narrow the range of their values. This departure\nfromcertainty and convergence towards probabilistic inference models for solar o bservations\nis, in our view, long overdue.\nHowever, despite this increase in interest in axisymmetric modes, th e relation that\napproximates their expected damping rate, see Eq. ( 66), requires knowledge of four\nparameters. Namely, the density and magnetic field contrast, the relative magnetic twist\nand the ratio of the thickness of the inhomogeneous layer versus t he tube radius, i.e.\n(χ,ζ,q=BϕA/BzA,ℓ/re). In contrast to the large body of observational evidence for th e\nkink mode, observations of sausage waves are relatively scarce. T his makes impossible\nan analysis similar to Verwichte et al. (2013) for these modes. Therefore, we adopt a\ndifferent approach, a probabilistic approach which is related to the u se of Bayesian inference\nsuggested by Arregui et al. (2015).– 36 –\nAs a first step towards improving this situation we provide a way to es timate the\nprobability that an observed sausage wave has a damping rate within a specified range,\ngiven that, one or more of the four parameters in Eq. ( 57) are known. The assumptions\nrequired for the validity of this estimate are the following:\n•The four parameters in Eq. ( 57) are independent, i.e. no parameter is a function of\nthe others.\n•The likelihood of any combination in the parameter space is the same. T hat is to say\nthat there exists no preferred combination of parameters.\nThese assumptions are difficult to prove, especially given that there exist no statistical\nanalyses of the properties of sausage waves and reliable estimates of all four parameters.\nSince we do not know if there is, in fact, a set of preferred paramet ers, these assumptions\nare required for an unbiased estimate. Acknowledging these uncer tainties, we make a\nfirst attempt in identifying the probability predicted by our model th at a wave with the\ncharacteristics described in this investigation is resonantly damped in the long wavelength\nlimit with a damping rate given by Eq. ( 66) for a given parameter combination.\nThe aforementioned probability can be estimated as follows. First, w e identify the\nparameters for which reasonably good estimates are available. The se parameters we refer\nto asfreeparameters denoted by f. The remaining parameters we refer to as integration\nparameters and are denoted by i. Subsequently, a domain is defined for the integration\nparameters. Then the probability of the damping rate being within th e open interval ( a,b)– 37 –\nis given by,\nP(a,b;f1,...,f n) =/integraltext\nCdi1...di4−nw(i1,...,i 4−n)I¯τd>a,¯τda,¯τda,¯τda,¯τda,¯τd0. Hereafter,\nthe superscript ”(0)” indicates the quantities at zero bias voltage . The magnetization in the\nreference layer is fixed to align in the positive zdirection.\nThe energy density of the FL is given by [26]\nE(mx,my,mz) =1\n2µ0M2\ns(Nxm2\nx+Nym2\ny+Nzm2\nz)\n+Ku(1−m2\nz)−µ0Msm·Hext, (1)\nwhere the demagnetization coefficients, Nx,Ny, andNz, are assumed to satisfy Nz≫Ny>\nNx.µ0is the vacuum permeability and Msis the saturation magnetization of the FL. The\nindex of the IP shape-anisotropy field is given by H(IP)\nk=Ms(Ny−Nx) [27].Kuis the\nuniaxial anisotropy constant. The value of Kucan be controlled by applying a bias voltage,\nV, through the VCMA effect, as shown in Fig. 1(b). Keffrepresents the effective anisotropy\nconstant Keff=Ku−(1/2)µ0M2\ns(Nz−Nx), andK(+V)\neffindicates the value of Keffduring\nthe voltage pulse.\nThe magnetization dynamics are simulated using the following Langevin equation [28]:\n(1+α2)dm\ndt=−γ0m×{(Heff+h)+α[m×(Heff+h)]}, (2)\nwheretis time,γ0is the gyromagnetic ratio, and αis the Gilbert-damping constant. h\nrepresents the thermal-agitation field satisfying the following relat ions:/an}bracketle{thι(t)/an}bracketri}ht= 0 and\n/an}bracketle{thι(t)hκ(t′)/an}bracketri}ht= [2αkBT/(γ0µ0MsΩ)]δικδ(t−t′), where /an}bracketle{t/an}bracketri}htrepresents the statistical mean,\nι,κ=x,y,z,kBis the Boltzmann constant, Tis the temperature, Ω represents the volume\n4of the FL, and δικis Kronecker’s delta. Heffis the effective magnetic field, defined as\nHeff=−1\nµ0Ms∂\n∂mE. (3)\nThe initial state of the simulation is prepared by relaxing the magnetiz ation from the\nequilibrium direction on the upper hemisphere ( mz>0) atKeff=K(0)\nefffor 10 ns. Then,\nthe magnetization dynamics are calculated while applying the voltage p ulse for a duration\noftp. During the pulse, Keffis reduced to K(+V)\neffthrough the VCMA effect as shown in Fig.\n1(b). After the pulse, the anisotropy constant rises to the initial value ofKeff=K(0)\neff. The\nsuccess or failure of switching is determined by the sign of mzafter 10 ns of relaxation.\nIII. RESULTS\nA. Magnetization dynamics in heavily damped precessional switching\nFigure 2(a) shows a typical example of a magnetization trajectory during heavily damped\nprecessional switching in the FL of the elliptical-cylinder MTJ at T= 300 K. We assume\nthatHext= 400 Oe, Ms= 1400 kA/m, α= 0.20,K(0)\neff= 70 kJ/m3, andK(+V)\neff= 10\nkJ/m3. The volume of the elliptical FL is assumed to be Ω = πrxryd= 123150 nm3,\nwhererx(ry) is half the length of the major (minor) axis of an ellipse with the aspec t ratio\nAR=rx/ry= 3, and d= 2 nm is the thickness of the FL. The demagnetizing constants of\nthe FL are Nx= 0.00535,Ny= 0.02574,Nz= 0.96891 [29], which give an IP anisotropy\nfield ofH(IP)\nk= 359 Oe.\nThe magnetization dynamics of heavily damped precessional switchin g in the elliptical\nFL are qualitatively the same as those in the circular FL [20, 21]. Star ting from the initial\nstate on the upper hemisphere ( mz>0), the magnetization precesses around the external\nmagneticfieldandrelaxestowardtheequilibriumdirectioninthelowerh emisphere( mz<0).\nThe temporal evolution of mx,my, andmzis shown in Fig. 2(b). It takes less than 2 ns\nto minimize mz. Aftermzis minimized, the magnetization does not return to the upper\nhemisphere but precesses around the energy minima on the lower he misphere. When the\nvoltage is turned off at any time after 2 ns, the magnetization relaxe s toward the equilibrium\ndirection to complete switching.\n5(a)\nmzmy\nmx-1 0 1-1 01 -1 10(b)\nt (ns)\nm\nx, m\ny, m\nz1\n-1 0\n10 8 6 4 2 0mx\nmzmy\nFIG. 2. (a) A typical example of a magnetization trajectory d uring heavily damped precessional\nswitching in the FL of the elliptical-cylinder MTJ at T= 300 K. (b) The temporal evolution of\nmx,my, andmz.\nB. Comparison of WER between circular and elliptical-cylinder MTJs\nThe WER of heavily damped precessional switching depends strongly on the shape of\nthe MTJ. Figure 3 shows the WER of the elliptical-cylinder MTJ (blue) an d that of the\ncircular-cylinder MTJ (red) reported in Ref. [20]. At tp= 10 ns, the WER of the elliptical-\ncylinder MTJ is 3 .1×10−6which is about 2 orders of magnitude lower than that of the\ncircular-cylinder MTJ (2 .1×10−4).\nAll of the parameters of the elliptical-cylinder MTJ are the same as th ose in Fig. 2. The\nvolume of the FL of the circular-cylinder MTJ is the same as that of th e elliptical-cylinder\nMTJ, i.e., Ω = πrxryd= 123150 nm3withrx=ry= 140 nm and d= 2 nm, which is also\nthe same as the FL in Ref. [20]. The demagnetization coefficients of th e circular FL are\nNx=Ny= 0.01325,Nz= 0.97350 [29] which yield H(IP)\nk= 0 Oe,α= 0.17, andK(+V)\neff= 33\nkJ/m3. The other parameters are the same as those of the elliptical FL.\n6Considering the temperature increase in some computing systems [3 0], we calculate the\nWER at tp= 10 ns and the temperature as 80◦C (T= 353 K). In the circular-cylinder\nMTJ, the WER at tp= 10 ns is 6 .1×10−4. In the elliptical-cylinder MTJ, the WER at\ntp= 10 ns is 1 .9×10−5. In both MTJs, the WER at tp= 10 ns increases, but the WER is\nstill less than 10−3[19].\nWe also conduct simulations adding the pulse-rise time ( tr) and the pulse-fall time ( tf)\n[31] to the parameters used in Fig. 3. In the circular-cylinder MTJ, t he WER at tp= 10 ns\nis insensitive to the introduction of tr= 70 ps, but it increases to 1 .7×10−2attr= 200 ps.\nIn the elliptical-cylinder MTJ, the WER at tp= 10 ns is insensitive to the introduction of\ntr= 40 ps, but it increases to 1 .3×10−2attr= 200 ps. To tr, the elliptical-cylinder MTJ\nis more sensitive than the circular-cylinder MTJ. To tf, for both the circular-cylinder MTJ\nand the elliptical-cylinder MTJ, the WER is insensitive even at tf= 1 ns.\nNote that, in practice, including the external IP magnetic field, whic h is perpendicular\nto the IP shape-anisotropy field, may be challenging on a chip. Compe tition between the\nfields can lead to nonuniform static distribution of the magnetization within the bit. In\naddition, the large size assumed in Figs. 2 and 3 may make the switching nonuniform and\nthedynamics might befarfromasingle-domainprecession asconside red inthemodel. Thus,\nwe conduct micromagnetic simulations, and the results are describe d in Appendix A. The\nresults support the validity of our analyses.\nEven in the case of smaller size, there remain technological challenge s. The elliptical\ngeometry is difficult to scale to small bit dimensions and increases bit-t o-bit variations\ncompared to the circular shape.\nIV. DETAILED ANALYSES OF MAGNETIZATION DYNAMICS\nBefore investigating the cause of the reduction in the WER, we cond uct detailed analyses\nof the magnetization dynamics in the elliptical FL. To save computatio nal time, we analyze\nthe smaller system with Ω = S×d= 15708 nm3. Regardless of ARranging from 1 to\n15, the area and the thickness of the FL are assumed to be S= 502πnm2andd= 2 nm,\nrespectively. Unless otherwise noted, Ms= 1400 kA/m , K(0)\neff= 200 kJ/m3,AR= 5, and\n(Nx,Ny,Nz) = (0.0075,0.0745, 0 .9180) are assumed.\n7t (ns) 0 2 4 6 8 101010-610-510-410-310-210-1100WER circular F L\nelliptical F L\n-7\np\nFIG. 3. The tpdependence of the write-error rate (WER). The red squares co nnected by red lines\nrepresent results for the circular FL. The blue circles conn ected by blue lines represent results for\nthe elliptical FL.\nA. Equilibrium direction of magnetization at T=0 andV=0\nTheequilibriumdirectionofmagnetizationat T= 0andV= 0(m(0))isobtainedbymin-\nimizingE. In this subsection and the next, we calculate m(0)and analyze the magnetization\ndynamics using the dimensionless energy density, ε, defined as follows [26]:\nε(mx,my,mz) =1\n2(Nxm2\nx+Nym2\ny+Nzm2\nz)\n+κ(1−m2\nz)−hextmy, (4)\n8ε(0) \n0.450.500.550.600.65\n1.00.50.00.51.0\n0 π3π/2 π/2 \nφ (rad) \n(b) (a)\nhextκeff κ(0) 0.10 \n0.05 \n0\n-0.05 \n0 0.05 0.10 0.15 0.20 C BA\n−π /2 mz\nFIG. 4. (a) The energy-density contour plot of Eq. (4) at 0 V in φ−mzspace.m(0)is indicated\nby open circles. (b) The classification of hext-κ(0)\neffspace for the calculation of m(0). In the shaded\nregionA,|m(0)\nz|>0. In the white regions, m(0)\nz= 0. In region B, 0< m(0)\ny<1. In region C,\nm(0)\ny= 1, and m(0)\nx= 0. In regions A,B, andC,m(0)has different analytical expressions.\nwhereε=E/(µ0M2\ns),κ=Ku/(µ0M2\ns), andhext=Hext/Ms. Without loss of generality,\nwe assume that hext>0. AtV= 0, the dimensionless anisotropy constant is κ=κ(0)=\nK(0)\nu/(µ0M2\ns). Theφandmzdependence of ε(0)atHext= 2000 Oe is shown in Fig. 4(a),\nwherem(0)= (m(0)\nx,m(0)\ny,m(0)\nz) = (0, 0.496, ±0.869) are indicated by open circles.\nTo derive the analytical expressions of m(0), we divide the hext-κ(0)\neffplane into three\nregions,A,B, andC, as shown in Fig. 4(b), where κeff=Keff/(µ0M2\ns) =κ−(1/2)(Nz−Nx).\nInregion A, indicated bytheshadedarea, the zcomponent of m(0)isnonzero, i.e. |m(0)\nz|>0.\nIn regions BandC, the magnetization is in the IP direction, i.e. mz= 0. Therefore, the\ninitial and final state of switching should be in region A.\nThe lower boundary of region Ais expressed as follows. For hext≤Ny−Nx,\nκeff>0. (5)\nForhext> Ny−Nx,\nκeff> κeff,c=1\n2(hext+Nx−Ny). (6)\n9In region A, the equilibrium directions of the magnetization are given by\nm(0)\nx= 0, (7)\nm(0)\ny=hext\n2κ(0)+Ny−Nz, (8)\nm(0)\nz=±/radicalBigg\n(2κ(0)+Ny−Nz)2−h2\next\n(2κ(0)+Ny−Nz)2. (9)\nBy substituting parameters used in Fig. 4(a) into Eqs. (7) - (9), we havem(0)= (m(0)\nx,m(0)\ny,\nm(0)\nz) = (0, 0.496, ±0.869), which is the same as the result of the numerical calculation.\nThe boundaries of region Bare given by\nhext≤Ny−Nx, (10)\nand\nκeff≤0. (11)\nIn region B, we have\nm(0)\nx=±/radicalBig\n1−(hext/(Ny−Nx))2, (12)\nm(0)\ny=hext/(Ny−Nx), (13)\nm(0)\nz= 0. (14)\nThe boundaries of region Care given by\nhext> Ny−Nx, (15)\nand\nκeff≤κeff,c=1\n2(hext+Nx−Ny). (16)\nIn region C, we have\nm(0)\nx= 0, (17)\nm(0)\ny= 1, (18)\nm(0)\nz= 0. (19)\n10mz1.00.50.00.51.0mz\nπ/2 0 π3π/2 π/2 \nφ (rad) \n0.450.500.550.60ε(+V) \n0.460.480.500.520.54ε(+V) \nκeff,U2 κeff,L2 =0 \n0.000 0.002 0.004 0.006 0.0080.00.10.20.30.4\nκeff κ(+V) α\n0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.000.010.020.030.040.050.06Ny-Nx\nhext κeff κ(+V) \n0.10\n0.05\n0\n-0.05 \n0 0.05 0.10 0.15 0.20Ny-Nx\nhext κeff κ(+V) κeff κ(0) \n(f)(c)\n(e)(b)\n(d)(a)\n0.020 0.025 0.030 0.035 0.040 0.045 0.0 0.2 0.4 0.6 0.8 κeff,U1 κeff,L1 \nκeff κ(+V) α\n1.00.50.00.51.0\n0 π3π/2 π/2 \nφ (rad) −π /2 \nFIG. 5. The magnetization dynamics at T= 0 in the elliptical FL with aspect ratio ( AR) 5 and\nK(0)\neff= 200 kJ/m3(κ(0)\neff= 0.0812). (a) The energy-density contour plot of Eq. (4) in φ−mzspace\nduring the application of a voltage for K(+V)\neff= 80 kJ/m3(κ(+V)\neff= 0.0325) and Hext= 2000 Oe\n(hext= 0.114> Ny−Nx= 0.0670). The open circles indicate m(0).m(0)is determined at κ(0)\neff.\nThe thick gray dotted curve represents the contour with the s ame energy density as ε(m(0)). The\nsolid green circles indicate new equilibrium directions ( m(+V)\neq) atκ(+V)\neff. (b) The region of heavily\ndamped precessional switching (shaded region) in κ(+V)\neff−αspace for the same parameters as (a).\n(c) Theregion ofheavily dampedprecessional switching(li ghter- anddarker-shadedregions)andits\nboundaryin hext−κ(+V)\neffspace. At given hext, dropping κefffromκ(0)\nefftoκ(+V)\neffin theshadedregions\ncan induce heavily damped precessional switching at an opti mal value of α. (d)The same plot as\nin (a) for K(+V)\neff= 15 kJ/m3(κ(+V)\neff= 0.00609) and Hext= 750 Oe ( hext= 0.0426< Ny−Nx).\n(e) The region of heavily damped precessional switching (sh aded region) in κ(+V)\neff−αspace for the\nsame parameters as (d). (f) An enlarged view of (c).\nB. Magnetization dynamics at T= 0\nThe application of a bias voltage modifies the anisotropy constants f romK(0)\nefftoK(+V)\neff\nand destabilizes the initial state. Under the optimal conditions of K(0)\neffandK(+V)\neff, the\nprecessional motion of magnetization around the IP magnetic field is induced [32]. For\n11example, in the case of the elliptical-cylinder MTJ with AR= 5 and Hext= 2000 Oe\n(hext= 0.114), a change from K(0)\neff= 200 kJ/m3toK(+V)\neff= 80 kJ/m3(fromκ(0)\neff= 0.0812\ntoκ(+V)\neff= 0.0325) induces the precession. In this case, the contour plot of ǫchanges from\nFig. 4(a) to Fig. 5(a). Because the energy contour (gray curve) including m(0)(open\ncircles) passes mz= 0, the magnetization can go down to the lower hemisphere to switch\nits direction. This condition yields an upper bound of κ(+V)\neff. We label this upper bound as\nκeff,U1, which is indicated by the dotted vertical line in Fig. 5(b).\nNote that in the condition of Fig. 5(a), the heavily damped precessio nal switching is\ninduced at relatively high α(0.11≤α≤0.30) while dynamic precessional switching is\ninduced at lower α(α <0.11). This is because, as shown in Fig. 5(a), the equilibrium\ndirections of matκ(+V)\neff,m(+V)\neq, indicated by the solid green circles exist on both the upper\nand the lower hemispheres. In such a case, the magnetization can r elax to the counterpart\nm(+V)\neqafter half a precession period even during the application of the bias voltage.\nIn Fig. 5(b), the values of ( κ(+V)\neffandα) that enable heavily damped precessional switch-\ning are indicated by the shaded region. The parameters are T= 0,K(0)\neff= 200 kJ/m3\n(κ(0)\neff= 0.0812),Hext= 2000 Oe ( hext= 0.114), and Ny−Nx= 0.0670(< hext). Similar to\nthe results for the circular MTJ reported in Refs. [20, 21], the shad ed region is triangular.\nWe label its lower bound κ(+V)\neffasκeff,L1. Atκ(+V)\neff≤κeff,L1, heavily damped precessional\nswitching cannot be induced because m(+V)\neqat suchκ(+V)\neffis only located at mz= 0.\nκeff,U1andκeff,L1are analytically calculated in the same way as in Refs. [20, 32] and their\nhextdependence is summarized in Figs. 5(c) and (f). Fig. 5(f) is an enlarg ed view of the\nlow-hextregion in Fig. 5(c). In both the lighter- and darker-shaded regions , heavily damped\nprecessional switching can be induced at appropriate values of α.\nForNy−Nx< hext<2κ(0)+Ny−Nz, the condition on κ(+V)\nefffor the heavily damped\nprecessional switching is\nκeff,L1< κ(+V)\neff< κeff,U1. (20)\nHere,\nκeff,L1=1\n2(hext+Nx−Ny). (21)\nThis lower bound can be obtained as κeffwhich yields mz= 0 in Eq. (9). κeff,L1is indicated\nby the solid blue curve in Figs. 5(c) and (f). This curve is the same ast he boundary between\nregionsAandCin Fig. 4(b).\n12The upper boundary is\nκeff,U1=hext\nm(0)\ny+1−Ny−Nx\n2, (22)\nwherem(0)\nyis given in Eq. (8). κeff,U1is indicated by the solid green curve in Figs. 5(c) and\n(f).\nFor 0< hext< Ny−Nx,\nκeff,L2< κ(+V)\neff< κeff,U2. (23)\nHere,\nκeff,L2= 0. (24)\nThis lower bound, κeff,L2, is indicated by the solid cyan line in Figs. 5(c) and (f). This line\nis the same as the boundary between regions AandBin Fig. 4(b). An example of κeff,L2is\nshown in Fig. 5(e), where Hextis 750 Oe ( hext= 0.0426< Ny−Nx= 0.0670).\nThe upper bound is\nκeff,U2=−1\n2(Nz−Nx)+\nh2\next−2hextm(0)\nyNyx+Nyx/bracketleftbigg\nNz−Nx−/parenleftBig\nm(0)\ny/parenrightBig2\n(Nz−Ny)/bracketrightbigg\n2/bracketleftbigg\n1−/parenleftBig\nm(0)\ny/parenrightBig2/bracketrightbigg\nNyx, (25)\nwhereNyx=Ny−Nx.κeff,U2is indicated by a solid red curve in Figs. 5(c) and (f). An\nexample of κeff,U2is shown in Fig. 5(e).\nAs seen in Figs. 5(c) and (f), the lower ( κeff,L2) and upper ( κeff,U2) bounds of κ(+V)\nefffor\nthe heavily damped precessional switching are different from κeff,L1andκeff,U1. Note that\nin the darker-shaded region of Figs. 5(c) and (f), there exist two contours at ε=ε(m(0)),\nas shown in Fig. 5(d), where K(+V)\neff= 15 kJ/m3(κ(+V)\neff= 0.00609),Hext= 750 Oe ( hext=\n0.0426< Ny−Nx= 0.0670). In Fig. 5(c), the bottom gray dotted-dashed curve show s that\nκ(+V)\neffless than the curve is too low to induce even dynamic switching stably b ecause the\nenergy contour including m(0)does not cross mz= 0 at such low κ(+V)\neff.\nC. Dependence of the WER on αandK(+V)\neff\nWe calculate the WER at Hext= 1000 Oe ( hext= 0.0568< Ny−Nx= 0.0670) and\nT= 300 K focusing on the range of κ(+V)\neffdescribed as Eq. (23). Figure 6(a) shows an\n13-5 0 5 10 15 20 25 30 35 00.1 0.2 0.3 0.4 0.5 0.6 α\nKeff (kJ/m 3) K(+V) WER at tp= 10 ns (b) (a)\n0 2 4 6 8 1010-310-210-1100WER \ntp(ns)10-310-210-1100\nFIG. 6. TheWERat 300 Kand Hext= 1000 Oe. (a) The tpdependenceoftheWER at K(+V)\neff= 10\nkJ/m3andα= 0.19. (b) The K(+V)\neffandαdependence of the WER at tp= 10 ns. The WER is\nat a minimum value, [WER] min, of 3.5×10−3atK(+V)\neff= 10 kJ/m3andα= 0.19.\nexample of the tpdependence of the WER calculated in the same way as in Fig. 3. Here,\nK(+V)\neff= 10 kJ/m3,α= 0.19, and the other parameters are the same as those in Fig. 4.\nThe WER is kept around 3 .5×10−3for the range of 1 .5≤tp≤10 ns due to the heavily\ndamped precessional switching.\nFigure 6(b) shows the color map of the WER at tp= 10 ns on the K(+V)\neff-αplane. The\nWER at tp= 10 ns is a minimum around the center of the trianglelike region, similarly to\nRef. [20, 21]. The minimum value of [WER] min= 3.5×10−3is obtained at K(+V)\neff= 10\nkJ/m3andα= 0.19. For example, experimentally, αhas been increased by using materials\nincluding Pt and Pd [33–37].\nD. Magnetic-field dependence of minimum value of WER\nThe minimum value of the WER, [WER] min, strongly depends on the magnitude of the\nexternal IP magnetic field, Hext. Because the precession period is inversely proportional\ntoHext, the disturbance due to the thermal-agitation field during precess ion increases as\nHextdecreases. As Hextapproaches 0, the WER approaches unity. Meanwhile, the energy\n14(b) (a)\ncircleAR 5AR 2\nAR 15 AR 10 \n10-410-310-210-1100[WER] min \n10-410-310-210-210-1100[WER] min \n1000 1500 2000 500\nHext (Oe) Hext (Oe)1000 1500 2000 2500 500\nFIG. 7. Theexternal in-planemagnetic fielddependenceof [W ER]minforvarious ARfrom1(circle)\nto 15: (a) at K(0)\neff= 200 kJ/m3(0.3Heff\nk≈900 Oe); (b) at K(0)\neff= 300 kJ/m3(0.3Heff\nk≈1300 Oe).\nbarrier between the equilibrium directions on the upper and lower hem ispheres decreases as\nHextincreases. Above a certain critical value of Hext, the WER increases as Hextincreases\nand approaches unity. Therefore, there is an optimal value of Hextat which the WER is\nminimized.\nTo determine the optimal value of Hext, we calculate the Hextdependence of [WER] min\nfor various values of ARranging from 1 (circle) to 15 as shown in Fig. 7(a). For the\ncircular MTJ (red open circles), [WER] minis minimized around Hext= 1000 Oe, where\nHext/Heff\nk≈0.3 [21]. Here Heff\nk= 2K(0)\neff/µ0Ms. AsARincreases, the minimum [WER] min\ndecreases and the optimal value of Hext(H(opt)\next) increases.\nTo investigate the effect of the inverse-bias method [38–40], we co nduct similar calcula-\ntions for a large anisotropy constant K(0)\neff= 300 kJ/m3, as shown in Fig. 7(b). For each\nAR, the minimum [WER] minin Fig. 7(b) is lower than that in Fig. 7(a). Also in Fig. 7(b),\nasARincreases, the minimum [WER] mindecreases and H(opt)\nextincreases.\nFrom the results shown in Figs. 7(a) (0 .3Heff\nk≈900 Oe) and 7(b) (0 .3Heff\nk≈1300\nOe), note that H(opt)\next≈0.3Heff\nkfor lowAR(≤2), where Ms(Ny−Nx)/lessorapproxeql0.3Heff\nk, and\n0.3Heff\nk< H(opt)\next< Ms(Ny−Nx) for high AR(≥5) where Ms(Ny−Nx)/greaterorapproxeql0.3Heff\nk. The IP\ndemagnetization fields for AR= 2, 5, 10, and 15 are H(IP)\nk= 516 Oe, 1178 Oe, 1691 Oe, and\n152014 Oe, respectively. These results indicate that the increase of the IP demagnetization\nfield causes the reduction in the WER for high AR.\nThe dependence of [WER] minon the angles ( φH) ofHextis also calculated for AR= 5,\nK(0)\neff= 200 kJ/m3, andHext= 1000 Oe. Here the definition of φHis the same as that of the\nφillustrated in Fig. 1(a). In the minor-axis direction, φH= 90◦, [WER] min= 3.5×10−3,\nas plotted in Fig. 7(a). At φH= 92◦, [WER] minreaches [WER] min= 2.9×10−2, which is\nhigher than [WER] min= 2.1×10−2forAR= 1 (circle), K(0)\neff= 200 kJ/m3, andHext= 1000\nOe.\n16V. EFFECT OF IP DEMAGNETIZATION FIELD ON THE WER\n(b) (a)\nTop view \nmy\nmx-1 0 1-1 01(d) \nTop view \nmy\nmx-1 0 1-1 01(c)-1 1 0\nmz-1 1 0\nmz1.11.21.3\n1.11.21.3ε(MJ/m 3)\nε(MJ/m 3)\nFIG. 8. (a) The mzdependence of the energy density (Eq. (1)) at mx= 0 for the circular FL. (b)\nThe same plot for the elliptical FL with AR= 5. (c) The distribution of the magnetization unit\nvector in the circular FL with α= 0.16 immediately before application of the voltage pulse, m(0)′,\natT= 300 K and V= 0. 105trials are conducted and each blue dot corresponds to mafter each\ntrial. Among the blue dots, the distribution of m(0)′that will result in a write error after voltage\napplication and subsequent relaxation for 10 ns is highligh ted as red dots. (d) The same plot for\nthe elliptical FL with AR= 5 and α= 0.19. In all panels, K(0)\n1,eff= 200 kJ/m3, andHext= 1000\nOe. The other parameters are the same as in Fig. 6.17TABLE I. A comparison between the circular free layer and the elliptical free layer with AR= 5\nin terms of the energy-barrier height in Figs. 8(a) and (b) an d the number and distribution of the\nblue and red dots in Figs. 8(c) and (d).\nGeometry of free layer (a) Circle, α= 0.16, error / all (b) AR= 5 ellipse, α= 0.16, all (c) AR= 5 ellipse, α= 0.19, error / all\nBarrier height (kJ/m3) 84.5 160 160\nNumber of dots 2318 (red dots) / 105(blue dots) 105350 (red dots) / 105(blue dots)\nStandard deviation of m(0)′\nz 0.01173 / 0.00967 0.00559 0.00744 / 0.00559\nStandard deviation of φ(0)′0.0697 / 0.0734 0.1031 0.0987 / 0.1031\nTo analyze the effect of the IP demagnetization field on the WER, we c ompare the mz\ndependence of the energy density (Eq. (1)) at mx= 0,K(0)\n1,eff= 200 kJ/m3, andHext= 1000\nOe between the circular FL (Fig. 8(a)) and the elliptical FL with AR= 5 (Fig. 8(b)). The\nother parameters are the same as in Fig. 6. The energy-barrier he ight indicated by the two-\nheaded arrow in Fig. 8(b) is higher than that in Fig. 8(a). The value of the energy-barrier\nheight is given in Table I. In the elliptical FL, the energy-barrier heigh t is enhanced by the\ndemagnetization energy. This enhancement is similar to the enhance ment from increasing\nK(0)\n1,eff. Therefore, the stability of mbefore the application of Vis expected to increase as\nARincreases.\nUsing the parameters in Figs. 8(a) and (b), the distribution of the in itial states ( m(0)′)\natT=300 K is compared between the circular FL (Fig. 8(c)) and the elliptic al FL with\nAR= 5 (Fig. 8(d)). The initial states are obtained by relaxing the magne tization from m(0)\nwithm(0)\nz>0 for 10 ns. The relaxation is conducted 105times.m(0)′after each simulation\nis plotted by the blue dots in Figs. 5(c) and (d). The standard deviat ion of the distribution\nin themzandφdirections (the standard deviation of m(0)′\nz,δz, and the standard deviation\nofφ(0)′,δφ) are also listed in Table I. In the elliptical FL, the standard deviation o fm(0)′\nzis\nsmaller than that in the circular FL, whereas the standard deviation ofφ(0)′is not. Note\nthatα= 0.16 in Fig. 8(c) and α= 0.19 in Fig. 8(d) because each αyields [WER] minat\nHext= 1000 Oe in Fig. 7(a).\nTo clarify the cause of the write error in the heavily damped precess ional switching, we\nplotm(0)′, which results in the write error as shown by the red dots in Figs. 8(c ) and (d).\nThe number of red dots and the corresponding standard deviation ofm(0)′\nzandφ(0)′(δzand\nδφ) are also listed in Table I. In both the circular FL and the elliptical FL, t heδzof the red\n18dots is larger than that of the blue dots, while δφof the red dots is smaller than that of the\nblue dots. This indicates that the cause of the write error is δzrather than δφin the heavily\ndamped precessional switching.\nIn the elliptical FL under external in-plane Hext, which is parallel to the minor axis of the\nellipse, the error in the heavily damped precessional switching is redu ced by the suppression\nofδz.δzis reduced by the energy barrier which is enhanced by the demagnet ization energy.\nNote that in the dynamic precessional switching, where the WER is se nsitive to tp, largeδφ\nalso leads to a high WER, because large δφyields the large distribution of optimal tpamong\nall of the trials [6–14, 32, 41].\nIt is expected that, in practice, the energy barrier can be enhanc ed more noticeably in\nsmaller FLs. This is because, in larger FLs, the energy barrier is decr eased by subvolume\nactivation effects. Thus, we perform simulations for the smaller FLs withS= 252πnm2\nand show the results in Appendix B. There, it is confirmed that the Hextdependence in the\ncase ofS= 252πnm2is qualitatively the same as that in the case of S= 502πnm2shown\nin Fig. 7.\nVI. CONCLUSIONS\nWe theoretically investigate heavily damped precessional switching in a perpendicularly\nmagnetized elliptical-cylinder voltage-controlled MTJ. We derive analy tical expressions of\nthe conditions of the parameters for heavily damped precessional switching. The simula-\ntions using the Langevin equation show that the WER in the elliptical FL can be several\norders of magnitude lower than that in the circular FL. From the dist ribution of the initial\nmagnetization state immediately before a voltage is applied, it is revea led that the error in\nthe heavily damped precessional switching is reduced by the suppre ssion of the distribution\nin thezdirection ( δz) andδzis reduced by the energy barrier, which is enhanced by the\ndemagnetization energy. The results provide a guide to designing hig h-density VCMRAM\nfor write-error-tolerant applications such as AI image recognition .\n19ACKNOWLEDGMENTS\nThis work is partly based on results obtained from a project, JPNP1 6007, commissioned\nby the New Energy and Industrial Technology Development Organiz ation (NEDO), Japan.\nAppendix A MICROMAGNETIC SIMULATIONS\nWe conduct the micromagnetic simulations by using the MuMax3 softw are package [42]\nand confirm the K(+V)\nu−αspace diagram as shown in Fig. 9. Here, an exchange stiffness\nconstant ( Aex) of 2×10−11J/m, a cell size of 2 nm ×2 nm,Ku= 1252.549 kJ/m3, and\nT= 0 K are assumed. The other parameters are the same as those in F ig. 3. In the circular\nFL, the region of heavily damped precessional switching (the area in gray) is quite narrow\nbecause of the multimagnetic domains nucleated during application of the voltage [20]. In\nthe elliptical FL, the region of heavily damped precessional switching (the area in cyan) is\nwider than that in the circular FL. The area in cyan qualitatively agree well with Fig. 5(e)\nwhich is analyzed in the macrospin model.\n20α\nKu (kJ/m3) K(+V)elliptical FL\ncircular FL\nFIG. 9. The K(+V)\neff−αspace diagram obtained by the micromagnetic simulations at 0 K with cell\nsize of 2 nm ×2 nm and an exchange stiffness constant ( Aex) of 2×10−11J/m.Ku= 1252.549\nkJ/m3is assumed and the other parameters are the same as those in Fi g. 3. The solid black circles\non black curves are boundaries for the circular FL (redrawn f rom Ref. [20]). The open blue circles\non blue curves are boundaries for the elliptical FL. The heav ily damped precessional switching\noccurs in the gray and cyan areas.\nAppendix B THE WER IN SMALLER JUNCTIONS\nFigure 10 shows the dependence of [WER] minon the external in-plane magnetic field\n(Hext) in the case of a smaller junction area, S= 252πnm2. 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Studenikina\naDepartment of Theoretical Physics,\nMoscow State University, 119991 Moscow, Russia\nE-mail: ar.popov@physics.msu.ru, studenik@srd.sinp.msu.ru\nAbstract. Neutrino propagation in the Galactic magnetic field is considered. To describe\nneutrino flavour and spin oscillations on the galactic scale baselines an approach using wave\npackets is developed. Evolution equations for the neutrino wave packets in a uniform and\nnon-uniform magnetic field are derived. Analytical expressions for neutrino flavour and spin\noscillations probabilities accounting for damping due to wave packet separation are obtained\nfor the case of uniform magnetic field. It is shown that for oscillations on magnetic frequencies\nωB\ni=µiB⊥the coherence lengths that characterizes the damping scale is proportional to\nthe cube of neutrino average momentum p3\n0. Probabilities of flavour and spin oscillations\nare calculated numerically for neutrino interacting with the non-uniform Galactic magnetic\nfield. Flavour compositions of high-energy neutrino flux coming from the Galactic centre are\ncalculated accounting for neutrino interaction with the magnetic field. It is shown that for\nneutrino magnetic moments ∼10−13µBand larger these flavour compositions significantly\ndiffer from ones predicted by the vacuum neutrino oscillations scenario.arXiv:2404.02027v1 [hep-ph] 2 Apr 2024Contents\n1 Introduction 1\n2 High-energy neutrinos flavour ratios: The Standard Model and beyond 3\n3 Neutrino oscillations in a magnetic field in the wave packet formalism 4\n3.1 Neutrino oscillations in a uniform magnetic field 4\n3.2 Neutrino oscillations in a non-uniform magnetic field 9\n4 Flavour compositions of astrophysical neutrinos 10\n4.1 Flavour compositions of neutrinos from point-like sources 10\n4.2 Flavour ratios of neutrinos from the Galactic centre 11\n5 Conclusion 12\n1 Introduction\nHigh-energy neutrino astronomy is one of the most rapidly developing areas of neutrino\nphysics. In 2013 the IceCube collaboration reported the discovery of extraterrestrial high-\nenergy neutrinos [1]. Subsequent years of data taking by the IceCube and Baikal-GVD col-\nlaborations confirmed existence of the isotropic diffuse flux of high-energy neutrinos [2–5].\nSources of high-energy neutrino emission remain largely unknown. Theoretical models\npropose numerous source candidates, such as Active Galactic Nuclei, Gamma-ray bursts, su-\npernovae and hypernovae, supenova remnants, binary systems, pulsars, magnetars and others\n[6–9]. Active galactic nuclei, in particular blazars and quasars, are among the most promising\nsources. Motivated by coincidence in direction and time of IceCube-170922A neutrino alert\nandagamma-rayflarefromtheblazarTXS0506+056, ananalysisof9.5yearsofIceCubedata\nprovided 3.5 σevidence of excess of high-energy neutrino events with respect to atmospheric\nneutrino background from direction of TXS 0506+056 [10]. This was a first ever evidence\nof a point source of high-energy neutrinos. Later studies greatly improved our knowledge of\nastrophysical neutrinos sources. The latest point sources search by IceCube [11] highlights\nthree possible sources of neutrino emission: blazars TXS 0506+056 and PKS 1424+240, and\nSeyfert II galaxy NGC 1068, where the latter achieves 4.2 σsignificance. Recent results by\nBaikal-GVD present additional evidence of neutrino emission from TXS 0506+056, as well as\nfrom other astrophysical sources [12]. Plavin et al.argue [13–15] that high-energy neutrinos\nmay originate from cores of radio blazars. Neutrino emission from the Galactic plane was\nidentified by IceCube with 4.5 σsignificance. After completion of Baikal-GVD [16, 17] and\nKM3NeT [18], as well as future neutrino telescopes, such as IceCube-Gen2 [19] and P-ONE\n[20], sensitivity to high-energy neutrinos sources will increase greatly.\nSources of high-energy neutrinos are believed to be connected with cosmic rays acceler-\nators. Within the hadronic ( pp) and photo-hadronic ( pγ) scenarios, it is expected that high-\nenergy protons and nuclei produced inside an astrophysical object interact with the ambient\ngas and radiation, creating charged and neutral pions. Neutral pions decay π0→γγemit\nhigh-energy gamma rays, while charged pions produce neutrinos and antineutrinos in the de-\ncay chain π+(−)→µ+(−)+νµ(¯νµ)followed by µ+(−)→e+(−)+ ¯νµ(νµ)+νe(¯νe)(see [21]). The\n– 1 –flavour ratios of neutrinos produced in such scenarios follows pattern νe:νµ:ντ= 1 : 2 : 0 .\nNote that in this paper we do not distinguish between neutrino and antineutrinos since they\nare not well discriminated by current detectors. However, in principle discrimination of νeand\n¯νeis possible via the Glashow resonance [22]. Alternatively to pions decay mechanism, the\nmuon decay, neutron decay and charm decay scenarios predict 0 : 1 : 0,1 : 0 : 0and1 : 1 : 0\nflavour ratios respectively [23]. Some models predict more complicated energy-dependent\nflavour ratios [24]. Due to phenomenon of neutrino oscillations flavour ratio observed by ter-\nrestrial neutrino telescopes is different from flavour ratios at source. This topic will be further\ndiscussed in Section 2.\nUnlike cosmic rays, neutrinos are not deflected by magnetic fields and travel through\ncosmic space without absorption. Thus, they are considered to be promising messengers\ncarrying information about processes inside astrophysical objects. However, even if neutrinos\nare considered to be electrically neutral particles, they still can interact with electromagnetic\nfields (see [25] for a review on neutrino electromagnetic interactions). Massive neutrinos\nmay possess electromagnetic properties, in particular nonzero anomalous magnetic moments\n[26, 27]. For the case of Dirac neutrinos diagonal magnetic moments, Standard Model predicts\nvalues\nµD\nii=3eGFmi\n8√\n2π2≈3.2×10−19\u0010mi\neV\u0011\nµB, (1.1)\nwhere µBis the Bohr magneton. A number of theories of physics beyond the Standard Model\npredict values of neutrino magnetic moments in the range 10−12÷10−17µB(see [25] for a\nreview). The most stringent terrestrial upper bound on neutrino effective magnetic moment\nis obtained by GEMMA [28] experiment and is on the level of 2.9×10−11µB. Solar neutrinos\nobservation by XENONnT provide the upper bound on the level of 6.4×10−12µB[29]. There\nare also a variety of astrophysical limits of order of 10−12µB[30].\nPhenomena of neutrino spin [31] and spin-flavour precession [32] arise due to interaction\nof neutrino magnetic moments with a magnetic field. Spin precession induced by neutrino\ninteractionwithcosmicmagneticfieldscansignificantlymodifytheflavourcontentofneutrino\nfluxes detected by the terrestrial neutrino telescopes.\nA variety of magnetic field strengths is found in different cosmic structures. Galactic\nmagnetic fields vary from 1÷10µG in the spiral galaxies to 50÷100µG in the starburst\ngalaxies. It follows from the radio synchrotron measurements that in Milky Way galaxy,\nthe magnetic field averaged over 1 kpc radius is about 6 µG [33]. Furthermore, within the\ngalaxy clusters a magnetic field of strength ∼1µG are found [34]. In particular, the Virgo\nSupercluster of galaxies can possess a magnetic field of ≈2µG [35]. Average intergalactic\nmagnetic field strength of 30÷60nG exist in cosmic filaments [36]. Magnetic fields in voids\nremains largely unconstrained. A lower bound for magnetic fields in the voids is estimated\nas10−15÷10−16Gauss [37, 38]. By relaxing assumptions, Dermer et al.[39] obtain more\nconservative values of 10−18Gauss. Recently, the upper bound of 10−10Gauss on the voids\nmagnetic field was obtained [40]. For a review on the galactic and extragalactic magnetic\nfields see [33, 34, 41, 42].\nNeutrino oscillations in the interstellar magnetic fields was studied before in several pa-\npers [43–48]. However, in these papers the probabilities of neutrino oscillations in a magnetic\nfield were calculated within the plane-wave approximation that does not account for possible\ndecoherence effects when neutrinos travel long distances.\nToexamineneutrinospropagationonthegalactic, galacticclustersorcosmologicalscales\n(∼10 kpc, ∼1 Mpc and ∼1 Gpc respectively) one has to use the wave packet approach\n– 2 –[49]. Previously the wave packet formalism was developed for the case of vacuum neutrino\noscillations [50–54], neutrino oscillations in matter [55, 56] and neutrino collective oscillations\n[57]. In the present paper the wave packet approach is extended to the case of neutrino\npropagating in a magnetic field.\nThis paper is organized as follows. In Section 2 we briefly review possible manifestations\nof BSM physics in flavour ratios of high-energy neutrinos. In Section 3 we develop a formalism\nfor calculation of the neutrino oscillation probabilities accounting for wave-packet separation\neffects. In Section 4 we discuss possible flavour compositions of neutrinos originating from the\nGalactic centre and propagating in the galactic magnetic field. Finally, Section 5 concludes\nour paper.\n2 High-energy neutrinos flavour ratios: The Standard Model and beyond\nThe standard picture of neutrino oscillations implies that at large distances neutrino flavour\noscillations probabilities average out to\nPstd\nαβ=3X\ni=1|Uαi|2|Uβi|2, (2.1)\nwhere Uαiare elements of Pontecorvo–Maki–Nakagawa–Sakata matrix and α, β =e, µ, τ.\nThis phenomenon occurs due to the following three reasons:\n•finite energy resolution of neutrino detectors,\n•oscillations damping due to wave packet separation,\n•finite size of neutrino production region.\nIn this case the flavour composition of neutrino flux measured by a terrestrial telescope\nis given by\nrα=X\nβr0\nβ3X\ni=1|Uαi|2|Uβi|2, (2.2)\nwhere r0\nβare the flavour ratios at the source. For the case of pure pion decay production, the\ninitial and final flavour ratios are (r0\ne:r0\nµ:r0\nτ) = (1 : 2 : 0) and(re:rµ:rτ)≈(1 : 1 : 1) ,\nrespectively. The predicted flavour composition is strongly effected by uncertainties in the\nmeasured neutrino mixing parameters [58, 59], especially for the values of the mixing angle\nsin2θ23and CP-violating phase δCP.\nDifferent models of physics beyond the Standard Model predict that the observed flavour\ncomposition significantly differ from one predicted by the standard vacuum oscillations. Var-\nious neutrino decay scenarios [59–62] predict that the observed flavour ratios are different for\nthe normal and inverted neutrinos mass orderings:\nνe:νµ:ντ=|Ue1|2:|Uµ1|2:|Uτ1|2,(NO) (2.3)\nνe:νµ:ντ=|Ue3|2:|Uµ3|2:|Uτ3|2,(IO) (2.4)\nwhere NO and IO stand for the normal and inverted mass orderings.\n– 3 –The quantum decoherence of neutrinos interacting with a dissipative environment pre-\ndicts the exponential damping of neutrino oscillations similar to the damping due the wave\npackets separation [63]. For example, the quantum decoherence in neutrino oscillations may\nbe a manifestation of quantum gravity [64].\nTheories with the Lorentz-violation predict modified dispersion relation for neutrinos,\nleadingtopeculiarpatternsinneutrinooscillationspotentiallyobservableinflavourratios[65].\nIn the pseudo-Dirac case, additional active-sterile neutrino oscillations occur with oscillations\nlength exceeding the standard flavour oscillations lengths by orders of magnitudes [66, 67].\nIn this case oscillations effects are not washed out even at cosmological distances and flavour\ncompositions of neutrinos coming from a point source become distance dependent. For review\nof possible BSM effects and capability of their detection in future neutrino telescopes see also\n[68–70].\n3 Neutrino oscillations in a magnetic field in the wave packet formalism\nIn this section we extend the approach to the problem of neutrino oscillations in a magnetic\nfield developed in [71] to account for wave packet separation effects. For illustrative purposes,\nwe start with analytical solution of neutrino wave packet evolution in a uniform magnetic\nfield. Next, we numerically solve evolution equation using a realistic model of the Galactic\nmagnetic field.\n3.1 Neutrino oscillations in a uniform magnetic field\nNeutrino evolution in a uniform magnetic field is described by the following Dirac equations\n(iγµ∂µ−mi)νi(x)−X\nkµikΣBνk(x) = 0 , (3.1)\nwhere i, k= 1,2,3. Neutrino magnetic moment matrices are defined by\nµD=\nµ11µ12µ13\nµ12µ22µ23\nµ13µ23µ33\n, µM=\n0 iµ12iµ13\n−iµ12 0iµ23\n−iµ13−iµ230\n (3.2)\nfor Dirac and Majorana neutrinos, respectively [25]. Here µikare the magnetic moments in\nthe neutrino mass basis.\nIn our previous paper [72], Eq. (3.1) was considered in the plane wave approximation\nthat does not account for potentially important decoherence effects in neutrino oscillations\nat long distances. To account for possible wave packet separation effects we transform Eq.\n(3.1) to the momentum representation. In the considered case Eq. (3.1) can be rewritten as\ni∂tνi(p, t) = [miγ0+γ0γ3p]νi(p, t) +X\nkµikγ0ΣBνk(p, t), (3.3)\nwhere the Fourier transform of the neutrino wave function is defined by\nνi(x, t) =Zdp\n(2π)1/2eipxνi(p, t). (3.4)\nHere we assume that the neutrino momentum pis directed along the nzaxis.\n– 4 –We assume that the initial neutrino wave function at t= 0in laboratory frame is\ndescribed by the Gaussian wave packet\nνi(p,0) = fi(p, p 0)u−\ni(p), (3.5)\nfi(p, p 0) =1\n(2πσ2p)1/4exp\u0012\n−(p−p0)2\n4σ2p\u0013\n, (3.6)\nwhere p0is the average wave packet momentum, σpis wave packet width and u−\niis a left-\nhanded solution of vacuum Dirac equation. For the sake of simplicity, we consider one-\ndimensional wave packets. Note that the three-dimensional wave packets without accounting\nfor a magnetic field presence are considered in [53, 73].\nIn what follows, we consider the case when neutrinos have only the diagonal magnetic\nmoments. In [43] it was shown that the transition magnetic moments affect patterns of\nneutrino oscillations in the interstellar magnetic fields for neutrino energies ∼100 EeV and\nhigher. Thus, the transition magnetic moments are irrelevant of neutrinos detected by Ice-\nCube, Baikal-GVD and KM3NeT. In particular, oscillations of high-energy Majorana neutri-\nnos in interstellar magnetic fields are described by vacuum oscillations probabilities. However,\ntransitionmagneticmomentsbecomerelevantforhigherneutrinoenergiesand/orhighermag-\nnetic field strength and can induce the resonant enhancement of neutrino oscillations [74].\nIn the case of absence of the transition magnetic moments, Eqs. (3.1) decouple and it\nbecomes possible to describe neutrino oscillations in a magnetic field as the solution of three\nindependent equations for the massive neutrino states:\ni∂tνi(p, t) = (miγ0+γ0γ3p+µiγ0ΣB)νi(p, t)≡Hi(p)νi(p, t), (3.7)\nwhere µi≡µiiare the diagonal magnetic moments of neutrinos. The form of this equation\nis similar to the one considered in [45, 71] that allows us to solve it using the same method.\nThe solutions of Eq. (3.7) with initial condition\nνh\ni(p,0) = fi(p, p 0)uh\ni(p) (3.8)\ncan be represented [45, 71] as a superposition of states with the definite helicity h′=±1:\nνh\ni(p, t) =X\ns,h′Chh′\nise−iEs\ni(p)tfi(p, p 0)uh′\ni, (3.9)\nwhere the dispersion relation is given by the eigenvalues of Hi(p)\nEs\ni(p) =±r\nm2\ni+p2+µ2\niB2+ 2sµiq\nm2\niB2+p2B2\n⊥, (3.10)\nwhere s=±1. Here we decompose the magnetic field Binto the transverse B⊥and the lon-\ngitudinal B∥components with respect to the neutrino momentum. Note that the interaction\nwith a magnetic field can induce transitions between the neutrino helicity states.\nThequantumnumber s=±1in(3.10)enumerateseigenstatesofneutrinosinamagnetic\nfields that are eigenvectors of the spin operator [75]\nSi=miq\nm2\niB2+p2B2\n⊥\u0014\nΣB−i\nmiγ0γ5[Σ×p]B\u0015\n, (3.11)\n– 5 –which commutes with the Hamiltonian Hi(p)introduced in (3.7).\nIn the ultrarelativistic limit that is obviously justified for high-energy neutrinos, we can\nneglect terms of order of mi/pand get\nu−\ni≈1√\n2\n0\n−1\n0\n1\n, u+\ni≈1√\n2\n1\n0\n1\n0\n. (3.12)\nIn this case, as it was shown in [71], the coefficients Chh′\nisare given by\nCLL\nis≈1\n2+O\u0010m2\ni\np2\u0011\n, CRL\nis≈ −s\n2+O\u0010m2\ni\np2\u0011\n. (3.13)\nThe dispersion relation (3.10) can be decomposed near the average momentum p0\nEs\ni(p) =E(p0) +vs\ni(p0)(p−p0) +O((p−p0)2), (3.14)\nwhere the neutrino wave packets group velocities are introduced\nvs\ni(p0) =∂Es\ni(p)\n∂p\f\f\f\np=p0=p0\nEs\ni(p0)\n1 +sµiB2\n⊥q\nm2\niB2+p2\n0B2\n⊥\n. (3.15)\nConsider a particular case of the transversal magnetic field B=B⊥. In this case the\ndispersion relation (3.10) takes the form\nEs\ni(p) =q\nm2\ni+p2+sµiB⊥. (3.16)\nIn this case the group velocities are given by\nvs\ni=pq\nm2\ni+p2(3.17)\nand coincide with the group velocities of neutrinos propagating in vacuum and do not depend\non the spin number s.\nThus, the separation of neutrino wave packets with different spin numbers sis caused\nby the longitudinal component of the magnetic field B∥.\nAfter substituting (3.14) into (3.9) and performing the Fourier transform back into the\ncoordinate space, we can obtain the following expression for the neutrino wave function:\nνh\ni(x, t) =1\nNX\ns,h′Chh′\nise−iEs\ni(p0)t+ip0xexp\u0010\n−(x−vs\nit)2\n4σ2x\u0011\nuh′\ni, (3.18)\nwhere handh′are the initial and final neutrino helicities, σx= 1/2σpis the wave packet\nwidth in the coordinate space and Nis the normalization factor.\nFinally, for the probabilities of the neutrino flavour and spin oscillations in a magnetic\nfield we get:\nPνhα→νh′\nβ(L, t) =\f\f\fX\niU∗\nβiUαi(uh′\ni)†νh\ni(L, t)\f\f\f2\n(3.19)\n=1\nN2X\ni,jX\ns,σ=±1U∗\nβiUαiUβjU∗\nαjChh′\nisChh′\njσe−iωsσ\nij(p0)texp \n−(ϕs\ni)2+ (ϕσ\nj)2\n4σ2x!\n,\n– 6 –where ϕs\ni=L−vs\ni(p0)t,ωsσ\nij(p0) =Es\ni(p0)−Eσ\nj(p0)are the frequencies of neutrino oscillations\nthat are given by\nωsσ\nij(p0)≈∆m2\nij\n2p0+ (µis−µjσ)B⊥. (3.20)\nNote that the oscillations on both the vacuum ωvac\nij= ∆m2\nij/2p0and magnetic ωB\ni=\nµiB⊥frequencies present in oscillations probabilities (3.19).\nSince the time tof the neutrino propagation from the source to the detector is not\nan observable quantity, we perform integration over time to obtain the final expression for\nprobabilities of high-energy neutrino oscillations in a magnetic field and get:\nPνhα→νh′\nβ(L) =X\ni,jX\ns,σU∗\nβiUαiUβjU∗\nαjChh′\nisChh′\njσexp\u0010\n−i2πL\nLijsσ\nosc\u0011\nexp\u0010\n−L2\n(Lijsσ\ncoh)2\u0011\n,(3.21)\nwhere the corresponding oscillations and coherence lengths are given by\nLijsσ\nosc=π\nωsσ\nij, (3.22)\nLijsσ\ncoh=2√\n2σx\nvs\ni−vσ\nj. (3.23)\nThe oscillation probabilities (3.21) generalize expressions we have obtained in our previ-\nous paper [45] and account for exponential damping of neutrino oscillations at large distances\ndue to the wave packets separation. Under realistic assumptions that p≫mi≫µiB, the\napproximate expressions for the coherence lengths can be obtained:\nLijss\ncoh≈4√\n2σxp2\n∆m2\nij, (3.24)\nLii−+\ncoh∼σxp3\nµiBm2\ni, (3.25)\nLij−+\ncoh≈Lijss\ncoh. (3.26)\nThe coherence lengths Lijss\ncohandLii−+\ncohcharacterize the damping scale of oscillations on vac-\nuum lengths Lijss\nosc=4πp\n∆m2\nijand magnetic lengths Lii−+\nosc=π\nµiB⊥correspondingly. Note that\nunlike the coherence lengths Lijss\ncohfor oscillations on vacuum frequencies ωvac\nij= ∆m2\nij/4p, the\ncoherence length Lii−+\ncohis proportional to p3. We obtain the following numerical estimations\nfor the coherence length:\nLijss\ncoh≈2√\n2\u0010p\n1TeV\u00112\u0010∆m2\nij\n10−5eV2\u0011−1\n1029σx, (3.27)\nLii−+\ncoh∼\u0010p\n1TeV\u00113\u0010mi\n1eV\u0011−2\u0010B\n1µG\u0011−1\u0012µi\n10−11µB\u0013−1\n1060σx. (3.28)\nThe dimensionless coherence lengths Lijss\ncoh/σxandLii+−\ncoh/σxare shown in Figure 1.\n– 7 –101102103104\nEnergy [TeV]1031103210331034103510361037L12ss\ncoh/σx\n101102103104\nEnergy [TeV]10661068107010721074Lii+−\ncoh/σxFigure 1 . Dimensionless coherence length Lcoh/σx. Left: coherence length of oscillations on vacuum\nfrequency L12ss\ncoh. Right: coherence length of oscillations on magnetic frequency Lii+−\ncoh.\nThere are various estimations for neutrino wave packet width σxin literature. Currently,\nthe best experimental limits on σxfor the reactor neutrinos are obtained by the Daya Bay\ncollaboration [76]. For the astrophysical high-energy neutrinos, it is shown in [77] that for\nneutrinos produced by the free π±orµ±decays σxare given by\nσx∼10−3\u001210TeV\nEν\u0013\ncm(forπ±decay ), (3.29)\nσx∼10−1\u001210TeV\nEν\u0013\ncm(forµ±decay ). (3.30)\nTheappearanceofthemagneticfieldintheneutrinosourcemaydecreasethewavepacket\nwidth by orders of magnitude [77]. In [57], the authors show that for neutrinos produces in\nthe supernova explosions σx∼10−12cm. Assuming that high-energy neutrinos are produced\nby the decay of free π±the coherence length is given by (3.29), and for 100TeV neutrinos we\nobtain L13ss\ncoh∼109pc and L12ss\ncoh∼1011pc.\nCurrently, the most distant source of neutrinos is the blazar TXS 0506+056 [10] located\nat approximately 1.7 Gpc. From our estimations it follows that oscillations on the vacuum\nfrequencies can disappear due to the wave packet separation for astrophysical high-energy\nneutrinos coming from the most distant sources in the Universe. Assuming B= 1µG,µi=\n10−12µBandmi= 0.8eV,forthecoherencelengthsofoscillationsonthemagneticfrequencies\nωB\ni=µiB⊥we obtain Lii−+\ncoh∼1046pc, that by orders of magnitudes exceeds the scale of\nobservable universe which is approximately 30 Gpc.\nThus, we conclude that for neutrinos propagating in the interstellar magnetic fields the\nwave packet separation does not lead to the disappearance of oscillations on the magnetic\nfrequencies ωB\ni=µiB⊥. This conclusion remains valid if one considers propagation of the\nsupernova neutrino, which are characterized by energy p0∼10 MeV in the Galactic magnetic\nfield. Note that in this paper we consider the evolution of the wave packets of ultra-relativistic\nneutrinos, calculations for non-relativistic relic neutrinos require a quite different formalism.\nThus, for the case of the neutrino propagation in a uniform magnetic field we obtain the\nfollowing expression for the probabilities of flavour and spin oscillations at distances exceeding\n– 8 –the coherence lengths of neutrino oscillations in vacuum Lvac\ncoh= max\u0000\nL12ss\ncoh, L13ss\ncoh\u0001\n:\nPαβ(L) =PνLα→νL\nβ(L)\f\f\f\nL≫Lvac\ncoh=3X\ni=1|Uαi|2|Uβi|2cos2(πL/LB\ni), (3.31)\nPνLα→νR(L)\f\f\f\nL≫Lvac\ncoh=3X\ni=1|Uαi|2sin2(πL/LB\ni). (3.32)\nThe oscillations lengths here are given by\nLii−+\nosc=π\nµiB⊥= 2.17·\u0012B\n1µG\u0013−1\u0012µi\n10−11µB\u0013−1\n103pc, (3.33)\nLijss\nosc=4πp\n∆m2\nij= 5.02· \n∆m2\nij\n10−5eV2!−1\u0010p\n1TeV\u0011\n·10−5pc. (3.34)\nNote that unlike the probabilities of neutrino oscillations in vacuum, probabilities given\nby (3.31) do not depend on the neutrino energy. Due to the unitarity of neutrino evolution\noperator, Eq.(3.31) satisfies the probability conservation relation,\nX\nβPαβ(L) +PνLα→νR(L) = 1 . (3.35)\n3.2 Neutrino oscillations in a non-uniform magnetic field\nIn previous section we considered the neutrino wave packet evolution in a uniform magnetic\nfield. In this case it is possible to obtain analytical expressions for the neutrino oscillations\nprobabilities. However, real astrophysical magnetic fields have the complicated spatial struc-\nture. In this section we consider the evolution of the neutrino wave packet in a non-uniform\nmagnetic field.\nThe neutrino wave function evolution in a magnetic field is described by the following\nDirac equation:\n(iγµ∂µ−mi)νi(x, t)−µiΣB(x)νi(x, t) = 0 . (3.36)\nAs it is shown in Section 3.1, for the case of a uniform magnetic field the partial differential\nequation (3.36) can be transformed into an ordinary differential equation (3.3) by the tran-\nsition to the momentum space. However, for the case of a non-uniform magnetic field in the\nmomentum space we obtain the integro-differential equation\niγ0∂tνi(p, t) = (γp+mi)νi(p, t) +µiZ\nΣB(p−q)νi(q, t)dq. (3.37)\nThe Eq. (3.37) can be simplified under the assumption that the spacial scale on which\nthe magnetic field varies (the so-called magnetic field coherence length1λ) is significantly\nlarger that the wave packet width σx.\nAs it is given by Eq.(3.18), the neutrino wave function has the following form\nνh\ni(x, t) =1\nNX\ns,h′Chh′\nise−iEs\ni(p0)t+ip0xexp\u0010\n−(x− ⟨xs\ni(t)⟩)2\n4σ2x\u0011\nuh′\ni, (3.38)\n1The magnetic field coherence length not to be confused with the neutrino oscillations coherence lengths\nthat characterise the damping of the oscillations\n– 9 –where ⟨xs\ni(t)⟩are the average coordinates of the corresponding wave packets for different\nmassive neutrino spin states. It is also shown in Section 3.1, that for the case of high-energy\nneutrinos and realistic values of the interstellar magnetic fields, the loss of the coherence of\ndifferent neutrino spin states does not occur even on the cosmological scales. Therefore, the\nfollowing condition\n|⟨xs\ni(t)⟩ − ⟨xσ\ni(t)⟩| ≪ σx (3.39)\nis satisfied even for the ultra-long baselines. Thus, it can be assumed that ⟨xs\ni(t)⟩=⟨xσ\ni(t)⟩=\n⟨xi(t)⟩in Eq.(3.38) for all neutrino spin numbers s, σ.\nNext, we decompose the strength of the magnetic field near the average coordinate\n⟨xi(t)⟩:\nB(x) =B(⟨xi(t)⟩) + (x− ⟨xi(t)⟩)·∂B(x)\n∂x\f\f\f\f\f\n⟨xi(t)⟩+... (3.40)\nIf the magnetic field remains approximately constant on the wave packet width, the term\ncontaining derivative can be neglected. Consider neutrino oscillations in the interstellar mag-\nnetic fields that have the coherence length λ≳10 parsec, which is many orders of magnitude\nlarger that the neutrino wave packet size. Under a realistic assumption that the magnetic\nfield does not vary significantly on the scale of the neutrino wave packet, i.e. λ≫σx, we\nperform the Fourier transform of (3.36) and arrive to the following equation:\niγ0∂tνi(p, t) = (γ3p+mi)νi(p, t) +µiΣB(⟨xi(t)⟩)νi(p, t). (3.41)\nEq.(3.41) can be further simplified if we suppose that the magnetic field Bremains\napproximately constant on the scale of the distance between the neutrino wave packets. For\nthe ultrarelativistic neutrinos the said distance can be estimated as\n∆xij=L\nc(vi−vj) =∆m2\nij\n2p2L\nc, (3.42)\nwhere Lis the distance from source. For 100 TeV neutrinos propagating for 10 kiloparsec we\nget:∆x12∼10−11cm and ∆x13∼10−9cm, while for 10 MeV supernova neutrinos ∆x12∼1\nkm and ∆x13∼10m. This values are significantly lower than the coherence length of the\ninterstellar magnetic field, and consequently we can assume B(⟨xi(t)⟩)≈B(t).\nThus, we arrive to the following evolution equation for the neutrino wave packet in a\nmagnetic field:\niγ0∂tνi(p, t) = (γ3p+mi)νi(p, t) +µiΣB(t)νi(p, t) = 0 . (3.43)\nInthenextsection, wenumericallysolveEq.(3.43)todescribetheneutrinofluxevolution\nin the Galactic magnetic field.\n4 Flavour compositions of astrophysical neutrinos\n4.1 Flavour compositions of neutrinos from point-like sources\nThe differential flux of the flavour neutrino νβ(β=e, µ, τ) observed by a terrestrial neutrino\ntelescope is given by\ndΦβ(L, E)\ndE=1\n4πL2X\nα=e,µ,τdΦ0\nα(E)\ndEPαβ(L), (4.1)\n– 10 –where dΦ0\nα/dEare the differential fluxes at the source and Lis the distance from the neutrino\nsource to the detector. The initial fluxes dΦ0\nα/dEare energy dependent (see [24]). Here we\nconsider the case of neutrino oscillations in a magnetic field and the oscillations probabilities\nare independent on the energy. As a consequence, the shape of the neutrino energy spec-\ntrum carries no information about the neutrino properties, in particular about the neutrino\nmagnetic moments. Thus, we introduce the integrated neutrino fluxes\nΦβ(L) =1\n4πL2X\nαΦ0\nαPαβ(L), (4.2)\nwhere\nΦβ(L) =ZEmax\nEmindΦβ(L, E)\ndEdE, Φ0\nβ=ZEmax\nEmindΦ0\nβ(E)\ndEdE. (4.3)\nForatypicalneutrinotelescopeweperformintegrationintheenergyrangebetween Emin≈30\nTeV and Emax≈10PeV.\nThe neutrino flavour compositions at the distance Lfrom the source accounting for the\neffects of neutrino oscillations in a magnetic field are given by\nrα(L) =Φα(L)P\nβΦβ(L)=P\nαr0\nαPαβ(L)P\nαβr0αPαβ(L)(4.4)\nwhere r0\nβ= Φ0\nβ/P\nαΦ0\nαare the flavour ratios at the source. To compute the flavour composi-\ntion we consider only the fluxes of the left-handed neutrino, since the right-handed states are\nsterile and invisible at the detectors. Note thatP\nβPαβ(L)≤1due to neutrino transitions\ninto the sterile states mediated by the interaction with a magnetic field.\nNow consider the case of a uniform magnetic field. For the special case µ1=µ2=µ3,\nEq. (4.4) yields\nrα=X\nβr0\nβX\ni|Uαi|2|Uβi|2, (4.5)\nwhich coincides with the flavour compositions predicted by vacuum neutrino oscillations case.\nThe numerical solution confirms that this feature retains for non-uniform magnetic fields.\nHowever, if at least two of the neutrino diagonal magnetic moments are not equal to each\nother and have sufficiently large values, the neutrino flavour compositions become distance\ndependent and in general differ from the vacuum oscillations case.\n4.2 Flavour ratios of neutrinos from the Galactic centre\nWe examine possible effects of nonzero neutrino magnetic moments on flavour composition\nof the high-energy neutrino fluxes coming from specific galactic sources. Although they have\nnot yet been observed, there are a number of proposed high-energy neutrino sources within\nour Galaxy, such as supermassive black hole Sagittarius A∗located at the Galactic center,\nsupernova remnant RX J1713.7-3946, star forming region Cygnus and others. Note that\nsince radii of these objects are approximately in the range ∼10÷100parsec and are much\nsmaller than the oscillations length in the galactic magnetic field, we can consider them as\npoint-like sources. For studying effects of the neutrino interaction with a magnetic field,\nwe are particularly interested in neutrinos coming from the supermassive black hole Sgr A∗\nsince prior to detection they propagate a long distance in the Galactic magnetic field, namely\nL= 8178 ±13stat±22sys[78].\n– 11 –To describe the magnetic field of the Galactic plane, we use the model from [79]. Here\nwe consider the neutrino interaction with the regular component of the magnetic field. The\ninteraction with a stochastic magnetic field for the solar neutrinos (see [80]) and for neutrinos\npropagatingininterstellarmagneticfield(see[81])leadstotheadditionaldampingofneutrino\noscillations. However, this effect arises due to the terms proportional to µ2\niand are small for\nneutrino interacting with the Galactic magnetic field.\nThe modeling of the high-energy neutrino flavour composition involves a sequence of the\nfollowing steps:\n1) the solution Eq. (3.43) numerically for different values of neutrino magnetic moments\nµiand different values of the magnetic field model parameters drawn from 3σintervals,\n2) the calculation of the probabilities of neutrino flavour oscillations using particular values\nof neutrino mixing parameters,\n3) the computation of the flavour ratios using Eq. (4.4) for a particular initial flavour\nratios r0.\nIn Figure 2 and Figure 3 we show the flavour compositions of high-energy neutrinos\npropagating in the Galactic magnetic field from the Galactic centre assuming π±decay neu-\ntrino production ( r0= (1/3, 2/3, 0)) and µ±decay neutrino production ( r0= (0, 1, 0))\nrespectively. It is assumed that neutrino mixing parameters sin2θ12,sin2θ13,sin2θ23and\nsinδCPare given by the best-fit values from the NuFIT 5.1 global fit [82]. We get that for\nboth cases of π±andµ±decay neutrinos, interaction with the Galactic magnetic field signif-\nicantly modifies the flavour composition predicted by vacuum neutrino oscillations ( µi= 0)\nwhen the neutrino magnetic moments µi∼10−13µB.\nPresently, neutrino mixing parameters sin2θ12,sin2θ13,sin2θ23andsinδCPare mea-\nsured with significant experimental errors. The predicted flavour compositions of high-energy\nneutrinos measured by the neutrino telescopes substantially depend on these errors [59]. In\nFigure 4 and Figure 5 the flavour compositions accounting for 3σuncertainties in neutrinos\nmixing parameters are shown. Accounting for uncertainties in the mixing parameters indeed\nmodifies the flavour compositions calculated using the best fit values and shown in Figure 2\nand 3.\n5 Conclusion\nIn this paper the neutrino oscillations in a magnetic field are considered within the wave\npacket formalism, that allowed us to account for the effects of decoherence due to wave\npacket separation.\nThe expressions for the probabilities of the neutrino flavour and spin oscillations in a\nuniform magnetic field are obtained. The coherence lengths for oscillations on the vacuum\nfrequencies ωvac\nij= ∆m2\nij/4p0and the magnetic frequencies ωB\ni=µiB⊥are calculated. It is\nshown that the coherence length of oscillations on the magnetic frequency is proportional to\np3\n0.\nThe evolution equation for the neutrino wave packet in a non-uniform magnetic field is\nderived. The numerical solution of this equation is obtained for the case of the high-energy\nneutrino propagation in the Galactic magnetic field. To describe the magnetic field of our\nGalaxy we use the model from [79]. We limit ourselves to the case of neutrino propagation in\nthe Galactic magnetic fields since the reliable models of this field exist. However, the results\n– 12 –Figure 2 . Flavour compositions of high-energy neutrinos produced by π±decay after propagating\nfrom the Galactic centre for different ranges of neutrino magnetic moments.\nFigure 3 . Flavour compositions of high-energy neutrinos produced by µ±decay after propagating\nfrom the Galactic centre for different ranges of the neutrino magnetic moments.\nof the present paper can be also applied to describe the neutrino evolution in the extragalactic\nmagnetic fields.\nThe flavour compositions of high-energy neutrinos observed by a terrestrial telescopes\naccounting for the effect of oscillations in the Galactic magnetic field are calculated. We\nassume that high-energy neutrinos are produced by the Galactic centre. Different high-energy\nneutrinos production mechanisms, namely π±decay and µ±decay, are considered. It is shown\nthat for the Dirac neutrinos interaction with a magnetic field can significantly modify the\n– 13 –Figure 4 . Flavour compositions of high-energy neutrinos produced by π±decay after propagating\nfrom the Galactic centre for different ranges of neutrino magnetic moments accounting for current\nexperimental errors of neutrino mixing parameters.\nFigure 5 . Flavour compositions of high-energy neutrinos produced by µ±decay after propagating\nfrom the Galactic centre for different ranges of neutrino magnetic moments accounting for current\nexperimental errors of neutrino mixing parameters.\n– 14 –flavour composition if neutrino magnetic moments are ∼10−13µBand higher, that is an order\nof magnitude smaller than the best current upper-bound. On the contrary, in the Majorana\nneutrino case the flavour composition is not altered by the interaction with a magnetic field.\nWhile computing the flavour compositions, we accounted for experimental uncertainties in\nneutrino mixing parameters and magnetic field model parameters. The results can be also\napplied to describe supernova neutrino fluxes measured by JUNO, Hyper-Kamiokande and\nDUNE (see also a recent paper [48]).\nThe high-energy neutrinos produced by the Galactic sources are not observed yet. Note\nthat the IceCube signal in the track channel from a source located in the southern sky, such\nas the Galactic center, is significantly contaminated by the atmospheric muons background.\nThus, the Galactic centre neutrino emission is currently probed with the cascade channel that\nhas higher angular uncertainty. The deployment of the neutrino telescopes in the Northern\nHemisphere, such as Baikal-GVD and KM3NeT, will make the Galactic centre neutrino emis-\nsion probes accessible via both the track and cascade channels. In [83] the authors estimate\nthat the neutrino telescope, such as KM3NeT and Baikal-GVD, will be able to accumulate\nseveral events a year.\nIn this paper we consider neutrinos coming from the point-like (or individual) sources.\nDue to averaging over the travelled by neutrinos distance, the effects of the neutrino interac-\ntion with a magnetic field are not pronounced in the diffuse high-energy neutrino or supernova\nneutrino fluxes.\nAcknowledgments\nThe work is supported by the Russian Science Foundation under grant No.24-12-00084. 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Chris Hammely\nDepartment of Physics, The Ohio State University, Columbus OH 43210, USA\n(Dated: August 4, 2018)\nWe observe a dependence of the damping of a con\fned mode of precessing ferromagnetic magne-\ntization on the size of the mode. The micron-scale mode is created within an extended, unpatterned\nYIG \flm by means of the intense local dipolar \feld of a micromagnetic tip. We \fnd that damping\nof the con\fned mode scales like the surface-to-volume ratio of the mode, indicating an interfa-\ncial damping e\u000bect (similar to spin pumping) due to the transfer of angular momentum from the\ncon\fned mode to the spin sink of ferromagnetic material in the surrounding \flm. Though unex-\npected for insulating systems, the measured intralayer spin-mixing conductance g\"#= 5:3\u00021019m\u00002\ndemonstrates e\u000ecient intralayer angular momentum transfer.\nSpin pumping driven by ferromagnetic resonance\n(FMR) is a powerful and well-established technique for\ngenerating pure spin currents in magnetic multilayers [1{\n4]. Understanding the mechanism that couples precess-\ning magnetization to spin transport is an important step\nin utilizing this phenomenon. In addition, probing the ef-\nfect of spin pumping on the damping of individual nanos-\ntructures is vital for the development of practical spin-\ntronic devices, such as spin-torque oscillators [5, 6]. Con-\nventional FMR studies at these sub-micron lengthscales\nbecome di\u000ecult due to the confounding e\u000bects arising\nfrom interfaces in multilayer materials and from sensi-\ntivity limitations in detecting lateral transport in single\ncomponent systems at these length scales. Recent stud-\nies have shown that individual nanoscale elements ex-\nhibit size-dependent e\u000bects, such as nonlocal damping\nfrom edge modes [7] and wavevector-dependent damping\nin perpendicular standing spin wave modes [8]. These\nexperiments have revealed the e\u000bect of damping due to\nintralayer spin pumping, which is the transfer of angu-\nlar momentum in systems with spatially-inhomogeneous\ndynamic magnetization.\nA primary challenge in these measurements is distin-\nguishing intralayer spin pumping from other mechanisms\nthat cause variations in linewidth from sample to sample,\nsuch as surface and edge damage [9, 10]. In this paper\nwe measure size-dependent angular momentum transport\nacross a clean interface without growth-de\fned defects or\nlithography-induced edge damage. This is achieved non-\ninvasively in a single sample by con\fning the magneti-\nzation precession to a mode within an area de\fned by\nthe controllable dipolar \feld from a nearby micron-sized\nmagnetic particle [11]. This enables a unique investiga-\ntion of changes in relaxation due to angular momentum\ntransfer across the \feld-de\fned interface between pre-\ncessing magnetization within a mode to the spin sink\nprovided by the surrounding quiescent material.\nWe investigate the size dependence of interfacial damp-\ning using the technique of localized mode ferromagnetic\nresonance force microscopy (FMRFM) [11]. By adjusting\nthe magnitude of the dipolar \feld from the probe we can\nn = 1 n = 2Uniform\nMode200\n150\n100\n50Cantilever amplitude (nm)\n3300 3200 3100 3000\nExternal Field Hext (Oe)Height = 7900 nm\nHeight = 5900 nm\nHeight = 4600 nm\nHeight = 3400 nm\nHeight = 2400 nmprobe\nfield welln=1n=2Field\nLateral positionFIG. 1. Localized mode FMRFM spectra for thin \flm YIG at\nseveral probe-sample separations. The dashed line indicates\nthe position of the uniform mode peak that does not shift\nwith probe-sample separation. As probe-sample separation is\nreduced the localized modes shift to higher \feld relative to\nthe uniform mode peak. Inset: transverse magnetization of\nthe \frst two spin wave modes con\fned by the magnetic \feld\nwell of the probe magnet. The energy of the con\fned modes\nis dictated by the depth of the \feld well.\ncontrol the con\fnement radius. Localized modes have\npreviously been observed in permalloy when the probe\n\feld is out-of-plane [11], in-plane [12] and at intermedi-\nate angles [13]. The azimuthal symmetry of the out-of-\nplane geometry permits simple numerical analysis based\non cylindrically symmetric Bessel function modes with\na well-de\fned localization radius [11], similar to those\nseen in perpendicularly magnetized dots [14]. In addi-\ntion, this geometry eliminates the e\u000bect of eigenmode\nsplitting, which can cause additional broadening [15].\nWe demonstrate the control of con\fnement radius byarXiv:1405.4203v2 [cond-mat.mtrl-sci] 7 Aug 20142\nthe observation of discrete modes in an FMRFM exper-\niment in the out-of-plane geometry in an unpatterned\nepitaxial yttrium iron garnet (YIG) \flm of thickness 25\nnm grown by o\u000b-axis sputtering [16] on a (111)-oriented\nGd3Ga5O12substrate. The probe \feld is provided by\na high coercivity Sm 1Co5particle that is milled to 1.75\n\u0016m after being mounted on an uncoated, diamond atomic\nforce microscope cantilever. The magnetic moment and\ncoercivity of the particle are measured by cantilever mag-\nnetometry to be 3 :9\u000210\u00009emu and 10 kOe respectively.\nWhen the applied \feld is anti-parallel to the tip mo-\nment, the tip creates a con\fning \feld well in the sample\nthat localizes discrete magnetization precession modes\nimmediately beneath it [11, 17], analogous to the discrete\nmodes in a quantum well [18]. The microwave frequency\nmagnetic \feld that excites the precession is provided by\nplacing the sample near a short in a microstrip trans-\nmission line. A force-detected ferromagnetic resonance\nspectrum is obtained by modulating the amplitude of\nthe microwaves at the cantilever frequency ( \u001918 kHz)\nand measuring the change in cantilever amplitude as a\nfunction of swept external magnetic \feld. Measurements\nwere made over a range of microwave frequencies: 2-6.5\nGHz.\nFigure 1 shows the evolution of the FMRFM spectra as\na function of tip-sample separation obtained at a particu-\nlar microwave frequency of 4 GHz. At large probe-sample\nseparation we observe a peak at the expected resonance\n\feld for the uniform mode in the out-of-plane geometry.\nAs expected, several discrete peaks emerge and shift to-\nward higher applied \feld as the probe-sample separation\ndecreases, thus increasing the (negative) probe \feld at\nthe sample, while the uniform mode stays at constant\nresonance \feld. The resonance frequency !of a con\fned\nmode for wavevectors kt\u001c1 is given by [11]\n!\n\r=Hext\u00004\u0019Ms+hHpi+\u0019Mskt+ 4\u0019Msaexk2(1)\nwhere\r= 2\u0019\u00022:8 MHz/Oe is the gyromagnetic ratio,\nHextis the external applied magnetic \feld, 4 \u0019Ms= 1608\nOe is the saturation magnetization, aex= 3:6\u000210\u000012\ncm2is the exchange constant of the material, kis the\nwavevector of the mode, tis the thickness of the \flm and\nhHpiis the spatial average of the dipole \feld from the\nprobe magnet weighted by the mode m\nhHpi=R\nSHp(r)m2(r) d2rR\nSm2(r) d2r(2)\nThe \flm is su\u000eciently thin relative to the size of the\nprobe particle that the dipole \feld is constant across the\nthickness of the \flm, and so the integration is performed\nover the sample surface S. Both the averaged probe \feld\nhHpiand the wavevector kare functions of the mode\nshape and mode radius R, so the frequency is obtained\nby numerical minimization with variation of radius [11].\nDue to cylindrical symmetry the magnetization pro\fle of\nMnM=M1\nMnM=M2\nMnM=M3\nMnM=M43300\n3250\n3200\n3150\n3100\n3050\n3000ResonanceMFieldM(Oe)\n10 8 6 4 2\nProbe-sampleMseparationM(mm)12\n10\n8\n6\n4\n2\n0ModeMRadiusM(mm)\n10 8 6 4 2\nProbe-sampleMseparationM(mm)MnM=M1\nMnM=M2\nMnM=M3FIG. 2. Resonance \felds of the \frst four localized modes\nas a function of probe-sample separation at 4 GHz. Filled\nmarkers indicate experimental peaks and solid lines indicate\nexpected resonance \feld obtained numerically. Inset: radius\nof the \frst three localized modes obtained from the numerical\nminimization procedure described in the text.\nthe mode can be described by a zeroth order Bessel func-\ntionm=J0(kr), with boundary conditions that de\fne\ndiscrete wavevectors kn=\u001fn=Rwhere\u001fnare the zeros\nof the Bessel function J0(\u001fn) = 0. The minimization of\nfrequency at \fxed \feld is equivalent to the maximiza-\ntion of \feld at \fxed frequency. Hence the deeper \feld\nwell shifts the modes to higher \feld when the microwave\nfrequency is \fxed, as seen in Fig. 1. This modeling pro-\ncedure provides both the resonance \feld and the radius\nof the mode, and these are given in Fig. 2. We see that\nthe resonance \felds of the experimental peaks are well\ndescribed by the model, con\frming the accuracy of the\ncalculated mode radius.\nTo measure damping of a con\fned mode we obtain\nFMRFM spectra for a \fxed mode radius Rat multiple\nfrequencies, one example of which can be seen in Fig. 3.\nThe \feld shift of the localized modes, relative to the uni-\nform mode Huniform =!\n\r+ 4\u0019Msis constant for a \fxed\nwavevector k=kn=\u001fn=R, independent of frequency !,\nas predicted by Equation (1).\nBy \ftting a Lorentzian lineshape to the n = 1 and\nn = 2 peaks we obtain the full-width at half-maximum\nlinewidth of the localized modes and plot this as a func-\ntion of microwave frequency to separate intrinsic and ex-\ntrinsic linewidth broadening mechanisms [19]. Following\nfrom the Landau-Lifshitz-Gilbert equation, the linewidth\n\u0001His given by\n\u0001H= \u0001H0+2\u000b!\n\r(3)3\nCantilever amplitude (nm)120\n100\n80\n60\n40\n20\n80 40 0 -40\nH-Huniform (Oe)2GHz\n4GHz\n6GHzUniform\nModen = 1 n = 2\nFIG. 3. FMRFM spectra at multiple microwave frequencies\nat a \fxed probe-sample separation of 3700 nm, equivalent to\na mode radius R= 1860 nm. Spectra are o\u000bset for clarity and\nthe external \feld His plotted relative to the uniform mode\nresonance \feld Huniform =!\n\r+ 4\u0019Ms.\nwhere the slope and intercept of the frequency-dependent\nlinewidth measure, respectively, the Gilbert damping pa-\nrameter\u000band inhomogeneous broadening \u0001 H0due to\nspatial variation of magnetic properties. We measure\nthis frequency dependence at several probe-sample sepa-\nrations corresponding to several mode radii R.\nThe key result of our study is the observation of en-\nhanced damping that is unambiguously dependent on the\nradius of the mode, as seen from the change in slope of the\n\frst localized mode linewidth with mode radius as seen\nin Fig. 4. The Gilbert damping parameter \u000b, for both\nthe \frst and second localized modes, shows a surprising\nlinear behavior when plotted against R\u00001, the recipro-\ncal of the mode radius, as seen in Fig. 5. An enhanced\ndamping is reminiscent of spin pumping observed when\na ferromagnetic layer is placed in contact with a normal\nmetal layer [1]. In this bilayer geometry the damping\nenhancement \u000bspscales inversely with thickness tof the\nFM \flm, which is equal to the ratio of the area of the\nferromagnet/metal interface to the volume of the ferro-\nmagnet, and is given by [20]\n\u000bsp=\r~g\"#\n4\u0019Ms1\nt(4)\nwhere ~is the reduced Planck constant and g\"#is the\nspin-mixing conductance parameter that describes the ef-\n\fciency of spin pumping. By analogy to this interfacial\ndamping due to spin pumping we suggest the possibility\nof an interfacial damping mechanism for con\fned modes\nFirst Localized Mode Linewidth DH (Oe)\nFrequency (GHz)10\n8\n6\n4\n2\n0\n8 6 4 2 0 R = 1600 nm | a = 1.3 x 10-3\n R = 1860 nm | a = 1.1 x 10-3\n R = 2230 nm | a = 1.0 x 10-3FIG. 4. Linewidths of \frst localized mode for mode radii R=\n1600 nm (red triangles), R= 1860 nm (blue squares) and R=\n2230 nm (green diamonds). Filled markers are experimental\nlinewidths and solid lines are linear \fts to the data. Gilbert\ndamping parameters \u000bare determined for each mode radius\nfrom the slope of the linear \ft.\nthat scales with the surface-to-volume ratio of the mode,\nwhere the volume of the on-resonant disk-like mode is\n\u0019R2tand relaxation to the surrounding material, which\nis o\u000b resonance, occurs through the curved surface 2 \u0019Rt\naround the edge of the disc. Hence, the enhanced damp-\ning of a con\fned mode with radius Ris\n\u000bsp=\r~g\"#\n4\u0019Ms2\nR(5)\nFrom Equation (5) and the linear \ft (solid black line)\nto the enhanced damping versus mode the reciprocal of\nthe mode radius, as shown in Fig. 5, we obtain g\"#=\n(5:3\u00060:2)\u00021019m\u00002for this system.\nIt is interesting and somewhat remarkable that we\nobserve angular momentum transport in this insulating\nsystem and that its e\u000eciency, characterized by g\"#, is\nlarger than the spin-mixing conductance measured in\nYIG-metal bilayers [16, 22, 23]. We suggest that g\"#\nmeasured in this study is an intralayer spin-mixing con-\nductance that describes a generalization of spin pump-\ning as the transport of energy and angular momentum\nfrom an on-resonance spin source to an o\u000b-resonance\nspin sink, even in the absence of both a material in-\nterface [7] and conduction electrons [24]. We describe\nthis e\u000bect as YIG-YIG intralayer spin pumping: the en-\nergy and angular momentum from the precessing con-\n\fned mode can be absorbed by the surrounding ferro-\nmagnetic material of the unpatterned \flm, as depicted4\n2.4x10-3\n2.0\n1.6\n1.2\n0.8\n0.4\n0.0\n20x103 15 10 5 0\nSurface/Volumed=d2/Rdbcm-1Ld1stdlocalizeddmode\nd2nddlocalizeddmode\ndk-dependentdintralayerdspindpumpingdb1stdModeL\ndk-dependentdintralayerdspindpumpingdb2nddModeL\ndconfined-modedintralayerdspindpumpingGilbertddampingda\n0mtmaxLocalized\nPrecession\nVolume\nAngular\nMomentum\nTransfer\nFIG. 5. Comparison of the measured size-dependent Gilbert\ndamping parameter \u000bof the \frst two localized modes with\ntheory. The solid black line is a linear \ft to con\fned-mode\nintralayer spin pumping that scales with the surface-volume\nratio of the mode as described by Eq. (5). The dashed red line\nis a \ft to the \frst localized mode linewidth using wavevector-\ndependent intralayer spin pumping theory [21] that scales as\nk2[7]. The dashed blue line is the prediction of wavevector-\ndependent intralayer spin pumping for the second localized\nmode. Inset: Cross-section showing intralayer angular mo-\nmentum transfer from the volume of the con\fned mode to\nsurrounding material through the surface of the mode. Color\nscale denotes magnitude of the transverse, precessing magne-\ntizationmt.\nin the inset of Fig. 5. The relatively large value of\ng\"#= (5:3\u00060:2)\u00021019m\u00002we obtain for YIG-YIG can\nbe compared to g\"#= (6:9\u00060:6)\u00021018m\u00002previously\nmeasured for YIG-Pt [23]. This enhancement may arise\nbecause the interface, rather than involving a material\ndiscontinuity, is de\fned by a magnetic \feld that occurs in\na uniform, essentially defect-free \flm leading to a strong\n\"interfacial\" coupling characterized by the YIG-YIG ex-\nchange interaction itself. In addition, it might be unex-\npected for the con\fned mode to relax via the surrounding\nmaterial where the lowest energy state, which is the uni-\nform mode, is well above the energy of the con\fned mode\ninside the well. However, previous experiments by Hein-\nrich et al. [4, 20] have shown that ferromagnets do act\nas good spin sinks when the precession frequency of the\nspin current source is not at a resonance frequency of the\nspin sink ferromagnet.\nWe consider the possible role of transverse spin di\u000bu-\nsion [21] used previously to describe enhanced damping\ndue to the interaction between itinerant electrons and\nspatially-inhomogeneous dynamic magnetization [7, 8].\nWe \fnd that the prediction of this wavevector-dependent\nintralayer spin pumping theory does not agree with ourexperimental data. In particular, this enhanced damp-\ning due to intralayer spin pumping is predicted [21] to\ndepend on wavevector k:\n\u000bsp=\u001bT\r\nMsk2(6)\nwhere\u001bTis the transverse spin conductivity and the\nwavevector k=\u001fn=Ris given by the Bessel zeros \u001f1\n= 2.405,\u001f2= 5.520. The spin conductivity due to itin-\nerant electrons is expected to be zero in YIG, but we\nnevertheless allow it to be a free parameter and \ft to the\n\frst localized mode linewidth; this \ft to the wavevector-\ndependent intralayer spin pumping theory is shown as\nthe red dashed line in Fig. 5. We \fnd that the spin con-\nductivity that describes this \ft, \u001bT= 1:5\u000210\u000022kg m/s,\nis two orders of magnitude larger than that measured in\na metallic ferromagnet [7]. In addition, using the same\nspin conductivity to estimate the linewidth of the second\nlocalized mode (blue dashed line) results in a prediction\nthat does not accurately describe the measured second\nmode linewidth (blue solid circles), while con\fned-mode\nintralayer spin pumping that scales as the surface-volume\nratio of the mode (black solid line) described by Eq. (5)\naccurately describes both sets of data. Hence our ob-\nservations do not follow the wavevector-dependent in-\ntralayer spin pumping theory observed previously [7, 8],\nbut manifests as a surface-volume intralayer relaxation\nspeci\fc to spatially-con\fned precession within an ex-\ntended \flm, previously predicted for nanocontact spin-\ntorque oscillators [25].\nOther mechanisms for linewidth broadening are ruled\nout by analysis of the phenomenology of our result. The\ndipolar \feld from the micromagnetic tip is a potential\nsource of linewidth broadening as it is produces an in-\nhomogneous \feld in the sample of several hundred gauss\nthat would dominate inhomogeneous spectral broaden-\ning in a paramagnetic sample [26, 27]. Inhomogeneous\nbroadening from the tip can be ruled out as the source of\nincreased damping in this study for two reasons. First,\nany inhomogeneous broadening would be frequency in-\ndependent, and hence would lead to a change in the in-\ntercept of the frequency-dependence of linewidth shown\nin Fig. 4, while the change in slope alone is a clear indi-\ncation of a Gilbert damping enhancement. Second, the\nferromagnetic resonance excitations of a ferromagnet are\neigenmodes [11, 26], in which the inhomogeneous \feld\nfrom the tip is cancelled by the dynamic \feld from the\nprecession. This allows the e\u000bective \feld to be equal at\nevery position inside the mode, and hence it can be de-\nscribed as an eigenmode with a single well-de\fned eigen-\nfrequency. Other well-established mechanisms for size-\nor wavevector-dependent relaxation can also be elimi-\nnated due to their insu\u000ecient magnitude and di\u000bering\nphenomenology; 3-magnon con\ruence [28, 29] manifests\nas a linewidth broadening that is linear in kbut indepen-\ndent of frequency, while 4-magnon scattering [30] scales\nask2.5\nTo conclude, we observe robust intralayer spin pump-\ning within an insulating ferromagnet, which manifests\nas enhanced damping of micrometer-scale con\fned spin\nwave modes. This result has consequences for devices\nthat induce spin precession in con\fned regions, such\nas spin-torque oscillators in the nanocontact geometry\n[31, 32]. In addition, our study highlights the power of\nlocalized mode FMRFM for illuminating local spin dy-\nnamics and in particular for spectroscopic studies of the\nimpact of mode relaxation across a controllable, \feld-\nde\fned interface.\nThe authors wish to thank Yaroslav Tserkovnyak for\nuseful discussions. This work was primarily supported\nby the U.S. Department of Energy (DOE), O\u000ece of Sci-\nence, Basic Energy Sciences (BES), under Award # DE-\nFG02-03ER46054 (FMRFM measurement) and Award #\nDE-SC0001304 (sample synthesis). This work was par-\ntially supported by the Center for Emergent Materials,\nan NSF-funded MRSEC under award # DMR-0820414\n(structural characterization). This work was supported\nin part by Lake Shore Cryotronics (magnetic charac-\nterization) and an allocation of computing time from\nthe Ohio Supercomputer Center (micromagnetic simu-\nlations). We also acknowledge technical support and as-\nsistance provided by the NanoSystems Laboratory at the\nOhio State University.\n\u0003fyyang@physics.osu.edu\nyhammel@physics.osu.edu\n[1] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[2] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Phys. Rev. B 66, 060404 (2002).\n[3] X. Wang, G. E. W. Bauer, B. J. van Wees, A. 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Nikoli´ c∗\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\n(Dated: December 15, 2023)\nUnderstanding spin wave (SW) damping, and how to control it to the point of being able to\namplify SW-mediated signals, is one of the key requirements to bring the envisaged magnonic tech-\nnologies to fruition. Even widely used magnetic insulators with low magnetization damping in their\nbulk, such as yttrium iron garnet, exhibit 100-fold increase in SW damping due to inevitable con-\ntact with metallic layers in magnonic circuits, as observed in very recent experiments [I. Bertelli\net al. , Adv. Quantum Technol. 4, 2100094 (2021)] mapping SW damping in spatially-resolved\nfashion. Here, we provide microscopic and rigorous understanding of wavevector-dependent SW\ndamping using extended Landau-Lifshitz-Gilbert equation with nonlocal damping tensor , instead\nof conventional local scalar Gilbert damping, as derived from Schwinger-Keldysh nonequilibrium\nquantum field theory. In this picture, the origin of nonlocal magnetization damping and thereby in-\nduced wavevector-dependent SW damping is interaction of localized magnetic moments of magnetic\ninsulator with conduction electrons from the examined three different types of metallic overlayers—\nnormal, heavy, and altermagnetic. Due to spin-split energy-momentum dispersion of conduction\nelectrons in the latter two cases, the nonlocal damping is anisotropic in spin and space, and it can\nbe dramatically reduced by changing the relative orientation of the two layers when compared to\nthe usage of normal metal overlayer.\nIntroduction. —Spin wave (SW) or magnon damping\nis a problem of great interest to both basic and ap-\nplied research. For basic research, its measurements [1–4]\ncan reveal microscopic details of boson-boson or boson-\nfermion quasiparticle interactions in solids, such as:\nmagnon-magnon interactions (as described by second-\nquantized Hamiltonians containing products of three or\nmore bosonic operators [5, 6]), which are frequently en-\ncountered in antiferromagnets [4, 5] and quantum spin\nliquids [7], wherein they play a much more important\nrole [8] than boson-boson interactions in other condensed\nphases, like anharmonic crystalline lattices or superflu-\nids [5]; magnon-phonon interactions [3], especially rel-\nevant for recently discovered two-dimensional magnetic\nmaterials [2]; and magnon-electron interactions in mag-\nnetic metals [1, 9–12]. For the envisaged magnon-\nbased digital and analog computing technologies [13–\n17], understanding magnon damping makes it possible\nto develop schemes to suppress [18] it, and, further-\nmore, achieve amplification of nonequilibrium fluxes of\nmagnons [19–22]. In fact, overcoming damping and\nachieving amplification is the keyto enable complex\nmagnon circuits where, e.g., a logic gate output must\nbe able to drive the input of multiple follow-up gates.\nLet us recall that the concept of SW was introduced by\nBloch [23] as a wave-like disturbance in the local mag-\nnetic ordering of a magnetic material. The quanta [6] of\nenergy of SWs of frequency ωbehave as quasiparticles\ntermed magnons, each of which carries energy ℏωand\nspin ℏ. As regards terminology, we note that in magnon-\nics [13] SW is often used for excitations driven by an-\ntennas [24–27] and/or described by the classical Landau-\nLifshitz-Gilbert (LLG) equation [9, 10, 28, 29], whereas\nmagnon is used for the quantized version of the same ex-\ne\nee\nee\neFIG. 1. (a) Schematic view of bilayers where a metallic over-\nlayer covers the top surface of magnetic insulator, as often\nencountered in spintronics and magnonics [13, 30]. Three\ndifferent energy-momentum dispersion of conduction elec-\ntrons at the interface are considered, with their Fermi sur-\nfaces shown in panel (b)—normal metal (NM); heavy metal\n(HM) with the Rashba SOC [31, 32], and altermagnetic metal\n(AM) [33, 34]—with the latter two being spin-split. The rel-\native alignment of the layers is labeled by an angle θ[33, 34],\nmeaning that the wavevector qof SWs within FI is at an an-\ngleθaway from the kx-axis.\ncitation [5], or these two terms are used interchangingly.\nIn particular, experiments focused on SW damp-\ning in metallic ferromagnets have observed [1] its de-\npendence on the wavevector qwhich cannot be ex-\nplained by using the standard LLG equation [28,\n29],∂tMn=−Mn×Beff\nn+αGMn×∂tMn(where ∂t≡\n∂/∂t), describing dynamics of localized magnetic mo-\nments (LMMs) Mnat site nof crystalline lattice (also\nused in atomistic spin dynamics [28]) viewed as classi-\ncal vectors of unit length. This is because αG, as the\nGilbert damping parameter [35, 36], is a local scalar (i.e.,\nposition-independent constant). Instead, various forms\nof spatially nonuniform (i.e., coordinate-dependent) and\nnonlocal (i.e., magnetization-texture-dependent) damp-\ning due to conduction electrons have been proposed [9,arXiv:2312.09140v1 [cond-mat.mes-hall] 14 Dec 20232\n10, 37–39], or extracted from first-principles calcula-\ntions [40], to account for observed wavevector-dependent\ndamping of SWs, such as ∝q2(q=|q|) measured in\nRef. [1]. The nonlocal damping terms require neither\nspin-orbit coupling (SOC) nor magnetic disorder scatter-\ning, in contrast to αGwhich is considered to vanish [41]\nin their absence.\nThus, in magnonics, it has been considered [30] that\nusage of magnetic insulators, such as yttrium iron gar-\nnet (YIG) exhibiting ultralow αG≃10−4(achieved on\na proper substrate [42]), is critical to evade much larger\nand/or nonlocal damping of SWs found in ferromagnetic\nmetals. However, very recent experiments [24–27] have\nobserved 100-fold increase of SW damping in the segment\nof YIG thin film that was covered by a metallic overlayer.\nSuch spatially-resolved measurement [24] of SW damp-\ning was made possible by the advent of quantum sensing\nbased on nitrogen vacancy (NV) centers in diamond [43],\nand it was also subsequently confirmed by other meth-\nods [25–27]. Since excitation, control, and detection of\nSWs requires to couple YIG to metallic electrodes [13],\nunderstanding the origin and means to control/suppress\nlarge increase in SW damping underneath metallic over-\nlayer is crucial for realizing magnonic technologies. To\nexplain their experiments, Refs. [24–27] have employed\nthe LLG equation with ad hoc intuitively-justified terms\n(such as, effective magnetic field due to SW induced eddy\ncurrents within metallic overlayer [24]) that can fit the\nexperimental data, which is nonuniversal and unsatisfac-\ntory (many other examples of similar phenomenological\nstrategy exist [1, 44]).\nIn contrast, in this Letter we employ recently derived\n∂tMn=−Mn×Beff\nn+Mn×X\nn′(αGδnn′+λR)·∂tMn′,\n(1)\nextended LLG equation with all terms obtained [45]\nmicroscopically from Schwinger-Keldysh nonequilibrium\nquantum field theory [46] and confirmed [45] via exact\nquantum-classical numerics [47–50]. It includes nonlo-\ncal damping as the third term on the right-hand side\n(RHS), where its nonlocality is signified by dependence\nonR=rn−rn′, where rnis the position vector of lat-\ntice site n. Equation (1) is applied to a setup depicted in\nFig. 1 where conduction electron spins from three differ-\nent choices for metallic overlayer are assumed to interact\nwith LMMs of ferromagnetic insulator (FI) at the inter-\nface via sdexchange interaction of strength Jsd, as well\nas possibly underneath the top surface of FI because of\nelectronic evanescent wavefunction penetrating into it.\nNote that FI/normal metal (NM) bilayer directly mod-\nels recent experiments [24] where FI was a thin film of\nYIG and NM was Au, and SW damping within FI was\nquantified using quantum magnetometry via NV centers\nin diamond. Next, the FI/heavy metal (HM) bilayer,\nsuch as YIG/Pt [18, 27], is frequently encountered in\n0 2 4\nK/J0.51.0q (1/a)\n kF= 0\n.92kF= 0\n.99kF= 1\n.08kF= 1\n.15kF= 1\n.22kF= 1\n.30kF= 1\n.38(a)\n1.0 1.2 1.4\nkF0.81.01.21.4qmax(1/a)\n∝kF(b)FIG. 2. (a) Wavevector qof SW generated by injecting spin-\npolarized current in TDNEGF+LLG simulations of NM over-\nlayer on the top of 1D FI [Fig. 1(a)] as a function of anisotropy\nK[Eq. (3)] for different electronic Fermi wavevectors kF. (b)\nMaximum wavevector qmaxof SWs that can be generated by\ncurrent injection [21, 57] before wavevector-dependent SW\ndamping becomes operative, as signified by the drop around\nkFin curves plotted in panel (a).\nvarious spintronics and magnonics phenomena [13, 30].\nFinally, due to recent explosion of interest in altermag-\nnets [33, 34], the FI/altermagnetic metal (AM) bilay-\ners, such as YIG/RuO 2, have been explored experimen-\ntally to characterize RuO 2as a spin-to-charge conver-\nsion medium [51]. The Schwinger-Keldysh field theory\n(SKFT), commonly used in high energy physics and cos-\nmology [52–54], allows one to “integrate out” unobserved\ndegrees of freedom, such as the conduction electrons in\nthe setup of Fig. 1, leaving behind a time-retarded dis-\nsipation kernel [48, 55, 56] that encompasses electronic\neffects on the remaining degrees of freedom. This ap-\nproach then rigorously yields the effective equation for\nLMMs only , such as Eq. (1) [45, 56] which bypasses the\nneed for adding [1, 24, 44] phenomenological wavevector-\ndependent terms into the standard LLG equation. In\nour approach, the nonlocal damping is extracted from\nthe time-retarded dissipation kernel [45].\nSKFT-based theory of SW damping in FI/metal bilay-\ners.—The nonlocal damping [45] λRin the third term\non the RHS of extended LLG Eq. (1) stems from back-\naction of conduction electrons responding nonadiabati-\ncally [48, 59]—i.e., with electronic spin expectation value\n⟨ˆsn⟩being always somewhat behind LMM which gener-\nates spin torque [60] ∝ ⟨ˆsn⟩×Mn—to dynamics of LMMs.\nIt is, in general, a nearly symmetric 3 ×3 tensor whose\ncomponents are given by [45]\nλαβ\nR=−J2\nsd\n2πZ\ndε∂f\n∂εTr\u0002\nσαAnn′σβAn′n\u0003\n. (2)\nHere, f(ε) is the Fermi function; α, β =x, y, z ;σα\nis the Pauli matrix; and A(ε) = i\u0002\nGR(ε)−GA(ε)\u0003\nis\nthe spectral function in the position representation ob-\ntained from the retarded/advanced Green’s functions\n(GFs) GR/A(ε) =\u0000\nε−H±iη\u0001−1. Thus, the calcula-\ntion of λRrequires only an electronic Hamiltonian H\nas input, which makes theory fully microscopic (i.e.,3\n−5 0 5\nX−505Y\nλNM\nR(a)NM\n−1 0 1\nλα\nR\n−5 0 5\nX−505Y\nλxx\nR(b)HM t SOC= 0.3t0\n−5 0 5\nX−505Y\nλzz\nR(c)\n−5 0 5\nX−505Y\nλ⊥\nR(d)AM t AM= 0.5t0\n0.0 0.5 1.0 1.5\nq (1/a)0123Γq≡Im(ωq) (J/¯ h)×10−1\nEq.(6)(e)\nη= 0.1\nη= 0\n−5 0 5\nX−505Y\nλyy\nR(f)\n−5 0 5\nX−505Y\nλxy\nR(g)\n−5 0 5\nX−505Y\nλ/bardbl\nR(h)\nFIG. 3. (a)–(d) and (f)–(h) Elements of SKFT-derived nonlocal damping tensor in 2D FI, λRwhere R= (X, Y, Z ) is the\nrelative vector between two sites within FI, covered by NM [Eq. (5)], HM [Eqs. (8)] or AM [Eqs. (9)] metallic overlayer. (e)\nWavevector-dependent damping Γ qof SWs due to NM overlayer, where the gray line is based on Eq. (6) in the continuous\nlimit [58] and the other two lines are numerical solutions of extended LLG Eq. (1) for discrete lattices of LMMs within FI. The\ndotted line in (e) is obtained in the absence of nonlocal damping ( η= 0), which is flat at small q.\nHamiltonian-based). Although the SKFT-based deriva-\ntion [45] yields an additional antisymmetric term, not\ndisplayed in Eq. (2), such term vanishes if the system\nhas inversion symmetry. Even when this symmetry is\nbroken, like in the presence of SOC, the antisymmet-\nric component is often orders of magnitude smaller [56],\ntherefore, we neglect it. The first term on the RHS of ex-\ntended LLG Eq. (1) is the usual one [28, 29], describing\nprecession of LMMs in the effective magnetic field, Beff\nn,\nwhich is the sum of both internal and external ( Bextez)\nfields. It is obtained as Beff\nn=−∂H/∂Mnwhere His\nthe classical Hamiltonian of LMMs\nH=−JX\n⟨nn′⟩Mn·Mn′+K\n2X\nn(Mz\nn)2−BextX\nnMz\nn.(3)\nHere we use g= 1 for gyromagnetic ratio, which sim-\nplifies Eq. (1); Jis the Heisenberg exchange coupling\nbetween the nearest-neighbors (NN) sites; and Kis the\nmagnetic anisotropy.\nWhen nonlocal damping tensor, λRis proportional\nto 3×3 identity matrix, I3, a closed formula for the\nSW dispersion can be obtained via hydrodynamic the-\nory [58]. In this theory, the localized spins in Eq. (1),\nMn= (Re ϕn,Imϕn,1−m)T, are expressed using com-\nplex field ϕnand uniform spin density m≪1. Then,\nusing the SW ansatz ϕn(t) =P\nqUqei(q·rn−ωqt), we ob-\ntain the dispersion relation for the SWs\nωq= (Jq2+K−B)\u0002\n1 +i(αG+˜λq)\u0003\n, (4)\nwhere qis the wavevector and ωis their frequency. Thedamping of the SW is then given by the imaginary part\nof the dispersion in Eq. (4), Γ q≡Imωq. It is comprised\nby contributions from the local scalar Gilbert damping\nαGand the Fourier transform of the nonlocal damping\ntensor, ˜λq=R\ndrnλrneiq·rn.\nResults for FI/NM bilayer. —We warm up by extract-\ning Γ qfor the simplest of the three cases in Fig. 1, a\none-dimensional (1D) FI chain under a 1D NM over-\nlayer with spin-degenerate quadratic electronic energy-\nmomentum dispersion, ϵkσ=t0k2\nx, where t0=ℏ2/2m.\nThe GFs and spectral functions in Eq. (2), can be\ncalculated in the momentum representation, yielding\nλ1D\nR=2J2\nsd\nπv2\nFcos2(kFR)I3, where vFis the Fermi velocity,\nR≡ |R|, and kFis the Fermi wavevector. Moreover, its\nFourier transform, ˜λq=2J2\nsd\nv2\nF[δ(q) +δ(q−2kF)/2], dic-\ntates additional damping to SWs of wavevector q=\n0,±2kF. Although the Dirac delta function in this ex-\npression is unbounded, this unphysical feature is an ar-\ntifact of the small amplitude, m≪1, approximation\nwithin the hydrodynamic approach [58]. The features of\nsuch wavevector-dependent damping in 1D can be cor-\nroborated via TDNEGF+LLG numerically exact simu-\nlations [47–50] of a finite-size nanowire, similar to the\nsetup depicted in Fig. 1(a) but sandwiched between two\nNM semi-infinite leads. For example, by exciting SWs\nvia injection of spin-polarized current into the metallic\noverlayer of such a system, as pursued experimentally in\nspintronics and magnonics [21, 57], we find in Fig. 2(a)\nthat wavevector qof thereby excited coherent SW in-4\ncreases with increasing anisotropy K. However, the max-\nimum wavevector qmaxis limited by kF[Fig. 2(b)]. This\nmeans that SWs with q≳kFare subjected to additional\ndamping, inhibiting their generation. Although our an-\nalytical results predict extra damping at q= 2kF, finite\nsize effects and the inclusion of semi-infinite leads in TD-\nNEGF+LLG simulations lower this cutoff to kF.\nSince SW experiments are conducted on higher-\ndimensional systems, we also investigate damping\non SWs in a two-dimensional (2D) FI/NM bilayer.\nThe electronic energy-momentum dispersion is then\nϵkσ=t0(k2\nx+k2\ny), and the nonlocal damping and its\nFourier transform are given by\nλNM\nR=k2\nFJ2\nsd\n2πv2\nFJ2\n0(kFR)I3, (5)\n˜λNM\nq=kFJ2\nsdΘ(2kF−q)\n2πv2\nFqp\n1−(q/2kF)2, (6)\nwhere Jn(x) is the n-th Bessel function of the first kind,\nand Θ( x) is the Heaviside step function. The nonlo-\ncal damping in Eqs. (5) and (6) is plotted in Fig. 3(a),\nshowing realistic decay with increasing R, in contrast to\nunphysical infinite range found in 1D case. Addition-ally, SW damping in Eq. (6) is operative for wavectors\n0≤q≤2kF, again diverging for q= 0,2kFdue to arti-\nfacts of hydrodynamic theory [58]. Therefore, unphysical\ndivergence can be removed by going back to discrete lat-\ntice, such as solid curves in Fig. 3(e) obtained for n=1–\n100 LMMs by solving numerically a system of coupled\nLLG Eq. (1) where λRin 2D is used [45]. In this numer-\nical treatment, we use kF= 0.5a−1where ais the lattice\nspacing; k2\nFJ2\nsd/2πv2\nF=η= 0.1;K= 0; Bext= 0.1J;\nandαG= 0.1.\nResults for FI/HM bilayer. —Heavy metals (such as of-\nten employed Pt, W, Ta) exhibit strong SOC effects due\nto their large atomic number. We mimic their presence at\nthe FI/HM interface [31] by using 2D energy-momentum\ndispersion ϵk=t0(k2\nx+k2\ny) +tSOC(σxky−σykx), which\nincludes spin-splitting due to the Rashba SOC [31, 32].\nUsing this dispersion, Eq. (2) yields\nλHM\nR=\nλxx\nRλxy\nR0\nλxy\nRλyy\nR0\n0 0 λzz\nR\n, (7)\nfor the nonlocal damping tensor. Its components are, in\ngeneral, different from each other\nλxx\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n+ cos(2 θ)\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8a)\nλyy\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n−cos(2 θ)\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8b)\nλzz\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n−\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8c)\nλxy\nR=−J2\nsdsin(2θ)\n4π\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\n, (8d)\nwhere kF↑andkF↓are the spin-split Fermi wavevec-\ntors [Fig. 1(b)], and θis the relative orientation angle\n[Fig. 1(b)] between the SW wavevector qand the kxdi-\nrection. Thus, the nonlocal damping tensor in Eq. (7)\ngenerated by HM overlayer is anisotropic in spin due to\nits different diagonal elements, as well as nonzero off-\ndiagonal elements. It is also anisotropic in space due to\nits dependence on the angle θ. Its elements [Eqs. (8)]\nare plotted in Figs. 3(b), 3(c), 3(f), and 3(g) using\ntSOC= 0.3t0. They may become negative, signifying the\npossibility of antidamping torque [21] exerted by conduc-\ntion electrons. However, the dominant effect of nearby\nLMMs and the presence of local scalar αGensures that\nLMM dynamics is damped overall. Although there is no\nclosed expression for the SW dispersion in the presence of\nanisotropic λHM\nR, we can still extract SW damping Γ qin-duced by an HM overlayer from the exponential decay of\nthe SW amplitude in numerical integration of extended\nLLG Eq. (1) using SW initial conditions with varying q.\nFor an HM overlayer with realistic [31, 32] tSOC= 0.1t0\nthe results in Fig. (4)(a) are very similar to those ob-\ntained for NM overlayer with the same Fermi energy.\nAlso, the spatial anisotropy of λHM\nRdid not translate into\nθ-dependence of the SW damping.\nResults for FI/AM bilayer. —Altermagnets [33, 34] are\na novel class of antiferromagnets with spin-split elec-\ntronic energy-momentum dispersion despite zero net\nmagnetization or lack of SOC. They are currently in-\ntensely explored as a new resource for spintronics [51,\n61, 62] and magnonics [63, 64]. A simple model for\nan AM overlayer employs energy-momentum dispersion\nϵkσ=t0(k2\nx+k2\ny)−tAMσ(k2\nx−k2\ny) [33, 34], where tAMis5\n0.0 0.5 1.0 1.5\nq (1/a)1.01.52.02.5Γq≡Im(ωq) (J/¯ h)×10−1\n(a)NM\nHM\n0.0 0.5 1.0 1.5\nq (1/a)1234Γq≡Im(ωq) (J/¯ h)×10−1\n(b)NM\nAM :θ= 45◦\nAM :θ= 0◦\nFIG. 4. (a) Wavevector-dependent damping Γ qof SWs under\nNM or HM overlayer with the Rashba SOC of strength tSOC=\n0.1t0. (b) Γ qof SWs under AM overlayer with tAM= 0.8t0\nand for different relative orientations of FI and AM layers\nmeasured by angle θ[Fig. 1]. All calculations employ η= 0.1\nand Fermi energy εF= 0.25t0.\nthe parameter characterizing anisotropy in the AM. The\ncorresponding λAM\nR= diag( λ⊥\nR, λ⊥\nR, λ∥\nR) tensor has three\ncomponents, which we derive from Eq. (2) as\nλ⊥\nR=J2\nsd\n4πA+A−\u0014\nJ2\n0\u0012rϵF\nt0R+\u0013\n+J2\n0\u0012rϵF\nt0R−\u0013\u0015\n,\n(9a)\nλ∥\nR=J2\nsd\n2πA+A−J0\u0012rϵF\nt0R+\u0013\nJ0\u0012rϵF\nt0R−\u0013\n, (9b)\nwhere A±=t0±tAMandR2\n±=X2/A±+Y2/A∓is\nthe anisotropically rescaled norm of R. They are plot-\nted in Figs. 3(d) and 3(h), demonstrating that λAM\nRis\nhighly anisotropic in space and spin due to the impor-\ntance of angle θ[61, 65, 66]. Its components can also\ntake negative values, akin to the case of λHM\nR. It is inter-\nesting to note that along the direction of θ= 45◦[gray\ndashed line in Figs. 3(d) and 3(h)], λ⊥\nR=λ∥\nRso that\nnonlocal damping tensor is isotropic in spin. The SW\ndamping Γ qinduced by an AM overlayer is extracted\nfrom numerical integration of extended LLG Eq. (1) and\nplotted in Fig. (4)(b). Using a relatively large, but real-\nistic [33], AM parameter tAM= 0.8t0, the SW damping\nfor experimentally relevant small wavevectors is reduced\nwhen compared to the one due to NM overlayer by up to\n65% for θ= 0◦[Fig. 4(b)]. Additional nontrivial features\nare observed at higher |q|, such as being operative for a\ngreater range of wavevectors and with maxima around\n|q|= 2p\nϵF/t0and|q|= 3p\nϵF/t0. Remarkably, these\npeaks vanish for wavevectors along the isotropic direction\nθ= 45◦[Fig. 4(b)].\nConclusions. —In conclusion, using SKFT-derived non-\nlocal damping tensor [45], we demonstrated a rigorous\npath to obtain wavevector damping of SWs in magnetic\ninsulator due to interaction with conduction electrons\nof metallic overlayer, as a setup often encountered in\nmagnonics [13–17, 30] where such SW damping was di-\nrectly measured in very recent experiments [24–27]. Ouranalytical expressions [Eqs. (5), (7), and (9)] for nonlo-\ncal damping tensor—using simple models of NM, HM,\nand AM overlayers as an input—can be directly plugged\ninto atomistic spin dynamics simulations [28]. For more\ncomplicated band structures of metallic overlayers, one\ncan compute λRnumerically via Eq. (2), including com-\nbination with first-principles calculations [40]. 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B 108, 054511 (2023)." }, { "title": "1607.01307v1.Magnetic_moment_of_inertia_within_the_breathing_model.pdf", "content": "Magnetic moment of inertia within the breathing model\nDanny Thonig,\u0003Manuel Pereiro, and Olle Eriksson\nDepartment of Physics and Astronomy, Material Theory, University Uppsala, S-75120 Uppsala, Sweden\n(Dated: June 20, 2021)\nAn essential property of magnetic devices is the relaxation rate in magnetic switching which\nstrongly depends on the energy dissipation and magnetic inertia of the magnetization dynamics.\nBoth parameters are commonly taken as a phenomenological entities. However very recently, a large\ne\u000bort has been dedicated to obtain Gilbert damping from \frst principles. In contrast, there is no\nab initio study that so far has reproduced measured data of magnetic inertia in magnetic materials.\nIn this letter, we present and elaborate on a theoretical model for calculating the magnetic moment\nof inertia based on the torque-torque correlation model. Particularly, the method has been applied\nto bulk bcc Fe, fcc Co and fcc Ni in the framework of the tight-binding approximation and the\nnumerical values are comparable with recent experimental measurements. The theoretical results\nelucidate the physical origin of the moment of inertia based on the electronic structure. Even though\nthe moment of inertia and damping are produced by the spin-orbit coupling, our analysis shows that\nthey are caused by undergo di\u000berent electronic structure mechanisms.\nPACS numbers: 75.10.-b,75.30.-m,75.40.Mg,75.78.-n,75.40.Gb\nThe research on magnetic materials with particular fo-\ncus on spintronics or magnonic applications became more\nand more intensi\fed, over the last decades [1, 2]. For\nthis purpose, \\good\" candidates are materials exhibiting\nthermally stable magnetic properties [3], energy e\u000ecient\nmagnetization dynamics [4, 5], as well as fast and stable\nmagnetic switching [6, 7]. Especially the latter can be\ninduced by i)an external magnetic \feld, ii)spin polar-\nized currents [8], iii)laser induced all-optical switching\n[9], or iv)electric \felds [10]. The aforementioned mag-\nnetic excitation methods allow switching of the magnetic\nmoment on sub-ps timescales.\nThe classical atomistic Landau-Lifshitz-Gilbert (LLG)\nequation [11, 12] provides a proper description of mag-\nnetic moment switching [13], but is derived within the\nadiabatic limit [14, 15]. This limit characterises the\nblurry boundary where the time scales of electrons and\natomic magnetic moments are separable [16] | usually\nbetween 10\u0000100 fs. In this time-scale, the applicabil-\nity of the atomistic LLG equation must be scrutinized\nin great detail. In particular, in its common formula-\ntion, it does not account for creation of magnetic inertia\n[17], compared to its classical mechanical counterpart of\na gyroscope. At short times, the rotation axis of the\ngyroscope do not coincide with the angular momentum\naxis due to a \\fast\" external force. This results in a\nsuperimposed precession around the angular-momentum\nand the gravity \feld axis; the gyroscope nutates. It is\nexpected for magnetisation dynamics that atomic mag-\nnetic moments behave in an analogous way on ultrafast\ntimescales [17, 18] (Fig. 1).\nConceptional thoughts in terms of \\magnetic mass\"\nof domain walls were already introduced theoretically by\nD oring [19] in the late 50's and evidence was found ex-\nperimentally by De Leeuw and Robertson [20]. More\nrecently, nutation was discovered on a single-atom mag-\nnetic moment trajectory in a Josephson junction [21{23]\nB\nprecession conenutation cone\nmFIG. 1. (Color online) Schematic \fgure of nutation in the\natomistic magnetic moment evolution. The magnetic moment\nm(red arrow) evolves around an e\u000bective magnetic \feld B\n(gray arrow) by a superposition of the precession around the\n\feld (bright blue line) and around the angular momentum\naxis (dark blue line). The resulting trajectory (gray line)\nshows an elongated cycloid.\ndue to angular momentum transfer caused by an elec-\ntron spin \rip. From micromagnetic Boltzman theory,\nCiornei et al. [18, 24] derived a term in the extended\nLLG equation that addresses \\magnetic mass\" scaled by\nthe moment of inertia tensor \u0013. This macroscopic model\nwas transferred to atomistic magnetization dynamics and\napplied to nanostructures by the authors of Ref. 17, and\nanalyzed analytically in Ref. 25 and Ref. 26. Even in the\ndynamics of Skyrmions, magnetic inertia was observed\nexperimentally [27].\nLike the Gilbert damping \u000b, the moment of inertia ten-\nsor\u0013have been considered as a parameter in theoretical\ninvestigations and postulated to be material speci\fc. Re-\ncently, the latter was experimentally examined by Li et\nal. [28] who measured the moment of inertia for Ni 79Fe21\nand Co \flms near room temperature with ferromagnetic\nresonance (FMR) in the high-frequency regime (aroundarXiv:1607.01307v1 [cond-mat.mtrl-sci] 5 Jul 20162\n200 GHz). At these high frequencies, an additional sti\u000b-\nening was observed that was quadratic in the probing fre-\nquency!and, consequently, proportional to the moment\nof inertia\u0013=\u0006\u000b\u0001\u001c. Here, the lifetime of the nutation \u001c\nwas determined to be in the range of \u001c= 0:12\u00000:47 ps,\ndepending not only on the selected material but also on\nits thickness. This result calls for a proper theoretical\ndescription and calculations based on ab-initio electronic\nstructure footings.\nA \frst model was already provided by Bhattacharjee\net al. [29], where the moment of inertia \u0013was derived\nin terms of Green's functions in the framework of the\nlinear response theory. However, neither \frst-principles\nelectronic structure-based numerical values nor a detailed\nphysical picture of the origin of the inertia and a poten-\ntial coupling to the electronic structure was reported in\nthis study. In this Letter, we derive a model for the\nmoment of inertia tensor based on the torque-torque cor-\nrelation formalism [30, 31]. We reveal the basic electron\nmechanisms for observing magnetic inertia by calculat-\ning numerical values for bulk itinerant magnets Fe, Co,\nand Ni with both the torque-torque correlation model\nand the linear response Green's function model [29]. In-\nterestingly, our study elucidate also the misconception\nabout the sign convention of the moment of inertia [32].\nThe moment of inertia \u0013is de\fned in a similar way\nas the Gilbert damping \u000bwithin the e\u000bective dissipation\n\feldBdiss[30, 33]. This ad hoc introduced \feld is ex-\npanded in terms of viscous damping \u000b@m=@tand magnetic\ninertia\u0013@2m=@t2in the relaxation time approach [32, 34]\n(see Supplementary Material). The o\u000b-equilibrium mag-\nnetic state induces excited states in the electronic struc-\nture due to spin-orbit coupling. Within the adiabatic\nlimit, the electrons equilibrate into the ground state at\ncertain time scales due to band transitions [35]. If this\nrelaxation time \u001cis close to the adiabatic limit, it will\nhave two implications for magnetism: i)magnetic mo-\nments respond in a inert fashion, due to formation of\nmagnetism, ii)the kinetic energy is proportional to mu2=2\nwith the velocity u=@m=@tand the \\mass\" m of mag-\nnetic moments, following equations of motion of classical\nNewtonian mechanics. The inertia forces the magnetic\nmoment to remain in their present state, represented in\nthe Kambersky model by \u000b=\u0000\u0013\u0001\u001c(Ref. 32 and 34);\ntheraison d'etre of inertia is to behave opposite to the\nGilbert damping.\nIn experiments, the Gilbert damping and the moment\nof inertia are measurable from the diagonal elements of\nthe magnetic response function \u001fvia ferromagnetic res-\nonance [31] (see Supplementary Material)\n\u000b=!2\n0\n!Mlim\n!!0=\u001f?\n!(1)\n\u0013=1\n2!2\n0\n!Mlim\n!!0@!<\u001f?\n!\u00001\n!0; (2)\nwhere!M=\rBand!0=\rB0are the frequencies re-lated to the internal e\u000bective and the external magnetic\n\feld, respectively. Thus, the moment of inertia \u0013is equal\nto the change of the FMR peak position, say the \frst\nderivative of the real part of \u001fwith respect to the prob-\ning frequency [29, 36]. Alternatively, rapid external \feld\nchanges induced by spin-polarized currents lead also to\nnutation of the macrospin [37].\nSetting\u001fonab-initio footings, we use the torque-\ntorque correlation model, as applied for the Gilbert\ndamping in Ref. 30 and 35. We obtain (see Supplemen-\ntary Material)\n\u000b\u0016\u0017=g\u0019\nmsX\nnmZ\nT\u0016\nnm(k)T\u0017\nnm(k)Wnmdk (3)\n\u0013\u0016\u0017=\u0000g~\nmsX\nnmZ\nT\u0016\nnm(k)T\u0017\nnm(k)Vnmdk; (4)\nwhere\u0016;\u0017 =x;y;z andmsis the size of the mag-\nnetic moment. The spin-orbit-torque matrix elements\nTnm=hn;kj[\u001b;Hsoc]jm;ki| related to the commuta-\ntor of the Pauli matrices \u001band the spin-orbit Hamilto-\nnian | create transitions between electron states jn;ki\nandjm;kiin bandsnandm. This mechanism is equal\nfor both, Gilbert damping and moment of inertia. Note\nthat the wave vector kis conserved, since we neglect non-\nuniform magnon creation with non-zero wave vector. The\ndi\u000berence between moment of inertia and damping comes\nfrom di\u000berent weighting mechanism Wnm;Vnm: for the\ndampingWnm=R\n\u0011(\")Ank(\")Amk(\")d\"where the elec-\ntron spectral functions are represented by Lorentzian's\nAnk(\") centred around the band energies \"nkand broad-\nened by interactions with the lattice, electron-electron\ninteractions or alloying. The width of the spectral func-\ntion \u0000 provides a phenomenological account for angular\nmomentum transfer to other reservoirs. For inertia, how-\never,Vnm=R\nf(\") (Ank(\")Bmk(\") +Bnk(\")Amk(\")) d\"\nwhereBmk(\") = 2(\"\u0000\"mk)((\"\u0000\"mk)2\u00003\u00002)=((\"\u0000\"mk)2+\u00002)3\n(see Supplementary Material). Here, f(\") and\u0011(\") are\nthe Fermi-Dirac distribution and the \frst derivative of it\nwith respect to \". Knowing the explicit form of Bmk, we\ncan reveal particular properties of the moment of inertia:\ni)for \u0000!0 (\u001c!1 ),Vnm=2=(\"nk\u0000\"mk)3. Sincen=m\nis not excluded, \u0013!\u00001 ; the perturbed electron system\nwill not relax back into the equilibrium. ii)In the limit\n\u0000!1 (\u001c!0), the electron system equilibrates imme-\ndiately into the ground state and, consequently, \u0013= 0.\nThese limiting properties are consistent with the expres-\nsion\u0013=\u0000\u000b\u0001\u001c. Eq. (4) also indicates that the time scale\nis dictated by ~and, consequently, on a femto-second\ntime scale.\nTo study these properties, we performed \frst-\nprinciples tight binding (TB) calculations [38] of the\ntorque-correlation model as described by Eq. (4) as well\nas for the Green's function model reported in Ref. 29.\nThe materials investigated in this letter are bcc Fe, fcc\nCo, and fcc Ni. Since our magnetic moment is \fxed3\n-1·10−3-5·10−405·10−41·10−3−ι(fs)\n10−110+0\nΓ (eV)Fe\nCo\nNiTorque\nGreen\n10−21α10−410−21Γ (eV)\nFIG. 2. (Color) Moment of inertia \u0013as a function of the band\nwidth \u0000 for bcc Fe (green dotes and lines), fcc Co (red dotes\nand lines), and fcc Ni (blue dotes and lines) and with two\ndi\u000berent methods: i)the torque-correlation method (\flled\ntriangles) and the ii)Greens function method [29](\flled cir-\ncles). The dotted gray lines indicating the zero level. The\ninsets show the calculated Gilbert damping \u000bas a function of\n\u0000. Lines are added to guide the eye. Notice the negative sign\nof the moment of inertia.\nin thezdirection, variations occur primarily in xory\nand, consequently, the e\u000bective torque matrix element is\nT\u0000=hn;kj[\u001b\u0000;Hsoc]jm;ki, where\u001b\u0000=\u001bx\u0000i\u001by. The\ncubic symmetry of the selected materials allows only di-\nagonal elements in both damping and moment of inertia\ntensor. The numerical calculations, as shown in Fig. 2,\ngive results that are consistent with the torque-torque\ncorrelation model predictions in both limits, \u0000 !0 and\n\u0000!1 . Note that the latter is only true if we assume\nthe validity of the adiabatic limit up to \u001c= 0. It should\nalso be noted that Eq. (4) is only valid in the adiabatic\nlimit (>10 fs). The strong dependency on \u0000 indicates,\nhowever, that the current model is not a parameter-free\napproach. Fortunately, the relevant parameters can be\nextracted from ab-initio methods: e.g., \u0000 is related ei-\nther to the electron-phonon self energy [39] or to electron\ncorrelations [40].\nThe approximation \u0013=\u0000\u000b\u0001\u001cderived by F ahnle et\nal. [32] from the Kambersk\u0013 y model is not valid for all\n\u0000. It holds for \u0000 <10 meV, where intraband transi-\ntions dominate for both damping and moment of inertia;\nbands with di\u000berent energies narrowly overlap. Here, the\nmoment of inertia decreases proportional to 1=\u00004up to a\ncertain minimum. Above the minimum and with an ap-\npropriate large band width \u0000, interband transitions hap-\npen so that the moment of inertia approaches zero for\nhigh values of \u0000. In this range, the relation \u0013=\u000b\u0001\u001c\nused by Ciornei et al [18] holds and softens the FMR res-\nonance frequency. Comparing qualitative the di\u000berence\n10−410−310−210−1−ι(fs)/α\n510+02510+12510+22\nτ(fs)\n5·10−310−22·10−23·10−2\nΓ (eV)−ι\nαFIG. 3. (Color online) Gilbert damping \u000b(red dashed line),\nmoment of inertia \u0013(blue dashed line), and the resulting nu-\ntation lifetime \u001c=\u0013=\u000b(black line) as a function of \u0000 in the\nintraband region for Fe bulk. Arrows indicating the ordinate\nbelonging of the data lines. Notice the negative sign of the\nmoment of inertia.\nbetween the itinerant magnets Fe, Co and Ni, we obtain\nsimilar features in \u0013and\u000bvs. \u0000, but the position of the\nminimum and the slope in the intraband region varies\nwith the elements: \u0013min= 5:9\u000110\u00003fs\u00001at \u0000 = 60 meV\nfor bcc Fe, \u0013min= 6:5\u000110\u00003fs\u00001at \u0000 = 50 meV for fcc\nCo, and\u0013min= 6:1\u000110\u00003fs\u00001at \u0000 = 80 meV for fcc Ni.\nThe crossing point of intra- and interband transitions for\nthe damping was already reported by Gilmore et al. [35]\nand Thonig et al. [41]. The same trends are also repro-\nduced by applying the Green's function formalism from\nBhattacharjee et al. [29] (see Fig. 2). Consequently, both\nmethods | torque-torque correlation and the linear re-\nsponse Green's function method | are equivalent as it\ncan also be demonstrated not only for the moment of\ninertia but also for the Gilbert damping \u000b(see Supple-\nmentary Material)[41]. In the torque-torque correlation\nmodel (4), the coupling \u0000 de\fnes the width of the en-\nergy window in which transitions Tnmtake place. The\nGreen function approach, however, provides a more ac-\ncurate description with respect to the ab initio results\nthan the torque-torque correlation approach. This may\nbe understood from the fact that a \fnite \u0000 broadens and\nslightly shifts maxima in the spectral function. In par-\nticular, shifted electronic states at energies around the\nFermi level causes di\u000berences in the minimum of \u0013in both\nmodels. Furthermore, the moment of inertia can be re-\nsolved by an orbital decomposition and, like the Gilbert\ndamping\u000b, scales quadratically with the spin-orbit cou-\npling\u0010, caused by the torque operator ^Tin Eq. (4). Thus,\none criteria for \fnding large moments of inertia is by hav-\ning materials with strong spin-orbit coupling.\nIn order to show the region of \u0000 where the approxi-\nmation\u0013=\u0000\u000b\u0001\u001cholds, we show in Fig. 3 calculated\nvalues of\u0013,\u000b, and the resulting nutation lifetime \u001cfor a\nselection of \u0000 that are below \u0013min. According to the data\nreported in Ref. 28, this is a suitable regime accessible4\nfor experiments. To achieve the room temperature mea-\nsured experimental values of \u001c= 0:12\u00000:47 ps, we have\nfurthermore to guarantee that \u0013 >> \u000b . An appropriate\nexperimental range is \u0000 \u00195\u000010 meV, which is realistic\nand caused, e.g., by the electron-phonon coupling. A nu-\ntation lifetime of \u001c\u00190:25\u00000:1 ps is revealed for these\nvalues of \u0000 (see Fig. 3), a value similar to that found in ex-\nperiment. The aforementioned electron-phonon coupling,\nhowever, is underestimated compared to the electron-\nphonon coupling from a Debye model (\u0000 \u001950 meV) [42].\nIn addition, e\u000bects on spin disorder and electron corre-\nlation are neglected, that could lead to uncertainties in\n\u0000 and hence discrepancies to the experiment. On the\nother hand, it is not excluded that other second order\nenergy dissipation terms, Bdiss, proportional to ( @e=@t)2\nwill also contribute [32] (see Supplementary material).\nThe derivation of the moment of inertia tensor from the\nKambersk\u0013 y model and our numerics corroborates that\nrecently observed properties of the Gilbert damping will\nbe also valid for the moment of inertia: i)the moment\nof inertia is temperature dependent [41, 43] and decays\nwith increasing phonon temperature, where the later usu-\nally increase the electron-phonon coupling \u0000 in certain\ntemperature intervals [42]; ii)the moment of inertia is\na tensor, however, o\u000b-diagonal elements for bulk mate-\nrials are negligible small; iii)it is non-local [36, 41, 44]\nand depends on the magnetic moment [45{47]. Note that\nthe sign change of the moment of inertia also e\u000bects the\ndynamics of the magnetic moments (see Supplementary\nMaterial).\nThe physical mechanism of magnetic moment of inertia\nbecomes understandable from an inspection of the elec-\ntron band structure (see Fig. 4 for fcc Co, as an example).\nThe model proposed here allows to reveal the inertia k-\nand band-index nresolved contributions (integrand of\nEq. (4)). Note that we analyse for simplicity and clarity\nonly one contribution, AnBm, in the expression for Vnm.\nAs Fig. 4 shows the contribution to Vnmis signi\fcant only\nfor speci\fc energy levels and speci\fc k-points. The \fg-\nure also shows a considerable anisotropy, in the sense that\nmagnetisations aligned along the z- or y-directions give\nsigni\fcantly di\u000berent contributions. Also, a closer in-\nspection shows that degenerate or even close energy levels\nnandm, which overlap due to the broadening of energy\nlevels, e.g. as caused by electron-phonon coupling, \u0000, ac-\ncelerate the relaxation of the electron-hole pairs caused\nby magnetic moment rotation combined with the spin\norbit coupling. This acceleration decrease the moment\nof inertia, since inertia is the tendency of staying in a\nconstant magnetic state. Our analysis also shows that\nthe moment of inertia is linked to the spin-polarization\nof the bands. Since, as mentioned, the inertia preserves\nthe angular momentum, it has largest contributions in\nthe electronic structure, where multiple electron bands\nwith the same spin-polarization are close to each other\n(cf. Fig. 4 c). However, some aspects of the inertia,\n-4-3-2-10E−EF(eV)\n-4-3-2-10E−EF(eV)\nι<0\nι>0\n-4-3-2-10E−EF(eV)\nΓ H N\nk(a−1\n0)(a)\n(b)\n(c)y\nz\nFIG. 4. (Color online) Moment of inertia in the electron band\nstructure for bulk fcc Co with the magnetic moment a) in y\ndirection and b) in zdirection. The color and the intensity\nindicates the sign and value of the inertia contribution (blue\n-\u0013 <0; red -\u0013 >0; yellow - \u0013\u00190). The dotted gray line\nis the Fermi energy and \u0000 is 0 :1 eV. c) Spin polarization of\nthe electronic band structure (blue - spin down; red - spin up;\nyellow - mixed states).\ne.g. being caused by band overlaps, is similar to the\nGilbert damping [48], although the moment of inertia is\na property that spans over the whole band structure and\nnot only over the Fermi-surface. Inertia is relevant in\nthe equation of motion [17, 35] only for \u001c&0:1 ps and\nparticularly for low dimensional systems. Nevertheless,\nin the literature there are measurements, as reported in\nRef. 37, where the inertia e\u000bects are present.\nIn summary, we have derived a theoretical model for\nthe magnetic moment of inertia based on the torque-\ntorque correlation model and provided \frst-principle\nproperties of the moment of inertia that are compared\nto the Gilbert damping. The Gilbert damping and the\nmoment of inertia are both proportional to the spin-\norbit coupling, however, the basic electron band struc-5\nture mechanisms for having inertia are shown to be dif-\nferent than those for the damping. We analyze details\nof the dispersion of electron energy states, and the fea-\ntures of a band structure that are important for having\na sizable magnetic inertia. We also demonstrate that\nthe torque correlation model provides identical results\nto those obtained from a Greens functions formulation.\nFurthermore, we provide numerical values of the moment\nof inertia that are comparable with recent experimen-\ntal measurements[28]. The calculated moment of inertia\nparameter can be included in atomistic spin-dynamics\ncodes, giving a large step forward in describing ultrafast,\nsub-ps processes.\nAcknowledgements The authors thank Jonas Frans-\nson and Yi Li for fruitful discussions. The support of\nthe Swedish Research Council (VR), eSSENCE and the\nKAW foundation (projects 2013.0020 and 2012.0031) are\nacknowledged. The computations were performed on re-\nsources provided by the Swedish National Infrastructure\nfor Computing (SNIC).\n\u0003danny.thonig@physics.uu.se\n[1] S. S. P. Parkin, J. X., C. Kaiser, A. Panchula, K. Roche,\nand M. Samant, Proceedings of the IEEE 91, 661 (2003).\n[2] Y. Xu and S. 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Appl.\nPhys. 103, 07D303 (2008)." }, { "title": "1911.02775v2.Quantum_Oscillations_of_Gilbert_Damping_in_Ferromagnetic_Graphene_Bilayer_Systems.pdf", "content": "arXiv:1911.02775v2 [cond-mat.mes-hall] 15 Apr 2020Quantum Oscillations of Gilbert Damping in Ferromagnetic/ Graphene Bilayer\nSystems\nYuya Ominato1and Mamoru Matsuo1,2\n1Kavli Institute for Theoretical Sciences, University of Ch inese Academy of Sciences, Beijing 100190, China and\n2CAS Center for Excellence in Topological Quantum Computati on,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n(Dated: April 16, 2020)\nWe study the spin dynamics of a ferromagnetic insulator on wh ich graphene is placed. We show\nthat the Gilbert damping is enhanced by the proximity exchan ge coupling at the interface. The\nmodulation of the Gilbert damping constant is proportional to the product of the spin-up and\nspin-down densities of states of graphene. Consequently, t he Gilbert damping constant in a strong\nmagnetic field oscillates as a function of the external magne tic field that originates from the Landau\nlevel structure of graphene. We find that a measurement of the oscillation period enables the\nstrength of the exchange coupling constant to be determined . The results theoretically demonstrate\nthat the ferromagnetic resonance measurements may be used t o detect the spin resolved electronic\nstructure of the adjacent materials, which is critically im portant for future spin device evaluations.\nIntroduction .—Graphene spintronics is an emergent\nfield aiming at exploiting exotic spin-dependent proper-\ntiesofgrapheneforspintronicsdevices[1]. Althoughpris-\ntinegrapheneisanon-magneticmaterial,therehavebeen\neffortstointroducemagnetismintographenetofindspin-\ndependent phenomena and to exploit its spin degrees of\nfreedom. Placing graphene on a magnetic substrate is\na reasonable way, which leads to magnetic proximity ef-\nfect and lifting of spin degeneracy [2, 3]. Subsequently,\nmagnetization was induced in graphene and spin depen-\ndent phenomena, such as the anomalous Hall effect [4, 5]\nand non-local spin transport [6, 7], were observed. In all\nthese experiments, a spin-dependent current was gener-\nated by an electric field. There is an alternative way to\ngenerate a spin current called spin pumping [8–12]. The\nproximity exchange coupling describes spin transfer at\nthe magnetic interface and a spin current is injected us-\ning ferromagnetic resonance (FMR) from ferromagnetic\nmaterials into the adjacent materials. The generation of\na spin current is experimentally detectable through both\nthe inverse spin Hall effect and modulation of the FMR,\nwhich were experimentally confirmed at magnetic inter-\nfaces between graphene and several magnetic materials\n[13–18].\nThe theory of spin transport phenomena at magnetic\ninterfaces has been formulated based on the Schwinger-\nKeldysh formalism [19], which is applicable to magnetic\ninterfaces composed of a variety of systems, such as a\nparamagnetic metal and a ferromagnetic insulator (FI)\n[20–23], a superconductor and FI [24, 25], and two FIs\n[26, 27]. The modulation of FMR has been investigated\nin several papers. The modulation of Gilbert damping\nwasfound to be proportionalto the imaginarypartofthe\ndynamical spin susceptibility [21, 23–25, 28, 29], which\nmeans that one can detect spin excitations and electronic\nproperties of adjacent materials through the FMR mea-\nsurements. This implies that the FMR measurements\nof FI/graphene bilayer systems allow us to access thespin-dependent properties of graphene in quantum Hall\nregime [30, 31]. However, the modulation of FMR at the\nmagnetic interface between a FI and graphene has not\nbeen investigated and the effect of Landau quantization\non the FMR signal is unclear.\nIn this work, we study the modified magnetization dy-\nnamics of a FI adjacent to graphene. Figure 1 (a) shows\naschematicofthe system. Microwavesareirradiatedand\nthe precession of localized spins is induced. Figure 1 (b)\nand (c) shows the electronic structure of graphene on the\nFI under aperpendicular magneticfield. The spin degen-\neracy is lifted by the exchange coupling at the interface\nand spin-split Landau levels are formed. The densities of\nstates for spin-up and spin-down are shown in the right\npanel; Landau level broadening is included. We find that\nthe modulation of Gilbert damping is proportionalto the\nproduct of the densities of states for spin-up and spin-\ndown, so that the FMR measurements may be used as\na probe of the spin-resolved densities of states. Owing\nto the peak structure of the density of states, the mod-\nulation of Gilbert damping exhibits peak structure and\nan oscillation as a function of Fermi level and magnetic\nfield, which reflects the Landau level structure. One may\ndetermine the exchange coupling constant by analyzing\nthe period of the oscillation.\nModel Hamiltonian .—The totalHamiltonian H(t)con-\nsists of three terms,\nH(t) =HFI(t)+HGr+Hex. (1)\nThe first term HFI(t) describes the bulk FI\nHFI(t) =/summationdisplay\nk/planckover2pi1ωkb†\nkbk−h+\nac(t)b†\nk=0−h−\nac(t)bk=0,(2)\nwhereb†\nkandbkdenote the creation and annihilation\noperators of magnons with momentum k. We assume a\nparabolic dispersion /planckover2pi1ωk=Dk2−/planckover2pi1γB, withγ(<0) the\nelectron gyromagnetic ratio. The coupling between the2\nMicrowaveB(a) System (b) spin splitting \nExchange \ncoupling\nB01230\n-1 \n-2 \n-3 \nDOSE\nup down(c) spin-split Landau level \nkx kyE\nE\nB123\n0-1 \n-2 \n-3 0\nkx kyE\nFIG. 1. (Color online) Schematic picture of the FMR measurem ent and the energy spectrum of graphene in a strong perpen-\ndicular magnetic field. (a) Graphene on a ferromagnetic insu lator substrate. The magnetic field perpendicular to graphe ne\nis applied and the microwave is irradiated to the FI. (b) The s pin degeneracy is lifted by the exchange coupling. (c) The\nperpendicular magnetic field leads to the spin-split Landau level structure. The density of states has a peak structure a nd the\nlevel broadening originating from disorder is included.\nmicrowave and magnons is given by\nh±\nac(t) =/planckover2pi1γhac\n2√\n2SNe∓iΩt, (3)\nwherehacand Ω are the amplitude and frequency of the\nmicrowave radiation, respectively, and Sis the magni-\ntude of the localized spin in the FI. The above Hamilto-\nnian is derived from a ferromagnetic Heisenberg model\nusing the Holstein-Primakoff transformation and the\nspin-wave approximation ( Sz\nk=S−b†\nkbk,S+\nk=√\n2Sbk,\nS−\n−k=√\n2Sb†\nk, whereSkis the Fourier transform of the\nlocalized spin in the FI).\nThe second term HGrdescribes the electronic states\naround the Kpoint in graphene under a perpendicular\nmagnetic field,\nHGr=/summationdisplay\nnXsεnc†\nnXscnXs, (4)\nwherec†\nnXsandcnXsdenote the creation and annihi-\nlation operators of electrons with Landau level index\nn= 0,±1,±2,···, guiding center X, and spin up s= +\nand spin down s=−. The eigenenergy is given by\nεn= sgn(n)√\n2e/planckover2pi1v2/radicalbig\n|n|B, (5)\nwherevis the velocity and the sign function is defined\nas\nsgn(n) :=\n\n1 (n >0)\n0 (n= 0)\n−1 (n <0). (6)\nIn the following, we neglect the Zeeman coupling be-\ntween the electron spin and the magnetic field because\nit is much smaller than the Landau-level separation and\nthe exchange coupling introduced below. In graphene,\nthere are two inequivalent valleys labelled KandK′. Inthis paper, we assume that the intervalley scattering is\nnegligible. This assumption is valid for an atomically flat\ninterface,whichisreasonablegiventherecentexperimen-\ntal setups [4, 17, 18]. Consequently, the valley degree\nof freedom just doubles the modulation of the Gilbert\ndamping.\nThe third term Hexis the exchange coupling at the\ninterface consisting of two terms\nHex=HZ+HT, (7)\nwhereHZdenotes the out-of-plane component of the\nexchange coupling and leads to the spin splitting in\ngraphene,\nHZ=−JS/summationdisplay\nnX/parenleftBig\nc†\nnX+cnX+−c†\nnX−cnX−/parenrightBig\n,(8)\nwithJthe exchange coupling constant. The z-\ncomponent of the localized spin is approximated as\n∝angbracketleftSz\nk∝angbracketright ≈S. The out-of-plane component HZis modeled\nas a uniform Zeeman-like coupling, although in general,\nHZcontains the effect of surface roughness, which gives\noff-diagonal terms. The Hamiltonian HTdenotes the in-\nplane component of the exchange coupling and describes\nspin transfer between the FI and graphene,\nHT=−/summationdisplay\nnX/summationdisplay\nn′X′/summationdisplay\nk/parenleftBig\nJnX,n′X′,ks+\nnX+,n′X′−S−\nk+h.c./parenrightBig\n,\n(9)\nwhereJnX,n′X′,kis the matrix element for the spin trans-\nfer processes and s+\nnX+,n′X′−is the spin-flip operator for\nthe electron spin in graphene.\nModulation of Gilbert Damping .—To discuss the\nGilbert damping, we calculated the time-dependent sta-\ntistical averageof the localized spin under the microwave\nirradiation. The first-order perturbation calculation\ngives the deviation from the thermal average,\nδ∝angbracketleftS+\nk=0(t)∝angbracketright=−h+\nac(t)GR\nk=0(Ω). (10)3\nThe retarded Green’s function is written as\nGR\nk(ω) =2S//planckover2pi1\nω−ωk+iαGω−(2S//planckover2pi1)ΣR\nk(ω),(11)\nwhere we have introduced the phenomenological dimen-\nsionlessdampingparameter αG, calledthe Gilbert damp-\ning constant, which originates from the magnon-phonon\nand magnon-magnon coupling, etc [32–34]. In this pa-\nper, we focus on the modulation of the Gilbert damping\nstemming from the spin transfer processes at the inter-\nface. The self-energy from the spin transfer processes at\nthe interface within second-order perturbation is given\nby\nΣR\nk(ω) =/summationdisplay\nnX/summationdisplay\nn′X′|JnX,n′X′,k|2χR\nn+,n′−(ω).(12)\nThe spin susceptibility is given by\nχR\nn+,n′−(ω) =fn+−fn′−\nεn+−εn′−+/planckover2pi1ω+i0,(13)\nwherefns= 1//parenleftbig\ne(εns−µ)/kBT+1/parenrightbig\nis the Fermi distribu-\ntion function and εns=εn−JSsis the spin-split Landau\nlevel. From the self-energy expression, one sees that the\nmodulation of the Gilbert damping reflects the property\nof the spin susceptibility of graphene. The modulation\nof the Gilbert damping under the microwave irradiation\nis given by [21, 23–25, 28, 29]\nδαK\nG=−2SImΣR\nk=0(ω)\n/planckover2pi1ω, (14)\nwhere the superscript Ksignifies the contribution from\ntheKvalley.\nTo further the calculation, we assume that the ma-\ntrix element JnX,n′X′,k=0is approximated by a constant\nJ0, including detail properties of the interface, that is,\nJnX,n′X′,k=0≈J0. Withthisassumption,theself-energy\nbecomes\nImΣR\nk=0(ω) =−|J0|2π/planckover2pi1ω/integraldisplay\ndε/parenleftbigg\n−∂f(ε)\n∂ε/parenrightbigg\nD+(ε)D−(ε),\n(15)\nwhereDs(ε) is the density of states for spin s=±\nDs(ε) =A\n2πℓ2\nB/summationdisplay\nn1\nπΓ\n(ε−εns)2+Γ2,(16)\nwith magnetic length ℓB=/radicalbig\n/planckover2pi1/(eB) and area of the\ninterface A. Here, we have introduced a constant Γ de-\nscribing level broadening arising from surface roughness\nand impurity scattering. This is the simplest approx-\nimation to include the disorder effect. The density of\nstates shows peaks at the Landau level, which is promi-\nnent when its separation exceeds the level broadening.\nLandau level (JS = 20 meV) δα G [δα 0 10-2 ]\n0.6\nB [T]1.0μ [meV] \n0.4 0.240 \n20 \n-20\n-400\n0.8s=-, n=0\ns=+, n=0123\n-1 \n-2 \n-3 Γ = 1 meV\nkBT = 1 meV (= 11 K)\n6\n03\n0.6\nB [T]1.0μ [meV] \n0.4 0.240 \n20 \n-20\n-400\n0.8\nFIG. 2. (Color online) Modulation of the Gilbert damping\nconstant δαGand spin-split Landau levels as a function of\nthe Fermi level µand the magnetic field B. The spin splitting\nJSis set to 20meV. In the left panel, δαGhas peaks at the\ncrossing points of spin-up and spin-down Landau levels. In\nthe right panel, the blue and red curves identify the spin-up\nand spin-down Landau levels, respectively.\nFinally, the modulation of the Gilbert damping constant\nδαGis derived as\nδαG= 2πgvS|J0|2/integraldisplay\ndε/parenleftbigg\n−∂f(ε)\n∂ε/parenrightbigg\nD+(ε)D−(ε),(17)\nwheregv= 2 denotes the valley degree of freedom.\nFrom this expression, one sees that the modulation of\nthe Gilbert damping is proportional to the product of\nthe densities of states for spin-up and spin-down. There-\nfore, combined with the density of states measurement,\nfor example, a capacitance measurement [35], the FMR\nmeasurement is used to detect the spin-resolved densities\nof states.\nFigure 2 shows the spin-split Landau levels and the\nmodulation of the Gilbert damping δαGas a function of\nthe Fermi level µand the magnetic field B. We use δα0\nas a unit of δαG\nδα0= 2πgvS|J0|2/parenleftbiggA\n2πℓ2\nB1\nmeV/parenrightbigg2\n.(18)\nWe note that δα0(∝B2) depends on the magnetic field.\nBoth the level broadening Γ and the thermal broadening\nkBTare set to 1meV, and JSis set to 20meV [2–4].\nδαGreflects the Landau level structure and has peaks at\ncrossing points of spin-up and spin-down Landau levels.\nThe peakpositions aredetermined bysolving εn+=εn′−\nand the inverse of the magnetic field at the peaks is given\nby\n1\nB=2e/planckover2pi1v2\n(2JS)2/parenleftBig/radicalbig\n|n|−/radicalbig\n|n′|/parenrightBig2\n. (19)\nThe peak structurebecomes prominent when the Landau\nlevel separation exceeds both level and thermal broaden-\ning.4\nΓ = 1 meV\nkBT = 1 meV (= 11 K)\n5 meV (= 57 K)\n10 meV (= 115 K)kBT = 1 meV (= 11 K)\nΓ = 1 meV \n2 meV \n4 meV μ = JS = 20 meV μ = JS = 20 meV (a) (b)\nΔ(1/B)\n3\n1/B [1/T]4δα G [δα 0 10 -2 ]\n2 18\n6\n4\n2\n0\n5 3\n1/B [1/T]4δα G [δα 0 10 -2 ]\n2 18\n6\n4\n2\n0\n5Δ(1/B)\nFIG. 3. (Color online) Quantum oscillation of the modulatio n\nof the Gilbert damping constant δαGas a function of the\ninverse of the magnetic field 1 /B. The Fermi level µand the\nmagnitude of the spin splitting JSare set to 20meV. (a)\nΓ = 1meV and δαGis plotted at several temperatures. (b)\nkBT= 1meV and δαGis plotted for several Γ’s. The period of\nthe oscillation ∆(1 /B) is indicated by double-headed arrows.\nFigure 3 shows the modulation of the Gilbert damping\nδαGas a function of the inverse of the magnetic field\n1/Bwith the Fermi level set to µ= 20meV, where the\nspin-down zeroth Landau level resides. δαGshows peak\nstructure and a periodic oscillatorybehavior. The period\nof the oscillation ∆(1 /B) is derived from Eq. (19) and is\nwritten as\n∆/parenleftbigg1\nB/parenrightbigg\n=2e/planckover2pi1v2\n(2JS)2. (20)\nThe above relation means that the magnitude of the spin\nsplitting JSis detectable by measuring the period of the\noscillation ∆(1 /B). For the peak structure to be clear,\nboth leveland thermalbroadeningmust to be sufficiently\nsmaller than the Landau level separation; otherwise, the\npeak structure smears out.\nDiscussion .—To observe the oscillation of Gilbert\ndamping, at least two conditions must be satisfied. First,\nthe well-separated landau levels have to be realized in\nthe magnetic field where the FMR measurements is fea-\nsible. Second, the FMR modulation caused by the ad-\njacent graphene have to be detectable. The graphene\nLandau levels are observed in recent experiments at 2T\n[36], andrecentbroadbandferromagneticresonancespec-\ntrometer enables the generation of microwaves with fre-\nquencies ≤40GHz and FMR measurements in a mag-\nnetic field ≤2T [37]. The modulation of the FMR\nlinewidth in Permalloy/Graphene [14, 16], yttrium iron\ngarnet/Graphene [17, 18] have been reported by sev-\neral experimental groups, although all of them were per-\nformed at room temperature. Therefore, the above two\nconditionsareexperimentallyfeasibleandourtheoretical\npredictions can be tested in an appropriate experimental\nsetup.Conclusion .—We have studied the modulation of the\nGilbert damping δαGin a ferromagnetic insulator on\nwhich graphene is placed. The exchange coupling at\nthe interface and the perpendicular magnetic field lead\nto the spin-split Landau levels in graphene. We showed\nthatδαGis proportional to the product of the densities\nof states for spin-up and spin-down electrons. Therefore,\nthe spin-resolved densities of states can be detected by\nmeasuring δαGand the total density of states. When the\nFermi level is fixed at a Landau level, δαGoscillates as a\nfunction of the inverse of the magnetic field. The period\nof the oscillation provides information on the magnitude\nof the spin splitting. Our main message is that the FMR\nmeasurement is a probe of spin-resolved electronic struc-\nture. In addition to spin current generation, one may use\ntheFMRmeasurementstodetectthe electronicstructure\nof adjacent materials.\nAcnowledgement WethankJ.Fujimoto, T.Kato,R.\nOhshima, and M. Shiraishi for helpful discussions. 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Shiraishi, Applied Physics Letters 110, 182402\n(2017)." }, { "title": "2201.09027v1.Effect_of_MagneticField_on_the_Damping_Behavior_of_a_Ferrofluid_based_Damper.pdf", "content": " Effect of Magnetic Field on the Damping Behaviour \nof a Ferrofluid Based Damper \nDurga N K P Rao Miriyala \nDepartment of Mechanical \nEngineering \nPillai College of Engineering \nNew Panvel – 410206, India \nmdurgarao@mes.ac.inP S Goyal \nPillai College of Engineering \nDepartment of Electronics \nNew Panvel – 410206, India \npsgoyal@mes.ac.in \n \nAbstract — This paper is an extension of our earlier work \nwhere we had reported a proof of concept for a ferrofluid \nbased damper. The damper used ferrofluid as damping \nmedium and it was seen that damping efficiency of the damper \nchanges on application of magnetic fie ld. The present paper \ndeals with a systematic study of the effect of magnetic field on \nthe damping efficiency of the damper. Results of these studies \nare reported. It is seen that damping ratio varies linearly with \nmagnetic field (ζ / H = 0.028 per kG) for magnetic field in \nrange of 0.0 to 4 .5 kG. \nIt may be mentioned that f errofluid is different from \nmagnetorheological fluid even though both of them are \nmagnetic field -responsive fluids. The ferrofluid -dampers are \nbetter suited than MR Fluid-dampers for the ir use in \nautomobiles. \n \nKeywords —active dampers, ferrofluids, damping ratio, \nmagnetic field \nI. INTRODUCTION \nCars, buses and other vehicles, often, experience random \noscillatory motions because of uneven roads or pot holes etc., \nand these oscillations are damped using shock absorbers or \ndampers. In general, above dampers work on piston -cylinder \nprinciple and their damping characteristics depend on the \ndamping medium (e.g., oil) [1]. Contrary to what are \navailable, one would like to have dam pers whose damping \nbehaviour can be controlled externally. This is because if the \nfact that damping requirements of a vehicle depend on road \ncondition and load etc. The development of \nMagnetorheological (MR) Dampers has been a major \nachievement in field of shock absorbers [2 , 3]. The damping \nbehaviour of above dampers can be controlled by applying \nexternal magnetic field. These dampers use MR fluid as \ndamping medium. The properties of MR fluids were not \nconducive to the working of viscous dampers and thus these \ndampers could not be exploited commercially. In a recent \nwork [ 4], we used a ferrofluid as a damping medium and \nshowed that it is possible to control damping behavior of a \nferrofluid based damper also using magnetic field. This paper \nreports results of a systematic study of the effect of damping \nefficiency of the damper as a function of magnetic field. It is \nbelieved that ferrofluid based dampers are better suited than \nMR dampers for regular use in buses and cars [4,5]. \n \nII. FERROFLUID BASED DAMPERS \nConve ntional dampers used in cars or buses are piston -\ncylinder type viscous dampers where the motion of piston is \ndamped by the viscous liquid present in the cylinder. The \ndamping behavior of above damper can be controlled by \nchanging the viscosity of damping m edium. It seems, \n \nThis research is financially supported by All India Council of Technical \nEducation (AICTE) —New Delhi , under Research Proposal Scheme (RPS) . rheology of ferrofluids can be controlled by applying \nmagnetic field [6] and thus these liquids are ideally suited for \nactive dampers. Ferrofluids should not be confused with \nmagnetorheological fluid (MR fluids), even though both of \nthem are magnetic field -responsive fluids. Unlike ferrofluids \nwhich consist of nano -sized particles, MR fluids consist of \nmicron sized particle suspended in a suitable oil. The \nmagnetic particles tend to self -aggregate and thus MR fluids \nare not s table. Moreover, they exhibit hysteresis. On the other \nhand, ferrofluids are much stable and ferrofluid dampers have \nadvantage of long life. The higher fluidity of ferrofluids as \ncompared to MR fluids makes them better suited in damper \napplications. \nIII. DAMPIN G EFFICIENCY OF A DAMPER \nThe piston of a piston -cylinder type of oscillator performs \nan oscillatory motion. The amplitude of oscillations is \nindependent of time in a free oscillator. However, amplitude \nof oscillations decreases with time in a damped oscill ator. \nDamping efficiency of a viscous damper is measured in \nterms of a dimensionless parameter (referred to as damping \nratio), which describes how rapidly the oscillations decay \nfrom one bounce to the next. The damping ratio ζ is defined \nin terms of a parameter (logarithmic decrement) such that \n = 1\n𝑛 ln(𝑥1\n𝑥𝑛+1) and = √2\n2+42 (1) \nwhere xn is the amplitude of oscillation at the end of nth cycle. \nDepending on the val ue of ζ, the oscillatory motion is referred \nto as undamped (ζ = 0), underdamped (ζ < 1), critically \ndamped (ζ = 1) or overdamped (ζ > 1). The measurement of \ndamper efficiency or damper ratio ζ involves monitoring the \ntime dependence of amplitude of oscillation. \nIV. EXPERIMENTAL DETAILS \nFig.1 is a schematic drawing of the piston -cylinder \nviscous damper used in present studies. Piston oscillates in a \ncylinder having diameter of 25.4 mm. The amplitude of \noscillation of the piston was monitored as a function of time, \nand these data are used to calculate the damping efficiency or \ndamping ratio of the damper. Measurements were made \nunder varying experimental conditions. First, data were \nrecorded for undamped damper when there wa s no viscous \nliquid i n the cylinder. The other standard run was tak en using \nwater as the damping medium. The main dat a were taken \nusing ferrofluid as damping medium and expos ing it to \nvarying magnetic fields . The damping medium was exposed \nto magnetic field using permanent neodymium magnets. \nThere was a provision to move the magnets laterally and \n thereby change the magnetic field. The strength of magnetic \nfield was measured using a hall probe. \n \nFig. 1. Piston -cylinder damper using ferrofluid as damping medium. The \nfluid is exposed to magnetic field using permanent mag nets. \nThe ferrofluid used for above studies was synthesized in \nour laboratory, the details of which are given in an earlier \npaper [ 4]. It consisted of a suspension of oleic acid coated \nFe3O4 nanoparticles in kerosene. The said nanoparticles were \nsynthesized using co -precipitation technique with FeCl 3 and \nFeCl 2 as starting materials . \nThe oscillation behavior of the piston was studied by \nconnecting the piston to the vibrating system (Fig. 2), which \nconsists of a beam pivoted at its one end and supported by a \nhelical spring at the other end. Vibrations are sensed using an \naccelerometer sensor and its associated electronics [ 4]. The \namplitude - time graphs are obtained from acceleration - time \ngraphs using standard integrations. This is illustrated in Fig.3 \nwhere amplitude - time graph is obtained from acceleration - \ntime graph for a typical data corresponding to magnetic field \nH = 3.2 kG . The effect of magnetic field H on the amplitude \n- time gra phs have been studied for several values of H in \nrange of 0.0 to 4. 5 kG. \n \nFig. 2. Vibrating system used for measuring amplitude -time graphs for \npiston -cylinder damper. The acceleration is measured using uni -axial \naccelerometer and corresponding electron ics. \n \nFig.3 LabVIEW readings obtained for Displacement, Velocity and \nAcceleration versus Time, for a typical data corresponding to m agnetic field \nH = 3.2 kG. \n \nV. RESULTS AND DISCUSSIONS \nThe magnetic field in region of ferrofluid depends on the \nposition of magnets or the gap size between the magnet and \nthe cylinder. Fig. 4 gives the variation of magnetic field as a \nfunction of gap size. This plot was obtained by measuring \nmagnetic field using a hall probe. In actual experiments, \nmeasurements were made for different gap sizes and the \nabove calibration curve was used to obtain the value of the \nmagnetic field. \n \nFig.4 Calibration curve between gap size and magnetic field . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 5. Displacement versus Time graphs for the piston in viscous medium \nunder different magnetic fields . \n \nFig. 5 shows the time dependence of amplitude of oscillations \nof the piston in above piston -cylinder dampe r corresponding \nto different magnetic fields (see Table 1). For the sake of \ncompleteness, amplitude - time graph of an undamped system \nis shown in Fig. 5 (i). It is noted that amplitude of vibrations \ndoes not change with time. Fig. 5 (ii) shows amplitude - time \ngraph for a damper that uses water as dampin g medium. The \namplitude of vibrations decreases slowly with time. Figures \n5 (iii) to 5 (v) show amplitude - time graphs for a damper that \nuses ferrofluid as damping medium. It is noted that damping \nof amplitude of vibration for ferrofluid -based damper is much \nmore than that for water -based damper . Figures 5 (iii), 5 (iv) \nand 5(v) correspond to measurements when damping medium \nwas exposed to magnetic fields of 0.0 kG, 3.2 kG and 4.5 kG \nrespectively. It is seen that the damping increases with \nincrease in the magnetic field. It may be noted that Fig. 5 shows typical data, but actual \nmeasurements have been made for a number of H values \n(Table 1). These data along were used to calculate damping \nratio ζ using the formulae given in Section III. Resu lts are \nshown in Table 1. Each measurement was repeated 5 times \nand average values of ζ are given in Table 1. Variation of ζ \nwith magnetic field is shown in Fig.6. It is interesting that ζ \nvaries linearly with H. Slope of curve suggests that rate of \nincre ase of damping efficiency is about 0.028 per kG. It \nseems the magnetic nanoparticles form chains in presence of \nmagnetic field and that gives rise to increase in viscosity and \nthe damping ratio. \n \nTABLE I \nEFFECT OF MAGNETIC FIELD ON DAMPING RATIO \n \n \n \n \n \nFig. 6. Variation of damping ratio with magnetic field for ferrofluid based \ndamper. \n \nSr. No. Gap Size \n(mm) Magnetic \nFlux \nDensity \n(kG) Average \nDamping Ratio \ncorresponding to \nfive readings \n1 100 mm 0 0.020 \n2 10 0.75 0.043 \n3 9 0.85 0.046 \n4 8 0.95 0.043 \n5 7 1.2 0.050 \n6 6 1.4 0.054 \n7 5 1.8 0.056 \n8 4 2.1 0.067 \n9 3 2.7 0.076 \n10 2 3.2 0.091 \n11 1 4 0.099 \n12 0 4.5 0.148 \n(i) Undamped (no damper connected) \n \n \n(ii) Damper with water as damping medium \n \n \n(iii) Ferrofluid based damper without magnetic field \n \n \n(iv) Ferrofluid based damper for H = 3.2 kG \n \n \n(v) Ferrofluid based damper for H = 4.5 kG) \nVI. CONCLUSIONS \nThis paper deals with ferrofluid based damper , where one \nuses ferrofluid as the damping medium. In an earlier work , \nwe had provided a proof of concept and show ed that the \ndamping behavior of the damper can be varied by applying \nmagnetic field to the damping medium. The present paper \nreports a systemat ic study of the variation of damping ratio ζ \nwith the magnetic field H. It is seen that ζ increases linearly \nwith magnetic field up to H = 4.5 kG. The damping \nefficiency increases by about 600 % as the magnetic field is \nchanged from H = 0.0 to H = 4.5 k G. \nThe fact that ferrofluids are more stable than MR fluids, \nit is expected that ferrofluid -dampers are better suited than \nMR Fluid-dampers for commercial exploitation. \n \nACKNOWLEDGMENT \nWe thank Biswajit Panda for help in synthesizing \nferrofluids and R I K Moorthy for useful discussions. \nEncouragement from Priam Pillai and Sandeep M Joshi is \nacknowledged. \n \nREFERENCES \n \n[1] J. C. Dixon, The Shock Absorber Handbook, 2nd Edition, John Wiley \nand Sons, 2007. \n[2] S. Sassi, K. Cherif, L. Mezghani, M. Thomas and A. Kotrane, \nInnovative Magneto -Rheological Damper for Automotive Suspension: \nFrom Design to Experimental Characterization, Smart Mater. Struct. \n14, pp. 811 -822, 2005. \n[3] B. K. Kumbhar and S. R. Patil , A Study on Properties and Selecti on \nCriteria for Magneto -Rheological (MR) Fluid Components, \nInternational Journal of Chem. Tech. Research 6, pp. 3303 -3306, 2014. \n[4] M. Durga Rao, P. S. Goyal, B. Panda and R. I. K. Moorthy, Ferrofluids \nfor Active Shock Absorbers, IOP Conference Series: Materi als Science \nand Engineering 360, 012002, 2018. \n[5] Chuan Huang et. al., Damping Applications of Ferrofluids: A Review, \nJournal of Magnetics 22, p.109, 2017. \n[6] J. Yao, D. Li, X. Chen, C. Huang and D. Xu, Damping Performance of \na Novel Ferrofluid Dynamic Vibratio n Absorber, Journal of Fluids and \nStructures 90, pp. 190 -204, 2019. \n[7] R. Patel, R. V. Upadhyay and R. V. Mehta, Viscosity Measurements of \na Ferrofluid: A Comparison with various Hydrodynamic Equations, \nJ. Coll. Int. Sci. 263, p. 661, 2003. \n " }, { "title": "1807.05254v3.Cyclotron_Damping_along_an_Uniform_Magnetic_Field.pdf", "content": "arXiv:1807.05254v3 [math.AP] 15 Apr 2019Cyclotron Damping along a Uniform Magnetic Field\nXixia Ma∗\nAbstract. We prove cyclotron damping for the collisionless Vlasov-Maxwell equa tions on T3\nx×R3\nvunder\nthe assumptions that the electric induction is zero and ( PSC) holds. It is a crucial step to solve the stability\nproblemoftheVlasov-Maxwellequations. Ourproofisbasedonane wdynamicalsystemoftheplasmaparticles,\noriginating from Faraday Law of Electromagnetic induction and Lenz ’s Law. On the basis of it, we use the\nimproved Newton iteration scheme to show the damping mechanism.\n1. Introduction\nIn this paper, it is assumed that the plasma system is collisionless, non relativistic, and hot. Cyclotron damp-\ning describes the phenomenon that a plasma with a prescribed zero- order distribution function, imbedded in a\nuniform magneticfield, which isassumedtobe perturbed byanelectr omagneticwavepropagatingparalleltothe\nfield. A number of treatments of the problem of cyclotron damping h ave appeared in the literature[18,20,24,25],\nbut there are few rigorous mathematical results except the rece nt results of Bedrossian and Wang [7] . The\nusual method that deals with this phenomenon is via the Vlasov equat ions. In this paper we study cyclotron\ndamping at the level of kinetic description based on the Vlasov equat ions from the mathematical view point.\nFirst, we analyze the Vlasov-Maxwell equations from perspective o f both equilibrium and stability theories.\nNow we give a detailed description of the Vlasov-Maxwell equations. W e denote the particles distribution\nfunction by f=f(t,x,v),and the electric and magnetic fields by E(t,x) andB(t,x),respectively. Then the\nVlasov equation says\n∂tf+v·∇xf+(E+v×B)·∇vf= 0. (0.1)\nThe electric and magnetic fields E(t,x) andB(t,x) in Eq.(0 .1) are determined from Maxwell’s equations:\n∇·E(t,x) =/integraldisplay\nR3f(t,x,v)dv,∇×B(t,x) =/integraldisplay\nR3vf(t,x,v)dv+∂E(t,x)\n∂t,\n∂B(t,x)\n∂t=−∇×E(t,x),∇·B(t,x) = 0. (0.2)\nNote that Eq.(0 .1) is nonlinear since E(t,x) andB(t,x) are determined in terms of f(t,x,v) from Maxwell’s\nequations (0 .2).\nAn equilibrium analysis of Eq.(0 .1) and Eqs.(0 .2) proceeds by setting∂\n∂t= 0 and looking for stationary\nsolutions, f0(x,v),E0(x),B0(x),that satisfy the equations\nv·∇xf0(x,v)+(E0+v×B0)·∇vf0(x,v) = 0,∇·B0(x) = 0,\n∇×B0(x) =/integraldisplay\nR3vf0(x,v)dv,∇·E0(x) =/integraldisplay\nR3f0(x,v)dv.\n(0.3)\nAn analysis of Eq.(0.3) reduces to a determination of the particle con stants of the motion in the equilibrium\nfieldsE0(x) andB0(x).In this paper, we assume that E0(x) = 0,namely,/integraltext\nR3f0(x,v)dv= 0.This implies that\nthere are no deviations from charge neutrality in equilibrium, B0(x) is produced by external current sources as\nwell as any equilibrium plasma currents.\nA stability analysis based on Eq.(0.1) and Eqs.(0.2) proceeds in the follo wing manner. The quantities\nf(t,x,v),E(t,x),andB(t,x) are expressed as the sum of their equilibrium values plus a time-depe ndent per-\nturbation:\nf(t,x,v) =f0(x,v)+δf(t,x,v), E(t,x) =E0(x)+δE(t,x), B(t,x) =B0(x)+δB(t,x).\n(0.4)\nThequantities f0(x,v),E0(x)andB0(x)satisfy(0.3). Thetimedevelopmentoftheperturbations δf(t,x,v),δE(t,x),\nandδB(t,x) is studied by using Eq.(0.1) and Eqs.(0.2). For small-amplitude pertur bations, the Vlasov-Maxwell\n∗Yau Mathematical Sciences Center, Tsinghua University. E- mail addresses:kfmaxixia@tsinghua.edu.cn\n1equations are linearized about the equilibrium f0(x,v),E0(x) andB0(x).This gives\n∂δf\n∂t+v·∇xδf(t,x,v)+(E0+v×B0)·∇vδf(t,x,v) =−(δE+v×δB)·∇vf0(x,v),\n∇·δB(t,x) = 0,∇×δB(t,x) =/integraldisplay\nR3vδf(t,x,v)dv+∂δE\n∂t,∇·δE(t,x) =/integraldisplay\nR3δf(t,x,v)dv.\n(0.5)\nIfthe perturbations δf(t,x,v),δE(t,x),andδB(t,x) grow,thenthe equilibriumdistribution f0(x,v) isunstable.\nOtherwise, the perturbations damp, so the system returns to eq uilibrium and is stable. We assume that the\nequilibrium f0is independent of space, namely, f0(x,v) =f0(v).\nFrom the above analysis and the form of Vlasov-Maxwellequations, it is obvious that when B≡0,cyclotron\ndamping is reduced to Landau damping. Hence, the method used is sim ilar to that employed by Mouhot\nand Villani [23]. However, compared with the electric field, a static mag netic field introduces a fascinating\ncomplication into the motion of charged particles. And the particles t rajectories become helices, spiraling\naround the magnetic lines of force. This severe alteration of the or bits tends to inhibit transport across the\nmagnetic field. The mechanism of Landau damping does depend on the transfer of electric field energy to\nparticles moving in phase with the wave. However, for cyclotron dam ping in electromagnetic plasmas, the\nelectric field of the wave is perpendicular to the direction of the magn etic field and the particle drifts and\naccelerates the particle perpendicular to the drift direction.\nIn the following we recall Landau damping through gathering lots of p hysical literature and results of\nmathematical articles. The existence of a damping mechanism by whic h plasma particles absorb wave energy\nwas found by L.D.Landau at the linear level, under the condition that t he plasma is not cold and the velocity\ndistribution is of finite extent. Next in linear case, many works from m athematical aspects found in [9,15,25,28]\ngave rigorousproofs under different assumptions. Later, a grou nd-breakingwork for Landau damping was made\nby Mouhot and Villani in the nonlinearcase. They gavethe first and rig oursproofof nonlinearLandau damping\nunder the assumption of the electric field. In this paper we will exten d their results and prove that cyclotron\ndamping in electromagnetic fields still occurs.\nNow we will make a brief statement about the connection and differen ce between the results of [23] and ours.\nFirst, in electric field case, Mouhot and Villani proved the existence o f Landau damping under assumption of\nthe (L) condition, that is expressed as follows:\n(L) There are constants C0,λ,κ >0 such that |ˆf0(η)| ≤C0e−2πλ|η|for anyη∈Rd; and for any ξ∈Cwith\n0≤Reξ < λ,\ninf\nk∈Zd|L(ξ,k)−1| ≥κ, (0.6)\nwhere we define a function L(ξ,k) =−4π2/integraltext∞\n0e2πξ∗|k|t/hatwiderW(k)ˆf0(kt)|k|2tdt,andξ∗is the complex conjugate to ξ.\nTosomeextent, (0.6)ofthe( L) conditionissimilartothe“SmallDenominators”conditioninKAMtheor yin\n[1], but is stronger. Herewe will considerthe conditionofcyclotrond amping froma totallydifferent perspective,\nin detail, we will give a physical condition that we call the “ Physical Stability Condition ”,in short, “ PSC”,\nwhich is stated in the following form (here we assume the background magnetic field B0along the ˆ zdirection):\n(PSC) : for any component velocity in the ˆ zdirection v3∈R,there exists some positive constant vTesuch\nthat ifv3=ω\nk3,ω,kare frequencies of time and space t,x,respectively, then |v3| ≫vTe.\n(PSC) tells us that the number of particles that the wave velocity greatly exceeds their velocity is much\nlarger than the number of particles’ velocity slower than the wave v elocity. And we will show that cyclotron\ndamping occurs under the above conditions, and that isn’t only cons istent with the physical observation, but\nalso is the same with the “Small Denominators” condition in KAM theory in [1] in some sense.\nSecond, compared with the electric field case, it is easy to find a new t ermv×Bin the electromagnetic field\nsetting. And this brings many difficulties because of the unboundedn ess ofv.Based on the physical facts of\nFaraday Law of Electromagnetic induction and Lenz’s Law, we know t hatδBgenerates the force that inhibits\nthe change of the electric field. This helps us estimate such term. An d the above fact leads us to study the\nfollowing dynamics of the particles trajectory:\n/braceleftbiggd\ndtXt,τ(x,v) =Vt,τ(x,v),d\ndtVt,τ(x,v) =Vt,τ(x,v)×B0+E[f](t,Xt,τ(x,v)),\nXτ,τ(x,v) =x, Vτ,τ(x,v) =v,(0.7)\nwhereB=B0+δB.In other words, we reduce inhomogeneous dynamical system to ho mogeneous dynamical\nsystem. Hence, based on the above dynamical system (0.7), we ca ll the adopted Newton iteration as the\nimproved Newton iteration.\nIndeed, there are many papers that contribute to Landau dampin g. Here we only list some results.\nBedrossian, Masmoudi, and Mouhot [4] provided a new, simple and sho rt proof of nonlinear Landau damp-\ning onTdin only electric field case that nearly obtains the “ critical ” Gevrey-1\nsregularity predicted in [23].\nAlthough their proofs have lots of the same ingredients as the proo f in [23] from a physical point of view, at a\n2mathematical level, the two proofs are quite different, they “mod o ut” by the characteristics of free transport\nand work in the coordinates z=x−vtwith (t,x,v)→(t,z,v).The evolution equation (0.1) ( B= 0) becomes\n∂tf+E(t,z+vt)·(∇v−t∇z)f+E(t,z+vt)·∇vf0= 0.\nFromthisformula,itiseasytoseethephasemixingmechanism. Andth iscoordinateshiftisrelatedtothenotion\nof “gliding regularity” used in [23]. One of the main ingredients of their p roof is to split nonlinear terms into the\ntransport structure term and “reaction” term in [23] by using par adifferential calculus . Bedrossian, Masmoudi\n[3] alsoused this method to provethe invisciddamping and asymptotic stability of2-DEulerequationsand later\nalso proved the stability threshold for the 3D Couette flow in Sobolev regularity in [5] and so on. They [6] also\nproved Landau damping for the collisionless Vlasov equation with a clas s ofL1interaction potentials on R3\nx×R3\nv\nfor localized disturbances of infinite, inhomogeneous background. Also, there are counterexamples that can be\nfound in [2,10,13], showing that there is in general no exponential dec ay without analyticity and confining.\nBedrossian stated one of these counterexamples by proving that the theorem of Mouhot and Villani on Landau\ndampingnearequilibriumfortheVlasov-Poissonequationson Tx×Rvcannot, ingeneral,beextendedtoSobolev\nspaces by constructing a sequence of homogeneous background distributions and arbitrarily small perturbations\ninHswhich deviates arbitrary far from free transport for a long time. L in and Zeng [10] also showed that there\nexist nontrivial traveling wave solutions to the Vlasov equation in Sob olev space Ws.p\nx,v(p >1,s <1+1\np) with\narbitrary traveling speed. This implies that nonlinear Landau damppin g is not true in Ws.p\nx,v(p >1,s <1+1\np)\nspace for any homogeneous equilibria and in any period box. In additio n, Deng and Masmoudi [13] showed\nthe instability of the Couette flow in low Gevrey spaces. In recent ye ars, there are also lots of results on the\nstability in other setting such as those in [12,14,15,28].\nThis paper is organized as follows. Section 1 mainly introduces hybrid a nalytic norms. In section 2, we will\nprove cyclotron damping at the linear level. We will state the new obse rvation and sketch the proof of main\ntheorem in section 3. Section 4 is dedicated to the deflection estimat es of the particles trajectory. Section 5 is\nthe key section, which states the phenomena of plasma echoes. We will control the error terms in section 6, and\ngive the iteration in section 7.\nBefore stating our main theorem, we assume that the electric induc tion is zero, then the Maxwell’s equations\nreduce to the following forms\n∇·E(t,x) =/integraldisplay\nR3f(t,x,v)dv,∇×B(t,x) = 0, ∂tB(t,x) =−∇×E(t,x),∇·B(t,x) = 0.(0.8)\nNow based on the assumption that the electric induction is zero, we fi rst give two results of the Vlasov equation\nwith the electric field E(t,x) and the magnetic field B(t,x) on both the linear and the nonlinear levels satisfying\nthe conditions E=W(x)∗ρ(t,x),∂tB=∇x×E,whereW(x) is a vector function and satisfies |/hatwiderW(k)| ≤1\n1+|k|γ.\nNow we state our main result as follows.\nTheorem 0.1 Letf0:R3→R+be an analytic velocity profile, and assume W(x) = (W1(x),W2(x),0) :T3→\nR3andW(x)is an odd function on x3satisfying |/hatwiderW(k)| ≤1\n1+|k|γ,γ >1.Further we assume that, for some\nconstant λ0such that λ0−B0>0,\nsup\nη∈R3e2π(λ0−B0)|η||˜f0(η)| ≤C0,/summationdisplay\nn∈N3\n0(λ0−B0)n\nn!/ba∇dbl∇n\nvf0/ba∇dblL1\ndv≤C0<∞. (0.9)\nAnd we consider the following system,\n\n\n∂tf+v·∇xf+q\nm(v×B0)·∇vf=−q\nm(E+v×B)·∇vf,\n∂tB=∇x×E,∇·B= 0,\nE=W(x)∗ρ(t,x),ρ(t,x) =/integraltext\nR3f(t,x,v)dv,\nf(0,x,v) =f0(x,v) =f0(x,v1,v2,vz),f0(v) =f0(v1,v2,vz),(0.10)\nthere isε=ε(λ0,µ0,β,γ,λ′\n0,µ′\n0)verifying the following property: f0=f0(x,v)is an initial data such that\nsup\nk∈Z3,η∈R3e2π(λ0−B0)|η|e2πµ0|k||f0−f0|+/integraldisplay\nT3/integraldisplay\nR3|f0−f0|eβ|v|dvdx≤ε, (0.11)\nwhere any β >0,λ0> λ′\n0> B0,µ0> µ′\n0>0.\nIn addition, we also assume that the (PSC)holds.\nThen there exists a unique classical solution (f(t,x,v),E(t,x),B(t,x))to the non-linear Vlasov system\n(0.12).\nMoreover, for any fixed η3,k3,∀r∈N,as|t| → ∞,we have\n|ˆf(t,k,η1+k1sinΩt\nΩ,η2+k2sinΩt\nΩ,η3)−ˆf0(k,η)| ≤e−(λ′\n0−B0)|η3+k3t|,/ba∇dblρ(t,·)−ρ0/ba∇dblCr(T3)=O(e−2π(λ′\n0−B0)|t|),\n/ba∇dblE(t,·)/ba∇dblCr(T3)=O(e−2π(λ′\n0−B0)|t|),/ba∇dblB(t,·)/ba∇dblCr(T3)=O(e−2π(λ′\n0−B0)|t|),\n(0.12)\n3whereρ0=/integraltext\nT3/integraltext\nR3f0(x,v)dvdx.\nNow we simply analyze the relation among the Vlasov-Poissonequation s, the Vlasov-Maxwell equations,and\nour model. If we assume that both the electric induction and the mag netic field are zero, then Vlasov-Maxwell\nequations reduce to the Vlasov-Poissonequations; if we only assum e that the electric induction is zero, then the\nVlasov-Maxwell equations reduce to our case. In other words, ou r case is the generalized case of the Vlasov-\nPoisson equations, and provides a new observation from the physic al viewpoint to solve the corresponding prob-\nlem of the Vlasov-Maxwell equations. However, we cannot still solve the Vlasov-Maxwell equations completely\nonly through this new observation. Therefore, the stability’s or un stability’s problem of the Vlasov-Maxwell\nequations is still open.\nIn the following we sketch the difficulties and methods in our paper’s se tting. The crucial estimates of\ncyclotron damping include the following two inequalities:\n•a control of ρ=/integraltext\nR3fdvinFλτ+µnorm, that is, supτ≥0/ba∇dblρτ/ba∇dblFλτ+µ<∞,\n•a control of fτ◦Ωt,τinZλ′(1+b),µ′;1\nτ−bt\n1+bnorm, where λ′< λ,µ′< µ.\nHowever, during the iteration scheme, for cyclotron damping, fro m the view of the original Newton iteration,\nthe characteristics are not only determined by the density ρn,but also related with the velocity at stage n.\nHowever, ρnis independent of the velocity and the key difficulty is that the velocity is unbounded. This makes\nthat we obtain the estimates of the associated deflection Ωnmore difficult. But this difficulty doesn’t exist for\nLandau damping in [23]. To overcome this difficulty, on the basis of a new observation from Lenz’s Law, we\nreduce the classical dynamical system to the improved dynamical s ystem (0.7). And the corresponding equation\nof the density ρ[hn+1] at stage n+1 becomes\nρ[hn+1](t,x) =/integraldisplayt\n0/integraldisplay\nR3−/bracketleftbigg\n(E[hn+1]◦Ωn\ns,t(x,v)·Gn\ns,t)−(B[hn+1]◦Ωn\ns,t(x,v)·Gn,v\ns,t)\n−(B[fn]◦Ωn\ns,t(x,v))·(∇′\nvhn+1×V0\ns,t(x,v))◦Ωn\ns,t(x,v)/bracketrightbigg\n(s,X0\ns,t(x,v),V0\ns,t(x,v))dvds+(terms from stage n),\nFrom the above equation, we see that, comparing with that in [23], th ere is a new term ( B[fn]◦Ωn\ns,t(x,v))·\n(∇′\nvhn+1×V0\ns,t(x,v)◦Ωn\ns,t(x,v)) that have the information of the stage n+ 1.Of course, this is due to the\nreason that we regard the perturbation of the magnetic field as a n egligible term. To get a self-consistent\nestimate, we have to deal with this term and have little choice but to c ome back the equation of hn+1. This\nleads to different kinds of resonances (in term of different norms), for example, v×B[fn]·∇vhn+1may generate\nresonance in Zλ,µ;1\nτnorm on hn+1,except in Fλτ+µnorm on ρ[hn+1],because both B[fn] andhn+1contain the\nspace variable. But there are no these problems in [23], and Landau d amping in [23] only has the resonances in\nFλτ+µnorm on ρ[hn+1].\nRemark 0.2 γ >1of Theorems 0.1 and 0.3 can be extended to γ≥1,the difference between γ >1andγ= 1\nis the proof of the growth integral in section 7. The proof of γ= 1is similar to section 7 in [23], here we omit\nthis case.\nRemark 0.3 Indeed, we don’t need to assume that W(x) = (W1(x),W2(x),0)andB0= (0,0,B0),only need\nto assume that the electric field E(t,x)is perpendicular to the background magnetic field B0.\nRemark 0.4 From the physics viewpoint, the condition of the damping is t hat the number of particles that the\nwave velocity greatly exceeds their velocity is much larger than the number of particles whose velocity is slower\nthan the wave velocity. The (PSC)condition in above theorem is in consistent with the stateme nt of the physics\nviewpoint. In fact, the (PSC)condition and (0.11) imply that the number of resonant parti cles is exponentially\nsmall and their effect corresponding is weak. From the conclu sion of Theorem (0.3), we know that, under this\ncondition, most particles absorb energy from the wave, and t hen the damping occurs.\nRemark 0.5 From the definition of the hybrid analytic norms and the proof of the following sections, there is\nstill the phenomena of cyclotron damping for the nonlinear V lasov-Maxwell equations, which is the same to that\nin the nonlinear Vlasov-Poisson equations but only in ˆzdirection, the position of the corresponding resonances\ntranslates 0 into B0,and in the horizon direction, the action of the particles mov es along circle that is the same\nto the linear case.\n1 Linear Cyclotron Damping\nIn this section, let us consider the following linear Vlasov equations in a uniform magnetic field, and recall\nthe equations: \n\n∂tf+v·∇xf+q\nm(v×B0)·∇vf=−q\nm(E+v×B)·∇vf0,\n∂tB=∇x×E,∇·B= 0,\nE=W(x)∗ρ(t,x),ρ(t,x) =/integraltext\nR3f(t,x,v)dv,\nf(0,x,v) =f0(x,v1,v2,vz),f0(v) =f0(v1,v2,vz),(1.1)\n4where the distribution function f=f(t,x,v) :R+×T3×R3→R,W(x) = (W1(x),W2(x),0) :T3→T3, B0is\na constant magnetic field along the ˆ zdirection, E=E(t,x) is the electric field, B0+Bis the magnetic field.\nTheorem 1.1 For any η,v∈R3,k∈N3\n0,we assume that the following conditions hold in equations (0 .9).\n(i)W(x)is an odd function on x3,|/hatwiderW(k)| ≤1\n1+|k|γ,γ >1,whereW(x) = (W1(x),W2(x),0);\n(ii)|ˆf0(η)| ≤C0e−2πλ0|η|,|∂η3ˆf0(η)| ≤C0e−2πλ0|η|,|f0(·,v3)| ≤C0e−2πα0|v3|,for some constants λ0,α0,C0>\n0;\n(iii)|ˆf0(k,η)| ≤C0e−2πλ0|η|for some constant C0>0,whereλ0is defined in (ii);\n(iv) In addition, (PSC)holds,\n(PSC) :for any component velocity in the ˆzdirection v3∈R,there exists some positive constant vTesuch\nthat ifv3=ω\nk3whenk3/\\e}atio\\slash= 0;ork3= 0whereω,kare frequencies of time and space t,x,respectively, then\n|v3| ≫vTe.\nThen for any fixed η3,k3,and for any λ′\n0< λ0,we have\n|ˆf(t,k,η)−ˆf0(k,η)| ≤e−2πλ′\n0|η3+k3t|,|ˆρ(t,k)−ˆρ0| ≤e−2πλ′\n0|ηk1|e−2πλ′\n0|ηk2|e−λ′\n0|k3|t,\n|ˆE(t,k)| ≤e−2πλ′\n0|ηk1|e−2πλ′\n0|ηk2|e−2πλ′\n0|k3|t,|ˆB(t,k)| ≤te−2πλ′\n0|ηk1|e−2πλ′\n0|ηk2|e−2πλ′\n0|k3|t.(1.2)\nwhereρ0=/integraltext\nT3/integraltext\nR3f0(x,v)dvdx,η k1=1\nΩ(−k2cosΩt+k2−k1sinΩt),ηk2=1\nΩ(−k2sinΩt−k1+k1cosΩt).\nRemark 1.2 In the linear case, from (1.2) and (1.24)-(1.26), it is easy t o observe that cyclotron damping\nis almost the same with Landau damping when B0tends to zero. However, when B0is fixed, in the horizon\ndirection there is no damping and the motion of particles is a circle; in the ˆzdirection, the damping still occurs.\nTherefore, there is no immediate tendency toward trapping. This is a crucial point for cyclotron damping. And\nin the latter case, the motion of plasma particles moves alon g spiral trajectories.\nBefore proving Theorem 1.1, we give a key lemma.\nLemma 1.3 Under the assumptions of Theorem 0.1, let Φ(t,k) =e2πλ′\n0|k3|t·e2πλ′\n0|ηk1|e2πλ′\n0|ηk2|ˆρ(t,k), A(t,k) =/integraltext\nT3/integraltext\nR3e−2πix·kf0(x′(0,x,v),v′(0,x,v))dv′dx,whereλ0> λ′\n0>0,λ0is defined in Theorem 0.1, we have\n/ba∇dblΦ/ba∇dblL∞(dt)≤/parenleftbigg\n1+C(k,W,Ω)\n(λ0−λ′\n0)3\n2/parenrightbigg\n/ba∇dbleλ0νkA/ba∇dblL∞(dt), (1.3)\nwhereηk1=1\nΩ(−k2cosΩt+k2−k1sinΩt),ηk2=1\nΩ(−k2sinΩt−k1+k1cosΩt),νk=|ηk1|+|ηk2|+|k3|t.\nProof.First, we consider the case that ω/\\e}atio\\slash= 0,k3/\\e}atio\\slash= 0,\n˜ρ(ω,k) =/integraldisplay\nR+/integraldisplay\nT3/integraldisplay\nR3e2πitωe−2πix·kf(t,x,v)dxdvdt\n=/integraldisplay\nR+/integraldisplay\nT3/integraldisplay\nR3e2πitωe−2πix·kf0(x′(0,x,v),v′(0,x,v))dvdxdt−q\nm/integraldisplay\nR+/integraldisplay\nT3/integraldisplayt\n0/integraldisplay\nR3\n·e2πitωe−2πix·k[(E+v′(τ,x,v)×B)·∇′\nvf0](τ,x′(τ,x,v),v′(τ,x,v))dvdxdτdt.\n(1.4)\nRecallE(t,x) =W(x)∗ρ(t,x), ∂tB(t,x) =∇x×E(t,x),then taking the Fourier transform in the variables t,x,\n/tildewideE(ω,k) =/hatwiderW(k)˜ρ(ω,k), ω/tildewideB(ω,k) =k×/tildewideE(ω,k).\nFurthermore, we get\nv×/tildewideB(ω,k) =1\nω[v×(k×/tildewideE(ω,k))]\n=1\nω/parenleftbigg\nv2(k1/hatwiderW2−k2/hatwiderW1)−v3k3/hatwiderW1,−v1(k1/hatwiderW2−k2/hatwiderW1)+v3k3/hatwiderW2,v1k3/hatwiderW1+v2k3/hatwiderW2/parenrightbigg\n˜ρ(ω,k).\n(1.5)\nCombining (1.3)-(1.4) and (2.3)-(2.4), and note that dv→dv′preserves the measure, we can change between\ndvanddv′whenever we need, but in order to simply the notations, we don’t diffe rentiate the notations dvand\ndv′in this paper, so we have\n˜ρ(ω,k) =/integraldisplay\nR+/integraldisplay\nT3/integraldisplay\nR3e2πitωe−2πix·kf0(x′(0,x,v),v′(0,x,v))dvdxdt\n5+q\nm˜ρ(ω,k)1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik3v3te−2πiηk2v′\n2e−2πiηk1v′\n1(k3v3)(/hatwiderW1∂v′\n1f0−/hatwiderW2∂v′\n2f0)dvdt\n+q\nm˜ρ(ω,k)1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik3v3te−2πiηk2v′\n2e−2πiηk1v′\n1(k1/hatwiderW2−k2/hatwiderW1)\n·(v2∂v′\n1f0−v1∂v′\n2f0)dvdt−q\nm˜ρ(ω,k)1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik3v3te−2πiηk2v′\n2e−2πiηk1v′\n1\n·(v1/hatwiderW1+v2/hatwiderW2)k3∂v′\n3f0dvdt−q\nm˜ρ(ω,k)/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik3v3te−2πiηk2v′\n2e−2πiηk1v′\n1\n·(/hatwiderW1∂v′\n1f0+/hatwiderW2∂v′\n2f0)dvdt,\n(1.6)\nwhereηk1=1\nΩ(−k2cosΩt+k2−k1sinΩt), ηk2=1\nΩ(−k2sinΩt−k1+k1cosΩt).\nLet\n˜L(ω,k) =q\nm1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik3v3te−2πiηk2v′\n2e−2πiηk1v′\n1(k3v3)(/hatwiderW1∂v′\n1f0−/hatwiderW2∂v′\n2f0)dvdt\n−q\nm/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik3v3t(/hatwiderW1∂v′\n1f0+/hatwiderW2∂v′\n2f0)·e−2πiηk2v′\n2e−2πiηk1v′\n1dvdt\n−q\nm1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik3v3te−2πiηk2v′\n2e−2πiηk1v′\n1(v1/hatwiderW1+v2/hatwiderW2)k3∂v′\n3f0dvdt\n+q\nm1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik3v3te−2πiηk2v′\n2e−2πiηk1v′\n1(k1/hatwiderW2−k2/hatwiderW1)(v2∂v′\n1f0−v1∂v′\n2f0)dvdt,\n(1.7)\nhence\n˜ρ(ω,k) =˜A(ω,k)+ ˜ρ(ω,k)˜L(ω,k). (1.8)\nThen taking the inverse Fourier transform in time t,we get ˆρ(t,k) =ˆA(t,k)+ ˆρ(t,k)∗ˆL(t,k),and\ne2πλ′\n0ηk1e2πλ′\n0ηk2e2πλ′\n0|k3|tˆρ(t,k) =e2πλ′\n0ηk1e2πλ′\n0ηk2e2πλ′\n0|k3|tˆA(t,k)+e2πλ′\n0ηk1e2πλ′\n0ηk2e2πλ′\n0|k3|tˆρ(t,k)∗ˆL(t,k).\n(1.9)\nLet Φ(t,k) =e2πλ′\n0ηk1e2πλ′\n0ηk2e2πλ′\n0|k3|tˆρ(t,k),A(t,k) =e2πλ′\n0ηk1e2πλ′\n0ηk2e2πλ′\n0|k3|tˆA(t,k),K0(t,k) =e2πλ′\n0ηk1\ne2πλ′\n0ηk2e2πλ′\n0|k3|tˆL(t,k),then from (1.8), we have /tildewideΦ(ω,k) =/tildewideA(ω,k)+/tildewideΦ(ω,k)/tildewideK0(ω,k).\nThen\n/ba∇dblΦ(t,k)/ba∇dblL2(dt)=/ba∇dbl/tildewideΦ(ω,k)/ba∇dblL2≤ /ba∇dbl/tildewideA(ω,k)/ba∇dblL2+/ba∇dbl/tildewideΦ(ω,k)/ba∇dblL2/ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞\n≤ /ba∇dble2πλ′\n0νkˆA(t,k)/ba∇dblL2+/ba∇dble2πλ′\n0νkˆρ(t,k)/ba∇dblL2/ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞,\n(1.10)\nwhereνk=|ηk1|+|ηk2|+|k3|t.\nNext we have to estimate /ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞.\nIndeed,\n/ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞≤sup\nωq\nm/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik3v3te2πλ′\n0νke−2πiηk2v′\n2e−2πiηk1v′\n1(k3v3)·(/hatwiderW1∂v′\n1f0\n−/hatwiderW2∂v′\n2f0)dvdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle+q\nm/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik3v3te2πλ0νk·(/hatwiderW1∂v′\n1f0+/hatwiderW2∂v′\n2f0)·e−2πiηk2v′\n2\ne−2πiηk1v′\n1dvdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle−q\nm(ω,k)1\nω/integraldisplay\nR+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3e2πitωe2πλ0νke−2πik3v3te−2πiηk2v′\n2e−2πiηk1v′\n1·(v1/hatwiderW1+v2/hatwiderW2)k3\n∂v′\n3f0dv/vextendsingle/vextendsingle/vextendsingle/vextendsingledt+q\nm(ω,k)1\nω/integraldisplay\nR+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3e2πitωe−2πik3v3te−2πiηk2v′\n2e2πλ0νke−2πiηk1v′\n1·(k1/hatwiderW2−k2/hatwiderW1)\n(v2∂v′\n1f0−v1∂v′\n2f0)dv/vextendsingle/vextendsingle/vextendsingle/vextendsingledt/bracketrightbigg\n=I+II+III+IV.\n(1.11)\n6In fact, we only need to estimate one term of (1.10) because of simila r processes of other terms. Without\nloss of generality, we give an estimate for I.In the same way, we only estimate one term of I,here we still\ndenoteI.\nI= sup\nωq\nm/vextendsingle/vextendsingle/vextendsingle/vextendsinglek3\nω/hatwiderW2/integraldisplay\nR+/integraldisplay\nRe2πitω(2πiηk1)e2πλ0|ηk1|e2πλ0|ηk2|e−2πik3v3te2πλ0|k3|t·/hatwidestv3f0(ηk1,ηk2,v3)dv3dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n= sup\nωq\nm/vextendsingle/vextendsingle/vextendsingle/vextendsinglek3\nω/hatwiderW2/integraldisplay\nR+/integraldisplay\nRe2πitω(2πiηk1)e2πλ0|ηk1|e2πλ0|ηk2|e−2πik3v3t·/summationdisplay\nn|2πiλ0|k3|t|n\nn!/hatwidestv3f0(ηk1,ηk2,v3)dv3dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n= sup\nωq\nm/summationdisplay\nnλn\nn!/vextendsingle/vextendsingle/vextendsingle/vextendsinglek3\nω/hatwiderW2/integraldisplay\nR+/integraldisplay\nR(2πiηk1)(−1)ne2πλ0|ηk1|e2πλ0|ηk2|e2πik3t(ω\nk3−v3)·∇n\nv3/hatwidestv3f0(ηk1,ηk2,v3)dv3dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n= sup\nωq\nm/summationdisplay\nnλn\n0\nn!/vextendsingle/vextendsingle/vextendsingle/vextendsinglek3\nω/hatwiderW2(2πiηk1)e2πλ0|ηk1|e2πλ0|ηk2|(−i∇ω\nk3)n/hatwidestv3f0(ηk1,ηk2,ω\nk3)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤q\nmvTee−c0vTe,\nwhere in the last inequality we use the facts that if v3=ω\nk3,thenv3≫vTe,and the assumption (i) and (iv).\nThen there exists some constant 0 < κ <1 such that /ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞≤κ.\nIn conclusion, we have /ba∇dble2πλ′\n0νkˆρ(t,k)/ba∇dblL2≤/bardble2πλ′\n0νkA/bardblL2(dt)\nκ.\nThen we get\n/ba∇dblΦ/ba∇dblL∞(dt)≤ /ba∇dble2πλ′\n0νkA/ba∇dblL∞(dω)+/ba∇dble2πλ′\n0νkL/ba∇dblL2(dt)/ba∇dble2πλ′\n0νkA/ba∇dblL2(dt)\nκ(1.12)\n/ba∇dble2πλ′\n0νkL/ba∇dbl2\nL2(dt)≤/integraldisplay∞\n0|e4πλ′\n0|ηk1|e4πλ′\n0|ηk2|e4πλ′\n0|k3|t/braceleftbiggq\nm/integraldisplayt\n0/bracketleftbigg\nk3/parenleftbigg\n(2πiηk1)/hatwiderW2∂η3ˆf0(ηk1,ηk2,k3τ)\n+(2πiηk2)/hatwiderW1∂η3ˆf0(ηk1,ηk2,k3τ)/parenrightbigg\n−q\nm/parenleftbigg\n/hatwiderv′\n1∂v′\n3f0(ηk1,ηk2,k3τ)/hatwiderW1+/hatwiderv′\n2∂v′\n3f0(ηk1,ηk2,k3τ)/hatwiderW2/parenrightbigg\n·k3+q\nm(k2/hatwiderW1+k1/hatwiderW2)/parenleftbigg\n/hatwiderv′\n2∂v′\n1f0(ηk1,ηk2,k3τ)−/hatwiderv′\n1∂v′\n2f0(ηk1,ηk2,k3τ)/parenrightbigg/bracketrightbigg\ndτ\n−q\nm/parenleftbigg\n/hatwiderW1(2πiηk1)ˆf0(ηk1,ηk2,k3t)−/hatwiderW2(2πiηk2)ˆf0(ηk1,ηk2,k3t)/parenrightbigg/bracerightbigg2\ndt.\n(1.13)\nUsing the conditions of Theorem 0.1, by the simple computation similar t oI,\n/ba∇dble2πλ′\n0νkL/ba∇dbl2\nL2(dt)≤C(W,k,Ω)/integraldisplay∞\n0e−4π(λ0−λ′\n0)|k3|tC(W,k,Ω)dt≤C(W,k,Ω)\n(λ0−λ′\n0). (1.14)\nNow we estimate /ba∇dble2πλ′\n0|ηk1|e2πλ′\n0|ηk2|e2πλ′\n0|k3|tA/ba∇dblL2(dt)as the above process,\n/ba∇dble2πλ′\n0|ηk1|e2πλ′\n0|ηk2|e2πλ′\n0|k3|tA/ba∇dblL2(dt)\n=/parenleftbigg/integraldisplay∞\n0|e2πλ′\n0|ηk1|e2πλ′\n0|ηk2|e2πλ′\n0|k3|tˆf0(k,ηk1,ηk2,k3t)|2dt/parenrightbigg1\n2\n≤C(Ω)\n(λ0−λ′\n0)1\n2/ba∇dble2πλ′\n0|ηk1|e2πλ′\n0|ηk2|e2πλ0|k3|tA/ba∇dblL∞(dt). (1.15)\nNow we consider k3= 0, k1k2/\\e}atio\\slash= 0,\n˜ρ(ω,k1,k2,0) =/integraldisplay\nR+/integraldisplay\nT3/integraldisplay\nR3e2πitωe−2πi(x1,x2)·(k1,k2)f(t,x1,x2,x3,v)dx1dx2dx3dvdt\n=/integraldisplay\nR+/integraldisplay\nT3/integraldisplay\nR3e2πitωe−2πi(x1,x2)·(k1,k2)f0(x′(0,x,v),v′(0,x,v))dvdx1dx2dx3dt−q\nm/integraldisplay\nR+/integraldisplay\nT3/integraldisplayt\n0/integraldisplay\nR3\n·e2πitωe−2πi(x1,x2)·(k1,k2)[(E+v′(τ,x,v)×B)·∇′\nvf0](τ,x′(τ,x,v),v′(τ,x,v))dvdxdτdt.\n(1.16)\nTaking the Fourier transform in the variables t,(x1,x2),\n/tildewideE(ω,k1,k2,0) =/hatwiderW(k1,k2,0)˜ρ(ω,k1,k2,0), ω/tildewideB(ω,k) = (k1,k2,∂x3)×/tildewideE(ω,k1,k2,0).\n7Furthermore, we get\nv×/tildewideB(ω,k1,k2,0) =1\nω[v×((k1,k2,∂x3)×/tildewideE(ω,k1,k2,0))]\n=1\nω/parenleftbigg\nv2(k1/hatwiderW2−k2ˆW1)−v3∂x3/hatwiderW1,−v1(k1/hatwiderW2−k2/hatwiderW1)+v3∂x3/hatwiderW2,v1∂x3/hatwiderW1+v2∂x3/hatwiderW2/parenrightbigg\n˜ρ(ω,k1,k2,0).\n(1.17)\n˜ρ(ω,k1,k2,0) =/integraldisplay\nR+/integraldisplay\nT3/integraldisplay\nR3e2πitωe−2πix12·k12f0(x′(0,x,v),v′(0,x,v))dvdx12dx3dt\n+q\nm˜ρ(ω,k12,0)1\nω/integraldisplay\nR+/integraldisplay\nR3/integraldisplay\nTe2πitωe−2πiηk2v′\n2e−2πiηk1v′\n1(v3∂x3)(/hatwiderW1∂v′\n1f0−/hatwiderW2∂v′\n2f0)dx3dvdt\n+q\nm˜ρ(ω,k12,0)1\nω/integraldisplay\nR+/integraldisplay\nR3/integraldisplay\nTe2πitωe−2πiηk2v′\n2e−2πiηk1v′\n1(k1/hatwiderW2−k2ˆW1)\n·(v2∂v′\n1f0−v1∂v′\n2f0)dx3dvdt−q\nm˜ρ(ω,k12,0)1\nω/integraldisplay\nR+/integraldisplay\nR3/integraldisplay\nTe2πitωe−2πiηk2v′\n2e−2πiηk1v′\n1\n·∂x3(v1ˆW1+v2/hatwiderW2)∂v′\n3f0dx3dvdt−q\nm˜ρ(ω,k12,0)/integraldisplay\nR+/integraldisplay\nR3/integraldisplay\nTe2πitωe−2πiηk2v′\n2e−2πiηk1v′\n1\n·(/hatwiderW1∂v′\n1f0+/hatwiderW2∂v′\n2f0)dx3dvdt,\n(1.18)\nwhereηk1=1\nΩ(−k2cosΩt+k2−k1sinΩt), ηk2=1\nΩ(−k2sinΩt−k1+k1cosΩt).\nLet\n˜L(ω,k12,0) =q\nm1\nω/integraldisplay\nR+/integraldisplay\nR3/integraldisplay\nTe2πitωe−2πiηk2v′\n2e−2πiηk1v′\n1(v3∂x3)(/hatwiderW1∂v′\n1f0−/hatwiderW2∂v′\n2f0)dx3dvdt\n−q\nm/integraldisplay\nR+/integraldisplay\nR3/integraltext\nTe2πitω(/hatwiderW1∂v′\n1f0+/hatwiderW2∂v′\n2f0)·e−2πiηk2v′\n2e−2πiηk1v′\n1dx3dvdt\n−q\nm1\nω/integraldisplay\nR+/integraldisplay\nR3/integraldisplay\nTe2πitωe−2πiηk2v′\n2e−2πiηk1v′\n1∂x3(v1/hatwiderW1+v2/hatwiderW2)∂v′\n3f0dx3dvdt\n+q\nm1\nω/integraldisplay\nR+/integraldisplay\nR3/integraldisplay\nTe2πitωe−2πiηk2v′\n2e−2πiηk1v′\n1(k1/hatwiderW2−k2/hatwiderW1)(v2∂v′\n1f0−v1∂v′\n2f0)dx3dvdt,\n(1.19)\nhence\n˜ρ(ω,k12,0) =˜A(ω,k12,0)+ ˜ρ(ω,k12,0)˜L(ω,k12,0). (1.20)\nThen taking the inverse Fourier transform in time t,we get ˆρ(t,k12,0) =ˆA(t,k12,0)+ ˆρ(t,k12,0)∗ˆL(t,k12,0),\nand\ne2πλ′\n0ηk1e2πλ′\n0ηk2ˆρ(t,k12,0) =e2πλ′\n0ηk1e2πλ′\n0ηk2ˆA(t,k12,0)+e2πλ′\n0ηk1e2πλ′\n0ηk2ˆρ(t,k12,0)∗ˆL(t,k12,0).(1.21)\nThen\n/ba∇dblΦ(t,k12,0)/ba∇dblL2(dt)=/ba∇dbl/tildewideΦ(ω,k12,0)/ba∇dblL2≤ /ba∇dbl/tildewideA(ω,k12,0)/ba∇dblL2+/ba∇dbl/tildewideΦ(ω,k12,0)/ba∇dblL2/ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞\n≤ /ba∇dble2πλ′\n0νk12ˆA(t,k12,0)/ba∇dblL2+/ba∇dble2πλ′\n0νk12ˆρ(t,k12,0)/ba∇dblL2/ba∇dbl/tildewideK0(ω,k12,0)/ba∇dblL∞,\n(1.22)\nwhereνk12=|ηk1|+|ηk2|.\nNext we have to estimate /ba∇dbl/tildewideK0(ω,k12,0)/ba∇dblL∞.\nIndeed,\n/ba∇dbl/tildewideK0(ω,k12,0)/ba∇dblL∞≤sup\nωq\nm/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nω/integraldisplay\nR+/integraldisplay\nR3/integraldisplay\nTe2πitωe2πλ′\n0νk12e−2πiηk2v′\n2e−2πiηk1v′\n1(v3∂x3)·(/hatwiderW1∂v′\n1f0\n−/hatwiderW2∂v′\n2f0)dx3dvdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle+q\nm/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR+/integraldisplay\nR3/integraldisplay\nTe2πitωe2πλ0νk12·(/hatwiderW1∂v′\n1f0+/hatwiderW2∂v′\n2f0)·e−2πiηk2v′\n2\n8e−2πiηk1v′\n1dx3dvdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle−q\nm(ω,k)1\nω/integraldisplay\nR+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3/integraldisplay\nTe2πitωe2πλ0νke−2πiηk2v′\n2e−2πiηk1v′\n1·(v1/hatwiderW1+v2/hatwiderW2)k3\n∂v′\n3f0dv/vextendsingle/vextendsingle/vextendsingle/vextendsingledt+q\nm(ω,k)1\nω/integraldisplay\nR+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3e2πitωe−2πiηk2v′\n2e2πλ0νke−2πiηk1v′\n1·(k1/hatwiderW2−k2/hatwiderW1)\n(v2∂v′\n1f0−v1∂v′\n2f0)dv/vextendsingle/vextendsingle/vextendsingle/vextendsingledt/bracketrightbigg\n=I+II+III+IV.\n(1.23)\nIn fact, we only need to estimate one term of (1.10) because of simila r processes of other terms. Without\nloss of generality, we give an estimate for I.In the same way, we only estimate one term of I,here we still\ndenoteI.\nI= sup\nωq\nm/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nω/integraldisplay\nR+/integraldisplay\nR/integraldisplay\nT/hatwiderW2(k12,0)e2πitω(2πiηk1)e2πλ0|ηk1|e2πλ0|ηk2|·∂x3/hatwidestv3f0(ηk1,ηk2,v3)dx3dv3dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤q\nmvTee−c0vTe,\nwhere in the last inequality we use the facts that if k3= 0,thenv3≫vTe,and the assumption (i) and (iv).\nThen there exists some constant 0 < κ <1 such that /ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞≤κ.\nFirst through Lemma1.1, we can study the asymptotic behavior of the electric field and th e magnetic field\nE(t,x),B(t,x).\nCorollary 1.4 Under the assumptions of Theorem 1.1, and E(t,x),B(t,x)satisfy the Maxwell equations of\n(1.1), then for any for any λ′′\n0< λ′\n0< λ0,we have\n|/hatwideE(t,k)| ≤e−2πλ′\n0|ηk1|e−2πλ′\n0|ηk2|e−2πλ′\n0|k3|t,|/hatwideB(t,k)| ≤te−2πλ′′\n0|ηk1|e−2πλ′′\n0|ηk2|e−2πλ′\n0|k3|t. (1.24)\nProof.Since∂t/hatwideB(t,k) =k×/hatwideE(t,k),/hatwideE(t,k) =/hatwiderW(k)ˆρ(t,k) = (ˆW1(k),/hatwiderW2(k),0)ˆρ(t,k),then∂t/hatwideB(t,k) =\n(−k3/hatwiderW2(k),k3/hatwiderW1(k),k1/hatwiderW2(k)−k2/hatwiderW1(k))ˆρ(t,k).By Lemma 1.1, |∂t/hatwideB(t,k)| ≤C(k,Ω,W)e−2πλ′\n0|ηk1|e−2πλ′\n0|ηk2|\ne−2πλ′\n0|k3|t.Hence, from /hatwideB(0,k) = 0,we have |/hatwideB(t,k)| ≤te−2πλ′\n0|ηk1|e−2πλ′\n0|ηk2|e−2πλ′\n0|k3|t.Then|/hatwideB(t,k)| ≤\ne−2πλ′′\n0|ηk1|e−2πλ′′\n0|ηk2|e−2πλ′′\n0|k3|t.\nProof of Theorem 0.1.From (1.1), we have\nf(t,x,v) =f0(x′(0,x,v),v′(0.x,v))−q\nm/integraldisplayt\n0[(E+v′×B)·∇′\nvf0](τ.x′(τ,x,v),v′(τ,x,v))dτ. (1.25)\nTaking the Fourier-Laplace transform in variables x,v,t,we find\n˜f(ω,k,η) =/integraldisplay\nR+/integraldisplay\nT3/integraldisplay\nR3e2πit·ωe−2πix·ke−2πiv·ηf0(x′(0,x,v),v′(0,x,v))dvdxdt\n−q\nm/integraldisplay\nR+/integraldisplay\nT3/integraldisplay\nR3e2πit·ωe−2πix·ke−2πiv·η/integraldisplayt\n0[B·∇′\nv×(f0v′)](τ,x′(τ,x,v),v′(τ,x,v))dτdvdxdt\n−q\nm/integraldisplay\nR+/integraldisplay\nT3/integraldisplay\nR3e2πit·ωe−2πix·ke−2πiv·η/integraldisplayt\n0[E·∇′\nvf0](τ,x′(τ,x,v),v′(τ,x,v))dτdvdxdt\n=I+II+III. (1.26)\nI=/integraldisplay\nR+/integraldisplay\nT3/integraldisplay\nR3e2πit·ωe−2πiv·η·f0(x,v′)exp(−2πi[x+(1\nΩ(v′\n2−v2),−1\nΩ(v′\n1−v1),v3t)]·k)dvdxdt\n=/integraldisplay\nR+/integraldisplay\nR3e2πit·ωe−2πiv3(η3+k3t)e−2πiv′\n2(η2+ηk2)e−2πiv′\n1(η1+ηk1)ˆf0(k,v′)dvdt.\n(1.27)\nAs the above process, we have\nII=−q\nm/integraldisplay\nR+/integraldisplay\nR3e2πit·ωe−2πi(k3t+η3)v3e−2πiv′\n2(η2+ηk2)e−2πiv′\n1(η1+ηk1)\n·/integraldisplayt\n0(ˆB(τ,k)·(v3∂v′\n2f0−v′\n2∂v3f0,v′∂v3f0−v3∂v′\n1f0,v′\n2∂v′\n1f0−v′\n1∂v′\n2f0)dτdvdt\n9=−q\nm/integraldisplay\nR+e2πiτ·ωˆB(τ,k)dτ·/integraldisplay\nR+e2πit·ω/integraldisplay\nR3e−2πi(k3t+η3)v3e−2πiv′\n2(η2+ηk2)e−2πiv′\n1(η1+ηk1)\n·(v3∂v′\n2f0−v′\n2∂v3f0,v′∂v3f0−v3∂v′\n1f0,v′\n2∂v′\n1f0−v′\n1∂v′\n2f0)dvdt\n=−q\nm˜ρ(ω,k)1\nω/integraldisplay\nR+e2πit·ω/integraldisplay\nR3e−2πi(kzt+η3)v3e−2πiv′\n2(η2+ηk2)e−2πiv′\n1(η1+ηk1)(k׈W(k))\n·(v3∂v′\n2f0−v′\n2∂v3f0,v′∂v3f0−v3∂v′\n1f0,v′\n2∂v′\n1f0−v′\n1∂v′\n2f0)dvdt.\n(1.28)\nSince/hatwiderW(k) = (/hatwiderW1(k),/hatwiderW2(k),0),thenk×/hatwiderW(k) = (−k3/hatwiderW2,k3/hatwiderW1,k1/hatwiderW2−k2/hatwiderW1),and\nII=q\nm˜ρ(ω,k)1\nω/integraldisplay\nR+e2πit·ω/integraldisplay\nRe−2πi(k3t+η3)v3e−2πiv′\n2(η2+ηk2)e−2πiv′\n1(η1+ηk1)·(k3/hatwiderW2(−v3∂v′\n2f0\n+v′\n2∂v3f0)+k3/hatwiderW1(v′\n1∂v3f0−v3∂v′\n1f0)+(k1/hatwiderW2−k2/hatwiderW1)(v′\n2∂v′\n1f0−v′\n1∂v′\n2f0))dvdt.\n(1.29)\nSimilarly,\nIII=−q\nm˜ρ(ω,k)/integraldisplay\nR+e2πit·ω/integraldisplay\nR3e−2πi(k3t+η3)v3e−2πiv′\n2(η2+ηk2)e−2πiv′\n1(η1+ηk1)(/hatwiderW1∂v′\n1f0+/hatwiderW2∂v′\n2f0)dvdt.\n(1.30)\nTherefore, from(1.17)-(1.21), we have\nω˜f(ω,k,v′,η3) =ω(I+II+III). (1.31)\nFurthermore,\n∂tˆf(t,k,η) =∂t/integraldisplay\nR3e−2πi(k3t+η3)v3e−2πiv′\n2(η2+ηk2)e−2πiv′\n1(η1+ηk1)ˆf0(k,v′)dv\n−q\nmˆρ(t,k)∗t/integraldisplay\nR3e−2πi(k3t+η3)v3e−2πiv′\n2(η2+ηk2)e−2πiv′\n1(η1+ηk1)·(k3/hatwiderW2(−v3∂v′\n2f0\n+v′\n2∂v3f0)+k3/hatwiderW1(v′\n1∂v3f0−v3∂v′\n1f0)+(k1/hatwiderW2−k2/hatwiderW1)(v′\n2∂v′\n1f0−v′\n1∂v′\n2f0))dvdt\n−q\nmˆρ(t,k)∗t∂t/integraldisplay\nR3e−2πi(k3t+η3)v3e−2πiv′\n2(η2+ηk2)e−2πiv′\n1(η1+ηk1)(/hatwiderW1∂v′\n1f0+/hatwiderW2∂v′\n2f0)dvdt\n=IV+V+VI. (1.32)\nNow we estimate IV,V,VI, respectively.\nIV=∂t/integraldisplay\nR3e−2πi(k3t+η3)v3e−2πiv′\n2(η2+ηk2)e−2πiv′\n1(η1+ηk1)ˆf0(k,v′)dv=∂tˆf0(k,η1+ηk1,η2+ηk2,k3t+η3)\n≤C(Ω,k)e−2πλ0|η1+ηk1|e−2πλ0|η2+ηk2|e−2πλ0|k3t+η3|, (1.33)\nhere we use the assumption that |ˆf0(k,η)| ≤C0e−2πλ0|η1|e−2πλ0|η2|e−2πλ0|η3|.\nV=−q\nmˆρ(t,k)∗t/integraldisplay\nR3e−2πi(k3t+η3)v3e−2πiv′\n2(η2+ηk2)e−2πiv′\n1(η1+ηk1)·(k3/hatwiderW2(−v3∂v′\n2f0\n+v′\n2∂v3f0)+k3/hatwiderW1(v′\n1∂v3f0−v3∂v′\n1f0)+(k1/hatwiderW2−k2ˆW1)(v′\n2∂v′\n1f0−v′\n1∂v′\n2f0))dv\n=−q\nmˆρ(t,k)∗t[k3/hatwiderW2(/hatwiderv′\n2∂v3f0−/hatwiderv3∂v′\n2f0)(η1+ηk1,η2+ηk2,k3t+η3)\n+k3/hatwiderW1(/hatwiderv′\n1∂v3f0−/hatwiderv3∂v′\n1f0)(η1+ηk1,η2+ηk2,k3t+η3)\n+(k1/hatwiderW2−k2/hatwiderW1)(/hatwiderv′\n2∂v1f0−/hatwiderv1∂v′\n2f0)(η1+ηk1,η2+ηk2,k3t+η3)]\n≤C/integraldisplayt\n0e−2πλ0|η1+ηk1,t−τ|e−2πλ0|η2+ηk2,t−τ|e−2πλ′\n0|ηk1,τ|e−2πλ′\n0|ηk2,τ|e−2πλ′\n0|k3|τ·|k3(t−τ)+η3|e−2πλ0|k3(t−τ)+η3|dτ\n≤C/integraldisplayt\n0e−2πλ0|η1+ηk1,t−τ|e−2πλ0|η2+ηk2,t−τ|e−2πλ′\n0|ηk1,τ|e−2πλ′\n0|ηk2,τ|e−2πλ′\n0|k3|τe−2π(λ0+λ′\n0)\n2|k3(t−τ)+η3|dτ\n≤C/parenleftbigg/integraldisplayt\n0e−2πλ0|η1+ηk1,t−τ|e−2πλ′\n0|ηk1,τ|dτ/parenrightbigg1\n3/parenleftbigg/integraldisplayt\n0e−2πλ′\n0|ηk2,τ|e−2πλ0|η2+ηk2,t−τ|dτ/parenrightbigg1\n3\ne−2πλ′′\n0|k3t+η3|\n10=C/parenleftbigg/integraldisplayt\n0e−2πλ0|η1+1\nΩ{k2−√\nk2\n1+k2\n2sin[Ω(t−τ)+ϕ1]}|e−2πλ′\n0|1\nΩ{k2−√\nk2\n1+k2\n2sin[Ωτ+ϕ1]}|dτ/parenrightbigg1\n3\n/parenleftbigg/integraldisplayt\n0e−2πλ0|η2+1\nΩ[−√\nk2\n1+k2\n2sin[Ω(t−τ)−ϕ2]−k1]|e−2πλ′\n0|1\nΩ[−√\nk2\n1+k2\n2sin[Ωτ−ϕ2]−k1]|dτ/parenrightbigg1\n3\ne−2πλ′′\n0|k3t+η3|,\n(1.34)\nwhereλ′′\n0=λ′\n0−1\n2(λ0−λ′\n0),tanϕ1=k2\nk1,tanϕ2=k1\nk2.\nVI=−q\nmˆρ(t,k)∗t∂t/parenleftbigg\nˆW1/hatwider∂v′\n1f0+ˆW2/hatwider∂v′\n2f0/parenrightbigg\n(η1+ηk1,η2+ηk2,k3t+η3)\n≤C/parenleftbigg/integraldisplayt\n0e−2πλ0|η1+1\nΩ{k2−√\nk2\n1+k2\n2sin[Ω(t−τ)+ϕ1]}|e−2πλ′\n0|1\nΩ{k2−√\nk2\n1+k2\n2sin[Ωτ+ϕ1]}|dτ/parenrightbigg1\n3\n/parenleftbigg/integraldisplayt\n0e−2πλ0|η2+1\nΩ[−√\nk2\n1+k2\n2sin[Ω(t−τ)−ϕ2]−k1]|e−2πλ′\n0|1\nΩ[−√\nk2\n1+k2\n2sin[Ωτ−ϕ2]−k1]|dτ/parenrightbigg1\n3\ne−2πλ′′\n0|k3t+η3|.\n(1.35)\nFrom (1.23)-(1.26) and Corollary 1.2, the results of Theorem 1.1 are obvious.\n2 Notation and Hybrid analytic norm\nIn order to prove our result in the nonlinear case, we have to introd uce the hybrid analytic norm that is one\nof the cornerstones of our analysis, because they will connect we ll to both estimates in xon the force field and\nuniform estimates in v.First, let us recall that the free transport equation in a constant magnetic field\n∂tf+v·∇xf+q\nm(v×B0)·∇vf= 0 (2.1)\nhas a strong mixing property: any solution of (1.1) converges weak ly in large time to a spatially homogeneous\ndistribution equal to the space-averaging of the initial datum. Let us sketch the proof.\nIffsolves (2.1) in T3×R3,with initial datum fin=f(τ,·),then\nf(t,x′\nt,v′\nt) =fin(x′\nτ−v′·(sinΩt\nΩ,sinΩt\nΩ,t),v′\nτ), (2.2)\nWe give a description on the motion of charged particles in the following way:\ndx′\ndt=v′,dv′\ndt=q\nmv′×B0, (2.3)\nwhereB0=B0ˆz.\nWe assume that x′(t,x,v) =x= (x1,x2,x3),v′(t,x,v) =v= (vx,vy,v3) = (v⊥cosθ,v⊥sinθ,v3) att=τ,\nthen the solution of Eq.(2.1) is obtained as follows:\nv′\nx(t) =v⊥cos(θ+Ω(t−τ)), v′\ny(t) =v⊥sin(θ+Ω(t−τ)), v′\n3=v3;\nx′\n1(t) =x1+v⊥\nΩ[sin(θ+Ω(t−τ))−sinθ],\nx′\n2(t) =x2−v⊥\nΩ[cos(θ+Ω(t−τ))−cosθ], x′\n3(t) =x3+v3(t−τ), (2.4)\nwhere Ω =qB0\nm,v⊥=/radicalbig\nv2\n1+v2\n2.From now on, without loss of generality, we assume m=q= 1.Also through\nsimple computation, it is easy to get\n\nv′\nt1\nv′\nt2\nv′\nt3\n=\ncosΩ(t−τ)−sinΩ(t−τ) 0\nsinΩ(t−τ) cosΩ( t−τ) 0\n0 0 1\n\nv′\nτ1\nv′\nτ2\nv′\nτ3\n/definesR(t−τ)v′\nτ, (2.5)\n\nx′\nt1\nx′\nt2\nx′\nt3\n=\nx1\nx2\nx3\n+\n1\nΩsinΩ(t−τ)1\nΩcosΩ(t−τ)−1\nΩ0\n−1\nΩcosΩ(t−τ)+1\nΩ1\nΩsinΩ(t−τ) 0\n0 0 ( t−τ)\n\nv1\nv2\nv3\n\n11/defines\nx1\nx2\nx3\n+M(t−τ)\nv1\nv2\nv3\n\n(x′\nt1,x′\nt2,x′\nt3) = (x′\nτ1,x′\nτ2,x′\nτ3)+/parenleftbiggv′\nτ1sinΩ(t−τ)\nΩ,v′\nτ2sinΩ(t−τ)\nΩ,v′\nτ3(t−τ)/parenrightbigg\n. (2.6)\nWe introduce an equivalence relation, that is, for any different veloc ityv′\nt1,v′\nt2,there exists an orthogonal\nmatrixO(t1,t2) such that v′\nt1=O(t1,t2)v′\nt2,we sayv′\nt1∼v′\nt2.And all elements satisfying the above equivalence\nrelation are denoted by [ v′\nt],for short v′\nt.\nDefinition 2.1 From the system (1.1)and the above equivalence relation, we can define the corresponding trans-\nform\nS0\nt,τ(x′,v′)/defines/parenleftbigg\nx′\nτ+/parenleftbiggv′\nτ1sinΩ(t−τ)\nΩ,v′\nτ2sinΩ(t−τ)\nΩ,v′\nτ3(t−τ)/parenrightbigg\n,v′\nτ/parenrightbigg\n,\nwhereM(t−τ),R(t−τ)are defined in (2.5).\nRemark 2.2 From the above equality (2.6), we can observe clearly the con nection and difference on between\nLandau damping and cyclotron damping. Indeed, it can be redu ced to Landau damping as B0→0.\nIn addition, from the dynamics system of the above order diffe rential system, it is known that S0\nt,τsatisfies\nS0\nt2,t3◦S0\nt1,t2=S0\nt1,t3.\nTo estimate solutions and trajectories of kinetic equations, maybe we have to work on the phase space\nT3\nx×R3\nv.And we also use the following three parameters: λ(gliding analytic regularity), µ(analytic regularity\ninx) andτ(time-shift along the free transport semigroup). From Remark 2.2, we know that the linear Vlasov\nequation has the property of the free transport semigroup. This property is crucial to our analysis. In this\npaper, one of the cornerstones of our analysis is to compare the s olution of the nonlinear case at time τwith\nthe solution of the linear case.\nNow we start to introduce the very important tools in our paper. Th ese are time-shift pure and hybrid\nanalytic norms. They are similar with those in the paper [23] written by Mouhot and Villani.\nFirst, we introduce some notations. We denote T3=R3/Z3.For function f(x′\nτ,v′\nτ),we define the Fourier\ntransform ˆf(k,η),where (k,η)∈Z3×R3,via\nˆf(k,η) =/integraldisplay\nT3×R3e−2iπx′\nτ·ke−2iπv′\nτ·ηf(x′\nτ,v′\nτ)dx′\nτdv′\nτ,\n˜f(ω,k,η) =/integraldisplay\nR+e2iπt·ω/integraldisplay\nT3×R3e−2iπx′\nτ·ke−2iπv′\nτ·ηf(x′\nτ,v′\nτ)dx′\nτdv′\nτdt.\nWe also write\nk= (k1,k2,k3) = (k⊥cosϕ,k⊥sinϕ,k3), η= (η1,η2,η3) = (η⊥cosγ,η⊥sinγ,η3).\nNow we define some notations\n˜f=˜f(ω,k,η),ˆf=ˆf(t,k,η),˜ρ= ˜ρ(ω,k),ˆρ= ˆρ(t,k).\nDefinition 2.3 (Hybrid analytic norms)\n/ba∇dblf/ba∇dblCλ,µ=/summationdisplay\nm,n∈N3\n0λn\nn!µm\nm!/ba∇dbl∇m\nx′τ∇n\nv′τf/ba∇dblL∞(T3\nx′×R3\nv′),/ba∇dblf/ba∇dblFλ,µ=/summationdisplay\nk∈Z3/integraldisplay\nR3|˜f(k,η)|e2πλ|η|e2πµ|k|dη,\n/ba∇dblf/ba∇dblZλ,µ=/summationdisplay\nl∈Z3/summationdisplay\nn∈N3\n0λn\nn!e2πµ|l|/ba∇dbl/hatwider∇n\nv′τf(l,v)/ba∇dblL∞(R3\nv′).\nDefinition 2.4 (Time-shift pure and hybrid analytic norms) For any λ,µ≥0,p∈[1,∞],we define\n/ba∇dblf/ba∇dblCλ,µ\nt,τ=/ba∇dblf◦S0\nt,τ(x′,v′)/ba∇dblCλ,µ=/summationdisplay\nm,n∈N3\n0λn\nn!µm\nm!/ba∇dbl∇m\nx′τ[∇v′τ+/parenleftbiggsinΩ(t−τ)\nΩ,sinΩ(t−τ)\nΩ,t−τ/parenrightbigg\n·∇x′τ]nf/ba∇dblL∞(T3x×R3v),\n/ba∇dblf/ba∇dblFλ,µ\nt,τ=/ba∇dblf◦S0\nt,τ(x′,v′)/ba∇dblFλ,µ=/summationdisplay\nk∈Z3/integraldisplay\nR3|˜f(k,η)|e2πλ|η+k·(sinΩ(t−τ)\nΩ,sinΩ(t−τ)\nΩ,(t−τ))|e2πµ|k|dη,\n/ba∇dblf/ba∇dblZλ,µ\nt,τ=/ba∇dblf◦S0\nt,τ(x,v)/ba∇dblZλ,µ=/summationdisplay\nl∈Z3/summationdisplay\nn∈N3\n0λn\nn!e2πµ|l|/ba∇dbl[∇v′τ+2iπ/parenleftbiggsinΩ(t−τ)\nΩ,sinΩ(t−τ)\nΩ,t−τ/parenrightbigg\n·l]nˆf(l,v)/ba∇dblL∞(R3v),\n12/ba∇dblf/ba∇dblZλ,µ;p\nt,τ=/summationdisplay\nl∈Z3/summationdisplay\nn∈N3\n0λn\nn!e2πµ|l|/ba∇dbl[∇v′\nτ+2iπ/parenleftbiggsinΩ(t−τ)\nΩ,sinΩ(t−τ)\nΩ,t−τ/parenrightbigg\n·l]nˆf(l,v)/ba∇dblLp(R3\nv′),\n/ba∇dblf/ba∇dblYλ,µ\nt,τ=/ba∇dblf/ba∇dblFλ,µ;∞\nt,τ= sup\nk∈Z3,η∈R3e2πµ|k|e2πλ|η+k·(sinΩ(t−τ)\nΩ,sinΩ(t−τ)\nΩ,(t−τ))||ˆf(k,η)|.\nFrom the above definitions, we can state some simple and important p ropositions, and the related proofs\ncan be found in [23], so we remove the proofs. From (2.6), we know th at the damping occurs only in the ˆ z\ndirection. Therefore, we will mainly focus on the third component of the “phase space” ( x,v) when referring to\nthe damping mechanics of the wave propagation.\nProposition 2.5 For any τ∈R,λ,µ≥0,\n(i) iffis afunction only of x,that is,t=τin the secondequality of (2.5), then /ba∇dblf/ba∇dblCλ,µ\nτ=/ba∇dblf/ba∇dblCλ|τ|+µ,/ba∇dblf/ba∇dblFλ,µ\nτ=\n/ba∇dblf/ba∇dblZλ,µ\nτ=/ba∇dblf/ba∇dblFλ|τ|+µ;\n(ii) iffis a function only of v,then/ba∇dblf/ba∇dblCλ,µ;p\nτ=/ba∇dblf/ba∇dblZλ,µ;p\nτ=/ba∇dblf/ba∇dblCλ,;p;\n(iii) for any λ >0,then/ba∇dblf◦(Id+G)/ba∇dblFλ≤ /ba∇dblf/ba∇dblFλ+ν,ν=/ba∇dblG/ba∇dbl˙Fλ;\n(iv) for any ¯λ > λ,p∈[1,∞],/ba∇dbl∇f/ba∇dblCλ;p≤1\nλelog(¯λ\nλ)/ba∇dblf/ba∇dblC¯λ;p,/ba∇dbl∇f/ba∇dblFλ;p≤1\n2πe(¯λ−λ)/ba∇dblf/ba∇dblF¯λ;p,\n(v) for any ¯λ > λ > 0,µ >0,then/ba∇dblvf/ba∇dblZλ,µ;1\nτ≤ /ba∇dblf/ba∇dblZ¯λ,µ;1\nτ;\n(vi) for any ¯λ > λ,¯µ > µ,/ba∇dbl∇vf/ba∇dblZλ,µ;p\nτ≤C(d)/parenleftbigg\n1\nλlog(¯λ\nλ)/ba∇dblf/ba∇dblZ¯λ,¯µ;p\nτ+τ\n¯µ−µ/ba∇dblf/ba∇dbl˙Z¯λ,¯µ;p\nτ/parenrightbigg\n;\n(vii) for any ¯λ > λ,/ba∇dbl(∇v+τ∇x)f/ba∇dblZλ,µ;p\nτ≤1\nC(d)λlog(¯λ\nλ)/ba∇dblf/ba∇dblZ¯λ,µ;p\nτ;\n(viii) for any ¯λ≥λ≥0,¯µ≥µ≥0,then/ba∇dblf/ba∇dblZλ,µ\nτ≤Z¯λ,¯µ\nτ.Moreover, for any τ,¯τ∈R, p∈[1,∞],we have\n/ba∇dblf/ba∇dblZλ,µ;p\nτ≤ /ba∇dblf/ba∇dblZλ,µ+λ|τ−¯τ|;p\n¯τ;\n(viiii)/ba∇dblf/ba∇dblYλ,µ\nτ≤ /ba∇dblf/ba∇dblZλ,µ;1\nτ;\n(ix) for any function f=f(x,v),/ba∇dbl/integraltext\nR3fdv/ba∇dblFλ|τ|+µ≤ /ba∇dblf/ba∇dblZλ,µ;1\nτ.\nProof.Here we only give the proofof (v). By the invariance under the actio n of free transport, it is sufficient\nto do the proof for t= 0.Applying the Fourier transform formula, we have\n∇m\nv(vˆf)(k,v) =/integraldisplay\nR∂η˜f(k,η)(2iπη)me2iπη·vdη,\nand therefore\n/summationdisplay\nm∈N0λm\nm!/integraldisplay\nR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR∂η˜f(k,η)(2iπη)me2iπη·vdη/vextendsingle/vextendsingle/vextendsingle/vextendsingledv≤/summationdisplay\nm∈N0λm\nm!sup\nη∈R|(2πη)m∂η˜f(k,η)|/integraldisplay\nR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRe2iπη·vdη/vextendsingle/vextendsingle/vextendsingle/vextendsingledv\n≤C(λ,¯λ)/summationdisplay\nm∈N0¯λm\nm!sup\nη∈R|(2πη)m˜f(k,η)| ≤C(λ,¯λ)/summationdisplay\nm∈N0¯λm\nm!/integraldisplay\nR|∇m\nvˆf(k,v)|dv,\nwhere the last second inequality uses the property (iv), then we ha ve\ne2πµ|k|/summationdisplay\nm∈N0λm\nm!/integraldisplay\nR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR∂η˜f(k,η)(2iπη)me2iπη·vdη/vextendsingle/vextendsingle/vextendsingle/vextendsingledv≤e2πµ|k|/summationdisplay\nm∈N0¯λm\nm!/integraldisplay\nR|∇m+1\nvˆf(k,v)|dv.\nThis establishes (v).\nProposition 2.6 For any X∈ {C,F,Z}and any t,τ∈R,\n/ba∇dblf◦S0\nτ/ba∇dblXλ,µ\nτ=/ba∇dblf/ba∇dblXλ,µ\nt+τ.\nLemma 2.7 Letλ,µ≥0,t∈R,and consider two functions F,G:T×R→T×R.Then there is ε∈(0,1\n2)\nsuch that if F,Gsatisfy\n/ba∇dbl∇(F−Id)/ba∇dblZλ′,µ′\nτ≤ε, (2.7)\nwhereλ′=λ+2/ba∇dblF−G/ba∇dblZλ,µ\nτ, µ′=µ+2(1+|τ|)/ba∇dblF−G/ba∇dblZλ,µ\nτ,thenFis invertible and\n/ba∇dblF−1◦G−Id/ba∇dblZλ,µ\nτ≤2/ba∇dblF−G/ba∇dblZλ,µ\nτ. (2.8)\n13Proposition 2.8 For any λ,µ≥0and any p∈[1,∞],τ∈R,σ∈R,a∈R\\{0}andb∈R,we have\n/ba∇dblf(x+bv′+X(x,v′),av′+V(x,v′))/ba∇dblZλ,µ;p\nτ≤ |a|−3\np/ba∇dblf/ba∇dblZα,β;p\nσ,\nwhereα=λ|a|+/ba∇dblV/ba∇dblZλ,µ\nτ, β=µ+λ|b+τ−aσ|+/ba∇dblX−σV/ba∇dblZλ,µ\nτ.\nLemma 2.9 LetG=G(x,v)andR=R(x,v)be valued in R,andβ(x) =/integraltext\nR(G·R)(x′,v′)dv′.Then for any\nλ,µ,t≥0and any b >−1,we have\n/ba∇dblβ/ba∇dblFλt+µ≤3/ba∇dblG/ba∇dblZλ(1+b),µ;1\nτ−bt\n1+b/ba∇dblR/ba∇dblZλ(1+b),µ\nτ−bt\n1+b.\n3 Linear cyclotron damping revisited\nIn this section, we recast the linear damping in the hybrid analytic nor ms.\nTheorem 3.1 For any η,v∈R3,k∈N3\n0,we assume that the following conditions hold in equations (0 .9).\n(i)W(x)is an odd function on x3,|/hatwiderW(k)| ≤1\n1+|k|γ,γ >1,whereW(x) = (W1(x),W2(x),0);\n(ii)/ba∇dbl∇vf0/ba∇dblCλ0;1≤C0;/ba∇dblf0/ba∇dblZλ0,µ0;1\n0≤δ0for some constants λ0,α0,C0>0,δ0>0;\n(iii) In addition, (PSC)holds,\n(PSC) :for any component velocity in the ˆzdirection v3∈R,there exists some positive constant vTesuch\nthat ifv3=ω\nk3whenk3/\\e}atio\\slash= 0;ork3= 0whereω,kare frequencies of time and space t,x,respectively, then\n|v3| ≫vTe.\nThen for any λ′\n0< λ0,we have\nsup\nt≥0/ba∇dblρ(t,·)/ba∇dblFλ′\n0t+µ0≤C(C0,λ0,λ′\n0,µ)δ0,\nsup\nt≥0/ba∇dblf−f0/ba∇dbl\nZλ′′\n0,µ;1\nt≤C(C0,λ0,λ′\n0,µ)δ0. (3.1)\nThe revisited proof of Lemma 1.1\nProof.Firstω/\\e}atio\\slash= 0,k3/\\e}atio\\slash= 0,we have\n˜ρ(ω,k) =/integraldisplay\nR+/integraldisplay\nT3/integraldisplay\nR3e2πitωe−2πix·kf0(x′(0,x,v),v′(0,x,v))dvdxdt\n+q\nm˜ρ(ω,k)1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik·(v′\nt1sinΩt\nt,v′\nt2sinΩt\nt,v3t)(k3v′\n3)(/hatwiderW1∂v′\n1f0−/hatwiderW2∂v′\n2f0)dvdt\n+q\nm˜ρ(ω,k)1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik·(v′\nt1sinΩt\nt,v′\nt2sinΩt\nt,v3t)(k1/hatwiderW2−k2/hatwiderW1)·(v2∂v′\n1f0−v1∂v′\n2f0)dvdt\n−q\nm˜ρ(ω,k)1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik·(v′\nt1sinΩt\nt,v′\nt2sinΩt\nt,v3t)·(v1/hatwiderW1+v2/hatwiderW2)k3∂v′\n3f0dvdt\n−q\nm˜ρ(ω,k)/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik·(v′\nt1sinΩt\nt,v′\nt2sinΩt\nt,v3t)(/hatwiderW1∂v′\n1f0+/hatwiderW2∂v′\n2f0)dvdt.\n(3.2)\nLet\n˜L(ω,k) =q\nm1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik·(v′\nt1sinΩt\nt,v′\nt2sinΩt\nt,v3t)(k3v3)(/hatwiderW1∂v′\n1f0−/hatwiderW2∂v′\n2f0)dvdt\n−q\nm/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik·(v′\nt1sinΩt\nt,v′\nt2sinΩt\nt,v3t)(/hatwiderW1∂v′\n1f0+/hatwiderW2∂v′\n2f0)·dvdt\n−q\nm1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik·(v′\nt1sinΩt\nt,v′\nt2sinΩt\nt,v3t)(v1/hatwiderW1+v2/hatwiderW2)k3∂v′\n3f0dvdt\n+q\nm1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe−2πik·(v′\nt1sinΩt\nt,v′\nt2sinΩt\nt,v3t)(k1/hatwiderW2−k2/hatwiderW1)(v2∂v′\n1f0−v1∂v′\n2f0)dvdt,\n(3.3)\n14hence\n˜ρ(ω,k) =˜A(ω,k)+ ˜ρ(ω,k)˜L(ω,k). (3.4)\nThen taking the inverse Fourier transform in time t,we get ˆρ(t,k) =ˆA(t,k)+ ˆρ(t,k)∗ˆL(t,k),and\ne2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|eµ|k|ˆρ(t,k) =e2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|eµ|k|ˆA(t,k)+e2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|eµ|k|ˆρ(t,k)∗ˆL(t,k).\n(3.5)\nLetΦ(t,k) =e2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|eµ|k|ˆρ(t,k),A(t,k) =e2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|eµ|k|ˆA(t,k),K0(t,k) =e2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|\n·ˆL(t,k),then from (3.4), we have /tildewideΦ(ω,k) =/tildewideA(ω,k)+/tildewideΦ(ω,k)/tildewideK0(ω,k).\nThen\n/ba∇dblΦ(t,k)/ba∇dblL2(dt)=/ba∇dbl/tildewideΦ(ω,k)/ba∇dblL2≤ /ba∇dbl/tildewideA(ω,k)/ba∇dblL2+/ba∇dbl/tildewideΦ(ω,k)/ba∇dblL2/ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞\n≤ /ba∇dble2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|eµ|k|ˆA(t,k)/ba∇dblL2+/ba∇dble2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|eµ|k|ˆρ(t,k)/ba∇dblL2/ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞.\n(3.6)\nNext we have to estimate /ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞.\nIndeed,\n/ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞≤sup\nωq\nm/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nω/integraldisplay\nR+/integraldisplay\nR3e2πitωe2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|e−2iπk·(v′\n1sinΩt\nt,v′\n2sinΩt\nt,v′\n3t)·(/hatwiderW1∂v′\n1f0\n−/hatwiderW2∂v′\n2f0)dvdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle+q\nm/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR+/integraldisplay\nR3e2πitωe2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|e−2iπk·(v′\n1sinΩt\nt,v′\n2sinΩt\nt,v′\n3t)(/hatwiderW1∂v′\n1f0+/hatwiderW2∂v′\n2f0)dvdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n−q\nm(ω,k)1\nω/integraldisplay\nR+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3e2πitωe2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|e−2iπk·(v′\n1sinΩt\nt,v′\n2sinΩt\nt,v′\n3t)(v1/hatwiderW1+v2/hatwiderW2)k3∂v′\n3f0dv/vextendsingle/vextendsingle/vextendsingle/vextendsingledt\n+q\nm(ω,k)1\nω/integraldisplay\nR+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3e2πitωe2πλ′\n0|k·(sinΩt\nt,sinΩt\nt,t)|e−2iπk·(v′\n1sinΩt\nt,v′\n2sinΩt\nt,v′\n3t)(k1/hatwiderW2−k2/hatwiderW1)(v2∂v′\n1f0−v1∂v′\n2f0)dv/vextendsingle/vextendsingle/vextendsingle/vextendsingledt/bracketrightbigg\n=I+II+III+IV.\n(3.7)\nIn fact, we only need to estimate one term of (3.6) because of similar processes of other terms. Without loss\nof generality, we give an estimate for I.In the same way, we only estimate one term of I,here we still denote I.\nI= sup\nωq\nm/vextendsingle/vextendsingle/vextendsingle/vextendsinglek3\nω/hatwiderW2/integraldisplay\nR+/integraldisplay\nRe2πitω/parenleftbigg\n2πik1sinΩt\nΩ/parenrightbigg\ne2πλ0|k·(sinΩt\nt,sinΩt\nt,t)|e−2πik3v3t·/hatwidestv3f0(k1sinΩt\nΩ,k2sinΩt\nΩ,v3)dv3dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n= sup\nωq\nm/vextendsingle/vextendsingle/vextendsingle/vextendsinglek3\nω/hatwiderW2/integraldisplay\nR+/integraldisplay\nRe2πitω/parenleftbigg\n2πik1sinΩt\nΩ/parenrightbigg\ne2πλ0|k·(sinΩt\nt,sinΩt\nt,t)|e−2πik3v3te−2πλ0|k3|t·/summationdisplay\nn|2πiλ0|k3|t|n\nn!\n·/hatwidestv3f0(k1sinΩt\nΩ,k2sinΩt\nΩ,v3)dv3dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n= sup\nωq\nm/summationdisplay\nnλn\n0\nn!/vextendsingle/vextendsingle/vextendsingle/vextendsinglek3\nω/hatwiderW2/integraldisplay\nR+/integraldisplay\nR/parenleftbigg\n2πik1sinΩt\nΩ/parenrightbigg\ne2πλ0|k·(sinΩt\nt,sinΩt\nt,t)|e−2πλ0|k3|te2πik3t(ω\nk3−v3)\n·∇n\nv3/hatwidestv3f0(k1sinΩt\nΩ,k2sinΩt\nΩ,v3)dv3dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n= sup\nωq\nm/summationdisplay\nnλn\n0\nn!/vextendsingle/vextendsingle/vextendsingle/vextendsinglek3\nω/hatwiderW2/parenleftbigg\n2πik1sinΩt\nΩ/parenrightbigg\ne2πλ0|k·(sinΩt\nt,sinΩt\nt,t)|e−2πλ0|k3|t(−i∇ω\nk3)n/hatwidestv3f0(k1sinΩt\nΩ,k2sinΩt\nΩ,ω\nk3)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤q\nmvTee−c0vTe,\nwhere in the last inequality we use the facts that if v3=ω\nk3,thenv3≫vTe,and the assumption (i) and (iv).\nThen there exists some constant 0 < κ <1 such that /ba∇dbl/tildewideK0(ω,k)/ba∇dblL∞≤κ.\nIn conclusion, we have\nsup\nt≥0/ba∇dblρ(t,·)/ba∇dblFλ′\n0t+µ′≤Csup\nt≥0/vextenddouble/vextenddouble/vextenddouble/vextenddoublef0◦S0\n−tdv/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nFλ0t+µ≤ /ba∇dblf0◦S0\n−t/ba∇dblZλ0,µ;1\nt=/ba∇dblf0/ba∇dblZλ0,µ;1\n0≤δ0.(3.8)\n15Proof of Theorem 3.1.From (2.1), we have\nf(t,x,v) =f0(x′(0,x,v),v′(0.x,v))−q\nm/integraldisplayt\n0[(E+v′×B)·∇′\nvf0](τ.x′(τ,x,v),v′(τ,x,v))dτ.(3.9)\nThen for any λ′′\n0< λ′\n0,we have, for all t≥0,\n/ba∇dblf/ba∇dbl\nZλ′′\n0,µ′;1\nt≤ /ba∇dblf0◦S0\n−t/ba∇dbl\nZλ′′\n0,µ′;1\nt+/integraldisplayt\n0/ba∇dbl(W(x)∗ρτ)◦S0\n−(t−τ)/ba∇dbl\nZλ′′\n0,µ′;∞\nt/ba∇dbl∇vf0/ba∇dbl\nZλ′′\n0,µ′;1\ntdτ\n+/integraldisplayt\n0/ba∇dblB◦S0\n−(t−τ)/ba∇dbl\nZλ′′\n0,µ′;∞\nt/ba∇dbl∇vvf0/ba∇dbl\nZλ′′\n0,µ′;1\ntdτ\n=/ba∇dblf0/ba∇dblZλ′′\n0,µ′;1+/integraldisplayt\n0/ba∇dbl(W(x)∗ρτ)/ba∇dblFλ′′\n0τ+µ′/ba∇dbl∇vf0/ba∇dblCλ′′\n0;1dτ+/integraldisplayt\n0/ba∇dblB/ba∇dblFλ′′\n0τ+µ′/ba∇dbl∇vvf0/ba∇dblCλ′′\n0;1dτ\n≤δ0+Cδ0/(λ0−λ′′\n0). (3.10)\n4 Nonlinear Cyclotron damping\nNext we give the proof of the main Theorem 0.3, stating the primary s teps as propositions which are proved\nin subsections.\n4.1 The improved Newton iteration\nThe first idea which may come to mind is a classical Newton iteration as d one by Mouhot and Villani [23]: Let\nf0=f0(v) be given ,\nand\nfn=f0+h1+...+hn,\nwhere\n/braceleftbigg∂th1+v·∇xh1+v×B0·∇vh1+(E[h1]+v×B[h1])·∇vf0= 0,\nh1(0,x,v) =f0−f0,(4.1)\nand now we consider the Vlasov equation in step n+1,n≥1,\n\n\n∂thn+1+v·∇xhn+1+v×B0·∇vhn+1+E[fn]·∇vhn+1+v×B[fn]·∇vhn+1\n=−E[hn+1]·∇vfn−v×B[hn+1]·∇vfn−E[hn]·∇vhn−v×B[hn]·∇vhn,\nhn+1(0,x,v) = 0,(4.2)\nthe corresponding dynamical system is described by the equations : for any ( x,v)∈T3×R3,let (Xn\nt,τ,Vn\nt,τ) as\nthe solution of the following ordinary differential equations\n/braceleftbiggd\ndtXn\nt,τ(x,v) =Vn\nt,τ(x,v),\nXn\nτ,τ(x,v) =x,\n/braceleftbiggd\ndtVn\nt,τ(x,v) =Vn\nt,τ(x,v)×(B0+B[fn](t,Xn\nt,τ(x,v)))+E[fn](t,Xn\nt,τ(x,v)),\nVn\nτ,τ(x,v) =v.(4.3)\nAt the same time, we consider the corresponding linear dynamics sys tem as follows,\n/braceleftbiggd\ndtX0\nt,τ(x,v) =V0\nt,τ(x,v),d\ndtV0\nt,τ(x,v) =V0\nt,τ(x,v)×B0,\nX0\nτ,τ(x,v) =x, V0\nτ,τ(x,v) =v.(4.4)\n16It is easy to check that\nΩn\nt,τ−Id/defines(δXn\nt,τ,δVn\nt,τ)◦(X0\nτ,t,V0\nτ,t) = (Xn\nt,τ◦(X0\nτ,t,V0\nτ,t)−Id,Vn\nt,τ◦(X0\nτ,t,V0\nτ,t)−Id).\nTherefore,inordertoestimate( Xn\nt,τ◦(X0\nτ,t,V0\nτ,t)−Id,Vn\nt,τ◦(X0\nτ,t,V0\nτ,t)−Id),weonlyneedtostudy( δXn\nt,τ,δVn\nt,τ)◦\n(X0\nτ,t,V0\nτ,t).\nFrom Eqs.(4.3) and (4.4),/braceleftbiggd\ndtδXn\nτ,t(x,v) =δVn\nτ,t(x,v),\nδXn\nτ,τ(x,v) = 0,\n/braceleftbiggd\ndtδVn\nτ,t(x,v) =δVn\nτ,t(x,v)×B0+E[fn](t,Xn\nτ,t(x,v))+(δVn\nτ,t(x,v)+V0\nτ,t(x,v))×B[fn](t,Xn\nτ,t(x,v)),\nδVn\nτ,τ(x,v) = 0,\n(4.5)\nand|V0\nτ,t(x,v)|=|v|.Sincev∈R3andB[fn](t,Xn\nτ,t(x,v)) independent of v,there is almost no hope to get a\n“good ” estimates of Ωn\nt,τ−Id.Furthermore, when k3= 0,because of /hatwiderE[fn](s,k1,k2,0) = 0,/hatwiderB[fn](s,k1,k2,0)/\\e}atio\\slash=\n0,the deflection estimates are in the absence of a decaying perturbe d magnetic field.\nTo circumvent these difficulties, we recall the basic physical Law on L enz’s Law:\nThe direction of current induced in a conductor by a changing magnetic field due to induction is such\nthat it creates a magnetic field that opposes the change that produced it.\nAccording to the statement of Lenz’s Law and Maxwell equations, b ased on the approximation equations\n(4.2), it is easy to know that we only need to consider the following dyn amical system\n/braceleftbiggd\ndtXn\nt,τ(x,v) =Vn\nt,τ(x,v),d\ndtVn\nt,τ(x,v) =Vn\nt,τ(x,v)×B0+E[fn](t,Xn\nt,τ(x,v)),\nXn\nτ,τ(x,v) =x,Vn\nτ,τ(x,v) =v,(4.6)\nthen we get\n/braceleftbiggd\ndtδXn\nt,τ(x,v) =δVn\nt,τ(x,v),d\ndtδVn\nt,τ(x,v) =δVn\nt,τ(x,v)×B0+E[fn](t,Xn\nt,τ(x,v)),\nδXn\nτ,τ(x,v) = 0,δVn\nτ,τ(x,v) = 0,(4.7)\nand we write the approximation equations (4.2) into the following form ,\n\n\n∂thn+1+v·∇xhn+1+v×B0·∇vhn+1+E[fn]·∇vhn+1\n=−E[hn+1]·∇vfn−v×B[hn+1]·∇vfn−E[hn]·∇vhn−v×B[hn]·∇vhn\n−v×B[fn]·∇vhn+1,\nhn+1(0,x,v) = 0.(4.8)\n4.2 Main challenges\nIntegrating (4.8) in time and hn+1(0,x,v) = 0,we get\nhn+1(t,Xn\nt,0(x,v),Vn\nt,0(x,v)) =/integraldisplayt\n0Σn+1(s,Xn\ns,0(x,v),Vn\ns,0(x,v))ds, (4.9)\nwhere\nΣn+1(t,x,v) =−E[hn+1]·∇vfn−v×B[hn+1]·∇vfn−E[hn]·∇vhn−v×B[hn]·∇vhn−v×B[fn]·∇vhn+1.\nBy the definition of ( Xn\nt,τ(x,v),Vn\nt,τ(x,v)),we have\nhn+1(t,x,v) =/integraldisplayt\n0Σn+1(s,Xn\ns,t(x,v),Vn\ns,t(x,v))ds=/integraldisplayt\n0Σn+1(s,δXn\ns,t(x,v)+X0\ns,t(x,v),δVn\ns,t(x,v)+V0\ns,t(x,v))ds.\nSince the unknown hn+1appears on both sides of (4.9), we hope to get a self-consistent es timate. For this,\nwe have little choice but to integrate in vand get an integral equation on ρ[hn+1] =/integraltext\nR3hn+1dv,namely\nρ[hn+1](t,x) =/integraldisplayt\n0/integraldisplay\nR3(Σn+1◦Ωn\ns,t(x,v))(s,X0\ns,t(x,v),V0\ns,t(x,v))dvds\n=/integraldisplayt\n0/integraldisplay\nR3−/bracketleftbigg\n(En+1\ns,t·Gn\ns,t)−(Fn+1\ns,t·Gn,v\ns,t)−(En\ns,t·Hn\ns,t)−(Fn\ns,t·Hn,v\ns,t)\n−(B[fn]◦Ωn\ns,t(x,v))·Hn+1,v\ns,t/bracketrightbigg\n(s,X0\ns,t(x,v),V0\ns,t(x,v))dvds\n=In+1,n+IIn+1,n+IIIn,n+IVn,n+Vn,n+1, (4.10)\n17where \n\nEn+1\ns,t=E[hn+1]◦Ωn\ns,t(x,v),En\ns,t=E[hn]◦Ωn\ns,t(x,v),\nGn\ns,t= (∇′\nvfn)◦Ωn\ns,t(x,v), Gn,v\ns,t= (∇′\nvfn×V0\ns,t(x,v))◦Ωn\ns,t(x,v),\nFn+1\ns,t=B[hn+1]◦Ωn\ns,t(x,v), Fn\ns,t=B[hn]◦Ωn\ns,t(x,v),\nHn\ns,t= (∇′\nvhn)◦Ωn\ns,t(x,v), Hn,v\ns,t= (∇′\nvhn×V0\ns,t(x,v))◦Ωn\ns,t(x,v).\nIt is obvious that Eq.(4.10) is not a closed equation, while ρ[hn+1](t,x) satisfies a closed equation written\nin [23] with the electric field case. To obtain a self-consistent estimat e, we go back the Vlasov equations (4.9),\ncomposing with (( Xn\n0,τ(x,v),Vn\n0,τ(x,v))),where 0≤τ≤t,this gives\nhn+1(t,Xn\nτ,t(x,v),Vn\nτ,t(x,v)) =/integraldisplayt\n0Σn+1(s,Xn\nτ,s(x,v),Vn\nτ,s(x,v))ds. (4.11)\nInordertoachievethegoal,wehavetocombineEq.(4.10)andEq.(4.1 1)toformaiteration,thenweobtainaself-\nconsistent estimate on ρ[hn+1].At the technical level, it is more difficult than that in Mouhot and Villani’pa per\n[23], since the new term v×B[fn]·∇vhn+1in Eqs.(4.9)-(4.10) brings different kinds of resonances (in terms of\ndifferent norms).\n4.3 Inductive hypothesis\nFor n=1, from (1.1), it is easy to see that (4.1) is a linear Vlasov equat ion. From section 3, we know that\nthe conclusions of Theorem 0.3 hold.\nNow for any i≤n,i∈N0,we assume that the following estimates hold,\nsup\nt≥0/ba∇dblρ[hi](t,·)/ba∇dblF(λi−B0)t+µi≤δi,\nsup\n0≤τ≤t/ba∇dblhi\nτ◦Ωi−1\nt,τ/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,sup\n0≤τ≤t/ba∇dbl(hi\nτv)◦Ωi−1\nt,τ/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,\n(4.12)\nthen we have the following inequalities, denote ( En) :\nsup\nt≥0/ba∇dblE[hi](t,·)/ba∇dblF(λi−B0)t+µi< δi,sup\nt≥0/ba∇dblB[hi](t,·)/ba∇dblF(λi−B0)t+µi< δi,\nsup\n0≤τ≤t/ba∇dbl∇x((hi\nτv)◦Ωi−1\nt,τ)/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,sup\n0≤τ≤t/ba∇dbl(∇x(hi\nτv))◦Ωi−1\nt,τ/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,\nsup\n0≤τ≤t/ba∇dbl∇x(hi\nτ◦Ωi−1\nt,τ)/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b��δi,sup\n0≤τ≤t/ba∇dbl(∇xhi\nτ)◦Ωi−1\nt,τ/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,\n/ba∇dbl(∇v+τ∇x)(hi\nτ◦Ωi−1\nt,τ)/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,/ba∇dbl((∇v+τ∇x)hi\nτ)◦Ωi−1\nt,τ/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,\n/ba∇dbl(∇v+τ∇x)(hi\nτv◦Ωi−1\nt,τ)/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,/ba∇dbl((∇v+τ∇x)hi\nτv)◦Ωi−1\nt,τ/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,\nsup\n0≤τ≤t/ba∇dbl(∇∇vhi\nτ)◦Ωi−1\nt,τ/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,sup\n0≤τ≤t/ba∇dbl(∇∇v(hi\nτv))◦Ωi−1\nt,τ/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,\nsup\n0≤τ≤t(1+τ)2/ba∇dbl(∇v×hi)◦Ωi−1\nt,τ−∇v×(hi◦Ωi−1\nt,τ)/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi,\nsup\n0≤τ≤t(1+τ)2/ba∇dbl(∇v×(hiv))◦Ωi−1\nt,τ−∇v×((hiv)◦Ωi−1\nt,τ)/ba∇dblZ(λi−B0)(1+b),µi;1\nτ−bt\n1+b≤δi.\nIt is easy to check that the first two inequalities of ( En) hold under our assumptions and (4.12), so we need\nto show that the other equalities of ( En) also hold, the related proofs are found in section 5.\n4.4 Local time iteration\nBefore working out the core of the proof of Theorem 0.1, we shall s how a short time estimate, which will\nplay a role as an initial data layer for the Newton scheme. The main too l in this section is given by the following\nlemma, which is through the direct computation from the definition of the corresponding norms. Therefore, we\nomit the proof.\nLemma 4.1 Letfbe an analytic function, λ(t) =λ−Ktandµ(t) =µ−Kt, K > 0,letT >0be so small\nthatλ(t)> λ′(t)>0,µ(t)> µ′(t)>0for0≤t≤T.Then for any τ∈[0,T]and any p≥1,\nd+\ndt|t=τ/ba∇dblf/ba∇dblZλ(t),µ(t);1\nτ≤ −K\n2(1+τ)/ba∇dbl∇f/ba∇dblZλ(τ),µ(τ);1\nτ−K\n2(1+τ)/ba∇dblv∇f/ba∇dblZλ′(τ),µ′(τ);1\nτ,\nwhered+\ndtstands for the upper right derivative.\n18Forn≥1,now let us solve\n∂thn+1+v·∇xhn+1+v×B0·∇vhn+1=/tildewideΣn+1,\nwhere\n/tildewideΣn+1=−E[fn]·∇vhn+1−E[hn+1]·∇vfn−v×B[hn+1]·∇vfn−E[hn]·∇vhn−v×B[hn]·∇vhn−v×B[fn]·∇vhn+1.\nHence\n/ba∇dblhn+1/ba∇dblZλn+1(t),µn+1(t);1\nt≤/integraldisplayt\n0/ba∇dbl/tildewideΣn+1\nτ◦S0\n−(t−τ)/ba∇dblZλn+1(t),µn+1(t);1\ntdτ≤/integraldisplayt\n0/ba∇dbl/tildewideΣn+1\nτ/ba∇dblZλn+1(t),µn+1(t);1\nτdτ,\nthen by Lemma 4.1,\nd+\ndt/ba∇dblhn+1/ba∇dblZλn+1(t),µn+1(t);1\nt≤ −K\n2/ba∇dbl∇xhn+1/ba∇dblZλn+1,µn+1;1\nt−K\n2/ba∇dbl∇vhn+1/ba∇dblZλn+1,µn+1;1\nt\n−K\n2/ba∇dblv∇xhn+1/ba∇dbl\nZλ′\nn+1,µ′\nn+1;1\nt−K\n2/ba∇dblv∇vhn+1/ba∇dbl\nZλ′\nn+1,µ′\nn+1;1\nt\n+/ba∇dblE[fn]/ba∇dblFλn+1t+µn+1/ba∇dbl∇vhn+1/ba∇dblZλn+1,µn+1;1\nt+/ba∇dblB[fn]/ba∇dblF¯λn+1t+¯µn+1/ba∇dbl∇vhn+1×v/ba∇dbl\nZλ′\nn+1,µ′\nn+1;1\nt\n+/ba∇dblE[hn+1]/ba∇dblFλn+1t+µn+1/ba∇dbl∇vfn/ba∇dblZλn+1,µn+1;1\nt+/ba∇dblB[hn+1]/ba∇dblFλn+1t+µn+1/ba∇dbl∇vfn×v/ba∇dblZλn+1,µn+1;1\nt\n+/ba∇dblE[hn]/ba∇dblFλn+1t+µn+1/ba∇dbl∇vhn/ba∇dblZλn+1,µn+1;1\nt+/ba∇dblB[hn]/ba∇dblFλn+1t+µn+1/ba∇dbl∇vhn×v/ba∇dblZλn+1,µn+1;1\nt,\nwhere the third term of the right-hand inequality uses (6.2) of Theo rem 6.1 in the following section 6. First, we\neasily get /ba∇dblE[hn]/ba∇dblFλn+1t+µn+1≤C/ba∇dbl∇hn/ba∇dblZλn+1,µn+1;1\nt,/ba∇dblB[hn]/ba∇dblFλn+1t+µn+1≤C/ba∇dbl∇hn/ba∇dblZλn+1,µn+1;1\nt.Moreover,\n/ba∇dbl∇vfn/ba∇dblZλn+1,µn+1;1\nt≤n/summationdisplay\ni=1/ba∇dbl∇vhi/ba∇dblZλn+1,µn+1;1\nt≤Cn/summationdisplay\ni=1/ba∇dblhi/ba∇dblZλi+1,µi+1;1\nt\nmin{λi−λn+1,µi−µn+1},\n/ba∇dbl∇vfn×v/ba∇dblZλn+1,µn+1;1\nt≤n/summationdisplay\ni=1/ba∇dbl∇vhi×v/ba∇dblZλn+1,µn+1;1\nt≤Cn/summationdisplay\ni=1/ba∇dblhi/ba∇dblZλi,µi;1\nt\nmin{λi−λn+1,µi−µn+1}.\nWe gather the above estimates,\nd+\ndt/ba∇dblhn+1/ba∇dblZλn+1(t),µn+1(t);1\nt≤/parenleftbigg\nCn/summationdisplay\ni=1δi\nmin{λi−λn+1,µi−µn+1}−K\n2/parenrightbigg\n/ba∇dbl∇hn+1/ba∇dblZλn+1,µn+1;1\nt\n+/parenleftbigg\nCn/summationdisplay\ni=1δi\nmin{λi−¯λn+1,µi−¯µn+1}−K\n2/parenrightbigg\n/ba∇dbl∇hn+1/ba∇dbl\nZλ′\nn+1,µ′\nn+1;1\nt+δ2\nn\nmin{λn−¯λn+1,µn−¯µn+1}.\nWe may choose\nδn+1=δ2\nn\nmin{λn−¯λn+1,µn−¯µn+1},\nif\nCmax/braceleftbiggn/summationdisplay\ni=1δi\nmin{λi−λn+1,µi−µn+1},n/summationdisplay\ni=1δi\nmin{λi−¯λn+1,µi−¯µn+1}/bracerightbigg\n≤K\n2(4.13)\nholds.\nWe choose λi−λi+1=µi−µi+1=Λ\ni2,where Λ >0 is arbitrarily small. Then for i≤n,λi−λn+1≥Λ\ni2,and\nδn+1≤δ2\nnn2/Λ.Next we need to check that/summationtext∞\nn=1δnn2<∞.In fact, we choose Klarge enough and T small\nenough such that λ0−KT≥λ∗,µ0−KT≥µ∗,and (4.13) holds, where λ0> λ∗,µ0> µ∗are fixed.\nIfδ1=δ,thenδn=n2δ2n\nΛn(22)2n−2(42)2n−2...((n−1)2)2n2.To prove the sequence convergence for δsmall\nenough, by induction that δn≤zan,wherezsmall enough and a∈(1,2).We claim that the conclusion holds\nforn+ 1.Indeed,δn+1≤z2an\nΛn2≤zan+1z(2−a)ann2\nΛ.Ifzis so small that z(2−a)an≤Λ\nn2for alln∈N,then\nδn+1≤zan+1,this concludes the local-time argument.\nRemark 4.2 It is worthy to note that there are resonances to occur in loca l time which are caused by the action\nof the magnetic field, in detail, the resonances are from the t ermv×B[fn]·∇vhn+1. This phenomenon is very\ndifferent from Landau damping in [23] in local time.\n194.5 Global time iteration\nBased on the estimates of local-time iteration, without loss of gener ality, sometimes we only consider the\ncaseτ≥bt\n1+b,wherebis small enough.\nFirst, we give deflection estimates to compare the free evolution wit h the true evolution from the particles\ntrajectories.\nProposition 4.3 Assume for any i∈N,0< i≤n,\nsup\nt≥0/ba∇dblE[hi](t,·)/ba∇dblF(λi−B0)t+µi< δi,sup\nt≥0/ba∇dblB[hi](t,·)/ba∇dblF(λi−B0)t+µi< δi.\nAnd there exist constants λ⋆> B0,µ⋆>0such that λ0−B0> λ′\n0−B0> λ1−B0> λ′\n1−B0> ... >\nλi−B0> λ′\ni−B0> ... > λ ⋆−B0, µ0> µ1> µ′\n1> ... > µ i> µ′\ni> ... > µ ⋆.\nThen we have\n/ba∇dblδXn+1\nt,τ◦(X0\nτ,t,V0\nτ,t)/ba∇dblZλ′n−B0,µ′n\nτ−bt\n1+b≤Cn/summationdisplay\ni=1δie−π(λi−λ′\ni)τmin/braceleftbigg(t−τ)2\n2,1\n2π(λi−λ′\ni)2/bracerightbigg\n,\n/ba∇dblδVn+1\nt,τ◦(X0\nτ,t,V0\nτ,t)/ba∇dblZλ′n−B0,µ′n\nτ−bt\n1+b≤Cn/summationdisplay\ni=1δie−π(λi−λ′\ni)τmin/braceleftbigg(t−τ)\n2,1\n2π(λi−λ′\ni)/bracerightbigg\n,\nfor0< τ < t, b =b(t,τ)sufficiently small.\nRemark 4.4 From Proposition 4.3, it is easy to know that the gliding anal ytic regularity λ > B 0in the cyclotron\ndamping, comparing with λ >0in Landau damping, here B0is called cyclotron frequency. In other words,\nresonances in cyclotron damping occur at the cyclotron freq uency, not zero frequency (Landau resonances occur\nat zero frquency).\nProposition 4.5 Under the assumptions of Proposition 4.3, then\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇Ωn+1Xt,τ−(Id,0)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nZ(λ′\nn+1−B0)(1−b),µ′\nn+1\ns+bt\n1−b0small. And there exist constants λ⋆> B0,µ⋆>0\nsuch that λ0−B0> λ′\n0−B0> λ1−B0> λ′\n1−B0> ... > λ i−B0> λ′\ni−B0> ... > λ ⋆−B0, µ0> µ1> µ′′\n1>\nµ′\n1> ... > µ i> µ′′\ni> µ′\ni> ... > µ ⋆.\nWe have\n/ba∇dbl¯IIn+1,n\ni(t,·)/ba∇dblF(λ′n−B0)t+µ′n\n≤C/integraldisplayt\n0Kn+1\n1(t,τ)/ba∇dbl∇′\nv×((hi\nτv)◦Ωi−1\nt,τ))−/a\\}b∇acketle{t∇′\nv×((hi\nτv)◦Ωi−1\nt,τ))/a\\}b∇acket∇i}ht/ba∇dbl\nZ(λ′\ni−B0)(1+b),µ′\ni;1\nτ−bt\n1+b\n/ba∇dblB[hn+1]/ba∇dblFνdτ+/integraldisplayt\n0Kn+1\n0(t,τ)/ba∇dbl/a\\}b∇acketle{t∇′\nv×((hi\nτv)◦Ωi−1\nt,τ))/a\\}b∇acket∇i}ht/ba∇dblC(λ′\ni−B0)(1+b);1·/ba∇dblB[hn+1]/ba∇dblFνdτ,\nwhere\nν= max/braceleftbigg\n(λ′\nn−B0)τ+µ′′\nn−1\n2(λ′\nn−B0)b(t−τ),0/bracerightbigg\n,\nKn\n0(t,τ) =e−π(λ′\ni−λ′\nn)(t−τ),\nKn+1\n1(t,τ) = sup\nk3,l3∈Ze−2π(µ′\ni−µ′\nn)|l3|e−π(λ′\ni−λ′\nn)|k3(t−τ)+l3τ|e−2π(λ′\nn\n2(τ−τ′)+µ′′\nn−µ′\nn)|k3−l3|.\n21Lemma 4.10 We have1\nτ/ba∇dblB[hn+1]/ba∇dblFν≤Csup0≤s≤τ/ba∇dblρ[hn+1]/ba∇dblF(λ′n−B0)s+µ′n, τ/ba∇dblhn/ba∇dbl\nZ(λ′n−B0)(1+b),µ′\ni;1\nτ−bt\n1+b≤C/ba∇dblhn/ba∇dbl\nZ(¯λ′n−B0)(1+b),¯µ′\ni;1\nτ−bt\n1+b,whereλ′\ni<¯λ′\ni< λi,µ′\ni<¯µ′\ni< µi,τ >0.\nProof.Since∂tB=∇ ×E, E=W(x)∗ρ(f), B=/integraltextt\n0∇ ×E(s,x)ds=/integraltextt\n0∇ ×(W(x)∗ρ(f))(s,x)ds,then\n/ba∇dblB[hn+1]/ba∇dblFν≤/integraltextτ\n0/ba∇dbl∇ ×E[hn+1]/ba∇dblFνds≤/integraltextτ\n0/ba∇dbl∇E[hn+1]/ba∇dblFνds≤τsup0≤s≤τ/ba∇dblρ[hn+1]/ba∇dblF(λ′n−B0)s+µ′n,where we\nuseν <(λ′\nn−B0)τ+µ′\nn.The last inequality can be obtained from the definition ofthe norm Zλ,µ;1\ntand (vi)-(vii)\nof Proposition 2.5.\nCorollary 4.11 From the above statement, we have\n/ba∇dbl¯IIn+1,n\ni(t,·)/ba∇dblF(λ′n−B0)t+µ′n≤/integraldisplayt\n0Kn+1\n0(t,τ)δi/ba∇dblρ[hn+1]/ba∇dblF(λ′n−B0)τ+µ′ndτ\n+/integraldisplayt\n0Kn+1\n1(t,τ)(1+τ)δi/ba∇dblρ[hn+1]/ba∇dblF(λ′n−B0)τ+µ′ndτ,\nwhereKn\n0(t,τ) =e−π(λ′\ni−λ′\nn)(t−τ),and\nKn+1\n1(t,τ) =e−2π(µ′\ni−µ′\nn)|l3|e−π(λ′\ni−λ′\nn)|k3(t−τ)+l3τ|e−2π((λ′n−B0)\n2(τ−τ′)+µ′′\nn−µ′\nn)|k3−l3|.\nProposition 4.12 (Error term I)\n/ba∇dblR0(t,·)/ba∇dblF(λ′n−B0)t+µ′n≤C/parenleftbigg\nC′\n0+n/summationdisplay\ni=1δi/parenrightbigg/parenleftbiggn/summationdisplay\ni=1δi\n(λi−λ′\ni)5/parenrightbigg/integraldisplayt\n0/ba∇dblρ[hn+1]/ba∇dblF(λ′n−B0)τ+µ′ndτ\n(1+τ)2.\nProposition 4.13 (Error term II)\n/ba∇dbl˜R0(t,·)/ba∇dblF(λ′n−B0)τ+µ′n≤/parenleftbigg\nC4\nω/parenleftbigg\nC′\n0+n/summationdisplay\ni=1δi/parenrightbigg/parenleftbiggn/summationdisplay\nj=1δj\n2π(λj−λ′\nj)6/parenrightbigg\n+n/summationdisplay\ni=1��i/parenrightbigg/integraldisplayt\n0/ba∇dblρ/ba∇dblF(λ′n−B0)τ+µ′n1\n(1+τ)2dτ\n=/integraldisplayt\n0˜Kn+1\n1/ba∇dblρ/ba∇dblF(λ′n−B0)τ+µ′n1\n(1+τ)2dτ.\nProposition 4.14 (Main term V)\n/ba∇dblV/ba∇dblF(λ′n−B0)t+µ′n≤/integraldisplayt\n0e−π(λn−λ′\nn)|k(t−s)+ls|/parenleftbiggn/summationdisplay\ni=1δi/parenrightbigg\n/ba∇dblhn+1◦Ωn\ns,t(x,v)/ba∇dbl\nZ(λ′n−B0)(1−1\n2b),µ′n;1\ns+bt\n1−bds.\nProposition 4.15 (Main term III)\nsup\n0≤s≤t/ba∇dblhn+1◦Ωn\nt,τ/ba∇dbl\nZ(λ′n−B0)(1−1\n2b),µ′n;1\ns+bt\n1−b≤δ2\nn+/parenleftbiggn/summationdisplay\ni=1δi/parenrightbigg\nsup\n0≤s≤t/ba∇dblρn+1/ba∇dblF(λ′n−B0)s+µ′n.\n4.6 The proof of main theorem\nStep 2.Note ifεin (0.13) is small enough, up to slightly lowering λ1,we may choose all parameters in such\na way that λk−B0,λ′\nk−B0→λ∞−B0> λ−B0andµk,µ′\nk→µ∞> µ,ask→ ∞; then we pick up\nD >0 such that µ∞−λ∞(1+D)D≥µ′\n∞> µ,and we let b(t) =D\n1+t.From the iteration, we have, for all k≥2,\nsup\n0≤τ≤t/ba∇dblhk\nτ◦Ωk−1\nt,τ/ba∇dblZλ∞(1+b),µ∞;1\nτ−bt\n1+b≤δk, (4.22)\nwhere/summationtext∞\nk=2δk≤Cδ.Choosing τ=tin (3.22) yields sup0≤τ≤t/ba∇dblhk\nτ/ba∇dblZ(λ∞−B0)(1+D),µ∞;1\nt−Dt\n1+D+t≤δk.This implies that\nsupt≥0/ba∇dblhk\nt/ba∇dblZ(λ∞−B0)(1+D),µ∞−λ∞(1+D)D;1\nt≤δk.In particular, we have a uniform estimate on hk\ntinZ(λ∞−B0),µ′\n∞;1\nt .\nSumming up over kyields for f=f0+/summationtext∞\nk=1hkthe estimate\nsup\nt≥0/ba∇dblf(t,·)−f0/ba∇dblZλ∞−B0,µ′∞;1\nt≤Cδ. (4.23)\n22From (viii) of Proposition 2.5, we can deduce from (4.21) that\nsup\nt≥0/ba∇dblf(t,·)−f0/ba∇dblYλ−B0,µ\nt≤Cδ. (4.24)\nMoreover, ρ=/integraltext\nR3fdvsatisfies similarly supt≥0/ba∇dblρ(t,·)/ba∇dblF(λ∞−B0)t+µ∞≤Cδ.It follows that |ˆρ(t,k)| ≤Cδ\ne−2π(λ∞−B0)|k3|te−2πµ∞|k|for anyk/\\e}atio\\slash= 0.On the one hand, by Sobolev embedding, we deduce that for any\nr∈N,\n/ba∇dblρ(t,·)−/a\\}b∇acketle{tρ/a\\}b∇acket∇i}ht/ba∇dblCr(T3)≤Crδe−2π(λ′−B0)t;\non the other hand, multiplying ˆ ρby the Fourier transform of W,and∂tB=∇×E,we see that the electric and\nmagnetic fields E,Bsatisfy\nsup\nt≥0/ba∇dblE(t,·)/ba∇dblF(λ′−B0)t+µ′≤δ,sup\nt≥0/ba∇dblE(t,·)/ba∇dblF(λ′−B0)t+µ′≤δ, (4.25)\nfor some λ0> λ′> λ, µ0> µ′> µ.\nNow, from (4.22), we have, for any fixed ( k3,η3)∈Z×Rand any t≥0,\n|ˆf(t,k,η1+k1sinΩt\nΩ,η2+k2sinΩt\nΩ,η3+k3t)−ˆf0(η)| ≤Cδe−2πµ′|k3|e−2π(λ′−B0)|η3|,(4.26)\nthis finishes the proof of Theorem 0.1.\n5 Dynamical behavior of the particles’ trajectory\nTo prove Proposition 4.3, by the classical Picard iteration, we only ne ed to consider the following equivalent\nequations\n\n\nd\ndtδXn+1\nt,τ(x,v) =δVn+1\nt,τ(x,v),\nd\ndtδVn+1\nt,τ(x,v) =δVn+1\nt,τ(x,v)×B0+E[fn](t,δXn\nt,τ(x,v)+X0\nt,τ(x,v)),\nδXn+1\nτ,τ(x,v) = 0,δVn+1\nτ,τ(x,v) = 0.(5.1)\nIt is easy to check that\nΩn+1\nt,τ−Id/defines(δXn+1\nt,τ,δVn+1\nt,τ)◦(X0\nτ,t,V0\nτ,t) = (Xn+1\nt,τ◦(X0\nτ,t,V0\nτ,t)−Id,Vn+1\nt,τ◦(X0\nτ,t,V0\nτ,t)−Id).\nTherefore, in order to estimate ( Xn+1\nt,τ◦(X0\nτ,t,V0\nτ,t)−Id,Vn+1\nt,τ◦(X0\nτ,t,V0\nτ,t)−Id),we only need to study\n(δXn+1\nt,τ,δVn+1\nt,τ)◦(X0\nτ,t,V0\nτ,t).\nNow we give a detailed proof of Proposition 4.3.\nProof.Ifn= 0,first, it is trivial that δV0\nt,τ(x,v) = 0,then Eqs.(5.1) reduces to the following equations\n/braceleftbiggd\ndtδX1\nt,τ(x,v) =δV1\nt,τ(x,v),d\ndtδV1\nt,τ(x,v) =δV1\nt,τ(x,v)×B0+E[f0](t,X0\nt,τ(x,v)),\nδX1\nτ,τ(x,v) = 0,δV1\nτ,τ(x,v) = 0.(5.2)\nThen we have\nδX1\nt,τ◦(X0\nτ,t,V0\nτ,t)(x,v) =/integraldisplayt\nτδV1\ns,τ◦(X0\nτ,s,V0\nτ,s)(x,v)ds,\nδV1\nt,τ◦(X0\nτ,t,V0\nτ,t)(x,v) =/integraldisplayt\nτeB0(t−s)E[f0](s,X0\ns,t(x,v))ds.\nBythedefinitionof E[f0],weknowthat /ba∇dblE[f0](s,·)/ba∇dbl\nZ(λ′\n0−B0)(1+b),µ′\n0\ns−bt\n1+b= 0,itistrivialthat /ba∇dblδV1\nt,τ◦(X0\nτ,t,V0\nτ,t)/ba∇dbl\nZ(λ′\n0−B0)(1+b),µ′\n0\nτ−bt\n1+b≤\nCδ0e−2π(λ0−λ′\n0)τmin/braceleftbigg\nt−τ\n2,1\n2π(λ0−λ′\n0)/bracerightbigg\n.\nSimilarly, /ba∇dblδX1\nt,τ◦(X0\nτ,t,V0\nτ,t)/ba∇dbl\nZ(λ′\n0−B0)(1+b),µ′\n0\nτ−bt\n1+b≤Cδ0e−2π(λ0−λ′\n0)τmin/braceleftbigg\n(t−τ)2\n2,1\n2π(λ0−λ′\n0)2/bracerightbigg\n.\nSuppose for n >1,both\n/ba∇dblδXn\nt,τ◦(X0\nτ,t,V0\nτ,t)/ba∇dbl\nZ(λ′\nn−1−B0)(1+b),µ′\nn−1\nτ−bt\n1+b≤Cn−1/summationdisplay\ni=1δie−2π(λi−λ′\ni)τmin/braceleftbigg(t−τ)2\n2,1\n2π(λi−λ′\ni)2/bracerightbigg\n,\n23and\n/ba∇dblδVn\nt,τ◦(X0\nτ,t,V0\nτ,t)/ba∇dbl\nZ(λ′\nn−1−B0)(1+b),µ′\nn−1\nτ−bt\n1+b≤Cn−1/summationdisplay\ni=1δie−2π(λi−λ′\ni)τmin/braceleftbigg(t−τ)\n2,1\n2π(λi−λ′\ni)/bracerightbigg\n.\nThen for n+1,since (δXn+1\nt,τ,δVn+1\nt,τ) satisfy\n\n\nd\ndtδXn+1\nt,τ(x,v) =δVn+1\nt,τ(x,v),\nd\ndtδVn+1\nt,τ(x,v) =δVn+1\nt,τ(x,v)×B0+E[fn](t,δXn\nt,τ(x,v)+X0\nt,τ(x,v)),\nδXn+1\nτ,τ(x,v) = 0,δVn+1\nτ,τ(x,v) = 0.(5.3)\nThenwehave δVn+1\nt,τ=/integraltextt\nτeB0(t−s)E[fn](s,δXn\ns,τ(x,v)+X0\ns,τ(x,v))ds,andδVn+1\nt,τ◦(X0\nτ,t,V0\nτ,t) =/integraltextt\nτeB0(t−s)[E[fn]◦\n(δXn\ns,τ◦(X0\nτ,s,V0\nτ,s))](s,X0\ns,t(x,v))ds.\nHence\n/ba∇dblδVn+1\nt,τ◦(X0\nτ,t,V0\nτ,t)/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b\n≤/integraldisplayt\nτeB0(t−s)/ba∇dbl[E[fn]◦(δXn\ns,τ◦(X0\nτ,s,V0\nτ,s))](s,X0\ns,t(x,v))/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+bds\n=/integraldisplayt\nτ/ba∇dblE[fn]◦(δXn\ns,τ◦(X0\nτ,s,V0\nτ,s))/ba∇dblZ(λ′n−B0)(1+b),µ′n\ns−bt\n1+bds=/integraldisplayt\nτ/ba∇dblE[fn](s,·)/ba∇dblFν′nds≤n/summationdisplay\ni=1/integraldisplayt\nτ/ba∇dblE[hi](s,·)/ba∇dblFν′nds,\nwhereν′\nn= (λ′\nn−B0)|s−b(t−s)|+µ′\nn+/ba∇dblδXn\ns,τ◦(X0\nτ,s,V0\nτ,s)/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b.\nNotethat /ba∇dblδXn\ns,τ◦(X0\nτ,s,V0\nτ,s)/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b≤ Cn\n1,ifs≥bt\n1+b,thenν′\nn≤(λ′\nn−B0)s+µ′\nn+Cn\n1≤(λi−B0)s+\nµi−(λi−λ′\nn)sas soon as Cn\n1≤(λi−B0)b(t−s)\n2(I); ifs≤bt\n1+b,thenν′\nn≤(λ′\nn−B0)bt+µ′\nn−(λ′\nn−B0)(1+b)s+Cn\n1≤\n(λ′\nn−B0)D+µ′\nn−(λi−λ′\nn)s+Cn\n1≤µ0−(λi−λ′\nn)sas soon as Cn\n1≤µ0−µ′\nn\n2(II).In order to the feasibility of\nthe conditions ( I) and (II),we only need to check that the following assumption ( I) holds\n2C1\nω/parenleftbiggn/summationdisplay\ni=1δi\n(2π(λi−λ′n))3/parenrightbigg\n≤min/braceleftbigg(λi−B0)b(t−s)\n6,µ0−µ′\nn\n2/bracerightbigg\n,\nsinceCn\n1≤ω1,2\n0,n= 2C1\nω/parenleftbigg/summationtextn\ni=1δi\n(2π(λi−λ′\nn))3/parenrightbigg\nmin{1\n2(t−s)2,1}\n1+s.\nWe can obtain the following conclusion,\n/ba∇dblδVn+1\nt,τ◦(X0\nτ,t,V0\nτ,t)/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b≤Cn/summationdisplay\ni=1δi/integraldisplayt\nτe−2π(λi−λ′\ni)sds≤Cn/summationdisplay\ni=1δie−2π(λi−λ′\ni)τmin/braceleftbigg(t−τ)\n2,1\n2π(λi−λ′\ni)/bracerightbigg\n,\nthen we have\n/ba∇dblδXn+1\nt,τ◦(X0\nτ,t,V0\nτ,t)/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b≤Cn/summationdisplay\ni=1δi/integraldisplayt\nτe−2π(λi−λ′\ni)sds≤Cn/summationdisplay\ni=1δie−2π(λi−λ′\ni)τmin/braceleftbigg(t−τ)2\n2,1\n2π(λi−λ′\ni)2/bracerightbigg\n.\nWe finish the proof of Proposition 4.3.\nIn the following we estimate ∇Ωn\nt,τ−Id.In fact, we write (Ωn\nt,τ−Id)(x,v) = (δXn\nt,τ,δVn\nt,τ)◦(X0\nτ,t,V0\nτ,t),\nwe get by differentiation ∇xΩn+1\nt,τ−(I,0) =∇x(δXn\nt,τ◦(X0\nτ,t,V0\nτ,t), δVn\nt,τ◦(X0\nτ,t,V0\nτ,t)),∇vΩn\nt,τ−(0,I) =\n(∇v+M(t−τ)∇x)(δXn\nt,τ◦(X0\nτ,t,V0\nτ,t),δVn\nt,τ◦(X0\nτ,t,V0\nτ,t)).\n\n\nd\ndt∇xδXi\nt,τ(x,v) =∇xδVi\nt,τ(x,v),\nd\ndt∇xδVi\nt,τ(x,v) =∇xδVi\nt,τ(x,v)×B0+∇xE[fi](t,δXi\nt,τ(x,v)+X0\nt,τ(x,v)),\nδXi\nτ,τ(x,v) = 0, δVi\nτ,τ(x,v) = 0.(5.4)\nUsing the same process in the proof of Proposition 4.3, we can obtain Proposition 4.5.\nTo establish a control of Ωi\nt,τ−Ωn\nt,τin norm Z(λ′\nn−B0)(1+b),µ′\nn\nτ−bt\n1+b,we start again from the differential equation\nsatisfied by δVi\nt,τandδVn\nt,τ:\n\n\nd\ndt(δXi\nt,τ−δXn\nt,τ)(x,v) =δVi\nt,τ(x,v)−δVn\nt,τ(x,v),\nd\ndt(δVi\nt,τ−δVn\nt,τ)(x,v) = (δVi\nt,τ(x,v)−δVn\nt,τ(x,v))×B0+E[fi−1](t,δXi−1\nt,τ(x,v)+X0\nt,τ(x,v))\n−E[fn−1](t,δXn−1\nt,τ(x,v)+X0\nt,τ(x,v)),\n(δXi\nt,τ−δXn\nt,τ)(x,v) = 0,(δVi\nt,τ−δVn\nt,τ)(x,v) = 0.(5.5)\n24So from (5.5), δVi\nt,τ−δVn\nt,τsatisfies the equation:\nd\ndt(δVi\nt,τ−δVn\nt,τ)(x,v) = (δVi\nt,τ(x,v)−δVn\nt,τ(x,v))×B0+E[fi−1](t,δXi−1\nt,τ(x,v)+X0\nt,τ(x,v))\n−E[fn−1](t,δXi−1\nt,τ(x,v)+X0\nt,τ(x,v))+E[fn−1](t,δXi−1\nt,τ(x,v)+X0\nt,τ(x,v))−E[fn−1](t,δXn−1\nt,τ(x,v)+X0\nt,τ(x,v)).\nUnder the assumption ( I),we can use the similar proof of Proposition 4.3 to finish Proposition 4.6.\nLetεbe the small constant appearing in Lemma 2.7.\nIf\n3Ci\n1+Ci\n2≤ε,for all i≥1,(II)\nthen/ba∇dbl∇Ωi\nt,τ/ba∇dbl\nZ(λ′\ni−B0)(1+b),µ′\ni\nτ−bt\n1+b≤ε; if in addition\n2(1+τ)(1+B)(3Ci,n\n1+Ci,n\n2)(τ,t)≤max{λ′\ni−λ′\nn,µ′\ni−µ′\nn},(III)\nfor alli∈ {1,...,n−1}and allt≥τ,then\n\n\nλ′\nn(1+b)+2/ba∇dblΩn−Ωi/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b≤λ′\ni(1+b),\nµ′\nn+2(1+|τ−bt\n1+b|)/ba∇dblΩn−Ωi/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b≤µ′\ni.(5.6)\nThen Lemma 2.7 and (5.6) yield Proposition 4.8.\nAs a corollary of Proposition 4.8 and Proposition 2.8, under the assum ption (IV) :\n4(1+τ)(Ci,n\n1+Ci,n\n2)≤min{λi−λ′\nn,µi−µ′\nn},\nfor alli∈ {1,...,n}and allτ∈[0,t],we have\nCorollary 5.1 under the assumption (4.12), we have\n/ba∇dblhi\nτ◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b≤δi,sup\n0≤τ≤t/ba∇dbl(hi\nτv)◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b≤δi,\nsup\n0≤τ≤t/ba∇dbl(∇x(hi\nτv))◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b≤δi,sup\n0≤τ≤t/ba∇dbl(∇xhi\nτ)◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b≤δi,\n/ba∇dbl(∇′\nv+τ∇x)(hi\nτ)◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b≤δi,/ba∇dbl((∇′\nv+τ∇x)hi\nτ)◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b≤δi.\n6 The estimates of main terms\nIn order to estimate these terms I,II,V,we have to make good understanding of plasma echoes. Therefore ,\nwe will try to explain plasma echoes and give the relevant mathematica l results.\n6.1 Plasma echoes\nThis section is one of the key sections in our paper. And from Theore m 6.1 in this section, we can see, it is\nreasonable that the influence of the magnetic field is regarded as an error term. First, we plan to briefly explain\nplasma echoes though a simple example from [17], then we control plas ma echoes (or obtain plasma echoes) in\ntime-shift pure and hybrid analytic norms.\nThe unusual non-linear phenomena that results from the undampe d oscillations of the distribution function\nfsatisfying the nonlinear Vlasov equation is called plasma echo. Let a pe rturbation be specified at the initial\ninstant, such that the distribution function δfis the perturbation of that of Maxwellian plasma f0(v)∼\nexp(−αv2),α >0 is a constant and varies periodically in the x−direction. Without loss of generality, we\nassumeδf=A1f0(v)cosk1xatt= 0; in this section, Aidenotes the amplitude and kidenotes the wave\nnumber for i= 1,2.The perturbation of the density, i.e. the integral/integraltext\nδfdv,varies in the same manner in the\nx−direction at t= 0.Subsequently, the perturbation of the distribution function varie s at time taccording to\nδf=A1f0(v)cosk1(x−vt),which corresponds to a free movement of each particle in the x−direction with its\n25own speed v.But the density perturbation is damped (in a time ∼1\nk1vT), because/integraltext\nδfdvis made small by the\nspeed-oscillatory factor cos k1(x−vt).The asymptotic form of the damping at times t≫1\nk1vTis given by\nδρ=/integraldisplay\nδfdv∝exp(−1\n2k2\n1v2\nTt2), (6.1)\nwhere the proof of (6.1) can be found in [17].\nNow let the distribution function be again modulates at a time t=τ≫1\nk1vT,with amplitude A2and a new\nwave number k2> k1.The resulting density perturbation is damped in time t∼1\nk2vT,but reappears at a time\nτ′=k2τ\nk2−k1,since the second modulation creates in the distribution function for t=τa second-order term of\nthe form\nδf(2)=A1A2f0(v)cosk1(x−vτ)cosk2x,\nwhose further development at t > τchanges into\nδf(2)=A1A2f0(v)cosk1(x−vτ)cosk2[x−v(t−τ)]\n=A1A2f0(v){cos[(k1−k2)(x−vt)+k2vτ]+cos[(k1+k2)(x−vt)+k2vτ]}.\nWe see that at t=τ′the oscillatory dependence of the first term on vdisappears, so that this term makes\na finite contribution to the perturbation of the density with wave nu mberk2−k1.The resulting echo is then\ndamped in a time ∼1\nvT(k2−k1),and the final stage of this damping follows a law similar to (6.1).\nFrom the above physical point of view, under the assumption of the stability condition, we are discovering\nthat, even in magnetic field case, echoes occurring at distinct freq uencies are asymptotically well separated. In\nthe following, through complicate computation, we give a detailed des cription by using mathematical tool. The\nsame to Section 1, since resonances only occur in the ˆ zdirection, in order to simplify the statement of the proof\nof the following theorem, we assume ( x,v) = (x3,v3)∈T×R.\nTheorem 6.1 Letλ,¯λ,µ,¯µ,µ′,ˆµbe such that 2λ≥¯λ > λ > 0,¯µ≥µ′> µ >ˆµ >0,and letb=b(t,s)>0,\nR=R(t,x),G=G(t,x,v)and assume /hatwideG(t,k1,k2,0,v) = 0,we have if\nσ(t,x,v) =/integraldisplayt\n0R(s,x+M(t−s)v)G(s,x+M(t−s)v,v)ds,\nσ1(t,x) =/integraldisplayt\n0/integraldisplay\nR3R(s,x+M(t−s)v)G(s,x+M(t−s)v,v)dvds.\nThen\n/ba∇dblσ(t,·)/ba∇dblZλ,µ;1\nt≤/integraldisplayt\n0sup\nk/ne}ationslash=l,k,l∈Z∗e−2π(¯µ−µ)|k−l|e−2π(¯λ−λ)|k−l|s/ba∇dblR/ba∇dblF¯λs+¯µ/ba∇dblG/ba∇dblZλ(1−b),ˆµ;1\nsds, (6.2)\n/ba∇dblσ(t,·)/ba∇dblZλ,µ;1\nt≤/integraldisplayt\n0sup\nk,l∈Z∗e−π(¯µ−µ)|l|e−π(¯λ−λ)|k(t−s)+ls|e−2π[µ′−µ+λb(t−s)]|k−l|·/ba∇dblR/ba∇dblFλs+µ′−λb(t−s)/ba∇dblG/ba∇dblZ¯λ(1+b),¯µ;1\ns−bt\n1+bds,\n(6.3)\n/ba∇dblσ1(t,·)/ba∇dblFλt+µ≤/integraldisplayt\n0sup\nk,l∈Z∗e−π(¯µ−µ)|l|e−π(¯λ−λ)|k(t−s)+ls|e−2π[µ′−µ+λb(t−s)]|k−l|·/ba∇dblR/ba∇dblFλs+µ′−λb(t−s)/ba∇dblG/ba∇dblZ¯λ(1+b),¯µ;1\ns−bt\n1+bds,\n(6.4)\n/ba∇dblσ1(t,·)/ba∇dblFλt+µ≤/integraldisplayt\n0sup\nk/ne}ationslash=l,k,l∈Z∗e−2π(¯λ−λ)|k−l|s/ba∇dblR/ba∇dblF¯λs+µ+λb(t−s)/ba∇dblG/ba∇dblZλ(1−b),µ;1\ns+bt\n1−bds. (6.5)\nProof.\n/ba∇dblσ(t,x,v)/ba∇dblZλ,µ;1\nt≤/integraldisplayt\n0/ba∇dbl(RG)◦S0\ns−t(s,·)/ba∇dblZλ,µ;1\ntds=/integraldisplayt\n0/ba∇dbl(RG)(s,·)/ba∇dblZλ,µ;1\nsds.\nLets′=s+b(t−s),b=Ds\nt(1+t),where some constant D >0 small enough. Note that\n/ba∇dbl(RG)(s,·)/ba∇dblZλ,µ;1\ns=/summationdisplay\nk∈Z/summationdisplay\nn∈N0λn\nn!e2πµ|k|/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbigg\n∇v+2iπks/bracketrightbiggn\n/hatwide(RG)(s,k,v)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1\ndv\n=/summationdisplay\nk∈Z/summationdisplay\nn∈N0λn\nn!e2πµ|k|/vextenddouble/vextenddouble/vextenddouble/vextenddoublee2iπv·k(t−s)/bracketleftbigg\n∇v+2iπks/bracketrightbiggn\n/hatwide(RG)(s,k,v)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1\ndv\n=/summationdisplay\nk∈Z/summationdisplay\nn∈N0λn\nn!e2πµ|k|/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbigg\n2iπk(t−s)+2iπks/bracketrightbiggn\ne2iπv·k(t−s)/hatwide(RG)(s,k,v)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1\ndv\n26=/summationdisplay\nk∈Z/summationdisplay\nn∈N0λn\nn!e2πµ|k|/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbigg\n2iπk(t−s′)+2iπk(s′−s)+2iπ(k−l(1−b))s+2iπls(1−b)/bracketrightbiggn\n·e2iπv·k(t−s)/summationdisplay\nlˆR(s,k−l)ˆG(s,l,v)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1\ndv\n≤/summationdisplay\nk∈Z/summationdisplay\nn∈N0λn\nn!e2πµ|k|/summationdisplay\nl|ˆR(s,k−l)|n/summationdisplay\nγ=0Cγ\nn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbigg\n2iπk(t−s′)+2iπls(1−b)/bracketrightbiggn−γ\n·e2iπv·k(t−s)ˆG(s,l,v)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1\ndv\n≤/summationdisplay\nk∈Z/summationdisplay\nn∈N0e2πµ|k|/summationdisplay\nl|ˆR(s,k−l)|n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλn−γ\n(n−γ)!/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbigg\n2iπk(t−s′)+2iπls(1−b)/bracketrightbiggn−γ\n·e2iπv·k(t−s)ˆG(s,l,v)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1\ndv\n≤/summationdisplay\nk∈Z/summationdisplay\nn∈N0e2πµ|k|/summationdisplay\nl|ˆR(s,k−l)|n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλn−γ(1−b)n−γ\n(n−γ)!/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbigg\n2iπk(t−s)+2iπls/bracketrightbiggn−γ\n·e2iπv·k(t−s)ˆG(s,l,v)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1\ndv\n≤/summationdisplay\nk∈Z/summationdisplay\nn∈N0e2πµ|k|/summationdisplay\nl|ˆR(s,k−l)|n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n·λn−γ(1−b)n−γ\n(n−γ)!/ba∇dbl[∇v+2iπls]n−γˆG(s,l,v)/ba∇dblL1\ndv.\nNow we will divide k,linto the following cases:\nCase 1. min {|k|,|l|}> k−l >0.We still decompose this case into two steps.\nStep 1. If min {|k|,|l|}> k−l >0, k <0,\nn/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2π(k−l)s+2iπkb(t−s)+2πlbs/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle=n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2π(k−l)s(1−b)+2πkbt/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nIfs≥−kDs\n(1+t)(k−l)(1−b),wehave/summationtextn\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/summationtextn\nγ=0λγ\nγ!/bracketleftbigg\n2π(k−l)s(1−b)+2πkbt/bracketrightbiggγ\n.\nThen\n/summationdisplay\nn∈N0e2πµ|k|/summationdisplay\nl|ˆR(s,k−l)|n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle·λn−γ(1−b)n−γ\n(n−γ)!/ba∇dbl[∇v+2iπls]n−γˆG(s,l,v)/ba∇dblL1\ndv\n≤/summationdisplay\nl/summationdisplay\nn∈N0e2πµ|k|e2π(k−l)s(1−b)+2πkbt|ˆR(s,k−l)|·λn(1−b)n\nn!/ba∇dbl[∇v+2iπls]nˆG(s,l,v)/ba∇dblL1\ndv\n≤/summationdisplay\nl/summationdisplay\nn∈N0e2π(µ+λbt)|k−l|e2πλ|k−l|s(1−b)|ˆR(s,k−l)|·e2π(µ−λbt)|l|λn(1−b)n\nn!/ba∇dbl[∇v+2iπls]nˆG(s,l,v)/ba∇dblL1\ndv.\nIfs≤−kDs\n(1+t)(k−l)(1−b)≤−kD\n(k−l)(1−b)≤t,for some constant 0 < ǫ0 k−l >0,k >0.Indeed, we only consider k−l >0,l >0,since min {|k|,|l|}> k−l >\n−min{|k|,|l|}.\nIt is easy to check that\nn/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle=n/summationdisplay\nγ=0λγ\nγ![2π(k−l)s′−2πlbt]γ.\nThis can be reduced to case 1, here we omit the details. We can obtain\n/summationdisplay\nn∈N0e2πµ|k|/summationdisplay\nl|ˆR(s,k−l)|n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle·λn−γ(1−b)n−γ\n(n−γ)!/ba∇dbl[∇v+2iπls]n−γˆG(s,l,v)/ba∇dblL1\ndv\n≤/summationdisplay\nl/summationdisplay\nn∈N0e2πµ|k−l|e2πλ|k−l|s′|ˆR(s,k−l)|·e2πˆµ|l|λn(1−b)n\nn!/ba∇dbl[∇v+2iπls]nˆG(s,l,v)/ba∇dblL1\ndv.\nCase 2. If −min{|k|,|l|}< k−l <0,the method of this case is the same to Case 1.\n/summationdisplay\nn∈N0e2πµ|k|/summationdisplay\nl|ˆR(s,k−l)|n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle·λn−γ(1−b)n−γ\n(n−γ)!/ba∇dbl[∇v+2iπls]n−γˆG(s,l,v)/ba∇dblL1\ndv\n≤/summationdisplay\nl/summationdisplay\nn∈N0e2πµ|k−l|e2πλ|k−l|s′|ˆR(s,k−l)|·e2πˆµ|l|λn(1−b)n\nn!/ba∇dbl[∇v+2iπls]nˆG(s,l,v)/ba∇dblL1\ndv.\nCase 3. When k−l >min{|k|,|l|},ork−l <−min{|k|,|l|},we only need to consider one of two cases.\nWithout loss of generality, we assume k−l >min{|k|,|l|}.\nn/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle=n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle2π(k−l)s(1−b)+2πkbt/vextendsingle/vextendsingle/vextendsingle/vextendsingleγ\n≤n/summationdisplay\nγ=0λγ\nγ!/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2π(k−l)s(1−b)/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle2πkbt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbiggγ\n≤n/summationdisplay\nγ=0λγ\nγ!/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2π(k−l)s(1−b)/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle2π(k−l)bt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbiggγ\n≤n/summationdisplay\nγ=0λγ\nγ!/bracketleftbigg\n2π(s(1−b)+bt)|k−l|/bracketrightbiggγ\n=n/summationdisplay\nγ=0λγ\nγ!/bracketleftbigg\n2π/parenleftbigg\n1−b+D\n1+t/parenrightbigg\n|k−l|s/bracketrightbiggγ\n.\n/summationdisplay\nn∈N0e2πµ|k|/summationdisplay\nl|ˆR(s,k−l)|n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle·λn−γ(1−b)n−γ\n(n−γ)!/ba∇dbl[∇v+2iπls]n−γˆG(s,l,v)/ba∇dblL1\ndv\n≤/summationdisplay\nl/summationdisplay\nn∈N0e2πµ|k−l|e2πλ(1−b+2D\n1+t)|k−l|s|ˆR(s,k−l)|·e2πˆµ|l|λn(1−b)n\nn!/ba∇dbl[∇v+2iπls]nˆG(s,l,v)/ba∇dblL1\ndv.\nCase 4. If t <|k|Ds\n(1+t)|k−l|(1−b)≤|k|D\n|k−l|(1−b)with min {|k|,|l|}> k−l >−min{|k|,|l|},k/\\e}atio\\slash=l.Without loss of\ngenerality, we assume that min {|k|,|l|}> k−l >0,\nn/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle=n/summationdisplay\nγ=0λγ\nγ!|[2π(k−l)s′−2πlbt]γ|.\nIfk−l >0, l >0,\n28n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2π(k−l)s′−2πlbt/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2π(k−l)·|k|Ds\n(1+t)|k−l|(1−b)−2πlbt/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2π·|k|Ds\n(1+t)(1−b)−2πlDs\n1+t/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤n/summationdisplay\nγ=0λγ\nγ!/bracketleftbigg\n2πD|k−l|\n(1+t)(1−b)s−2πǫ0lDs\n1+t/bracketrightbiggγ\n≤n/summationdisplay\nγ=0λγ\nγ!/bracketleftbigg\n2π|k−l|s−2πǫ0|l|Ds\n1+t/bracketrightbiggγ\n.\nIfk−l >0,l <0,this is equal to k−l >0,k <0,since min {|k|,|l|}> k−l >−min{|k|,|l|}.\nn/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=n/summationdisplay\nγ=0λγ\nγ!/bracketleftbigg\n2π(k−l)s(1−b)+2πkbt/bracketrightbiggγ\n≤n/summationdisplay\nγ=0λγ\nγ!/bracketleftbigg\n2π|k−l|s−2πǫ0|l|Ds\n1+t/bracketrightbiggγ\n.\nIn summary,\n/summationdisplay\nk,l∈Z∗/summationdisplay\nn∈N0e2πµ|k||ˆR(s,k−l)|n/summationdisplay\nγ=0λγ\nγ!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n2iπk(s′−s)+2iπ(k−l(1−b))s/bracketrightbiggγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle·λn−γ(1−b)n−γ\n(n−γ)!/ba∇dbl[∇v+2iπls]n−γˆG(s,l,v)/ba∇dblL1\ndv\n≤sup\nk/ne}ationslash=l,k,l∈Z∗e−2π(¯µ−µ)|k−l|e−2π(¯λ−λ)|k−l|s/ba∇dblR/ba∇dblF¯λs+¯µ/ba∇dblG/ba∇dblZλ(1−b),ˆµ;1\ns,\nthen\n/ba∇dblσ(t,·)/ba∇dblZλ,µ;1\nt≤/integraldisplayt\n0sup\nk/ne}ationslash=l,k,l∈Z3∗e−2π(¯µ−µ)|k−l|e−2π(¯λ−λ)|k−l|s/ba∇dblR/ba∇dblF¯λs+¯µ/ba∇dblG/ba∇dblZλ(1−b),ˆµ;1\nsds.\nNow we estimate the second inequality of Theorem 6.1.\n/ba∇dblσ(t,x,v)/ba∇dblZλ,µ;1\nt≤/integraldisplayt\n0/ba∇dbl(RG)◦S0\ns−t(s,·)/ba∇dblZλ,µ;1\ntds=/integraldisplayt\n0/ba∇dbl(RG)(s,·)/ba∇dblZλ,µ;1\nsds.\nNote that\n/ba∇dbl(RG)(s,·)/ba∇dblZλ,µ;1\ns=/summationdisplay\nk∈Z3/summationdisplay\nn∈N3\n0λn\nn!e2πµ|k|/ba∇dbl[∇v+2iπks]n/hatwide(RG)(s,k,v)/ba∇dblL1\ndv\n=/summationdisplay\nk∈Z/summationdisplay\nn∈N0λn\nn!e2πµ|k|/ba∇dble2iπv·k(t−s)[∇v+2iπks]n/hatwide(RG)(s,k,v)/ba∇dblL1\ndv\n=/summationdisplay\nk∈Z/summationdisplay\nn∈N0λn\nn!e2πµ|k|/ba∇dbl[2iπk(t−s)+2iπks]ne2iπv·k(t−s)/hatwide(RG)(s,k,v)/ba∇dblL1\ndv\n≤/summationdisplay\nk,l∈Ze2πλ|k|te2πµ|k||ˆR(s,k−l)|/ba∇dble2iπv·k(t−s)ˆG(s,l,v)/ba∇dblL1\ndv.\nThe proof of the inequality (6.3) is as follows, here we sketch the main steps. We let s′=s−b(t−s) and write\ne2π(λt+µ)|k|≤e−2π(¯µ−µ)|l|e−2πλ(s−s′)|k−l|e−2π(µ′−µ)|k−l|e−2π(¯λ−λ)|k(t−s′)+ls′|\n×e2π¯µ|l|e2π(λs+µ′)|k−l|e2π¯λ|k(t−s′)+ls′|\n≤e−2π(¯µ−µ)|l|e−π(¯λ−λ)|k(t−s)+ls|e−2π[µ′−µ+λ(s−s′)/2]|k−l|\n×e2π¯µ|l|e2π[λs+µ′−λ(s−s′)/2]|k−l|/summationdisplay\nn∈N0|2iπ¯λ(k(t−s′)+ls′)|n\nn!,\nthen we can deduce that\n/ba∇dblσ(t,·)/ba∇dblZλ,µ;1\nt≤/integraldisplayt\n0sup\nk,l∈Z∗e−π(¯µ−µ)|l|e−π(¯λ−λ)|k(t−s)+ls|e−2π[µ′−µ+λb(t−s)]|k−l|·/ba∇dblR/ba∇dblFλs+µ′−λb(t−s)/ba∇dblG/ba∇dblZ¯λ(1+b),¯µ;1\ns−bt\n1+bds.\nIn the following we estimate the norm Fλt+µof the function σ1(t,x),\nσ1(t,x) =/integraldisplayt\n0/integraldisplay\nRR(s,x+M(t−s)v)G(s,x+M(t−s)v,v)dvds.\n29We write\nˆσ1(t,k) =/integraldisplayt\n0/summationdisplay\nl∈Z/integraldisplay\nRˆR(s,k−l)ˆG(s,l,v)e2πiv·k(t−s)dvds,\n|ˆσ1(t,k)| ≤/integraldisplayt\n0/parenleftbigg/summationdisplay\nl∈Z/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRˆG(s,l,v)e2πiv·k(t−s)dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle|ˆR(s,k−l)|/parenrightbigg\nds.\nNext,\ne2π(λt+µ)|k|≤e2πλ(s+b(t−s))|k−l|e2πλ(1−b)|k(t−s)+l(s+bt\n1−b)|e2πµ|k−l|e2πµ|l|\n≤e2π¯λ|k−l|se2πλ(1−b)|k(t−s)+l(s+bt\n1−b)|e2π(µ+λb(t−s))|k−l|e2πµ|l|e−2π(¯λ−λ)|k−l|s.\nHence\n/ba∇dblσ1(t,·)/ba∇dblFλt+µ≤/integraldisplayt\n0sup\nk/ne}ationslash=l,k,l∈Z3∗e−2π(¯λ−λ)|k−l|s/ba∇dblR/ba∇dblF¯λs+¯µ/ba∇dblG/ba∇dblZλ(1−b),µ;1\ns+bt\n1−bds,\nwhere ¯µ=µ+λb(t−s).\n6.2 Estimates of main terms\nIn the following we estimate ¯IIn+1,n\ni(t,x).Note that their zero modes vanish. For any n≥i≥1,\n/hatwider¯IIn+1,n\ni(t,k) =/integraldisplayt\n0/integraldisplay\nT3/integraldisplay\nR3e−2πik·x/parenleftbigg\nB[hn+1]·(∇′\nv×((hi\nτv)◦Ωi−1\nt,τ)/parenrightbigg\n(τ,x′(τ,x,v),v′(τ,x,v))dv′dxdτ,\n|/hatwider¯IIn+1,n\ni(t,k)| ≤/integraldisplayt\n0/parenleftbigg/summationdisplay\nl∈Z3\n∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3e−2πik3·(v3(t−τ))e−2πiηk1·v′\n1e−2πiηk2·v′\n2\n·(/hatwider∇′v×((hiτv)◦Ωi−1\nt,τ))(τ,l,v′)dv′/vextendsingle/vextendsingle/vextendsingle/vextendsingle|/hatwiderB[hn+1](τ,k−l)|/parenrightbigg\ndτ\n=/integraldisplayt\n0/parenleftbigg/summationdisplay\nl∈Z3∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRe−2πik3·(v3(t−τ))e−2πiηk1·v′\n1e−2πiηk2·v′\n2\n·(/hatwider∇′v×((hiτv)◦Ωi−1\nt,τ))(τ,l,ηk1,ηk2,v3)dv3/vextendsingle/vextendsingle/vextendsingle/vextendsingle|/hatwiderB[hn+1](τ,k−l)|/parenrightbigg\ndτ.\n(6.6)\nFrom (6.3) of Theorem 6.1 and (6.6), we can get Proposition 4.9.\nTo finish Propositions 4.11-4.12, we shall again use the Vlasov equatio n. We rewrite it as\nhn+1(t,Xn\nτ,t(x,v),Vn\nτ,t(x,v)) =/integraldisplayt\n0Σn+1(s,Xn\nτ,s(x,v),Vn\nτ,s(x,v))ds.\nThen we get\n/ba∇dblhn+1(t,Xn\nτ,t(x,v),Vn\nτ,t(x,v))/ba∇dblZ(λ′n−B0)(1−b),µ′n;1\nt+bt\n1−b\n≤/integraldisplayt\n0/ba∇dblΣn+1(s,Xn\nτ,s(x,v),Vn\nτ,s(x,v))/ba∇dblZ(λ′n−B0)(1−b),µ′n;1\nt+bt\n1−b(ηk1,ηk2)ds\n=/integraldisplayt\n0/ba∇dblΣn+1(s,Ωn\nτ,s(x,v))/ba∇dblZ(λ′n−B0)(1−b),µ′n;1\ns+bt\n1−bds\n≤/integraldisplayt\n0sup\nk,l∈Z3∗e−π(µ′′\nn−µ′\nn)|l|e−π(λn−λ′\nn)|k(t−s)+l(s+bt\n1−b)|e−2π[µ′′\nn−µ′\nn+(λ′\nn−B0)b(t−s)]|k−l|\n·/ba∇dblEn+1\ns,t/ba∇dblFς′n/ba∇dblGn\ns,t/ba∇dblZ(λn−B0)(1+b),µ′′n;1\ns+bt\n1−b−bt\n1+bds+/integraldisplayt\n0sup\nk,l∈Z3∗e−π(µ′′\nn−µ′\nn)|l|e−π(λ′′\nn−λ′\nn)|k(t−s)+l(s+bt\n1−b)|\n·e−2π[µ′′\nn−µ′\nn+(λ′\nn−B0)b(t−s)]|k−l|/ba∇dblFn+1\ns,t/ba∇dblFς′n/ba∇dblGn,v\ns,t/ba∇dblZ(λ′′n−B0)(1+b),µ′′n;1\ns+bt\n1−b−bt\n1+bds\n+/integraldisplayt\n0/bracketleftbigg\n/ba∇dblHn\ns,t/ba∇dblZ(λ′n−B0)(1−b),µ′n;1\ns+bt\n1−b/ba∇dblEn\ns,t/ba∇dblZ(λ′n−B0)(1−b),µ′n\ns+bt\n1−b+/ba∇dblHn,v\ns,t/ba∇dblZ(λ′n−B0)(1−b),µ′n;1\ns+bt\n1−b\n30·/ba∇dblFn\ns,t/ba∇dblZ(λ′n−B0)(1−b),µ′n\ns+bt\n1−b/bracketrightbigg\nds+/integraldisplayt\n0sup\nk/ne}ationslash=l,k,l∈Z3∗e−2π(µn−µ′\nn)|k−l|e−2π(λn−λ′\nn)|k−l|s\n·/ba∇dblB[fn]◦Ωn\nt,s/ba∇dblFς′′n/ba∇dblHn+1,v\ns,t/ba∇dbl\nZ(λ′n−B0)(1−b)2,ˆµ′n;1\ns+bt\n1−bds,\nwhereς′\nn= (λ′\nn−B0)(1−b)s+µ′′\nn−(λ′\nn−B0)(1−b)b(t−s), ς′′\nn=λ′′\nns+µ′+(λ′\nn−B0)b(t−s).\nThen\nsup\n00 sufficiently small,\n/ba∇dbl(∇Ωn\nt,s)−1−Id/ba∇dblZ(λn−B0)(1−b),µ′n\ns+bt\n1−b< ε.\nIf the claim holds, then\n/ba∇dblV/ba∇dblF(λ′n−B0)t+µ′n\n≤/integraldisplayt\n0sup\nk/ne}ationslash=l,k,l∈Z3∗e−2π(λn−λ′′\nn)|k−l|s/ba∇dblB[fn]/ba∇dblFν′n·/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbigg/parenleftbigg\n(∇v×(hn+1v))◦Ωn\ns,t(x,v)/parenrightbigg\n−/parenleftbigg\n(∇v×/bracketleftbigg\n(hn+1v)◦Ωn\ns,t(x,v)/bracketrightbigg/parenrightbigg/bracketrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nZ(λ′n−B0)(1−b),µ′n;1\ns+bt\n1−b(ηk1,ηk2)ds+/integraldisplayt\n0sup\nk/ne}ationslash=l,k,l∈Z3∗e−2π(λn−λ′′\nn)|k−l|s\n·/ba∇dblB[fn]/ba∇dblFν′n·/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇v×/bracketleftbigg\n(hn+1v)◦Ωn\ns,t(x,v)/bracketrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nZ(λ′n−B0)(1−b),µ′n;1\ns+bt\n1−b(ηk1,ηk2)ds\n≤/integraldisplayt\nτe−π(λn−λ′\nn)|k(t−s)+ls|/ba∇dblB[fn]/ba∇dblFν′n/ba∇dblhn+1◦Ωn\ns,t(x,v)/ba∇dbl\nZ(λ′n−B0)(1−1\n2b),µ′n;1\ns+bt\n1−b(ηk1,ηk2)ds,\nwhere\nν′\nn= (λ′′\nn−B0)s+µ′\nn+(λ′\nn−B0)b(t−s)+/ba∇dblΩn\nt,s−Id/ba∇dblZλ′′n−B0,µ′n+(λ′n−B0)b(t−s)\ns\n≤(λ′′\nn−B0)s+µ′\nn+(λ′\nn−B0)b(t−s)+/ba∇dblΩn\nt,s−Id/ba∇dblZ(λ′′n−B0),µ′n+2(λ′n−B0)b(t−s)\ns−bt\n1+b\n31≤(λ′′\nn−B0)s+µ′\nn+(λ′\nn−B0)b(t−s)+/ba∇dblΩn\nt,s−Id/ba∇dblZλ′′n−B0,µ′′n\ns−bt\n1+b,\nbyProposition 4.3, we have ν′\nn≤(λn−B0)s+µn−(λn−λ′′\nn)sas soon as\n4C1\nωn/summationdisplay\ni=1δi\n(2π(λi−λ′\ni))3≤λ′\n∞D\n3(V).\nSo\n/ba∇dblV/ba∇dblF(λ′n−B0)t+µ′n≤/integraldisplayt\nτe−π(λn−λ′\nn)|k(t−s)+ls|/parenleftbiggn/summationdisplay\ni=1δi/parenrightbigg\n/ba∇dblhn+1◦Ωn\ns,t(x,v)/ba∇dbl\nZ(λ′n−B0)(1−1\n2b),µ′n;1\ns+bt\n1−bds.\nWe finish the proof of Proposition 4.12.\n7 Estimates of error terms\nIn the following we estimate one of the error terms R0.\nRecall\nR0(t,x) =/integraldisplayt\n0/integraldisplay\nR3/parenleftbigg/parenleftbigg\nB[hn+1]◦Ωn\nt,s(x,v)−B[hn+1]/parenrightbigg\n·Gn,v\nt,s/parenrightbigg\n(s,X0\nt,s(x,v),V0\nt,s(x,v)))dvds.\nFirst,\n/ba∇dblR0(t,·)/ba∇dblF(λ′n−B0)t+µ′n\n≤/integraldisplayt\n0/ba∇dblB[hn+1]◦Ωn\nt,s(x,v)−B[hn+1]/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b/ba∇dblGn,v\nt,s/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+bds.\nNext,\n/ba∇dbl(B[hn+1]◦Ωn\nt,s(x,v)−B[hn+1]/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b\n≤ /ba∇dblΩn\nt,τ−Id/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b/integraldisplay1\n0/ba∇dbl∇B[hn+1]((1−θ)Id+θΩn\nt,s)/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+bdθ\n≤ /ba∇dbl∇B[hn+1]/ba∇dblFν′n/ba∇dblΩn\nt,τ−Id/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b,\n(7.1)\nwhereν′\nn= (λ′\nn−B0)(1+b)/vextendsingle/vextendsingle/vextendsingle/vextendsingleτ−bt\n1+b/vextendsingle/vextendsingle/vextendsingle/vextendsingle+µ′\nn+/ba∇dblΩnXt,τ−x′/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b.\nHere we only focus on the case τ≥bt\n1+b,then we need to show /ba∇dbl∇B[hn+1]/ba∇dblFν′n≤ /ba∇dblρ[hn+1]/ba∇dblF(λ′n−B0)τ+µ′n.\nFor that, we have to prove ν′\nn<(λ′\nn−B0)τ+µ′\nn−ι,for some constant ι >0 sufficiently small.\nIndeed,\nν′\nn≤(λ′\nn−B0)τ+µ′\nn−(λ′\nn−B0)b(t−τ)+Cn/summationdisplay\ni=1δie−π|k3|(λi−(λ′\ni))t·min/braceleftbigg(t−τ)2\n2,1\n2π(λi−λ′\ni)2/bracerightbigg\n≤(λ′\nn−B0)τ+µ′\nn−(λ′\nn−B0)B(t−τ)\n1+t+C/parenleftbiggn/summationdisplay\ni=1δi\n(λi−λ′\ni)3/parenrightbiggmin{t−τ,1}\n1+τ.\nNote thatmin{t−τ,1}\n1+τ≤3t−τ\n1+t.In the following we also need to show that\nCn/summationdisplay\ni=1δi\n(λi−λ′\ni)3≤λ∗B\n3−ι,(VI)\nNow we assume that (7.1) holds, then\n/ba∇dbl(B[hn+1]◦Ωn\nt,s(x,v)−B[hn+1]/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b≤C/parenleftbiggn/summationdisplay\ni=1δi\n(λi−λ′\ni)5/parenrightbigg1\n(1+τ)3/ba∇dblρ[hn+1]/ba∇dblF(λ′n−B0)τ+µ′n.\n32SinceGn,v\nt,s= (∇′\nvfn×V0\nt,s(x,v))◦Ωn\nt,s(x,v),\n/ba∇dblGn,v\nt,s/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b≤ /ba∇dbl(∇′\nvf0×V0\nt,s(x,v))◦Ωn\nt,s(x,v)/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b\n+n/summationdisplay\ni=1/ba∇dbl(∇′\nvhi\nτ×V0\nt,s(x,v))◦Ωn\nt,s(x,v)/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b\n≤C′\n0+/parenleftbiggn/summationdisplay\ni=1δi/parenrightbigg\n(1+τ).\nWe can conclude\n/ba∇dblR0(t,·)/ba∇dblF(λ′n−B0)t+µ′n≤C/parenleftbigg\nC′\n0+n/summationdisplay\ni=1δi/parenrightbigg/parenleftbiggn/summationdisplay\ni=1δi\n(λi−λ′\ni)5/parenrightbigg/integraldisplayt\n0/ba∇dblρ[hn+1]/ba∇dblF(λ′n−B0)τ+µ′ndτ\n(1+τ)2.\nIn order to finish the control of ˜R0,we still need the estimate of /ba∇dblGn,v\nt,s−¯Gn,v\nt,s/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b.\nIn fact,\n/ba∇dblGn,v\nt,s−¯Gn,v\nt,s/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b\n≤ /ba∇dbl(∇v×(f0v))◦Ωn\nt,τ−∇v×(f0v)/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b\n+n/summationdisplay\ni=1/ba∇dbl(∇v×(hiv))◦Ωn\nt,τ−(∇v×(hiv))◦Ωi−1\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b\n+n/summationdisplay\ni=1/ba∇dbl(∇v×(hiv))◦Ωi−1\nt,τ−∇v×((hiv)◦Ωi−1\nt,τ)/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b.\nNow on the one hand, we treat the second term\nn/summationdisplay\ni=1/ba∇dbl(∇v×(hiv))◦Ωn\nt,τ−(∇v×(hiv))◦Ωi−1\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b\n≤/integraldisplay1\n0/ba∇dbl∇∇vhi\nτ((1−θ)Ωn\nt,τ+θΩi−1\nt,τ)/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b·/ba∇dblΩn\nt,τ−Ωi−1\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+bdθ\n≤2/ba∇dbl∇∇vhi\nτ◦Ωi−1\nt,τ/ba∇dbl\nZ(λ′\ni−B0)(1+b),µ′\ni;1\nτ−bt\n1+b/ba∇dblΩn\nt,τ−Ωi−1\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b\n≤4Cδi/parenleftbiggn/summationdisplay\nj=iδj\n(λj−λ′\nj)6/parenrightbigg1\n(1+τ)2,\nwhere from Proposition 4.6, we know\n/ba∇dblΩnXt,τ−Ωi−1Xt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b≤2Ri−1,n\n2(t,τ),\n/ba∇dblΩnVt,τ−Ωi−1Vt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b≤ Ri−1,n\n1(t,τ)+Ri−1,n\n2(t,τ)\nwith\nRi−1,n\n1(t,τ) =/parenleftbiggn/summationdisplay\nj=iδje−2π(λj−λ′\nj)τ\n2π(λj−λ′\nj)/parenrightbigg\nmin{t−τ,1},Ri−1,n\n2(t,τ) =/parenleftbiggn/summationdisplay\nj=iδje−2π(λj−λ′\nj)τ\n(2π(λj−λ′\nj))2/parenrightbigg\nmin/braceleftbigg(t−τ)2\n2,1/bracerightbigg\n.\nOn the other hand, by the induction hypothesis, since Zλ,µ\nτnorms are increasing as a function of λandµ,if\nfixedτ,\nn/summationdisplay\ni=1/ba∇dbl(∇vhi\nτ)◦Ωi−1\nt,τ−∇v(hi\nτ◦Ωi−1\nt,τ)/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b≤/parenleftbiggn/summationdisplay\ni=1δi/parenrightbigg1\n(1+τ)2.\nSo we have\n/ba∇dbl˜R0(t,·)/ba∇dblF(λ′n−B0)τ+µ′n≤/parenleftbigg\nC4\nω/parenleftbigg\nC′\n0+n/summationdisplay\ni=1δi/parenrightbigg/parenleftbiggn/summationdisplay\nj=1δj\n2π(λj−λ′\nj)6/parenrightbigg\n+n/summationdisplay\ni=1δi/parenrightbigg/integraldisplayt\n0/ba∇dblρ/ba∇dblF(λ′n−B0)τ+µ′n1\n(1+τ)2dτ\n=/integraldisplayt\n0˜Kn+1\n1/ba∇dblρ/ba∇dblF(λ′n−B0)τ+µ′n1\n(1+τ)2dτ.\n(7.2)\n33Up to now, we finish the estimates of error terms.\n8 Iteration\nNow let us first deal with the source term\nˆIIIn,n(t,k)+ˆIVn,n(t,k) =−/integraldisplayt\n0/integraldisplay\nT3/integraldisplay\nR3e−2πik·x(En\nt,s·Hn\nt,s)(s,X0\nt,s(x,v),V0\nt,s(x,v))\ndvdxds−/integraldisplayt\n0/integraldisplay\nT3/integraldisplay\nR3e−2πik·x(Fn\nt,s·Hn,v\nt,s)(s,X0\nt,s(x,v),V0\nt,s(x,v))dvdxds,\n(8.1)\nthen\n/ba∇dblIII(t,·)/ba∇dblF(λ′n−B0)t+µ′n+/ba∇dblIV(t,·)/ba∇dblF(λ′n−B0)t+µ′n\n≤/integraldisplayt\n0/ba∇dblEn\nτ,s/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b/ba∇dblHn\nτ,s/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b+/ba∇dblFn\nτ,s/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b\n·/ba∇dblHn,v\nτ,s/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+bdτ\n≤/integraldisplayt\n0/ba∇dblρn\nτ,s/ba∇dblFνn+1/ba∇dblHn\nτ,s/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+bdτ+/integraldisplayt\n0/ba∇dblρn\nτ,s/ba∇dblFν′n·/ba∇dblHn,v\nτ,s/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+bdτ\n≤/integraldisplayt\n0/ba∇dblρn\nτ,s/ba∇dblFν′n(1+τ)δndτ≤/integraldisplayt\n0/ba∇dblρn\nτ,s/ba∇dblF(λ′n−B0)τ+µ′ne−2πτ(λn−λ′\nn)(1+τ)δndτ≤Cδ2\nn\n(λn−λ′n)2.\n(8.2)\nFrom Propositions 4.9-4.12, combining (4.10), we conclude\n/ba∇dblρ[hn+1](t,·)/ba∇dblF(λ′n−B0)t+µ′n\n≤Cδ2\nn\n(λn−λ′n)2+/integraldisplayt\n0|Kn\n1(t,τ)|(1+τ)n/summationdisplay\ni=1δi/ba∇dblρ[hn+1]/ba∇dblF(λ′n−B0)τ+µ′ndτ+2/integraldisplayt\n0|Kn\n0(t,τ)|\n·n/summationdisplay\ni=1δi/ba∇dblρ[hn+1]/ba∇dblF(λ′n−B0)τ+µ′ndτ+/integraldisplayt\n0(˜Kn+1\n0+˜Kn+1\n1)\n(1+τ)2/ba∇dblρ/ba∇dblF(λ′n−B0)τ+µ′ndτ,\nwhereKn\n0(t,τ),Kn\n1(t,τ) are defined in Proposition 4.9, and\n˜Kn+1\n0/defines2C/parenleftbigg\nC0+n/summationdisplay\ni=1δi/parenrightbigg/parenleftbiggn/summationdisplay\ni=1δi\n(2π(λi−λ′\ni))5/parenrightbigg\n,\n˜Kn+1\n1/defines/parenleftbigg\nC4\nω/parenleftbigg\nC′\n0+n/summationdisplay\ni=1δi/parenrightbigg/parenleftbiggn/summationdisplay\nj=1δj\n2π(λj−λ′\nj)6/parenrightbigg\n+n/summationdisplay\ni=1δi/parenrightbigg\n.\nProposition 8.1 From the above inequality, we obtain the following integral inequality:\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleρ[hn+1](t,x)−/integraldisplayt\n0/integraldisplay\nR3/parenleftbigg\nE[hn+1]+v×B[hn+1]/parenrightbigg\n(τ,x+M(t−τ)v)·∇vf0dvdτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nF(λ′n−B0)t+µ′n\n≤Cδ2\nn\n(λn−λ′n)2+/integraldisplayt\n0/parenleftbigg\nK′n\n1(t,τ)+K′n\n0(t,τ)+cn\n0\n(1+τ)2/parenrightbigg\n/ba∇dblρ[hn+1](τ,·)/ba∇dblF(λ′n−B0)τ+µ′ndτ, (8.3)\nwhereK′n\n1(t,τ) =|Kn\n1(t,τ)|(1+τ)/summationtextn\ni=1δi, K′n\n0(t,τ) =|Kn\n0(t,τ)|/summationtextn\ni=1δi, cn\n0=˜Kn+1\n0+˜Kn+1\n1.\n348.1 Exponential moments of the kernel\nBefore doing the iteration, based on the analysis and computation in Sections 4-7, we explain the connection\nand difference in between electromagnetic field case and electric field case. First, we state the connection,\nfrom (6.3)-(6.4) of Theorem 6.1 in section 6.1, the kernel is the same under two different norms Zλ,µ;1\ntand\nFλt+µ, in other words, there are no new echoes to generate in different n ormsZλ,µ;1\ntandFλt+µ,this is a\nkey point for us, it implies that the first-order term of the magnetic field has no influence on the resonances\nof the plasma particles, that is, the first-order term of the magne tic field can be regarded as an error term\nand doesn’t play an important role in dynamical behavior of the partic les’ trajectory. The difference is to\nadd the new term V=−/integraltextt\n0/integraltext\nR3[(B[fn]◦Ωn\ns,t(x,v))·((∇vhn+1×v)◦Ωn\ns,t(x,v))](s,X0\ns,t(x,v),V0\ns,t(x,v))dvdsin\nthe density equation on ρ[hn+1].In order to deal with this term, we have to go back to the Vlasov equ ation\nonhn+1(s,Xn\ns,t(x,v),Vn\ns,t(x,v)),and then there are new echoes to appear during estimating hn+1(s,Xn\ns,t(x,v),\nVn\ns,t(x,v)),but the echoes’ form is still the same with that generated by the Vla sov-Poissonequation, the details\ncan be found in section 5.1. From the inequalities (6.3)-(6.4), we know that the reason is that echoes don’t\nchange as the norm changes. In summary, in order to iterate on th e density ρ[hn+1],we only need to estimate\nthe same kernel with Landau damping in electric field case. The followin g theorems are the same with the\nresults in [23] and the proofs can also be found in Section 7 in [23], so we sketch the proof.\nProposition 8.2 (Exponential moments of the kernel) Let γ∈(1,∞)be given. For any α∈(0,1),letK(α),γ\nbe defined\nK(α),γ(t,τ) = (1+ τ) sup\nk,l∈Z∗e−α|l|e−α(t−τ)|k−l|\nte−α|k(t−τ)+lτ|\n1+|k−l|γ.\nThen for any γ <∞,there is¯α= ¯α(γ)>0such that if α≤¯αandε∈(0,1),then for any t >0,\ne−εt/integraldisplayt\n0K(α),γ(t,τ)eετdτ≤C/parenleftbigg1\nαεγtγ−1+1\nαεγtγlog1\nα+1\nα2ε1+γt1+γ+/parenleftbigg1\nα3+1\nα2εlog1\nα/parenrightbigg\ne−εt\n4+e−αt\n2\nα3/parenrightbigg\n,\nwhereC=C(γ).\nIn particular, if ε≤α,thene−εt/integraltextt\n0K(α),γ(t,τ)eετdτ≤C(γ)\nα3ε1+γtγ−1.\nProof.Without loss of generality, we shall set d= 1 and first consider τ≤1\n2t.We can write\nK(α)(t,τ)≤(1+τ) sup\nl∈Z,k∈Ze−α|l|e−α|k−l|/2e−α|k(t−τ)+lτ|.\nBy symmetry, we may also assume that k >0.\nExplicit computations yield\n/integraldisplayt\n2\n0e−α|k(t−τ)+lτ|(1+τ)dτ≤\n\n1\nα(l−k)+1\nα2(l−k)2,ifl > k,\ne−αkt(t\n2+t2\n8),ifl=k,\ne−α(k+l)t/2\nα|k−l|(1+t\n2),if−k≤l < k,\n2\nα|k−l|+2kt\nα|k−l|2+1\nα2|k−l|2,ifl <−k.(8.4)\nSo from (8.4), we have\ne−εt/integraldisplayt\n2\n0e−α|k(t−τ)+lτ|(1+τ)eετdτ\n≤e−εt\n4/parenleftbigg3\nα|k−l|+1\nα2|k−l|2+8z\nαε|k−l|/parenrightbigg\n1k/ne}ationslash=l+e−tα\n2/parenleftbiggz\nα+8z2\nα2/parenrightbigg\n1l=k,\nwherez= supxxe−x=e−1.\nUsing the bounds (for α∼0+)\n/summationdisplay\nl∈Ze−αl=O/parenleftbigg1\nα/parenrightbigg\n,/summationdisplay\nl∈Ze−αl\nl=O/parenleftbigg\nlog1\nα/parenrightbigg\n,/summationdisplay\nl∈Ze−αl\nl2=O(1),\nwe end up, for α≤1\n4,with a bound like\ne−εt/integraldisplayt\n2\n0K(α)(t,τ)eετdτ≤C/bracketleftbigg\ne−εt\n4/parenleftbigg1\nα3+1\nα2ε/parenrightbigg\n+e−αt/2\nα3/bracketrightbigg\n.\nNext we turn to the more delicate contribution of τ≥1\n2t.We write\nK(α)(t,τ)≤(1+τ) sup\nl∈Z∗e−α|l|sup\nk∈Ze−α|k(t−τ)+lτ|\n1+|k−l|γ. (8.5)\n35Without loss of generality, we restrict the supremum l >0.The function x→(1+|x−l|γ)−1e−α|x(t−τ)+lτ|\nis decreasing for x≥l,increasing for x≤ −lτ/(t−τ); and on the interval [ −lτ/(t−τ),l],its logarithmic\nderivative goes from/parenleftbigg\n−α+γ/lt\n1+((t−τ)/lt)γ/parenrightbigg\n(t−τ) to−α(t−τ).It is easy to check that a given integer koccurs\nin the supremum only for some times τsatisfying k−1<−lτ/(t−τ)< k+ 1.We can assume k≥0,then\nk−11\n2timplies that\nk≥l.Thus, for t≥γ\nα,we have\ne−εt/integraldisplayt\nt\n2K(α)(t,τ)eετdτ≤e−εt∞/summationdisplay\nl=1e−αl∞/summationdisplay\nk=l/integraldisplay(k+1)t\nk+l+1\n(k−1)t\nk+l−1(1+τ)eα|k(t−τ)−lτ|eετ\n1+|k+l|γdτ. (8.6)\nFort≤γ\nα,it is easy to check that e−εt/integraltextt\nt\n2K(α)(t,τ)eετdτ≤γ\n2αholds. Next we shall focus on (8 .6).According\ntoτsmaller or larger than kt/(k+l),we separate the integral in the right-hand side of (8.6) into two par ts,\nand by simple computation, we get the explicit bounds\ne−εt/integraldisplaykt/(k+l)\n(k−l)t/(k+l−1)(1+τ)e−α|k(t−τ)−lτ|eετdτ≤e−εlt\nk+l/parenleftbigg1\nα(k+l)+kt\nα(k+l)2/parenrightbigg\n,\ne−εt/integraldisplay(k+1)t\nk+l+1\nkt\nk+l(1+τ)e−α|k(t−τ)−lτ|eετdτ≤e−εlt\nk+l+1/parenleftbigg1\nα(k+l)+kt\nα(k+l)2+1\nα2(k+l)2/parenrightbigg\n.\nHence, (8.6) is bounded above by\nC∞/summationdisplay\nl=1e−αl∞/summationdisplay\nk=l/parenleftbigg1\nα2(k+l)2+γ+1\nα(k+l)1+γ+kt\nα(k+l)2+γ/parenrightbigg\ne−εlt\nk+l. (8.7)\nWe consider the first term I(t) of (8.7) and change variables ( x,y)/ma√sto→(x,u),whereu(x,y) =εxt\nx+y,then we\ncan find that\nI(t) =1\nα2ε1+γt1+γ/integraldisplay∞\n1e−αx\nx1+γdx/integraldisplayεt/2\n0e−uuγdu=O/parenleftbigg1\nα2ε1+γt1+γ/parenrightbigg\n.\nThe same computation for the second integral of (8.7) yields\n1\nαεγtγ/integraldisplay∞\n1e−αx\nxγdx/integraldisplayεt/2\n0e−uuγ−1du=O/parenleftbigg1\nαεγtγ/parenrightbigg\n.\nFinally,weestimatethelasttermof(8.7)thatistheworst. Ityieldsa contributiont\nα/summationtext∞\nl=1e−αl/summationtext∞\nk=le−εltk/(k+l)\n(k+l)2+γ.\nWe compare this with the integralt\nα/integraltext∞\n1e−αx/integraltext∞\nxe−εltx/(x+y)\n(x+y)2+γdydx,and the same change of variables as before\nequates this with\n1\nαεγtγ−1/integraldisplay∞\n1e−αx\nxγdx/integraldisplayεt\n2\n0e−uuγ−1du−1\nαε1+γtγ/integraldisplay∞\n1e−αx\nxγdx/integraldisplayεt\n2\n0e−uuγdu=O/parenleftbigg1\nαεγtγ−1/parenrightbigg\n.\nThe proof of Proposition 8.2 follows by collecting all these bounds and keeping only the worst one. To finish\nthe growth control, we have to prove the following result.\nProposition 8.3 With the same notations as in Proposition 8.2, for any γ >1,we have\nsup\nτ≥0eετ/integraldisplay∞\nτe−εtK(α),γ(t,τ)dt≤C(γ)/parenleftbigg1\nα2ε+1\nαεγ/parenrightbigg\n. (8.8)\nProof.We first still reduce to d= 1,and split the integral as\neετ/integraldisplay∞\nτe−εtK(α),γ(t,τ)dt=eετ/integraldisplay∞\n2τe−εtK(α),γ(t,τ)dt+eετ/integraldisplay2τ\nτe−εtK(α),γ(t,τ)dt=I1+I2.\nFor the first term I1,we haveK(α),γ(t,τ)≤(1+τ)/summationtext∞\nk=2/summationtext\nl∈Z∗e−α|l|−α|k−l|/2≤C(1+τ)\nα2,and thus eετ/integraltext∞\nτe−εt\nK(α),γ(t,τ)dt≤C\nεα2.\nWe treat the second term I2as in the proof of Proposition 8.2:\neετ/integraldisplay∞\nτe−εtK(α),γ(t,τ)dt≤eετ(1+τ)∞/summationdisplay\nl=1e−αl∞/summationdisplay\nk=l/integraldisplay(k+l−1)τ\nk−1\n(k+l+1)τ\nk+1e−α|k(t−τ)−lτ|\n1+(k+l)γe−εtdt≤C\nαεγ,\nwhere the last inequality is obtained by the same method in Proposition 8.2 with the change of variable u=εxτ\ny.\n368.2 Growth control\nFrom now on, we will state the main result of this section that is the sa me with section 7.4 in [23]. We define\n/ba∇dblΦ(t)/ba∇dblλ=/summationtext\nk∈Z3∗|Φ(k,t)|e2πλ|k|.\nTheorem 8.4 Assume that f0(v),W=W(x)satisfy the conditions of Theorem 0.1, and the (PSC)condition\nholds. Let A≥0,µ≥0andλ∈(0,λ∗]with0< λ∗< λ0.Let(Φ(k,t))k∈Z3∗,t≥0be a continuous functions of\nt≥0,valued in CZ3\n∗,such that for all t≥0,\n/ba∇dblΦ(t)−/integraldisplayt\n0K0(t−τ)Φ(τ)dτ/ba∇dblλt+µ≤A+/integraldisplayt\n0(K0(t,τ)+K1(t,τ)+c0\n(1+τ)m)/ba∇dblΦ(τ)/ba∇dblλτ+µdτ,(8.9)\nwherec0≥0,m >1,andK0(t,τ),K1(t,τ)are non-negative kernels. Let ϕ(t) =/ba∇dblΦ(t)/ba∇dblλt+µ.Then we have the\nfollowing:\n(i) Assume that γ >1andK1=cK(α),γfor some c >0,α∈(0,¯α(γ)),whereK(α),γ,¯α(γ)are the same with\nthat defined by Proposition 8.2. Then there are positive cons tantsC,χ,depending only on γ,λ∗,λ0,κ,c0,CWand\nm,uniform as γ→1,such that if supt≥0/integraltextt\n0K0(t,τ)dτ≤χandsupt≥0(/integraltextt\n0K0(t,τ)2dτ)1\n2+supt≥0/integraltext∞\ntK0(t,τ)dt≤\n1,then for any ε∈(0,α),for allt≥0,\nϕ(t)≤CA1+c2\n0√εeCc0(1+c\nαε)eCTeCc(1+T2)eεt, (8.10)\nwhereTε=Cmax/braceleftbigg/parenleftbigg\nc2\nα5ε2+γ/parenrightbigg1\nγ−1\n,/parenleftbigg\nc\nα2ε1\n2+γ/parenrightbigg1\nγ−1\n,/parenleftbigg\nc2\n0\nε/parenrightbigg1\n2m−1/bracerightbigg\n.\n(ii) Assume that K1=/summationtextN\nj=1cjK(αj,1)for some αj∈(0,¯α(γ)),where¯α(γ)also appears in proposition 7.2;\nthen there is a numeric constant Γ>0such that whenever 1≥ε≥Γ/summationtextN\nj=1cj\nα3\nj,with the same notation as in\n(I), for all t≥0,one has,\nϕ(t)≤CA1+c2\n0√εeCc0(1+c\nαε)eCTeCc(1+T2)eεt, (8.11)\nwherec=/summationtextN\nj=1cjandT= max/braceleftbigg\n1\nε2/summationtextN\nj=1cj\nα3\nj,/parenleftbigg\nc2\n0\nε/parenrightbigg1\n2m−1/bracerightbigg\n.\nProof.Here we only prove (i), the proof of (ii) is similar. We decompose the pr oof into three steps.\nStep1. Crude pointwise bounds. From (8.9), we have\nϕ(t) =/summationdisplay\nk∈Z3∗|Φ(k,t|e2π(λt+µ)|k|≤A+/summationdisplay\nk∈Z3∗/integraldisplayt\n0|K0(k,t−τ)|e2π(λt+µ)|k||Φ(t,τ)|dτ\n+/integraldisplayt\n0(K0(t,τ)+K1(t,τ)+c0\n(1+τ)m)ϕ(τ)dτ\n≤A+/integraldisplayt\n0(K0(t,τ)+K1(t,τ)+c0\n(1+τ)m+ sup\nk∈Z3∗|K0(k,t−τ)|e2πλ(t−τ)|k|)ϕ(τ)dτ.\nWe note that for any k∈Z3\n∗andt≥0,\n|K0(k,t−τ)|e2πλ|k|(t−τ)≤4π2|/hatwiderW(k)|C0e−2π(λ0−λ)|k|t|k|2t≤CC0CW\nλ0−λ,\nwhere (here and below) Cstands for a numeric constant which may change from line to line. Ass uming that/integraltextt\n0K0(t,τ)dτ≤1\n2,we deduce that\nϕ(t)≤A+1\n2sup\n0≤τ≤tϕ(τ)+C/integraldisplayt\n0/parenleftbiggC0CW\nλ0−λ+c(1+τ)+c0\n(1+τ)m/parenrightbigg\nϕ(τ)dτ,\nand, by Gr¨ onwall’s lemma,\nϕ(t)≤2AeC(C0CWt/(λ0−λ)+c(t+t2)+c0Cm), (8.12)\nwhereCm=/integraltext∞\n0(1+τ)−mdτ.\nStep2.L2bound. For all k∈Z3\n∗andt≥0,we define Ψ k(t) =e−εtΦ(k,t)e2π(λt+µ)|k|,K0\nk(t) =e−εtK0(k,t)\ne2π(λt+µ)|k|, Rk(t) =e−εt/parenleftbigg\nΦ(k,t)−/integraltextt\n0K0(k,t−τ)Φ(k,τ)dτ/parenrightbigg\ne2π(λt+µ)|k|= (Ψk−Ψk∗K0\nk)(t),and we extend\n37all these functions by 0 for negative values of t.Taking Fourier transform in the time-variable yields ˆRk=\n(1−/hatwideK0\nk)/hatwiderΨk.Since the ( PSC) condition implies that |1−/hatwideK0\nk| ≥κ,we can deduce that /ba∇dblˆΨk/ba∇dblL2≤κ−1/ba∇dblˆRk/ba∇dblL2,\ni.e.,/ba∇dblΨk/ba∇dblL2≤κ−1/ba∇dblRk/ba∇dblL2.So we have\n/ba∇dblΨk−Rk/ba∇dblL2(dt)≤κ−1/ba∇dblK0\nk/ba∇dblL1(dt)/ba∇dblRk/ba∇dblL2(dt)for all k∈Z3\n∗. (8.13)\nThen\n/ba∇dblϕ(t)e−εt/ba∇dblL2(dt)=/ba∇dbl/summationdisplay\nk∈Z3|Ψk|/ba∇dblL2(dt)≤ /ba∇dbl/summationdisplay\nk∈Z3|Rk|/ba∇dblL2(dt)+/summationdisplay\nk∈Z3/ba∇dblRk−Ψk/ba∇dblL2(dt)\n≤ /ba∇dbl/summationdisplay\nk∈Z3|Rk|/ba∇dblL2(dt)(1+1\nκ). (8.14)\nNext, we note that\n/ba∇dblK0\nk/ba∇dblL1(dt)≤4π2|/hatwiderW(k)|/integraldisplay∞\n0C0e−2π(λ0−λ)|k|t|k|2tdt≤4π|/hatwiderW(k)|C0\n(λ0−λ)2,\nso/summationtext\nk∈Z3∗/ba∇dblK0\nk/ba∇dblL1(dt)≤4π(/summationtext\nk∈Z3∗|/hatwiderW(k)|)C0\n(λ0−λ)2.Furthermore, we get\n/ba∇dblϕ(t)e−εt/ba∇dblL2(dt)≤/parenleftbigg\n1+CC0CW\nκ(λ0−λ)2/parenrightbigg\n/ba∇dbl/summationdisplay\nk∈Z3∗/ba∇dblL2(dt)\n≤/parenleftbigg\n1+CC0CW\nκ(λ0−λ)2/parenrightbigg/parenleftbigg/integraldisplay∞\n0e−2εt/parenleftbigg\nA+/integraldisplayt\n0/parenleftbigg\nK1+K0+c0\n(1+τ)m/parenrightbigg\nϕ(τ)dτ/parenrightbigg2/parenrightbigg1\n2\n. (8.15)\nBy Minkowski’sinequality, we separate (8.15) into variouscontributio ns which we estimate separately. First,/parenleftbigg/integraltext∞\n0e−2εtA2dt/parenrightbigg1\n2\n=A√\n2ε.Next, for any T≥1,by Step 1 and/integraltextt\n0K1(t,τ)dτ≤Cc(1+t)\nα,we have\n/parenleftbigg/integraldisplayT\n0e−2εt/parenleftbigg/integraldisplayt\n0K1(t,τ)ϕ(τ)/parenrightbigg2/parenrightbigg1\n2\n≤( sup\n0≤t≤Tϕ(t))/parenleftbigg/integraldisplayT\n0e−2εt/parenleftbigg/integraldisplayt\n0K1(t,τ)/parenrightbigg2/parenrightbigg1\n2\n≤CAeC(C0CWT/(λ0−λ)+c(T+T2))c\nα/parenleftbigg/integraldisplay∞\n0e−2εt(1+t)2dt/parenrightbigg1\n2\n≤CAc\naε3\n2eC(C0CWT/(λ0−λ)+c(T+T2)). (8.16)\nInvoking Jensen’s inequality and Fubini’s theorem, we also have\n/integraldisplay∞\nTe−2εt/parenleftbigg/integraldisplayt\n0K1(t,τ)ϕ(τ)dτ/parenrightbigg2\ndt/parenrightbigg1\n2\n=/integraldisplay∞\nT/parenleftbigg/integraldisplayt\n0K1(t,τ)e−2ε(t−τ)e−2ετϕ(τ)dτ/parenrightbigg2\ndt/parenrightbigg1\n2\n≤/integraldisplay∞\nT/parenleftbigg/integraldisplayt\n0K1(t,τ)e−ε(t−τ)dτ/parenrightbigg/parenleftbigg/integraldisplay∞\nT/parenleftbigg/integraldisplayt\n0K1(t,τ)e−ε(t−τ)e−2ετϕ(τ)2dτ/parenrightbigg\ndt/parenrightbigg1\n2\n≤/parenleftbigg\nsup\nt≥T/integraldisplayt\n0K1(t,τ)e−ε(t−τ)dτ/parenrightbigg1\n2/parenleftbigg/integraldisplay∞\nT/parenleftbigg/integraldisplayt\n0K1(t,τ)e−ε(t−τ)e−2ετϕ(τ)2dτ/parenrightbigg\ndt/parenrightbigg1\n2\n=/parenleftbigg\nsup\nt≥T/integraldisplayt\n0K1(t,τ)e−ε(t−τ)dτ/parenrightbigg1\n2/parenleftbigg/integraldisplay∞\n0/integraldisplay∞\nmax{τ,T}K1(t,τ)e−ε(t−τ)e−2ετϕ(τ)2dtdτ/parenrightbigg1\n2\n≤/parenleftbigg\nsup\nt≥T/integraldisplayt\n0K1(t,τ)e−ε(t−τ)dτ/parenrightbigg1\n2/parenleftbigg\nsup\nτ≥0/integraldisplay∞\nτK1(t,τ)e−ε(t−τ)e−2ετdt/parenrightbigg1\n2/parenleftbigg/integraldisplay∞\n0ϕ(τ)2e−2ετdτ/parenrightbigg1\n2\n.\n(8.17)\nSimilarly,\n/integraldisplay∞\nTe−2εt/parenleftbigg/integraldisplayt\n0K0(t,τ)ϕ(τ)dτ/parenrightbigg2\ndt/parenrightbigg1\n2\n≤/parenleftbigg\nsup\nt≥T/integraldisplayt\n0K0(t,τ)dτ/parenrightbigg1\n2/parenleftbigg\nsup\nτ≥0/integraldisplay∞\nτK0(t,τ)dt/parenrightbigg1\n2/parenleftbigg/integraldisplay∞\n0ϕ(τ)2dτ/parenrightbigg1\n2\n.\n(8.18)\n38The last term is also split, this time according to τ≤Torτ > T:\n/parenleftbigg/integraldisplay∞\n0e−2εt/parenleftbigg/integraldisplayT\n0c0ϕ(τ)\n(1+τ)mdτ/parenrightbigg2\ndt/parenrightbigg1\n2\n≤c0( sup\n0≤τ≤Tϕ(τ))/parenleftbigg/integraldisplay∞\n0e−2εt/parenleftbigg/integraldisplayT\n0dτ\n(1+τ)m/parenrightbigg2\ndt/parenrightbigg1\n2\n≤c0CA√εeC(C0CWT/(λ0−λ)+c(T+T2))Cm, (8.19)\nand\n/parenleftbigg/integraldisplay∞\n0e−2εt/parenleftbigg/integraldisplayt\nTc0ϕ(τ)\n(1+τ)mdτ/parenrightbigg2\ndt/parenrightbigg1\n2\n≤c0/parenleftbigg/integraldisplay∞\n0e−2εtϕ(t)2/parenrightbigg1\n2/parenleftbigg/integraldisplay∞\n0/integraldisplayt\nTe−2ε(t−τ)\n(1+τ)2mdτdt/parenrightbigg1\n2\n=c0/parenleftbigg/integraldisplay∞\n0e−2εtϕ(t)2/parenrightbigg1\n2/parenleftbigg/integraldisplay∞\nTdτ\n(1+τ)2m/parenrightbigg1\n2/parenleftbigg/integraldisplay∞\n0e−2εsds/parenrightbigg1\n2\n=C1\n2\n2mc0\nTm−1\n2√ε/parenleftbigg/integraldisplay∞\n0e−2εtϕ(t)2/parenrightbigg1\n2\n.(8.20)\nGathering estimates (8.16)-(8.20), we deduce from (8.15) that\n/ba∇dblϕ(t)e−εt/ba∇dblL2(dt)≤/parenleftbigg\n1+CC0CW\nκ(λ0−λ)2/parenrightbiggCA√ε/parenleftbigg\n1+c\naε+c0Cm/parenrightbigg\neC(C0CWT/(λ0−λ)+c(T+T2))\n+a/ba∇dblϕ(t)e−εt/ba∇dblL2(dt), (8.21)\nwhere\na=/parenleftbigg\n1+CC0CW\nκ(λ0−λ)2/parenrightbigg/bracketleftbigg/parenleftbigg\nsup\nt≥T/integraldisplayt\n0e−εtK1(t,τ)eετdτ/parenrightbigg1\n2/parenleftbigg\nsup\nτ≥0/integraldisplay∞\nτeετK1(t,τ)e−εtdt/parenrightbigg1\n2\n+/parenleftbigg\nsup\nt≥T/integraldisplayt\n0K0(t,τ)dτ/parenrightbigg1\n2/parenleftbigg\nsup\nτ≥0/integraldisplay∞\nτK0(t,τ)dt/parenrightbigg1\n2\n+C1\n2\n2mc0\nTm−1\n2√ε/bracketrightbigg\n.\nUsing Propositions 8.2 and 8.3, together with the assumptions of The orem 8.4, we see that a≤1\n2forχ\nsufficiently small. Then we have\n/ba∇dblϕ(t)e−εt/ba∇dblL2(dt)≤/parenleftbigg\n1+CC0CW\nκ(λ0−λ)2/parenrightbiggCA√ε/parenleftbigg\n1+c\naε+c0Cm/parenrightbigg\neC(C0CWT/(λ0−λ)+c(T+T2)).\nStep3. Fort≥T,using (8.9), we get\ne−εtϕ(t)≤Ae−εt+/bracketleftbigg/parenleftbigg/integraldisplayt\n0/parenleftbigg\nsup\nk∈Z3∗|K0(k,t−τ)|e2πλ(t−τ)|k|/parenrightbigg2\ndτ/parenrightbigg1\n2\n+/parenleftbigg/integraldisplayt\n0K0(t,τ)2dτ/parenrightbigg1\n2\n+/parenleftbigg/integraldisplay∞\n0c2\n0\n(1+τ)2mdτ/parenrightbigg1\n2\n+/parenleftbigg/integraldisplayt\n0e−2εtK1(t,τ)2e2ετdτ/parenrightbigg1\n2/bracketrightbigg/parenleftbigg/integraldisplay∞\n0ϕ(τ)e−ετdτ/parenrightbigg1\n2\n.\n(8.22)\nWe note that, for any k∈Z3\n∗,(|K0(k,t)|e2πλ|k|t)2≤Cπ4|/hatwiderW(k)|2|ˆf0(kt)|2|k|4t2≤CC0\n(λ0−λ)2C2\nWe−2π(λ0−λ)t,so\nwe get/integraltextt\n0/parenleftbigg\nsupk∈Z3\n∗|K0(k,t−τ)|e2πλ(t−τ)|k|/parenrightbigg2\ndτ≤CC2\n0C2\nW\n(λ0−λ)3.\nFrom Proposition 8.2, (8.22), the conditions of Theorem 8.4 and Step 2, the conclusion is finished.\nCorollary 8.5 Assume that f0=f0(v),under the assumptions of Theorem 0.1, we pick up λ∗\nn−B0< λ′\nn−B0\nsuch that 2π(λ′\nn−λ∗\nn)≤αn,chooseε= 2π(λ′\nn−λ∗\nn);recalling that ˆρ(t,0) = 0,our conditions imply an upper\nbound on cnandcn\n0,we have the uniform control,\n/ba∇dblρ[hn+1](t,x)/ba∇dblF(λ∗n−B0)t+µ′n≤Cδ2\nn(1+cn\n0)2\n√ε(λn−λ′n)2/parenleftbigg\n1+1\nαn(λ′n−λ∗n)3\n2/parenrightbigg\neCT2\nn,\nwhereTn=C/parenleftbigg\n1\nα5n(λ′n−λ∗n)/parenrightbigg1\nγ−1\n.\nProof.From Propositions 8.1-8.3, we know that\n/integraldisplayt\n0Kn\n0(t,τ)dτ≤CWn/summationdisplay\ni=1δi\nπ(λi−λ′n),/integraldisplay∞\nτKn\n0(t,τ)dτ≤CWn/summationdisplay\ni=1δi\nπ(λi−λ′n),\n/parenleftbigg/integraldisplayt\n0Kn\n0(t,τ)2dτ/parenrightbigg1\n2\n≤CWn/summationdisplay\ni=1δi/radicalbig\n2π(λi−λ′n).\n39Hereαn=πmin{(µn−µ′\nn),(λn−λ′\nn)},and assume that αnis smaller than ¯ α(γ) in Theorem 8.4, and that\n/parenleftbigg\nC4\nω/parenleftbigg\nC′\n0+n/summationdisplay\ni=1δi+1/parenrightbigg/parenleftbiggn/summationdisplay\nj=1δj\n2π(λj−λ′\nj)6/parenrightbigg\n≤1\n8,(VII)\nCWn/summationdisplay\ni=1δi/radicalbig\n2π(λi−λ′n)≤1\n4,n/summationdisplay\ni=1δi\nπ(λi−λ′n)≤max{χ,1\n8}.(VIII)\nApplying Theorem 8.4, we can deduce that for any ε∈(0,αn) andt≥0,\n/ba∇dblρ[hn+1](t,x)/ba∇dblF(λ∗n−B0)t+µ′n≤e−2π(λ′\nn−λ∗\nn)t/ba∇dblρ[hn+1](t,x)/ba∇dblF(λ′n−B0)t+µ′n\n≤Cδ2\nn(1+cn\n0)2\n√ε(λn−λ′n)2/parenleftbigg\n1+1\nαn(λ′n−λ∗n)3\n2/parenrightbigg\neCT2\nn,\nwhereTn=C/parenleftbigg\n1\nα5n(λ′n−λ∗n)/parenrightbigg1\nγ−1\n.\n8.3 Estimates related to hn+1(t,Xn\nτ,t(x,v),Vn\nτ,t(x,v)))\nNext we show the control on hi.\nLemma 8.6 For any n≥i≥1,\n/ba∇dbl(∇′\nv×((hi\nτv)◦Ωi−1\nt,τ)−/a\\}b∇acketle{t∇′\nv×((hi\nτv)◦Ωi−1\nt,τ)/a\\}b∇acket∇i}ht)/ba∇dbl\nZ(λ′\ni−B0)(1+b),µ′\ni;1\nτ−bt\n1+b≤(1+τ)δi.\nProof.First, we consider i= 1.\nIn fact,\n/ba∇dbl∇′\nv×(h1\nτv)−/a\\}b∇acketle{t∇′\nv×((h1\nτv)/a\\}b∇acket∇i}ht/ba∇dbl\nZ(λ′\n1−B0)(1+b),µ′\n1;1\nτ−bt\n1+b\n≤ /ba∇dbl∇′\nv×(h1\nτv)/ba∇dbl\nZ(λ′\n1−B0)(1+b),µ′\n1;1\nτ−bt\n1+b+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nT3∇′\nv×((h1\nτv)dx/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC(λ′\n1−B0)(1+b);1\n≤ /ba∇dbl∇′\nv(h1\nτv)/ba∇dbl\nZ(λ′\n1−B0)(1+b),µ′\n1;1\nτ−bt\n1+b≤ /ba∇dblh1\nτ/ba∇dblZ(λ1−B0)(1+b),µ1;1\nτ−bt\n1+b≤(1+τ)δ1,\nwhere we use the property ( v) of Proposition 2.5.\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nT3∇′\nv×(h1\nτv)dx/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC(λ′\n1−B0)(1+b);1=/ba∇dbl/a\\}b∇acketle{t∇′\nv×(h1\nτv)/a\\}b∇acket∇i}ht/ba∇dblC(λ′\n1−B0)(1+b);1\n=/ba∇dbl/a\\}b∇acketle{t(∇′\nv+τ∇x)×(h1\nτv)/a\\}b∇acket∇i}ht/ba∇dblC(λ′\n1−B0)(1+b);1≤ /ba∇dbl(∇′\nv+τ∇x)×(h1\nτv)/ba∇dbl\nZ(λ′\n1−B0)(1+b),µ′\n1;1\nτ−bt\n1+b≤δ1,\nwhere we use ( vi) of Proposition 2.5.\nSuppose that i=k,the conclusion holds, that is,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇′\nv×((hk\nτv)◦Ωk−1\nt,τ)−/a\\}b∇acketle{t∇′\nv×((hk\nτv)◦Ωk−1\nt,τ)/a\\}b∇acket∇i}ht/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nZ(λ′\nk−B0)(1+b),µ′\nk;1\nτ−bt\n1+b\n≤/vextenddouble/vextenddouble/vextenddouble/vextenddoublehk\nτ◦Ωk−1\nt,τ−/a\\}b∇acketle{thk\nτ◦Ωk−1\nt,τ/a\\}b∇acket∇i}ht/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nZ(λk−B0)(1+b),µk;1\nτ−bt\n1+b≤(1+τ)δk.\nWeneedtoshowthattheconclusionstillholdsfor i=k+1.Wecangettheestimatefor hk+1(t,Xk\nt,τ(x,v),Vk\nt,τ(x,v))\nfrom (4.11).\nNote that\n/braceleftbigg(∇h)◦Ω = (∇Ω)−1∇(h◦Ω),\n(∇2h)◦Ω = (∇Ω)−2∇2(h◦Ω)−(∇Ω)−1∇2Ω(∇Ω)−1(∇h◦Ω).(8.23)\nTherefore, from (8.23), we get\n/ba∇dbl(∇hn+1\nτ)◦Ωn\nt,τ/ba∇dbl\nZ(λ′\nn+1−B0)(1+b),µ′\nn+1;1\nτ−bt\n1+b≤C(d)/ba∇dbl∇(hn+1\nτ◦Ωn\nt,τ)/ba∇dbl\nZ(λ′\nn+1−B0)(1+b),µ′\nn+1;1\nτ−bt\n1+b\n≤C(d)(1+τ)\nmin{λb\nn+1−λ′\nn+1,µn+1−µ′\nn+1}/ba∇dblhn+1\nτ◦Ωn\nt,τ/ba∇dbl\nZ(λb\nn+1−B0)(1+b),µn+1;1\nτ−bt\n1+b, (8.24)\n40and\n/ba∇dbl(∇2hn+1\nτ)◦Ωn\nt,τ/ba∇dbl\nZ(λ′\nn+1−B0)(1+b),µ′\nn+1;1\nτ−bt\n1+b\n≤C(d)/bracketleftbigg\n/ba∇dbl∇2(hn+1\nτ◦Ωn\nt,τ)/ba∇dbl\nZ(λ′\nn+1−B0)(1+b),µ′\nn+1;1\nτ−bt\n1+b\n+/ba∇dbl∇2Ωn\nt,τ/ba∇dbl\nZ(λ′\nn+1−B0)(1+b),µ′\nn+1\nτ−bt\n1+b/ba∇dbl(∇hn+1\nτ)◦Ωn\nt,τ/ba∇dbl\nZ(λ′\nn+1−B0)(1+b),µ′\nn+1;1\nτ−bt\n1+b/bracketrightbigg\n≤C(d)(1+τ)\nmin{λb\nn+1−λ′\nn+1,µn+1−µ′\nn+1}/ba∇dblhn+1\nτ◦Ωn\nt,τ/ba∇dbl\nZ(λb\nn+1−B0)(1+b),µn+1;1\nτ−bt\n1+b\n≤C(d)(1+τ)\nmin{λb\nn+1−λ′\nn+1,µn+1−µ′\nn+1}/ba∇dblhn+1\nτ◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1−b),µ′n;1\nτ+bt\n1−b. (8.25)\nWe first write\n∇(hn+1\nτ◦Ωn\nt,τ)−(∇hn+1\nτ)◦Ωn\nt,τ=∇(Ωn\nt,τ−Id)·[(∇hn+1\nτ)◦Ωn\nt,τ],\nand we get\n/ba∇dbl∇(hn+1\nτ◦Ωn\nt,τ)−(∇hn+1\nτ)◦Ωn\nt,τ/ba∇dbl\nZ(λ′†\nn−B0)(1+b),µ′†\nn;1\nτ−bt\n1+b\n≤ /ba∇dbl∇(Ωn\nt,τ−Id)/ba∇dbl\nZ(λ′†\nn−B0)(1+b),µ′†\nn\nτ−bt\n1+b/ba∇dbl(∇hn+1\nτ)◦Ωn\nt,τ/ba∇dbl\nZ(λ′†\nn−B0)(1+b),µ′†\nn;1\nτ−bt\n1+b\n≤C/parenleftbigg1+τ\nmin{λ′n−λ′†\nn,µ′n−µ′†\nn}/parenrightbigg2\n/ba∇dblΩn\nt,τ−Id/ba∇dblZ(λ′n−B0)(1+b),µ′n\nτ−bt\n1+b/ba∇dblhn+1\nτ◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b\n≤CC4\nω\nmin{λ′n−λ′†\nn,µ′n−µ′†\nn}2/parenleftbiggn/summationdisplay\nk=1δk\n(2π(λk−λ′\nk))6/parenrightbigg\n(1+τ)−2/ba∇dblhn+1\nτ◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b,\nthe above inequality implies ∇(hn+1\nτ◦Ωn\nt,τ)≃(∇hn+1\nτ)◦Ωn\nt,τasτ→ ∞.\nSince\n/ba∇dbl∇(hn+1\nτ◦Ωn\nt,τ)/ba∇dbl\nZ(λ′†\nn−B0)(1+b),µ′†\nn;1\nτ−bt\n1+b≤C/parenleftbigg1+τ\nmin{λ′n−λ′†\nn,µ′n−µ′†\nn}/parenrightbigg2\n/ba∇dblhn+1\nτ◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b,\nand\n/ba∇dbl∇x(hn+1\nτ◦Ωn\nt,τ)/ba∇dbl\nZ(λ′†\nn−B0)(1+b),µ′†\nn;1\nτ−bt\n1+b+/ba∇dbl(∇x+τ∇v)(hn+1\nτ◦Ωn\nt,τ)/ba∇dbl\nZ(λ′†\nn−B0)(1+b),µ′†\nn;1\nτ−bt\n1+b\n≤C\nmin{λ′n−λ′†\nn,µ′n−µ′†\nn}/ba∇dblhn+1\nτ◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b,\nwe have\n/ba∇dbl(∇xhn+1\nτ)◦Ωn\nt,τ/ba∇dbl\nZ(λ′†\nn−B0)(1+b),µ′†\nn;1\nτ−bt\n1+b+/ba∇dbl((∇x+τ∇v)(hn+1\nτ)◦Ωn\nt,τ/ba∇dbl\nZ(λ′†\nn−B0)(1+b),µ′†\nn;1\nτ−bt\n1+b\n≤C/parenleftbiggC4\nω\nmin{λ′n−λ′†\nn,µ′n−µ′†\nn}2/parenleftbiggn/summationdisplay\nk=1δk\n(2π(λk−λ′\nk))6/parenrightbigg\n+1\nmin{λ′n−λ′†\nn,µ′n−µ′†\nn}/parenrightbigg\n/ba∇dblhn+1\nτ◦Ωn\nt,τ/ba∇dblZ(λ′n−B0)(1+b),µ′n;1\nτ−bt\n1+b.\n8.4 The Choice of δn+1\nNowweseethatthe corresponding n+1thstepconclusionoftheinductivehypothesishasallbeen estab lished\nwith\nδn+1=CF(1+CF)(1+C4\nω)eCT2\nn\nmin{λ∗n−λn+1,µ∗n−µ9\nn+1}max/braceleftbigg\n1,n/summationdisplay\ni=1δk/bracerightbigg/parenleftbigg\n1+n/summationdisplay\ni=1δi\n(2π(λi−λ∗\ni))6/parenrightbigg\nδ2\nn.\nFor anyn≥1,we setλn−λ∗\nn=λ∗\nn−λn+1=µn−µ∗\nn=µ∗\nn−µn+1=Λ\nn2for some Λ >0.By choosing Λ\nsmall enough, we can make sure that the conditions 2 π(λk−λ∗\nk)<1 and 2 π(µk−µ∗\nk)<1 are satisfied for\nallk,as well as the other smallness assumptions made throughout this se ction. We also have λk−λ∗\nk≥Λ\nk2.\n(I)−(VIII) will be satisfied if we choose constants Λ ,ω >0 such that/summationtext∞\ni=1i12δi≤Λ6ω.\n41Then we have that Tn≤Cγ(n2/Λ)7+γ\nγ−1,so the induction relation on δngivesδ1≤Cδandδn+1=\nC(n2\nΛ)9eC(n2/Λ)(14+2γ)/(γ−1)δ2\nn.\nTo make this relation hold, we alsoassume that δnis bounded below by the errorcoming from the short-time\niteration; but this follows easily by construction, since the constra ints imposed on δnare much worse than those\non the short-time iteration.\nHaving fixed Λ ,we will check that for δsmall enough, the above relation for δn+1holds and there is a fast\nconvergence of {δi}∞\ni=1.The details are similar to that of the local-time case,and it can be also fo und in [23],\nhere we omit it.\nAcknowledgements: I would like to thank professor Pin Yu in YMSC, Ts inghua University who gave me lots\nof advice. 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Anal.PDE\n10,no.6, 2017.\n[31] D.Wei, Z.Zhang, W.Zhao, Linear inviscid damping for a class of monot one shear flow in Sobolev spaces,\nComm.Pure Appl.Math.71(2018), no.4, 617-687.\n43" }, { "title": "0904.3150v2.Tensor_damping_in_metallic_magnetic_multilayers.pdf", "content": "Tensor damping in metalli c magnetic multilayers \n \nNeil Smith \nSan Jose Research Center, Hitachi Globa l Storage Technologies, San Jose, CA 95135 \n \nThe mechanism of spin-pumping, described by Tserkovnyak et al. , is formally analyzed in the general \ncase of a magnetic multilayer consisting of two or more metallic ferromagnetic (FM) films separated \nby normal metal (NM) layers. It is shown that the spin-pumping-induced dynamic coupling between \nFM layers modifies the linearized Gilbert equations in a way that replaces the scalar Gilbert damping \nconstant with a nonlocal matrix of Cartesian dampi ng tensors. The latter are shown to be methodically \ncalculable from a matrix algebra solution of the Valet- Fert transport equations. As an example, explicit \nanalytical results are obtained for a 5-layer (spi n-valve) of form NM/FM/NM'/FM/NM. Comparisons \nwith earlier well known results of Tserkovnyak et al. for the related 3-layer FM/NM/FM indicate that \nthe latter inadvert ently hid the tensor character of the damping, and instea d singled out the diagonal \nelement of the local damping tens or along the axis normal to the plane of the two magnetization \nvectors. For spin-valve devices of technological interest, the influen ce of the tensor components of the \ndamping on thermal noise or spin -torque critical currents are st rongly weighted by the relative \nmagnitude of the elements of the nonlocal, anisotropic stiffness-fiel d tensor-matrix, and for in-plane \nmagnetized spin-valves are generally more sensitive to the in-plane element of the damping tensor. I. INTRODUCTION \n For purely scientific r easons, as well as technological applica tions such as magnetic field sensors \nor dc current tunable microwave oscillator s, there is significant present interest1 in the magnetization \ndynamics in current-perpendicular-to-plane (CPP) metallic multilayer devices comprising multiple \nferromagnetic (FM) films separated by normal meta l (NM) spacer layers. The phenomenon of spin-\npumping, described earlier by Tserkovnyak et al.2,3 introduces an additional source of dynamic \ncoupling, either between the magnetization of a single FM layer and its NM elect ronic environment, or \nbetween two or more FM layers as mediated through their NM spacers. In the former case,2 the effect \ncan resemble an enhanced magnetic damping of an individual FM layer, whic h has important practical \napplication for substantially increasing the spin-t orque critical currents of CPP spin-valves employed \nas giant-magnetoresistiv e (GMR) sensors for read head applications.4 Considered in this paper is a \ngeneral treatment in the case of two or more FM layers in a CPP stack, where it will be shown in Sec. \nII that spin-pumping modifies the linearized equations of motion in a way that replaces a scalar \ndamping constant with a nonlocal matrix of Cartesian damping tensors.5 Analytical results for the case \nof a 5-layer spin-valve stack of the form NM/FM/NM'/FM/NM are discussed in de tail in Sec. III, and \nare in Sec. IV compared and contrasted with the early well-know n results of Tserkovnyak et al..3, as \nwell as some very recent results of that author and colleagues.6 In the case of CPP-GMR devices of \ntechnological interest, the relativ e importance of the different elements of the damping tensor on \ninfluencing measureable thermal fluctuations or spin-t orque critical currents is shown to be strongly \nweighted by the anisotropic nature of the stiffness-field tensor-matrix. \n \nII. SPIN-PUMPING AN D TENSOR DAMPING \n As discussed by Tserkovnyak et al,2,3 the spin current pumpI flowing into the normal metal (NM) \nlayer at an FM/NM interface (Fig. 1) due to the spin-pumping effect is described the expression \n \n⎥⎦⎤\n⎢⎣⎡− ×π=↑↓ ↑↓\ndtdgdtdgm mm IˆIm )ˆˆ( Re4pumph (1) \n \nwhere is a dimensionless mixing conductance, and m is the unit magnetization vector. In this \npaper, for any ferromagnetic (FM) layer is treated as a uniform macrospin. A restatement of (1) in \nterms more natural to Valet-Fert↑↓g ˆ\nmˆ\n7form of transport equations is di scussed in Appendix A. With the \nnotational conversion , where A is the cross sectional area of the film stack, \nequation (1), for the case , simplifies to pump pump) 2 / (J I A eh− →\n↑↓ ↑↓> > g gIm Re \ninterface NM/FM for \" \" interface, FM/NM for \" \"ReIm,ˆ ˆˆ) 2 / (\n22\npump\n+ −≡ ε⎟\n⎠⎞⎜\n⎝⎛ε + ×π≅↑↓↑↓\n↑↓rr\ndtd\ndtd\nre h e m mm Jm (2) \n \nwhere is the inverse mixing conductance (with dimensions of resistance-area), and \nis the well known inverse conductance quantum ↑↓ ↑↓≅ r rRe22 /e h\n) k 9 . 12 (Ω≅ . In the present notation, all spin current \ndensities have the same dimensions as electron charge current density , and for conceptual \nsimplicity are defined with a parallel (i.e., ) rather than anti-parallel alignment with \nmagnetization . Positive J is defined as electrons flowing to the right (along in Fig. 1.). spinJeJ\nm Jˆ ˆspin+ =\nmˆ yˆ+\n For a FM layer sandwiched by tw o NM layers in which the FM layer is the layer of a \nmultilayer thj ) 0 (≥j\nfilm stack, spin-pumping contributions at the interface, i.e., either left or right thi ) (j i= ) 1 (+=j i \nFM-NM interfaces, (2) can be expressed as \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ε + ×−=\n↑↓−\n+ =dtd\ndtd\nr ej\nij\nj\nij i\nj j im m\nm Jˆ ˆ\nˆ) 1 (\n2pump\n1 ,h (3) \n \n The physical picture to now be invoked is that of small (thermal) fluctuations of m about \nequilibrium giving rise to the terms in (2). Since ˆ\n0ˆm dt d/ˆm 1ˆ≡m , the three vector components of \n and/or are not linearly independent. To remove this interdependency, as well as higher order mˆ dt d/ˆmFig. 1 Cross section cartoon of an N-layer multilayer stack with N-1 interior nterfaces of FM-NM or NM-NM type, \nsuch as found in CPP-GMR pillars sandwiched between conductive leads of much larger cross section. In the \nexample shown, the jth layer is FM, sandwiched by NM layers, with spin pumping contributions at the ith (NM/FM) \nand ( i+1)st (FM/NM) interfaces located at iy y= and 1+iy (with i=j for the labeling scheme shown). j=0j-1j=1\ni=0 i=1 i-1j\ni+1j+1-Jpump\ni Jpump\ni+1\ni=jNM NM FMmj\nj=N-1z\ny\nlead lead\ni=N N-1terms in (3) it thus is useful to work in a primed coordinate system where , through use of a \n Cartesian rotation matrix such that 0ˆ ˆm z=′\n3 3× )ˆ(0mℜ m mˆ ˆ′⋅ℜ= .8 To first order in linearly independent \nquantities , ) , (y xm m′ ′ m m m′⋅ ℜ + =~ˆ ˆ0 , where , and where ℜ ⎟⎠⎞⎜⎝⎛≡′\n′′′′\nymxmm~ denotes the matrix from \nthe first two (i.e., x and y ) columns of 2 3×\nℜ. Replacing z m′→ˆ ˆ0 , and _ˆ×′z with matrix multiplication, \nthe linearized form of (3) becomes \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n′′\n⋅⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nε− ε⋅ ℜ−=\n′′−\n+ = ↑↓ dt m ddt m d\nr e2h\ny jx j\nii\nj\nij i\nj j i//\n11 ~ ) 1 ( pump\n1 ,J (4) \n \n Using the present sign convention, j i s j A t Mm S ˆ/ ) (γ = is the spin angular momentum of the \nFM layer with saturation magnetization-thickness product , and is the gyromagnetic ratio. \nTaking thj\nj st M) ( 0> γ\nsM=M as constant, it follows by angu lar momentum conservation that3 \n \n∑+\n=−× × − = ⇔γ1\nˆ ˆ ) 1 (21 ˆ ) (NMj\nj ij i jj i j j j s\ne dtd\nA dtd t M\nm J mS mh (5) \n \nis the contribution to due to the net transverse spin current entering the FM layer (Fig. 1). \nIn (5), denotes the spin-current density in the NM layer at the FM-NM interface. Taking the \ncross product on both sides of (5), transforming to pr imed coordinates by matrix-multiplying by \n, and employing similar linearization as to obtain (4), one finds to first order that dt dj/ˆmthj\nNM\niJthi\n×mˆ\nTℜ = ℜ−1\n \n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n− ≡ Δ ℜ ⋅⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−=′\n×′ ∑+\n=−⋅1spin NM) 1 (~\n0 11 0\n21ˆj\nj iij i\nj jj\nje dtd\nAJ JS\nzT h (6) \n \nwhere Tℜ~ is the matrix transpose of ℜ 3 2×~. By definition, 0 ˆ~\n0= ⋅ ℜj jmT. \n The quantities in (6) are not known a priori , but must be determined after solution of the \nappropriate transport equations (e.g. , Appendix B). Even in the absence of charge current flow (i.e., \n as considered here, the are nonzero due to the set of in (4) which appear as \nsource terms in the boundary conditions (A 9) at each FM-NM interface. Given the linear relation of \n(4), one can now apply linear superposition to express spin\njJΔ\n) 0=eJspin\njJΔpump\niJ \n∑ ∑+\n=↑↓ ↑↓ ↑↓≡′\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nε− ε⋅ ℜ ⋅ = Δ1spin 1\n21 1,11 ~ 1\n2k\nk ii k kk\nk jk\nkjr r dtdC\nremJt h (7) \n \nin terms of the set of 3-D dimensionless Cartesian tensor jkCt\n. The jkCt\n are convenient for formal \nexpressions such as (9), or for analytical work in algebraically simple cases, such as exampled in \nSec.III. However, they are also subject to met hodical computation. For the kth magnetic layer, the \ncolumns of each are the dimensionless vectors simultaneously obtainable \nfor all magnetic layers j from a matrix solutionrd nd st3 or , 2 , 1jkCtspin\njJΔ\n9 of the Valet-Fert7 transport equatio ns with nonzero \ndimensionless spin-pump vectors )ˆor ,ˆ,ˆ)( / ( ) 1 (pump\n1 ,z y x J↑↓ ↑↓ −\n+ =− =i kk i\nk k ir r . \n To include spin currents via (5) into the magnetization dynamics, the conventional Gilbert \nequations of motion for can be amended as ) (ˆtm\n \ndtd\nA t M dtd\ndtdj\nj sj\nj j j ij S m\nm H mm 1\n) (ˆ\nˆ ) ˆ(ˆG eff γ+ × α + × γ − = (8) \n \nwhere is the usual (scalar) Gilbert damping paramete r. From (6) and (7), one can deduce that the \nrightmost term in (8) will scale linearly with G\njα\ndt d/m′, as does the conventiona l Gilbert damping term. \nCombining these terms together after applying the analogous linearization procedure to (8) as was \ndone in going from (5) to (6), one obtains \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nε− ε⋅ ℜ ⋅ ⋅ ℜ ⋅⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n− πγ= α′α′+ δ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nαα≡ α′′⋅ α′−′⋅ − ⋅ ℜ γ =′\n×′\n↑↓∑\n11 ~ ~\n0 11 0 2 /\n) 4 (00] ) ˆ(~[ ˆ\n2\npumppumpeff eff\nGG\nk jk j\nj j sjkjk jk\njj\njkk\nkjk j j j j jj\nj\nC\nre h\nt Mdtd\ndtd\nt h tt tt\nTT mm H m Hm\nz\n (9) \nwhere Kronecker delta k j k jjk jk ≠=δ= = δ if 0 and , if 1 . \n In (9), is a 2-dimensional Cartesian \"damping tensor \" expressed in a coordinate system where \n, while is a \"nonlocal tensor\" spanning two such coordinate systems. This formalism \nfollows naturally from the lineariza tion of the equations of motion for non-collinear macrospins, and is \nparticularly useful for describing the influence of \"tensor damping\" on the thermal fluctuations and/or j jα′t\nj jz m′=′ ˆ ˆ0 j k j≠α′tspin-torque critical currents of su ch multilayer film structures (e.g., as described further in Sec. IV.). \nDue to the spin-pumping contribution pump\njkα′t, the four individual (with v u\njk′ ′α′ y x v u′′=′ ′ or , ) are in \ngeneral nonzero with , reflecting the true tensor na ture of the damping in this \ncircumstance, which is additionally nonl ocal between magnetic layers (i.e., ). The are \nsomewhat arbitrary to the extent that one may replace y y\njkx x\njk′ ′ ′ ′α′≠ α′\n0≠ α′′ ′\n≠v u\nj jkv u\njk′ ′α′\n2~ ~ℜ ⋅ ℜ ↔ ℜ in (9), where 2ℜ is the \nmatrix representation of any rotation about the 2 2× z′ˆaxis. \n It is perhaps tempting to contemplate an \"inverse linearization\" of (9 ) to obtain a 3D nonlinear \nGilbert equation with a fully 3D damping tensor T\nk jk j jkℜ ⋅ α′⋅ ℜ = αt t. However, (9) has a null zˆ′ \ncomponent, and contains no information rega rding the heretofore undefined quantities or . \nFor local, isotropic/scal ar Gilbert damping, one can independen tly argue on spatial symmetry grounds \nthat . However, the analogous extension is not so obviously available for z u\njk′ ′α′z z\njk′ ′α′\nG G G α = α′= α′′ ′ ′ ′u u z z pump\njkα′t, \ngiven the intrinsically nonlocal, anisotropic natu re of spin-pumping. The proper general equation \nremains that of (8), with the rightmost term given by that in (5), or its equivalent. \n \nIII. EXAMPLE: FIVE LAYER SYSTEM \n \n \n Fig. 2 shows a 5-layer system with 2 FM layers resembling a CPP-GMR spin-valve, to be used as a Fig. 2. Cartoon of a prototypical 5-layer CPP-GMR stack (leads not shown) with two FM layers (#1 and #3),\nsandwiching a central NM spacer layer (#2 ), and with outer NM cap layers (#0 and #4). For discussion purpose\n \nprototype. Although the full genera lization is straightforward, the material properties and layer s\ndescribed in text, the magnetization vectors 1ˆm and 3ˆm can be considered to lie in the film plane ( zx- plane). m1\nNMj=0 j=1 23 4\nFM NM' FM NMm3z\nxθ\ny\ny1y0 y2y3y4y5thickness will be assumed symmetric about the centr al (#2) normal metal spacer layer, which will \nadditionally be taken to have a large spin-diffusion length (with the thickness of the \nlayer), such that the \"ballistic\" approximation (B3) applies. The inverse mixing conductances \nwill also be assumed to be real. Using the outer boundary conditions described by (B5), one \nfinds for the FM-NM interfaces at that 2 2t l> >>jtthj\n↑↓\n=4 1-ir\n, and4 1y y y=\n \n] )) / hyp( ( [ˆ\nNMFM NM\n1 1pump\n1\n3 , 1 4 , 1l t l r rrJi\nj i iρ + ≡ ′+ =\n↑↓ ↑↓↑↓\n= =Jm J (10a) \n \nFM\nNMFM\n4 1 1 4 21] )) / hyp( ( [J l t l r r Vi ρ + ≡′= Δ −= (10b) \n \nwhere , , and subscript \"NM\" refers to either outer layer 0 or 4. In (10) and below, \n are used interchangeably. Inside FM la yer 3, (B1,2) have solution expressible as 4 1r r=↑↓↑↓=4 1r r\nj j0ˆ ˆm m↔\n \n3 1 1 33 3 3 3spin\n33 3 3 3 4 3 3\n] ) / tanh( ) /[( ] ) / tanh( ( [() / ) sinh(( ) / ) cosh(( [ ) /( 1 ) () / ) cosh(( 2 ) / ) sinh(( 2 ) (\nFM FM FMFM FM FMFM FM\nA l t r l l t l r Bl y y B l y y A l y Jl y y B l y y A y y y V\n′+ ρ ρ +′− =− + − ρ =− + − = ≤ ≤ Δ\n (11) \n \nwhere the expression for follows from (10b). Subscript \"FM\" re fers to either layers 1 or 3. The \nboundary conditions (A5) and (A9) applied to the FM/NM boundary at 3B\n3y y= yield \n \n) ( ˆ]ˆ ) /( )[ ( ) 2 (pump\n3spin\n2 2 3 3spin\n2 3 2 2 2 3 21\nFM J J m m J V − + ⋅ = ρ − = −↑↓ ↑↓r l A r r BΔ (12) \n \nwhere , . The \"ballistic\" values 3 2r r=↑↓↑↓=3 2r r2VΔ and are constant inside central layer 2. \nUsing (11) to eliminate coefficient in (12), the latter may be rewritten as spin\n2J\n3B\n \n⎥⎦⎤\n⎢⎣⎡\nρ ′+ρ +′+ − ≡− ⋅ ⋅ + = −\n↑↓\n↑↓↑↓\nFMFM\n)] /( ) / [tanh( 1)) / tanh( (\n21] )ˆ ˆ 2 1 [(\n11\n2 2\n2pump\n3spin\n2 3 3 2 2 21\nl l t rl t l rr r\nrqq r J J m m VTt\nΔ\n (13) \n \nwhere is the 3-D identity tensor, and denotes the 3-D tensor formed from the vector outer -\nproduct of with itself. 1tT\n3 3ˆ ˆm m⋅\n3ˆm Working through the equivalent comput ations applied now to the NM/FM interface at 2y y= , one \nfinds the analogous result: \n \n] )ˆ ˆ 2 1 [(pump\n2spin\n2 1 1 2 2 21J J m m V − ⋅ ⋅ + = +↑↓ Tq rt\nΔ (14) \n \nEliminating between (13) and (14) derives the remaining needed result for : 2VΔspin\n2J\n \n1\n3 3 1 1pump\n3pump\n2 21 spin\n2)]ˆ ˆ ˆ ˆ ( 1 [ ), (−⋅ + ⋅ + ≡ + ⋅ = ⋅T Tm m m m J J J q Q Qtt t\n (15) \n \ntreating tensor Qt\n as the matrix inverse of the [ ]-bracketed te nsor in (15). Using (10a) and (15) to \ncompute , then additional use of (4) and (6), allow computation of the 3 3×\nNM\n4 1-=iJjkCt\n defined in (7): \n \n) / 1 / 1 ( / 1 ; 2 / , /, 1\n2 1 212 131 13 33 11\n↑↓ ↑↓ ↑↓ ↑↓↑↓ ↑↓ ↑↓+ = ≡′≡− = = + = =\nr r r r r b r r aQ b C C Q b a C Cttttttt\n (16) \n \n For explicit evaluation of pump\njkα′t, it is convenient to assume a choice of 3 , 1~\n=ℜj for which 3 1ˆ ˆy y′=′ , \nsuch that and lie in the plane. To simplify the inte rmediate algebra to obtain Q03ˆm01ˆm z x′ ′-t\n from \n(15), one can consider \"in-plane\" magnetizations (Fig.2), taking z mˆ ˆ03=, and in the x-z plane \n( ). This allows a particularly easy determination of 01ˆm\nθ = ⋅cosˆ ˆ01z mjℜ~ for which : y y yˆ ˆ ˆ3 1=′=′\n \n0 , ;0 1 0sin 0 cos ~\n3 1 3 , 1 = θ θ = θ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ θ − θ= ℜ=j j\njT (17) \n \nUsing (16) and (17) with (9) allows explicit solution for the pump\njkα′t: \n \nθ + +θ + −= =\nθ + +θ + += =⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+ δ− δ + δ\nπγ= α′\n↑↓\n2 231 132 22\n33 112\npump\nsin 2 1cos ) 2 1 (,\nsin 2 1cos 100 ) 1 2 ( 2 /\n) 4 (\nq qqd d\nq qq qd dd b ab a\nre h\nt M jk jkjk jk\nj j sjkh t\n (18) \n \nTaking , (18) holds for arbi trary orientation of and , provided the flexibility \nin choosing the 03 01ˆ ˆ cos m m⋅ = θ01ˆm03ˆm\n3 , 1~\n=ℜj is used to maintain 3 1ˆ ˆy y′=′ . However, for multilayer film stacks with three or \nmore magnetic layers with magnetizations that do not all lie in a singl e plane, it wi ll generally be \nthe case that some of the off-diagonal elements of the j0ˆm\npump\njkα′t will be nonzero. \n \n \nIV. DISCUSSION \n \n It is perhaps instructive to compare and contrast the results of (9) and (18). with the prior results of \nRef. 3. The latter are for a a trilaye r stack, corresponding most directly to taking ∞→ ρNM in the \npresent model, whereby . It is also effectively equivalent to the 5-layer case with \ninsulating outer boundaries in the limit , whereby but due to \nperfect cancellation by the spin current reflected from the boundaries without intervening spin-\nflip scattering. Either way, it corresponds to in (10) and in (16) and (18). 0NM\n4 , 1pump\n4 , 1= == = i iJ J\n0 ) / (NM→ l t 0pump\n4 , 1≠=iJ 0NM\n4 , 1→=iJ\n5 , 0=iy\n∞ →′ ′↑↓\n1 1,r r 0→a\n However, a more interesting difference is that Ref. 3 treats as stationary (hence , \nand as undergoing a perfectly circular precession about with a possibly large cone angle 3ˆm ) 0pump\n3= J\n1ˆm3ˆm θ. \nBy contrast, the present analysis treats and equally as quasi-stationary vectors which undergo \nsmall but otherwise random fluctuations about their equilibrium positions and , with \n. To further elucidate this distinction, one can assume the aforementioned physical \nmodel of Ref. 3, and reanalyz e that situation in terms of the present formalism. With \n, and by explicitly inserting the condition (e.g., from (3)) that , an \nexplicit solution of (15) can be expressed in the form: 1ˆm3ˆm\n01ˆm03ˆm\nθ = ⋅cos ˆ ˆ03 03m m\npump\n3 3 0 /J m = = dt d 0 ˆ1pump\n2≡ ⋅m J\n \n]ˆ\ncos ) 1 (ˆ) 1 ( ˆ cos[3pump\n2 2 2 23 12\npump\n2 212NMm Jm mJ J ⋅\nθ − ++ − θ+ =\nq qq q q (19) \n \nCombining (19) with the earlier re sult from (5) and then (3) (with ) 0=ε , it is readily found that \n \ndtd\nq qq q\nre h\nt Mq qq q\nt M et M e dtd\nA t M dtd\nsss s\n1\n2 2 22\n22\n11 32 2 2pump\n2 3 pump\n2 1\n12 1\n11\n1\n11\n1\nˆ\ncos ) 1 (sin ) 1 (12 /\n) 8 (ˆ ˆ\ncos ) 1 () ˆ)( 1 (ˆ\n) ( 4ˆ\n) ( 21ˆ\n) (ˆˆNM\nmm mJ mJ mJ mSmmm\n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nθ − +θ +−πγ− =⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n×\nθ − +⋅ ++ ×γ− =×γ− = ×γ⇔ ×\n↑↓hhh\n (20) \n The last result in (20) uses from (3), and the fact that pump\n2J θ = ×sin ˆ ˆ1 3m m , and that and \n are parallel vectors in the case of steady circular precession of about a fixed . It is the \ndirect equivalent of Eq. (9) of Ref. 3 with the identification dt d/ˆ1m\n1 3ˆ ˆm m×1ˆm3ˆm\n) 1 /(+⇔ν q q . \n Although the final expression in (20) is azimuthally invariant with vector orientation of , it is \nmost convenient to compare it with (18) at that instant where is \"in-plane\" as shown in Fig. 2. At \nthat orientation,1ˆm\n1ˆm\ndt m d dt dm dt dy y / / /ˆ1 1 1′= → m , and it is immediately confirmed from (9) and (18) \n(with ) that the [ ]-term in (20) is simply the tensor element 0→ay y′′α′11 of pump\n11α′t. It is now seen that \nthe analysis of Ref. 3 happens to mask the tensor nature of the spin-pump damping by its restricting \nattention a specific form of the mo tion of the magnetization vectors, which in this case singles out the \nsingle diagonal element of the pump\n11α′t tensor along the axis perpendicular to the plane formed by \nvectors and . The very recent results of Ref. 6 do addr ess this deficiency of generality, and \nreveal the tensor nature of 1ˆm3ˆm\npump\n11α′t with specific results for ππ=θ and , 2 / , 0 . The present Sec. III \nadditionally includes the nonlocal tensors pump\n31pump\n13α′= α′t t, as well as diagonal terms jkaδ in (18) \n(and the variation in parameter q) when it is not the case that )/ hyp( ) (NM NM NM FM NM l t l r ρ<<- in \nboundary condition (B4). The latter condition will likely apply in the case of the technological \nimportant example of CPP-GMR spin-valves. \n Speaking of such, two important practical i ssues for these devices involve thermal magnetic noise \nand spin-torque induced oscillat ions. As described previously8, an explicit linearization of the effH \nterm in (9) about equilibrium state that is a minimum of the free energy 0ˆm E leads to the following \nmatrix form of the linearized Gilber t equation including spin-pumping (with : )0=eJ\n \nmA t M\npE\nmH H Hp p p\nGp p\nDt p t HdtdG D\nj s\nj\njjk jk j jk\nkj\njk j j jkkj k jk j\njkj\njkkj k jk j\njkj j j j\nkk jkk\nkjk jk\nΔ≡∂∂\nΔ−≡ℜ ⋅ ⋅ ℜ ≡ ′\n∂∂\n− δ ⋅ ≡⎥\n⎦⎤\n⎢\n⎣⎡\nγα′− α′\n+ δγ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−≡′\nγα′+ α′\n≡′⋅ ℜ ≡ ′= ⋅′⋅′+′⋅′+′ ∑ ∑\n) (\n,ˆ)ˆ( 1~ ~,ˆ1 ) ˆ(2 0 11 0,2) (~) ( ) (\n0 eff\n0eff\n0 eff\n0 0\nmmHmH\nH mh h mm\nt t t tt ttt ttt tt\nTT\n (21) \n where the are small perturbation fields. The form of ) (tjhjkDt\n′ and jkG′t\n in (21) is chosen so that they \nretain the original delineation8 as symmetric and antisymmetic tensors regardless of the symmetry of \n. By use of a fixed \"reference-moment\" jkα′tmΔ in the definition of , the \"stiffness-field\" tensor-\nmatrix is symmetric positive-definite, and eff\njH\nv k u jv u\njk m m E H′ ′′ ′′∂′∂ ∂ ∝ ′ / ∑ ⋅ − =δj j j j sδ t M A E m h ) ( \n∑′⋅′Δ − =j j j m m h has the proper conjugate form so that (21) are now ready to directly apply \nfluctuation-dissipation expressions specifically suited to such linear matrix equations of motion.8 \nTreating the fields now as thermal fluctuat ion fields driving the ) (tjh′ ) (tjm′-fluctuations, \n \nv u\njkB\nh hv u\njkB\nv k u j DmT kS DmT kh hv k u j′ ′\n′ ′′ ′\n′ ′ ′\nΔ= ω′⇔ τ δ′\nΔ= 〉′τ′〈′ ′2) ( ) (2) 0 ( ) ( (22) \n \nare the time-correlation or cross power spectral density (PSD) F ourier transform pairs. Through their \nrelationship described in (21), the nonlocal, tensor nature of the spin-pumping contribution pump\njkα′t to \njkα′t is directly translated into those of the FM FM2 2N N× system \"damping tensor-matrix\" v u\njkD′ ′′↔′Dt\n, \nwhere is the number of FM layers in the multila yer film stack. The cross-PSD tensor-matrix FMN\n) ( ) ( ω′↔ ω′′ ′′ ′ ′v k u jm mSm mSt\n for the m-fluctuations can then be expressed as′8 \n \n1)] ( [ ) () ( ) ( )] ( ) ( [ ) (\n−′ ′ ′ ′\n′+′ω −′≡ ω′ω′⋅′⋅ ω′→ ω′− ω′\nΔ ω= ω′\nD G HS Sh h m m\nttt tttt t t t\nim iT kB\nχχ χ χ χ@ @\n (23) \n \nwhere is the complex susceptibility tensor-matrix for the ) (ω′χt} , {h m′′ system, and ) (ω′@χt its \nHermitian transpose. It has been theoretically argued10 that (22), and thus the second expression in (23), \nremain valid when , despite spin-torque contributions to resulting in an asymmetric 0≠eJeff\njH H′t\n \n(e.g., see (25)) that violates the condition of therma l equilibrium implicitly a ssumed for the fluctuation \ndissipation relations. \n Since is in general a fully nonlocal with anisotropic/tensor character, any additional tensor \nnature of H′t\nDt\n will likely be altered or muted as to the influence on the detectable -fluctuations. As an \nexample, one can again consider th e situation depicted in Fig. 2, applied to the case of a CPP-GMR \nspin-valve with typical in-plane magnetization. The device's output noise PSD will reflect fluctuations m′in . Taking to again play the simplifying role of an ideal fixed (or pinned) reference layer \n(i.e., ), the PSD will be proportional to . As was also shown \npreviously,3 1ˆ ˆm m⋅3ˆm\n0 /ˆ3→dt dm ) ( sin1 12ω′θ′ ′′x xm mS\n11 it follows from (23) (and assuming azimuthal symmetry 011 11=′=′′ ′ ′′ x y y xH H ) that \n \nx x y y y y x x x x y yy y x x y y x x\nsB\nm m\nH H H HH H\ntA MT kSx x\n′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′′ ′ ′′′′′′\n′ ′\n′α′+′α′= ω Δ′ ′ γ = ωω Δ ω + ω − ωω α′+ ω′ ′ α′γ≅ ω′′\n11 11 11 11 11 11 02 2 2\n022\n112\n0 11 11 11\n1 ) ( ) () / (\n) (2) (1 1 (24) \n \ntreating . The tensor influence of the is seen to be weighted by the relative size \nof the stiffness-field matrix elements . For the thin film geometries 111 11 < < < α′α′′ ′′ ′ y y x x u u′ ′α′11\nv vH′ ′′11 A t< < typical of such \ndevices, out-of-plane demagnetization fi eld contribution typically result in that are an order of \nmagnitude larger than . Since y yH′ ′′11\nx xH′ ′′11x x y y ′ ′ ′′α′≤ α′11 11 from (18), it follows that the linewidth ωΔ and the \nPSD in the spectral range of practical interest will both be expected to be determined \nprimarily by . ) (01 1ω ≤ ω ′′ ′′x xm mS\nx x′ ′α′11\n A similar circumstance also applies to the im portant problem of critical currents for spin-torque \nmagnetization excitation in CPP-GMR spin valves with 0≠eJ . Consider the same example as above, \nagain treating as stationary, and seeking nontri vial solutions of (21) (with of the form \n. Summarizing results obtainable from (5), (8), and (21) 3ˆm ) 0 ) (=′th\nste t−∝′) (1m\n \n0 detˆ\n) (2 /\n11 11 1111 11 111 2\n10eff\n1eff\n1NM\n=⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nα′ ′−′ ′ −′′+′ α′ ′−′× + =\n′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′=\ny y y y x yy x x x x xsJ\ns H s Hs H s Ht Me\nem J H Hh\n (25) \n \nwhere , as in (18),and where in (25) is now the solution of the transport \nequations with but . The cross-product form of the spin-torque contribution to \nexplicitly yields an asymmetr ic/nonreciprocal contribution γ =′ /s sv u′ ′α′11 eJ∝NM\n2J\n0pump= J 0≠eJeff\n1H\nex y y xJ H H∝′−′′′′′\n11 11 to . The critical \ncurrent density is that value of where becomes negative. Given th e basic stability criterion \nthat , the spin-torque critical condition from (25) can be expressed as Ht\n′\neJ sRe\n0 det11>′Ht \nx y y x y y x x x x y yH H H H′′′′′′′ ′ ′ ′ ′ ′′−′=′α′+′α′11 11 11 11 11 11 (26) \n \nLike for thermal noise, the spin-torque critical point should again be determined primarily by for \nin-plane magnetized CPP-GMR spin-valves with typical x x′ ′α′11\nx x y yH H′ ′ ′′′> >′11 11. This simply reflects the fact \nthat the (quasi-uniform) modes of thermal fluctuati on or critical-point spin-torque oscillation tend to \nexhibit rather \"elliptical\", mostly in-plane motion when x x y yH H′ ′ ′′′> >′11 11. This is obviously different \nthan the steady, pure circular precession described in Ref. 3, which contrastingly highlights the \ninfluence of , along with its inte resting, additional y y′ ′α′11θ-dependence. \n \nAPPENDIX A \n \n The well known \"circuit theory\" formulation12 of the boundary conditions for the electron charge \ncurrent density and the (dimensionally equiva lent) spin current density at a FM/NM interface \ncan (taking ) be expressed as eJspin\nNMJ\nm V ˆFM FM VΔ =Δ\n) ˆ )( ( ) )( (FM NM FM NM 21V G G V V G G Je Δ − ⋅ − + − + =↓ ↑ ↓ ↑m VΔ (A1) \n \n)ˆ ( Im ) ˆ ˆ( Reˆ)] ˆ )( ( ) )( [(\nNM NMFM NM FM NM NM 21 spin\nm V m V mm m V J\n× + × × +Δ − ⋅ + + − − =\n↑↓ ↑↓↓ ↑ ↓ ↑\nΔ ΔΔ\nG GV G G V V G G\n (A2) \nin terms of spin-indepe ndent electric potential V and accumulation VΔ (Δμe= ). Setting 0=eJ in \n(A1) and substituting into (A2), one obtains in the limit the result 0 Im→↑↓G\n \n)ˆ ˆ( ˆ) ˆ (2\nNM FM NM NM0spinm V m m m V J × × + Δ − ⋅\n+=↑↓\n↓ ↑↓ ↑\n=Δ Δ G V\nG GG G\neJ (A3) \n \nComparing with Eq. (4) of Tserkovnyak et al.3 (with )sμΔ⇔V and remembering the present \nconversion of , one immediately make s the identification spin 1 spin\nNM NM ) 2 / (I J−− ↔ e Ah\n \n↑↓ ↑↓= G e h A g) 2 / ( 22 (A4) \n \nrelating dimensionless in (1) to , the conventional mixing conductance (per area). ↑↓g↑↓G\n The common approximations that inside all FM layers, and that longitudinal spin \ncurrent density is conserved at FM/NM interfaces, yields the us ual interface boundary condition m J ˆspin spin\nFM FM J=spin spin\nFM NMˆJ= ⋅m J (A5) \nSolving for from (A2) then leads (with (A1)) to a second scalar boundary condition: m Jˆspin\nNM⋅\nspin\nFM FM NM4 4J\nG GG GJ\nG GG GV Ve ↓ ↑↓ ↑\n↓ ↑↓ ↑−−+= − (A6) \nEquation (A6) is identical in form with the standard (collinear) Valet-Fert model,7 and immediately \nyields the following identifications \n↓ ↑↓ ↑\n↓ ↑↓ ↑\n+−= γ+=\nG GG G\nG GG Gr ,\n4 (A7) \nfor the conventional Valet-Fert interface parameters . γandr\n The three vector terms on th e right of (A2) are mutually orthogona l. Working in a rotated (primed) \ncoordinate system where , (A1) and (A2) can be similarly inverted to solve for the three \ncomponents of the vector m z′=′ˆˆ\n)ˆ (FM NM m V ′Δ −′ VΔ in terms of , , and . A final transformation \nback to the original (umprimed) coordinate s yields the vector interface-boundary condition spin\nNMJ′spin\nFMJeJ\n \n) / /( ) 2 / ( ) 2 /( 1ˆ Im Reˆ] ) Re [( ) ˆ (\n2spin spin spin\n21\nNM NM FM FM NM\nA g e h G rr r J r J r r Ve\n↑↓ ↑↓ ↑↓↑↓ ↑↓ ↑↓\n= ≡× + + γ − − = Δ − J m J m m VΔ\n (A8) \n \nCombined with (A4), the last relation in (A8) yields (2). Equation (A8) is a generalization of Valet-\nFert to the non-collinear case. \n As noted by Tserkovnyak et al.,3 boundary conditions (A3) do not di rectly include spin-pumping \nterms, but instead involve only \"backflow\" terms in the NM layer. With spin-pumping \nphysically present, arises as the response to the spin accumulation back spin\nNM NM J J↔\nback\nNMJNMVΔ created by . It \nfollows that , where is henceforth the total spin current in the NM layer. \nThus, including spin-pumping in Va let-Fert transport equations is then a matter of replacing \n in (A8). The modified form of (A 8), for a FM/NM interface, becomes: pumpJ\npump spin back\nNM NM J J J− =spin\nNMJ\npump spin spin\nNM NM J J J− →\n \n) (ˆ Im ) ( Reˆ] ) Re [( ) ˆ (\npump spin pump spinspin\n21\nNM NMFM FM NM\nJ J m J Jm m V\n− × + − +γ − − = Δ −\n↑↓ ↑↓↑↓\nr rJ r J r r Ve Δ\n (A9) \n \nFor an NM/FM interface, the sign is flippe d on the left sides of (A6) and (A9). \n APPENDIX B \nFor 1-D transport (flow along the y-axis), the quasi-static Valet-Fert7 (drift-diffusion, quasi-static) \ntransport equations can be written as9 \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∂∂+∂∂βρ==⎥\n⎦⎤\n⎢\n⎣⎡\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∂∂⋅ β +∂∂\nρ=∂∂=\n∂∂\ny yVy yVJy l ye\nVm JVmV V\nΔΔ Δ Δ\n21ˆ1with along0 ˆ21 1,\nspin2 22\n (B1) \n \nwhere = bulk resistivityρ13, l = spin diffusion length, and β = bulk/equilibrium spin current \npolarization in FM layers ( in NM layers). The solution for any one layer has the form 0≡ β\n \nm B m AB A V m V\nˆ ,ˆ : layers FM for,ˆ/ /\n21\nB Ae e C y J Vl y l y\ne\n= =+ = ⋅ β − + ρ =−Δ Δ (B2) \n \nIn the case where film thickness, one may employ an al ternative \"ballistic\" approximation: > > >l\n \nC V , = = = ,spinB J A V Δ (B3) \n \nIt is not necessary to solve for the V and/or the C-coefficients using (A6) if only and are \nrequired. The remaining coefficients are determin ed by the interface boundary conditions (A5), (A6,7) \nand (A9), and external boundary conditions at the outer two surfaces of the film stack. VΔspinJ\n Regarding the latter, one approximation is to treat the external \"leads\" (with quasi-infinite cross \nsection) as equilibrium reservoirs and set 0 ) (, 0→==N i y y VΔ at the outermost (i =0, N) lead-stack \ninterfaces of an N -layer stack (Fig. 1). The complimentary approximation is of an insulating boundary, \nwith . . For the case (such as in Sec. III) where the outer ( j=0, N-1) layers are \nNM, and the adjacent inner ( j=1, N-2) layers are FM, it is readily found using (B1) and (B2) that 0 ) (, 0spin→==N i y y J\n \nNM NM) / hyp( ) ( 21 , 0 1 , 1 i j j N j N i l t l J V− = − =ρ ± = Δ (B4) \n \nwhere hyp( ) = tanh( ) or coth( ) for equipotential, or insulating boundaries, respectively. Combining \n(B4) with (A9), and neglecting ↑↓rIm , one finds for 0=eJ that \n ) / hyp( ) (ˆ)] / hyp( ) ( [\npump1 , 0 1 , 1 21\nFM NMFM FM\nj j j ii i\ni ii j j N j i N i\nl t l rrJJ l t l r V\nρ ++ =ρ + = Δ ±\n↑↓↑↓− = − =\nJm J (B5) \n \nACKNOWLEDGMENT \n \nThe author would like to thank Y. Tserkovnya k for bringing Ref. 6 to his attention. \n \nREFERENCES \n \n1. D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mat. 320, 1190 (2008) and many re ferences therein. \n2. Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B, 66, 224403 (2002). \n3. Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B, 67, 140404 (2003). \n4. S. Maat, N. Smith, M. J. Carey, and J. R. Childress, Appl. Phys. Lett., 93, 103506 (2008). \n5. J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W.Bauer, Phys. Rev. B 78, 140402 (2008). This \npaper describes a damping mechanism distinct from Refs. 6, or this work, where nonlocal/tensor \nproperties arise from a strong magnetization gr adient in a single FM film or wire. \n6. J. Foros, A. Brataas, G. E. W.Bauer, and Y. Tserkovnyak, arXiv:con-mat/0902.3779. \n7. T. Valet and A. Fert, Phys. Rev. B, 48, 7099 (1993). \n8. N. Smith, J. Appl. Phys. 92, 3877 (2002); N. Smith, J. Magn. Magn. Mater. 321, 531 (2009) \n9. N. Smith, J. Appl. Phys., 99, 08Q703, (2006). \n10. R. Duine, A.S. Nunez, J. Si nova, A.H. MacDonald, Phys. Rev. B 75, 214420 (2007) \n11. N. Smith, J. Appl. Phys. 90, 5768 (2001). \n12. A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur. Phys. J. B 22, 99, (2001). \n13. Some poor choice of words in the appendix of Ref. 9 c onfused the bulk resistivity, ρ, with the \nValet-Fert7 parameter . *ρ\n \n \n " }, { "title": "1004.1548v2.Oscillations_of_weakly_viscous_conducting_liquid_drops_in_a_strong_magnetic_field.pdf", "content": "arXiv:1004.1548v2 [physics.flu-dyn] 3 Nov 2010Under consideration for publication in J. Fluid Mech. 1\nOscillations of weakly viscous conducting\nliquid drops in a strong magnetic field\nBy J ¯ANIS PRIEDE\nApplied Mathematics Research Centre, Coventry University\nPriory Street, Coventry CV1 5FB, UK\n(Received 29 October 2018)\nWe analyse small-amplitude oscillations of a weakly viscou s electrically conducting liquid\ndrop in a strong uniform DC magnetic field. An asymptotic solu tion is obtained showing\nthat the magnetic field does not affect the shape eigenmodes, w hich remain the spherical\nharmonics as in the non-magnetic case. Strong magnetic field , however, constrains the\nliquid flow associated with the oscillations and, thus, redu ces the oscillation frequencies\nby increasing effective inertia of the liquid. In such a field, liquid oscillates in a two-\ndimensional (2D) way as solid columns aligned with the field. Two types of oscillations\nare possible: longitudinal and transversal to the field. Suc h oscillations are weakly damped\nby a strong magnetic field – the stronger the field, the weaker t he damping, except for the\naxisymmetric transversal and inherently 2D modes. The form er are overdamped because\nof being incompatible with the incompressibility constrai nt, whereas the latter are not\naffected at all because of being naturally invariant along th e field. Since the magnetic\ndamping for all other modes decreases inversely with the squ are of the field strength, vis-\ncous damping may become important in a sufficiently strong mag netic field. The viscous\ndamping is found analytically by a simple energy dissipatio n approach which is shown\nfor the longitudinal modes to be equivalent to a much more com plicated eigenvalue per-\nturbation technique. This study provides a theoretical bas is for the development of new\nmeasurement methods of surface tension, viscosity and the e lectrical conductivity of liq-\nuid metals using the oscillating drop technique in a strong s uperimposed DC magnetic\nfield.\n1. Introduction\nShape oscillations of levitated metal droplets can be used t o measure the surface tension\nand viscosity of liquid metals (Rhim et al. 1999; Egry et al. 2005). Theoretically, the\nformer determines the frequency, while the latter accounts for the damping rate of oscil-\nlations. In the reality, experimental measurements may be a ffected by several side-effects.\nFirstly, levitated drops may be significantly aspherical an d the oscillations amplitudes\nnot necessarily small, whereas the classical theories desc ribing the oscillation frequencies\n(Rayleigh 1945) and damping rates (Lamb 1993; Chandrasekha r 1981; Reid 1960)\nassume small-amplitude oscillations about an ideally sphe rical equilibrium shape. Cor-\nrections due to the drop asphericity have been calculated by Cummings & Blackburn\n(1991) and Suryanarayana & Bayazitoglu (1991). Bratz & Egry (1995) find the same\norder correction to the damping rate resulting also from AC- magnetic field. The ef-\nfect of a moderate amplitude on the oscillations of inviscid drops has been analysed by\nTsamopoulos & Brown (1983) who find that the oscillation freq uency decreases with\nthe square of the amplitude. Using a boundary-integral meth od, Lundgren & Mansour2 J. Priede\n(1991) show that small viscosity has a relatively large effec t on the resonant-mode cou-\npling phenomena in the nonlinear oscillations of large axia lly symmetric drops in zero\ngravity. Numerical simulation of large-amplitude axisymm etric oscillations of viscous liq-\nuid drop by Basaran (1992), who uses the Galerkin/finite-ele ment technique, shows that\na finite viscosity results in a much stronger mode coupling th an predicted by the small-\nviscosity approximation.\nSecondly, the measurements may strongly be disturbed by AC- driven flow in the\ndrop. The mode coupling by the internal circulation in axisy mmetrically oscillating drop\nhas been studied numerically by Mashayek & Ashgriz (1998) us ing the Galerkin/finite-\nelement technique. To reduce the strength of the AC field nece ssary for the levitation\nand, thus, to minimise the flow, experiments may be conducted under the microgravity\nconditions during parabolic flights or on the board of space s tation (Egry et al. 1999).\nA cheaper alternative might be to apply a sufficiently strong D C magnetic field that can\nnot only stabilise AC-driven flow but also suppress the conve ctive heat and momentum\ntransport responsible for the mode coupling under the terre strial conditions as origi-\nnally shown by Shatrov et al. (2003). Such an approach has been implemented first by\nYasuda et al. (2004) on the electromagnetically levitated drops of Coppe r and Nickel\nwhich were submitted to a DC field of the induction up to 10T.The only motion of Cu\ndrops observed to persist in magnetic field above 1Twas a solid-body rotation about\nan axis parallel to the magnetic field. No shape oscillations , usually induced by the AC-\ndriven flow fluctuations, were observed. Note that this impli es only the suppression of\nAC-driven flow but not of the shape oscillations themselves w hich require an external\nexcitation to be observable. Yasuda et al. (2005) study the effect of suppression of the\nmelt flow on the structure of various alloys obtained by the el ectromagnetic levitation\nmelting technique in a strong superimposed DC magnetic field . The use of high magnetic\nfields in various material processing applications is revie wed by Yasuda (2007).\nNote that a similar suppression of AC-driven flow can also be a chieved by a fast\nspinning of the drop (Shatrov et al. 2007) that may be driven by an electromagnetic\nspin-up instability (Priede & Gerbeth 2000, 2006). The effec ts of both the drop rotation\nand AC-driven flow on the frequency spectrum of shape oscilla tions have been modelled\nnumerically by Bojarevics & Pericleous (2009). Watanabe (2 009) demonstrates numer-\nically that a large enough oscillation amplitude can compen sate for the effect of rotation\non the frequency shift.\nA novel method of measuring thermal conductivity of liquid s ilicon using the elec-\ntromagnetic levitation in a strong superimposed DC magneti c has been introduced by\nKobatake et al. (2007). Subsequent numerical modelling by Tsukada et al. (2009) shows\nthat applying a DC magnetic field of 4Tcan suppress convection in molten silicon\ndroplet enough to measure its real thermal conductivity. La ter on this method has\nbeen extended to the measurements of heat capacity of molten austenitic stainless steel\n(Fukuyama et al. 2009) and also that of supercooled liquid silicon (Kobatake et al.\n2010).\nIn order to determine the surface tension and viscosity or th e electrical conductivity\none needs to relate the observed surface oscillations with t he relevant thermophysical\nproperties of the liquid. General small-amplitude shape os cillations of conducting drop\nin a uniform DC magnetic field have been analysed first by Gaili tis (1966). Although\na magnetic field of arbitrary strength is considered, the sol ution is restricted to inviscid\ndrops. Moreover, only the frequency spectrum and magnetic d amping rates are found\nbut not the associated shape eigenmodes, which may be useful for experimental iden-\ntification of the oscillation modes. Energy dissipation by a xisymmetric oscillations of a\nconducting drop in a weak DC magnetic field is considered by Za mbran (1966), who findsOscillations of weakly viscous conducting liquid drops in a strong magnetic field 3\nθz\nγ\nxy\nrφBR\nηρσ\nFigure 1. Sketch to the formulation of problem.\nthe magnetic damping rates in agreement with more general re sults of Gailitis (1966).\nAxisymmetric oscillations of an electromagnetically levi tated drop of molten Alin a\nsuperimposed DC magnetic field are modelled numerically by B ojarevics & Pericleous\n(2003). A moderate DC magnetic field is shown to stabilise AC- driven flow and, thus, to\neliminate the associated shape oscillations. A three-dime nsional numerical simulation of\nan oscillating liquid metal drop in a uniform static magneti c field has been carried out\nby Tagawa (2007). The numerical results show that vertical m agnetic field effectively\ndamps the flow, while horizontal field tries to render the flow t wo-dimensional.\nIn the present paper, we analyse free oscillations of a visco us electrically conducting\ndrop in a homogeneous DC magnetic field. In contrast to Gailit is (1966), we assume the\nviscosity to be small but non-zero and the magnetic field to be strong. This allows us\nto obtain an asymptotic solution to the eigenvalue problem f or general small-amplitude\n3D shape oscillations including the eigenmodes left out by G ailitis (1966), which are\nnecessary for the subsequent determination of the viscous d amping. Firstly, we show that\nthe eigenmodes of shape oscillations are not affected by stro ng magnetic field. Namely,\nthey remain the spherical harmonics as in the non-magnetic c ase. The magnetic field,\nhowever, changes the internal flow associated with the surfa ce oscillations and, thus,\nthe frequency spectrum. As the drop oscillates in a strong ma gnetic field, the liquid\nmoves as solid columns aligned with the field. Two types of suc h oscillations are possible:\nlongitudinal and transversal to the magnetic field. The osci llations are weakly damped by\na strong magnetic field, except for both the axisymmetric tra nsversal and inherently 2D\nmodes. The former are magnetically overdamped because the i ncompressibility constraint\ndoes not permit an axially uniform radial flow. The latter, wh ich are transversal modes\ndefined by the spherical harmonics with equal degree and orde r,l=m, are not affected\nat all because these modes are naturally invariant along the field. Because the magnetic\ndamping for all other modes decreases inversely with the squ are of the field strength, the\nviscous damping may become important in a sufficiently strong magnetic field.\nThe paper is organised as follows. The problem is formulated in§2. Section 3 presents\nan inviscid asymptotic solution which yields the shape eige nmodes and frequency spec-\ntrum of longitudinal and transversal oscillations. Magnet ic damping is found in §3.2 as a\nnext-order asymptotic correction to the frequency. Viscou s damping rates are calculated\nin§3.3 first by the eigenvalue perturbation technique for the lo ngitudinal modes and\nthen by an energy dissipation approach for both of the oscill ation modes. The paper is\nconcluded by a summary and discussion of the results in §4.4 J. Priede\n2. Problem formulation\nConsider a spherical non-magnetic drop of an incompressibl e liquid with radius R0,\ndensityρ,surface tension γ,electrical conductivity σ,and a small dynamic viscosity η\nperforming small-amplitude shape oscillations in a strong uniform DC magnetic field B\nas illustrated in figure 1. The velocity of liquid flow vand the pressure distribution pare\ngoverned by the Navier-Stokes equation with electromagnet ic body force\nρ∂tv=−∇p+η∇2v+j×B, (2.1)\nwhere the induced current follows from Ohm’s law for a moving medium\nj=σ(E+v×B). (2.2)\nOwing to the smallness of oscillation amplitude, the nonlin ear term in (2.1) as well as\nthe induced magnetic field are both negligible. In addition, the characteristic oscillation\nperiodτ0is supposed to be much longer than the magnetic diffusion time µ0σR2\n0,where\nµ0is the permeability of vacuum. This leads to the quasi-stati onary approximation ac-\ncording to which ∇×E= 0andE=−∇ϕ,whereϕis the electric potential. The\nincompressibility constraint ∇·v= 0and the charge conservation condition ∇·j= 0\napplied to (2.1) and (2.2) result, respectively, in\n∇2p=σ(B·∇)(B·v), (2.3)\n∇2ϕ=B·∇×v. (2.4)\nFor a uniform Bunder consideration here, applying the operators ∇×∇×,(B·∇)B·\nand(B·∇)B·∇×to (2.1) and taking into account ∇×∇×(j×B) =σ(B·∇)2v\ntogether with (2.3) and (2.4), we obtain\n/bracketleftbig\nρ∇2∂t+σ(B·∇)2−η∇4/bracketrightbig\n{p,ϕ,v}= 0. (2.5)\nAlthough the equation above applies to p, ϕandvseparately, these variables are not\nindependent of each other. Firstly, owing to the incompress ibility constraint, only two\nvelocity components are mutually independent. Secondly, v elocity is related to the pres-\nsure and electric potential by (2.1), which can be used to rep resentvin terms of pand\nϕas done in the following.\nBoundary conditions are applied at the drop surface Sdefined by its spherical radius\nR=R0+R1(θ,φ,t),whereR1is a small perturbation, which depends on the poloidal\nand azimuthal angles, θandφ,and the time t. The radial velocity at the surface is related\nto the radius perturbation by the kinematic constraint\nvR|S=∂tR1. (2.6)\nThe normal component of the current at the drop surface, whic h is assumed to be sur-\nrounded by vacuum or insulating gas, vanishes, i.e., jn|S= 0.In addition, there is no\ntangential stress at the free surface:\nn·∂τv+τ·∂nv|S= 0, (2.7)\nwhile the normal stress component is balanced by the capilla ry pressure\np0+p−2η∂nvn=γ∇·n, (2.8)\nwherep0= 2γ/R0is the constant part of pressure, τis a unit tangent vector and n=\n∇(R−R1)/|∇(R−R1)|is the outward surface normal. For small-amplitude oscilla tions\ndefined by R1≪R0,we have n≈eR−∇R1.\nHenceforth, we proceed to dimensionless variables by choos ing the radius R0and theOscillations of weakly viscous conducting liquid drops in a strong magnetic field 5\ncharacteristic capillary pressure P0=γ/R0as the length and pressure scales. The char-\nacteristic period of capillary oscillations is determined by the balance of inertia and\npressure, which yields the time scale τ0=/radicalbig\nR3\n0ρ/γ.The velocity and potential scales are\nchosen as v0=R0/τ0andϕ0=v0BR0,respectively, where B=|B|.In the dimensionless\nvariables, (2.5) takes the form\n/bracketleftbig\n∇2∂t+Cm(ǫ·∇)2−Ca∇4/bracketrightbig\n{p,ϕ,v}= 0, (2.9)\nwhereǫ=B/Bis a unit vector in the direction of the magnetic field and Ca=η/√R0ργ\nandCm=σB2R2\n0/√R0ργare the conventional and magnetic capillary numbers, respe c-\ntively. They are the ratios of the capillary oscillation tim eτ0defined above and the viscous\nand magnetic damping times, which are τv=ρR2\n0/ηandτm=ρ/(σB2),respectively. In\nthe dimensionless form, the normal stress balance conditio n (2.8) reads as\n(∇2+2)R1+p−2Ca∂RvR/vextendsingle/vextendsingle\nR=1= 0. (2.10)\nIn the following, we assume viscosity to be small but the magn etic field strong so that\nCa≪1andCm≫1,which means that the second and third terms in (2.9) are much\ngreater and much smaller, respectively, than the first one. T hus, we first focus on the\neffect of the magnetic field and ignore that of viscosity, whic h is considered later in §3.3.\n3. Inviscid asymptotic solution\nHere we ignore viscosity that allows us to formulate the prob lem in terms of p, ϕand\nR1.Projecting the dimensionless counterpart of (2.1), which t akes the form\nCmv+∂tv=−∇p+Ca∇2v+Cm[ǫ×∇ϕ+ǫ(ǫ·v)], (3.1)\nontoeRandǫ,and putting Ca= 0,we obtain\nCmvR+∂tvR=−eR·∇p+Cm[eR×ǫ·∇ϕ+eR·ǫv/parallelshort], (3.2)\n∂tv/parallelshort=−ǫ·∇p, (3.3)\nwherev/parallelshort=ǫ·vis the velocity component along the magnetic field. Different iating (3.2)\nwith respect to tand substituting ∂tv/parallelshortfrom (3.3), we represent (2.6) in terms of pandϕ\nCm∂2\ntR1+∂3\ntR1= [Cm(eR×ǫ·∇∂tϕ−(eR·ǫ)ǫ·∇p)−eR·∇∂tp]R=1.(3.4)\nVelocity has to be eliminated also from the electric boundar y condition given by the\nradial component of Ohm’s law\njR|R=1=−eR·[∇ϕ+ǫ×v]R=1= 0. (3.5)\nFirstly, applying (Cm+∂t)to (3.5) and then using (3.1), we obtain\n[Cm(eR·ǫ)ǫ·∇ϕ−eR×ǫ·∇p+eR·∇∂tϕ]R=1= 0. (3.6)\nIn the inviscid approximation, (2.10) reduces to\np|R=1=−(∇2+2)R1. (3.7)\nIn the following, besides the spherical coordinates (R,θ,φ),we will be using also the\ncylindrical ones (r,φ,z)with the axis aligned along the magnetic field so that ǫ=ez.\nSolution is sought in the normal mode form {p,ϕ,R 1}={ˆp,ˆϕ,ˆR}(r)eβt+imφ,whereˆp,\nˆϕandˆRare axisymmetric amplitude distributions, mis the azimuthal wave number, and\nβis a generally complex temporal variation rate which has to b e determined depending6 J. Priede\nonm,CmandCa.Then boundary conditions (3.4), (3.6) and (3.7) for the osci llation\namplitudes at R= 1take the form\nβ2ˆR+imβˆϕ+z∂zˆp=−Cm−1(β3ˆR+β∂Rˆp), (3.8)\nz∂zˆϕ=−Cm−1(imˆp+β∂Rˆϕ), (3.9)\nˆp=−(Lz+2)ˆR, (3.10)\nwhereLz≡d\ndz/parenleftbig\n(1−z2)d\ndz/parenrightbig\n−m2\n1−z2is the angular part of the Laplace operator in the\nspherical coordinates for the azimuthal mode mwritten in terms of z= cosθ.Further,\nit is important to note that\nLzPm\nl(z) =−l(l+1)Pm\nl(z), (3.11)\nwherePm\nl(z),the associated Legendre function of degree land order m,is an eigenfunc-\ntion ofLzwith eigenvalue −l(l+1)(Abramowitz & Stegun 1972). Equation (2.9) for ˆp\nandˆφcan be written as\n/bracketleftbig\n∂2\nz+Cm−1(Lr+∂2\nz)(β−Ca(Lr+∂2\nz))/bracketrightbig\n{ˆp,ˆϕ}= 0, (3.12)\nwhereLr≡∂2\nr+r−1∂r−m2/r2is the radial part of the Laplace operator in the cylindrical\ncoordinates for the azimuthal mode m.Here we put Ca= 0,supposeCm≫1,and search\nfor an asymptotic solution in the terms of a small parameter Cm−1as\n{ˆp,ˆϕ,ˆR,β} ∼ {ˆp0,ˆϕ0,ˆR0,β0}+Cm−1{ˆp1,ˆϕ1,ˆR1,β1}+···.\nNote that although (3.12) admits solutions with β∼Cmfound by Gailitis (1966),\nsuch quickly relaxing modes cannot be related with the surfa ce deformations. From the\nphysical point of view, drop is driven to its equilibrium sha pe by the surface tension,\nand the magnetic field can only oppose but not to accelerate th e associated liquid flow.\nFrom the mathematical point of view, β∼Cm≫1applied to (3.8) results in ˆR0= 0,\nwhich means no surface deformation at the leading order in ag reement with the previous\nphysical arguments. Consequently, these fast modes repres ent internal flow perturbations\nwhich are not relevant for the shape deformations under cons ideration here.\n3.1.Oscillation frequencies\nAt the leading order, (3.12) reduces to ∂2\nz{ˆp0,ˆϕ0}= 0,whose general solution is\n{ˆp0,ˆϕ0}(r,z) ={ˆp+\n0,ˆϕ+\n0}(r)+z{ˆp−\n0,ˆϕ−\n0}(r), (3.13)\nwhere the first pair of particular solutions are the function s ofronly, while the second pair\nis linear in zbut general in r.Owing to the z-reflection symmetry of the problem these\ntwo types particular of solutions do not mix and, thus, they a re subsequently considered\nseparately. We refer to these solutions in accordance to the irz-parity as even and odd\nones using the indices eando.As shown below, the odd and even solutions describe\nlongitudinal and transversal oscillation modes, respecti vely.\n3.1.1. Longitudinal modes\nFor the odd solutions {ˆpo\n0,ˆϕo\n0}(r,z) =z{ˆp−\n0,ˆϕ−\n0}(r),boundary condition (3.9), which at\nthe leading order reads as z∂zˆϕ0= 0,results in ˆϕ−\n0(r) = 0.The two remaining boundary\nconditions (3.8) and (3.10) take the form\nβo\n02ˆRo\n0=−zˆp−\n0, (3.14)\n(Lz+2)ˆRo\n0=−zˆp−\n0. (3.15)Oscillations of weakly viscous conducting liquid drops in a strong magnetic field 7\nB\nxz\n(a)Bz\nx\n(b)B\nxz\n(c)Bz\nx\n(d)\nFigure 2. Shapes and the associated liquid oscillations in the (x,z)-plane parallel to the mag-\nnetic field for the first four longitudinal oscillation modes with indices ( l,m) = (2,1)(a),(3,0)\n(b),(3,2)(c), and (4,1) (d).\nEliminating the pressure term between the equations above, we obtain an eigenvalue\nproblem in β2\n0forˆRo\n0\n(Lz+2−βo\n02)ˆRo\n0= 0, (3.16)\nwhich is easily solved by using (3.11) as\nˆRo\n0(z) =Ro\n0Pm\nl(z), (3.17)\nβo\n0=±i/radicalbig\n(l−1)(l+2), (3.18)\nwhereRo\n0is a small amplitude of oscillations and l−mis an odd positive number. Note\nthat imaginary βo\n0describes constant-amplitude harmonic oscillations with the circular\nfrequency |βo\n0|which differs from the corresponding non-magnetic result on ly by the\nfactor of√\nl(Lamb 1993), and coincides with the result stated by Gailiti s (1966). Thus,\nstrong magnetic field changes only the eigenfrequencies but not the eigenmodes of shape\noscillations which, as without the magnetic field, are repre sented by separate spherical\nfunctions (associated Legendre functions with integer ind ices) (Abramowitz & Stegun\n1972). Similarly to the non-magnetic case, the frequency sp ectrum for odd modes is\ndegenerate because it depends only on the degree lbut not on the order m.Thus, for\neachl,there are [l/2]odd modes with different m.\nTaking into account that z=√\n1−r2at the surface, the radial pressure distribution\nis obtained from (3.14) as\nˆp−\n0(r) =−β2\n0ˆRo\n0(/radicalbig\n1−r2)//radicalbig\n1−r2. (3.19)\nAccording to (3.3), this pressure distribution is associat ed with the axial velocity com-\nponent\nˆwo\n0(r) =−β−1\n0ˆp−\n0(r), (3.20)\nwhile two other velocity components transversal to the magn etic field are absent in the\nleading-order approximation. Thus, the liquid effectively oscillates in solid columns along\nthe magnetic field as illustrated in figure 2 for the first four l ongitudinal oscillation modes\ndefined by the indices (l,m) = (2,1),(3,0),(3,2),and(4,1).Since such a flow does not\ncross the flux lines, the oscillations are not damped by the ma gnetic field in the leading-\norder approximation.\n3.1.2. Transversal modes\nFor the even solutions {ˆpe\n0,ˆϕe\n0}(r,z) ={ˆp+\n0,ˆϕ+\n0}(r),the leading-order boundary condi-\ntion (3.9) is satisfied automatically. The two remaining con ditions (3.8) and (3.10) then\ntake the form\nβe\n0ˆRe\n0+imˆϕ+\n0= 0, (3.21)8 J. Priede\n(Lz+2)ˆRe\n0=−ˆp+\n0. (3.22)\nIn contrast to the longitudinal modes considered above, now we have two equations (3.21)\nand (3.22) but three unknowns. To solve this problem, we need to consider the first-order\nsolution to (3.12) which now takes the form ∂2\nz{ˆpe\n1,ˆϕe\n1}=−β0Lr{ˆp+\n0,ˆϕ+\n0}and yields\n{ˆpe\n1,ˆϕe\n1}(r,z) ={ˆp+\n1,ˆϕ+\n1}(r)−1\n2βe\n0z2Lr{ˆp+\n0,ˆϕ+\n0}. (3.23)\nThen boundary condition (3.9) results in imp+\n0−βe\n0(z2Lr−r∂r)ˆϕ+\n0= 0.Combining this\nwith (3.21) and (3.22) and taking into account\nz2Lr−r∂r/vextendsingle/vextendsingle\nR=1≡Lz+m2, (3.24)\nwe obtain/bracketleftbig\nLz+2+(βe\n0/m)2(Lz+m2)/bracketrightbigˆRe\n0= 0. (3.25)\nUsing (3.11), we readily obtain\nˆRe\n0(z) =Re\n0Pm\nl(z), (3.26)\nβe\n0=±im/radicalBigg\n(l−1)(l+2)\nl(l+1)−m2, (3.27)\nwhereRe\n0is a small oscillation amplitude and l−mis an even non-negative number.\nThe result above again agrees with the asymptotic solution g iven by Gailitis (1966).\nSimilarly to the odd solutions found in the previous section , even eigenmodes are repre-\nsented by separate spherical functions, and the oscillatio ns are not damped at the leading\norder. In contrast to the odd modes as well as to the non-magne tic case, the frequency\nspectrum (3.27) is no longer degenerate and frequencies var y with the azimuthal wave\nnumberm. In particular, there are two important results implied by ( 3.27). Firstly, the\noscillation frequency for the axisymmetric modes specified bym= 0is zero. This means\nthat these modes are over-damped and do not oscillate at all. Secondly, the oscillation\nfrequency for the modes with m=lis exactly the same as without the magnetic field, i.e.,/radicalbig\nl(l−1)(l+2).This is because the liquid flow associated with these oscilla tion modes\nis inherently invariant along the field and, thus, not affecte d by the last (Gailitis 1966).\nThe electric potential and pressure distributions follow f rom (3.21) and (3.22) as\nˆϕe\n0(r) =im−1βe\n0ˆRe\n0(/radicalbig\n1−r2), (3.28)\nˆpe\n0(r) = (l−1)(l+2)ˆRe\n0(/radicalbig\n1−r2). (3.29)\nThe associated velocity distribution is obtained from (3.1 ). Firstly, equation (3.3) implies\nthat the liquid oscillations are purely transversal to the m agnetic field. In the leading-\norder terms, we obtain from (3.1)\nve\n0(r,φ) =ez×∇ϕe\n0(r,φ), (3.30)\nwhich shows that the velocity is not only transversal but als o invariant along the magnetic\nfield. Thus, the liquid again oscillates as solid columns, bu t in this case transversely to\nthe field which has no effect on such a flow. This is because the e. m.f induced by the flow,\nwhich is invariant along the magnetic field, is irrotational , i.e.,∇×(v×B) = (B·∇)v≡0,\nand, thus, unable to drive current circulation in a closed li quid volume.\nNote that for the axisymmetric modes (m= 0),the potential (3.28) and the associated\nvelocity (3.30) take an indeterminate form. Namely, for m= 0,boundary condition (3.21),\nwhich in this is case straightforwardly implies a zero frequ ency, is satisfied by an arbitraryOscillations of weakly viscous conducting liquid drops in a strong magnetic field 9\nBy\nx\n(a)By\nx\n(b)B\nxy\n(c)B\nxy\n(d)\nFigure 3. Shapes and the associated liquid flows in the horizontal mid- plane(z= 0)perpen-\ndicular to the magnetic field for the first four transversal os cillation modes defined by indices\n(l,m) = (2,2)(a),(3,1)(b), (3,3) (c) and (4,2)(d).\npotential distribution independent of the radius perturba tion. As seen from (3.28), a non-\nzero axisymmetric potential is associated with a purely azi muthal velocity. Consequently,\nthis mode is irrelevant and can subsequently be neglected be cause it represents an internal\nflow perturbation which is just compatible but not coupled wi th axisymmetric shape\ndeformations similarly to the fast modes discussed at the en d of§3. Moreover, this is\nconsistent with (3.27) according to which axisymmetric tra nsversal modes are static in\nthe leading-order approximation that implies a zero veloci ty and, consequently, a zero\nassociated potential.\nExpression (3.30) implies that the velocity streamlines co incide with the isolines of\nϕ0,which, thus, represents a stream function for the flow oscill ations. Figure 3 shows\nthe shapes and streamlines of the associated liquid flow in th e horizontal mid-plane for\nthe first four transversal oscillation modes. Note that the fi rst and the third mode with\nthe indices (l,m) = (2,2)and(3,3)are both naturally invariant in the direction of the\nmagnetic field and, thus, effectively non-magnetic. The seco nd mode with (l,m) = (3,1)\ncorresponds to the drop oscillating in such a way that horizo ntal cross-sections remain\ncircular in the small-amplitude limit under consideration while the whole shape deforms\nbecause of vertical offset of the cross-sections.\n3.2.Magnetic damping\n3.2.1. Longitudinal modes\nIn order to determine the magnetic damping rates for longitu dinal modes, we have to\nconsider the first-order solution governed by\n∂2\nz{ˆpo\n1,ˆϕo\n1}=−βo\n0zLr{ˆp−\n0,0},\nwhich yields\n{ˆpo\n1,ˆϕo\n1}(r,z) =z{ˆp−\n1,ˆϕ−\n1}(r)−1\n6βo\n0z3Lr{ˆp−\n0,0}.\nThen (3.9) and (3.14) applied consecutively result in zˆϕ−\n1=−imzˆp−\n0,which combined\nwith (3.8), (3.10) and (3.14) yields\n(Lz+2−βo\n02)ˆRo\n1=1\n3βo\n03z/bracketleftbig\nz2Lr−3r∂r/bracketrightbig\nz−1ˆRo\n0+βo\n0(2βo\n1−m2βo\n02)ˆRo\n0.\nAfter some algebra, we obtain z/bracketleftbig\nz2Lr−3r∂r/bracketrightbig\nz−1/vextendsingle/vextendsingle\nR=1≡Lz+2+m2,and, consequently,\n(Lz+2−βo\n02)ˆRo\n1=1\n3βo\n0/bracketleftBig\nβo\n02(Lz−2m2+2)+6βo\n1/bracketrightBig\nˆRo\n0. (3.31)10 J. Priede\nThe l.h.s. operator above is the same as that in (3.16) which h asˆRo\n0as its eigensolution\nwith a zero eigenvalue. Owing to (3.11) and (3.17), ˆRo\n0is an eigensolution of the r.h.s\noperator of (3.31), too. Thus, for (3.31) to be solvable, its r.h.s has to be free of the terms\nproportional to ˆRo\n0,that yields\nβo\n1=−1\n6(l−1)(l+2)((l−1)(l+2)+2m2). (3.32)\nNote that conversely to the frequency for longitudinal osci llation modes (3.18), the mag-\nnetic damping rate above is not degenerate and varies with m.\n3.2.2. Transversal modes\nSimilarly to the oscillation frequency considered above, b oundary conditions (3.10)\nand (3.8) applied to the first-order solution (3.23) result i n\nβ0ˆRe\n1+imˆϕ+\n1=/bracketleftbigg\nβe\n02/parenleftbigg\nm−2(z2Lr−r∂r)2−1\n2z2Lr−1/parenrightbigg\n−βe\n1/bracketrightbigg\nˆRe\n0,(3.33)\n(Lz+2)ˆRe\n1=−ˆpe\n1. (3.34)\nTo solve this first-order problem, we again need a second-ord er solution governed by\n∂2\nz{ˆpe\n2,ˆϕe\n2}=−βe\n0(Lr+∂2\nz){ˆpe\n1,ˆϕe\n1}−βe\n1Lr{ˆp+\n0,ˆϕ+\n0},\nwhich, by taking into account (3.23), yields\nˆϕe\n2(r,z) = ˆϕ+\n2(r)−1\n2βe\n0z2Lrˆϕ+\n1+βe\n02/bracketleftbiggz2\n2Lr+z4\n4!L2\nr/bracketrightbigg\nˆϕ+\n0−1\n2βe\n1z2Lrˆϕ+\n0, (3.35)\nThen (3.10) results in\nimˆpe\n1−βe\n0(z2Lr−r∂r)ˆϕ+\n1=im−1/bracketleftbigg\nβe\n0(z2Lr−r∂r)−1\n6βe\n02z2(z2Lr−3r∂r)Lr/bracketrightbigg\nˆRe\n0.\n(3.36)\nSubstituting ˆϕ+\n1andˆpe\n1from (3.33) and (3.36) into (3.34) and using\nz2(z2Lr−3r∂r)Lr−3(z2Lr−r∂r)z2Lr/vextendsingle/vextendsingle\nR=1≡2m2−2(Lz+m2)2,\nafter some algebra we obtain an equation for ˆRe\n1,which is the same as (3.25) for ˆRe\n0,\nexcept for the r.h.s. that now reads as\nβe\n0\n3m2/bracketleftbig\n(βe\n0/m)2/parenleftbig\n(Lz+m2)2−m2/parenrightbig\n(3Lz+2m2)−6βe\n1(Lz+m2)/bracketrightbigˆRe\n0\nBy the same arguments as for (3.31), the solvability conditi on applied to the expression\nabove results in\nβe\n1=−(l−1)(l+2)(l2−m2)((l+1)2−m2)(3l(l+1)−2m2)\n6(l(l+1)−m2)2, (3.37)\nwhich again coincides with the corresponding result of Gail itis (1966).\n3.3.Weak viscous damping\nThere are three effects due to viscosity in this problem. Firs tly, viscosity appears in\nthe normal stress balance condition (2.10) as a O(Ca)correction to the inviscid solu-\ntion obtained above. Secondly, viscosity also appears as a s mall parameter Cain (3.12)\nwhich again implies the same order correction when the leadi ng-order inviscid solution is\nsubstituted into this term. Thirdly, viscosity enters the p roblem implicitly through theOscillations of weakly viscous conducting liquid drops in a strong magnetic field 11\nfree-slip boundary condition (2.7) which was ignored by the inviscid solution but needs\nto be satisfied when viscosity is taken into account. To satis fy this condition, the leading-\norder solution needs to be corrected by the viscous term in (3 .12), where Caappears as\na small parameter at the higher-order derivative. For this s mall viscous term to become\ncomparable with the dominating magnetic term at the surface , the expected correction\nhas to vary over the characteristic length scale δ∼/radicalbig\nCa/Cm=Ha−1,which is defined\nby the Hartmann number Ha=B0R0/radicalbig\nσ/(ρν). Moreover, for the viscous correction of\nthe tangential velocity ˜vτin the Hartmann layer to compensate for a O(1)tangential\nstress due to the leading-order inviscid solution, ˜vτ∼Ha−1is required. Then the incom-\npressibility constraint implies an associated normal velo city component of an order in δ\nsmaller than ˜vτ,i.e.,˜vn∼Ha−2.This normal velocity correction is subsequently neg-\nligible. But this not the case for the tangential velocity co rrection ˜vτ,which according\nto (2.3) is expected to produce a pressure correction ˜p∼Cm/Ha2∼Ca.The last is\ncomparable with the normal viscous stress produced by the le ading-order inviscid flow.\nTaking into account the estimates above and ˜ϕ∼δ˜vφ∼Ha−2,which follows from (2.4),\nwe search for a viscous correction as\n{ˆp,ˆϕ,ˆv} ∼ {ˆp0,ˆϕ0,ˆv0}+Ca{ˆp01,ˆϕ01,ˆv01}+{Ca˜p,Ha−2˜ϕ,Ha−1˜v}···,\n{ˆR,β} ∼ {ˆR0,β0}+Ca{ˆR01,β01}+···,\nwhere the terms with the tilde account for a Hartmann layer so lution localised at the\nsurface.\n3.4.Eigenvalue perturbation for longitudinal modes\nWe start with the core region, where the additive boundary la yer corrections are supposed\nto vanish. The first-order viscous corrections for the press ure and potential {ˆpo\n01,ˆϕo\n01}(r,z) =\nz{ˆp−\n01,ˆϕ−\n01}(r)are obtained similarly to the leading-order inviscid solut ion (3.13). Now,\ninstead of the kinematic and electric boundary conditions ( 3.4) and (3.6) derived in the\ninviscid approximation, we have to use the original ones (2. 6) and (3.5) containing the\nvelocity, which again follows from the Navier-Stokes equat ion (3.1) including the viscous\nterm∼Ca.\nFor the longitudinal modes, described by the odd solutions, (3.1) yields\nβo\n0ˆwo\n01+βo\n01ˆwo\n0=−ˆp−\n01+Lrˆwo\n0, (3.38)\nˆuo\n01=ez×Dˆϕo\n01, (3.39)\nwhereˆwandˆuare the velocity components parallel and perpendicular, re spectively, to\nthe field direction ez,andD≡e−imφ∇eimφis a spectral counterpart of the nabla oper-\nator for the azimuthal mode m.Since, as shown above, both the potential and velocity\nperturbations in the Hartmann layer are higher-order small quantities and, thus, negli-\ngible with respect to the core perturbation, the electric bo undary condition (3.5) can be\napplied at R= 1directly to the first-order core solution as ∂Rˆϕo\n01=−eR·ez׈uo\n01.\nTaking into account (3.39), this yields ˆϕ−\n01≡0and, hence, ˆuo\n01≡0.Consequently, the\nfirst-order velocity perturbation in the core for the odd mod es is again purely longitu-\ndinal. Then, the kinematic constraint (2.6) for the leading - and first-order terms takes,\nrespectively, the form\nzˆwo\n0=βo\n0ˆRo\n0,\nzˆwo\n01=βo\n0ˆRo\n01+βo\n01ˆRo\n0.\nThese expressions combined with (3.38) result in\nβo\n0(βo\n0ˆRo\n01+2βo\n01ˆRo\n0−zLrz−1ˆRo\n0) =−ˆpo\n01, (3.40)12 J. Priede\nwhich defines the first-order core pressure perturbation at R= 1.In addition, we need\nalso the Hartmann layer pressure correction which accordin g to the estimates above is of\nthe same order of magnitude as the core one.\nTo resolve the Hartmann layer, we introduce a stretched coor dinate˜R= (1−R)/δ\n(Hinch 1991), where δ=Ha−1is the characteristic Hartmann layer thickness. In the\nHartmann layer variables, (3.12) takes the form\n(Cm−1β0+z2−∂2\n˜R)∂2\n˜R{˜p,˜ϕ,˜v}= 0. (3.41)\nForCm≫1, the inertial term ∼Cm−1is negligible in (3.41) with respect to the magnetic\none∼z2,except for |z|/lessorsimilarCm−1/2.First, ignoring this term, which, as shown below,\ngives a next-order small correction, the solution of (3.41) vanishing outside the Hartmann\nlayer can be written as\n{˜p,˜ϕ,˜v}={˜ps,˜ϕs,˜vs}(z)e−|z|˜R, (3.42)\nwhere the index sdenotes the surface distribution of the corresponding quan tity. Then\nthe free-slip boundary condition (2.7) results in\n˜vs\nφ=−|z|−1(imr−1ˆv0,R+∂R(ˆv0,φ/R)), (3.43)\n˜vs\nθ=−|z|−1(∂θˆv0,R+∂R(ˆv0,θ/R)). (3.44)\nFor the longitudinal modes, defined by the odd solutions, the leading-order inviscid ve-\nlocity is purely axial\nˆvo\n0=ezˆw0(r) =−ezβo\n0−1ˆp−\n0(r). (3.45)\nSubstituting this into (3.44) and taking into account that t he radial pressure distribution\nat the surface is related to the radius perturbation by (3.14 ), we obtain\n˜vs\nθ=β0r(z2−r2)\nz|z|d\ndzˆRo\n0\nz. (3.46)\nPressure is related to the velocity by (2.3), which in the dim ensionless form reads as\n∇2p=Cm∂zvz.In the Hartmann layer variables, this equation takes the for m\n∂2\n˜R˜p=rz∂˜R˜vθ. (3.47)\nSubstituting the general solutions for pressure and veloci ty given by (3.42) into (3.47)\nand using (3.46), we find\n˜ps=−rz|z|−1˜vs\nθ. (3.48)\nSubstituting the normal component of viscous stress\n−2∂Rˆvo\n0,R= 2βo\n0r2d\ndzˆRo\n0\nz\ntogether with the core and boundary layer pressure contribu tions defined by (3.40) and\n(3.48) into the normal stress balance condition (2.10), we fi nally obtain\n(Lz+2−βo\n02)ˆRo\n01=βo\n0/bracketleftBigg\n2βo\n01ˆRo\n0−z−2(Lz+m2+2)ˆRo\n0−2(1−z−2)d\ndzˆRo\n0\nz/bracketrightBigg\n.(3.49)\nThe sought for viscous damping rate is obtained in the usual w ay by applying the solv-\nability condition to (3.49) that after some algebra results in\nβo\n01=−(2l+1)(l−m)!\n(l+m)!/integraldisplay1\n0/bracketleftbigg1\n2(l(l+1)−m2−2)Pm\nl(z)\nz(3.50)Oscillations of weakly viscous conducting liquid drops in a strong magnetic field 13\nm= 0 1 2 3 4 5 6\nl= 25\n2\n335\n37\n451\n227\n2\n5154\n344 22\n6169\n2403\n665\n27135 125 95 45\nTable 1. The viscous damping rates −βo\n01for the first 6 longitudinal oscillation modes.\n−(z−z−1)d\ndzPm\nl(z)\nz/bracketrightbiggPm\nl(z)\nzdz=−(2l+1)/bracketleftbigg1\n2(l(l+1)−m2)−1−Im\nl/bracketrightbigg\n,\nwhere\nIm\nl=(l−m)!\n(l−m)!/integraldisplay1\n0Pm\nl(z)\nz(z−z−1)d\ndzPm\nl(z)\nzdz\n=((l−1)2−m2)Im\nl−2+(2l−1)(l(l−1)−m2)\nl2−m2(3.51)\ncan be calculated from the above recurrence relation starti ng withl=m+1and taking\ninto account that Im\nl= 0forl < m. For the modes with m=l−1, we have βo\n01=\n1\n2(2l+1)(l−1),which is the half of the corresponding viscous damping rate w ithout the\nmagnetic field (Lamb 1993). Although the viscous damping rat e increases for smaller m,\nas seen from the numerical values of −βo\n01for the first 7 longitudinal oscillation modes\ncalculated by the Mathematica (Wolfram 1996) and shown in ta ble 1, it remains below\nits non-magnetic counterpart up to l= 5modes.\nNote that the r.h.s of (3.49) has a simple pole (z−1)singularity at z= 0,which is due to\nthe neglected inertial term in (3.41). As discussed above, t his term becomes relevant for\n|z|/lessorsimilarCm−1/2,where it cuts off the singularity at z−1∼Cm1/2.This cut-off integrated\nin (3.50) over |z|/lessorsimilarCm−1/2,wherePm\nl(z)∼zfor the odd modes, results in the damping\nrate correction O(Cm−1/2),which is a higher-order small quantity.\n3.5.Viscous energy dissipation\nViscous damping rate can be found in an alternative much simp ler way by considering\nthe energy balance following from the dot product of (3.1) an dv,which integrated over\nthe drop volume yields\n1\n2∂t/integraldisplay\nVv2dV+/integraldisplay\nS(∇·n)v·ds=−/integraldisplay\nV(2Caε2+Cmj2)dV, (3.52)\nwhere the first and second term on the l.h.s. stand for the time -variation of kinetic and\nsurface energies, while the terms on the r.h.s. with the rate -of-strain tensor (ε)i,j=\n1\n2(vi,j+vj,i)and the dimensionless current density j=−∇ϕ+v×ǫaccount for the\nviscous and ohmic dissipations, respectively. As estimate d above, viscosity gives rise to\nthe tangential current density ∼Ha−1in the Hartmann layer of the thickness ∼Ha−1\nthat according to (3.52) produces the ohmic dissipation ∼Cm/Ha3∼Ca/Hawhich for\nHa≫1is negligible with respect to the viscous dissipation ∼Ca.Note that although\nthe contribution of the Hartmann layer to the normal stress b alance is important, its14 J. Priede\nm= 1 2 3 4 5 6 7\nl= 2 5\n370\n1114\n4135\n827\n51232\n29220\n744\n61339\n1950 65\n71350\n114930\n472250\n3190\nTable 2. The viscous damping rates −βe\n01for the first 6 transversal oscillation modes.\ncontribution to the energy dissipation is still negligible . This fact results in a substantial\nsimplification of the solution procedure for the viscous dam ping rate.\nThus, neglecting the ohmic dissipation and averaging the re st of (3.52) over the period\nof oscillation and taking into account that the mean kinetic and surface energies for small\namplitude harmonic oscillations are equal, we obtain a simp le expression for the viscous\ndamping rate in terms of inviscid leading order solution (La ndau & Lifshitz 1987)\nβ01=−/integraldisplay\nV|ˆε0|2dV//integraldisplay\nV|ˆv0|2dV. (3.53)\nFor the longitudinal modes, this equation takes the form\nβe\n01=−/integraltext1\n0[(rz−1∂zˆwo\n0)2+(mˆwo\n0/r)2]z2dz\n/integraltext1\n0ˆwo\n02z2dz. (3.54)\nSubstituting ˆwo\n0(z) =βo\n0Ro\n0Pm\nl(z)/zfrom (3.20) into (3.54), after some algebra the last\ncan be shown to be equivalent to (3.50).\nThis approach is particularly useful for the transversal mo des for which the conven-\ntional eigenvalue perturbation solution becomes excessiv ely complicated and, thus, it is\nomitted here. In this case, using (3.30) we can represent (3. 53) in terms of scalar potential\nβe\n01=−/integraltext1\n0/bracketleftBig/parenleftbig\nr∂r(r−1∂rˆϕe\n0)+m2ˆϕe\n0/r/parenrightbig2+(2m∂r(ˆϕe\n0/r))2/bracketrightBig\nz2dz\n/integraltext1\n0[(∂rˆϕe\n0)2+(mˆϕe\n0/r)2]z2dz. (3.55)\nSubstituting ˆϕe\n0(z) =im−1βe\n0Re\n0Pm\nl(z)from (3.28) into (3.55), after a lengthy algebra\nwe obtain\nβe\n01=−(2l+1)l(l+1)(l−2)−m2(l−3)+(l2−m2)Im\nl−1\n2(l(l+1)−m2), (3.56)\nwhereIm\nl−1is defined by (3.51). Note that for 2D modes, defined by m=l,which\nare not affected by the magnetic field, we recover the well-kno wn non-magnetic result\nβe\n01=−(2l+1)(l−1)(Lamb 1993). For other indices, (3.56) can be verified by a dir ect\nintegration of (3.55) using the Mathematica (Wolfram 1996) . As seen from the numerical\nvalues shown in table 2, the next even mode with m=l−2has the viscous damping\nrate which is by the factor of (l−2)/(l−4/5)lower than the non-magnetic counterpart\ngiven by m=l. Only for the modes with m≤l−4,the viscous damping rate in the\nmagnetic field becomes higher than that without the field.\nThe approach above is not directly applicable to the axisymm etric transversal modes\nwhich, as discussed at the end of §3.1.2, are stationary in the leading-order inviscid ap-Oscillations of weakly viscous conducting liquid drops in a strong magnetic field 15\nproximation. For these overdamped modes, a flow with the velo city∼1/Cmrelative\nto the leading-order radius perturbation appears only in th e first-order approximation,\nwhich according to (3.52) produces the same order ohmic diss ipation. In this case, dissi-\npation takes place on the account of the surface energy reduc tion, while that of the kinetic\nenergy is negligible because it is by ∼1/Cm2smaller than the former. The contribution\nof the viscous dissipation in (3.52) is ∼Ca/Cm2,which for a low viscosity and a high\nmagnetic field is much smaller than the ohmic dissipation ∼1/Cm,and, thus negligible\nwith respect to the latter.\n4. Conclusion\nIn the present study, we have considered small-amplitude os cillations of a conducting\nliquid drop in a uniform DC magnetic field. Viscosity was assu med to be small but\nthe magnetic field strong. Combining the regular and matched asymptotic expansion\ntechniques we obtained a relatively simple solution to the a ssociated eigenvalue problem.\nFirstly, we showed that the eigenmodes of shape oscillation s are not affected by strong\nmagnetic field. Namely, they remain the spherical harmonics as in the non-magnetic case.\nStrong magnetic field, however, constrains the liquid flow as sociated with the oscillations\nand, thus, reduces the oscillations frequency by increasin g apparent inertia of the liquid.\nIn such a field, liquid oscillates in a two-dimensional (2D) w ay as solid columns aligned\nwith the field. Two types of oscillations are possible: longi tudinal and transversal to the\nfield. Such oscillations are weakly damped by strong magneti c field – the stronger the\nfield, the weaker the damping, except for the axisymmetric tr ansversal and 2D modes.\nThe former are magnetically overdamped because the incompr essibility constraint does\nnot permit an axially uniform radial flow. The latter, which a re transversal modes defined\nby the spherical harmonics with equal degree and order, l=m, are not affected by the\nmagnetic field because these modes are naturally invariant a long the field. In a uniform\nmagnetic field, no electric current is induced and, thus, no e lectromagnetic force acts\non such a 2D transversal flow because the associated e.m.f. is irrotational. Because the\nmagnetic damping for all other modes decreases inversely wi th the square of the field\nstrength, the viscous damping may become important in a suffic iently strong magnetic\nfield. Consequently, the relaxation of axisymmetric transv ersal modes, whose viscous\ndamping is negligible relative to the magnetic one, can be us ed to determine the electrical\nconductivity, while the damping of l=mmodes can be used to determine the viscosity.\nThe damping of all other modes is affected by both the viscous a nd ohmic dissipations.\nAlthough the latter reduces inversely with the square of the field strength while the\nformer stays constant, an extremely strong magnetic field ma y be required for the viscous\ndissipation to be become dominant.\nAs an example, let us consider a drop of Nickel of 1cmin diameter (R0= 5×10−3m)\nwhich, at the melting point (1455◦C),has the surface tension γ= 1.8N/m, density\nρ= 7.9×103kg/m3,the dynamic viscosity η= 4.9×10−3Ns/m2and the elec-\ntrical conductivity σ= 1.2×106S/m(Gale & Totemeier 2004). The capillary time\nscale and frequency of non-magnetic fundamental mode (l= 2) for such a drop are\nτ0=/radicalbig\nR3\n0ρ/γ≈23msandf=/radicalbig\nl(l−1)(l+2)/(2πτ0)≈19Hz,respectively. The vis-\ncous damping time without the magnetic field (Lamb 1993) is τv/((2l+1)(l−1))≈8s,\nwhereτv=ρR2\n0/η≈40sis the viscous time scale. Note that weak-viscosity approx-\nimation is applicable in this case because Ca=τ0/τv= 5.8×10−4is small. In the\nmagnetic field of B= 5T,for which Cm=σB2R2\n0/√ργR0≈87≫1,the oscillation\nfrequency of longitudinal fundamental mode (l,m) = (2,1)drops according to equation\n(3.18) to fo\n2,1=/radicalbig\n(l−1)(l+2)/(2πτ0)≈14Hz.The corresponding viscous damping16 J. Priede\ntime increases by the factor of two to −τν/βo\n01≈16s,where−βo\n01= 5/2according to\ntable 1. The magnetic damping time of this mode, for which (3. 32) yields βo\n1=−4,is\n−τ0Cm/βo\n1≈0.5s.According to this formula, for the magnetic damping time to e xceed\nthe viscous one, a magnetic field of B/greaterorsimilar20Tis necessary. The relaxation time for the\naxisymmetric fundamental mode (l,m) = (2,0),which is magnetically over-damped, is\n−τ0Cm/βe\n1≈28ms,whereβe\n1= 72follows from (3.37). The magnetic field affects neither\nthe frequency nor the damping rate of (l,m) = (2,2)transversal oscillation mode, which\nis naturally invariant along the field. For the same reason, t here is no magnetic damp-\ning of this mode either. The first oscillatory transversal mo de is(l,m) = (3,1)whose\nfrequency drops according to (3.27) from fl=/radicalbig\nl(l−1)(l+2)/(2πτ0)≈38Hzwithout\nthe magnetic field to fe\n3,1=/radicalBig\n(l−1)(l+2)\nl(l+1)−m2/(2πτ0)≈5Hzin a strong magnetic field. The\nmagnetic damping time for this mode in a 5Tmagnetic field is −τ0Cm/βe\n1≈36ms,\nwhereβe\n1= 6800/121follows from (3.37). The viscous damping time for this mode i s\n−τν/βe\n01≈6s,whereβe\n01= 70/11follows from table 2. The viscous damping is small\nrelative to the magnetic one for this mode, and a magnetic fiel d of about 65Twould be\nnecessary for the magnetic damping time to become as long as t he viscous one.\nIn conclusion, this theoretical model provides a basis for t he development of new mea-\nsurement method of surface tension, viscosity and electric al conductivity of liquid metals\nusing oscillating drop technique in a strong superimposed D C magnetic field.\nThe author would like to thank Agris Gailitis and Raúl Avalos -Zúñiga for constructive\ncomments and stimulating discussions.\nREFERENCES\nAbramowitz, A. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.\nBasaran, O. A. 1992 Nonlinear oscillations of viscous liqui d drops. J. 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Box 3049, 67663 Kaiserslautern, Germany\n(Dated: June 4, 2018)\nWe present a microscopic calculation of magnetization damping for a magnetic \\toy model.\"\nThe magnetic system consists of itinerant carriers coupled antiferromagnetically to a dispersionless\nband of localized spins, and the magnetization damping is due to coupling of the itinerant carriers\nto a phonon bath in the presence of spin-orbit coupling. Using a mean-\feld approximation for\nthe kinetic exchange model and assuming the spin-orbit coupling to be of the Rashba form, we\nderive Boltzmann scattering integrals for the distributions and spin coherences in the case of an\nantiferromagnetic exchange splitting, including a careful analysis of the connection between lifetime\nbroadening and the magnetic gap. For the Elliott-Yafet type itinerant spin dynamics we extract\ndephasing and magnetization times T1andT2from initial conditions corresponding to a tilt of the\nmagnetization vector, and draw a comparison to phenomenological equations such as the Landau-\nLifshitz (LL) or the Gilbert damping. We also analyze magnetization precession and damping for\nthis system including an anisotropy \feld and \fnd a carrier mediated dephasing of the localized spin\nvia the mean-\feld coupling.\nPACS numbers: 75.78.-n, 72.25.Rb, 76.20.+q\nI. INTRODUCTION\nThere are two widely-known phenomenological ap-\nproaches to describe the damping of a precessing mag-\nnetization in an excited ferromagnet: one introduced\noriginally by Landau and Lifshitz1and one introduced\nby Gilbert,2which are applied to a variety of prob-\nlems3involving the damping of precessing magnetic mo-\nments. Magnetization damping contributions and its in-\nverse processes, i.e., spin torques, in particular in thin\n\flms and nanostructures, are an extremely active \feld,\nwhere currently the focus is on the determination of novel\nphysical processes/mechanisms. Apart from these ques-\ntions there is still a debate whether the Landau-Lifshitz\nor the Gilbert damping is the correct one for \\intrin-\nsic\" damping, i.e., neglecting interlayer coupling, inter-\nface contributions, domain structures and/or eddy cur-\nrents. This intrinsic damping is believed to be caused\nby a combination of spin-orbit coupling and scattering\nmechanisms such as exchange scattering between s and d\nelectrons and/or electron-phonon scattering.4{6Without\nreference to the microscopic mechanism, di\u000berent macro-\nscopic analyses, based, for example, on irreversible ther-\nmodynamics or near equilibrium Langevin theory, prefer\none or the other description.7,8However, material param-\neters of typical ferromagnetic heterostructures are such\nthat one is usually \frmly in the small damping regime so\nthat several ferromagnetic resonance (FMR) experiments\nwere not able to detect a noticeable di\u000berence between\nLandau Lifshitz and Gilbert magnetization damping. A\nrecent analysis that related the Gilbert term directly to\nthe spin-orbit interaction arising from the Dirac equa-\ntion does not seem to have conclusively solved this dis-\ncussion.9\nThe dephasing term in the Landau-Lifshitz form isalso used in models based on classical spins coupled\nto a bath, which have been successfully applied to\nout-of-equilibrium magnetization dynamics and magnetic\nswitching scenarios.10The most fundamental of these\nare the stochastic Landau-Lifshitz equations,10{13from\nwhich the Landau-Lifshitz Bloch equations,14,15can be\nderived via a Fokker-Planck equation.\nQuantum-mechanical treatments of the equilibrium\nmagnetization in bulk ferromagnets at \fnite temper-\natures are extremely involved. The calculation of\nnon-equilibrium magnetization phenomena and damp-\ning for quantum spin systems in more than one dimen-\nsion, which include both magnetism and carrier-phonon\nand/or carrier-impurity interactions, at present have to\nemploy simpli\fed models. For instance, there have been\nmicroscopic calculations of Gilbert damping parameters\nbased on Kohn-Sham wave functions for metallic ferro-\nmagnets16,17and Kohn-Luttinger p-dHamiltonians for\nmagnetic semiconductors.18While the former approach\nuses spin density-functional theory, the latter approach\ntreats the anti-ferromagnetic kinetic-exchange coupling\nbetween itinerant p-like holes and localized magnetic\nmoments originating from impurity d-electrons within a\nmean-\feld theory. In both cases, a constant spin and\nband-independent lifetime for the itinerant carriers is\nused as an input, and a Gilbert damping constant is ex-\ntracted by comparing the quantum mechanical result for\n!!0 with the classical formulation. There have also\nbeen investigations, which extract the Gilbert damping\nfor magnetic semiconductors from a microscopic calcula-\ntion of carrier dynamics including Boltzmann-type scat-\ntering integrals.19,20Such a kinetic approach, which is of\na similar type as the one we present in this paper, avoids\nthe introduction of electronic lifetimes because the scat-\ntering is calculated dynamically.arXiv:1405.2347v1 [cond-mat.mtrl-sci] 9 May 20142\nThe present paper takes up the question how the spin\ndynamics in the framework of the macroscopic Gilbert\nor Landau-Lifshitz damping compare to a microscopic\nmodel of relaxation processes in the framework of a rel-\natively simple model. We analyze a mean-\feld kinetic\nexchange model including spin-orbit coupling for the itin-\nerant carriers. Thus the magnetic mean-\feld dynamics is\ncombined with a microscopic description of damping pro-\nvided by the electron-phonon coupling. This interaction\ntransfers energy and angular momentum from the itin-\nerant carriers to the lattice. The electron-phonon scat-\ntering is responsible both for the lifetimes of the itiner-\nant carriers and the magnetization dephasing. The lat-\nter occurs because of spin-orbit coupling in the states\nthat are connected by electron-phonon scattering. To be\nmore speci\fc, we choose an anti-ferromagnetic coupling\nat the mean-\feld level between itinerant electrons and\na dispersion-less band of localized spins for the magnetic\nsystem. To keep the analysis simple we use as a model for\nthe spin-orbit coupled itinerant carrier states a two-band\nRashba model. As such it is a single-band version of the\nmulti-band Hamiltonians used for III-Mn-V ferromag-\nnetic semiconductors.18,21{24The model analyzed here\nalso captures some properties of two-sublattice ferrimag-\nnets, which are nowadays investigated because of their\nmagnetic switching dynamics.25,26The present paper is\nset apart from studies of spin dynamics in similar mod-\nels with more complicated itinerant band structures19,20\nby a detailed comparison of the phenomenological damp-\ning expressions with a microscopic calculation as well as\na careful analysis of the restrictions placed by the size\nof the magnetic gap on the single-particle broadening in\nBoltzmann scattering.\nThis paper is organized as follows. As an extended\nintroduction, we review in Sec. II some basic facts con-\ncerning the Landau-Lifshitz and Gilbert damping terms\non the one hand and the Bloch equations on the other.\nIn Sec. III we point out how these di\u000berent descriptions\nare related in special cases. We then introduce a micro-\nscopic model for the dephasing due to electron-phonon\ninteraction in Sec. IV, and present numerical solutions\nfor two di\u000berent scenarios in Secs. V and VI. The \frst\nscenario is the dephasing between two spin subsystems\n(Sec. V), and the second scenario is a relaxation process\nof the magnetization toward an easy-axis (Sec. VI). A\nbrief conclusion is given at the end.\nII. PHENOMENOLOGIC DESCRIPTIONS OF\nDEPHASING AND RELAXATION\nWe summarize here some results pertaining to a single-\ndomain ferromagnet, and set up our notation. In equilib-\nrium we assume the magnetization to be oriented along\nits easy axis or a magnetic \feld ~H, which we take to\nbe thezaxis in the following. If the magnetization\nis tilted out of equilibrium, it starts to precess. As\nillustrated in Fig. 1 one distinguishes the longitudinal\nFIG. 1. Illustration of non-equilibrium spin-dynamics in pres-\nence of a magnetic \feld without relaxation (a) and within\nrelaxation (b).\ncomponent Mk, inzdirection, and the transverse part\nM?\u0011q\nM2\u0000M2\nk, precessing in the x-yplane with the\nLarmor frequency !L.\nIn connection with the interaction processes that re-\nturn the system to equilibrium, the decay of the trans-\nverse component is called dephasing. There are three\nphenomenological equations used to describe spin de-\nphasing processes:\n1. The Bloch(-Bloembergen) equations27,28\n@\n@tMk(t) =\u0000Mk(t)\u0000Meq\nT1(1)\n@\n@tM?(t) =\u0000M?(t)\nT2(2)\ndescribe an exponential decay towards the equilib-\nrium magnetization Meqinzdirection. The trans-\nverse component decays with a time constant T2,\nwhereas the longitudinal component approaches its\nequilibrium amplitude with T1. These time con-\nstants may be \ft independently to experimental\nresults or microscopic calculations.\n2. Landau-Lifshitz damping1with parameter \u0015\n@\n@t~M(t) =\u0000\r~M\u0002~H\u0000\u0015~M\nM\u0002\u0000~M\u0002~H\u0001\n(3)\nwhere\ris the gyromagnetic ratio. The \frst term\nmodels the precession with a frequency !L=\rj~Hj,\nwhereas the second term is solely responsible for\ndamping.\n3. Gilbert damping2with the dimensionless Gilbert\ndamping parameter \u000b\n@\n@t~M(t) =\u0000\rG~M\u0002~H+\u000b\u0010~M\nM\u0002@t~M\u0011\n(4)\nIt is generally accepted that \u000bis independent of\nthe static magnetic \felds ~Hsuch as anisotropy\n\felds,18,29and thus depends only on the material\nand the microscopic interaction processes.3\nThe Landau-Lifshitz and Gilbert forms of damping are\nmathematically equivalent2,7,30with\n\u000b=\u0015\n\r(5)\n\rG=\r(1 +\u000b2) (6)\nbut there are important di\u000berences. In particular, an in-\ncrease of\u000blowers the precession frequency in the dynam-\nics with Gilbert damping, while the damping parameter\n\u0015in the Landau-Lifshitz equation has no impact on the\nprecession. In contrast to the Bloch equations, Landau-\nLifshitz and Gilbert spin-dynamics always conserve the\nlengthj~Mjof the magnetization vector.\nAn argument by Pines and Slichter,31shows that there\nare two di\u000berent regimes for Bloch-type spin dynamics\ndepending on the relation between the Larmor period and\nthe correlation time. As long as the correlation time is\nmuch longer than the Larmor period, the system \\knows\"\nthe direction of the \feld during the scattering process.\nStated di\u000berently, the scattering process \\sees\" the mag-\nnetic gap in the bandstructure. Thus, transverse and\nlongitudinal spin components are distinguishable and the\nBloch decay times T1andT2can di\u000ber. If the correlation\ntime is considerably shorter than the Larmor period, this\ndistinction is not possible, with the consequence that T1\nmust be equal to T2. Within the microscopic approach,\npresented in Sec. IV D, this consideration shows up again,\nalbeit for the energy conserving \u000efunctions resulting from\na Markov approximation.\nThe regime of short correlation times has already been\ninvestigated in the framework of a microscopic calcula-\ntion by Wu and coworkers.32They analyze the case of\na moderate external magnetic \feld applied to a non-\nmagnetic n-type GaAs quantum well and include di\u000ber-\nent scattering mechanisms (electron-electron Coulomb,\nelectron-phonon, electron-impurity). They argue that\nthe momentum relaxation rate is the crucial time scale\nin this scenario, which turns out to be much larger than\nthe Larmor frequency. Their numerical results con\frm\nthe identity T1=T2expected from the Pines-Slichter\nargument.\nIII. RELATION BETWEEN\nLANDAU-LIFSHITZ, GILBERT AND BLOCH\nWe highlight here a connection between the Bloch\nequations (1, 2) and the Landau-Lifshitz equation (3).\nTo this end we assume a small initial tilt of the mag-\nnetization and describe the subsequent dynamics of the\nmagnetization in the form\n~M(t) =0\n@\u000eM?(t) cos(!Lt)\n\u000eM?(t) sin(!Lt)\nMeq\u0000\u000eMk(t)1\nA (7)\nwhere\u000eM?and\u000eMjjdescribe deviations from equilib-\nrium. Putting this into eq. (3) one gets a coupled set ofequations.\n@\n@t\u000eM?(t) =\u0000\u0015HMeq\u0000\u000eMk(t)\nj~M(t)j\u000eM?(t) (8)\n@\n@t\u000eMk(t) =\u0000\u0015H1\nj~M(t)j\u000eM2\n?(t) (9)\nEq. (8) is simpli\fed for a small deviation from equilib-\nrium, i.e.,\u000eM(t)\u001cMeqandj~M(t)j\u0019Meq:\n\u000eM?(t) =Cexp(\u0000\u0015Ht) (10)\n\u000eMk(t) =C2\n2Meqexp(\u00002\u0015Ht) (11)\nwhereCis an integration constant. For small excitations\nthe deviations decay exponentially and Bloch decay times\nT1andT2result, which are related by\n2T1=T2=1\n\u0015H: (12)\nOnly this ratio of the Bloch times is compatible with a\nconstant length of the magnetization vector at low exci-\ntations. By combining Eqs. (12) and (5) one can connect\nthe Gilbert parameter \u000band the dephasing time T2\n\u000b=1\nT2!L: (13)\nIf the conditions for the above approximations apply, the\nGilbert damping parameter \u000bcan be determined by \ft-\nting the dephasing time T2and the Larmor frequency !L\nto computed or measured spin dynamics. This dimen-\nsionless quantity is well suited to compare the dephasing\nthat results from di\u000berent relaxation processes.\nFigure 2 shows the typical magnetization dynamics\nthat results from (3), i.e., Landau-Lifshitz damping. As\nan illustration of a small excitation we choose in Fig. 2(a)\nan angle of 10\u000efor the initial tilt of the magnetization,\nwhich results in an exponential decay with 2 T1=T2.\nFrom the form of Eq. (3) it is clear that this behavior\npersists even for large !Land\u0015. Obviously the Landau-\nLifshitz and Gilbert damping terms describe a scenario\nwith relatively long correlation times (i.e., small scat-\ntering rates), because only in this regime both decay\ntimes can di\u000ber. The microscopic formalism in Sec. IV\nworks in the same regime and will be compared with\nthe phenomenological results. For an excitation angle\nof 90\u000e, the Landau-Lifshitz dynamics shown in Fig. 2(b)\nbecome non-exponential, so that no well-de\fned Bloch\ndecay times T1,T2exist.\nIV. MICROSCOPIC MODEL\nIn this section we describe a microscopic model that in-\ncludes magnetism at the mean-\feld level, spin-orbit cou-\npling as well as the microscopic coupling to a phonon\nbath treated at the level of Boltzmann scattering inte-\ngrals. We then compare the microscopic dynamics to4\n0 5000.51δM⊥/Meq\ntime (ps)0 5000.51\ntime (ps)δM/bardbl/Meq0 5000.010.02δM/bardbl/Meq\ntime (ps)0 5000.10.2\ntime (ps)δM⊥/Meq\n \nT1= 5.02 ps(a)\n(b)T2= 10.04 ps\nFIG. 2. Dynamics of \u000eM?and\u000eMkcomputed using to\nLandau-Lifshitz damping ( !L= 1 ps\u00001,H= 106A\nm\u0019\n1:26\u0001104Oe,\u0015= 10\u00007m\nA ps). (a) An angle of 10\u000eleads to\nexponential an exponential decay with well de\fned T1andT2\ntimes. (b). For an angle of 90\u000e, the decay (solid line) is not\nexponential as comparison with the exponential \ft (dashed\nline) clearly shows.\nthe Bloch equations (1), (2), as well as the Landau-\nLifshitz (3) and Gilbert damping terms (4). The mag-\nnetic properties of the model are de\fned by an anti-\nferromagnetic coupling between localized magnetic im-\npurities and itinerant carriers. As a prototypical spin-\norbit coupling we consider an e\u000bectively two-dimensional\nmodel with a Rashba spin-orbit coupling. The reason\nfor the choice of a model with a two-dimensional wave\nvector space is not an investigation of magnetization dy-\nnamics with reduced dimensionality, but rather a reduc-\ntion in the dimension of the integrals that have to be\nsolved numerically in the Boltzmann scattering terms.\nSince we treat the exchange between the localized and\nitinerant states in a mean-\feld approximation, our two-\ndimensional model still has a \\magnetic ground state\"\nand presents a framework, for which qualitatively dif-\nferent approaches can be compared. We do not aim at\nquantitative predictions for, say, magnetic semiconduc-\ntors or ferrimagnets with two sublattices. Finally, we\ninclude a standard interaction hamiltonian between the\nitinerant carriers and acoustic phonons. The correspond-\ning hamiltonian reads\n^H=^Hmf+^Hso+^He\u0000ph+^Haniso: (14)\nOnly in Sec. VI an additional \feld ^Haniso is included,\nwhich is intended to model a small anisotropy.A. Exchange interaction between itinerant carriers\nand localized spins\nThe \\magnetic part\" of the model is described by the\nHamiltonian\n^Hmf=X\n~k\u0016~2k2\n2m\u0003^cy\n~k\u0016^c~k\u0016+J^~ s\u0001^~S: (15)\nwhich we consider in the mean-\feld limit. The \frst term\nrepresents itinerant carriers with a k-dependent disper-\nsion relation. In the following we assume s-like wave\nfunctions and parabolic energy dispersions. The e\u000bective\nmass is chosen to be m\u0003= 0:5me, wheremeis the free\nelectron mass, and the ^ c(y)\n~k\u0016operators create and annihi-\nlate carriers in the state j~k;\u0016iwhere\u0016labels the itinerant\nbands, as shown in Fig. 3(a).\nThe second term describes the coupling between itiner-\nant spins~ sand localized spins ~Svia an antiferromagnetic\nexchange interaction\n^~ s=1\n2X\n~kX\n\u0016\u00160h~k;\u00160j^~ \u001bj~k;\u0016i^cy\n~k\u0016^c~k\u00160 (16)\n^~S=1\n2X\n\u0017\u00170h\u00170j^~ \u001bj\u0017iX\n~K^Cy\n~K\u0017^C~K\u00170 (17)\nHere, we have assumed that the wave functions of the lo-\ncalized spins form dispersionless bands, i.e., we have im-\nplicitly introduced a virtual-crystal approximation. Due\nto the assumption of strong localization there is no or-\nbital overlap between these electrons, which are therefore\nconsidered to have momentum independent eigenstates\nj\u0017iand a \rat dispersion, as illustrated in Fig. 3(a). The\ncomponents of the vector ^~ \u001bare the Pauli matrices ^ \u001biwith\ni=x;y;z , and ^C(y)\n~K\u0017are the creation and annihilation op-\nerators for a localized spin state.\nWe do notinclude interactions among localized or itin-\nerant spins, such as exchange scattering. For simplicity,\nwe assume both itinerant and localized electrons to have\na spin 1=2 and therefore \u0016and\u0017to run over two spin-\nprojection quantum numbers \u00061=2. In the following we\nchosse an antiferromagnetic ( J > 0) exchange constant\nJ= 500 meV, which leads to the schematic band struc-\nture shown in Fig. 3(b).\nIn the mean \feld approximation used here, the itiner-\nant carriers feel an e\u000bective magnetic \feld ^Hloc\n~Hloc=\u0000J\u0016B\u0016\ng~S (18)\ncaused by localized moments and vice versa. Here \u0016B\nis the Bohr magneton and g= 2 is the g-factor of the\nelectron. The permeability \u0016is assumed to be the vac-\nuum permeability \u00160. This time-dependent magnetic\n\feld~Hloc(t) de\fnes the preferred direction in the itiner-\nant sub-system and therefore determines the longitudinal\nand transverse component of the itinerant spin at each\ntime.5\nr#k (a) (b) E(k) \nk \n\u0010\nk \n\u000e\u0010,k\n\u000e,kE(k) \nEF \nFIG. 3. Sketch of the band-structure with localized (\rat\ndispersions) and itinerant (parabolic dispersions) electrons.\nAbove the Curie-Temperature TCthe spin-eigenstates are de-\ngenerate (a), whereas below TCa gap between the spin states\nexists.\nB. Rashba spin-orbit interaction\nThe Rashba spin-orbit coupling is given by the Hamil-\ntonian\n^Hso=\u000bR(^\u001bxky\u0000^\u001bykx) (19)\nA Rashba coe\u000ecient of \u000bR= 10 meV nm typical for semi-\nconductors is chosen in the following calculations. This\nvalue, which is close to the experimental one for the\nInSb/InAlSb material system,33is small compared to the\nexchange interactions, but it allows the exchange of an-\ngular momentum with the lattice.\nC. Coherent dynamics\nFrom the above contributions (15) and (19) to the\nHamiltonian we derive the equations of motion contain-\ning the coherent dynamics due to the exchange interac-\ntion and Rashba spin-orbit coupling as well as the inco-\nherent electron-phonon scattering. We \frst focus on the\ncoherent contributions. In principle, one has the choice\nto work in a basis with a \fxed spin-quantization axis or\nto use single-particle states that diagonalize the mean-\n\feld (plus Rashba) Hamiltonian. Since we intend to use\na Boltzmann scattering integral in Sec. IV D we need to\napply a Markov approximation, which only works if one\ndeals with diagonalized eigenenergies. In our case this is\nthe single-particle basis that diagonalizes the entire one-\nparticle contribution of the Hamiltonian ^Hmf+^Hso. In\nmatrix representation this one-particle contribution for\nthe itinerant carriers reads:\n^Hmf+^Hso= \n~2k2\n2m\u0003+ \u0001loc\nz(\u0001loc\n++R~k)\u0003\n\u0001loc\n++R~k~2k2\n2m\u0003\u0000\u0001loc\nz!\n(20)\nwhere we have de\fned \u0001loc\ni=J1\n2h^SiiandR~k=\n\u0000i\u000bRkexp(i'k) with'k= arctan(ky=kx). The eigenen-\nergies are\n\u000f\u0006\n~k=~2k2\n2m\u0003\u0007q\nj\u0001loczj2+jR~k+ \u0001loc\n+j2: (21)and the eigenstates\nj~k;+i=\u0012\n1\n\u0018~k\u0013\n;j~k;\u0000i=\u0012\u0000\u0018\u0003\n~k\n1\u0013\n(22)\nwhere\n\u0018~k=\u0001loc\n++R~k\n\u0001locz+q\nj~\u0001locj2+jR~kj2(23)\nIn this basis the coherent part of the equation of mo-\ntion for the itinerant density matrix \u001a\u0016\u00160\n~k\u0011 h^cy\n~k\u0016^c~k\u00160i\nreads\n@\n@t\u001a\u0016\u00160\n~k\f\f\f\ncoh=i\n~\u0000\n\u000f\u0016\n~k\u0000\u000f\u00160\n~k\u0001\n\u001a\u0016\u00160\n~k: (24)\nNo mean-\feld or Rashba terms appear explicitly in these\nequations of motion since their contributions are now hid-\nden in the time-dependent eigenstates and eigenenergies.\nSince we are interested in dephasing and precessional\ndynamics, we assume a comparatively small spin-orbit\ncoupling, that can dissipate angular momentum into the\nlattice, but does not have a decisive e\u000bect on the band-\nstructure. Therefore we use the spin-mixing only in the\ntransition matrix elements of the electron-phonon scat-\nteringM~k0\u00160\n~k\u0016(31). For all other purposes we set R~k= 0.\nIn particular, the energy-dispersion \u000f\u0006\n~kis assumed to be\nuna\u000bected by the spin-orbit interaction and therefore it\nis spherically symmetric.\nWith this approximation the itinerant eigenstates are\nalways exactly aligned with the e\u000bective \feld of the local-\nized moments ~Hloc(t). Since this e\u000bective \feld changes\nwith time, the diagonalization and a transformation of\nthe spin-density matrix in \\spin space\" has to be re-\npeated at each time-step. This e\u000bort makes it easier\nto identify the longitudinal and transverse spin compo-\nnents with the elements of the single-particle density\nmatrix: The o\u000b-diagonal entries of the density matrix\n\u001a\u0006\u0007\n~k, which precess with the k-independent Larmor fre-\nquency!L= 2\u0001loc=~, always describe the dynamics of\nthe transverse spin-component. The longitudinal compo-\nnent, which does not precess, is represented by the diag-\nonal entries \u001a\u0006\u0006\n~k. Since both components change their\nspatial orientation continuously, we call this the rotating\nframe. The components of the spin vector in the rotating\nframe are\nh^ski=1\n2X\n~k\u0000\n\u001a++\n~k\u0000\u001a\u0000\u0000\n~k\u0001\n(25)\nh^s?i=X\n~k\f\f\u001a+\u0000\n~k\f\f (26)\nThe components in the \fxed frame are obtained from\nEq. (16)\nh^~ si=1\n2X\n~kX\n\u0016\u00160h~k;\u00160j^~ \u001bj~k;\u0016i\u001a\u0016\u00160\n~k(27)6\nIn this form, the time-dependent states carry the infor-\nmation how the spatial components are described by the\ndensity matrix at each time step. No time-independent\n\\longitudinal\" and \\transverse\" directions can be identi-\n\fed in the \fxed frame.\nIn a similar fashion, the diagonalized single-particle\nstates of the localized spin system are obtained. The\neigenenergies are\nE\u0006=\u0007\f\f~\u0001itin\f\f (28)\nwhere \u0001itin\ni=J1\n2h^siiis the localized energy shift caused\nby the itinerant spin component si. The eigenstates are\nagain always aligned with the itinerant magnetic mo-\nment. In this basis the equation of motion of the localized\nspin-density matrix \u001a\u0017\u00170\nloc\u0011P\n~Kh^Cy\n~K\u0017^C~K\u00170iis simply\n@\n@t\u001a\u0017\u00170\nloc=i\n~(E\u0017\u0000E\u00170)\u001a\u0017\u00170\nloc (29)\nand does not contain explicit exchange contributions.\nEqs. (25), (26), and (27) apply in turn to the components\nhSkiandhS?iof the localized spin and its spin-density\nmatrix\u001a\u0017\u00170\nloc.\nD. Electron-phonon Boltzmann scattering with\nspin splitting\nRelaxation is introduced into the model by the interac-\ntion of the itinerant carriers with a phonon bath, which\nplays the role of an energy and angular momentum sink\nfor these carriers. Our goal here is to present a derivation\nof the Boltzmann scattering contributions using stan-\ndard methods, see, e.g., Refs. 34 and 36. However, we\nemphasize that describing interaction as a Boltzmann-\nlike instantaneous, energy conserving scattering process\nis limited by the existence of the magnetic gap. Since we\nkeep the spin mixing due to Rashba spin-orbit coupling\nonly in the Boltzmann scattering integrals, the resulting\ndynamical equations describe an Elliott-Yafet type spin\nrelaxation.\nThe electron-phonon interaction Hamiltonian reads34\n^He\u0000ph=X\n~ q~!ph\nq^by\n~ q^b~ q\n+X\n~k~k0X\n\u0016\u00160\u0000\nM~k0\u00160\n~k\u0016^cy\n~k\u0016^b~k\u0000~k0^c~k0\u00160+ h.c.\u0001(30)\nwhere ^b(y)\n~ qare the bosonic operators, that create or an-\nnihilate acoustic phonons with momentum ~ qand linear\ndispersion!ph(q) =cphj~ qj. The sound velocity is taken\nto becph= 40 nm/ps and we use an e\u000bectively two-\ndimensional transition matrix element35\nM~k0\u00160\n~k\u0016=Dq\nj~k\u0000~k0jh~k;\u0016j~k0;\u00160i (31)\nwhere the deformation potential is chosen to be D=\n60 meVnm1=2. The scalar-product between the initialstatej~k0;\u00160iand the \fnal state j~k;\u0016iof an electronic\ntransition takes the spin-mixing due to Rashba spin-orbit\ncoupling into account.\nThe derivation of Boltzmann scattering integrals for\nthe itinerant spin-density matrix (24) leads to a memory\nintegral of the following shape\n@\n@t\u001aj(t)\f\f\f\ninc=1\n~X\nj0Zt\n\u00001ei(\u0001Ejj0+i\r)(t\u0000t0)Fjj0[\u001a(t0)]dt0;\n(32)\nregardless whether one uses Green's function36or\nequation-of-motion techniques.34Since we go through a\nstandard derivation here, we highlight only the impor-\ntant parts for the present case and do not write the equa-\ntions out completely. In particular, for scattering process\nj0=j\u00160;~k0i!j=j\u0016;~ki, we useFjj0[\u001a(t0)] as an abbre-\nviation for a product of dynamical electronic spin-density\nmatrix elements \u001a, evaluated at time t010\u000e) on the spin dynamics in\nthe microscopic calculation. Apart from this the initial\ncondition of the dynamics is the same as before, in par-\nticular, the itinerant spin is tilted such that the absolute\nvalue of the spin is unchanged.\nFigure 11 shows the time development of the skand\ns?components of the itinerant spin in the rotating frame\nfor an initial tilt angle \f= 140\u000e. While the transverse\ncomponent s?in the rotating frame can be well described\nby an exponential decay, the longitudinal component sk\nshows a di\u000berent behavior. It initially decreases with a\ntime constant of less than 1 ps, but does not reach its\nequilibrium value. Instead, the eventual return to equi-\nlibrium takes place on a much longer timescale, during\nwhich the s?component is already vanishingly small.\nThe long-time dynamics are therefore purely collinear.\nFor the short-time dynamics, the transverse component\ncan be \ft well by an exponential decay, even for large ex-\ncitation angles. This behavior is di\u000berent from Landau-\nLifshitz and Gilbert dynamics, cf. Fig. 2, which both ex-\nhibit non-exponential decay of the transverse spin com-\nponent.\nIn Fig. 12 the dependence of T2on the excitation an-\ngle is shown. From small \fup to almost 180\u000e, the decay\ntime decreases by more than 50%. This dependence is\nexclusively due to the \\excitation condition,\" which in-\n0 1 2 3 4−0.100.1\ntime (ps)s/bardbl\n0 1 2 3 400.050.1\ntime (ps)s⊥FIG. 11. Dynamics of the longitudinal and transverse itiner-\nant spin components in the rotating frame (solid lines) for a\ntilt angle of \f= 140\u000e, together with exponential \fts toward\nequilibrium (dashed lines). The longitudinal equilibrium po-\nlarization is shown as a dotted line.\nvolves only spin degrees of freedom (\\tilt angle\"), but no\nchange of temperature. Although one can \ft such a T2\ntime to the transverse decay, the overall behavior with\nits two stages is, in our view, qualitatively di\u000berent from\nthe typical Bloch relaxation/dephasing picture.\nTo highlight the similarities and di\u000berences from the\nBloch relaxation/dephasing we plot in Fig. 13 the mod-\nulus of the itinerant spin vector j~ sjin the rotating\nframe, whose transverse and longitudinal components\nwere shown in Fig. 11. Over the 2 ps, during which the\ntransverse spin in the rotating frame essentially decays,\nthe modulus of the spin vector undergoes a fast initial\ndecrease and a partial recovery. The initial length of ~ s\nis recovered only over a much larger time scale of several\nhundred picoseconds (not shown). Thus the dynamics\ncan be seen to di\u000ber from a Landau-Lifshitz or Gilbert-\nlike scenario because the spin does not precess toward\nequilibrium with a constant length. Additionally they\ndi\u000ber from Bloch-like dynamics because there is a com-\nbination of the fast and slow dynamics that cannot be\ndescribed by a single set of T1andT2times. We stress\nthat the microscopic dynamics at larger excitation angles\nshow a precessional motion of the magnetization with-\nout heating and a slow remagnetization. This scenario is\nsomewhat in between typical small angle-relaxation, for\nwhich the modulus of the magnetization is constant and\nwhich is well described by Gilbert and Landau-Lifshitz\ndamping, and collinear de/remagnetization dynamics.\nVI. EFFECT OF ANISOTROPY\nSo far we have been concerned with the question\nhow phenomenological equations describe dephasing pro-\ncesses between itinerant and localized spins, where the11\n0 50 100 1500.40.60.81\nβ(◦)T2(ps)\nFIG. 12.T2time extracted from exponential \ft to s?dynam-\nics in rotating frame for di\u000berent initial tilting angles \f.\n0 0.5 1 1.5 20.040.060.080.10.120.14\ntime (ps)|s|\n \n10°\n50°\n90°\n140°\nFIG. 13. Dynamics of the modulus j~ sjof the itinerant spin\nfor di\u000berent initial tilt angles \f. Note the slightly di\u000berent\ntime scale compared to Fig. 11.\nmagnetic properties of the system were determined by a\nmean-\feld exchange interaction only. Oftentimes, phe-\nnomenological models of spin dynamics are used to de-\nscribe dephasing processes toward an \\easy axis\" deter-\nmined by anisotropy \felds.29\nIn order to capture in a simple fashion the e\u000bects of\nanisotropy on the spin dynamics in our model, we sim-\nply assume the existence of an e\u000bective anisotropy \feld\n~Haniso, which enters the Hamiltonian via\n^Haniso =\u0000g\u0016B\u0016^~ s\u0001~Haniso (40)\nand only acts on the itinerant carriers. Its strength is\nassumed to be small in comparison to the \feld of the\nlocalized moments ~Hloc. This additional \feld ~Haniso has\nto be taken into account in the diagonalization of the\ncoherent dynamics as well, see section IV C.\nFor the investigation of the dynamics with anisotropy,\nwe choose a slightly di\u000berent initial condition, which is\nshown in Fig. 14. In thermal equilibrium, both spins\nare now aligned, with opposite directions, along the\nanisotropy \feld ~Haniso, which is assumed to point in the\nzdirection. At t= 0 they are both rigidly tilted by an\n5&:P; \nO&:P; U T \nV E *_lgqm FIG. 14. Dynamics of the localized spin ~Sand itinerant spin\n~ s. Att= 0, the equilibrium con\fguration of both spins is\ntilted (\f= 10\u000e) with respect to an anisotropy \feld ~Haniso.\nThe anisotropy \feld is only experienced by the itinerant sub-\nsystem.\n01002003004005006000.490.4950.5\ntime(ps)Sz(t)\n010020030040050060000.050.1\ntime(ps)/radicalBig\nS2x(t)+S2y(t)\n \nFIG. 15. Relaxation dynamics of the localized spin toward the\nanisotropy direction for longitudinal component Szand the\ntransverse componentp\nS2x+S2y. An exponential \ft yields\nBloch decay times of Taniso\n1 = 67:8 ps andTaniso\n2 = 134:0 ps.\nangle\f= 10\u000ewith respect to the anisotropy \feld.\nFigure 14 shows the time evolution of both spins in the\n\fxed frame, with zaxis in the direction of the anisotropy\n\feld for the same material parameters as in the previous\nsections and an anisotropy \feld ~Haniso =\u0000108A\nm\u0001~ ez.\nThe dynamics of the entire spin-system are somewhat\ndi\u000berent now, as the itinerant spin precesses around the\ncombined \feld of the anisotropy and the localized mo-\nments. The localized spin precesses around the itinerant\nspin, whose direction keeps changing as well.\nFigure 15 contains the dynamics of the components\nof the localized spin in the rotating frame. Both com-\nponents show an exponential behavior that allows us to\nextract well de\fned Bloch-times Taniso\n1 andTaniso\n2. Again\nwe \fnd the ratio of 2 Taniso\n1\u0019Taniso\n2, because the abso-\nlute value of the localized spin does not change, as it is\nnot coupled to the phonon bath.\nIn Fig. 16 the Larmor-frequency !aniso\nL, which is the\nprecession frequency due to the anisotropy \feld, and the\nBloch decay times Taniso\n2 are plotted vs. the strength of\nthe anisotropy \feld ~Haniso. The Gilbert damping pa-12\n0 5 10 150510\nHaniso(107A/m)ωaniso\nL (ps−1)\n0 5 10 1505001000\nHaniso(107A/m)Taniso\n2 (ps)\n0 5 10 1501020\nHaniso(107A/m)αaniso (10−4)\nFIG. 16. Larmor frequency !aniso\nL and Bloch decay time Taniso\n2\nextracted from the spin dynamics vs. anisotropy \feld Haniso,\nas well as the corresponding damping parameter \u000baniso.\nrameter\u000baniso for the dephasing dynamics computed via\nEq. (13) is also presented in this \fgure.\nThe plot reveals a decrease of the dephasing time Taniso\n2\nand a almost linear increase of the Larmor frequency\n!aniso\nL with the strength of the anisotropy \feld Haniso.\nThe Gilbert damping parameter \u000baniso shows only a neg-\nligible dependence on the anisotropy \feld Haniso. This\ncon\frms the statement that, in contrast to the dephas-\ning rates, the Gilbert damping parameter is independent\nof the applied magnetic \feld. In the investigated range\nwe \fnd an almost constant value of \u000baniso'9\u000210\u00004.\nThe Gilbert damping parameter \u000baniso for the de-\nphasing toward the anisotropy \feld is about 4 times\nsmaller than \u000biso, which describes the dephasing between\nboth spins. This disparity in the damping e\u000eciency\n(\u000baniso< \u000b iso) is obviously due to a fundamental di\u000ber-\nence in the dephasing mechanism. In the anisotropy case\nthe localized spin dephases toward the zdirection with-\nout being involved in scattering processes with itinerant\ncarriers or phonons. The dynamics of the localized spins\nis purely precessional due to the time-dependent mag-\nnetic moment of the itinerant carriers ~Hitin(t). Thus,\nonly this varying magnetic \feld, that turns out to be\nslightly tilted against the localized spins during the en-\ntire relaxation causes the dephasing, in presence of the\ncoupling between itinerant carriers and a phonon bath,\nwhich acts as a sink for energy and angular momentum.\nThe relaxation of the localized moments thus occurs only\nindirectly as a carrier-meditated relaxation via their cou-\npling to the time dependent mean-\feld of the itinerant\nspin.\nNext, we investigate the dependence of the Gilbert pa-\nrameter\u000baniso on the bath coupling. Fig. 17 shows that\n0 50 100 150 20000.0040.0080.012\nD(meV√nm)αaniso\n FIG. 17. Damping parameter \u000baniso vs. coupling constant D\n(black diamonds). The red line is a quadratic \ft, indicative\nof\u000baniso/D2.\n\u000baniso increases quadratically with the electron-phonon\ncoupling strength D.\nSince Fig. 9 establishes that the spin-dephasing rate\n1=T2for the fast dynamics discussed in the previous sec-\ntions, is proportional to D2, we \fnd\u000baniso/1=T2. We\nbrie\ry compare these trends to two earlier calculations\nof Gilbert damping that employ p-dmodels and assume\nphenomenological Bloch-type rates 1 =T2for the dephas-\ning of the itinerant hole spins toward the \feld of the\nlocalized moments. In contrast to the present paper, the\nlocalized spins experience the anisotropy \felds. Chovan\nand Perakis38derive a Gilbert equation for the dephasing\nof the localized spins toward the anisotropy axis, assum-\ning that the hole spin follows the \feld ~Hlocof the localized\nspins almost adiabatically. Tserkovnyak et al.39extract\na Gilbert parameter from spin susceptibilities. The re-\nsulting dependence of the Gilbert parameter \u000baniso on\n1=T2in both approaches is in qualitative accordance and\nexhibits two di\u000berent regimes. In the the low spin-\rip\nregime, where 1 =T2is small in comparison to the p-dex-\nchange interaction a linear increase of \u000baniso with 1=T2\nis found, as is the case in our calculations with micro-\nscopic dephasing terms. If the relaxation rate is larger\nthan thep-ddynamics,\u000baniso decreases again. Due to\nthe restriction (36) of the Boltzmann scattering integral\nto low spin-\rip rates, the present Markovian calculations\ncannot be pushed into this regime.\nEven though the anisotropy \feld ~Haniso is not cou-\npled to the localized spin ~Sdirectly, both spins precess\naround the zdirection with frequency !aniso\nL. In analogy\nto Sec. V B we study now the in\ruence of the damping\nprocess on the precession of the localized spin around\nthe anisotropy axis and compare it to the behavior of\nLandau-Lifshitz and Gilbert dynamics. Fig. 18 reveals a\nsimilar behavior of the precession frequency as a function\nof the damping rate 1 =Taniso\n2 as in the isotropic case. The\nmicroscopic calculation predicts a distinct drop of the\nLarmor frequency !aniso\nL for a range of dephasing rates\nwhere the precession frequency is unchanged according\nto the Gilbert and Landau-Lifshitz damping models. Al-\nthough Gilbert damping eventually leads to a change in\nprecession frequency for larger damping, this result shows\na qualitative di\u000berence between the microscopic and the13\n0 0.02 0.04 0.06 0.087.127.167.27.24\n1/Taniso\n2(ps−1)ωaniso(ps−1)\n \nGilbert\nLL\nMicroscopic\nFIG. 18. Precession frequency of the localized spin around\nthe anisotropy \feld vs. Bloch decay time 1 =Taniso\n2.\nphenomenological calculations.\nVII. CONCLUSION AND OUTLOOK\nIn this paper, we investigated a microscopic descrip-\ntion of dephasing processes due to spin-orbit coupling\nand electron-phonon scattering in a mean-\feld kinetic\nexchange model. We \frst analyzed how spin-dependent\ncarrier dynamics can be described by Boltzmann scat-\ntering integrals, which leads to Elliott-Yafet type relax-\nation processes. This is only possible for dephasing rates\nsmall compared to the Larmor frequency, see Eq. (36).\nThe microscopic calculation always yielded Bloch times\n2T1=T2for low excitation angles as it should be due\nto the conservation of the absolute value of the mag-\nnetization. A small decrease of the e\u000bective precession\nfrequency occurs with increasing damping rate, which is\na fundamental di\u000berence to the Landau-Lifshitz descrip-\ntion and exceeds the change predicted by the Gilbert\nequation in this regime.We modeled two dephasing scenarios. First, a relax-\nation process between both spin sub systems was studied.\nHere, the di\u000berent spins precess around the mean-\feld of\nthe other system. In particular, for large excitation an-\ngles we found a decrease of the magnetization during the\nprecessional motion without heating and a slow remag-\nnetization. This scenario is somewhat in between typi-\ncal small angle-relaxation, for which the modulus of the\nmagnetization is constant and which is well described\nby Gilbert and Landau-Lifshitz damping, and collinear\nde/remagnetization dynamics. Also, we \fnd important\ndeviations from a pure Bloch-like behavior.\nThe second scenario deals with the relaxation of the\nmagnetization toward a magnetic anisotropy \feld expe-\nrienced by the itinerant carrier spins for small excitation\nangles. The resulting Gilbert parameter \u000baniso is inde-\npendent of the static anisotropy \feld. The relaxation of\nthe localized moments occurs only indirectly as a carrier-\nmeditated relaxation via their coupling to the time de-\npendent mean-\feld of the itinerant spin.\nTo draw a meaningful comparison with Landau-\nLifshitz and Gilbert dynamics we restricted ourselves\nthroughout the entire paper to a regime where the elec-\ntronic temperature is equal to the lattice temperature Tph\nat all times. In general our microscopic theory is also ca-\npable of modeling heat induced de- and remagnetization\nprocesses. We intend to compare microscopic simulations\nof hot electron dynamics in this model, including scat-\ntering processes between both types of spin, with phe-\nnomenological approaches such as the Landau-Lifshitz-\nBloch (LLB) equation or the self-consistent Bloch equa-\ntion (SCB)40.\nWe \fnally mention that we derived relation (13) con-\nnecting the Bloch dephasing time T2and the Gilbert\ndamping parameter \u000b. 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Phys. 113, 163911 (2013)." }, { "title": "2402.01570v1.Controllable_frequency_tunability_and_parabolic_like_threshold_current_behavior_in_spin_Hall_nano_oscillators.pdf", "content": "Controllable frequency tunability and parabolic-like threshold current behavior in\nspin Hall nano-oscillators\nArunima TM1and Himanshu Fulara1,∗\n1Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, India.\n(Dated: February 5, 2024)\nWe investigate the individual impacts of critical magnetodynamical parameters—effective mag-\nnetization ( µ0Meff) and magnetic damping ( α)—on the auto-oscillation characteristics of nano-\nconstriction-based Spin Hall Nano-Oscillators (SHNOs). Our micromagnetic simulations unveil a\ndistinctive non-monotonic relationship between current and auto-oscillation frequency in out-of-\nplane magnetic fields. The influence of effective magnetization on frequency tunability varies with\nout-of-plane field strengths. At large out-of-plane fields, the frequency tunability is predominantly\ngoverned by effective magnetization, achieving a current tunability of 1 GHz/mA—four times larger\nthan that observed at the lowest effective magnetization. Conversely, at low out-of-plane fields,\nalthough a remarkably high-frequency tunability of 4 GHz/mA is observed, the effective magnetiza-\ntion alters the onset of the transition from a linear-like mode to a spin-wave bullet mode. Magnetic\ndamping primarily affects the threshold current with negligible impact on auto-oscillation frequency\ntunability. The threshold current scales linearly with increased magnetic damping at a constant out-\nof-plane field but exhibits a parabolic behavior with variations in out-of-plane fields. This behavior\nis attributed to the qualitatively distinct evolution of the auto-oscillation mode across different\nout-of-plane field values. Our study not only extends the versatility of SHNOs for oscillator-based\nneuromorphic computing with controllable frequency tunability but also unveils the intricate auto-\noscillation dynamics in out-of-plane fields.\nI. INTRODUCTION\nSpin Hall nano-oscillators (SHNOs) [1–9] are promi-\nnent spintronic devices that leverage the spin Hall effect\n(SHE) phenomenon [10–12] within the heavy metal layer\ntoconvertalateraldirectchargecurrentintoatransverse\npure spin current. This spin current then drives mag-\nnetization auto-oscillations in nanoscopic regions within\nthe adjacent ferromagnetic layer through the transfer of\nspin angular momentum [1, 4, 6, 13]. Recently, SHNOs\nhave attracted significant attention due to their advan-\ntages in nanofabrication, rapid and ultra-broadband mi-\ncrowave frequency tunability, compatibility with com-\nplementary metal-oxide-semiconductor (CMOS) technol-\nogy, and emerging applications in neuromorphic comput-\ning and magnonics [4, 5, 13–18]. Notably, among var-\nious SHNO device geometries, nano-constriction-based\nSHNOs offer distinct advantages, including flexible de-\nvice layouts, direct optical access to auto-oscillating re-\ngions, and voltage control of the constriction region [4, 6,\n8,19–21]. Theirhighlynon-linearpropertiesandin-plane\ncurrent flow facilitate the mutual synchronization of mul-\ntiple SHNOs in both linear chains and two-dimensional\narrays,enablinghigh-qualitymicrowavesignalgeneration\nand scaling neuromorphic computing to large oscillator\nnetworks [6, 19, 20, 22]. The precise control of auto-\noscillation frequency over a wide range plays a pivotal\nrole in controlling the dynamical couplings between oscil-\nlators, offering a potential avenue for facilitating synap-\ntic communication among artificial neurons. To achieve\n∗himanshu.fulara@ph.iitr.ac.inlearning in oscillator-based neuromorphic computing, ro-\nbust frequency tunability is crucial, especially at higher\noperating frequencies [23–25]. The agility to control\nfrequency tunability is also beneficial for communication\nsystems where specific frequency bands are allocated for\ndifferent applications. The degree of frequency tunabil-\nity depends on magnetodynamical properties, drive cur-\nrent, and the orientation and strength of the applied\nmagnetic field [26, 27]. In a recent study, Haidar et\nal. [28] reported the impact of NiFe alloy composition on\nthe auto-oscillation properties of nano-constriction-based\nNi100-xFex/Pt SHNOs under in-plane fields. The study\nrevealed that Fe-rich devices require higher current den-\nsities for driving auto-oscillations at higher frequencies,\nraising a fundamental question about how damping and\nsaturation magnetization independently influence these\nauto-oscillation properties. Despite numerous studies on\ntheauto-oscillationbehaviorofnanoconstrictionSHNOs,\na comprehensive understanding of the independent roles\nof key magnetodynamical parameters, such as effective\nmagnetization ( µ0Meff) and magnetic damping ( α) on\nthe auto-oscillation properties remains elusive.\nIn this study, we systematically investigate the in-\ndividual impacts of effective magnetization and mag-\nnetic damping on the magnetization auto-oscillations of\nSHNOs. Our observations reveal qualitatively similar\nauto-oscillation behavior for different effective magne-\ntization values at a fixed out-of-plane field. However,\nthe frequency tunability controlled by effective magne-\ntization exhibits significant variations at different out-\nof-plane fields. Conversely, magnetic damping primarily\ninfluences the threshold current with minimal impact on\nfrequency tunability. Additionally, the threshold current\nof auto-oscillation at different out-of-plane fields exhibitsarXiv:2402.01570v1 [cond-mat.mes-hall] 2 Feb 20242\nFIG. 1. Device schematic, current density, Oersted field,\nand demagnetizing field (a) Schematic of a nano-constriction-\nbased Pt(6 nm)/Pt(5 nm) spin Hall nano-oscillator (SHNO),\nwith a constriction width of w= 150 nm. (b) Distribution of\nlateral charge current density in the Pt/Py nano-constriction\nfor an input current, Idc= 1 mA. The inset illustrates the\nsimulated current density in the individual Pt and Py layers.\n(c) Estimation of the out-of-plane component of the Oersted\n(Oe) field in the Py layer for Idc= 1 mA. (d) Distribution of\ndemagnetizing field in the Py layer for Idc= 0 mA.\nanintriguingparabolic-likebehavior, indicativeofaqual-\nitatively different spatial evolution of the auto-oscillating\nmode within distinct field energy landscapes.\nII. MICROMAGNETIC SIMULATIONS\nHere, we simulate a Pt(6 nm)/Py(5 nm) nano-\nconstriction-based SHNO with an optimized constriction\nwidthof150nm, asillustratedschematicallyinFig. 1(a).\nThe device geometry is initially modeled using the COM-\nSOL [29] Multiphysics Software. Simulations consider\nthe full-scale Pt/Py bilayer with an input current, Idc\n= 1 mA, and utilize resistivity values of 32.6 µΩ-cm for\nPy and 11.2 µΩ-cm for Pt, as reported in experimental\nstudy [4]. Figure 1(b) shows the charge current density\ndistribution along the y-axis in a 150 nm SHNO obtained\nfrom COMSOL simulations. Notably, there is a markedincrease in local current density at the constriction re-\ngion. Due to the disparate electrical resistances of the Pt\nand Py layers, the current predominantly flows through\nthe Pt metal, as illustrated in the inset of Fig. 1(b). As a\nresult, in the micromagnetic simulations, we neglect the\ncontribution of the charge current passing through the\nPy layer to influence the magnetization dynamics. Given\nthat the spin current injected into Py is directly propor-\ntional to the current density in Pt, JS= Θ SHJC, the\nnano-constriction region, characterized by a large cur-\nrent density, defines the active device area, wherein the\ninjected spin current is large enough to excite magneti-\nzation dynamics. Figure 1(c) shows the simulated out-\nof-plane component of the current-induced Oersted (Oe)\nfield profile in the Py layer for an input current, Idc= 1\nmA. Using MUMAX3 [30], we then modeled a NiFe layer\ncharacterized by an exchange stiffness ( Aex) of 10 pJ/m\nand a gyromagnetic ratio ( γ/2π) of 29.53 GHz/T [6]\nwithin a mesh area measuring 2000 x 2000 nm ², divided\ninto a grid size of 512 x 512. The externally applied mag-\nnetic field has fixed in-plane ( ϕ) and out-of-plane angles\n(θ) of 24 and 80 degrees, respectively.\nThe spatial distribution of demagnetizing fields in the\nnano-constriction area is illustrated in Fig. 1(d), ob-\ntainedthroughmicromagneticsimulationsusingtheMU-\nMAX3 solver [30]. The calculations reveal that the nano-\nconstriction edges produce an Oe field and a demagne-\ntizing field opposing the external static magnetic field,\nleading to a localized decrease in the internal field within\nthe nano-constriction area of the Py film. Our methodol-\nogy involves running current sweep simulations to inves-\ntigate the influence of independently varying Meffand\nαon the magnetization auto-oscillation frequency under\nboth lower and higher applied magnetic fields. Subse-\nquently, Fast Fourier Transform (FFT) calculations were\nappliedtothetime-dependentandspace-dependentmag-\nnetization data to extract frequencies and spatial profiles\nof the auto-oscillating modes.\nIII. MAGNETODYNAMICAL NON-LINERAITY\nTheintricatenon-linearmagneto-dynamicswithinpat-\nterned magnetic thin films can be analytically character-\nized by a non-linearity coefficient, denoted as N[31].\nThe magnitude and sign of Nplay a pivotal role in de-\ntermining the strength and nature of magnon-magnon\ninteractions, where positive and negative values corre-\nspond to magnon repulsion and attraction, respectively\n[27, 31]. A negative non-linearity leads to a reduction\nin the auto-oscillation frequency with increasing ampli-\ntude, ultimately causing it to enter the spin-wave band\ngap. This leads to the self-localization of spin-waves,\ntriggering the excitation of solitonic modes such as spin-\nwave bullets in in-plane magnetized films [32, 33], and\nmagnetic droplets in films exhibiting very large perpen-\ndicular magnetic anisotropy (PMA) [34, 35]. Conversely,\nlarge positive non-linearity results in an increase in the3\nFIG. 2. Contour plot illustrating the analytically determined\nnon-linearity coefficient ( N) for a thin ferromagnetic film,\nplotted as a function of saturation magnetization and exter-\nnally applied out-of-plane magnetic field ( θ= 80◦). The dot-\nted black line shows the N= 0 region. The black arrow\nsignifies the enhancement in the precession amplitude with\napplied current, resulting in an effective reduction of MSvia\nthe projection of magnetization ( Mβ=MScosβ) along the\nstatic field direction. Here, βrepresents the precession angle\nof magnetization.\nauto-oscillationfrequencywithamplitude, surpassingthe\nferromagnetic resonance (FMR) frequency and leading to\nthe excitation of propagating spin waves[17, 36, 37].\nThe magnetodynamical non-linearity is primarily gov-\nernedbymaterialparameterssuchassaturationmagneti-\nzation ( MS), magnetic anisotropy, and the strength and\norientation of applied magnetic fields. To analytically\ncalculate the nonlinear coefficient Nfor an in-plane mag-\nnetized film, shown in Fig. 2, we adopted the methodol-\nogy outlined in Ref. [31, 38]. The results are plotted as a\nfunction of MSand externally applied out-of-plane mag-\nnetic fields ( θ= 80◦). As shown in Fig. 2, Npredomi-\nnantly exhibits negative values (depicted by red regions)\nat low applied magnetic fields, regardless of variations in\nMS. However, at higher applied fields, Nundergoes a\nmonotonic transition from negative (red regions) to pos-\nitive values (illustrated by blue regions), depending on\nthe strength of MS. Notably, it crosses through zero at\na specific field strength (indicated by the black dotted\nline), the precise location of which is influenced by MS.\nIn other words, the reduction in MSshifts the point of\nzero non-linearity towards lower values of applied fields.\nTailoringnon-linearitythroughthemanipulationofeffec-\ntive magnetization and applied magnetic fields in SHNOs\nshould, in principle, offer precise control over frequency\ntunability and magnetization auto-oscillation dynamics.IV. RESULTS AND DISCUSSION\nFigure 3(a-h) presents micromagnetically simulated\ncurrent sweep auto-oscillations under two qualitatively\ndifferentoperatingregimesforfourdistincteffectivemag-\nnetization values, which are practically tunable through\nthe optimization of Fe composition in the Ni 100-xFex\nlayer [28]. The objective was to discern the influence\nof effective magnetization on auto-oscillation properties\nwhile maintaining other magnetodynamical parameters\nconstant. In Fig. 3(a-d), at the low applied field of µoH\n= 0.3 T, the auto-oscillations display a typical redshifted\nfrequency vs. current behavior, accompanied by a lin-\near rise in threshold frequency with effective magnetiza-\ntion. A similar red-shifted frequency behavior is experi-\nmentally observed in earlier studies on SHNOs [33, 39].\nThe frequency of auto-oscillations remains nearly con-\nstant at low current values, but at a specific effective\nmagnetization-governed current, there is an abrupt jump\nin the frequency indicative of the so-called spin-wave bul-\nlet—a non-topological self-localized mode that emerges\nin large negative non-linearity regions [27, 31, 33], as\nshown in Fig. 2.\nIn the presence of large out-of-plane field of µoH=\n0.7 T, as shown in Fig. 3(e-h), auto-oscillations exhibit a\nnonmonotonic frequency vs. current behavior, character-\nized by a red-shifted frequency at lower currents followed\nbyablue-shiftedoneathighercurrentvalues[6,27]. This\ndistinct non-monotonic frequency behavior has been ex-\ntensively reported in recent experimental investigations\non SHNOs [6, 17, 39–41]. For in-plane magnetized Py\nthin films, the non-linearity at high applied fields is pre-\ndominantly negative, and therefore, the auto-oscillation\nfrequency initially experiences a redshift with increasing\ndrive current [27, 40]. As the drive current increases, the\nprecession amplitude increases, and the demagnetizing\nfield along the static field direction reduces, leading to a\nsituation where the non-linearity becomes zero (refer to\nFig. 2). Consequently, the redshifting of the frequency\nstops, and the auto-oscillation frequency passes through\na minimum where the non-linearity is zero. With any\nfurther increase in the drive current, the non-linearity\nbecomes positive, inducing a blue shift in the auto-\noscillation frequency [17, 27, 40]. Similar to the low-field\nauto-oscillation behavior, this nonmonotonic frequency\nresponseathigherfieldsremainsessentiallyunchangedin\nnature across all four distinct µ0Meffvalues, with only a\nminor increase in onset auto-oscillation frequency. How-\never, the frequency tunability demonstrates a significant\nboost at higher effective magnetization values, achiev-\ning a fourfold increase in current tunability at µ0Meff=\n0.8 T. As illustrated in Fig. 3(i), at low fields, effective\nmagnetization primarily influences the onset of the bullet\nmode by significantly enhancing negative non-linearity\nwhile moderately adjusting frequency tunability, reach-\ning a maximum current tunability of 4 GHz/mA. Con-\nversely, at high fields, frequency tunability experiences\na fourfold increase with increasing effective magnetiza-4\nFIG. 3. Power spectral density obtained from micromagnetically simulated data, illustrating its variation with current for\nfour distinct effective magnetization values under out-of-plane magnetic fields ( θ=80◦,φ=24◦): (a-d) µoH= 0.3 T, (e-h) µoH\n= 0.7 T. Comparison of auto-oscillation frequency tunability ( d f/dI) as a function of drive current for four distinct effective\nmagnetization values at (i) µoH= 0.3 T, (j) µoH= 0.7 T\ntion values (refer to Fig. 3(j)). Our results indicate the\npivotal role of effective magnetization in tailoring the\nmagnetodynamic non-linearity at the active dynamical\nregion, thereby influencing both the frequency tunability\nand the onset of the bullet mode.\nSubsequently, we explore the critical influence of\nmagnetic damping on the auto-oscillation properties of\nSHNOs, while maintaining a fixed effective magneti-\nzation value of µ0Meff= 0.74 T. This investigation\nis confined to high out-of-plane fields, µoH= 0.7 T\n(θ=80◦,φ=24◦), where significant variations in effective\nmagnetization-controlled frequency tunability were ob-\nserved (refer to Fig. 3(j)). In Fig. 4(a-d), we present\ncurrent sweep auto-oscillations at four distinct magnetic\ndamping values ranging from α= 0.02 to 0.05. As shown\nin Fig. 4(e), magnetic damping predominantly influ-\nences the threshold current and onset frequency, with\nnegligible impact on auto-oscillation frequency tunabil-\nity. Thequalitativelysimilarfrequencytunabilityimplies\nthat the fundamental nature of auto-oscillations remains\nunaffected by variation in damping. As illustrated in theinset of Fig. 4(e), the onset frequency demonstrates a\nlinear decrease with an increase in α, while the threshold\ncurrent increases with increasing αvalues. Note that we\nestimated the threshold current by monitoring the mag-\nnetization behavior over time. The initial 5 ns were ex-\ncluded to mitigate transient effects, and the subsequent\n15 ns were observed to detect the magnetization auto-\noscillations. If the auto-oscillations sustain over time,\nwe identify it as the onset of auto-oscillation. The ob-\nserved enhancement in threshold current is slightly below\nthe anticipated value of 2.5 times, indicating that the in-\ncrease in damping is linked with the stronger localization\nof auto-oscillating mode due to a higher Oe field. The Oe\nfield effectively suppresses the spin-wave well on one side\nof the nano-constriction while enhancing the localization\nof the mode on the opposite side [27].\nTo delve deeper into the impact of magnetic damping\nacross various operational regimes, micromagnetic simu-\nlations were conducted, maintaining a constant effective\nmagnetization of µ0Meff= 0.74 T while varying out-of-\nplane magnetic fields ( µ0H). Figure 5(a) summarizes5\nFIG. 4. Micromagnetically simulated auto-oscillation fre-\nquency vs. drive current for four damping values at a fixed\nµ0Meff= 0.74 T : (a) α= 0.02, (b) α= 0.03, (c) α= 0.04,\n(d)α= 0.05. (e) Comparison of auto-oscillation frequencies\nas a function of drive current at various damping magnitudes.\nThe inset shows the variation of threshold frequency (blue\nsquares) and threshold current (black squares) as a function\nof damping, with red lines indicating linear fits.\nthe variation in threshold currents as a function of out-\nof-plane field strengths at three distinct damping levels.\nThe plots of Ithversus µ0Hexhibit a parabolic-like non-\nmonotonic trend for all three damping values. In general,\nthe threshold current ( Ith) is expected to increase mono-\ntonically with the applied magnetic field strength [27].\nHowever, in our case, an intriguing deviation is observed,\nwherein a reduction in Ithoccurs at intermediate field\nstrengths. This reduction reaches a minimum value be-\nfore the threshold current begins to increase at strong\nout-of-plane fields. A similar parabolic-like behavior has\nbeen reported by Yin et al. [42] in Py 100-x-yPtxAgy/Pt-\nbased nanoconstriction SHNOs. This intriguing non-\nmonotonic behavior observed in the field dependence of\nthreshold current underlines a more in-depth investiga-\ntion of the dynamics of auto-oscillating modes in the con-\nstriction region.\nTo elucidate this behavior, we performed simulations\nof current sweep auto-oscillations at three different oper-\nating regimes depicted in Fig. 5(a). Figures 5(b-d) re-\nveal three distinct characteristics of auto-oscillation fre-\nquency vs. current, highlighting the distinct evolution of\nFIG. 5. (a) Variation of threshold current ( Ith) as a function\nof applied out-of-plane fields ( θ=80◦,φ=24◦), calculated for\nthree different damping values. Micromagnetically simulated\ncurrentsweepauto-oscillationsfor α=0.03subjectedtothree\ndistinct out-of-plane field strengths: (b) µoH= 0.1 T, (c)\nµoH= 0.4 T, and (d) µoH= 0.8 T. The inset illustrates the\nspatial profiles of auto-oscillations simulated at the onset of\nauto-oscillations for each field strength.\nauto-oscillating modes. In weak fields, such as µ0H=\n0.1 T, the auto-oscillation frequency exhibits negligible\nchange with current, and the mode remains quasi-linear,\nas illustrated in Fig. 5(b). The spatially simulated pro-\nfile, depicted as the inset of Fig. 5(b), confirms that the\nmode originates at the constriction edges due to inhomo-\ngeneous current distributions, as shown in Fig. 1(b). A\nsmall negative non-linearity (refer to Fig. 2) governs this\nquasi-linear mode at weak out-of-plane fields [17, 27, 31].\nAs the applied field strength increases, the non-linearity6\nbecomes more negative (see Fig. 2), leading to a stronger\nlocalization of the auto-oscillating mode at the constric-\ntion edges [31, 43]. Consequently, the mode undergoes a\ntransition from a quasi-linear behavior to a strongly lo-\ncalized spin-wave bullet behavior, resulting in an abrupt\ndrop in frequency, as shown in Figure 5(c). With a fur-\nther increase in the field strength, the non-linearity ini-\ntially increases from a negative value at low currents,\npasses through a zero value, and eventually transitions\nto a positive value at higher currents [27, 39] (refer to\nFig. 2). This behavior arises due to the magnetization\nvectorundergoingprecessionatanincreasinglylargeran-\ngle as the current magnitude rises. Consequently, there\nis an effective reduction of MSthrough the projection of\nmagnetization ( Mβ=MScosβ) along the applied field\ndirection [17, 44], resulting in a shift of the non-linearity\nfrom a negative value to a positive one, as illustrated by\nblack arrow in Fig. 2. This transition occurs as the auto-\noscillatingmodedetachesfromtheconstrictionedgesand\ntransforms into a bulk type. As shown in the inset of\nFig. 5(d), the auto-oscillating mode shifts inwards into\nthe interior of the constriction and expands away from\nthe constriction as the current increases.\nTo look forward, controllable frequency tunability in\nSHNOs can enhance their adaptability and versatility,\nmaking them suitable for a wide range of practical ap-\nplications in communication, adaptive signal processing,\ninformation storage, and computing [8, 20, 24, 45]. De-\npending on the desired functionality, the SHNOs can\nbe tuned to operate in different frequency ranges (re-\nfer to Fig. 3), enabling their use in a variety of appli-\ncations within a single device. In the realm of neuro-\nmorphic computing, the ability to adjust the oscillation\nfrequency of SHNOs becomes particularly valuable. This\ntunability can be leveraged to emulate synaptic plastic-\nity, where frequency modulation represents variations in\nsynaptic strength [24, 25]. This, in turn, facilitates the\nimplementation of learning and memory functions within\nneuromorphic circuits. Moreover, the tunable frequen-\ncies of SHNOs enable them to replicate the dynamic be-\nhavior of neurons in response to changing input condi-\ntions. The capability to dynamically adjust frequencieson the fly allows for swift modifications to neural connec-\ntions, facilitating continuous learning in neuromorphic\nsystems. With controllable frequency tunability, SHNOs\nexhibitthecapabilitytodynamicallyadapttheirfrequen-\ncies based on the computational task at hand. This not\nonly optimizes energy consumption but also contributes\nto the overall efficiency of the system.\nV. CONCLUSION\nIn summary, we have studied the individual influence\nof key magnetodynamical parameters, namely effective\nmagnetization ( µ0Meff) and magnetic damping ( α), on\nthe auto-oscillation characteristics of nano-constriction-\nbased SHNOs. Our findings reveal a significant vari-\nation in the impact of effective magnetization on fre-\nquency tunability, particularly in response to applied\nfield strengths. Conversely, magnetic damping primarily\ninfluences the threshold current, exerting minimal effects\non frequency tunability. We observed a linear increase\nin the threshold current with increasing magnetic damp-\ning under a constant magnetic field. However, when ex-\nploring the interplay of threshold current with applied\nfield strengths, a distinct parabolic-like trend emerges.\nThis behavior suggests qualitatively different origins of\nauto-oscillating modes across different field landscapes,\nproviding valuable insights into the intricate magneto-\ndynamics of SHNOs in nano-constriction geometry. Our\nstudy not only contributes to oscillator-based neuromor-\nphiccomputingthroughcontrollablefrequencytunability\nbutalsoprovidesvaluableinsightsintotheintricateauto-\noscillation dynamics of nano-constriction-based SHNOs.\nAcknowledgements\nThis work is supported by the Faculty Initiation Grant\n(FIG) sponsored by SRIC, IIT Roorkee. We thankfully\nacknowledge the Institute Computer Center (ICC), IIT\nRoorkee for providing a high-end computational facility\nto run simulations.\n[1] V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich,\nA.Slavin,D.Baither,G.Schmitz, andS.O.Demokritov,\nNature Mater. 11, 1028 (2012).\n[2] L. 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Lett. 94, 102507 (2009)." }, { "title": "1506.05622v2.The_absence_of_intraband_scattering_in_a_consistent_theory_of_Gilbert_damping_in_metallic_ferromagnets.pdf", "content": "arXiv:1506.05622v2 [cond-mat.str-el] 23 Oct 2015The absence of intraband scattering in a consistent theory o f Gilbert damping in\nmetallic ferromagnets\nD M Edwards\nDepartment of Mathematics, Imperial College London, Londo n SW7 2BZ, United Kingdom\nDamping of magnetization dynamics in a ferromagnetic metal , arising from spin-orbit coupling,\nis usually characterised by the Gilbert parameter α. Recent calculations of this quantity, using\na formula due to Kambersky, find that it is infinite for a perfec t crystal owing to an intraband\nscattering term which is of third order in the spin-orbit par ameterξ. This surprising result conflicts\nwith recent work by Costa and Muniz who study damping numeric ally by direct calculation of\nthe dynamical transverse susceptibility in the presence of spin-orbit coupling. We resolve this\ninconsistency by following the approach of Costa and Muniz f or a slightly simplified model where\nit is possible to calculate αanalytically. We show that to second order in ξone retrieves the\nKambersky result for α, but to higher order one does not obtain any divergent intrab and terms.\nThe present work goes beyond that of Costa and Muniz by pointi ng out the necessity of including\nthe effect of long-range Coulomb interaction in calculating damping for large ξ. A direct derivation\nof the Kambersky formula is given which shows clearly the res triction of its validity to second order\ninξso that no intraband scattering terms appear. This restrict ion has an important effect on the\ndamping over a substantial range of impurity content and tem perature. The experimental situation\nis discussed.\nI. INTRODUCTION\nMagnetization dynamics in a ferromagnetic metal is central to the fi eld of spintronics with its many applications.\nDamping is an essential feature of magnetization dynamics and is usu ally treated phenomenologically by means\nof a Gilbert term in the Landau-Lifshitz-Gilbert equation [1, 2]. For a system with spin-rotational invariance the\nuniform precession mode of the magnetization in a uniform external magnetic field is undamped and the fundamental\norigin of damping in ferromagnetic resonance is spin-orbit coupling (S OC). Early investigations of the effect include\nthose of Kambersky [3–5] and Korenman and Prange [6]. Kambersky ’s [4] torque-correlation formula for the Gilbert\ndamping parameter αhas been used by several groups [7–14], some of whom have given alt ernative derivations.\nHowever the restricted validity of this formula, as discussed below, has not been stressed. In this torque-correlation\nmodel contributions to αof both intraband and interband electronic transitions are usually c onsidered. The theory\nis basically developed for a pure metal with the effect of defects and /or phonons introduced as phenomenological\nbroadening of the one-electron states. Assuming that the electr on scattering-rate increases with temperature T due\nto electron-phonon scattering the intraband and interband tran sitions are found to play a dominant role in low and\nhigh T regimes, respectively. The intraband(interband) term is pre dicted to decrease(increase) with increasing T and\nto be proportional to ξ3(ξ2) whereξis the SOC parameter. Accordingly αis expected to achieve a minimum at an\nintermediate T. This is seen experimentally in Ni and hcp Co [15] but not in Fe [15] and FePt [16]. The ξ2dependence\nofαis well-established at high T [17, 18] but there seems to be no experim ental observation of the predicted ξ3\nbehaviourat lowT. The interband ξ2term in Kambersky’stheory canbe givenaverysimple interpretation in termsof\nsecond-orderperturbation theory [5]. A quite different phenomen ologicalapproach, not applicable in some unspecified\nlow scattering-rate regime, has been adopted to try and find a phy sical interpretation of the intraband term [5, 8].\nNo acceptable theoretical treatment of damping in this low scatter ing regime is available because the intraband term\nof Kambersky’s theory diverges to infinity in the zero-scattering lim it of a pure metal with translational symmetry at\nT=0 [9, 11, 13]. We consider it essential to understand the pure met al limit before introducing impurity and phonon\nscattering in a proper way.\nCosta and Muniz [19] recently studied damping numerically in this limit by direct calculation of the dynamical\nspin susceptibility in the presence of SOC within the random phase app roximation (RPA). They determine αfrom\nthe linewidth of the uniform (wave-vector q= 0) spin-wave mode which appears as a resonance in the transvers e\nsusceptibility. One of the main objects of this paper is to establish so me degree of consistency between the work of\nKambersky and that of Costa and Muniz. We follow the approach of t he latter authors for a slightly simplified model\nwhere it is possible to calculate αanalytically. We show that to second order in ξone retrieves the Kambersky result,\nbut to higher order no intraband terms occur, which removes the p roblem of divergent α. To confirm this point, in\nAppendix A we derive the Kambersky formula directly in a way that mak es clear its restriction to second order in\nξto which order the divergent terms in αarising from intraband transitions do not appear. This throws open the\ninterpretation of the minimum observed in the temperature depend ence ofαfor Ni and Co.\nAt this point we may mention an alternative theoretical approach to the calculation of Gilbert damping using2\nscattering theory [20, 21]. Starikov et al [21] find that, for Ni 1−xFexalloys at T=0, αbecomes large near the pure\nmetal limits x=0,1. They attribute this to the Kambersky intraband c ontribution although no formal correspondence\nis made between the two approaches.\nThe work of Costa and Muniz [19] follows an earlier paper [22] where it is shown that SOC has the effect of\ncoupling the transverse spin susceptibility to the longitudinal spin su sceptibility and the charge response. It is known\nthat a proper calculation of these last two quantities in a ferromagn et must take account of long-range Coulomb\ninteractions [23–27]. The essential role of these interactions is to e nsure conservation of particle number. Costa et\nal [19, 22] do not consider such interactions but we show here that this neglect is not serious for calculating αwith\nsufficiently small SOC. Howeverin the wider frameworkof this paper, where mixed charge-spinresponse is also readily\nstudied, long-rangeinteractions are expected to sometimes play a role. They also come into play, even to second order\ninξ, when inversion symmetry is broken.\nIn section II we establish the structure of spin-density response theory in the presence of SOC by means of exact\nspin-density functional theory in the static limit [28]. In section III w e introduce a spatial Fourier transform and an\napproximation to the dynamical response is obtained by introducing the frequency dependence of the non-interacting\nsusceptibilities. The theory then has the same structure as in the R PA. Section IV is devoted to obtaining an explicit\nexpression for the transverse susceptibility in terms of the non-in teracting susceptibilities. Expressions for these, in\nthe presence of SOC, are obtained within the tight-binding approxim ation in section V. In section VI we consider\nthe damping of the resonance in the q=0 transverse susceptibility a nd show how the present approach leads to the\nKambersky formula for the Gilbert damping parameter αwhere this is valid, namely to second order in the SOC\nparameter ξ. We do not give an explicit formula for αbeyond this order but it is clear that no intraband terms appear.\nIn section VII some experimental aspects are discussed with sugg estions for future work. The main conclusions are\nsummarized in section VIII.\nII. SPIN-DENSITY FUNCTIONAL THEORY WITH SPIN-ORBIT COUPLI NG\nThe Kohn-Sham equation takes the form\n/summationdisplay\nσ′[−δσσ′(/planckover2pi12/2m)∇2+Veff\nσσ′(r)+Hso\nσσ′]φnσ′(r) =ǫnφnσ(r) (1)\nwith the spin index σ=↑,↓corresponding to quantization along the direction of the ground-s tate magnetization in a\nferromagnet. This may be written in 2 ×2 matrix form with eigenvectors ( φn↑,φn↓)T. The density matrix is defined\nin terms of the spin components φnσ(r) of the one-electron orbitals by\nnσσ′=/summationdisplay\nnφnσ(r)φnσ′(r)∗θ(µ0−ǫn) (2)\nwhereθ(x) is the unit step function and µ0is the chemical potential. The electron density is given by\nρ(r) =/summationdisplay\nσnσσ(r) =/summationdisplay\nnσ|φnσ(r)|2θ(µ0−ǫn) (3)\nand the effective potential in (1) is\nVeff\nσσ′(r) =wσσ′(r)+δσσ′/integraldisplay\nd3r′ρ(r′)v(r−r′)+vxc\nσσ′(r) (4)\nwherewσσ′(r) is the external potential due to the crystal lattice and any magn etic fields and v(r) =e2/|r|is the\nCoulomb potential. The exchange-correlation potential vxc\nσσ′(r) is defined as δExc/δnσσ′(r), a functional derivative of\nthe exchange-correlation energy Exc. The term Hso\nσσ′in (1) is the SOC energy. A small external perturbation δwσσ′,\nfor example due to a magnetic field, changes the effective potential toVeff+δVeff, giving rise to new orbitals and\nhence to a change in density matrix δnσσ′. The equation\nδnσσ′(r) =−Ω−1/summationdisplay\nσ1σ′\n1/integraldisplay\nd3r1χ0\nσσ′σ1σ′\n1(r,r1)δVeff\nσ1σ′\n1(r1), (5)\nwhere Ω is the volume of the sample, defines a non-interacting respo nse function χ0and the full response function χ\nis defined by\nδnσσ′(r) =−Ω−1/summationdisplay\nττ′/integraldisplay\nd3r′χσσ′ττ′(r,r′)δwττ′(r′). (6)3\nFrom (4)\nδVeff\nσ1σ′\n1(r1) =δwσ1σ′\n1(r1)+/summationdisplay\nσ2σ′\n2/integraldisplay\nd3r2[v(r1−r2)δσ1σ′\n1δσ2σ′\n2+δvxc\nσ1σ′\n1(r1)\nδnσ2σ′\n2(r2)]δnσ2σ′\n2(r2) (7)\nand we may write\nδvxc\nσ1σ′\n1(r1)\nδnσ2σ′\n2(r2)=δ2Exc\nδnσ2σ′\n2(r2)δnσ1σ′\n1(r1)=Kσ1σ′\n1σ2σ′\n2(r1,r2). (8)\nCombining (5) - (8) we find the following integral equation for the spin -density response function χσσ′ττ′(r,r′):\nχσσ′ττ′(r,r′) =χ0\nσσ′ττ′(r,r′)−(Ω)−1/summationdisplay\nσ1σ′\n1/summationdisplay\nσ2σ′\n2/integraldisplay\nd3r1/integraldisplay\nd3r2χ0\nσσ′σ1σ′\n1(r,r1)[v(r1−r2)δσ1σ′\n1δσ2σ′\n2+Kσ1σ′\n1σ2σ′\n2(r1,r2)]\nχσ2σ′\n2ττ′(r2,r′).\n(9)\nThis equation is a slight generalisation of that given by Williams and von Ba rth [28]. In the static limit it is formally\nexact although the exchange-correlation energy Excis of course not known exactly. In the next section we generalise\nthe equation to the dynamical case approximately by introducing th e frequency dependence of the non-interacting\nresponse functions χ0, and also take a spatial Fourier transform. In the case where SOC is absent the result is directly\ncompared with results obtained using the RPA.\nIII. DYNAMICAL SUSCEPTIBILITIES IN THE PRESENCE OF SPIN-OR BIT COUPLING AND\nLONG-RANGE COULOMB INTERACTION\nIn general the response functions χ(r,r′) are not functions of r−r′and a Fourier representation of (9) for a\nspatially periodic system involves an infinite number of reciprocal latt ice vectors. There are two cases where this\ncomplication is avoided. The first is a homogeneous electron gas and t he second is in a tight-binding approximation\nwith a restricted atomic basis. We may then introduce Fourier trans forms of the form χ(r) =/summationtext\nqχ(q)eiq·ror\nχ(q) = (Ω)−1/integraltext\nd3rχ(r)e−iq·rand write (9) as\nχσσ′ττ′(q,ω) =χ0\nσσ′ττ′(q,ω)−/summationdisplay\nσ1σ′\n1/summationdisplay\nσ2σ′\n2χ0\nσσ′σ1σ′\n1(q,ω)Vσ1σ′\n1σ2σ′\n2(q)χσ2σ′\n2ττ′(q,ω), (10)\nwhere we have also introduced the ωdependence of χas indicated at the end of the last section. Here V(q) is an\nordinary Fourier transform, without a factor (Ω)−1, so that\nVσ1σ′\n1σ2σ′\n2(q) =v(q)δσ1σ′\n1δσ2σ′\n2+Kσ1σ′\n1σ2σ′\n2, (11)\nwherev(q) = 4πe2/q2is the usual Fourier transform of the Coulomb interaction and the s econd term is independent\nofqsinceKis a short-range spatial interaction. In the gas case it is proportio nal to a delta-function δ(r−r′) in the\nlocal-density approximation (LDA) [28] and in tight-binding it can be t aken as an on-site interaction. In both cases\nKmay be expressed in terms of a parameter Uas\nKσ1σ′\n1σ2σ′\n2=−U[δσ1σ′\n1δσ2σ′\n2δσ1σ2+δσ1σ′\n1δσ2σ′\n2δσ′\n1σ2] (12)\nwhereσ=↓,↑forσ=↑,↓. in the tight-binding case this form of Kcorresponds to a simple form of interaction\nwhich leads to a rigid exchange splitting of the bands ( [29], [22]). This is only appropriate for transition metals in a\nmodel with d bands only, hybridization with s and p bands being neglect ed. We adopt this model in order to obtain\ntransparent analytic results as far as possible. Although not as re alistic as ”first-principles” models of the electronic\nstructure it has been used, even with some quantitative success, in treating the related problem of magnetocrystalline\nanisotropy in Co/Pd structures as well as pure metals [30]. In (10) t he response functions χare per unit volume\nin the gas case but, more conveniently, may be taken as per atom in t he tight-binding case with v(q) modified to\nv(q) = 4πe2/(q2Ωa) where Ω ais the volume per atom.4\nTo show how equations (10) - (12) are related to RPA we examine two examples in the absence of SOC. First\nconsider the transverse susceptibility χ↓↑↑↓(q,ω) which is more usually denoted by χ−+(q,ω). Equation (10) becomes\nχ↓↑↑↓=χ0\n↓↑↑↓−χ0\n↓↑↑↓V↑↓↓↑χ↓↑↑↓ (13)\nand, from (11) and (12), V↑↓↓↑=K↑↓↓↑=−U. Hence\nχ↓↑↑↓=χ0\n��↑↑↓(1−Uχ0\n↓↑↑↓)−1(14)\nwhich is just the RPA result of Izuyama et al [31] for a single-orbital Hubbard model and of Lowde and Windsor [32]\nfor a five-orbital d-band model. Clearly in the absence of SOC the Co ulomb interaction v(q) plays no part in the\ntransverse susceptibility, as is well-known. A more interesting case is the longitudinal susceptibility denoted by χmm\nin the work of Kim et al ( [26], [27]) and in [28]. This involves only the respon se functions χσσττwhich we abbreviate\ntoχστ. In fact [28]\nχmm=χ↑↑+χ↓↓−χ↑↓−χ↓↑. (15)\nWithout SOC χ0\nστtakes the form χ0\nσδστand (10) becomes\nχστ=χ0\nσδστ−/summationdisplay\nσ2χ0\nσVσσ2χσ2τ (16)\nwithVσσ2=v(q)−Uδσσ2. On solving the 2 ×2 matrix equation (16) for χστ, and using (15), we find the longitudinal\nsusceptibility in the form\nχmm=χ0\n↑+χ0\n↓−2[U−2v(q)]χ0\n↑χ0\n↓\n1+(χ0\n↑+χ0\n↓)[v(q)−U]+U[U−2v(q)]χ0\n↑χ0\n↓(17)\nwhich agrees with the RPA result that Kim et al ( [26], [27])found for a s ingle-orbitalmodel. The Coulomb interaction\nv(q) is clearly important, particularly for the uniform susceptibility with q =0, where v→ ∞. It plays an essential\nrole in enforcing particle conservation and hence in obtaining the cor rect result of Stoner theory. In view of the\ncorrespondence between our approach and the RPA method it see ms likely that when SOC is included our procedure\nusing equations (10) - (12) should be almost equivalent to that of Co sta and Muniz [19] in the case of a model with\nd-bands only. However our inclusion of the long-range Coulomb inter action will modify the results.\nIV. AN EXPLICIT EXPRESSION FOR THE TRANSVERSE SUSCEPTIBILI TY\nIn this section we obtain an explicit expression for the transverses usceptibility χ↓↑↑↓in terms of the non-interacting\nresponse functions χ0. We consider equation (10) as an equation between 4 ×4 matrices where σσ′=↓↑,↑↓,↑↑,↓↓\nlabels the rows in that order and ττ′labels columns similarly. The formal solution of (10) is then\nχ= (1+χ0V)−1χ0. (18)\nThis expression could be used directly as the basis of a numerical inve stigation similar to that of Costa and Muniz.\nHowever we wish to show that the present approach leads to a Gilber t damping parameter αin agreement with the\nKambersky formula, to second order in the SOC parameter ξwhere Kambersky’s result is valid. This requires some\nquite considerable analytic development of (18).\nFirst we partition each matrix in (18) into four 2 ×2 matrices. Thus from (11) and (12)\nV=/parenleftbigg\nV10\n0V2/parenrightbigg\n(19)\nwith\nV1=/parenleftbigg\n0−U\n−U0/parenrightbigg\n, V2=/parenleftbigg\nv−U v\nv v−U/parenrightbigg\n. (20)\nAlso\nχ=/parenleftbigg\nχ11χ12\nχ21χ22/parenrightbigg\n(21)5\nand similarly for χ0. If we write\n1+χ0V=/parenleftbigg\n1+χ0\n11V1χ0\n12V2\nχ0\n21V11+χ0\n22V2/parenrightbigg\n=/parenleftbigg\nA B\nC D/parenrightbigg\n(22)\n(18) becomes\nχ=/parenleftbigg\nS−1−S−1BD−1\n−D−1CS−1D−1+D−1CS−1BD−1/parenrightbigg/parenleftbigg\nχ0\n11χ0\n12\nχ0\n21χ0\n22/parenrightbigg\n(23)\nwhere\nS=A+BD−1C. (24)\nThe transverse susceptibility χ↓↑↑↓in which we are interested is the top right-hand element of χ11so this is the\nquantity we wish to calculate. From (23)\nχ11=S−1(χ0\n11−BD−1χ0\n21) (25)\nand, from (24) and (22),\nS= 1+χ0\n11V1−χ0\n12(V−1\n2+χ0\n22)−1χ0\n21V1. (26)\nThe elements of the 2 ×2 matrix S are calculated by straight-forward algebra and\nS11= 1−Uχ0\n↓↑↑↓+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↑↓χ0\n↓↑↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↑↓χ0\n↓↑↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↑↓χ0\n↓↑↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↑↑χ0\n↓↑↓↓](27)\nwhere\nX= [v−U]/[U(U−2v)], Y=−v/[U(U−2v)] (28)\nand\nΛ = (X+χ0\n↑↑↑↑)(X+χ0\n↓↓↓↓)−(Y+χ0\n↑↑↓↓)(Y+χ0\n↓↓↑↑) (29)\nThe other three elements of Sare given in Appendix B. The transverse susceptibility is obtained fro m (25) as\nχ↓↑↑↓= [S22(χ0\n11−BD−1χ0\n21)12−S12(χ0\n11−BD−1χ0\n21)22]/(S11S22−S12S21) (30)\nand\nBD−1=χ0\n12(V−1\n2+χ0\n22)−1. (31)\nA comparison of the fairly complex equation above for the transver se susceptibility with the simple well-known result\n(14) showsthe extent ofthe new physicsintroducedby SOC.This is due tothe coupling ofthe transversesusceptibility\nto the longitudinal susceptibility and the charge response, both of which involve the long-range Coulomb interaction.\nTo proceed further it is necessary to specify the non-interacting response functions χ0\nσσ′σ1σ′\n1which occur throughout\nthe equations above.\nV. THE NON-INTERACTING RESPONSE FUNCTIONS\nIn the tight-binding approximation the one-electron basis function s are the Bloch functions\n|kµσ∝angb∇acket∇ight=N−1/2/summationdisplay\njeik·Rj|jµσ∝angb∇acket∇ight (32)\nwherejandµare the site and orbital indices, respectively, and Nis the number of atoms in the crystal. The\nHamiltonian in the Kohn-Sham equation now takes the form\nHeff=/summationdisplay\nkµνσ(Tµν(k)+Veff\nσδµν)c†\nkµσckνσ+Hso(33)6\nwhereTµνcorresponds to electron hopping and\nVeff\nσ=−(σ/2)(∆+bex) (34)\nwhereσ= 1,−1 for spin ↑,↓respectively. Here ∆ = 2 U∝angb∇acketleftSz∝angb∇acket∇ight/NwhereSzis the total spin angular momentum, in\nunits of /planckover2pi1, and the Zeeman splitting bex= 2µBBex, whereBexis the external magnetic field and µBis the Bohr\nmagneton. The spin-orbit term Hso=ξ/summationtext\njLj·Sjtakes the second-quantized form\nHso= (ξ/2)/summationdisplay\nkµν[Lz\nµν(c†\nkµ↑ckν↑−c†\nkµ↓ckν↓)+L+\nµνc†\nkµ↓ckν↑+L−\nµνc†\nkµ↑ckν↓] (35)\nwhereLz\nµν,L±\nµνare matrix elements of the atomic orbital angular momentum operat orsLz,L±=Lx±iLyin units\nof/planckover2pi1. Within the basis of states (32) eigenstates of Hefftake the form\n|kn∝angb∇acket∇ight=/summationdisplay\nµσaσ\nnµ(k)|kµσ∝angb∇acket∇ight, (36)\nand satisfy the equation\nHeff|kn∝angb∇acket∇ight=Ekn|kn∝angb∇acket∇ight. (37)\nThus\nc†\nkµσ=/summationdisplay\nnaσ\nnµ(k)∗c†\nkn(38)\nwherec†\nkncreates the eigenstate |kn∝angb∇acket∇ight.\nThenon-interactingresponsefunction χ0\nσσ′σ1σ′\n1(q,ω) isconvenientlyexpressedastheFouriertransformofaretarde d\nGreen function by the Kubo formula\nχ0\nσσ′σ1σ′\n1(q,ω) =/summationdisplay\nk∝angb∇acketleft∝angb∇acketleft/summationdisplay\nµc†\nk+qµσckµσ′;/summationdisplay\nνc†\nkνσ1ck+qνσ′\n1∝angb∇acket∇ight∝angb∇acket∇ight0\nω (39)\nwhere the right-hand side is to be evaluated using the one-electron Hamiltonian Heff. Consequently, using (38), we\nhave\nχ0\nσσ′σ1σ′\n1(q,ω) =/summationdisplay\nkµν/summationdisplay\nmnaσ\nmµ(k+q)∗aσ′\nnµ(k)aσ1nν(k)∗aσ′\n1mν(k+q)∝angb∇acketleft∝angb∇acketleftc†\nk+qmckn;c†\nknck+qm∝angb∇acket∇ight∝angb∇acket∇ight0\nω\n=N−1/summationdisplay\nkµν/summationdisplay\nmnaσ\nmµ(k+q)∗aσ′\nnµ(k)aσ1\nnν(k)∗aσ′\n1mν(k+q)fkn−fk+qm\nEk+qm−Ekn−/planckover2pi1ω+iη.(40)\nThe last step uses the well-known form of the response function pe r atom for a non-interacting Fermi system (e.g. [33])\nandηis a small positive constant which ultimately tends to zero. The occup ation number fkn=F(Ekn−µ0) where\nFis the Fermi function with chemical potential µ0. Clearly for q= 0 the concept of intraband transitions ( m=n),\nfrequently introduced in discussions of the Kambersky formula, ne ver arises for finite ωsince the difference of the\nFermi functions in the numerator of (40) is zero. Equation (40) ma y be written in the form\nχ0\nσσ′σ1σ′\n1(q,ω) =N−1/summationdisplay\nkmnBσσ′\nmn(k,q)Bσ1′σ1\nmn(k,q)∗fkn−fk+qm\nEk+qm−Ekn−/planckover2pi1ω+iη(41)\nwhere\nBσσ′\nmn(k,q) =/summationdisplay\nµaσ\nmµ(k+q)∗aσ′\nnµ(k). (42)7\nVI. FERROMAGNETIC RESONANCE LINEWIDTH; THE KAMBERSKY FORM ULA\nWe now consider the damping of the ferromagnetic resonance in the q= 0 transverse susceptibility. The present\napproach, like the closely-related one of Costa and Muniz [19], is valid f or arbitrary strength of the SOC and can\nbe used as the basis of numerical calculations, as performed by the latter authors. However it is important to show\nanalytically that the present method leads to the Kambersky [4] for mula for the Gilbert damping parameter where\nthis is valid, namely to second order in the SOC parameter ξ. This is the subject of this section.\nIt is useful to consider first the case without SOC ( ξ= 0). The eigenstates nofHeffthen have a definite spin and\nmay be labelled nσ. It follows from (40) that χ0\nσσ′σ1σ′\n1∝δσσ′\n1δσ′σ1. Henceχ0\n12= 0 and, from (31), BD−1= 0. Also,\nfrom Appendix B, S12=S21= 0. Thus, (30) reduces to (14) as it should. Considering χ0\n↓↑↑↓(0,ω), given by (40)\nand (41), we note that state mis pure↓spin, labelled by m↓, andnis pure↑, labelled by n↑. Hence for ξ= 0\nB↓↑\nmn(k,0) =/summationdisplay\nµ∝angb∇acketleftkm|kµ∝angb∇acket∇ight∝angb∇acketleftkµ|kn∝angb∇acket∇ight=δmn (43)\nfrom closure. Thus\nχ0\n↓↑↑↓(0,ω) =N−1/summationdisplay\nknfkn↑−fkn↓\nEkn↓−Ekn↑−/planckover2pi1ω+iη(44)\nand it follows from (34) that Eknσmay be written as\nEknσ=Ekn−(σ/2)(∆+bex). (45)\nHence we find from (14) that for ξ= 0\nχ↓↑↑↓(0,ω) = (2∝angb∇acketleftSz∝angb∇acket∇ight/N)(bex−/planckover2pi1ω+iη)−1. (46)\nThus, as η→0,ℑχ↓↑↑↓(0,ω) has a sharp delta-function resonance at /planckover2pi1ω=bexas expected.\nWhen SOC is included /planckover2pi1ωacquires an imaginary part that corresponds to damping. We now pr oceed to calculate\nthis imaginary part to O(ξ2). To do this we can take ξ= 0 in the numerator of (30) so that\nχ↓↑↑↓(0,ω) =χ0\n↓↑↑↓(0,ω)/(S11−S12S21/S22) (47)\nIn factS12andS21are both O(ξ2) whileS22isO(1). Thus to obtain /planckover2pi1ωtoO(ξ2) we need only solve S11= 0.\nFurthermore all response functions such as χ0\n↑↑↑↓, with all but one spins in the same direction, are zero for ξ= 0 and\nneed only be calculated to O(ξ) in (27). We show below that to this order they vanish, so that to O(ξ2) the last term\ninS11is zero and we only have to solve the equation 1 −Uχ0\n↓↑↑↓= 0 for/planckover2pi1ω. This means that to second order in ξ\nthe shift in resonance frequency and the damping do not depend on the long-range Coulomb interaction.\nTo determine χ0\n↑↑↑↓(0,ω) to first order in ξfrom (40) we notice that states nmust be pure ↑spin, that is |kn∝angb∇acket∇ight=\n|kn↑∝angb∇acket∇ight, while states mmust be calculated using perturbation theory. The latter states m ay be written\n|km1∝angb∇acket∇ight=|km↑∝angb∇acket∇ight−ξ/summationdisplay\npσ∝angb∇acketleftkpσ|hso|km↑∝angb∇acket∇ight\nEkm↑−Ekpσ|kpσ∝angb∇acket∇ight (48)\n|km2∝angb∇acket∇ight=|km↓∝angb∇acket∇ight−ξ/summationdisplay\npσ∝angb∇acketleftkpσ|hso|km↓∝angb∇acket∇ight\nEkm↓−Ekpσ|kpσ∝angb∇acket∇ight, (49)\nwhere we have put Hso=ξhso, and to first order in ξ,\nχ0\n↑↑↑↓=1\nN/summationdisplay\nkµν/summationdisplay\nmn(a↑\nm1µ∗anµa∗\nnνa↓\nm1νfkn↑−fkm↑\nEkm↑−Ekn↑−/planckover2pi1ω+iη+a↑\nm2µ∗anµa∗\nnνa↓\nm2νfkn↑−fkm↓\nEkm↓−Ekn↑−/planckover2pi1ω+iη) (50)\nwithaσ\nmsµ=∝angb∇acketleftkµσ|kms∝angb∇acket∇ight,s= 1,2,andanµ=∝angb∇acketleftkµ|kn∝angb∇acket∇ightis independent of spin. Since a↓\nm1ν∼ξwe takea↑\nm1µ=amµin\nthe first term of (50). Also/summationtext\nµa∗\nmµanµ=δmnby closure so that the first term of (50) vanishes since the differen ce\nof Fermi functions is zero. Only the second term of χ0\n↑↑↑↓remains and this becomes, by use of (49),\nχ0\n↑↑↑↓=−ξ/summationdisplay\nkµν/summationdisplay\nmnp∝angb∇acketleftkp↑ |hso|km↓∝angb∇acket∇ight∗\nEkm↓−Ekp↑a∗\npµanµa∗\nnνamνfkn↑−fkm↓\nEkm↓−Ekn↑−/planckover2pi1ω+iη. (51)8\nAgain using closure only terms with p=m=nsurvive and the matrix element of hsobecomes\n∝angb∇acketleftkn↑ |/summationdisplay\njLj·Sj|kn↓∝angb∇acket∇ight=1\n2∝angb∇acketleftkn|L−|kn∝angb∇acket∇ight= 0 (52)\ndue to the quenching of total orbital angular momentum L=/summationtext\njLj[30]. Thus, to first order in ξ,χ0\n↑↑↑↓(0,ω), and\nsimilar response functions with one reversed spin, are zero. Hence we have only to solve 1 −Uχ0\n↓↑↑↓= 0 to obtain\nℑ(/planckover2pi1ω) toO(ξ2). Here we assume the system has spatial inversion symmetry witho ut which the quenching of orbital\nangular momentum, as expressed by (52), no longer pertains [30]. We briefly discuss the consequences of a breakdown\nof inversion symmetry at the end of this section.\nOn introducing the perturbed states (48) and (49) we write (41) in the form\nχ0\n↓↑↑↓(0,ω) =1\nN/summationdisplay\nkmn(|B↓↑\nm1n1|2fkn1−fkm1\nEkm1−Ekn1−/planckover2pi1ω+iη+|B↓↑\nm1n2|2fkn2−fkm1\nEkm1−Ekn2−/planckover2pi1ω+iη\n+|B↓↑\nm2n1|2fkn1−fkm2\nEkm2−Ekn1−/planckover2pi1ω+iη+|B↓↑\nm2n2|2fkn2−fkm2\nEkm2−Ekn2−/planckover2pi1ω+iη).(53)\nClearlyB↓↑\nm1n1andB↓↑\nm2n2are of order ξ,B↓↑\nm1n2isO(ξ2) andB↓↑\nm2n1isO(1). We therefore neglect the term |B↓↑\nm1n2|2\nand, using (48) and (49), we find\nB↓↑\nm1n1=−B↓↑\nm2n2=ξ\n2∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight\nEkn↓−Ekm↑. (54)\nThe evaluation of |B↓↑\nm2n1|2requires more care. It appears at first sight that to obtain this to O(ξ2) we need to include\nsecond order terms in the perturbed eigenstates given by (48) an d (49). However it turns out that these terms do not\nin fact contribute to |B↓↑\nm2n1|2toO(ξ2) so we shall not consider them further. Then we find\nB↓↑\nm2n1=δmn−ξ∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight\nEkn−Ekm−ξ2\n4/summationdisplay\np∝angb∇acketleftkm|Lz|kp∝angb∇acket∇ight∝angb∇acketleftkp|Lz|kn∝angb∇acket∇ight\n(Ekm−Ekp)(Ekn−Ekp)(55)\nand hence to O(ξ2)\n|B↓↑\nm2n1|2=δmn(1−ξ2\n2/summationdisplay\np|∝angb∇acketleftkm|Lz|kp∝angb∇acket∇ight|2\n(Ekm−Ekp)2)+ξ2|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2\n(Ekn−Ekm)2(56)\nThe contribution of this quantity to χ0\n↓↑↑↓(0,ω) in (53) may be written to O(ξ2) as\n1\nN/summationdisplay\nkmn|B↓↑\nm2n1|2fkn1−fkm2\nEkm2−Ekn1−bex+iη(1−bex−/planckover2pi1ω\nEkm2−Ekn1−bex+iη). (57)\nThisisobtainedbyintroducingtheidentity −/planckover2pi1ω=−bex+(bex−/planckover2pi1ω)intherelevantdenominatorin(53),andexpanding\nto first order in bex−/planckover2pi1ωwhich turns out to be O(ξ2). The remaining factors of this second term in (57) may then\nbe evaluated with ξ= 0, as at the beginning of this section, so that this term becomes ( /planckover2pi1ω−bex)/(2U2∝angb∇acketleftSz∝angb∇acket∇ight). By\ncombining equations (53), (54) and (57), and ignoring some real te rms, we find that the equation 1 −Uχ0\n↓↑↑↓(0,ω) = 0\nleads to the relation\nℑ(/planckover2pi1ω) =πξ2/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm[(fkn↑−fkm↓)|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2δ(Ekm↓−Ekn↑−bex)\n+(1/4)(fkn↑+fkn↓−fkm↑−fkm↓)|∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight|2δ(Ekm−Ekn−bex)](58)\nThe Gilbert damping parameter αis given by ℑ(/planckover2pi1ω)/bex(e.g. [39]) and in (58) we note that\n(fkn↑−fkm↓)δ(Ekm↓−Ekn↑−bex) = [F(Ekn↑−µ0)−F(Ekn↑+bex−µ0)]δ(Ekm↓−Ekn↑−bex)\n=bexδ(Ekn↑−µ0)δ(Ekm↑−µ0)(59)\nto first order in bexat temperature T= 0. Similarly\n(fknσ−fkmσ)δ(Ekm−Ekn−bex) =bexδ(Eknσ−µ0)δ(Ekmσ−µ0). (60)9\nThus from (58)\nα=πξ2/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2δ(Ekn↑−µ0)δ(Ekm↓−µ0)\n+πξ2/(8∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknmσ|∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight|2δ(Eknσ−µ0)δ(Ekmσ−µ0)(61)\ncorrect to O(ξ2). We note that there is no contribution from intraband terms since ∝angb∇acketleftkn|L|kn∝angb∇acket∇ight= 0. It is straight-\nforward to show that to O(ξ2) this is equivalent to the expression\nα=π/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm/summationdisplay\nσσ′|Amσ,nσ′(k)|2δ(Ekmσ−µ0)δ(Eknσ′−µ0) (62)\nwhere\nAmσ,nσ′(k) =ξ∝angb∇acketleftkmσ|[S−,hso]|knσ′∝angb∇acket∇ight (63)\nandS−is the total spin operator/summationtext\njS−\njwithS−\nj=Sx\nj−iSy\nj. This may be written more concisely as\nα=π/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm|Amn(k)|2δ(Ekm−µ0)δ(Ekn−µ0) (64)\nwith\nAmn(k) =ξ∝angb∇acketleftkm|[S−,hso]|kn∝angb∇acket∇ight (65)\nand the understanding that the one-electron states km,knare calculated in the absence of SOC. Equation (64) is the\nstandard form of the Kambersky formula ( [4], [9]) but in the literatur e SOC is invariably included in the calculation\nof the one-electronstates. This means that the intraband terms withm=nno longer vanish. They involve the square\nof a delta-function and this problem is always addressed by invoking t he effect of impurity and/or phonon scattering\nto replace the delta-functions by Lorentzians of width proportion al to an inverse relaxation time parameter τ−1. Then\nas one approaches a perfect crystal ( τ→ ∞) the intraband contribution to αtends to infinity. This behaviour is\nillustrated in many papers ( [7], [10], [13], [14]). In fig. 1 of [14] it is shown c learly that αremains finite if one does\nnot include SOC in calculating the one-electron states. However the effect of not including SOC is not confined to\ntotal removal of the intraband contribution. The remaining interb and contribution is increased considerably in the\nlow scattering rate regime, by almost an order of magnitude in the ca se of Fe. This makes αalmost independent of\nscattering rate in Fe which may relate to its observed temperature independence [15]. The corresponding effect in\nCo is insufficient to produce the increase of αat low scattering rate inferred from its temperature dependence . The\nnon-inclusion of SOC in calculating the one-electron states used in th e Kambersky formula clearly makes a major\nqualitative and quantitative change in the results. This occurs as so on as intraband terms become dominant in\ncalculations where they are included. For Fe, Co and Ni this corresp onds to impurity content and temperature such\nthat the scattering rate 1 /τdue to defects and/or phonons is less that about 1014sec−1( [7, 14]). Typically these\nmetals at room temperature find themselves well into the high scatt ering-rate regime where the damping rate can\nbe reliably estimated from the Kambersky interband term, with or wit hout SOC included in the band structure [35].\nThe physics at room temperature is not particularly interesting. On e needs to lower the temperature into the low\nscattering-rate regime where intraband terms, if they exist, will d ominate and lead to an anomalous ξ3dependence of\nthe damping on spin-orbit parameter ξ( [4], [8], [14]). The origin of this behaviour is explained in [4], [14]. It arises\nin theksum of (64) from a striplike region on the Fermi surface around a line where two different energy bands cross\neach other in the absence of SOC. The strip width is proportional to ξ, or more precisely |ξ|. SinceAnn(k) is of\norderξthe contribution of intraband terms in (64) is proportional to |ξ|3. Thus the intraband terms lead to terms\ninαwhich diverge in the limit τ−1→0 and are non-analytic functions of ξ. The calculation of αin this section can\nbe extended to higher powers of ξthan the second. No intraband terms appear and the result is an an alytic power\nseries containing only even powers of ξ.\nThe interband term in Kambersky’s formula can be given a very simple in terpretation in terms of Fermi’s ”golden\nrule” for transition probability [5]. This corresponds to second orde r perturbation theory in the spin-orbit interaction.\nThe decay of a uniform mode ( q= 0) magnon into an electron-hole pair involves the transition of an ele ctron from\nan occupied state to an unoccupied state of the same wave-vecto r. This is necessarily an interband transition and\nthe states involved in the matrix element are unperturbed, that is c alculated in the absence of SOC. A quite different\napproach has been adopted to try and find a physical interpretat ion of Kambersky’s intraband term ( [5, 8]). This\nemploys Kambersky’s earlier ”breathing Fermi surface” model ( [3, 34]) whose range of validity is uncertain.10\nWe now briefly discuss the consequences of a breakdown of spatial inversion symmetry so that total orbital angular\nmomentum is not quenched. In general response functions such a sχ0\n↑↑↑↓(0,ω) with one reversed spin are no longer\nzero to first order in ξ. HenceS11is not given to order ξ2just by the first two terms of (27) but involves further terms\nwhich depend explicitly on the long-range Coulomb interaction. Conse quentlyαhas a similar dependence which\ndoes not emerge from the torque-correlation approach. In Appe ndix A it is pointed out how the direct proof of the\nKambersky formula breaks down in the absence of spatial inversion symmetry.\nVII. EXPERIMENTAL ASPECTS\nThe inclusion of intraband terms in the Kambersky formula, despite t heir singular nature, has gained acceptance\nbecause they appear to explain a rise in intrinsic damping parameter αat low temperature which is observed in some\nsystems [15]. The calculated intraband contribution to αis proportional to the relaxation time τand it is expected\nthat, due to electron-phonon scattering, τwill increase as the temperature is reduced. This is in qualitative agre ement\nwith data [15] for Ni and hcp Co. Also a small 10% increase in αis observed in Co 2FeAl films as the temperature is\ndecreased from 300 K to 80 K [36]. However in Fe the damping αis found to be independent of temperature down to\n4 K [15]. Very recent measurements [16] on FePt films, with varying an tisite disorder xintroduced into the otherwise\nwell-ordered structure, show that αincreases steadily as xincreases from 3 to 16%. Hence αincreases monotonically\nwith scattering rate 1 /τas expected from the Kambersky formula in the absence of intraba nd terms. Furthermore\nforx= 3% it is found that αremains almost unchanged when the temperature is decreased fro m 200 to 20 K. Ma et\nal [16] therefore conclude that there is no indication of an intraban d term in α. From the present point of view the\norigin of the observed low temperature increase of αin Co and Ni is unclear. Further experimental work to confirm\nthe results of Bhagat and Lubitz [15] is desirable.\nThe second unusual feature of the intraband term in Kambersky’s formula for αis its|ξ|3dependence on the SOC\nparameter ξ. This contrasts with the ξ2dependence of the interband contribution which has been observe d in a\nnumber of alloys at room temperature [17]. Recently this behaviour has been seen very precisely in FePd 1−xPtxalloys\nwhereξcan be varied over a wide range by varying x[18]. Unfortunately this work has not been extended to the\nlow temperature regime where the |ξ|3dependence, if it exists, should be seen. It would be particularly inte resting to\nsee low temperature data for NiPd 1−xPtxand CoPd 1−xPtxsince it is in Ni and Co where the intraband contribution\nhas been invoked to explain the low temperature behaviour of α. From the present point of view, with the intraband\nterm absent, one would expect ξ2behaviour over the whole temperature range.\nVIII. CONCLUSIONS AND OUTLOOK\nIn this paper we analyse two methods which are used in the literature to calculate the damping in magnetization\ndynamics due to spin-orbit coupling. The first common approach is to employ Kambersky’s[4] formula for the Gilbert\ndamping parameter αwhich delivers an infinite value for a pure metal if used beyond second order in the spin-orbit\nparameter ξ. The second approach [19] is to calculate numerically the line-width of the ferromagnetic resonance seen\nin the uniform transverse spin susceptibility. This is always found to b e finite, corresponding to finite α. We resolve\nthis apparent inconsistency between the two methods by an analyt ic treatment of the Costa-Muniz approach for the\nsimplified model of a ferromagnetic metal with d-bands only. It is sho wn that this method leads to the Kambersky\nresult correct to second order in ξbut Kambersky’s intraband scattering term, taking the non-analy tic form |ξ|3, is\nabsent. Higher order terms in the present work are analytic even p owers of ξ. The absence of Kambersky’s intraband\nterm is the main result of this paper and it is in agreement with the conc lusion that Ma et al [16] draw from their\nexperiments on FePt films. Further experimental work on the depe ndence of damping on electron scattering-rate and\nspin-orbit parameter in other systems is highly desirable.\nA secondaryconclusionis that beyond second orderin ξsome additional physics ariseswhich has not been remarked\non previously. This is the role of long-range Coulomb interaction which is essential for a proper treatment of the\nlongitudinal susceptibility and charge response to which the transv erse susceptibility is coupled by spin-orbit interac-\ntion. Costa and Muniz [19] stress this coupling but fail to introduce t he long-range Coulomb interaction. Generally,\nhowever, it seems unnecessary to go beyond second order in ξ[17, 18] and for most bulk systems Kambersky’s for-\nmula, with electron states calculated in the absence of SOC, should b e adequate. However in systems without spatial\ninversion symmetry, which include layered structures of practical importance, the Kambersky formulation may be\ninadequate even to second order in ξ. The long-range Coulomb interaction can now play a role.\nAn important property of ferromagnetic systems without inversio n symmetry is the Dzyaloshinskii-Moriya inter-\naction (DMI) which leads to an instability of the uniform ferromagnet ic state with the appearance of a spiral spin\nstructure or a skyrmion structure. This has been studied extens ively in bulk crystals like MnSi [37] and in layered11\nstructures [38]. The spiral instability appears as a singularity in the t ransverse susceptibility χ(q,0) at a value of q\nrelated to the DMI parameter. The method of this paper has been u sed to obtain a novel closed form expression for\nthis parameter which will be reported elsewhere.\nIn this paper we have analysed in some detail the transverse spin su sceptibility χ↓↑↑↓but combinations of some of\nthe 15 other response functions merit further study. Mixed char ge-spin response arising from spin-orbit coupling is\nof particular interest for its relation to phenomena like the spin-Hall effect.\nAppendix A: A direct derivation of the Kambersky formula\nIn this appendix we give a rather general derivation of the Kambers ky formula for the Gilbert damping parameter\nαwith an emphasis on its restriction to second order in the spin-orbit in teraction parameter ξ.\nWe consider a general ferromagnetic material described by the ma ny-body Hamiltonian\nH=H1+Hint+Hext (A1)\nwhereH1is a one-electron Hamiltonian of the form\nH1=Hk+Hso+V. (A2)\nHereHkis the total kinetic energy, Hso=ξhsois the spin-orbit interaction, Vis a potential term, Hintis the\nCoulomb interaction between electrons and Hextis due to an external magnetic field Bexin thezdirection. Thus\nHext=−Szbexwherebex= 2µBBex, as in (34), and Szis thezcomponent of total spin. Both HsoandVcan contain\ndisorderalthough in this paper we consider a perfect crystal. Follow ingthe general method of Edwardsand Fisher [40]\nwe use equations of motion to find that the dynamical transverse s usceptibility χ(ω) =χ−+(0,ω) satisfies [39]\nχ(ω) =−2∝angb∇acketleftSz∝angb∇acket∇ight\n/planckover2pi1ω−bex+ξ2\n(/planckover2pi1ω−bex)2(χF(ω)−ξ−1∝angb∇acketleft[F−,S+]∝angb∇acket∇ight) (A3)\nwhere\nχF(ω) =/integraldisplay\n∝angb∇acketleft∝angb∇acketleftF−(t),F+∝angb∇acket∇ight∝angb∇acket∇ighte−iωtdt (A4)\nwithF−= [S−,hso]. This follows since S−commutes with other terms in H1and with Hint. For small ω,χis\ndominated by the spin wave pole at /planckover2pi1ω=bext+/planckover2pi1δωwhereδω∼ξ2, so that\nχ(ω) =−2∝angb∇acketleftSz∝angb∇acket∇ight\n/planckover2pi1(ω−δω)−bex. (A5)\nFollowing [39] we compare (A3) and (A5) in the limit /planckover2pi1δω≪/planckover2pi1ω−bexto obtain\n−2∝angb∇acketleftSz∝angb∇acket∇ight/planckover2pi1δω=ξ2(χF(ω)−ξ−1∝angb∇acketleft[F−,S+]∝angb∇acket∇ight) =ξ2[ lim\n/planckover2pi1ω→bexχξ=0\nF(ω)−lim\nξ→0(1\nξ∝angb∇acketleft[F−,S+]∝angb∇acket∇ight)] (A6)\ncorrect to order ξ2. It is important to note that the limit ξ→0 within the bracket must be taken before putting\n/planckover2pi1ω=bex. If we put /planckover2pi1ω=bexfirst it is clear from (A3) that the quantity in brackets would vanish, giving the incorrect\nresultδω= 0. Furthermore it may be shown [M. Cinal, private communication] th at the second term in the bracket\nis real. Hence\nℑ(/planckover2pi1ω) =−ξ2\n2∝angb∇acketleftSz∝angb∇acket∇ightlim\n/planckover2pi1ω→bexℑ[χξ=0\nF(ω)]. (A7)\nKambersky [4] derived this result, using the approach of Mori and K awasaki ( [41] [42]), without noting its restricted\nvalidity to second order in ξ. This restriction is crucial since, as discussed in the main paper, it av oids the appearance\nof singular intraband terms. Oshikawa and Affleck emphasise strong ly a similar restriction in their related work on\nelectron spin resonance (Appendix of [43]).\nEquation (A7) is an exact result even in the presence of disorder in t he potential and spin-orbit terms of the\nHamiltonian. In the following we assume translational symmetry.12\nTo obtain the expression (61) for α=ℑ(/planckover2pi1ω)/bex, which is equivalent to Kambersky’s result (62), it is necessary to\nevaluate the response function χξ=0\nF(ω) in tight-binding-RPA. Using (35)we find\nF−=/summationdisplay\nkµν[Lz\nµνc†\nkµ↓ckν↑+(1/2)L−\nµν(c†\nkµ↓ckν↓−c†\nkµ↑ckν↑)]. (A8)\nHence\nχξ=0\nF=/summationdisplay\nµν/summationdisplay\nαβ[Lz\nµνLz\nβαGµ↓ν↑,β↑α↓+(1/4)L−\nµνL+\nβα(Gµ↓ν↓,β↓α↓+Gµ↑ν↑,β↑α↑−Gµ↓ν↓,β↑α↑−Gµ↑ν↑,β↓α↓)] (A9)\nwhere\nGµσνσ′,βτατ′=∝angb∇acketleft∝angb∇acketleft/summationdisplay\nkc†\nkµσckνσ′;/summationdisplay\nuc†\nuβτcuατ′∝angb∇acket∇ight∝angb∇acket∇ightω. (A10)\nThe Green function Gis to be calculated in the absence of SOC ( ξ= 0). Within RPA it satisfies an equation of the\nform\nGµσνσ′,βτατ′=G0\nµσνσ′,βτατ′−/summationdisplay\nµ1σ1ν1σ′\n1/summationdisplay\nµ2σ2ν2σ′\n2G0\nµσνσ′,µ1σ1ν1σ′\n1Vµ1σ1ν1σ′\n1,µ2σ2ν2σ′\n2Gµ2σ2ν2σ′\n2,βτατ′ (A11)\nwhereG0is the non-interacting (Hartree-Fock) Green function and\nVµ1σ1ν1σ′\n1,µ2σ2ν2σ′\n2=Vσ1σ′\n1σ2σ′\n2(q)δµ1ν1δµ2ν2 (A12)\nwithV(q) given by (11) and (12). Hence\nGµσνσ′,βτατ′=G0\nµσνσ′,βτατ′−/summationdisplay\nµ1σ1σ′\n1/summationdisplay\nµ2σ2σ′\n2G0\nµσνσ′,µ1σ1µ1σ′\n1Vσ1σ′\n1σ2σ′\n2Gµ2σ2µ2σ′\n2,βτατ′. (A13)\nThe form of the interaction Vgiven in (A12) is justified by the connection between (A13) and (10) , withq= 0. To\nsee this connection we note that χσσ′ττ′=/summationtext\nµνGµσµσ′,ντντ′and that (A13) then leads to (10) which is equivalent to\nRPA. On substituting (A13) into (A9) we see that the contributions from the second term of (A13) contain factors\nof the form\n/summationdisplay\nµνµ1Lz\nµνG0\nµ↓ν↑,µ1↑µ1↓,/summationdisplay\nµνµ1L−\nµνG0\nµσνσ,µ 1σ1µ1σ1. (A14)\nWe now show that such factors vanish owing to quenching of orbital angular momentum in the system without SOC\n(ξ= 0). Hence the Green functions Gin (A9) can be replaced by the non-interacting ones G0. The non-interacting\nGreen functions G0are of a similar form to χ0in (40) and for ξ= 0 may be expressed in terms of the quantities\nanµ=∝angb∇acketleftkµ|kn∝angb∇acket∇ightwhere|kn∝angb∇acket∇ightis a one-electron eigenstate as introduced in section VI. Hence we fi nd, in the same way\nthat (44) emerged,\n/summationdisplay\nµνµ1Lz\nµνG0\nµ↓ν↑,µ1↑µ1↓=/summationdisplay\nµν/summationdisplay\nknLz\nµνa∗\nnµanνfkn↑−fkn↓\n∆+bex−/planckover2pi1ω+iη. (A15)\nAlso by closure\n/summationdisplay\nµνLz\nµνa∗\nnµanν=∝angb∇acketleftkn|Lz|kn∝angb∇acket∇ight= 0, (A16)\nthe last step following from quenching of total orbital angular mome ntum. The proof that the second expression\nin (A14) vanishes is very similar.\nHence we can insert the non-interacting Green functions G0in (A9) and straight-forwardalgebra, with use of (A7),\nleads to (58). At the end of section VI this is shown to be equivalent t o the Kambersky formula for α. We emphasize\nagain that the present proof is valid only to order ξ2so that the one-electron states used to evaluate the formula\nshould be calculated in the absence of SOC.\nThis proof relies on the quenching of orbital angular momentum which does not occur in the absence of spatial\ninversion symmetry. When this symmetry is broken it is not difficult to s ee that the second term of (A13) gives a\ncontribution to the first term on the right of (A9) which contains th eq= 0 spin-wave pole and diverges as /planckover2pi1ω→bex.\nHence the proof of the torque-correlationformula (A7) collapses . The method of section VI must be used as discussed\nat the end of that section.13\nAppendix B: Elements of S\nThe element S11of the matrix Sis given in (27). The remaining elements are given below.\nS12=−Uχ0\n↓↑↓↑+(U/Λ)[(X+χ0\n↓↓↓↑)χ0\n↑↑↓↑χ0\n↓↑↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↓↑χ0\n↓↑↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↓↑χ0\n↓↑↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↓↑χ0\n↓↑↓↓](B1)\nS21=−Uχ0\n↑↓↑↓+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↑↓χ0\n↑↓↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↑↓χ0\n↑↓↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↑↓χ0\n↑↓↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↑↓χ0\n↑↓↓↓](B2)\nS22= 1−Uχ0\n↑↓↓↑+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↓↑χ0\n↑↓↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↓↑χ0\n↑↓↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↓↑χ0\n↑↓↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↓↑χ0\n↑↓↓↓](B3)\nAcknowledgement\nMy recent interest in Gilbert damping arose through collaboration wit h O. Wessely, E. Barati, M. Cinal and A.\nUmerski. I am grateful to them for stimulating discussion and corre spondence. The specific work reported here arose\ndirectly from discussion with R.B.Muniz and I am particularly grateful t o him and his colleague A. T. Costa for this\nstimulation.\nReferences\n[1] Landau L D, Lifshitz E M and Pitaevski L P 1980 Statistical Physics part2 (Oxford: Pergamon)\n[2] Gilbert T L 1955 Phys. 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Lundqvist S and March N H\n(Plenum)14\n[29] Edwards D M 1984, in Moment Formation in Solids NATO Advanced Study Institute, Series B: Physics Vol. 117, e d.\nBuyers W J L (Plenum)\n[30] Cinal M and Edwards D M 1997 Phys. Rev B553636\n[31] Izuyama T, Kim D J and Kubo R 1963 J. Phys. Soc. Japan 181025\n[32] Lowde R D and Windsor C G 1970 Adv. Phys. 19813\n[33] Doniach S and Sondheimer E H 1998 Green’s Functions for Solid State Physicists Imperial College Press\n[34] Kunes J and Kambersky V 2002 Phys. Rev B65212411\n[35] Gilmore K, Garate I, MacDonald AH and Stiles MD Phys. Rev B84224412\n[36] Yuan HC, Nie SH, Ma TP, Zhang Z, Zheng Z, Chen ZH, Wu YZ, Zha o JH, Zhao HB and Chen LY 2014 Appl. Phys. Lett.\n105072413\n[37] Grigoriev SV, Maleyev SV, Okorokov AI, Chetverikov Yu. O, B¨ oni P, Georgii R, Lamago D, Eckerlebe H and Pranzas K\n2006Phys. Rev. B74214414\n[38] von Bergmann K, Kubetzka A, Pietzsch O and Wiesendanger R 2014J. Phys. Condens. 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B65134410" }, { "title": "1809.11020v1.Isotropic_non_local_Gilbert_damping_driven_by_spin_currents_in_epitaxial_Pd_Fe_MgO_001__films.pdf", "content": "Isotropic non -local Gilbert damping d riven by spin current s in epitaxial \n Pd/Fe/MgO(001) film s \nYan Li1,2,Yang Li1,2,Qian Liu3, Zhe Yuan3, Wei He1,Hao -Liang Liu1, Ke Xia3,Wei Yu1, \nXiang- Qun Zhang1, and Zhao- Hua Cheng1,2, * \n1State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed \nMatter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, \nChina \n2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing \n100049, China \n3The Center for Advanced Quantum Studies and Department of Physics, Beijing \nNormal University, 100875 China \nABSTRACT \nAlthough both theoretical predications and experimental observations \ndemonstrate d that the damping factor is anisotropic at ferromagne t/semiconductor \ninterface with robust interfacial spin- orbit coupling , it is not well understood whether \nnon-local Gilbert damping driven by spin current s in heavy metal /ferromagnetic metal \n(HM/FM) bilayers is anisotropic or not. H ere, we investigated the in -plane angular - \nand frequenc y- dependen ce of magnetic relaxation of epitaxial Fe /MgO(001) films \nwith different capping layers of Pd and Cu. After disentangl ing the parasitic \ncontributions, such as two -magnon scattering (TMS) , mosaicity, and field-dragging \neffect, we unambiguously observed that both local and non- local Gilbert damping are \nisotropic in Fe(001) plane , suggest ing that the pure spin current s absorption is \nindependent of Fe magnetization orientation in the epitaxial Pd/Fe heterostructure. \nFirst principles calculation reveal s that the effective spin mixing conductance of Pd/Fe interface is nearly invariant for different magnetization directions in good \nagreement with the experimental observation s. These results offer a valuable insight \ninto the transmission and absorption of pure spin currents, and facilitate us to utilize \nnext-generation spintronic devices. \nPACS number s: 72.25.Mk, 75.78.- n, 76.50.+g \n*Corresponding author \nE-mail: zhcheng@iphy.ac.cn \n \n I. INTRODUCTION \nThe rapid development of spintronic devices inquires deeper understanding of \nthe magnetization relaxation mechanism1-3. The Gilbert damping factor, one of key \nparameter s in spin dynamics , characterizes the energy transfer from the spin \nsubsystem to the lattice and governs the magnetization switching time and the critical \ncurrent density in spin transfer torque devices4-6. Since the shape of Fermi surface \ndepends on the orientation of the mag netization direction due to the spin- orbit \ninteraction, an anisotropic Gilbert damping is expected in single crystal ultrathin \nfilms7-10. Chen et al. discove red an anisotropic damping in the Fe/GaAs(001) ultrathin \nfilms where an robust interfacial spin -orbit field exists , due to GaAs substrate . The \nmagnitude of damping anisotropy , however, decreases with increasing Fe thickness , \nand disappears when the Fe thickness is larger than 1.9 nm11-13. \nBesides intrinsic Gilbert damping in ferromagnetic materials (FM) , spin current s \nsink into heavy metal s (HM) or other magnetic layer s importing non -local Gilbert \ndamping in HM/FM bilayer s or spin valve structure according to spin pumping \nmodel14-16. Although anisotropic magnetization relaxation in ferromagnetic \nmultilayers w as observed, it is debated whether the absorption of pure spin currents is \nanisotropic or isotropic in ferromagnetic multilay ers17-21. This is because the \nfrequency - and angular -dependent ferromagnetic resonance (FMR) linewidth results \nare often contaminated by parasitic contributions , such as two -magnon scattering \n(TMS), mosaicity, and field -dragging effect . Li et al. found that nearly isotropic \nabsorption of pure spin current in Co in Py1-xCux/Cu(5 nm)/Co(5 nm) trilayers using spin pumping technique22. Meanwhile, Baker et al. found an anisotropic absorption of \npure spin currents in Co 50Fe50/Cr/Ni 81Fe19 spin valves with variable Cr thickness, \nwhile the anisotropy is suppressed above the spin diffusion length23. Here, we \ninvestigated spin pumping and clarified the dependence of diverse magnetic \nrelaxations on Fe magnetic orientation using Vector Network Analyzer ferromagnetic \nresonance (VNA- FMR) of epitaxial Fe /MgO(001) films capped by Pd and Cu layers. \nSimple FM/HM bilayers would be a more convincing candidate to explore the \nnon-local relaxation mechanism. Exclu ding the misleading dragging effect and the \ndeceitful extrinsic terms, we unambiguously observed that both local and non- local \nGilbert damping are isotropic in Fe(001) plane . The i sotropic non- local Gilbert \ndamping suggest s that the pure spin current s abso rption is independent of Fe \nmagnetization orientation , which is supported by the first principle s calculation. \nII. EXPERIMENTS \nSample s were prepared in molecular beam epitaxy chambers with a basic \npressure-102 10× mbar24. Prior to deposition, MgO(001) substrate was annealed at 700 ℃ \nfor 2 hours, and then 6 nm Fe film was deposited on a MgO(001) substrate using \nelectron -beam gun , and finally 5 nm Pd w as covered on Fe films . The crystalline \nquality and epitaxial relationship was confirmed by high- resolution transmissio n \nelectron micro scopy (HRTEM), as shown in Fig . 1(a) and (b). It has been revealed \nthat the films were grown with the epitaxial relationship Pd(001)<110>||Fe(001)<100>||MgO(001)<110> (see the inset of Fig . 1(b)). For \ncomparison, Cu(3.5 nm)/Fe (6 nm)/MgO(001) sample was also prepared. In-plane VNA- FMR measurements were performed by facing the sample down on employing \na co-planar waveguide (CPW) and recording the transmission coefficient S 2125-27. All \ndepositions and measurements were performed at room temperature. \nIII. RESULTS AND DISCUSSI ON \nFig. 2(a) s hows schematically the stacked sample and the measured \nconfigurat ion. The representative FMR spectra at fixed frequency 13.4 GHz and \nvarious magnetic field angle s Hϕ are illustrated in Fig. 2(b). T he FMR signal (the \ntransmission parameter S 21) is a superposition of symmetric and antisymmetric \nLorentzian functions . The following equation could be used to extract the resonance \nfield H r and the resonance line width H∆: \n2\n21 0 22 22( / 2)( ) ( / 2)Re ( ) +( ) ( / 2) ( ) ( / 2)r\nrrH HH HSH SL DHH H HH H∆ − ∆= −− +∆ − +∆. (1) \nHere, Re S21, S0, H, L and D are the real part of transmission parameter, the offset, the \nexternal magnetic field, the symmetric and antisymmetric magnitude , respectively25-27\n. \nThe resonance frequency f is given by Kittel formula28 \n0=2RR\nab f HHγµ\nπ (2) \nwith2\n42 cos( ) (3 cos 4 ) / 4 sin ( 45 )R\nar MH M M d H H HH H ϕϕ ϕ ϕ = −++ + − −a, \n42 cos( ) cos 4 sin 2R\nb r MH M M HH H H ϕϕ ϕ ϕ= −+ − and \n02=out\nds\nsKHMMµ− . Here, γ and \n0µ are the gyromagnetic ratio and the vacuum permeability. H , H2, H4 and Ms are the \napplied magnetic field, the uniaxial and four-fold magnetic anisotropy field s and \nsaturation magnetization , respectively. outK is the out -of-plane uniaxial magnetic \nanisotropy constant. The equilibrium azimuthal angle of magnetization Mϕis determined by the following equation: \n42 sin( ) ( / 4)sin 4 ( / 2)cos 2 0r MH M M H HHϕϕ ϕ ϕ −+ + = . (3) \nThe angular dependent FMR measurements were performed by rotating the \nsamples in plane while sweeping the applied magnetic field. At a fixed frequen cy of \n13.4 GHz, the angular dependence of H r can be derived from Eq. (2) and plotted in \nFig. 2(c) and 2(d) for Fe/MgO(001) sample s capped by Pd and Cu, respectively . It can \nbe seen clearly that the angular dependence of H r demonstrates a four -fold symmetry \nand the values of 2=0H Oe, 4=625H Oe and 0 2.0dHµ= T for Pd/ Fe/MgO(001) \nand2=0H Oe, 4=625H Oe and 0 1.9dHµ= T for Cu /Fe/MgO(001) , respectivel y. \nCompar ing to the sample with Cu c apping l ayer, Pd/Fe interface modifies the \nout-of-plane uniaxial magnetic anisotropy, and has a negligible contribution to the \nin-plane uniaxial magnetic anisotropy. \nIn cont rast to the four -fold symmetry of H r, the angul ar dependence of H∆for \nthe samples with Pd and Cu capping layers indicates to be superposition of four-fold \nand quasi -eight -fold contributions , as shown in Fig. 3(a) and 3(b) , respectively . In fact, \nthe quasi -eight -fold broadening also represent s a four-fold symmetry with multiple \nextreme value point s. In the case of the sample with Pd capping layer, H∆exhibits \ntwo peaks around the hard magnetization direction s Fe<1 10>, and the values of H∆ \nfor Fe<100> and Fe<110 > direction s are almost the same (58 Oe). On the other hand, \na larger difference in the magnitude of H∆ was observed along these two directions \nof Cu/Fe/MgO(001) sample , i.e. 71 Oe and 4 9 Oe for Fe<100> and Fe<110> axes, \nrespectively . In order to understand the mechanism of anisotropic magnetic relaxation, we \nmust take both intrinsic and extrinsic contributions into account29-34. H∆ is follow ed \nby the expression32\n: \n_ =mosaicity TMS Gilbert dragging HH H H∆ ∆ +∆ +∆ . ( 4) \nThe first term denotes TMS, represent ing that a uniform prerecession magnon ( 0k=) \nis scattered into a degenerate magno n ( 0k≠) due to imperfect crystal structure. \nTherefore, the contribution of TMS to the linewidth reli es heavily on the symmetrical \ndistribution of defects and manifest s anisotropic feature accordingly . The second term \ndescribes the mo saicity contribution in a film plane, which is caused by a slightly \nspread of magnetic parameters on a very large scale. The last term _ Gilbert draggingH∆ is \nthe Gilbert damping contribution with field -dragging . \nIn the case of Fe/ MgO( 001) epitaxial film, the contribution of TMS to FMR \nlinewidth composes of numerous two-fold and four -fold TMS channel s31-34, \nj,max 4 j,max j,max 2 j,maxcos ( ) cos 2( )TMS twofold M twofold fourfold M fourfold\njjH ϕϕ ϕϕ ∆ =Γ − +Γ − ∑∑ . ( 5) \nHere, j,max\ntwofoldϕ and j,max\nfourfoldϕ represent angle of the maximum scattering rate in \ntwo-fold and four -fold scatterings along the direction j. However, the same values of \nH∆between Fe<100> and Fe<110> directions suggest that the TMS can be neglected \nin Pd/Fe/MgO(001 ) epitaxial film. On the other hand, the larger difference in the \nmagnitude of H∆ was observed along these two directions , suggesting that either \nsignificant TMS contribution or anisotropic Gilbert damping exists in \nCu/Fe/MgO(001) s ample13, 32, 33. \nThe angular dependence of mosaicity contribution can be described as32, 34 =r\nmosaicity H\nHHH ϕϕ∂∆∆∂, ( 6) \nwhere Hϕ∆ represents an in plane variation of mosaicity. 0mosaicityH∆= Oe should \nbe hold along easy magnetization direction s and hard directions where =0r\nHH\nϕ∂\n∂. \nDue to magnetocrystalline anisotropy, magn etization would not always align at \nthe direction of the applied field when the field is weaker than the saturation field. We \nevaluate the field -dragging effect during rotation of the sample or frequency -swept \nbased on the numerical calculation using Eq. ( 3). Fig. 4(a) shows Hϕdependence on \nHϕ at 13.4 GHz. The relation reveals a conspicuous dragging effect with a four-fold \nsymmetry. At 25Hϕ=a, HMϕϕ− is as high as 12a. Fig. 4(b) sh ows Mϕ \ndependence on f at various Hϕ. When the magnetic field is applied along Fe<100> or \nFe<110> directions , the magnetization is always aligned along the applied magnetic \nfield. However, there is a conspicuous angle between the magnetization and the \nmagnetic field with the field along intermediate axis. Owing to the angle between \nmagnetization and applied field , H∆ corresponding to Gilbert contributio n with the \nfield-dragging could be disclose d according to the following equations12, 13 \n_ = [Im( )]Gilbert draggingH χ ∆∆ ( 7) \nand 22 2[]Im( )( ) ()RR RR\ne f f ab a a ab s\nRR RR\na b ab e f f ab a bHH H H HH M\nH H HH HH H+ Haχa+=−+ , (8) \nwhere aH and bH are R\naH and R\nbH in non- resonance condition. The effective \nparameter effa consist s of the intrinsic Gilbert damping an d the non- local one driven \nby spin currents . \nGenerally , effa was obtained by the slope of the linear dependence of H∆ on frequency f along the directions without field -dragging 28: \n0\n04efffHHπa\nµγ∆ = +∆ , (9) \nwhere 0H∆ is inhomogeneous non- Gilbert linewidth at zero -frequency25-27. Fig. 5 \nshows H∆ dependence on frequency at various Hϕ. Obviously , H∆ versus f can \nbe fitted linear ly with 3\n/ 6.0 10Pd Fea−= × and 3\n/=4.2 10Cu Fea−× for magnetic field \nalong easy axes Fe<1 00> or hard axes Fe<1 10> of the samples with Pd and Cu \ncapping layers, respectively , indicating isotropic damping (Fig. 5(f) and 5( j)). By \nusing the aforementioned isotropic damping factor s, the contributions of TMS, \nmosaicity, and field -dragging effect are separated from the angular dependence of \nH∆ (Fig. 3 a-b). Table I summar izes the fitted parameters in the two samples. \nCompared with Cu/Fe sample , one observes a significant reduction of mosaicity \nbroadening and a negligible TMS term in Pd/Fe bilayers. In fact, due to high mobilit y, \nthe capping layer Cu forms nanocrystallites on Fe film, which causes interfacial \ndefects dependence on the crystallographic ax es35-38. The interfacial defects will \nimpact a four-fold linewidth broadening due to TM S. In contrast , the excellent \nepitaxial quality at Pd/Fe interface not only ensures a sharp interfacial structure , but \nalso reduces defect density to decrease TMS contribution. Moreover, the mosaicity \ncontribution, indicat ing the fluctuation of the magne tic anisotropy field, could be \nstrengthen by the interfacial stacking faults. C onsequently , a fully epitaxial structure \ncould significantly decrease the extrinsic contributions, especially TMS and mosaicity \nterms . \nTaking these contributions to magnetizatio n relaxation into account, the frequency dependence of H∆ at various directions can be well reproduced, as shown \nin Fig. 5(f)- (j). For other directions rather than Fe<1 00> and Fe<1 10>, nonlinear \nrelationship between H∆ and f are evident and illustrated in Fig . 5(g-i). At =20Hϕa, \nthe H∆ vs f curve brings out a slight bump comparing to the linear ones along hard \nor easy ax es. At =27Hϕa, H∆ has a rapid decrease after H∆ experiencing an \nabrupt enhancement . At =33Hϕa, H∆ decreases more sharply after 11 GHz. The \nnonlinearity can be ascrib ed to the parasitic contrib utions, such as TMS, mosaicity, \nand field -dragging effect . It is virtually impossible to stem from TMS for the d istorted \ncurves because a nonlinear linewidth broadening due to TMS increases as frequency \nincreases, and approach es to saturation at high frequency31. According to the \ncalculation in Fig. 4(b), there is a huge field -dragging effect except the applied \nmagnetic field H along hard and easy ax es. The field -dragging will make H∆ vs f \ndeviate from the linear relationship . As expected , we could effectively fit the \nexperimental data H∆ vs f using the following equation in association with the \noriginal formula s (7), \n0 [Im( )] HH χ ∆ =∆ +∆ . ( 10) \nEq. (10) converges to the Eq. (9) with the applied magnetic field along the directions \nwithout field -dragging, i.e. easy axes Fe<1 00> or hard axes Fe<1 10>13. \nAfter distinguish ing the contributions of extrinsic terms and field -dragging effect, \nthe Gilbert damping factors effa along various direc tions are show n in Fig. 6(a). \nAccording to the classical spin pumping model14, precess ional magnetization in FM \nlayer will pump spins into adjacent nonmagnetic metals across interface. Cu with only s conduction band has a smaller spin- flip probability and a large r spin diffusion length \nthan 500 nm39, therefore, the reference sample Cu/Fe cannot increase the Gilbert \ndamping due to a capp ing layer Cu. In contrast , Pd-layer with stron g spin- orbit \ncoupling has a larger spin- flip probability , the injected spin currents are dissipated in \nPd-layer , and enhance the intrinsic Gilbert damping of Fe film. The enhancement of \nthe Gilbert damping allows us to comprehend the non- local relaxation m echanism. \nObviously, it can be seen from Fig. 6(a) that there is no strong relation between the \nnon-local Gilbert damping and the magnetization orientation in epitaxial film Pd/Fe. \nThe parameters-3\n/=4.2 10Cu Fea × and -3\n/=6.0 10Pd Fea × are the Gilbert damping of \nPd/Fe and Cu/Fe , respectively. The non-local Gilbert damping could be evaluated \nusing the effective spin mixing conductance effg↑↓14 \n// =4B\nPd Fe Cu Fe eff\ns FeggMtµaa aπ↑↓∆ −= . ( 11) \nThe obtained isotropic value 19 2=1.23 10effgm↑↓ −× is comparable to the literature s40-42. \nIn order to theoretically investigate the dependence of the non- local Gilbert \ndamping on the magnetization orientation, the first principles calculation was \nperformed to calculate the total Gilbert damping of the Pd /Fe/Pd multilayer on the \nbasis of the scattering theory43-45. The electronic structure of the Pd/Fe interface was \ncalculated self -consistently using the surface Green’s function technique implemented \nwith the tight- binding linearized muffin -tin orbitals method. Within the atomic sphere \napproximation, the charge and spin densities and the effective Kohn- Sham potentials \nwere evaluat ed inside atomic spheres. The total Gilbert damping was then calculated \nusing the scattering theory of magnetization dissipation45. We simulate d the room tempe rature via introducing frozen thermal lattice disorder into a 5x5 lateral \nsupercell43. The root -mean -squared displacement of the atoms is determined by the \nDebye model with the Debye temperature 470 K. A 28x 28 k-mesh is used to sample \nthe two -dimensional Brillouin zone and five different configurations of disorder have \nbeen calculated for each Fe thickness. The total Gilbert damping exhibits a linear \ndependence on the length of Fe and the intercept of the linear function can be \nextracted corresponding to the contribution of the spin pumping at the Pd/Fe \ninterface44. The interfacial contribution i s converted to the effective spin mixing \nconductance, plotted in Fig. 6( b) as a function of the magnetization orientation. It can \nbe seen that the effective spin mixing conductance across Pd/Fe interface \n19 2=1.29 10effgm↑↓ −× is independent of the magnet ization direction , and is in very \ngood agreement with the experimental value19 21.23 10 m−× . According to the \nElliott- Yafet mechanism in a nonmagnetic metal , spins relax indiscriminately energy \nand momentum along all orientation in Pd-layer since a cubic metal is expected to \npossess a weak anisotropy of the Elliott -Yafet parameter46. Incidentally , the fitting \nerror will mislead an aniso tropic Gilbert damping if ones use Eq. ( 9) to fit the entire \ncurves H∆ vs f. Besides , an epitaxial magnetic film integrated into a pseudo spin \nvalve could lead to an anisotropic absorption of spin current based on s pin transfer \ntorque mechanism since it is demanding to drag magnetization parallel ing to the \napplied field11. \n I V. CONCLUSIONS \nIn summary, a non-local Gilbert damping is induced by the spin pumping in \nPd/Fe bilayers as spin currents transfer angular momentum into the Pd- layer . Due to \nstrong magnetocrystalline anisotropy, the field -dragging effect makes the line width \nversus frequency deviate from the linear relationship except magnetic field along hard \nor easy ax es. Extrinsic relaxation , such as TMS and mosaicity, relies heavil y on \nmagnetization orientation. Howeve r, an epitaxial interface could significantly \ndecrease and minimize the extrinsic contributions, especially TMS and mosaicity. It is \nnoteworthy that an isotropic non- local Gilb ert damping factor is clarified after ruling \nout the misleading field-dragging effect and the deceitful extrinsic contributions. \nMagnetization orientation has a negligible contribution to the non- local Gilbert \ndamping based on both theoretical and experimental results , manifesting that the \nabsorption of pure spin currents across interface Pd(100)[110]/Fe(001)[100] is \nindependent of Fe magnetization orientation. Our works provide deeper i nsight into \nthe non- local Gilbert damping mechanism. \n ACKNOWLEDGMENTS \nThis work is supported by the National Key Research Program of China (Grant Nos. \n2015CB921403, 2016YFA0300701, and 2017YFB0702702), the National Natural \nSciences Foundation of China (Grant Nos. 51427801,1187411 ,51671212, and \n11504413) and the Key Research Program of Frontier Sciences, CAS (Grant Nos. \nQYZDJ -SSW -JSC023, KJZD -SW-M01 and ZDYZ2012- 2). The work at Beijing \nNormal University is partly supported by the National Natural Sciences Foundation of \nChina (Grant Nos. 61774017, 61704018, and 11734004), the R ecruitment Program of \nGlobal Youth Experts and the Fundamental Research Funds for the Central Universities (Grant No. 2018EYT03). REFERENCES \n1. I. Zutic, J. Fabian and S. Das Sarma, Rev. Mod. Phys. 76 (2), 323 -410 (2004). \n2. K. Ando, S . Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa and E. Saitoh, Phys. Rev. Lett. 101 \n(3), 036601 (2008). \n3. J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back and T. Jungwirth, Rev. Mod. Phys. 87 (4), \n1213- 1260 (2015). \n4. V. Kambersky, Czech. J. P hys. 26 (12), 1366- 1383 (1976). \n5. A. B. Cahaya, A. O. Leon and G. E. W. Bauer, Phys. Rev. B 96 (14) (2017). \n6. D. Thonig, Y . Kvashnin, O. Eriksson and M. Pereiro, Phys. Rev. Materials 2 (1) (2018). \n7. D. Steiauf and M. Fähnle, Phys. Rev. B 72 (6) (2005). \n8. J. Seib, D. Steiauf and M. Fähnle, Phys. Rev. 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Shaw, H. T. Nembach and T. J. Silva, Phys. Rev. B 85 (5) (2012). \n43. Y . Liu, A. A. Starikov, Z. Yuan and P . J. Kelly, Phys. Rev. B 84 (1) (2011). \n44. Y . Liu, Z. Yuan, R. J. Wesselink, A. A. Stari kov and P . J. Kelly, Phys. Rev. Lett. 113 (20), 207202 \n(2014). \n45. A. A. Starikov, Y . Liu, Z. Yuan and P . J. Kelly, Phys. Rev. B 97 (21) (2018). \n46. B. Zimmermann, P . Mavropoulos, S. Heers, N. H. Long, S. Blugel and Y . Mokrousov, Phys. Rev. Lett. \n109 (23), 236603 (2012). \n FIGURE CAPTIONS \nFig. 1 ( Color online) (a) Dark field scanning high -resolution transmission electron \nmicroscopy image and ( b) selected area electron diffraction pattern of \nPd/Fe/MgO(001). The inset of Fig. 1(b) shows a schematic of the ep itaxial \nrelationship . \nFig. 2 ( Color online) (a) A schematic illustration of the stacked sample \nPd/Fe/MgO(001). The sample is placed on the CPW for FMR measurement, and \ncould be rotated in plane . (b) Typical real FMR spectra of Pd/Fe at fixed frequency \n13.4 GHz at various magnetic field angle sHϕ. Magnetic field angle Hϕ dependen ce \nof the resonanc e field H r at a fixed frequency 13.4 GHz for Pd/Fe (c) and Cu /Fe (d) . \nThe red curves are fit to Kittel’s formula (2). (In order to show clearly the tendency , \nwe show the data at 45 225Hϕ−≤≤aa, the same below ) \nFig. 3 ( Color online) The measured linewidth H∆ as a function of Hϕat 13.4 GHz \nfor Pd /Fe (a) and Cu/Fe (b). The line width H∆ is superimposed by several terms, \nsuch as TMS, mosaicity and Gilbert contribution with field- dragging. \nFig. 4 (Color online) Field -dragging effect for Pd/Fe. (a) The green line denotes the \nequilibrium direction of magnetization as a function of magnetic field angleHϕ at \n13.4 GHz. T he red line indicates the misalignment between the magnetization and the \napplied magnetic field according ly. (b) The equilibrium direction of the magnetization \nin the frequency -swept mode at variousHϕ. \nFig. 5 ( Color online) Frequency dependence of the resonance field Hr (a-e) and \nfrequency dependence of the resonance line width H∆ (f-j) for Pd/Fe at variousHϕ. The blue solid squares and curves in (f) and (j) corresponding to frequency \ndependence of H∆ at 0Hϕ=a and 45Hϕ=a for Cu/Pd. \nFig. 6 (Color online) Angular dependent Gilbert damping and first principles \ncalculation. ( a) The opened and solid green squares represent the obtained Gilbert \ndamping for Pd/Fe and Cu/Fe films, respectively. The red and blue lines are guide to \nthe eyes. ( b) The experimental and calculated spin mixing conductance as a function \nof the orientation of the equilibrium magnetization. \nTable I The fitted magnetic anisotropy parameters and magnetic relaxation \nparameters in Pd /Fe and Cu/Fe films . \n \n Fig.1 \n \n \n \nFig. 2 \n \n \nFig. 3 \n \n \nFig. 4 \n \n \nFig. 5 \n \nFig. 6 \n \n \n \nTable I The fitted m agnetic anisotropy parameters and magnetic relaxation \nparameters in Pd /Fe and Cu/Fe films in Fig. 3. \nSample 4H(Oe) 2H(Oe) 0 dHµ (T) effa 100γ<>Γ (710Hz) ϕ∆(deg.) \nPd/Fe 625 0 2.0 0.0060 0 0.23 \nCu/Fe 625 0 1.9 0.0042 58 1.26 \n \n " }, { "title": "2403.01625v1.Magnonic___varphi__Josephson_Junctions_and_Synchronized_Precession.pdf", "content": "arXiv:2403.01625v1 [cond-mat.mes-hall] 3 Mar 2024Magnonic ϕJosephson Junctions and Synchronized Precession\nKouki Nakata,1Ji Zou,2Jelena Klinovaja,2and Daniel Loss2,3\n1Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai, Ibaraki 319-1195, Japan\n2Department of Physics, University of Basel, Klingelbergst rasse 82, 4056 Basel, Switzerland\n3Center for Emergent Matter Science, RIKEN, 2-1 Hirosawa, Wa ko-shi, Saitama 351-0198, Japan\n(Dated: March 5, 2024)\nThere has been a growing interest in non-Hermitian physics. One of its main goals is to engineer\ndissipation and to explore ensuing functionality. In magno nics, the effect of dissipation due to\nlocal damping on magnon transport has been explored. Howeve r, the effects of non-local damping\non the magnonic analog of the Josephson effect remain missing , despite that non-local damping is\ninevitable and has been playing a central role in magnonics. Here, we uncover theoretically that\na surprisingly rich dynamics can emerge in magnetic junctio ns due to intrinsic non-local damping,\nusing analytical and numerical methods. In particular, und er microwave pumping, we show that\ncoherent spin precession in the right and left insulating fe rromagnet (FM) of the junction becomes\nsynchronized by non-local damping and thereby a magnonic an alog of the ϕJosephson junction\nemerges, where ϕstands here for the relative precession phase of right and le ft FM in the stationary\nlimit. Remarkably, ϕdecreases monotonically from πtoπ/2as the magnon-magnon interaction,\narising from spin anisotropies, increases. Moreover, we al so find a magnonic diode effect giving rise\nto rectification of magnon currents. Our predictions are rea dily testable with current device and\nmeasurement technologies at room temperatures.\nI. INTRODUCTION\nRecently, non-Hermitian physics [1] has been attract-\ning growing interest from both fundamental science and\napplications such as energy-efficient devices. One of its\nmain themes is to engineer dissipation and to explore the\nresulting functionality. Since magnons are intrinsically\ndamped in magnetic systems, the goal of non-Hermitian\nphysics aligns well with magnonics [2–4], which aims\nat efficient transmission and processing of information\nfor computing and communication technologies using\nmagnons as its carrier in units of the Bohr magneton µB.\nTo this end, establishing methods for the control and ma-\nnipulation of magnon transport subjected to dissipation\nis crucial. In magnonics, the effect of dissipation due\nto local (Gilbert) damping on magnon transport, such as\nthe magnonic analog of the Josephson effect [5], has been\nexplored [6–8], where the macroscopic coherent magnon\nstate, the key ingredient for the magnonic Josephson ef-\nfect, realizes an oscillating behavior of magnon trans-\nport. However, the effect of non-local damping on the\nmagnonic Josephson effect and the resulting function-\nality remain missing, despite that non-local damping is\ninevitable and has been playing a central role in non-\nHermitian magnonics [9–11].\nHere, we fill this gap. We find that the inherent non-\nlocal damping leads to rich dynamics, and, interestingly,\ncan be utilized to realize a magnonic analog of the ϕ\nJosephson junction [12–31]. Under microwave pumping,\ncoherent spin precession in each insulating ferromagnet\n(FM) of the magnetic junction (see Fig. 1) is synchro-\nnized by non-local damping as time advances and gives\nrise of a magnonic ϕJosephson junction, where ϕstands\nfor the stationary value of the relative precession phase\nof the left and right FMs. Interestingly, we find that ϕ\ndecreases monotonically from πtoπ/2as the magnon-magnon interaction, arising from spin anisotropies across\nthe junction, increases. Applying microwaves to each\nFM continuously, the junction reaches the nonequilib-\nrium steady state where the loss of magnons due to dissi-\npation is precisely balanced by the injection of magnons.\nI (t)L I (t)Rθ (t)L θ (t)R\n θθ θθ\n((\ntt\n))LL \nLeft FM Right FM\nƤ\nγ γƤMicrowave Microwave\n((\n(\n(\ntt\n)\n)R\nƤƤ\nƤ\nMicrowave\nI I\nLI I \neft FM\nt\n)\nRight FM\n))\n(< 0) (> 0)\nII\nθθ\n ())L\nL\nNonmagnetic\nspacer layerV(/g2324,/g2324 , G)*\nFIG. 1. Magnonic ϕJosephson junction, formed by a ferro-\nmagnetic bilayer coupled by a non-magnetic spacer. Magnons\nare subjected to local damping at rate γ >0. The spacer\nlayer-mediated interaction Vbetween the two FMs consists\nof coherent coupling ( J) and non-local damping ( G). Under\nmicrowave pumping, the spatially uniform mode of magnons\nis injected into each FM at rate P, generating coherent spin\nprecessions. The loss of magnons due to dissipation is ex-\nactly balanced by the injection of magnons. In the steady\nstate, the spin precessions in each FM get synchronized due\nto non-local damping and a relative precession angle emerge s\nϕ= lim t→∞[θR(t)−θL(t)]that depends on the magnon-\nmagnon interaction. The associated magnon currents IL,R\nget rectified when G/negationslash= 0.2\nHence, spins in each FM continue to precess coherently,\nand the synchronized precession of the left and the right\nmagnetization remains stable even at room temperature.\nFinally, we show that the magnetic junction exhibits rec-\ntification and acts as a diode for the magnon current.\nII. MAGNONIC JOSEPHSON JUNCTIONS\nWe consider a magnetic junction, shown in Fig. 1, con-\nsisting of a bilayer of two insulating ferromagnetic lay-\ners, where the two FMs are separated by a nonmagnetic\nspacer layer, thereby weakly spin-exchange coupled. Ap-\nplying microwaves to each FM and tuning the microwave\nfrequency Ω>0of GHz to the magnon energy gap, ferro-\nmagnetic resonance is generated for each FM separately,\nwhere spins precess coherently. Under this microwave\npumping, the zero wavenumber mode (i.e., spatially uni-\nform mode) of magnons is excited and injected into each\nFM at the rate of P>0[32]. The magnons are subjected\nto local Gilbert damping at the rate γ >0in each FM.\nIn addition, there is non-local damping [9, 10] that\nis mediated by the spacer interaction between the two\nFMs, see Fig. 1. Using the Holstein-Primakoff transfor-\nmation [33], this coupling term becomes to leading or-\nder [11]V= (J−i/planckover2pi1G/2)a†\nLaR+(J∗−i/planckover2pi1G/2)aLa†\nR, where\na(†)\nL(R)represents the magnon annihilation (creation) op-\nerator for the zero wavenumber mode, which satisfies\nthe bosonic commutation relation. Here, J ∈C, with\nits complex conjugate J∗, describes coherent coupling,\nwhile/planckover2pi1G∈Rdescribes the non-local damping (or dissi-\npative coupling) [11]. The real part of coherent coupling,\nRe(J) =JRe, arises from the symmetric spin-exchange\ninteraction between the two FMs, while its imaginarypart [5, 32, 34, 35], Im (J) =JIm, is induced, for exam-\nple, by the Dzyaloshinskii-Moriya interaction when the\nspatial inversion symmetry of the nonmagnetic spacer\nlayer is broken [36]. Each component of J//planckover2pi1can reach\nMHz in experiments [37, 38]. The values of γandG\nare typically within the MHz regime [39], and the con-\ndition,/planckover2pi1Ω≫ |J|,/planckover2pi1γ,|/planckover2pi1G|, is satisfied. The condition\n|G| ≤2γensures the complete positivity of the system\ndynamics [11, 40, 41], and we focus on this regime hence-\nforth. We note that the coupling term Vbecomes non-\nHermitian due to non-local damping, i.e., V∝negationslash=V†for\nG∝negationslash= 0.\nThe magnon-magnon interaction in the left (right) FM,\nUL(R), arises from anisotropies of spin, where UL=\n(U/2)a†\nLa†\nLaLaL,UR= (U/2)a†\nRa†\nRaRaR. Here, Ucan\ntake both positive and negative values, depending on the\ncombination of anisotropies such as the spin anisotropy\nalong the quantization axis and the anisotropy of the\nspin-exchange interaction in each FM [5, 32].\nIn this study, we envisage to continuously apply mi-\ncrowaves to each FM, which results in spins exhibit-\ning macroscopic coherent precession characterized as\n∝angbracketleftaL(R)(t)∝angbracketright ∝negationslash= 0 [32]. Therefore, assuming a macro-\nscopic coherent magnon state, thereby using the semi-\nclassical approximation, we replace the operators aL(t)\nandaR(t)by their expectation values as ∝angbracketleftaL(t)∝angbracketright=/radicalbig\nNL(t)eiθL(t)and∝angbracketleftaR(t)∝angbracketright=/radicalbig\nNR(t)eiθR(t), respectively,\nwhereNL(R)(t)>0represents the number of coherent\nmagnons for each site in the left (right) FM and θL(R)is\nthe phase (see Fig. 1). Defining the relative precession\nphase as θ(t) =θR(t)−θL(t), each time evolution [e.g.,\n˙θ(t)represents the time derivative of θ(t)] is described as\n(see the Supplemental Material (SM) for details [42])\n˙θ(t) =JRe\n/planckover2pi1cosθ(t)/parenleftBigg/radicalBigg\nNR(t)\nNL(t)−/radicalBigg\nNL(t)\nNR(t)/parenrightBigg\n+sinθ(t)/bracketleftBigg/parenleftbiggJIm\n/planckover2pi1+G\n2/parenrightbigg/radicalBigg\nNL(t)\nNR(t)−/parenleftbiggJIm\n/planckover2pi1−G\n2/parenrightbigg/radicalBigg\nNR(t)\nNL(t)/bracketrightBigg\n+u[NL(t)−NR(t)],\n(1a)\n˙NL/R(t) =−2γNL/R(t)±2/bracketleftbiggJRe\n/planckover2pi1sinθ(t)+/parenleftbiggJIm\n/planckover2pi1∓G\n2/parenrightbigg\ncosθ(t)/bracketrightbigg/radicalbig\nNL(t)NR(t)+P, (1b)\nwhereu=U//planckover2pi1. The second term on the right-hand\nside of Eq. (1b) describes the nonmagnetic spacer layer-\nmediated transport of coherent magnons in the junction.\nThe transport of coherent magnons in this junction\nis analogous to the Josephson effect [43] in the sense\nthat the current arises as the term of the order of V,\nO(V), for the interaction V=V(J,J∗,G)between the\ntwo FMs (see Fig. 1) and is characterized by the rel-\native precession phase θ(t). For these reasons [36], we\nrefer to the transport of coherent magnons and the junc-\ntion as the magnonic Josephson effect and the magnonicJosephson junction, respectively. Note that, in contrast,\nthe current for incoherent magnons arises from a term of\nO(V2)[44, 45].\nIII. SYNCHRONIZED PRECESSION\nTo seek for a junction setup that exhibits synchroniza-\ntion and rectification effects, we consider the case where\nJRe= 0andJIm//planckover2pi1=G/2>0. (2)3\nπ\n-π θ(t) \n2\n1\n-2 -1 0\n-3 3\n3 9 5 7t [μs]θ(0) > 0 \nθ(0) < 0 θ(0) = 0 \n1\nFIG. 2. Plots of the relative phase θ(t)as a function of time\nin the absence of the magnon-magnon interaction, i.e., u= 0,\nfor the initial condition θ(0) = 0 and|θ(0)|=π/2nwith\nn= 0,1,...,7obtained by numerically solving Eqs. (3a)-(3c)\nfor the parameter values JRe= 0,JIm//planckover2pi1=G/2 = 0.5MHz,\nγ= 1.1MHz,P= 2γ= 2.2MHz, and NL(0) =NR(0) =\n10−6. As time advances, coherent spin precession in each FM\nis synchronized and forms the magnonic πJosephson junction\nasϕ=θ(t→ ∞) =±πforθ(0)/negationslash= 0. The initial condition\nθ(0) = 0 foru= 0results in ϕ= 0.\nUnder these assumptions, Eqs. (1a)-(1b) become\n˙θ(t) =G/radicalbig\nNL/NRsinθ+u[NL−NR], (3a)\n˙NL=−2γNL+P, (3b)\n˙NR=−2γNR−2G/radicalbig\nNLNRcosθ+P,(3c)\nwhere we suppressed for brevity the explicit time-\ndependence of the quantities θ(t)andNL/R(t). Under\nmicrowave pumping P, the nonequilibrium steady state\n˙θ(t) =˙NL(t) =˙NR(t) = 0 is realized, where θ(t),NL(t),\nandNR(t)approach their stationary values as time ad-\nvances, i.e., ϕ=θ(t→ ∞)andNL(R)(t→ ∞) =N∞\nL(R).\nUsing Eqs. (3a)-(3c), we find the following relations be-\ntween these asymptotic quantities:\ncosϕ= (γ/G)(N∞\nL−N∞\nR)//radicalbig\nN∞\nRN∞\nL,(4a)\ntanϕ=−uN∞\nR/γ, (4b)\nwhereN∞\nL=P/2γis a constant, while N∞\nRbecomes\na function of the relative precession phase ϕ. In what\nfollows, we will show that there is a unique solution to\nthe system of Eqs. (4a) and (4b). Thereby, a magnonic\nanalog of the ϕJosephson junction [12–31] is realized.\nA. Absence of magnon interaction: u= 0\nFirst, we study the behavior of ϕin the absence of the\nmagnon-magnon interaction, u= 0. From Eq. (4b), weimmediately find that ϕcan only be equal to ±πor0.\nOne of these three values is chosen based on the initial\ncondition θ(0). Figure 2 shows the plots of the relative\nphaseθ(t)for several initial conditions θ(0)as a func-\ntion of time obtained by numerically solving Eqs. (3a)-\n(3c). If θ(0) = 0 , the relative phase stays constant,\nθ(t) = 0 . If the symmetry is broken, i.e., θ(0)∝negationslash= 0,\nwe have ϕ=πsgnθ(0). This shows that the coherent\nspin precessions in each FM get synchronized with each\nother as time advances. This asymptotic locking of the\nspin precessions is a direct consequence of the dissipative\ncoupling term G. The junction behavior represents a\nmagnonic analog of the well-known πJosephson junction\neffect in superconductors [12–17].\nFigure 2 also shows that the point θ(0) = 0 is unstable\nin the sense that ϕ= 0for the initial condition θ(0) = 0 ,\nwhereas ϕ=±πforθ(0)∝negationslash= 0. However, to realize such\na special condition, θ(0) = 0 withu= 0, will be out of\nexperimental reach. Moreover, in what follows, we will\nshow that any finite value of uresults in ϕ∝negationslash= 0, no matter\nwhat initial value we choose for θ(0)including the fine-\ntuned value θ(0) = 0 , see Fig. 3.\nWe emphasize that our results are independent of the\ninitial values NL(0)andNR(0)and do not depend on the\nassumption that both FMs are pumped at the same rate\nP. A detailed discussion is available in the SM [42]. We\nalso remark that the synchronized precession of the left\nand the right magnetization, ˙θ(t) = 0, remains valid even\nif the parameter values slightly deviate from Eq. (2). See\nthe SM [42] for the plots, where it is shown numerically\nthat, although the value of ϕslightly changes depend-\ning on the magnitude of the deviation, the synchronized\nprecession is robust against such perturbations. We note\nthat under the initial condition θ(0) = 0 , we have ϕ∝negationslash= 0\nforJRe∝negationslash= 0, whereas ϕ= 0forJRe= 0. Also in this\nsense, the point θ(0) = 0 is unstable. See the SM [42] for\nmore details.\nB. Finite magnon interaction: u/negationslash= 0\nNext, we study the behavior of ϕin the presence of the\nmagnon-magnon interaction, u∝negationslash= 0, and determine ϕas a\nfunction of u. First, from Eq. (4b), we deduce that ϕ(u)\nis an odd function of u, i.e.,ϕ(u) =−ϕ(−u)(mod2π).\nHere, we used that N∞\nR(ϕ) =N∞\nR(−ϕ), which follows\nfrom Eq. (4a). In Fig. 3, we solve numerically Eqs. (3a)-\n(3c) and confirm that ϕ(u)is odd. Remarkably, even if\nthe initial condition is chosen as θ(0) = 0 , foru∝negationslash= 0, the\nmagnitude of ϕmonotonically decreases from |ϕ|=π\nto|ϕ|=π/2as the magnitude of the magnon-magnon\ninteraction increases. We note that this condition is con-\nsistent with the requirement that the direction of magnon\npropagation in the junction, see Fig. 1, is chosen to be\nfrom left to right (see also below).\nIntroducing the rescaled magnon-magnon interaction\n˜u= (u/γ)N∞\nLand combining Eqs. (3a)-(3c), we get the4\n20 π\n2\n1\n03\n10 π/2 \n30 |u|/ γ|φ|\nθ(0) = 0 θ(0) ≠ 0 \n0(c) (b)\n|φ|\n0.5 3 4 5 2 11.92.02.12.22.32.42.52.6\n0.51.01.52.0\n|u|/ γG/γ\n1.8\n0.0120 π\n-π θ(t) \nt [μs]2\n1\n-2 -1 0\n-3 3\n10 π/2 \n-π /2 15 5(a)\nu= 10 ~ -6 \nu= 10 ~ -2 \nu= 0.5~\nu= 1~\nu= 3~\nu=-10 ~ -6 u=-10 ~ -2 u=-0.5~u=-1~u=-3~u > 0\nu < 0\nFIG. 3. (a) Plots of the relative phase θ(t)as a function of time for several values of ˜uobtained by numerically solving Eqs. (3a)-\n(3c) for the same parameter values as in Fig. 2 as well as for θ(0) = 0 . The stationary relative phase, ϕ=θ(t→ ∞), decreases\nfrom|ϕ|=πto|ϕ|=π/2as the magnon-magnon interaction |˜u|increases, where we set |˜u|= 10−6,10−2,0.5,1,3,5,10. We\nnote that sgn (ϕ) =sgn(u). (b,c) The relative phase |ϕ|decreases as |u|/γandG/γincrease. These plots are obtained by\nnumerically solving Eq. (5) for the same parameter values as in Fig. 3(a). (c) Plot of |ϕ|as a function of |u|/γ, showing that |ϕ|\ndecreases monotonically from πtoπ/2as|u|/γincreases. For u/negationslash= 0, the value of |ϕ|does not depend on the initial conditions.\nForu= 0,|ϕ|depends on the initial condition θ(0)as|ϕ|=πforθ(0)/negationslash= 0, whereas ϕ= 0forθ(0) = 0 (see Fig. 2).\nfollowing implicit equation on tanϕ:\ntan4ϕ+2˜utan3ϕ+(1+ ˜u2)tan2ϕ\n+/bracketleftBig\n2+(G/γ)2/bracketrightBig\n˜utanϕ+ ˜u2= 0,(5)\nwhose solution gives us ϕas function of ˜u. Forϕ∈\n[−π,π], there are generally four solutions for ϕ. The two\nsolutions with cosϕ >0are excluded as they will result\nin an unphysical direction of the current IRthat is set\nby the nonlocal interaction term V(see below). Finally,\none more solution is eliminated as it breaks the condition\ntanϕ <−˜u, which follows from Eqs. (4a) and (4b) for\n˜u >0. That leaves us with a unique solution for ϕ.\nIn Fig. 3(b), we investigate |ϕ|as a function of both\n|u|/γandG/γby numerically solving Eq. (5). For a given\n˜u(G),|ϕ|decreases monotonically from πtoπ/2asG\n(˜u) increases, see Fig. 3(b,c). The asymptotic expressions\nforϕfor the two limiting cases of weak and strong inter-\nactions can be obtained also analytically. Indeed, from\nEqs. (5) and (4a) and for strong interactions, ˜u≫1, we\nget in leading order in 1/˜u\nϕ≃π/2+1/˜u, (6a)\nN∞\nR/N∞\nL≃1+G/˜uγ. (6b)\nIn the opposite limit of weak interactions, 0<˜u≪1, we\nobtain, keeping leading corrections in ˜uin each quantity,\nϕ≃π−c˜u, (7a)\nN∞\nR/N∞\nL≃c/parenleftBig\n1−G(c˜u)2//radicalbig\nG2+4γ2/parenrightBig\n(7b)wherec= (/radicalBig\n1+(G/2γ)2+G/2γ)2. These equations\ntogether with Fig. 3(c) are one of the main results of this\nstudy. Remarkably, both limiting values for ϕ, namely\nπ/2andπ, are obviously universal, i.e., independent of\nany material properties as well as of initial conditions.\nThis underlines the robust and universal property of\nthe synchronization of precessions brought about by the\nnonlocal damping in magnetic junctions driven by mi-\ncrowaves.\nIV. RECTIFICATION\nThe magnonic ϕJosephson junction can be regarded\nas a magnonic analog of the Josephson diode [46–48],\nin the sense that it exhibits a rectification effect for the\nmagnon currents, as we will show next. From Eq. (1b)\none gets that the current of coherent magnons that flows\nfrom the nonmagnetic spacer layer into the left (right)\nFM (see Fig. 1), IL(R)(t) =O(V), is given by\nIL/R=±2/bracketleftBigJRe\n/planckover2pi1sinθ+/parenleftBigJIm\n/planckover2pi1∓G\n2/parenrightBig\ncosθ/bracketrightBig/radicalbig\nNLNR,\n(8)\nin units of gµBfor theg-factorgof the constituent spins,\nand where we suppressed for brevity the explicit time-\ndependence of the quantities θ(t)andNL/R(t). The con-5\ndition specified in Eq. (2) results in\nIL(t) =0, (9a)\nIR(t) =−2G/radicalbig\nNL(t)NR(t)cosθ. (9b)\nSince the nonequilibrium steady state is realized under\nmicrowave pumping P, the current IR(t)continues to\nflow while keeping the rectification effect characterized\nbyIL(t) = 0 in the magnonic ϕJosephson junction\n(see Fig. 1). We emphasize that the rectification of the\nmagnon current holds also in the presence of the magnon-\nmagnon interaction.\nNote that our analytical solutions for ϕmust satisfy\ncosϕ <0and thus the magnitude of ϕis bounded as\nπ/2<|ϕ| ≤πto ensure limt→∞IR(t)≥0. We re-\ncall that the non-local damping Gprovides the non-\nHermitian property to the nonmagnetic spacer layer-\nmediated interaction described by V. Under the special\nconditions stated in Eq. (2) one gets V=−i/planckover2pi1GaLa†\nR.\nThis shows that propagation of magnons becomes chiral\nand is allowed only in the direction from left to right in\nthe junction. This is consistent with the sign choice of\nthe current and its direction as shown in Fig. 1.\nIn the absence of non-local damping, G= 0, the\ncurrent of coherent magnons propagates in both direc-\ntions through the nonmagnetic spacer layer, from the\nleft to the right FM and vice versa [5]. This results in\nIR(t) =−IL(t)forG= 0[see Eqs. (8)]. Thus, we see\nthat rectification only occurs in the presence of non-local\ndamping when G∝negationslash= 0.\nIf instead of Eq. (2), one uses the condition\nJRe= 0andJIm//planckover2pi1=−G/2, (10)\nwe get from Eq. (8) that\nIL(t) =−2G/radicalbig\nNL(t)NR(t)cosθ(t), (11a)\nIR(t) =0. (11b)\nThus, the rectification effect changes its direction. We\nconclude that the polarity of the rectification effect is\ndetermined by the sign of G.\nV. EXPERIMENTAL FEASIBILITY\nThe key ingredient for the magnonic Josephson effect\nis the coherent magnon state ∝angbracketleftaL(R)(t)∝angbracketright ∝negationslash= 0(i.e., co-\nherent spin precession), and it can be realized through\nmicrowave pumping [32]. Since each component of co-\nherent coupling Jis tunable by adjusting the thickness\nof the nonmagnetic spacer layer [49–52] or applying an\nelectric field [53–56], the magnonic ϕJosephson junction\nis realizable by tuning coherent coupling appropriately.\nMoreover, applying microwave to each FM continuously,\nthe loss of magnons due to dissipation is precisely bal-\nanced by the injection of magnons achieved through mi-\ncrowave pumping. Therefore, spins in each FM continueto precess coherently, and the synchronized precession\nof the magnonic ϕJosephson junctions remains stable.\nThus, our theoretical prediction is within experimental\nreach with current device and measurement techniques\nthrough magnetization measurement.\nVI. CONCLUSION\nWe have investigated the effect of non-local damp-\ning on magnetic junctions and found that it serves as\nthe key ingredient for the synchronized precession and\ngives rise to a magnonic ϕJosephson junction. The\nspacer layer-mediated interaction between the two FMs\nin the junctions consists of coherent coupling and non-\nlocal damping, and it becomes non-Hermitian due to non-\nlocal damping. Tuning them appropriately, coherent spin\nprecession in each FM is synchronized by non-local damp-\ning as time advances and forms a ϕJosephson junction,\nwhere the relative precession angle ϕdecreases mono-\ntonically from |ϕ|=πto|ϕ|=π/2as the magnitude\nof the magnon-magnon interaction increases, with both\nlimiting values being entirely universal. The magnon cur-\nrents in the junction exhibits rectification and gives rise\nto a magnonic diode effect. Applying microwaves to each\nFM continuously, the junction reaches the nonequilib-\nrium steady state where the loss of magnons due to dissi-\npation is precisely balanced by the injection of magnons\nachieved through microwave pumping. Hence, spins in\neach FM continue to precess coherently, and the synchro-\nnized precession of the left and the right magnetization\nremains stable.\nACKNOWLEDGMENTS\nThis work was supported by the Georg H. Endress\nFoundation and by the Swiss National Science Founda-\ntion, and NCCR SPIN (grant number 51NF40-180604).\nK.N. acknowledges support by JSPS KAKENHI Grants\nNo. JP22K03519.6\nSupplemental Material for “Magnonic ϕJosephson Junctions and Synchronized Precession”\nIn this Supplemental Material, we provide details on the der ivation of the magnonic Josephson equations, on the\nequation for ϕas a function of the magnon-magnon interaction, and on the pa rameter dependence of our results.\nAppendix S-I: Magnonic Josephson equations\nIn this section, we provide details on the derivation of the m agnonic Josephson equations. The effective non-\nHermitian Hamiltonian Hfor the magnonic Josephson junction is given as [5, 11, 32] H=HL+HR+V+UL+UR\nwith (see also main text) HL=/planckover2pi1(Ω−iγ)a†\nLaLandHR=/planckover2pi1(Ω−iγ)a†\nRaR, where a(†)\nL(R)represents the magnon\nannihilation (creation) operator for the zero wavenumber m ode (i.e., spatially uniform mode) in the left (right) FM.\nThis provides the time evolution of each operator as\ni/planckover2pi1˙aL(t) =/planckover2pi1(Ω−iγ)aL(t)+(J −i/planckover2pi1G/2)aR(t)+Ua†\nL(t)aL(t)aL(t), (S1a)\ni/planckover2pi1˙aR(t) =/planckover2pi1(Ω−iγ)aR(t)+(J∗−i/planckover2pi1G/2)aL(t)+Ua†\nR(t)aR(t)aR(t). (S1b)\nHere, assuming a macroscopic coherent magnon state, thereb y using the semiclassical approximation, we replace\nthe operators aL(t)andaR(t)by their expectation values as ∝angbracketleftaL(t)∝angbracketright=/radicalbig\nNL(t)eiθL(t)and∝angbracketleftaR(t)∝angbracketright=/radicalbig\nNR(t)eiθR(t),\nrespectively, where NL(R)(t)∈Rrepresents the number of coherent magnons for each site in th e left (right) FM and\nθL(R)(t)∈Ris the phase. Defining the relative phase as\nθ(t) =θR(t)−θL(t), (S2)\nEqs. (S1a) and (S1b) for NL(R)(t)∝negationslash= 0become\ni/planckover2pi1/parenleftBig1\n2˙NL\nNL+i˙θL/parenrightBig\n=/planckover2pi1(Ω−iγ)+UNL+/bracketleftBig\nJRe+i/parenleftBig\nJIm−/planckover2pi1\n2G/parenrightBig/bracketrightBig/radicalbigg\nNR\nNLeiθ, (S3a)\ni/planckover2pi1/parenleftBig1\n2˙NR\nNR+i˙θR/parenrightBig\n=/planckover2pi1(Ω−iγ)+UNR+/bracketleftBig\nJRe−i/parenleftBig\nJIm+/planckover2pi1\n2G/parenrightBig/bracketrightBig/radicalbigg\nNL\nNRe−iθ, (S3b)\nwhere Re (J) =JRe∈Rand Im(J) =JIm∈R. Real and imaginary parts of Eqs. (S3a) and (S3b) provide\n−/planckover2pi1˙θL=/planckover2pi1Ω+UNL+/bracketleftBig\nJRecosθ−/parenleftBig\nJIm−/planckover2pi1\n2G/parenrightBig\nsinθ/bracketrightBig/radicalbigg\nNR\nNL, (S4a)\n/planckover2pi1\n2˙NL\nNL=−/planckover2pi1γ+/bracketleftBig\nJResinθ+/parenleftBig\nJIm−/planckover2pi1\n2G/parenrightBig\ncosθ/bracketrightBig/radicalbigg\nNR\nNL, (S4b)\n−/planckover2pi1˙θR=/planckover2pi1Ω+UNR+/bracketleftBig\nJRecosθ−/parenleftBig\nJIm+/planckover2pi1\n2G/parenrightBig\nsinθ/bracketrightBig/radicalbigg\nNL\nNR, (S4c)\n/planckover2pi1\n2˙NR\nNR=−/planckover2pi1γ−/bracketleftBig\nJResinθ+/parenleftBig\nJIm+/planckover2pi1\n2G/parenrightBig\ncosθ/bracketrightBig/radicalbigg\nNL\nNR, (S4d)\nwhere Eq. (S4a) is the real part of Eq. (S3a), Eq. (S4b) is the i maginary part of Eq. (S3a), Eq. (S4c) is the real part\nof Eq. (S3b), and Eq. (S4d) is the imaginary part of Eq. (S3b). Taking the difference between Eqs. (S4a) and (S4c),\nthose are rewritten as\n˙θ(t) =JRe\n/planckover2pi1cosθ/parenleftBig/radicalbigg\nNR\nNL−/radicalbigg\nNL\nNR/parenrightBig\n+sinθ/bracketleftBig/parenleftBigJIm\n/planckover2pi1+G\n2/parenrightBig/radicalbigg\nNL\nNR−/parenleftBigJIm\n/planckover2pi1−G\n2/parenrightBig/radicalbigg\nNR\nNL/bracketrightBig\n+u(NL−NR), (S5a)\n˙NL(t) =−2γNL+2/bracketleftBigJRe\n/planckover2pi1sinθ+/parenleftBigJIm\n/planckover2pi1−G\n2/parenrightBig\ncosθ/bracketrightBig/radicalbig\nNLNR, (S5b)\n˙NR(t) =−2γNR−2/bracketleftBigJRe\n/planckover2pi1sinθ+/parenleftBigJIm\n/planckover2pi1+G\n2/parenrightBig\ncosθ/bracketrightBig/radicalbig\nNLNR. (S5c)\nIn this study, we assume to continuously apply microwaves to each FM. The coherent magnon state, the key ingredient\nfor the magnonic Josephson effect, is realized by microwave p umping. Under microwave pumping, coherent magnons\nare injected into each FM at the rate of P[32]. Taking this effect of the magnon injection through micr owave pumping7\ninto account, Eqs. (S5b) and (S5c) become\n˙NL(t) =−2γNL+2/bracketleftBigJRe\n/planckover2pi1sinθ+/parenleftBigJIm\n/planckover2pi1−G\n2/parenrightBig\ncosθ/bracketrightBig/radicalbig\nNLNR+P, (S6a)\n˙NR(t) =−2γNR−2/bracketleftBigJRe\n/planckover2pi1sinθ+/parenleftBigJIm\n/planckover2pi1+G\n2/parenrightBig\ncosθ/bracketrightBig/radicalbig\nNLNR+P. (S6b)\nFinally, the magnonic Josephson equations under microwave pumping in the presence of non-local damping are\nsummarized as\n˙θ(t) =JRe\n/planckover2pi1cosθ(t)/parenleftBig/radicalBigg\nNR(t)\nNL(t)−/radicalBigg\nNL(t)\nNR(t)/parenrightBig\n+sinθ(t)/bracketleftBig/parenleftBigJIm\n/planckover2pi1+G\n2/parenrightBig/radicalBigg\nNL(t)\nNR(t)−/parenleftBigJIm\n/planckover2pi1−G\n2/parenrightBig/radicalBigg\nNR(t)\nNL(t)/bracketrightBig\n+u[NL(t)−NR(t)],\n(S7a)\n˙NL(t) =−2γNL(t)+2/bracketleftBigJRe\n/planckover2pi1sinθ(t)+/parenleftBigJIm\n/planckover2pi1−G\n2/parenrightBig\ncosθ(t)/bracketrightBig/radicalbig\nNL(t)NR(t)+P, (S7b)\n˙NR(t) =−2γNR(t)−2/bracketleftBigJRe\n/planckover2pi1sinθ(t)+/parenleftBigJIm\n/planckover2pi1+G\n2/parenrightBig\ncosθ(t)/bracketrightBig/radicalbig\nNL(t)NR(t)+P. (S7c)\nAppendix S-II: Dynamics with different pumping rates\nIn this section, we consider the scenario where JRe= 0,JIm//planckover2pi1=G/2, andu= 0, with different pumping rates for\ntwo FMs. By solving the dynamics analytically, we demonstra te that our results do not rely on the initial conditions.\nLet us first only pump the left FM with rate P. In this case, the coupled dynamics is governed by the follow ing\nequations:\n˙θ(t) =G/radicalbigg\nNL\nNRsinθ,\n˙NL=−2γNL+P,\n˙NR=−2γNR−2G/radicalbig\nNLNRcosθ.(S1)\nWe now solve these equations analytically. We first introduc enL/R(t) =e2γtNL/R(t), which allows us to rewrite the\nequations into the following form:\n˙nL=e2γtP,˙θ=G/radicalbiggnL\nnRsinθ,˙nR=−2G√nRnLcosθ. (S2)\nThe first equation can be solved:\nnL(t) =NL(0)+P\n2γ(e2γt−1). (S3)\nThe last two equations leads to\ndθ\ndnR=−1\n2tanθ\nnR−→ |sinθ(t)|/radicalbig\nnR(t) =|sinθ(0)|/radicalbig\nNR(0). (S4)\nWe then obtain the equation for θ(t)under the initial condition θ(0)∝negationslash= 0andθ(0)∝negationslash=±π:\ndθ\ndt=G/radicalbig\nnL(t)\n|sinθ(0)|/radicalbig\nNR(0)sinθ|sinθ|. (S5)\nWe point out that the above equation can be solved analytical ly since the integral of/radicalbig\nnL(t)can be performed\nanalytically. But we will focus on the case where t≫1/(2γ)sonL(t)≈(P/2γ)e2γtfor our purpose.\nLet us also focus on the case of sinθ(0)>0and look for the solution that satisfies sinθ(t)>0. One can similarly8\nsolve for the case when sinθ(0)<0. We have:\ntanθ(t) =1\ncotθ(0)−αeγt−→θ(t) = arctan/bracketleftbigg1\ncotθ(0)−αeγt/bracketrightbigg\n, (S6)\nwhereαis a constant:\nα≡(G/γ)/radicalbig\nP/(2γ)\nsinθ(0)/radicalbig\nNR(0). (S7)\nIn the large t→ ∞ limit, we can drop cotθ(0)in the expression. Then the expression of θ(t)is reduced to:\nθ(t) =π−1\nαe−γt. (S8)\nOne can also easily write down the expression of nR(t):\n/radicalbig\nnR(t) =α|sinθ(0)|/radicalbig\nNR(0)e2γt. (S9)\nAt large t→ ∞, we have\n/radicalBigg\nN∞\nR\nN∞\nL=G\nγ. (S10)\nIt is interesting to note that this final value is independent of the initital condition and also the pumping rate P.\nFinally, when we pump the two FMs at the same time, the physics is very similar to the case that we discussed\nabove. The only difference is that N∞\nRis different. The equations are given by\n˙θ(t) =G/radicalbigg\nNL\nNRsinθ,\n˙NL=−2γNL+PL,\n˙NR=−2γNR−2G/radicalbig\nNLNRcosθ+PR.(S11)\nHerePLandPRare the pumping rates of the two FMs. We introduce the ratio\nβ≡/radicalBigg\nN∞\nR\nN∞\nL. (S12)\nThen it is determined by the equation\nβ2−G\nγβ−PR\nPL= 0−→β=G/γ+/radicalbig\n(G/γ)2+4PR/PL\n2, (S13)\natt→ ∞. Here, let us again focus on the case of sinθ(0)>0and look for the solution that satisfies sinθ(t)>0.\nSimilar to our previous conclusion, θ(t)also approaches πexponentially fast but now with a different exponent:\nπ−θ(t)∝e−(G/β)t. (S14)\nNote that in the absence of the pumping of the right FM PR= 0, we have β=G/γandπ−θ(t)∝e−γt, which is\nwhat we obtained before.\nAppendix S-III: Robustness of synchronized precession\nIn the main text, to seek for the junction that exhibits a rect ification effect characterized by IL(t) = 0, we consider\nthe case\nJRe= 0andJIm//planckover2pi1=G/2>0. (S1)9\nπ\n-π θ(t) \n2\n1\n-2 -1 0\n-3 3\n10 30 20 t [μs]θ(0) > 0 \nθ(0) < 0 θ(0) = 0 (a)\nπ\n-π θ(t) \n2\n1\n-2 -1 0\n-3 3\n10 30 20 t [μs]θ(0) > 0 \nθ(0) < 0 θ(0) = 0 (b)\nFIG. S1. Plots of the relative phase θ(t)as a function of time in the absence of the magnon-magnon inte raction, i.e., u= 0,\nfor the initial condition θ(0) = 0 and|θ(0)|=π/2nwithn= 0,1,...,5obtained by numerically solving Eqs. (S7a)-(S7c). The\nparameter values are the same as in Fig. 2 of the main text, e.g .,G= 1MHz, except for (a) C1=C2= 10 kHz and (b)\nC1= 100 kHz and C2= 10kHz. Even in the presence of such perturbation C1(2), the synchronized precession of the left and\nthe right magnetization, ˙θ(t) = 0, remains valid. The value of ϕslightly changes depending on the magnitude of the deviatio n\n(C1andC2). Under the initial condition θ(0) = 0 , it becomes ϕ/negationslash= 0forC1/negationslash= 0, whereas ϕ= 0forC1= 0(see Fig. 2). Also in\nthis sense, the point θ(0) = 0 is instable.\nIn this section, although the rectification effect ceases to w ork and it becomes IL(t)∝negationslash= 0, we numerically show that the\nsynchronized precession of the left and the right magnetiza tion,˙θ(t) = 0, remains valid even if the parameter values\nslightly deviate from Eq. (S1). For this, we consider a case w ith\nJRe//planckover2pi1=C1andJIm//planckover2pi1=G/2+C2, (S2)\nwhere each constant C1(2)satisfies|C1(2)| ≪G. Figure S1 shows that the synchronized precession of the lef t and the\nright magnetization is robust against such perturbation C1(2).\nAppendix S-IV: Nonequilibrium steady state for finite magno n interaction\nIn this section, we provide details on the derivation of the e quation for ϕas a function of the magnon-magnon\ninteraction. For\nJRe= 0andJIm//planckover2pi1=G/2, (S1)\nEqs. (S7a)-(S7c) become\n˙θ(t) =G/radicalBigg\nNL(t)\nNR(t)sinθ(t)+u[NL(t)−NR(t)], (S2a)\n˙NL(t) =−2γNL(t)+P, (S2b)\n˙NR(t) =−2γNR(t)−2G/radicalbig\nNL(t)NR(t)cosθ(t)+P. (S2c)\nUnder microwave pumping P, the nonequilibrium steady state ˙θ(t) =˙NL(t) =˙NR(t) = 0 is realized, where θ(t),\nNL(t), andNR(t)approach asymptotically to time-independent constant as t ime advances, ϕ= lim t→∞θ(t)and10\nN∞\nL,R= limt→∞NL,R(t). Thus, from Eqs. (S2a), (S2b) and (S2c), for u∝negationslash= 0, we get\nN∞\nL−N∞\nR=−G\nu/radicalBigg\nN∞\nL\nN∞\nRsinϕ, (S3a)\nN∞\nL=P\n2γ, (S3b)\nN∞\nL−N∞\nR=G\nγ/radicalbig\nN∞\nLN∞\nRcosϕ, (S3c)\nFrom Eq. (S3c), we get the ratio between the number of coheren t magnons in the left and right FMs,\n/radicalBigg\nN∞\nR\nN∞\nL=/radicalBigg\n1+/parenleftbiggG\n2γcosϕ/parenrightbigg2\n−G\n2γcosϕ. (S4)\nCombining Eq. (S3a) and Eq. (S3c), we obtain\ntanϕ=−u\nγN∞\nR, (S5)\n(N∞\nL−N∞\nR)2=−G2\nγuN∞\nLsin(2ϕ)\n2. (S6)\nIfu >0, the solution exists only for tanϕ <0. Substituting N∞\nRfrom Eq. (S5) into Eq. (S6), we arrive at the\nimplicit equation for tanϕ:\n/parenleftBigγ\nutanϕ+N∞\nL/parenrightBig2\n=−G2\nγuN∞\nLtanϕ\n1+tan2ϕ. 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Riego, J. A. Arregi, and A. Berger,\nPhys. Rev. Lett. 122, 257202 (2019), arXiv:1803.10570." }, { "title": "1007.4752v2.Alfvèn_wave_phase_mixing_and_damping_in_the_ion_cyclotron_range_of_frequencies.pdf", "content": "Astronomy & Astrophysics manuscript no. 15479 c\rESO 2021\nNovember 14, 2021\nAlfv´en wave phase-mixing and damping in the ion cyclotron range\nof frequencies\nJ. Threlfall1, K. G. McClements2, and I. De Moortel1\n1School of Mathematics and Statistics, University of St. Andrews, St. Andrews, Fife, KY16 9SS, U.K. e-mail:\njamest@mcs.st-and.ac.uk;ineke@mcs.st-and.ac.uk\n2EURATOM /CCFE Fusion Association, Culham Science Centre, Abingdon, Oxfordshire, OX14 3DB, U.K. e-mail:\nk.g.mcclements@ccfe.ac.uk\nABSTRACT\nAims. To determine the e \u000bect of the Hall term in the generalised Ohm’s law on the damping and phase mixing of Alfv ´en waves in the\nion cyclotron range of frequencies in uniform and nonuniform equilibrium plasmas.\nMethods. Wave damping in a uniform plasma is treated analytically, whilst a Lagrangian remap code (Lare2d) is used to study Hall\ne\u000bects on damping and phase mixing in the presence of an equilibrium density gradient.\nResults. The magnetic energy associated with an initially Gaussian field perturbation in a uniform resistive plasma is shown to decay\nalgebraically at a rate that is una \u000bected by the Hall term to leading order in k2\u000e2\niwhere kis wavenumber and \u000eiis ion skin depth.\nA similar algebraic decay law applies to whistler perturbations in the limit k2\u000e2\ni\u001d1. In a nonuniform plasma it is found that the\nspatially-integrated damping rate due to phase mixing is lower in Hall MHD than it is in MHD, but the reduction in the damping rate,\nwhich can be attributed to the e \u000bects of wave dispersion, tends to zero in both the weak and strong phase mixing limits.\nKey words. Plasmas - Magnetohydrodynamics (MHD) - Waves - Sun: flares - Sun: chromosphere\n1. Introduction\nThe interaction between Alfv ´en waves and plasma inhomo-\ngeneities forms a well-studied and important area of research, for\nboth laboratory and astrophysical plasmas. One process which\narises as a result of this interaction is Alfv ´en wave phase-mixing .\nEarly studies of Alfv ´en wave phase-mixing demonstrated a po-\ntential for significantly enhanced plasma heating. In particular,\nHeyvaerts & Priest (1983) proposed the phase-mixing of Alfv ´en\nwaves as a potential solar coronal heating mechanism through\nenhanced wave-dissipation. They outlined that for a magneto-\nhydrodynamic (MHD) treatment of initially planar shear-Alfv ´en\nwaves, propagating independently on individual magnetic sur-\nfaces, large di \u000berences in phase are quickly generated between\nwaves on neighbouring field lines, as a result of variation in\nAlfv ´en speed across the field. These phase-di \u000berences generate\nprogressively smaller scales, and dramatically enhance the ef-\nfects of viscous and Ohmic dissipation, in the locations where\nthe Alfv ´en speed gradient is steepest.\nThis initial concept has been subsequently adapted for a va-\nriety of problems using MHD, based on the premise that the\nwaves in question arise as a result of an infinite series of bound-\nary motions at the photosphere. For example, this treatment has\nbeen used to investigate the heating of open magnetic field lines\nunder various conditions, e.g. by Parker (1991); Hood et al.\n(1997); De Moortel et al. (1999, 2000). Phase-mixing has alsobeen studied as a source of non-linear coupling to other wave\nmodes (see, e.g. Nakariakov et al. 1997; Botha et al. 2000).\nOf particular interest for this paper is the work of Hood et al.\n(2002). They note that an infinite series of boundary motions\nis unrealistic and instead investigate the e \u000bect on the amplitude\ndamping rate due to phase-mixing when the waves are driven by\nonly one or two initial impulsive motions at the boundary.\nRecent studies have also begun to move away from the orig-\ninal MHD treatment, instead focussing on full kinetic descrip-\ntions of a plasma undergoing phase-mixing in a collisionless\nregime, as a potential mechanism for electron acceleration (see,\ne.g. G ´enot et al. 2004; Tsiklauri et al. 2005; Tsiklauri & Haruki\n2008; Bian & Kontar 2010). On the assumption that the wave-\nlengths of interest ( \u0015) are small compared to the particle mean\nfree path\u0015m f p, Tsiklauri et al. (2005) and Bian & Kontar (2010)\nmodel the corona as a collisionless plasma and cite Landau\ndamping as their primary wave dissipation mechanism. Such\ndamping is strongly suppressed if \u0015\u001d\u0015m f p(Ono & Kulsrud\n1975), as in the case of propagating EUV disturbances with pe-\nriods of tens or hundreds of seconds (see e.g. De Moortel 2009,\nand references therein), and a fluid model is then appropriate for\nthe coronal plasma. On the other hand for waves with frequen-\ncies approaching the ion cyclotron frequency, typical coronal pa-\nrameters correspond to classical (Spitzer) collisional mean free\npaths that exceed \u0015, suggesting instead the validity of the col-\n1arXiv:1007.4752v2 [astro-ph.SR] 8 Oct 2010J. Threlfall et al.: Alfv ´en wave phase-mixing and damping in the ion cyclotron range of frequencies\nlisionless approach. However, Craig & Litvinenko (2002) have\nshown that it is not appropriate to use classical resistivity under\nflaring conditions because it implies current scale lengths that\nare several orders of magnitude shorter than \u0015m f p, and moreover\nis inconsistent with the electric fields required to account for the\nobserved acceleration of protons to tens of MeV on timescales\nof the order of one second (Hamilton et al. 2003). Craig &\nLitvinenko proposed that the e \u000bective resistivity under flaring\nconditions (specifically in a reconnecting current sheet) is de-\ntermined by turbulence arising from electron-ion drift (e.g. ion\nacoustic) instabilities, and deduced that this e \u000bective resistivity\ncould exceed the classical value by a factor of around 106. Under\nthese circumstances a fluid model can be appropriate even for\nrelatively high frequency waves. This may also be true in the up-\nper chromosphere, where the plasma is both cooler and denser\n(and consequently much more collisional) than in the corona.\nIt is well known that the electron inertia term in the gener-\nalised Ohm’s law becomes comparable to the MHD terms when\nthe system lengthscale approaches the electron skin depth, \u000ee,\nwhich in a low beta plasma can exceed the ion Larmor radius.\nMoreover for perturbations with frequencies approaching the ion\ncyclotron frequency \ni, the Hall term in the generalised Ohm’s\nlaw becomes as important as the MHD terms when the length-\nscale of the system approaches the ion skin depth \u000ei(\u001d\u000ee).\nWhen the introduction of the Hall term into Ohm’s law is the\nonly modification made to the otherwise standard set of MHD\nequations, we may refer to this as a Hall MHD system.\nHall MHD has been found to be important for a number\nof fundamental plasma processes. For example, in magnetic re-\nconnection studies, Birn et al. (2001) found that all models\nwhich included Hall dynamics returned indistinguishable recon-\nnection rates, concluding that the inclusion of the Hall term is\nthe minimum requirement for fast reconnection (for a summary\nof Geospace Environmental Modelling (GEM) challenge results,\nsee Birn & Priest (2007)).\nHigh frequency waves (i.e. with frequencies !\u0018\ni), have\nbeen observed in a range of astrophysical plasma systems, for\nexample in the solar corona (summarised in Marsch (2006)) and\nin situ, at the Earth’s bow shock (Sckopke et al. 1990). When\noscillations in this frequency range are excited in collision-\ndominated plasmas, such that the collisional mean free path is\nless than the wavelength, it is then appropriate to use a Hall\nMHD model.\nThe goal of our work is to determine the extent to which\nthe main consequences of phase-mixing (wave dissipation and\nplasma heating) are a \u000bected solely by the addition of the Hall\nterm to Ohm’s law. To do this, we first investigate the damping\nrate of a uniform plasma using Hall MHD (Section 2). Phase-\nmixing is then included, by allowing the equilibrium density to\nvary (Section 2.3). Numerical simulations of a Hall MHD sys-\ntem, with various density profiles and Hall term strengths, are\ndescribed in Section 3. We interpret these results and present\nconclusions in Section 4.2. Wave Damping Analysis\nThe Hall MHD form of Ohm’s Law is\nE+v\u0002B=\u0011j+1\nnej\u0002B;\nwhere EandBare electric and magnetic fields, vis the single-\nfluid plasma velocity, jis current density, \u0011is electrical resis-\ntivity, nis plasma number density and \u0000eis electron charge.\nLinearising the induction equation corresponding to this form\nof Ohm’s law, together with the equation of motion (neglecting\nthe pressure gradient force), for a plasma with a uniform equilib-\nrium field B0=B0ˆyfor a constant B0, uniform number density\nand resistivity, and zero equilibrium flow, gives:\n@B1\n@t=r\u0002(v\u0002B0)+\u0011\n\u00160r2B1\u00001\n\u00160ner\u0002[(r\u0002B1)\u0002B0](1)\n@v\n@t=1\n\u001a0\u00160(r\u0002B1)\u0002B0; (2)\nwhere B1[=(Bx;0;Bz)] now represents a transverse perturbation\nto the equilibrium field, \u001a0is equilibrium mass density, and \u00160is\nthe permeability of free space. Note that although, for simplicity,\nwe neglect plasma pressure, by assuming \f=0 in this analyt-\nical treatment (as Alfv ´en, whistler and ion cyclotron waves are\nall incompressible in a linear regime), the numerical simulations\npresented in Section 3 incorporate plasma pressure, i.e. \f,0.\nBy inserting (2) into the time derivative of (1) in the usual man-\nner, the linearised Alfv ´en wave equation is then modified:\n@2Bc\n@t2=cA2@2Bc\n@y2+ \u0011\n\u00160\u0000icA\u000ei!@2\n@y2 @Bc\n@t!\n; (3)\nwhere we have expressed transverse field perturbations in the\nform of a complex variable Bc=Bz+iBx, and introduced\nthe Alfv ´en speed, cA=B0=p\u001a0\u00160, and the ion skin depth,\n\u000ei=c=!pi=cp\nmi\u000f0=ne2, where miis the ion mass, cis the\nspeed of light, and \u000f0is the permittivity of free space.\nSeeking wave-like solutions of the form exp\u0000i\u0002ky\u0000!t\u0003\u0001allows\nus to form a dispersion relation to express perturbation frequen-\ncies,!, as a function of wavenumber k. By considering only\nthe regime of weak damping ( \u00112k2=\u00162\n0cA2\u001c1), we obtain two\nseparate solutions depending on the size of the parameter k2\u000e2\ni.\nTaking first the case of k2\u000e2\ni\u001c1, we make use of a simple Taylor\nexpansion to find (to leading order in k2\u000e2\ni):\n!=\u0006cAk\u0006cA\u000eik2\n2\u0000i\u0011k2\n2\u00160; (4)\nwhich describes a shear Alfv ´en wave, modified by Hall e \u000bects,\nand subject to resistive damping.\nConsidering the opposite case, when k2\u000e2\ni\u001d1, we find:\n!=\u0006cA\u000eik2\n2\u0006cA\u000eik2\n2s\n1+2i\u0011\n\u00160cA\u000ei+4\nk2\u000e2\ni\u0000i\u0011k2\n2\u00160:\nIncluding the next term in the Taylor expansion, we find two\ndistinct forward propagating solutions:\n!+=cA\n\u000ei\u0000i\u0011\n\u00160\u000e2\ni; (5a)\n2J. Threlfall et al.: Alfv ´en wave phase-mixing and damping in the ion cyclotron range of frequencies\n!\u0000=cA\u000eik2\u0000i\u0011k2\n\u00160; (5b)\nwhere in this limit, we now obtain a combination of whistler\nand ion cyclotron (i.c.) waves, both subject to a form of resistive\ndamping.\n2.1. Long Wavelength Hall MHD Regime (Uniform)\nWe can examine the e \u000bect of this di \u000berence in behaviour of\nboth k2\u000e2\niregimes, by focussing on the evolution of an initially\nGaussian pulse (of width \u001b, and amplitude Bi), which is allowed\nto travel along the equilibrium field, taking the form:\nBc(y;0)=Bz+iBx=Biexp \n\u0000y2\n2\u001b2!\n: (6)\nOur complex variable can be interpreted as a Fourier integral,\nevolving in time as:\nBc=Bc(y;t)=Z1\n\u00001f(k)expfi(ky\u0000!t)gdk; (7)\nwith f(k) determined by the initial conditions:\nf(k)=1\n4\u0019Z1\n\u00001Bc(y;0) exp (iky)dy=Bi\u001b\n2p\n2\u0019exp \n\u0000k2\u001b2\n2!\n;(8)\nwhere we have used the standard result (Abramowitz & Stegun\n1964, Eq. 7.4.6):\nZ1\n0exp\u0010\n\u0000\fx2\u0011\ncos(bx)dx=1\n2r\u0019\n\fexp \n\u0000b2\n4\f!\n: (9)\nWe also use this result to evaluate the integral in Eq. (7) in the\nlimit k2\u000e2\ni\u001c1, finding:\nBc=X\n+;\u0000Biq\n1+\u0010\u0011\n\u00160\u0000icA\u000ei\u0011t\n\u001b2exp0BBBBBBB@\u0000(y\u0006cAt)2\n2h\n\u001b2+\u0010\u0011\n\u00160\u0000icA\u000ei\u0011\nti1CCCCCCCA(10)\nwhere the summation is over forward- and backward-\npropagating waves. Eq. (10) describes a pair of pulses travelling\nin opposite direction at approximately the Alfv ´en speed, which\nare damped by finite resistivity and circularly polarised. We can\nalso calculate the contribution to the total energy per unit area in\n(x,z),EBTOT, made by the magnetic energy per unit area in ( x,z)\nof both pulses (EBc) as follows:\nEBTOT=1\n2\u00160Z1\n\u00001(B0+B1)2dy=B2\n0\n2\u00160+EBc:\nSince the magnetic perturbation is transverse to the equilibrium\nfield, B0\u0001B1=0 and hence\nEBc=1\n2\u00160Z\nB12dy=1\n2\u00160Z1\n\u00001BcB\u0003\ncdy: (11)\nMany of the factors in BcB\u0003\ncwill cancel upon integration, hence\nthe energy associated with the pulse ( EBc) evolves as:\nEk2\u000e2\ni\u001c1\nBc=p\u0019\u001bB2\ni\n4\u00160\u00001+\u0011t=\u00160\u001b2\u00011=2(\n1+exp \n\u0000cA2t2\n\u001b2+\u0011t=\u00160!)\n:(12)Thus after a short initial transient phase (essentially the time\ntaken for an Alfv ´en wave to travel a distance equal to the ini-\ntial pulse width, \u001b), we recover a power law decay ( /t\u00001=2for\nt\u001d\u00160\u001b2=\u0011) in the energy associated with the pulse. This ex-\npression (Eq. 12) is compared with several numerical simulation\nresults in Section 3, and can be seen in Fig. 2. It is straightfor-\nward to show that the expression given by Eq. (12) for the Hall\nMHD long wavelength ( k2\u000e2\ni\u001c1) regime is identical to that\nfound for an initially Gaussian pulse in the MHD limit.\n2.2. Short Wavelength Hall MHD Regime (Uniform)\nTurning to the opposite limit, k2\u000e2\ni\u001d1, the perturbation fre-\nquencies (5) comprise of a combination of resistively damped\nwhistler and ion cyclotron waves. We can again describe the\npulse evolution associated with each separate wave branch in this\nlimit, in the manner described previously (Section 2.1), again for\nan initially Gaussian perturbation. Beginning with (7), and with\nthe same initial conditions (8) as the previous limit, we find that\nthe whistler wave calculation proceeds similarly to that of the\nprevious section, however the i.c. wave (being independent of\nwavenumber) di \u000bers somewhat:\nBw\nc=Bi\u001bp\n2\u0019Z1\n0eikyexp \n\u0000\u001b2k2\n2\u0000\u0011k2t\n\u00160\u0000icA\u000eik2t!\ndk; (13a)\nBic\nc=Bi\u001bp\n2\u0019exp0BBBB@\u0000icAt\n\u000ei\u0000\u0011t\n\u00160\u000e2\ni1CCCCAZ1\n0eikyexp \u0000k2\u001b2\n2!\ndk:(13b)\nEvaluating the integrals in Eq. (13), using Eq. (9), we obtain:\nBw\nc=Biq\n1+2\u0010\u0011\n\u00160\u0000icA\u000ei\u0011t\n\u001b2exp0BBBBBBB@\u0000y2\n2h\n\u001b2+2\u0010\u0011\n\u00160\u0000icA\u000ei\u0011\nti1CCCCCCCA(14a)\nBic\nc=Biexp0BBBB@\u0000icAt\n\u000ei\u0000\u0011t\n\u00160\u000e2\ni1CCCCAexp \n\u0000y2\n2\u001b2!\n: (14b)\nIn this short wavelength ( k2\u000e2\ni\u001d1) regime, the peak of the\npulse now no longer propagates, but decreases in amplitude. The\nright circularly polarised component of the pulse rapidly broad-\nens, due to the high whistler speed, and damps algebraically at a\nrate similar to that found in both the MHD and long wavelength\n(k2\u000e2\ni\u001c1) Hall MHD regimes. The left circularly polarised (ion\ncyclotron wave) component, on the other hand, damps exponen-\ntially. It should be noted that this damping arises from resistive\ndissipation, and as such should be distinguished from the kinetic\nion cyclotron damping arising from wave-particle interactions.\nWe may, again, calculate the energy associated with each\nsolution (14), using (11), where we still only obtain transverse\nperturbations, and hence B1\u0001B0still makes no contribution to\nthe energy. In this limit, we obtain an expression for the energy\nassociated with the individual whistler and i.c. wave branches:\nEw\nBc=B2\ni\u001bp\u0019\n2\u00160 \n1+2\u0011t\n\u00160\u001b2!\u00001=2\n; (15a)\n3J. Threlfall et al.: Alfv ´en wave phase-mixing and damping in the ion cyclotron range of frequencies\nEic\nBc=B2\ni\u001bp\u0019\n2\u00160exp0BBBB@\u00002\u0011t\n\u00160\u000e2\ni1CCCCA: (15b)\nThus, for waves in the k2\u000e2\ni\u001d1 regime, we no longer see\nthe initial transient phase seen previously in (12), and for long\ntimescales the algebraically-damped whistler contribution to the\nwave energy is dominant over the exponentially-damped contri-\nbution from the ion cyclotron wave.\n2.3. Wave Damping and Phase-Mixing in a Non-Uniform\nPlasma\nBy now allowing the equilibrium plasma density to vary in a\ndirection perpendicular to both the direction of the equilibrium\nfield ( y) and the direction of initial perturbation ( z), we can in-\nvestigate what e \u000bect the Hall term has on the dissipation rate\nin a non-uniform plasma. When the gradients in the x-direction\nare su \u000eciently large, and the e \u000bects of viscosity are negligible,\nthe linearised Alfv ´en wave equation in the MHD limit takes the\nform [Hood et al. (2002)]:\n@2Bz\n@t2=cA2(x)@2Bz\n@y2+\u0011\n\u00160@2\n@x2 @Bz\n@t!\n: (16)\nThe variation in Alfv ´en speed cA(x)=B0=p\n\u00160\u001a(x) causes\nsteep gradients to build up in the direction of the inhomogene-\nity which, in turn, significantly enhances resistive damping in\nthe regions where the inhomogeneity is greatest. Hood et al.\n(2002) used a multiple time-scale analysis to derive from Eq.\n(16) a one-dimensional di \u000busion equation whose solutions can\nbe expressed in terms of the Alfv ´en speed gradient cA0(x)=\ndcA(x)=dx. For the case of the initially Gaussian pulse defined\nby Eq. (6), the forward-propagating solution takes the form:\nBz=Bi\n2p\n1+cA02\u0011t3=3\u00160\u001b2exp \n\u0000(y\u0000cAt)2\n2\u0002\u001b2+cA02\u0011t3=3\u00160\u0003!\n:(17)\nWe can evaluate the perturbed magnetic field energy per unit\nlength in the z-direction EH\nBcfor this case by integrating B2\nz=2\u00160\nover a finite distance x0\u00182Jeff.\n•Amplitude fluctuation: The amplitude fluctuations play\na crucial role in broadening the lineshape at weak cou-\npling, where the fluctuation width varies as \u0018p\nT=U .\nWhile we do not capture the real ”amplitude mode” at\n!\u0018Uwe can access amplitude fluctuation effects on\nthe spin waves at !\u0018Jeff.\nII. On the triangular lattice:\n•Broad regimes: The triangular lattice has a finite critical\ninteraction for the MIT, with Uc\u00185t. We restrict our-\nselves toU=twhere the 120\u000eordered state is the ground\nstate. The typical lineshape is two-peak in this case.\nThe thermal crossover scales are inferred from the be-\nhaviour of the peak which broadens quicker with re-\nspect toT. The behaviour of Tcr\n1andTcr\n2with respect\ntoUis similar to what is observed in the square lattice,\nwith the distinction that their maxima occur at larger U\nand the scales are\u00180:5their square lattice values.\n•Dispersion and damping: Due to emergence of longer\nrange couplings, the low Tdispersion along \u0000\u0000K\nshows a larger curvature at lower U=t. The damping is\nalso much larger, compared to the square lattice, at sim-\nilar values of T=Jeff. AtU=t\u001810, whereJeff=t\u0018\n0:04the crossover scales are just Tcr\n1=Jeff\u00180:4and\nTcr\n2=Jeff\u00180:8.\n•Fluctuation: The role of amplitude fluctuations in\ndamping the modes is enhanced at a given Uand the\nsameT=Jeff, due to the finite Ucand mild frustration.\nII. MODEL AND METHOD\nWe work with the single band, repulsive Hubbard model\non square and triangular lattice geometries. The Hamiltonian\nreads-\nH=\u0000X\n\u001btij(cy\ni\u001bcj\u001b+h:c:) +UX\nini\"ni#\u0000\u0016X\ni\u001bni\u001b\nThe hopping amplitude tijis chosen to be non-zero only\namongst nearest neighbours for the square case and has a uni-\nform valuet= 1:0. On adding the next-nearest neighbourcouplingt0= 1:0on top of this along one diagonal in each\nsquare motif , we get the triangular lattice.\nFirst, the interaction term is decoupled using a Hubbard-\nStratonovich transformation to obtain a spin-fermion model-\nHSF=\u0000X\n\u001btij(cy\ni\u001bcj\u001b+h:c:)\u0000UX\nimi:\u001bi+UX\nijmij2\nWe solve for the finite Tdynamics miusing the following\nequation of motion45:\ndmi\ndt=\u0000mi\u0002@hHSFi\n@mi\u0000\r@hHSFi\n@mi+~\u0018i (1)\nThe noise is specified through-\nh\u0018\u0016\ni(t)i= 0 (2)\nh\u0018\u0016\ni(t)\u0018\u0017\nj(t0)i= 2\rkBT\u000eij\u000e\u0016\u0017\u000e(t\u0000t0) (3)\nHere\ris a dissipation parameter. Within our scheme, it’s\nvalue can’t be determined from first principles. To calculate\nit, one has to evaluate the imaginary part of the Keldysh po-\nlarizability ( Im\u0005K(q;!)) at low frequencies. We comment\nthat in the deep Mott phase, this contribution is vanishingly\nsmall due to the gapped single electron spectrum. However,\non moving to lower Uvalues, this quantity picks up weight at\nfinite temperature. The evolution equation has a phenomeno-\nlogical justification as well as a a microscopic basis. We touch\non these briefly.\nI. First the phenomenological motivation45,78. One starts\nfrom the Heisenberg limit with moments of fixed magnitude.\nThe torque term comes from evaluating the Poisson brack-\nets in the semiclassical equation of motion. The damping is\ntaken to be proportional to the angular momentum, following\nan analogy with the particle Langevin equation. Lastly, the\nnoise is chosen so as to satisfy the fluctuation-dissipation re-\nlation, ensuring that one captures the Boltzmann distribution\nin the long-time limit78,79. The additive form of the damp-\ning and noise allows for longitudinal relaxation of the mag-\nnetic moments. This approach does not determine the value\nof the dissipation coefficient \r. In our treatment, we fix the\n\rvalue by comparing our static results with a Monte Carlo\n(MC) method and ensuring a decent match. The MC strategy\nis briefly discussed in Appendix B.\nAlternately, II. One starts from a model of a spin coupled\nlinearly to a bosonic bath and integrates out the bath degrees\nof freedom to obtain an effective equation of motion for the\nspin, it has been shown80that under certain conditions, a\nLandau-Lifshitz-Gilbert-Bloch (LLGB) equation81emerges.\nThe derivation may also be done in presence of conduction\nelectrons82or both phonons and electrons83. This equation\nexplicitly conserves spin magnitudes. Our equation also re-\nduces to the LLGB form upon constraining the spins on the\nunit sphere78.\nFinally, III. One may also try to derive the present equa-\ntion starting from the Keldysh action of the Hubbard model.\nFirst, one introduces auxiliary fields to decouple the interac-\ntion term and subsequently assumes them to be slow com-\npared to the electrons. This allows one to write an effectiveequation of motion for them. Upon doing certain simplifica-\ntions, this equation can be mapped on to Eq.1. We briefly\nallude to this in subsection E of our Discussion section.\nThe typical timescale for magnon oscillations is \u001cmag\u0018\n1=Jeff. We set an ”equilibration time” \u001ceq= 100\u001cmag be-\nfore saving data for the power spectrum. The outer timescale,\n\u001cmax\u001810\u001ceq. The ”measurement time” \u001cmeas =\u001cmax\u0000\u001ceq,\nand the number of sites is N. Some details regarding the nu-\nmerical solution of Eq.1 are given in Appendix A.\nWe calculate the following from the time series m(ri;t):\n1. Dynamical structure factor, D(q;!) =jm(q;!)j2\nwhere\nm(q;!) =X\niZ\u001cmax\n\u001ceqdteiq:rie\u0000i!tm(ri;t) (4)\n2. The instantaneous structure factor\nS(q;t) =1\nN2X\nijeiq:(ri\u0000rj)m(ri;t):m(rj;t) (5)\nThe corresponding time averaged structure factor is\n\u0016S(q) =1\n\u001cmeasZ\u001cmax\n\u001ceqdtS(q;t) (6)\n3. The distribution of moment magnitudes:\nP(jmj) =1\nN\u001cmeasX\niZ\u001cmax\n\u001ceqdt\u000e(jmj\u0000jmi(t)j)(7)\n4. Dispersion \nqand damping \u0000q:\n\nq=Z!max\n0d!!D (q;!)\n\u00002\nq=Z!max\n0d!(!\u0000\nq)2D(q;!)\nIII. BENCHMARKS AND OVERALL FEATURES\nA. Fixing the Langevin parameters\nWe do a bechmarking of the Langevin scheme using the\nsquare lattice as a test case. Three coupling regimes are\nexplored- weak ( U=t= 3:0), intermediate ( U=t= 6:0) and\nstrong (U=t= 10:0). The statics is quantified through two\nquantities- the structure factor S(\u0019;\u0019)and the moment mag-\nnitude distribution P(jmj). The former shows the correlation\ntemperatures ( Tcorr), below which the correlation length ap-\nproaches the system size. The latter details the longitudinal\nfluctuations of local moments. The alternate technique used\nto compute these quantities is a Monte Carlo calculation done\nassuming the auxiliary mifield to be classical and using the\nsum of electronic free energy and the stiffness cost (last term\ninHSF) as the sampling weight16(see Appendix B for more\ndetails).(a)\n(b)FIG. 1. \u0016S(\u0019;\u0019)(a) andP(jmj)(b) for the square lattice Hubbard\nmodel atU=t= 6:0. Solid lines denote answers obtained using the\npresent LD method and open circles indicate MC data. We observe a\nreasonable agreement between the two methods.\nThe method of fixing \rwas the following. We started with\na low value (motivated by its vanishing magnitude at strong\ncoupling, and the fact that we should get undamped spin\nwaves at low enough T) at a fixed coupling and run length.\nNext, we increased the \rat that coupling in steps till the match\nwith MC results on temperature dependence became reason-\nable, while ensuring that the low Tspin waves remain sharp\nenough. Results for a typical coupling are quoted above.\nFig.1(a) shows a comparison of \u0016S(\u0019;\u0019)atU=t= 6, with\na reasonable match. The dissipative coefficients are \r= 0:05\nand\r= 0:1. In Fig.1(b), the P(jmj)distributions also show\nreasonable agreement (for \r= 0:05). We’ve used \r= 0:025\nto generate the bulk of our final dynamics results, which\nroughly corresponds to a relaxation timescale \u001crel\u001840\u001cmag.\nWe will later quantify the increasing relevance of magnitude\nfluctuations on decreasing coupling, which is an important\npiece of the non-Heisenberg physics.\nB. Magnetic scales for varying U=t\nAt low temperature, our dynamical equation (Eq.1) gives\nrise to weakly damped, dispersive spin wave excitations.\nFrom the obtained spectrum, we extract two scales- (i) the\nspin-wave stiffness, Jeff, and (ii) the magnon bandwidth,\nWmag. The first is computed from the spin wave velocity of\nthe linear branch near the respective Goldstone modes on the\nsquare and triangular lattice. The latter requires knowledge of\nthe full magnon band structure. We plot these quantities for\nboth the square and triangular lattice in Fig.2.\nIn Fig.2(a), we find a monotonic decrease of JeffwithU=t\nin the square lattice case, with a 1=Uasymptote at strong\ncoupling. The value at U=t = 20:0matches the expected\nJeff= 4t2=U, indicating that one has reached the Heisen-\nberg limit. On the triangular lattice, the stiffness goes to zero\nforU=t= 6:0, indicating a breakdown of the 120\u000eordered\nstate. The scale then rises and finally falls as \u00181=Uat strong\ncoupling. In Appendix C, we compare the extracted spin wave\nvelocities with those obtained from RPA41.\nThe magnon bandwidths Fig.2(b) feature a non-\nmonotonicity in the square case, with a maximum around\n0.8\n0.6\n0.4\n0.2\n0.0\n(a)\n(b)FIG. 2. (a): The dimensionless effective exchange ( Jeff=t), cal-\nculated from the spin wave velocity, for the square and triangular\nlattice Hubbard models at various U=t values. We see a mono-\ntonic behaviour for the square lattice and a non-monotonic behaviour\nfor the triangular lattice case. Moreover, the scale vanishes around\nU=t = 6:0for the latter, signalling a breakdown of 120\u000eor-\nder. (b):The spin wave bandwidth ( Wmag), calculated from the full\nmagnon dispersion, for the square and triangular cases. Here, we see\na non-monotonicity in the square lattice, and a gradual decrease in\nthe triangular lattice.\nU=t= 6:0.Wmag increases on lowering Uon the triangle,\nrising to 0:6tbefore the ordered state breaks down.\nC. Comparison with Heisenberg as U=t!1\nWe compare the Hubbard results at U=t= 20 on the square\nlattice with the Heisenberg model with J= 1 . The for-\nmer effectively reduces to the latter with Jeff= 4t2=Uand\njmij= 1=2. First, in 3(a), the low Tdispersions are com-\npared, with both being scaled by Wmag, the spin wave band-\nwidth. There’s a nearly perfect agreement.\nThe Heisenberg model features three broad thermal\nregimes. These are- (i) weakly damped ( T\u001cJ), where we\nobtain dispersive excitations with low damping, (ii) strongly\ndamped (T\u0018 O (J)), where there’s significant mode cou-\npling among spin waves, but dispersion is still discernable,\nand (iii) diffusive ( T\u001dJ), where mode frequencies col-\nlapse to zero and the dampings are comparable to Wmag. In\nthese regimes, we compare the lineshapes of the Heisenberg\nmodel at q= (\u0019=2;\u0019=2)with those of the large UHubbard\nmodel in Fig.3(b). In regime (i), a sharp lineshape centered\naround \nq= 4Jis seen, which picks up significant damping\nin regime (ii), before becoming diffusive in (iii). A quantita-\ntive agreement is seen between the Hubbard and Heisenberg\nresults. The frequencies are scaled by Jeffin the Hubbard\ncase, andJin the Heisenberg one.\nD. General features of dynamics in the Mott phase\nWe first comment on the broad dynamical regimes obtained\non the square and triangular lattice problems. This is charac-\nterized by the the number of peaks, their location, and width.(a)\n(b)\n(c)FIG. 3. (a): Comparison of dispersions \nqalong theK\u0000\u0000di-\nrection of the Brillouin Zone (BZ) between the square lattice Hub-\nbard model at U=t = 20:0and the Heisenberg model with J=\n1. One gets a near perfect agreement on scaling the former by\nJeff= 4t2=U. (b),(c): Lineshapes at three characteristic temper-\naturesT=J = 0:01;0:5;1:5for the Heisenberg model (in (b)) and\ntheU=t= 20:0Hubbard model (in (c)). There’s again a marked\nagreement.\nAs mentioned earlier, we find three broad dynamical\nregimes on analyzing the data- (i) weakly damped, where the\nlinewidth for a generic momentum \u0000q\u001cWmag, (ii) strongly\ndamped, where \u0000q\u0018 O (Wmag)and (iii) diffusive, where\n\u0000q\u0018O(Wmag)and\nq!0.\nOn the square lattice (Figs.4(a) and 4(c)), the low Tline-\nshapes are unimodal. There is a gradual crossover to regimes\n(ii) and (iii) at Tcr\n1(U)andTcr\n2(U)respectively. The win-\ndow of regime (ii) is maximum around U=t = 6:0. The\ncrossover lines behave \u00181=Uasymptotically, but have a max-\nimum around U=t= 10:0. Below this coupling, the amplitude\nfluctuation effect dominates and consequent excess thermal\ndampings cause a downward trend. This non-Heisenberg fea-\nture is much better highlighted in 2(c), where both Tcr\n1=Jeff\nandTcr\n2=Jeffdecrease markedly on lowering U. At weak\ncoupling, both these scales collapse quickly.\nThe loss of antiferromagnetic correlations at finite temper-\nature is characterized through a temperature scale Tcorr, ex-\ntracted from S(\u0019;\u0019). The crossover lines have a similarity\nto the locus of this Tcorr(U)84, which also coincides with\nthe metal-insulator transition line at weak coupling. How-\never, there are quantitative differences. the peak location in\nour dynamical phase diagram Fig.4(a) is at \u0018U=t = 10 ,\nahigher coupling compared to the peak location in Tcorr at\n\u0018U=t= 4. We emphasize that our focus is on the ”local\nmoment” regime, i.e, intermediate to strong coupling. Our\nmethod can address the weak coupling Slater regime as well\n0510152025U/ t0.00.10.2T/ t\ndi\u0001usivestronglydampedweaklydamped(a)\n(a)\n0 5 10 15 20 25\nU/ t0.00.10.2 T/ t\n(b)\nstrongly\ndampeddiffusive\nweakly dampedmetal\n161116t/ Jef f012T/ Jef f\ndi\u0001usivestronglydampedweakly\ndamped\n(c)damped\n5101520t/ Jef f\ndi\u0001usive\nstronglydampedweaklydamped\n(d)\n012T/ Jef f(d)FIG. 4. Magnon phase diagrams for square ((a) and (c)) and trian-\ngular ((b) and (d)) lattice Hubbard models at half-filling. The top\nrow features the U=t\u0000T=t phase diagrams, while the bottom one\nexhibits theT=Jeff\u0000t=Jeffplots. We broadly observe three ”dy-\nnamical regimes”- (i) weakly damped (where \u0000q<\u00180:2Wmag), (ii)\n”strongly damped” (where \u0000q\u0018 O (Wmag)) and (iii) ”diffusive”\n(where \u0000q\u0018O(Wmag)and\nq!0). The metallic region in (b)\nis not tackled by our approach. Vertical sections indicate couplings\nused in actual simulations.\nbut that regime is dominated by amplitude fluctuations and\nalso requires larger system size.\nIn Section VI (subsection C), we discuss an effective clas-\nsical moment model which actually interpolates between the\nHeisenberg and Slater limits, borrowing a few parameters\nfrom the Hubbard mean field and RPA results. This captures\nthe low temperature dynamics of the Hubbard problem fairly\nwell at allU=t, and the Heisenberg limit at all temperatures.\nMoreover, the non-monotonicity of Tcorras a function of U=t\nand the qualitative behaviour of the thermal regimes are also\ncaptured by the effective model.\nIn the triangular case (Fig.4(b) and 4(d)), the generic low T\nlineshapes is two-peak. The crossover regimes (ii) and (iii) oc-\ncur at much lower temperatures compared to the square case,\nowing to mild geometric frustration and consequently frag-\nile magnetic order. The fall of the crossover scales on de-\ncreasingU(belowU=t= 10:0, say) is also sharper than the\nformer. Close to the transition ( U=t\u00186) the lineshapes be-\ncome diffusive even at very low temperatures ( T=t\u00180:01).\nThe scaled phase diagram (4(d)) reveals a minimum in the\ncrossover scales around t=Jeff\u001812:5. This is related to the\nnon-monotonic behaviour of Jeffitself, shown in Fig.2.\nWe comment that our scheme at weak coupling generates a\npeak centered at zero frequency for all momenta, exclusively\ndue to amplitude fluctuations. This arises from an oversimpli-Γ X K Γ\n0.00.40.8ω/tT/t=0.001\nΓ X K Γ\nT/t=0.050\nΓ X K Γ\nT/t=0.100\nΓ X K Γ\nT/t=0.150\n0.0000.0020.004\n(a)U/t=20.0\nΓ X K Γ\n0.00.40.8ω/tT/t=0.001\nΓ X K Γ\nT/t=0.050\nΓ X K Γ\nT/t=0.100\nΓ X K Γ\nT/t=0.150\n0.0000.0020.004\n(b)U/t=10.0\nΓ X K Γ\n0.00.40.8ω/tT/t=0.001\nΓ X K Γ\nT/t=0.050\nΓ X K Γ\nT/t=0.100\nΓ X K Γ\nT/t=0.150\n0.0000.0020.004\n(c)U/t=6.0\nΓ X K Γ\n0.00.40.8ω/tT/t=0.001\nΓ X K Γ\nT/t=0.050\nΓ X K Γ\nT/t=0.100\nΓ X K Γ\nT/t=0.150\n0.0000.0020.004\n(d)U/t=3.0FIG. 5. Power spectrum of magnetization field D(q;!)for the Hubbard model on the square lattice for U=t= 20;10;6;3respectively. The\ntrajectory chosen in Brillouin Zone is \u0000\u0000X\u0000K\u0000\u0000. Temperatures are scaled by electron hopping t. We observe a resemblance of the\nstrong coupling Hubbard spectrum with that of the Heisenberg model with Jeff= 4t2=U. At lower couplngs, the dispersion changes at low\nT, owing to longer-range spin couplings. Thermal damping is more prominent at weaker couplings, as the stiffness for amplitude fluctuation\ndecreases.\nfication of our equations of motion. However, the fraction of\nthis weight isn’t visible on a linear scale above U=t\u00184on\nthe square. Moreover, if we ignore the near-zero energy part\nof the magnon spectrum (upto some cutoff \u00180:05Wmag), the\nrest of it doesn’t have any spurious features. We still capture\nthe impact of magnitude fluctuations on the damping of spin\nwaves, which reside at higher energies.\nNext, we present detailed numerical results on the dynamics\nof square and triangular lattice Hubbard models found using\nour scheme. The focus is on deviations from the Heisenberg\nlimit, quantified through finite temperature behaviour of the\ndamping of spin waves.IV . DYNAMICS ON THE SQUARE LATTICE\nIn this section, we first show the spectral maps of D(q;!)\nacross a section of the Brillouin Zone (BZ) for four repre-\nsentative couplings, starting from the Heisenberg limit. Next,\nwe extract the mode energies and magnon damping from the\ndata and plot their variation with respect to Tandqrespec-\ntively. Finally, a comparison of actual lineshapes for a generic\nwavevector q= (\u0019=2;\u0019=2)is featured.A. Spectral maps for varying U=t and temperature\nThe dynamical structure factor maps are exhibited in Fig.5.\nThe top row shows results for a U=t= 20:0Hubbard model\n(the Heisenberg limit) in various temperature regimes. The\nfirst column corresponds to the lowest T. Here, we see sharply\ndefined spin waves, with Goldstone modes at both (0;0)and\n(\u0019;\u0019)and a characteristic antiferromagnetic dispersion. At\nintermediate temperatures ( T=t= 0:05), the bandwidth re-\nduces and the spin waves broaden. On further increase in\nT, the correlations weaken to give a diffusive spectrum, with\nprominent low-energy weight close to (\u0019;\u0019). Ultimately, the\nmomentum dependence is also lost for T=t= 0:15.\nThe lower panels show results on the Hubbard model for\nthree successively lower couplings- strong ( U=t= 10:0), in-\ntermediate (U=t= 6:0) and weak ( U=t= 3:0) respectively.\nAt strong coupling, the behaviour is Heisenberg-like, with\nJeff\u0018t2=U, with small deviations. The spectrum remains\nmostly coherent till T\u0018Jeff, with momentum dependent\nthermal damping. The Goldstone mode at (\u0019;\u0019)survives as a\nbroad low-energy feature till T\u00182Jeff.\nAt intermediate coupling ( U=t= 6:0), the bandwidth in-\ncreases compared to the earlier case and the low Tdisper-\nsion changes in shape. This owes its origin to the emergence\nof multi-spin couplings. There’s also a faint, momentum-\nindependent low-energy band, more clearly visible in a log-\narithmic color scale. This band arises from longitudinal fluc-\ntuations of moments within our scheme, which is controlled\nby the local stiffness. Thermal fluctuations broaden the spin\nwaves gradually, with the dispersion being discernable even at\nT\u00180:1t.\nThe bottom row features weak coupling ( U=t= 3:0) re-\nsults, where the low energy band gains more weight (now vis-\nible on a linear scale) and the bandwidth shortens again. Ther-\nmal effects are stronger, as amplitude fluctuations are more\nprominent here.\n0.0 0.2 0.4 0.6 0.8 1.0\nT/¯J024Ω(q)/˜J(a)\nq=(π/2,π/2)U/t=20.0\nU/t=10.0\nU/t=6.0\nU/t=3.0\n0.0 0.2 0.4 0.6 0.8 1.0\nT/¯J0.00.51.0(Γq−Γ0\nq)/˜J(b) U/t=20.0\nU/t=10.0\nU/t=6.0\nU/t=3.0\nFIG. 6. Fitted dispersions ( \nq) and intrinsic thermal dampings\n(\u0000q\u0000\u00000\nq) as functions of T, extracted from the dynamical spec-\ntra in the square lattice. The temperature axes are scaled by \u0016J=\nJeffjmHFj2, while the frequencies are scaled by ~J=JeffjmHFj\nvalues for the various couplings studied. The dispersions soften\nslowly with increasing T, while one clearly observes the onset of\nnon-Heisenberg behaviour in (b) for lower Uvalues, with large\ndampings showing up much below T=\u0016J= 1.B. Variation of mode energy and damping with T\nFig.6 highlights the evolution of mean frequency ( \nq) and\nthermally induced linewidth ( \u0000q\u0000\u00000\nq) with temperature at\na generic wavevector q= (\u0019=2;\u0019=2). The former mono-\ntonically falls with increasing T, as seen in 6(a). The rate\nof decrease speeds up around successively lower fractions of\n~J=JeffjmHFjon moving to lower couplings. In 6(b), we\nsee that the rise in thermal damping has an initially quadratic\ntrend at large Uand lowT, which then changes to a linear\none one moving to lower couplings, and becomes T\u000bwith\n1< \u000b < 2on raisingT. A somewhat sharper fall is seen\nin the ”onset temperature” for strongly damped behaviour on\nloweringU=t, compared to the trend followed by the mean.\nC. Momentum dependence of energy and damping with\nchanging temperature\nIn Fig.7, we concentrate on the momentum dependence of\nthe same two quantities in the three broad thermal regimes,\ndiscussed before. We firstly see a monotonic behaviour of\nthe peak frequency (at q= (\u0019=2;\u0019=2)), as well as the fi-\nniteTbandwidth (scaled by ~J), on lowering Uin the weakly\ndamped regime. The linewidths here are very small. In the\nstrongly damped regime (green curves), the peak location of\nmean frequency shifts to slightly lower qat weak coupling,\nwhile the peak in magnon damping shifts towards higher q\nvalues. Finally, even in the diffusive regime, a residual mo-\nmentum dependence can be observed in the linewidth plots\n((d)-(f)).\nK Γ\nq0246Ωq/˜J(a)U/t=10.0T/t=0.001\nT/t=0.100\nT/t=0.200\nK Γ\nq0246Ωq/˜J(b)U/t=6.0T/t=0.001\nT/t=0.100\nT/t=0.200\nK Γ\nq0246Ωq/˜J(c)U/t=3.0T/t=0.001\nT/t=0.100\nT/t=0.200\nK Γ\nq012(Γq−Γ0\nq)/˜J(d)U/t=10.0\nK Γ\nq012(Γq−Γ0\nq)/˜J(e)U/t=6.0\nK Γ\nq012(Γq−Γ0\nq)/˜J(f)U/t=3.0\nFIG. 7. Fitted dispersions ( \nq) in (a)-(c) and intrinsic thermal damp-\nings ( \u0000q\u0000\u00000\nq) in (d)-(f), plotted against qalong theK\u0000\u0000trajectory\nin three thermal regimes- (i) weakly damped, (ii) strongly damped\nand (iii) diffusive. The couplings chosen are U=t= 3;6;10and the\nabsolute temperatures are T=t= 0:001;0:1;0:2. We observe a non-\nmonotonicity in the peak frequency, and a mild shift of this peak to\nlowerqon heating up. The bottom row reveals a residual momentum\ndependence of magnon damping even in the diffusive regime.0.0 0.5 1.0 1.5\nω/Wmag024D(q,ω)q=(π/2,π/2)(a)U/t=20.0\nq=(π/2,π/2)(a)U/t=20.0\nq=(π/2,π/2)(a)U/t=20.0\nq=(π/2,π/2)(a)U/t=20.0T/Wmag=0.002\nT/Wmag=0.05\nT/Wmag=0.10\nT/Wmag=0.25\n0.0 0.5 1.0 1.5\nω/Wmag024D(q,ω)(b)U/t=10.0 (b)U/t=10.0 (b)U/t=10.0 (b)U/t=10.0T/Wmag=0.002\nT/Wmag=0.05\nT/Wmag=0.10\nT/Wmag=0.25\n0.0 0.5 1.0 1.5\nω/Wmag024D(q,ω)(c)U/t=6.0 (c)U/t=6.0 (c)U/t=6.0 (c)U/t=6.0T/Wmag=0.002\nT/Wmag=0.05\nT/Wmag=0.10\nT/Wmag=0.25\n0.0 0.5 1.0 1.5\nω/Wmag024D(q,ω)(d)U/t=3.0 (d)U/t=3.0 (d)U/t=3.0 (d)U/t=3.0T/Wmag=0.002\nT/Wmag=0.05\nT/Wmag=0.10\nT/Wmag=0.25FIG. 8. Lineshapes at q= (\u0019=2;\u0019=2)for the Hubbard model (a-d) for U=t= 20;10;6;3respectively. We see a clear deviation from\nHeisenberg-like behaviour in the thermal trends on decreasing coupling. Frequencies and temperatures are scaled by the respective bandwidths\n(Wmag) of the magnetization spectrum.\nD. Lineshapes on the square lattice\nFig.8 highlights the behaviour of a specific high-\nmomentum lineshape (at q= (\u0019=2;\u0019=2)) as a function of\nfrequency for several temperatures. Fig.8(a) is the Heisen-\nberg limit (U=t= 20:0) result. We see sharp mode gradually\nbroadening and developing a tail-like feature upto T=Wmag=\n0:1on increase in T. Finally, a diffusive lineshape emerges at\nhigh temperature ( T=Wmag= 0:25). The plots for U=t=\n10:0shares most of these qualitative features. However, the\nextent of broadening at intermediate temperatures is much\nmore at the same scaled temperatures for U=t= 6:0. There’s\na zero frequency feature for weaker couplings, most promi-\nnent forU=t= 3:0. As discussed already, this is an artifact of\nthe present method and shouldn’t be taken seriously.\nWe next move on to an example of a weakly frustrated sys-\ntem, the Hubbard model on the isotropic triangular lattice.\nThis system has a finiteUc\u00184:5tand features 120\u000eordered\nground states for U>\u00186t. We focus our attention to the lat-\nter coupling regime. First, the spectral maps are exhibited,\nfollowed by lineshapes at two specific momenta.\nV . DYNAMICS ON THE TRIANGULAR LATTICE\nA. Spectral maps for varying U=t and temperature\nFig.9 exhibits the spectral maps for the triangular lattice,\nin the same layout as in the square case. The four cou-plings represent ”Heisenberg” ( U=t= 20:0), ”strong” (U=t=\n10:0), ”intermediate” ( U=t = 8:0) and ”close to the tran-\nsition” (U=t = 6:0) regimes. The non-Heisenberg features\nlike amplitude fluctuations and multi-spin couplings increase\ncolumn-wise.\nThe spectrum in the Heisenberg limit is much more compli-\ncated than in the square case, as the background order corre-\nsponds to q= (2\u0019=3;2\u0019=3)due to the effect of mild frustra-\ntion. We plot the spectrum along \u0000\u0000K\u0000M\u0000\u0000trajectory in\nthe Magnetic Brillouin Zone (MBZ). There are two bands at\na generic wavevector. The magnetic order is fragile, as indi-\ncated by the reduced bandwidth compared to the square case.\nEven on mild increase in T(T=Wmag= 0:2), the multi-band\nstructure becomes fuzzy and large linewidths develop in the\nM\u0000\u0000region. Further increase in Tmakes most of the spec-\ntrum incoherent, apart from the Goldstone mode at the order-\ning wavevector.\nMoving to the lower coupling counterparts, the strong cou-\npling spectrum at low Tis similar to the Heisenberg result,\nwithJeff\u0018t2=U. The dip near Mpoint is more promi-\nnent. Thermal effects are also Heisenberg-like. On decreas-\ning the coupling to U=t= 8:0, the curvature of the \u0000\u0000K\nbranch increases at low T, as does the dip. Amplitude fluctua-\ntions induce more dramatic damping of the spin-wave modes\nat comparable temperatures. Finally, close to the Mott transi-\ntion (U=t= 6:0), even the low- Tspectrum is incoherent. Soft\nmodes are visible in a wide region of momentum space. In\nAppendix D, we show the gradual evolution of the low tem-\nperature spectrum as one approaches the Mott transition, stay-\ning within the 120\u000eordered family of states.Γ KM Γ\n0.00.40.8ω/tT/t=0.001\nΓ KM Γ\nT/t=0.020\nΓ KM Γ\nT/t=0.050\nΓ KM Γ\nT/t=0.100\n0.0000.0020.004\n(a)U/t=20.0\nΓ KM Γ\n0.00.40.8ω/tT/t=0.001\nΓ KM Γ\nT/t=0.020\nΓ KM Γ\nT/t=0.050\nΓ KM Γ\nT/t=0.100\n0.0000.0020.004\n(b)U/t=10.0\nΓ KM Γ\n0.00.40.8ω/tT/t=0.001\nΓ KM Γ\nT/t=0.020\nΓ KM Γ\nT/t=0.050\nΓ KM Γ\nT/t=0.100\n0.0000.0020.004\n(c)U/t=8.0\nΓ KM Γ\n0.00.40.8ω/tT/t=0.001\nΓ KM Γ\nT/t=0.020\nΓ KM Γ\nT/t=0.050\nΓ KM Γ\nT/t=0.100\n0.0000.0020.004\n(d)U/t=6.0FIG. 9. Power spectrum of magnetization field D(q;!)for the Hubbard model on the triangular lattice for U=t= 20;10;8;6respectively.\nThe trajectory chosen in Brillouin Zone is \u0000\u0000K\u0000M\u0000\u0000. Temperatures are scaled by electron hopping t. Again, we observe a similarity\nof the strong coupling Hubbard spectrum with the Heisenberg case. The lower branch between \u0000\u0000Kin the Heisenberg limit develops a\nprominent dip for lower Uvalues. The thermal dampings are stronger on moving to weaker couplings compared to the square case.\nB. Lineshapes on the triangular lattice\nFig.10 elaborates the comparison of detailed lineshapes of\nthe Hubbard model with those of the Heisenberg in the tri-\nangular case. The two rows feature lineshapes for q=\n(\u0019=3;\u0019=3)andq= (\u0019;\u0019)respectively. Once again, the fre-\nquencies and temperatures are scaled with respect to the low\nTbandwidth. The leftmost columns represent the Heisen-\nberg limit (U=t = 20 ) results. We observe that for both\nwavevectors, a bimodal spectrum is obtained at low T, which\ngradually broadens on increasing temperature. Even upto\nT=Wmag\u00180:1, the spectra retain two distinct peaks.\nMoving to the Hubbard results, we see that the strong cou-\npling results ( U=t= 10:0) bear a striking resemblance to theHeisenberg case, as expected. However, even at moderately\nhigh coupling ( U=t = 8:0), the thermal damping results in\ndiffusive behaviour even at T=Wmag\u00180:05. On going closer\nto the Mott transition ( U=t= 6:0), even the low Tlineshapes\nsignificantly change their character, with prominent zero fre-\nquency weights cropping up in both the wavevectors. Diffu-\nsive behaviour sets in immediately on increasing T.\nVI. DISCUSSION\nWe have tried to organise the results in this paper in terms\nof three dynamical regimes and then quantified the detailed\nresponse on these regimes in terms of the lineshape, the mode0.0 0.5 1.0 1.5\nω/Wmag0246D(q,ω)\nq=(π/3,π/3)(a)\nU/t=20.0\nq=(π/3,π/3)(a)\nU/t=20.0\nq=(π/3,π/3)(a)\nU/t=20.0\nq=(π/3,π/3)(a)\nU/t=20.0T/Wmag=0.003\nT/Wmag=0.03\nT/Wmag=0.06\nT/Wmag=0.09\n0.0 0.5 1.0 1.5\nω/Wmag0246D(q,ω)(b)\nU/t=10.0(b)\nU/t=10.0(b)\nU/t=10.0(b)\nU/t=10.0\n0.0 0.5 1.0 1.5\nω/Wmag0246D(q,ω)(c)\nU/t=8.0(c)\nU/t=8.0(c)\nU/t=8.0(c)\nU/t=8.0\n0.0 0.5 1.0 1.5\nω/Wmag0246D(q,ω)(d)\nU/t=6.0(d)\nU/t=6.0(d)\nU/t=6.0(d)\nU/t=6.0\n0.0 0.5 1.0 1.5\nω/Wmag02468D(q,ω)\nq=(π,π)(e)\nU/t=20.0\nq=(π,π)(e)\nU/t=20.0\nq=(π,π)(e)\nU/t=20.0\nq=(π,π)(e)\nU/t=20.0\n0.0 0.5 1.0 1.5\nω/Wmag02468D(q,ω)(f)\nU/t=10.0(f)\nU/t=10.0(f)\nU/t=10.0(f)\nU/t=10.0\n0.0 0.5 1.0 1.5\nω/Wmag02468D(q,ω)(g)\nU/t=8.0(g)\nU/t=8.0(g)\nU/t=8.0(g)\nU/t=8.0\n0.0 0.5 1.0 1.5\nω/Wmag02468D(q,ω)(h)\nU/t=6.0(h)\nU/t=6.0(h)\nU/t=6.0(h)\nU/t=6.0FIG. 10. Triangular lattice: lineshapes at q= (\u0019=3;\u0019=3)(a-d) and q= (\u0019;\u0019)(e-h) for the Hubbard model for U=t= 20;10;6;3respec-\ntively. We see a clear deviation from Heisenberg-like behaviour in the thermal trends on decreasing coupling. Frequencies and temperatures\nare scaled by the respective bandwidths ( Wmag) of the magnetization spectrum.\nenergy and the damping. In what follows we shall try to\nprovide the analytic basis of some of the results seen in the\nLangevin simulations, also point out some of the limitations\nof our approach. The main effect observed in this paper is the\nenhancement of thermal damping of magnons as one moves\naway from the Heisenberg limit. We argue this effect maybe\nminimally captured by a simpler classical toy model, which\nallows for amplitude fluctuations and approaches the classical\nHeisenberg limit upon tuning a single parameter.\nA. Classification of non-Heisenberg effects at finite U=t\nWe first comment that there exists a two-particle contin-\nuum of excitations, originating from particle-hole processes,\nmissed out by the present scheme. This is accessed by a quan-\ntum RPA calculation done on the mean-field ordered states on\nsquare and triangular geometries. However, this continuum\nis energetically well separated from the spin wave spectrum\nat strong coupling and hence don’t influence each other at the\ntemperature scales of interest. But, this argument breaks down\nat weak coupling (e.g. U=t= 3:0), where indeed there’s ap-\npreciable mixing even at low temperature, and our dynamical\nresults are indeed imperfect, except near special, symmetry-\nprotected wavevectors like (0;0)or(\u0019;\u0019). In what follows,\nwe only underline the non-Heisenberg features observed in\nthe spin wave part.\nIn the full Hubbard problem, at intermediate U=t values,\nthere are two main non-Heisenberg features- (i) the ordered\nstate and the low Tdispersion are modified, and (ii) the mo-\nment magnitudes are no longer fixed but are reduced at lowTand also fluctuate thermally. We’ll discuss the impact of\nthe second class of features in detail in the upcoming subsec-\ntions. To obtain the effects of the first class systematically\nat lowT, one does an expansion about the mean-field state,\nwhich may (as in the square lattice case) or may not (as in the\ntriangular one) have the same ordering as in the Heisenberg\nlimit, with a reduced moment value. The effective Hamilto-\nnian for mi’s, obtained through integrating out the electrons\nperturbatively in t=U, now involves longer range, multi-spin\nterms58,59. The couplings are decided by the electronic band\nstructure on the mean field state. However, we should remem-\nber that our model is composed of classical moments. Hence,\nthe coefficients don’t match with those in the actual quantum\nmodel.\nThese coefficients depend non-trivially on U=t. As a result,\nthe crossover lines between the thermal regimes are modified\nwith respect to the Heisenberg case.\nTo lowest order, a linear theory maybe written down for the\nfluctuations, which has an analytic solution. We’ll discuss this\nsubsequently in subsection C. The contribution to the effective\nfield (@\n@mi) coming from the leading non-Heisenberg term,\nexpanded uptoO(\u000emi)in fluctuations, looks like-\nX\nijklKijkl(m0\nj(m0\nk:\u000eml+\u000emk:m0\nl) +\u000emj(m0\nk:m0\nl))\nThe coupling Kijkl has a lowest order contribution of\nO(t4=U3), as maybe motivated from a perturbative argument,\nstarting from the strong coupling limit. One now puts this\nexpression back in the first and second terms of Eq.1, along\nwith the Heisenberg term 4t2=UP\nmjand the stiffness(a)\n0.0 0.2 0.4\nω/t00.10.2D(q,ω)\nq=(π/2,π/2)\n(b)q=(π/2,π/2)\n(b)q=(π/2,π/2)\n(b)q=(π/2,π/2)\n(b)T/t=0.001\nT/t=0.05\nT/t=0.10\nT/t=0.15FIG. 11. (a): Fitted standard deviations ( \u0001jmj) fromP(jmj)distri-\nbutions, plotted against temperature for three couplings in the square\nlattice case. Blue open circles denote actual data points, while solid\nlines are fits using a square root function. The trends indicate the\nincreasing importance of amplitude fluctuations at weaker couplings\nand a square root dependence, expected of a ”soft spin” Heisenberg\nmodel. (b): Lineshapes at q= (\u0019=2;\u0019=2)for the amplitude fluc-\ntuations atU=t= 6:0, indicating a diffusive mode centered at zero\nenergy.\ncontribution ( U(jmij\u00001=2)2), and solves the resulting equa-\ntion via Fourier transformation. From the poles of the ensuing\npower spectrum, one gets the low Tdispersion, which con-\ntains the leading non-Heisenberg effects.\nB. Quantifying amplitude fluctuations\nIn this subsection, we quantify the extent and intrinsic dy-\nnamical signature of fluctuations in the moment magnitude,\nbefore launching into the construction of an effective model to\ndescribe them. Fig.11(a) focusses on the longitudinal fluctua-\ntions of the magnetic moments. These are, of course, frozen in\nthe Heisenberg limit. We fit the P(jmj)distributions, shown\nearlier in Fig.1, to Gaussians and extracted the correspond-\ning standard deviations. These are plotted as functions of\ntemperature for various coupling values in the square lattice\ncase. In a ”soft spin” Heisenberg model, where the intersite\nterm is Heisenberg but longitudinal fluctuations are allowed,\nthe behaviour should be \u0018p\nT. However, we observe devia-\ntions from this trend at lower Uvalues. The coefficient of the\nsquare root fits is exactly 1=p\nUat strong coupling. Even at\nweaker couplings, the deviations are small. Hence, the ampli-\ntude fluctuations can be effectively captured by a local term\nHamp=P\niU(jmij\u00001=2)2.\nThe spectral signature of these fluctuations is a diffusive\nmode centered at zero frequency, shown in Fig.11(b). This\nis obvious from the locality of Hamp, which deactivates the\ntorque term in Eq.1. The width is regulated by \r. Interest-\ningly, the weight at low frequency shows a non-monotonic be-\nhaviour with T. This behaviour, however, doesn’t capture the\ntrue physics of the amplitude mode, which should have a sig-\nnature at!\u0018U. For that, one needs to incorporate quantum\nfluctuations of the magnetization field in the effective equation\nof motion. We’ll discuss this briefly in subsection E.C. Construction of an effective model\nIn the following, we describe the construction of an effec-\ntive ”classical moment” model, which essentially captures the\nqualitative features of the full Hubbard model calculation at\nallU=t. The model reads-\nHeff=JeffX\nmi:mj+Keff\n2X\ni(jmij\u0000jmHFj)2\n\u00002JeffX\nijmij2(8)\nThe first term encapsulates an ”effective” nearest neigh-\nbour exchange between the local moments mi, the second\nterm is an amplitude stiffness which regulates the thermally\ninduced fluctuations of the moment magnitude and the third\nterm is a counterterm that fixes the low Tmoment size to\nexactlyjmHFj, the Hartree-Fock value. The parameters\nJeffandKeffare extracted, respectively, from the low T\nRPA spin wave velocity (fitted to a nearest-neighbour Heisen-\nberg model) and the “curvature” of the Hartree-Fock energy,\n@2EHF=@m2. Fig.12 illustrates the behaviour of the above\nparameters for various U=tvalues.\nThe model is constructed based on a strong coupling expan-\nsion argument. At large U=t, the Hubbard model reduces to a\nspin model of the following form-\nHeff=Hloc+Hcoup\nHloc=U(jmij\u00001\n2)2+:::\nHcoup=J2X\nmi:mj+J4X\nijklf[mi;:::ml] +:::(9)\nHlocis basically the HF energy in terms of moment mag-\nnitude, expanded to quadratic order in the deviations. Hcoup\nreduces to the first term with J2= 4t2=UasU=t!1 . This\ncan be shown explicitly by expanding about the U=t!1\nlocal limit. On including further terms in the expansion (sub-\nleading int=U), one gets longer range, multi-spin couplings.\nWe lump the effect of all non local terms into an equiva-\nlent nearest neighbour coupling Jeffand retain the local am-\nplitude stiffness in our simplified model. The strong cou-\npling limit is also correctly recovered as Jeff!4t2=U,\nKeff!2Uandjmij! 1=2asU=t!1 in our model.\nThe result of the aforesaid construction is that it reproduces\nthe thermal physics of the classical Heisenberg model at all\nT=t for largeU=t. At weaker couplings, the T= 0 state is\ncaptured with the correct (mean-field) moment value and the\nlow-energy spin wave excitations (in particular their velocity\nvSW) are also correctly captured by construction.\nAs regards the results obtained using the above model, we\nfirst compare the static indicators, in particular the low tem-\nperature structure factor S(\u0019;\u0019)between the original Hub-\nbard model and this effective model at various U=t values.\nTo minimize parametric dependencies, the comparison was\ndone using the Monte Carlo technique, elaborated in Ap-\npendix B. The results for the correlation temperatures ( Tcorr)0 5 10 15 20\nU/t0.00.20.40.60.81.0Jeff/t\n0 5 10 15 20\nU/t051015202EHF\nm2\n0 5 10 15 20\nU/t0.00.10.20.30.40.5|mHF|FIG. 12. The effective exchange Jeff, second derivative of Hartree-Fock energy with respect to moment magnitude (@2EHF\n@m2), which is\nproportional to the amplitude stiffness Keffand Hartree-Fock moment value ( jmHFj), as determined from HF and RPA calculations, for\nvariousU=t values on the square lattice Hubbard model.\nare shown in Fig.13(a). The basic observation is that the non-\nmonotonicity of this scale as a function of U=t, is succes-\nfully captured by the effective model, albeit the maximum is\nslightly shifted to higher U=t. TheTcorr within the effec-\ntive model scales roughly as \u0018jmj2\nHFJefffor largeU=t, but\ncrashes faster at lower Udue to the effect of Keff.\nTo further simplify the three parameter effective model of\nEq.8, we scaled the effective couplings JeffandKeffby the\nmoment valuejmHFjappropriately and reduced Eq.8 to an\n”equivalent one-parameter” model of the following form-\nH1par=JX\nmi:mj+K\n2X\ni(jmij\u00001)2\u00002JX\nijmij2\n(10)\nwhereJis set to 1 and K=J is varied to mimic the be-\nhaviour of the earlier model. The moment magnitudes fluc-\ntuate about unity for all couplings in this model. The results\nobtained using Eq.10 agree quantitatively with those originat-\ning from Eq.8, which is formally equivalent.\n0 5 10 15 20U/t0.000.040.080.12Tcorr\nHubbard\nEffective model (a)\n11\n012\n30 10 20diffusive\nstrongly damped\nweakly dampedT/JT/J\nK/J(b)\nFIG. 13. Left: Comparison of the correlation temperatures ( Tcorr),\nextracted from the respective structure factors S(\u0019;\u0019)of the full\nHubbard (blue curves) and effective model (green curves) obtained\nusing Monte Carlo (MC) method described in the paper. One ob-\nserves that the non-monotonicity is well captured by the former\nmodel. Right: Thermal regimes obtained using Langevin dynam-\nics of the effective model (Eq.10) with varying K=J . A qualitative\nresemblance with the square lattice Hubbard results (Fig.4(c)) is ap-\nparent.Next, we move to the dynamics. The thermal regimes in\nthe dynamics of the effective model (Eq.10) are depicted in\nFig.13(b). They qualitatively resemble the scaled phase dia-\ngram (Fig.4(c)) of the full Hubbard problem. This corrobo-\nrates the usefulness of the effective model, not only to under-\nstand the static properties, but also dynamical features.\nAfter comparing the gross features of the dynamics, we\nalso examined whether the same effective model (Eq.10) can\nmimic the changing low Tbehaviour of the damping in the\nfull Hubbard problem. We extracted the excess damping at fi-\nniteTand plotted it for the generic q= (\u0019=2;\u0019=2)as a func-\ntion ofT=J. One finds that empirically one may fit this excess\ndamping \u0000q\u0000\u00000\nqto a polynomial of the form \u000bT+\fT2, with\nthe coefficients depending on K=J .\nUpon examining the fitting parameters, one observes that\nthe\u000b/1=Kat lowKand decreases to zero in the fixed\nmoment limit ( K=J! 1 ). The quadratic coefficient \f\nis roughly constant at large K. The results are shown in\nFig.14(a) and 14(c). Such features are also observed qualita-\ntively in the full Hubbard calculation, where the normalizing\nenergy scale is chosen as Jeff= 4t2=U. These results are\nshown in Fig.14(b).\nWe next try to find an a posteriori justification for the ris-\ning linear coefficient and rise in damping as on reduces the\namplitude stiffness by imagining undamped spin wave modes\ngetting affected by amplitude disorder. If one is at sufficiently\nlow temperature, the equation of motion (Eq.1) maybe lin-\nearized in terms of deviation from the ground state configura-\ntion. On the square lattice, for instance, one simply expands\nthemias\nmi=m0\ni+\u000emi\nm0\ni= (\u00001)ix+iy^z (11)\nKeeping upto the linear order in fluctuations \u000emigives us an\nanalytically solvable starting point. The effective equation is-\nd\u000emi\ndt+J(m0\ni\u0002X\n\u000emj\u0000X\nm0\nj\u0002\u000emi)\n+\r(JX\n\u000emj+KX\ni(\u00001)i\u000emz\ni^z) =~\u0018i(12)\nThe transverse and longitudinal modes gets decoupled at\nthis order. On Fourier transforming this equation and solv-0.0 0.1 0.2 0.3\nT/J0.00.10.20.3(Γq−Γ0\nq)/J(a)K/J=1000\nK/J=100\nK/J=50\nK/J=20\nK/J=10\n0.00 0.05 0.10\nT/Jeff0.00.10.2(Γq−Γ0\nq)/Jeff(b)U/t=20.0\nU/t=15.0\nU/t=10.0\nU/t=6.0\n(c)FIG. 14. (a): The excess thermal damping ( \u0000q\u0000\u00000\nq), plotted as a function of T=J for various stiffness values in the approximate J\u0000K\nmodel for q= (\u0019=2;\u0019=2). One notes that the low Tlinear regime shrinks on increasing K=J and the behaviour turns to parabolic. (b): The\nsame quantity extracted from the full Hubbard model calculation at various U=t values. Similar qualitative features are observed. (c): Plot of\nfitting parameters \u000band\ffor the approximate model, showing the quadratic to linear crossover on decreasing K=J .\ning for the power spectrum, one finds the usual dispersion of\nthe antiferromagnetic classical Heisenberg model, while the\ndamping of transverse spin wave modes is limited by \rJ. The\nlongitudinal modes generally give rise to a diffusive lineshape,\nand freeze for K=J! 1 . On top of this low tempera-\nture, purely transverse theory, one may switch-on amplitude\nfluctuations perturbatively. The width of these fluctuations is\n/1=K. On treating them as static, uncorrelated disorder, they\ncause the eigenmodes of the linear theory to scatter. In the\nlowest order Born approximation, this generates a self-energy,\nwhose imaginary part translates to an additional contribution\nto the magnon linewidth. This has a prefactor Tcoming from\nthe propagator of transverse fluctuations. In the static limit,\nthe coefficient of this correction is thus proportional to T=K .\nHence asKis reduced from infinity, the linear Tcorrection\nto spin wave damping increases as 1=K, as is seen in the nu-\nmerical data.\nThe aforesaid argument doesn’t include the effect of non-\nlinear interactions among the transverse fluctuations. To eval-\nuate their effect, one expands upto second order in the devia-\ntion field, which generates a \u000emq\u0002\u000emq0contribution in the\nequation of motion. If one substitutes the lowest order solu-\ntion in this and averages over the noise, this correction term\nvanishes, owing to the fact that the noise is uncorrelated be-\ntween different Cartesian axes. Hence, no O(T)contribution\nis found for the damping of transverse fluctuations. The low-\nest order correction is of ( O(T2)), as is found in the extensive\nliterature85,86. This becomes the leading term when ampli-\ntude fluctuations are completely restricted (in the K=J!1\nlimit).\nD. Computational issues for frustrated systems\nOne would want to ultimately apply this formalism to\nstudy the Hubbard model on fully frustrated geometries (e.g.\nKagome in 2d and pyrochlore in 3d). The rich spin dynam-\nics, with the moment softening and multipsin coupling effects\npresent beyond the Heisenberg limit, should be accessible at\nfinite temperature. However, there are some tough compu-tational difficulties associated with this attempt. Briefly, the\nissues are-\n• Extracting even the static properties correctly (vis-a-vis\nMonte Carlo) requires much longer run lengths com-\npared to the square or triangular case. This occurs due\nto the rugged free energy landscape associated with the\nproblem. Novel strategies, involving simultaneous up-\ndation of multiple moments, ameliorate the situation in\nspecific cases.\n• The numerical implementation of the Langevin dy-\nnamics scheme, using Suzuki-Trotter decomposition,\nbreaks down when the systematic torque on a site be-\ncomes identically zero. This happens, for instance, for\nthe Heisenberg model on the 2d Kagome lattice. Hence,\na more complicated discretization strategy is called for.\nE. Adiabaticity and thermal noise\n1. The adiabatic assumption\nOur approach has assumed that the characteristic timescale\nfor magnetic fluctuations is much greater than electronic\ntimescales, in analogy with the electron-phonon problem87.\nIn such a situation (i) the electronic energy depends only on\nthe instantaneous magnetic configuration, and (ii) the leading\ncontribution to electronic correlators can be computed without\ninvoking retardation effects. This argument holds good in the\nstrong coupling regime, where the magnetic fluctuations oper-\nate on a scale of Jeff\u0018t2=Uand the electrons are gapped at\na scale\u0018U. However, as U=treduces, the former scale rises\nand the latter diminishes due to closing of the gap. So, the\nargument isn’t very good. We also comment that the auxiliary\nfield correlator, which we computed, reproduces the essential\nfeatures of the real spin-spin correlator h\u001bi(t):\u001bj(t0)i, mea-\nsured in INS experiments as long as the adiabaticity assump-\ntion holds good. This happens because the auxiliary field dy-\nnamics basically follows the \u001bifield, with the distinction thatits magnitude is not strictly bounded between 0 and 1. As a\nresult, the respective intensities are different.\n2. The noise driving the dynamics\nThe present method for accessing spin dynamics excludes\nthe effect of quantum fluctuations. This firstly results in the\nunphysical freezing of the moments at T= 0 and makes the\nmethod unable to access the ground state magnon spectrum.\nFurthermore, this feature limits the viability of the scheme at\nlow temperatures for frustrated geometries, where order by\ndisorder phenomena are observed. To remedy this, the noise\nhas to be consistently generated with respect to the polariz-\nability of the problem, which itself will depend on the mi(t)\ntrajectories.\nUsing a Keldysh formulation of the original Hubbard\nmodel, and decomposing the interaction term using an aux-\niliary vector field mi, we may subsequently assume this field\nto be slow with respect to the electrons. This enables one to\nwrite an effective equation of motion for mi;clof the follow-\ning form-\n=h\nTr\u0010\n^GK\nii(t;t)~ \u001b\u0011i\n=mi;cl(t) +~\u0018i(t)\nh\u0018a\ni(t)\u0018b\nj(t0)i=h\n^\u0005K(t;t0)iab\nij(13)\nHereGKand\u0005Kare the Keldysh Green’s function and\n(spin-dependent) polarizability of the electrons respectively.\nIn the adiabatic limit, each of these maybe expanded in a\nKramers-Moyal series88. On assuming that the coefficients\ndon’t have any spatial dependence and the temperature is high\nenough compared to characteristic frequency scale of these,\none arrives at a much simpler equation of the LLG form,\nwhich upon neglecting certain multiplicative noise terms re-\nduces to Eq.1.\nTo include the effect of quantum fluctuations, the high\nTapproximations done on the coefficients of the Kramers-\nMoyal expansion need to be relaxed. Basically, if the temper-\nature approaches the energy scale of two-particle excitations,\nthe memory-less assumption on the noise becomes unjustified.\nVII. CONCLUSIONS\nWe’ve studied the dynamics of magnetic moments in the\nMott insulating phase of the half-filled Hubbard model on\nsquare and triangular lattice geometries, using a Langevin dy-\nnamics based real time technique. The method reproduces\nknown results on the Heisenberg model in the strong cou-\npling limit, and the RPA based low-energy dispersion at low\nTfaithfully. We observe three broad regimes in the dynamics-\n(i) weakly damped, where spin waves are dispersive and\ndampings are small, (ii) strongly damped, where one can see\nsignificant broadening due to mode coupling, but the disper-\nsive character survives, and (iii) diffusive, where the mode fre-\nquencies collapse to zero and the dampings span the full band-width. The main results are twofold- (a) we obtain the devi-\nation of low temperature dispersion from the Heisenberg re-\nsults, and (b) we observe the onset of the thermal crossovers at\nsignificantly lower values of T=Jeff, compared to the Heisen-\nberg case. One also captures the effect of mild geometric frus-\ntration on the mode damping, on going from the square to the\ntriangle. The method maybe applied to study equilibrium dy-\nnamics in fully frustrated lattices (e.g. pyrochlore) in near\nfuture.\nWe acknowledge use of the High Performance Computing\nFacility at HRI.\nAPPENDIX A: NUMERICAL DETAILS OF THE LANGEVIN\nSCHEME\nAll of our Langevin dynamics simulations are done by dis-\ncretizing Eq.1 in real time and implemented in a Cartesian\ncoordinate scheme. The particular technique used to solve the\nequations is the Euler-Maruyama method89. The time step is\nchosen to be 0:01\u001cmag. At each step, the derivatives appearing\nin the RHS of Eq.1 are computed through exact diagonaliza-\ntion of the electronic problem. The derivative@hHSFi\n@mifor our\nmodel is just U(mi\u0000h\u001bii). Typically, the simulations are ran\nfor3\u0002106steps. We gave parallel runs for each temperature\npoint, with the Hartree-Fock (HF) state as the initial condition\nfor each value of the Hubbard coupling. The lattice size for\nthe results shown for both the square and triangular cases is\n18\u000218.\nAPPENDIX B: NUMERICAL DETAILS OF THE MONTE\nCARLO SCHEME\nTo benchmark the static properties obtained via the\nLangevin scheme, we used a competing Monte Carlo (MC)\nmethod. One first writes the Hubbard model in the Matsub-\nara formalism and then decouples the quartic interaction in\n2 7 12 17 22\nU/t0.20.40.60.81.0\nvSW(LD)/tavSW(RPA)/ta\nFIG. 15. Comparison of spin wave velocities ( vSW) computed using\nour Langevin dynamics (LD) technique and the random phase ap-\nproximation (RPA) on the square lattice. We observe similar trends\nand quantitatively lower values in LD compared to RPA. This is due\nto our assumption of classical spins.Γ KM Γ\n0.00.40.8ω/tU/t=10.0\nΓ KM Γ\nU/t=9.0\nΓ KM Γ\nU/t=8.0\nΓ KM Γ\nU/t=7.0\n0.0000.0020.004FIG. 16. Low temperature spectra on the triangular lattice on gradually lowering U, approaching the Mott transition. All the couplings shown\ndisplay order at zero temperature, with progressively smaller moment magnitudes. One observes a dramatic softening of modes along the\n\u0000\u0000Kregion in momentum space, albeit with a robust magnon bandwidth.\nterms of the mifield. Next, only the zero Matsubara mode\nof this field is retained, assuming T>\u0018Jeffand temporal\nfluctuations of the field can be neglected. However, the ther-\nmal fluctuations and the associated spatial correlations are\ntreated non-perturbatively. This enables one to write an ef-\nfective Hamiltonian for the auxiliary fields as-\nHeff=\u00001\n\flogTre\u0000\fHel+UX\nijmij2(14)\nHel=\u0000X\n\u001btij(cy\ni\u001bcj\u001b+h:c:)\u0000UX\nimi:\u001bi\nFinally, configurations of the mifield are sampled using\nP(mi) =Trccye\u0000\fHeffas the sampling weight. These con-\nfigurations are used for computing static structure factors and\ndistribution of moment magnitudes, defined in Eq.5 and Eq.7\nrespectively and shown in Fig.1(a) and 1(b). We also mention\nthat the correlation temperatures in 2(a) are size-dependent,\nand will ultimately collapse logarithmically with system size.\nHowever, we’ve still compared the MC and Langevin answers\nfor the same system size to ensure that the latter method faith-\nfully reproduces the static properties.\nAPPENDIX C: COMPARISON OF LOW TEMPERATURE\nSPECTRUM WITH RPA\nWe compare the low temperature spectra obtained using\nour technique with the standard spin wave theory (RPA) re-\nsults for the square lattice in Fig.15. The spin wave velocities\nare quoted from the work of Singh et. al.41. One observes\na fair agreement in terms of the trends. The RPA values are\nslightly higher. We ascribe this discrepancy to our assump-\ntion of classical magnetic moments. However, since our main\nfocus is on the finite temperature dynamics, the quantitative\nmismatch isn’t very important. The agreement improves as\none approaches the Heisenberg limit.\nAPPENDIX D: APPROACHING THE MOTT TRANSITION\nIn the triangular lattice, there’s a finite Uc\u00184:5tfor\nthe Mott transition. Close to the transition, one observescomplex large-period order19. However, staying within the\n120\u000eordered state (restricting ourselves to large enough U=t\nvalues where the ground state is the former), we observe sig-\nnatures of proximity to Ucin the spectrum. Fig.16 shows a\nmarked softening of magnetic modes along the \u0000\u0000Ktrajec-\ntory and a gradual linear trend of the dispersion along K\u0000M\nas the coupling is lowered. We’ve already shown the spectra at\nU=t= 6in the main text, which is the lowest coupling we’ve\nexplored within the 120\u000eordered family. Ideally, the complex\ndynamics in the vicinity of the transition should also be cap-\nturable using our strategy, but requires considerably more nu-\nmerical effort, as one needs to do a thermal annealing to even\nfix the initial state for the dynamics.\nAPPENDIX E: REAL TIME DYNAMICS\nIn Fig.17, we show the trajectory of the real part of mz\nqfor\na generic wavevector, q= (\u0019=2;\u0019=2), in real time for the\nthree representative regimes- (i) weakly damped, (ii) strongly\ndamped and (iii) diffusive. These are results for the square\nlattice Hubbard model at U=t = 10:0. We’ve also scaled\nthe y-axis byp\nT, to gauge out the dominant part of am-\nplitude fluctuations. At the lowest T, we see oscillatory be-\nhaviour, modified by weak noise. The characteristic timescale\n0 30 60\nt0.10\n0.05\n0.000.05Re(mz\nq)(t)//radicalbig\nTT/t=0.001\nq=(π/2,π/2)\n0 30 60\ntT/t=0.060\n0 30 60\ntT/t=0.200\nFIG. 17. 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Platen, Numerical Solution of Stochastic Dif-\nferential Equations , Springer, Berlin (1992)." }, { "title": "1206.3468v1.Comment_on__Anisotropic_Critical_Magnetic_Fluctuations_in_the_Ferromagnetic_Superconductor_UCoGe_.pdf", "content": "arXiv:1206.3468v1 [cond-mat.str-el] 15 Jun 2012Comment on ”Anisotropic Critical Magnetic Fluctuations in the Ferromagnetic\nSuperconductor UCoGe”\nV.P.Mineev and V.P.Michal\nService de Physique Statistique, Magn´ etisme et Supracond uctivit´ e,\nInstitut Nanosciences et Cryog´ enie, UMR-E CEA/UJF-Greno ble1, F-38054 Grenoble, France\n(Dated: August 16, 2018)\nPACS numbers: 75.40.Gb, 74.70.Tx, 75.50.Cc\nThe results of neutron scattering measurements of\nmagnetic fluctuations in weakly ferromagnetic supercon-\nductor UCoGe have been reported in a recent Letter1.\nThere was observed finite attenuation of excitations at\nzero wave vector earlier found also in another related fer-\nromagnet UGe 22that has been interpreted by the au-\nthors as ”strong non-Landau damping of excitations” .\nHere we point out that revealed phenomenon can be\ntreated as the Landau damping corresponding to the in-\ntersection of Fermi surfaces relating to different bands.\nThe intensity of neutron scattering is proportional to\nthe imaginary part of susceptibility (see for instance2).\nIn the isotropic case the susceptibility is scalar and its\nimaginary part is given by\nχ′′(q,ω)\nω=χ(q)Γq\nω2+Γ2q, (1)\nχ(q)∝χpk2\nF\nξ−2+q2, (2)\nHere,χpis the Pauli susceptibility and the line width is\ndetermined by equality\nΓqχ(q) =χpω(q) (3)\nwhereω(q) is the Landau damping frequency.\nTakinginto accountthe possibilityofbandintersection\none can define the Landau damping frequency through\nthe imaginary part of bubble diagram with one electron\nGreen’s functions\nN0ω\nωνν′(q)∝ImT/summationdisplay\nk,ωnGν(k,ωn)Gν′(k+q,ωn+νm)|iνm→ω+i0,\n(4)whereνandν′are the band indices and N0is average\ndensity of states at the Fermi surface. For the intraband\ncaseν=ν′the Landau damping frequency ω(q)≈vFq\nvanishes linearly at q→0. For the case when the Fermi\nsurfaces of two bands intersect each other along a line l\nthe Landau damping at q= 0 acquires finite value3\nN0\nω12(q= 0)≈/contintegraldisplaydl\n(2π)3|v1×v2|. (5)\nHere vectors v1andv2are the Fermi velocities on the\nFermi sheets 1 and 2 at point klat linel.\nAtq= 0 one can estimate the product (3) as fol-\nlows Γ qχ(q)|q=0=χpω12(q)|q=0≈χpεF. Numerically\nthis value is of the order of 10−2Kthat is in correspon-\ndence with the experimentally found values of 0 .7µeV\nand 0.4µeV in UGe 22and UCoGe1correspondingly.\nThe intersection of the different band Fermi surfaces\ncan be established from the ab initio calculations. The\nlatter have been already performed4–7although without\nspecial attention to this problem.\nWe have presented the potential explanation of nonva-\nnishing at q= 0 Landau damping measured experimen-\ntally in ferromagnetic compounds UGe 22and UCoGe1\nbased on possible intersection of the Fermi sheets cor-\nresponding different bands. Quite large and nonvanish-\ning atq= 0 value of the Landau damping means that\nthe amplitude of pairing interaction is determined by fre-\nquency independent susceptibility. The latter of course\nshould not be taken as it is in the isotropic case given by\nequation (2).\n1C. Stock, D. A. Sokolov, P. Bourges, P. H. Tobash, K. God-\nfryk, F. Ronning, E. D. Bauer, K. C. Rule, and A. D. Hux-\nley, Phys. Rev. Lett. 107, 187202 (2011).\n2A.D. Huxley, S. Raymond, and E. Ressouche, Phys. Rev.\nLett.91, 207201 (2003).\n3M. F. Smith, Phys. Rev. B 74, 172403 (2006).\n4M. Biasini, R. Troc, Phys. Rev. B 68, 245118 (2003).5M. Divis Physica B 403, 2505 (2008).\n6P. de la Mora and O. Navarro, J. Phys.: Condens. Matter\n20, 285221 (2008).\n7M. Samsel-Czekala, S. Elgazzar, P.M. Oppeneer, E. Tallik,\nW. Walerczyk and R.Troc, J. Phys.: Condens. Matter 22,\n015503 (2010)." }, { "title": "1610.02799v1.A_Five_Freedom_Active_Damping_and_Alignment_Device_Used_in_the_Joule_Balance.pdf", "content": "IEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 1\nA Five-Freedom Active Damping and Alignment\nDevice Used in the Joule Balance\nJinxin Xu, Qiang You, Zhonghua Zhang, Zhengkun Li and Shisong Li, Member, IEEE\nAbstract —Damping devices are necessary for sup-\npressing the undesired coil motions in the watt/joule\nbalance. In this paper, an active electromagnetic damp-\ning device, located outside the main magnet, is in-\ntroduced in the joule balance project. The presented\ndamping device can be used in both dynamic and static\nmeasurement modes. With the feedback from a detec-\ntion system, five degrees of freedom of the coil, i.e. the\nhorizontal displacement x,yand the rotation angles \u0012x,\n\u0012y,\u0012z, can be controlled by the active damping device.\nHence, two functions, i.e. suppressing the undesired coil\nmotions and reducing the misalignment error, can be\nrealized with this active damping device. The principle,\nconstruction and performance of the proposed active\ndamping device are presented.\nIndex Terms —watt balance, joule balance, the\nPlanck constant, electromagnetic damping, misalign-\nment error.\nI. Introduction\nSeveral national metrology institutes (NMIs) across the\nworld are working hard on redefining one of the seven\nSI base units, the kilogram (kg), in terms of the Planck\nconstanth, by means of either the watt/joule balance\n[1-9] or the X-ray crystal density (XRCD) method [10].\nThe watt balance, which was proposed by Kibble in 1975\n[11], has been adopted by the majority of NMIs. The\nNational Institute of Metrology (NIM, China) is focusing\non the joule balance, which can be seen as an alternative\nrealization of the watt balance [12].\nThe unwanted coil motions in the watt/joule balance,\ne.g., horizontal and rotational movements, will lower the\nsignal to noise ratio in the measurement. With these\nmotions, the measurement would take a much longer time\nin order to achieve the type A relative uncertainty at\nan order of 10\u00008. Moreover, the noticeable undesired coil\nmotions will introduce a systematic bias [13]. Therefore,\ndamping devices are necessary to be employed to suppress\nthese undesired coil motions in the watt/joule balance. In\nwatt balances, e.g., NIST-4 at the National Institute of\nStandards and Technology (NIST, USA) [14], an active\ndamping system has been used during the measurement.\nThis work is supported by the China National Natural Science\nFoundation (Grant Nos.91536224, 51507088) and the China National\nKey Research and Development Plan (Grant No. 2016YFF0200102).\nJ. Xu, Q. You are with Tsinghua University, Beijing 100084, China.\nZ. Zhang and Z. Li are with the National Institute of Metrology\n(NIM), Beijing 100029, China and the Key Laboratory for the\nElectrical Quantum Standard of AQSIQ, Beijing 100029, China.\nS. Li is currently with the International Bureau of Weights and\nMeasures (BIPM), Pavillon de Breteuil, F-92312 S` evres Cedex,\nFrance. E-mail:leeshisong@sina.com.The damping coils in such systems are conventionally\nlocated on the framework of the suspended coil. To avoid\nadditional force in the weighing mode and unwanted in-\nduced voltage in the velocity mode, the damping device\nis turned off during both measurement modes. During the\ndynamic measurement mode, the undesired coil motions,\nhowever, is not taken care of. In the joule balance at NIM\nto suppress unwanted motions during the measurement\nwithout introducing any flux into the main magnet, a novel\nactive electromagnetic damping device, located outside\nthe main magnet, is designed and used. The merit of\nthe system is that the field of the damping device has\nno effect on the main field, and hence can be used in\nboth the dynamic and static measurement modes. With\nthe feedback from a detection system, five degrees of\nfreedom of the coil, i.e. the horizontal displacement x,y\nand the rotation angles \u0012x,\u0012y,\u0012z, can be controlled by\nthis device. Two functions, i.e. suppressing the undesired\ncoil motions and reducing the misalignment error, then\ncan be realized in the measurements. The original idea was\npresented in [15]. In this paper, the principle, construction\nand performance of the active electromagnetic damping\ndevice are presented.\nThe rest of this article is organized as follows: the\nprinciple of the joule balance is introduced in section\n2; the principle and construction of the active damping\ndevice are presented in section 3; the performance of the\nactive damping device in the joule balance is presented\nin section 4; some potential systematic effects using the\nactive damping device, the vertical force and the flux\nleakage at the mass weighing position, are discussed in\nsection 5.\nII. Principles of Joule Balance\nThe joule balance is the integral of the watt balance. The\nmathematical details are presented in [12]. The equation\nis expressed as\nZ\nLF\u0001dl+Z\nL\u001c\u0001d\u0012=I[ (B)\u0000 (A)]; (1)\nwhereFdenotes the magnetic force, \u001cthe torque relative\nto the mass center of the coil, Lthe vector trajectory\nwhen the coil is moved from position A to B, Ithe current\nthrough the coil, (B)\u0000 (A)the flux linkage difference\nof the position B and A.\nOn the right side of equation (1), the product of the flux\nlinkage difference (B)\u0000 (A)and the current Iis the\nmagnetic energy change. On the left side of equation (1),arXiv:1610.02799v1 [physics.ins-det] 10 Oct 2016IEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 2\nFig. 1. The schematic diagram of the misalignment errors.\nthe two integral terms are the work done by the magnetic\nforce and torque. They can be expressed as\nwAB=Z\nLFzdz+Z\nLFxdx+Z\nLFydy\n+Z\nL\u001czd\u0012z+Z\nL\u001cxd\u0012x+Z\nL\u001cyd\u0012y:(2)\nNote that in equation (2) only the vertical force, i.e.\nFz, can be measured precisely. The integral terms of the\nhorizontal forces Fx,Fyand torques \u001cx,\u001cy,\u001czare parts\nof the misalignment errors in the joule balance. Other\nparts of the misalignment errors come from the position\nchange of the coil between the two measurement modes\nin the joule balance. In equation (1), it is assumed that\nthe positions A and B in the measurement of the flux\nlinkage difference are the same as the starting point and\nending point of the integral trajectory L. However, it is\nnot the case in the actual measurement. As shown in Fig.\n1, when current pass through the coil, the positions A and\nB will actually shift to A’ and B’ . Then, equation (1) can\nbe rewritten as\nwAA0+wA0B0+wB0B=I[ (B)\u0000 (A)]; (3)\nwherewAA0,wA0B0,wB0Bare the work done by the mag-\nnetic force and torque when the coil is respectively moved\nfrom position A to A’, A’ to B’ and B’ to B. It can be seen\nfrom equation (3) that the misalignment error includes\nthree parts: wAA0,wB0Band the integral terms of the\nhorizontal forces and torques in wA0B0.\nThe active damping device used in the joule balance\nis able to produce auxiliary horizontal forces and torques\non the coil to change the position of the coil. Therefore,\nwith the feedback from the detection system of the five\ndirections of displacement, the position of the coil can\nbe easily controlled. As a result, the positions A and B\ncan be adjusted to be the same as A’ and B’ by the\nfeedback of the damping device. In the meanwhile, the\nhorizontal displacement x,yand the rotation angles \u0012x,\n\u0012y,\u0012z, can be kept unvaried when the coil is moved along\nthe vector trajectory Lin the weighting mode of the\nFig. 2. Construction of the damping device. (a) The overall structure\nof the damping device with the top cover 1 open. (b) The permanent\nmagnets (5, 6, 7, 11,12, 13) and the inner yokes (8, 9, 10). (c) Com-\nbination of auxiliary coils 14 and 15. (d) Combination of auxiliary\ncoils 14 and 16.\njoule balance. These two improvements will significantly\nreduce the misalignment error. For an ideal case, i.e. the\nalignment is adjusted to be good enough, only the work\ndone by the vertical force in wA0B0is left on the left side\nof equation (3) and other terms can be ignored. Then\nequation (3) can be rewritten as\nZB0\nA0Fzdz=I[ (B)\u0000 (A)]: (4)\nIII. Principle and Construction of the Damping\nDevice\nA. The practical structure\nFig. 2 shows the practical structure of the active damp-\ning device used in the joule balance.The outer yokes 1, 2,\n3 and the inner yokes 8, 9,10 are made of soft iron with\nhigh permeability. The component 4 shown in Fig. 2(a)\nis made of aluminum, for mechanically supporting three\ninner yokes. The component 17 in Fig. 2(c) and Fig. 2(d)\nis the framework of the auxiliary coils which is made up\nof polysulfone.\nComponents 5, 6, 7, 11, 12 and 13 are permanent\nmagnets magnetized along the zaxis. It should be noted\nthat the magnetization direction of the three permanent\nmagnets 5, 12, 13 is opposite to the magnetization direc-\ntion of the other three permanent magnets 6, 7, 11. As a\nresult, in this magnetic circuit, the magnetic flux in the\nair gap originates from the yoke 8 and enters the yokes 9,\n10.IEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 3\nComponents 14, 15 and 16 are three orthogonal aux-\niliary coils. Fig. 2(c) and Fig. 2(d) show two different\ncombinations of the auxiliary coils. Coils 14 and 15 are\nglued together in Fig. 2(c) and coils 14 and 16 are glued\ntogether in Fig. 2(d). Since there are considerable mag-\nnetic gradients in x,yand the azimuth \u0012, torques and\nforces will be produced on the auxiliary coils when current\npasses through the auxiliary coils.\nThe cross sectional view of the inner yokes and the\nauxiliary coils in the active damping device is shown in\nFig. 3. The dotted lines denote the magnetic flux in the\nair gap. The coil 14 in the magnetic field shown in Fig.\n3(a) will produce a torque \u001cxaroundxaxis, relative to\nthe mass center. Coil 15 in Fig. 3(b) will produce a force\nFyalong theyaxis and coil 16 in Fig. 3(c) will produce\na forceFxalong thexaxis. The forces and torques of the\nthree coils along other directions are theoretically small.\nThe schematic diagram of the damping device is shown\nin Fig. 3(d), containing 3 damping segments. The coil is\nsuspended from the spider by three rods. Each damping\nsegment is fixed on a support with the auxiliary coils\nmechanically connected to the coil spider rod. In two\nof them, the combination of the auxiliary coils 14 and\n15 shown in Fig. 2(c) is used. In the other one, the\ncombination of the auxiliary coils 14 and 16 shown in Fig.\n2(d) is used. Hence, the torque produced by coil 14 in all\nthe three damping segments is used to adjust the rotation\nangle\u0012x,\u0012yof the suspended coil. The force of coil 15 in\ntwo segments points to the center of the suspended coil\nand can be applied to adjust the horizontal movement x,\ny. The force of coil 16 in the third segment is along the\ntangential of the suspended coil and can be employed to\nadjust the rotation along zaxis, i.e.\u0012z.\nThe auxiliary coils are connected to the current sources\nthrough the suspension system by fine wires called hair-\nsprings. To reduce the number of the hairsprings and keep\nthe symmetry of the suspended coil, six auxiliary coils are\ninstalled in the practical structure instead of nine in the\ninitial design [15]. Fig. 4(a) shows the six auxiliary coils\ninstalled on the suspended system. In theory, to suppress\nthe motions of the five degrees of freedom, the minimum\nnumber of auxiliary coils is five. Hence, only two of the\nthree coils 14 are connected to the current sources in\npractice. The three damping devices are placed at angles\nof 120 degrees around the magnet in the joule balance as\nshown in Fig. 4(b). The pallets of the damping devices are\nfixed with respect to the marble shelf in the joule balance\nwhich will be kept static.\nIn the joule balance, the suspended coil is kept static\nwhile the magnet is moved by a linear translation stage.\nFor the active damping device used in the joule balance,\nthe auxiliary coils are always kept static and do not require\ntoo much moving space. If such a damping device is used\nin the velocity mode of watt balances, the length of the\ninner yokes must be long enough for the movement of the\nauxiliary coils.\nFig. 3. The cross sectional view of the inner yokes and the auxiliary\ncoils. (a) The torque of coil 14. (b) The force of coil 15. (c) The force\nof coil 16. (d) The schematic diagram of three damping devices.\nFig. 4. (a) The auxiliary coils installed on the suspended system. (b)\nMechanical assembling of the damping device.\nB. The feedback control\nThe entire control circuit is shown in Fig. 5. The PID\ncontroller is realized with a LabVIEW program. Five\nchannels of a high-speed analog output (NI 6733) are used\nas the voltage control of the current sources. With an\nexternal voltage reference, the range of the analog output\nvoltage is \u00061 V. The circuit of a current source is shown in\nFig. 6, performing the U=Iconverter. A 10 \nfour-terminal\nresistor with a low temperature coefficient ( <1 ppm/\u000eC)\nis used as the sense resistor in the current source. Hence,\nthe range of the output current is \u0006100 mA.\nThe detection system of the relative position between\nthe coil and the main magnet is composed of a laser\ninterferometer and position sensitive devices (PSD). The\nrotation angles \u0012x,\u0012yare obtained with the laser interfer-\nometer. The resolution of \u0012xand\u0012yis less than 1 \u0016rad.\nThe horizontal displacements x,yand rotation angle \u0012zIEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 4\nFig. 5. The block diagram of the active control circuit.\nFig. 6. The circuit of the current source.\nFig. 7. The setup of auxiliary coils and the coordinate axis of the\ndetection system.\nare obtained from four PSDs. The resolution in xandyis\nabout 2\u0016m and the resolution of \u0012zis about 5\u0016rad.\nThe five auxiliary coils and the coordinate axis of the\ndetection system are shown in Fig. 7. The large circle\ndenotes the suspended coil in joule balance. Five auxiliarycoils are set concentric to the suspended coil with every 120\ndegrees. Since the horizontal displacement and rotation\nangle of the suspended coil are in a small range, the\nrelationship between the detection and the auxiliary coil\ncurrent is close to be linear. In order to check the depen-\ndence between the coil motion and the current through\nthe auxiliary coils, the transfer matrix Mis measured by\napplying a 50 mA current individually to each of the five\nauxiliary coils. The measurement is written as\n2\n66664dx\ndy\nd\u0012x\nd\u0012y\nd\u0012z3\n77775=M2\n66664I16\nI15\u00002\nI15\u00001\nI14\u00002\nI14\u000013\n77775\n=2\n666640:12 0:32\u00001:04 \u00000:20 0:16\n0:16\u00001:44 0:88 \u00000:24\u00000:24\n0:10 0:12 0:16 1:78 1:30\n0:04 0:10\u00000:08\u00001:16 1:78\n6:30 0:20\u00000:30 0:00 0:003\n777752\n66664I16\nI15\u00002\nI15\u00001\nI14\u00002\nI14\u000013\n77775;\n(5)\nwhere dx,dy,d\u0012x,d\u0012yandd\u0012zare the variations of the\ndetection system with units of \u0016m,\u0016m,\u0016rad,\u0016rad,\u0016rad\nrespectively, I16,I15\u00002,I15\u00001,I14\u00002andI14\u00001the currents\npassing through the five auxiliary coils with unit of mA.\nThe determined matrix Min equation (5) matches the\nanalysis of the above section: The horizontal displacement\nx,yis mainly controlled by coils 15-1 and 15-2. The\nrotation angle \u0012x,\u0012yis mainly adjusted by coils 14-1 and\n14-2. The rotation angles \u0012zis mainly controlled by coil\n16. It can be seen that due to some imperfection of the\nmagnetic field and the misalignment of five auxiliary coils,\nthere are some weak couplings between different channels.\nIn the control, we should use the inverse of the transfer\nmatrix to decouple the correlation of different loops, i.e.\n2\n66664I16\nI15\u00002\nI15\u00001\nI14\u00002\nI14\u000013\n77775=M\u000012\n66664dx\ndy\nd\u0012x\nd\u0012y\nd\u0012z3\n77775\n=2\n66664\u00000:03 0:02 0:00 0:00 0:16\n\u00000:78\u00000:87\u00000:16 0:07 0:04\n\u00001:22\u00000:26\u00000:08 0:14 0:03\n0:12 0:04 0:39\u00000:29\u00000:01\n0:07 0:06 0:26 0:37\u00000:013\n777752\n66664dx\ndy\nd\u0012x\nd\u0012y\nd\u0012z3\n77775:\n(6)\nWith a known difference of the detection and the target,\nthe output voltage of auxiliary coils will be determined\nby both the PID controller and the inverse matrix M\u00001.\nNote that since the response of auxiliary coils can easily\nexcite the suspension mechanism, fast and large feedback\ncurrents should be avoided.\nIV. Experimental Results\nA. Damping performance\nWhen the suspended coil receives external shocks such\nas loading of the test mass, ground vibration, etc., theIEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 5\nenergy blue transfer out through the suspension system is\nvery slow due to a high Qfactor. The suspended coil will\nswing for a long time without damping. Fig. 8 shows the\nmotions of the suspended coil when the mass is put on the\nmass pan with and without the proposed damper. Without\nany damping device, the maximal motion amplitudes of\nthe five degrees of freedom x,y,\u0012x,\u0012y,\u0012zare 250\u0016m,\n150\u0016m, 500\u0016rad, 550\u0016rad, 300\u0016rad respectively. After\nabout 5 minutes, the motion amplitudes will decrease\nto 10\u0016m, 10\u0016m, 15\u0016rad, 15\u0016rad, 50\u0016rad.The damping\nfactor is about 0.11 kg \u0001s\u00001. The steady state deviations\nbetween the mass on state and the mass off state of the\nfive degrees of freedom are 10 \u0016m, 35\u0016m, 50\u0016rad, 20\u0016rad,\n30\u0016rad respectively.\nWhen the active damping device is used, the motions of\nthe suspended coil will decay rapidly. As shown in Fig.\n8,x,y,\u0012x,\u0012y,\u0012zwill reduce to 8 \u0016m, 8\u0016m, 10\u0016rad,\n10\u0016rad, 20\u0016rad in 30 seconds. The damping factor is\nabout 1.1 kg \u0001s\u00001, which is about 10 times larger than that\nwithout the damper. The steady state deviations between\nthe mass on state and the mass off state of the five degrees\nof freedom are less than 2 \u0016m, 2\u0016m, 1\u0016rad, 1\u0016rad, 5\u0016rad.\nB. Reduction of the current dependence of the coil position\nWhen current passes through the suspended coil, the\npositions A and B in the flux linkage difference mea-\nsurement will change to A’ and B’, yielding the errors\nwAA0andwB0B. To generate a 5 N magnetic force, the\nsuspended coil current is about 7.5 mA. Fig. 9 shows the\nchanges of the five degrees of freedom x,y,\u0012x,\u0012y,\u0012z\nwhen the suspended coil current is reduced from 7.5 mA\nto 0 mA slowly. Without the active damping device, the\nmagnitude changes are 25 \u0016m, 5\u0016m, 15\u0016rad, 15\u0016rad,\n70\u0016rad respectively. Due to the restriction of the narrow\nair gap and the laser alignment in the present construction\nof the joule balance, it is very difficult to further adjust\nthe concentricity and horizontality of the suspended coil\nto reduce the magnitudes of the coil position change.\nTo reduce the errors wAA0andwB0B, the positions of A\nand B can be adjusted to be the same as A’ and B’ by the\nactive damping device. As shown in Fig. 9, when the active\ndamping device is added, the magnitude changes of x,y,\n\u0012x,\u0012y,\u0012zare less than 2 \u0016m, 2\u0016m, 1\u0016rad, 1\u0016rad, 5\u0016rad\nrespectively, which on average is one magnitude improved\nthan that without the electromagnetic damper.\nC. Improvement of the vertical movement\nIn both the weighing and dynamic modes of the joule\nbalance, the magnet is moved in the vertical direction zby\na translation stage [17]. However, due to the distortion of\nthe guide rail in the translation stage and other mechanical\nexcitations, the other five degrees of freedom x,y,\u0012x,\u0012y,\n\u0012zwill also change with the movement of the magnet.\nTherefore, the horizontal forces Fx,Fyand torques \u001cx,\n\u001cy,\u001czwill contribute to in the misalignment errors.\nFig. 10 shows the value of the five degrees of freedom\nx,y,\u0012x,\u0012y,\u0012zwith different zduring the movement of\nFig. 8. Coil motions when test mass is put on the mass pan with\nand without the proposed damper. The red lines (–) are without the\ndamper while the blue dash lines (- -) are with the damper.IEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 6\nFig. 9. Coil motions when the coil current is reduced from 7.5 mA\nto 0 mA slowly. The red lines (–) are without the damper while the\nblue dash lines (- -) are with the damper.\nFig. 10. Coil motions with different zduring the movement of the\nmagnet. The red lines (–) are without the damper while the blue dash\nlines (- -) are with the damper.IEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 7\nTABLE I\nThe unwanted vertical force of the five auxiliary coils\nwith 100 mA excitation.\nCoil No. Vertical force /mg uncertainty /mg\n16 0.08 0.01\n15-1 0.06 0.01\n15-2 0.06 0.01\n14-1 1.04 0.01\n14-2 0.85 0.01\nthe magnet. Without the active damping device, the peak-\nto-peak values of the five degrees of freedom are 15 \u0016m,\n15\u0016m, 10\u0016rad, 8\u0016rad, 40\u0016rad respectively. When the\nactive damping device is added during the movement, the\nunwanted motions of the magnet can be compensated by\nthe motions of the suspended coil. Hence, the changes of\nthe relative value of x,y,\u0012x,\u0012y,\u0012zcan be reduced. As\nshown in Fig. 10, the peak-to-peak values of the five de-\ngrees of freedom are reduced to 6 \u0016m, 6\u0016m, 4\u0016rad, 4\u0016rad,\n25\u0016rad with the active damping device. The measurement\nshows that all curves of the five degrees of freedom are\nnearly flat. Therefore, the misalignment errors, resulting\nfrom the work done by the horizontal forces and torques\ninwA0B0, are greatly reduced.\nV. Consideration of Potential Systematic\nEffects\nA. The vertical force\nWhen the active electromagnetic damping device is used\nin the weighing mode of the joule balance, any additional\nvertical force caused by the five auxiliary coils should be\nconsidered. Here the mass comparator has been used to\nmeasure the vertical force. With 100 mA current in the\nauxiliary coils, the measurement results are shown in Table\nI. As discussed in [15], the vertical forces of coil 16 and 15\nare very small when compared with that of coil 14. Since\nthe working current in the auxiliary coils is much less than\n100 mA in the weighing mode, the vertical forces will be\nmuch smaller than the results in Table I.\nWhen the force of the suspended coil is measured in the\nweighing mode, the suspended coil is static and the current\nin the auxiliary coils of the active damping device is nearly\nconstant. As the magnetic force is proportional to the cur-\nrent, the additional vertical force of the auxiliary coil can\nbe determined by measuring the current in the auxiliary\ncoil. Note that in this measurement, the uncertainty of the\nmass comparator is about 0.01 mg. A correction may then\nbe made and the effect of the unwanted vertical forces can\nbe reduced to several parts in 108.\nB. Magnetic flux leakage\nThe distance between the active damping devices and\nthe test mass is about 160 mm. Hence, the leakage mag-\nnetic field of the active damping devices should be consid-\nered. The interaction between the magnetic field and the\nmass is discussed in [16]. The equation of the vertical force\nFig. 11. The absolute magnitude of the magnetic field at the location\nof the mass. The measurement is fitted by a quadratic function.\non the mass with a volume susceptibility \u001fand permanent\nmagnetization Mis given by [18]\nFz=\u0000\u00160\n2@\n@zZ\n\u001fH\u0001HdV\u0000\u00160@\n@zZ\nM\u0001HdV: (7)\nThe magnetic field at the location of the mass is\nmeasured by a gauss meter. Fig. 11 shows the absolute\nmagnitude of the magnetic field as a function of the\ndistance from the top surface of the mass pan. Note that\nalthough the measured field is comparable to the earth’s\nmagnetic field, it produces a much larger field gradient,\nabout 1.3\u0016T/mm. The employed 500 g test mass is about\n50 mm in height and 40 mm in diameter. The magnetic\nsusceptibility \u001fof the mass material is less than 6\u000210\u00004\nand the permanent magnetization Mis about 0.1 A/m.\nWith the method used in [16], the calculated magnetic\nforce of the mass is less than 10 \u0016g. Hence, the relative\nsystematic error caused by the leakage magnetic field is\nless than 2\u000210\u00008.\nVI. Conclusions\nThe active electromagnetic damping device presented\nin this paper can be used in both the dynamic and\nstatic measurement modes of the joule balance. Five de-\ngrees of freedom of the suspended coil, i.e. the horizontal\ndisplacement x,yand the rotation angles \u0012x,\u0012y,\u0012z,\ncan be actively controlled by the damping device. The\nexperimental results show that the active damping device\nwell meets the design targets. The damping factor of the\nsystem is increased by a factor of 10 and the alignment\nis greatly improved by compensating for the non-vertical\nmovement of the coil. Two potential systematic effects, i.e.\nthe unwanted vertical force and the leakage magnetic field\nof the active damping device, are analyzed. It is shown that\nthese effects are small and compensable within few parts\nin108. At present, the misalignment error is about several\nparts in 107due to the restriction of the narrow air gap\nand the laser alignment in the joule balance project. The\npresented active damping device in this case is powerful\nto reduce misalignment uncertainties.IEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 8\nReferences\n[1] Z. Zhang, Q. He and Z. Li, ”An approach for improving the\nwatt balance” in Digest of Conf. on Precision Electromagnetic\nMeasurement , Torino, Italy, Jul. 9-14, 2006, pp. 126-127.\n[2] A. Robinson, B. P. Kibble, ”An initial measurement of Planck’s\nconstant using the NPL Mark II watt balance. ” Metrologia ,\nvol.44, no. 6, pp.427-440, 2007.\n[3] A. G. Steele, J. Meija, C. A. Sanchez, et al, ”Reconciling Planck\nconstant determinations via watt balance and enriched-silicon\nmeasurements at NRC Canada.” , Metrologia , vol.48, pp.L8-L10,\n2012.\n[4] A. Picard, H. Fang, A. Kiss, et al, ”Progress on the BIPM watt\nbalance.” IEEE Transactions on Instrumentation and Measure-\nment , vol.58, pp.924-929, 2008.\n[5] S. Schlamminger et al, ”Determination of the Planck constant\nusing a watt balance with a superconducting magnet system at\nthe National Institute of Standards and Technology”, Metrologia ,\nvol.51, pp15-24, 2014.\n[6] A. Eichenberger, H. Baumann, B. Jeanneret, B. Jeckelmann, P.\nRichard,and W. Beer, ”Determination of the Planck constant\nwith the METAS wattbalance,” Metrologia , vol. 48, no. 3, pp.\n133ÂĺC141,2011.\n[7] M. Thomaset al, ”First determination of the Planck constantus-\ning the LNE watt balance”, Metrologia , vol.52, pp433-443, 2015.\n[8] M.C.Sutton, T.M.Clarkson. ”A magnet system for the MSL watt\nbalance.” Metrologia , vol.51, pp. 101-106,2014.\n[9] D.Kim, B.C.Woo, K.C.Lee, et al, ”Design of the KRISS watt\nbalance.” Metrologia , vol.51, pp. 96-100, 2014.\n[10] Y. Azumaet al, ”Improved measurement results forthe Avo-\ngadro constant using a28Si-enriched crystal”, Metrologia , vol.52,\npp360-375, 2015.\n[11] B. P. Kibble, ”A measurement of the gyromagnetic ratio of\nthe proton by the strong field method”, Atomic Masses and\nFundamental Constants5 , Springer US, pp545-551, 1976.\n[12] J. Xu et al, ”A determination of the Planck constant by the gen-\neralized joule balance method with a permanent-magnet system\nat NIM”, Metrologia , vol.53, pp86-97, 2016.\n[13] S. Li et al, ”Coil motion effects in watt balances: a theoretical\ncheck”, Metrologia , vol.53, pp817-828, 2016.\n[14] D. Haddadet al, ”A precise instrument to determine the Planck\nconstant, and the future kilogram”, Review of Scientific Instru-\nments ,vol.87, pp1-14, 2016.\n[15] J. Xu et al, ”A Magnetic Damping Device for Watt and Joule\nBalances”, To be published inProc. CPEM Dig., Jul. 2016 .\n[16] Seifert F, Panna A, Li S, et al,”Construction, Measurement,\nShimming, and Performance of the NIST-4 Magnet System”,\nIEEE Transactions on Instrumentation and Measurement ,vol.63,\npp.3027-3038, 2014.\n[17] Zhengkun Li, et al, ”The Improvement of Joule Balance NIM-\n1 and the Design of New Joule Balance NIM-2”, IEEE Trans-\nactions on Instrumentation and Measurement , vol.64, pp.1676-\n1684, 2015.\n[18] R. S. Davis, ”Determining the magnetic properties of 1 kg mass\nstandards”, J. Res. Nat. Inst. Standards Technol. , vol.100, no.3,\npp.209ÂĺC226, May. 1995.\nJinxin Xu was born in Yancheng, Jiangshu\nprovince, China, in 1990. He received the B.S.\ndegree from Harbin Institute of Technology,\nHarbin, China, in 2012. He is currently pursu-\ning the Ph.D. degree at Tsinghua University,\nBeijing, China. His dissertation will be a part\nof the joule Balance at the National Institute\nof Metrology, China.\nQiang You was born in Shandong province,\nChina. He is currently pursuing the Ph.D. de-\ngree with Tsinghua University, Beijing, China.\nHis dissertation will be a part of the joule bal-\nance with the National Institute of Metrology,\nBeijing, China.\nZhonghua Zhang was born in Suzhou,\nJiangshu, China in July 1940. He received his\nB.S. degree and M.S. degree in Electrical En-\ngineering from Tsinghua University, Beijing,\nChina separately in 1965 and 1967. In 1995, he\nbecame a member of the Chinese Academy of\nEngineering. In 1967, Zhonghua Zhang joined\nthe Electromagnetic Division of National In-\nstitute of Metrology (NIM), Beijing, China.\nSince then he has been involved in the re-\nsearch work on the Cross Capacitor standard,\nsuperconducting high magnetic field standard and Quantum Hall\nResistance standard. In 2006, he proposed the joule balance and led\nthe research work since then.\nZhengkun Li was born in Henan Province,\nChina in 1977. He received the B.S. degree in\ninstrument science and technology from China\nInstitute of Metrology, Hangzhou, China, in\n1999. He received the M.S. degree in quantum\ndivision of National Institute of Metrology\n(NIM), China in 2002. He received his Ph.D.\ndegree from Xi’an Jiaotong University, Xi’an,\nChina in 2006. In 2002, he became a perma-\nnent staff of NIM and worked on the establish-\nment of Quantum Hall Resistance standard.\nSince 2006, he has been involved in the research work on joule balance\nproject at NIM and focuses on the electromagnetic measurement\nincluding mutual inductance measurement for the joule balance.\nShisong Li (M’15) got the Ph.D. degree in\nTsinghua University, Beijing, China in July,\n2014. He then joined the Department of Elec-\ntrical engineering, Tsinghua University during\nJuly, 2014-September, 2016. He has been a\nguest researcher at the National Institute of\nMetrology in China since 2009, and he worked\nat the National Institute of Standards and\nTechnology, United States for 14 months dur-\ning 2013-2016. He is currently with the In-\nternational Bureau of Weights and Measures\n(BIPM), France. His research interests include modern precision\nelectromagnetic measurement and instrument technology." }, { "title": "1102.0810v2.Damping_of_Electron_Density_Structures_and_Implications_for_Interstellar_Scintillation.pdf", "content": "arXiv:1102.0810v2 [astro-ph.GA] 9 Feb 2011Damping of Electron Density Structures and Implications fo r\nInterstellar Scintillation\nK.W. Smith and P.W. Terry\nCenter for Magnetic Self Organization in Laboratory and Ast rophysical Plasmas and\nDepartment of Physics, University of Wisconsin-Madison, M adison, WI 53706\nkwsmith1@wisc.edu\nABSTRACT\nThe forms of electron density structures in kinetic Alfv´ en wave tu rbulence\nare studied in connection with scintillation. The focus is on small scales\nL∼108−1010cm where the Kinetic Alfv´ en wave (KAW) regime is active in\nthe interstellar medium, principally within turbulent HII regions aroun d bright\nstars. MHD turbulence converts to a KAW cascade, starting at 10 times the ion\ngyroradiusandcontinuing tosmaller scales. Thesescalesareinferr edtodominate\nscintillation in the theory of Boldyrev et al.(Boldyrev & Gwinn 2003a,b, 2005;\nBoldyrev & Konigl 2006). From numerical solutions of a decaying kine tic Alfv´ en\nwaveturbulencemodel, structuremorphologyrevealstwotypeso flocalizedstruc-\ntures, filaments and sheets, and shows that they arise in different regimes of re-\nsistive and diffusive damping. Minimal resistive damping yields localized cu rrent\nfilaments that form out of Gaussian-distributed initial conditions. W hen resis-\ntive damping is large relative to diffusive damping, sheet-like structur es form.\nIn the filamentary regime, each filament is associated with a non-loca lized mag-\nnetic and density structure, circularly symmetric in cross section. Density and\nmagnetic fields have Gaussian statistics (as inferred from Gaussian -valued kur-\ntosis) while density gradients are strongly non-Gaussian, more so t han current.\nThisenhancement ofnon-Gaussianstatistics inaderivative fieldisex pected since\ngradientoperationsenhancesmall-scale fluctuations. Theenhanc ement ofdensity\ngradient kurtosis over current kurtosis is not obvious, yet it sugg ests that modest\nfluctuation levels in electron density may yield large scintillation events during\npulsarsignalpropagationintheinterstellar medium. Inthesheet re gimethesame\nstatistical observations hold, despite the absence of localized filam entary struc-\ntures. Probability density functions are constructed from statis tical ensembles in\nboth regimes, showing clear formation of long, highly non-Gaussian t ails.\nSubject headings: ISM: electron density −ISM: general −MHD−Turbulence– 2 –\n1. Introduction\nModels of scintillation have a long history. Many (Lee & Jokipii 1975a,b; Sutton 1971)\ncarry an implicit or explicit assumption of Gaussian statistics, applying to either the electron\ndensity field itself or its autocorrelation function (herein referred to as “Gaussian Models”).\nLee & Jokipii(1975a)isarepresentativeapproach. Thestatistics ofthetwo-pointcorrelation\nfunction of the index of refraction ǫ(r),A(ρ) =/integraltext\ndz′∝angb∇acketleftǫ(x,z)ǫ(x+ρ,z′)∝angb∇acket∇ightdetermines, among\nother effects, the scaling of pulsar signal width τwith dispersion measure DM. The index\nof refraction ǫ(r) is a function of electron density fluctuation n(r). The quantity A(0) enters\nthe equations, representing the second moment of the index of re fraction. If the distribution\nfunction of ǫ(r) has no second-order moment (as in a L´ evy distribution) A(0) is undefined.\nThe assumption of Gaussian statistics leads to a scaling of τ∼DM2, which contradicts\nobservation for pulsars with DM >30 cm−3pc. For these distant pulsars, τ∼DM4\n(Sutton 1971; Boldyrev & Gwinn 2003a,b).\nTo explain the anomalous DM4scaling, Sutton (1971) argued that the pulsar signal\nencounters strongly scattering turbulent regions for longer lines of sight, essentially arguing\nthat the statistics, as sampled by a pulsar signal, are nonstationar y. Considering the pulse\nshapeintime, Williamson(1972,1973,1974)isunabletomatchobserv ationswithaGaussian\nModelofscintillationunlessthescatteringregionisconfinedto1 /4ofthelineofsightbetween\nthe pulsar and Earth. These assumptions may have physical basis, since the ISM may not be\nstatistically stationary, being composed of different regions with va rying turbulence intensity\n(Boldyrev & Gwinn 2005).\nThe theory of Boldyrev & Gwinn (2003a,b, 2005); Boldyrev & Konigl ( 2006) takes a\ndifferent approach to explain the anomalous DM4scaling by considering L´ evy statistics for\nthe density difference (defined below). L´ evy distributions are cha racterized by long tailswith\nno defined moments greater than first-order [i.e., A(0) is undefined for a L´ evy distribution].\nThe theory recovers the τ∼DM4relation with a statistically stationary electron density\nfield. Thistheoryalsodoesnotconstrainthescatteringregiontoa fractionoftheline-of-sight\ndistance.\nThe determinant quantity in the theory of Boldyrev et al.is the density difference,\n∆n=n(x1,z)−n(x2,z). According to this model, if the distribution function of ∆ nhas a\npower-law decay as |∆n| → ∞and has no second moment, then it is possible to recover the\nτ∼DM4scaling (Boldyrev & Gwinn 2003b). Assuming sufficiently smooth fluctu ations,\n∆ncan be expressed in terms of the density gradient, σ(z):n(x1)−n(x2)≃σ(z)·(x1−x2).\nPerhaps more directly, the density gradient enters the ray tracin g equations [Eqns. (7) in\nBoldyrev & Gwinn (2003a)], and is seen to be central to determining t he resultant pulsar\nsignal shape and width. This formulation of a scintillation theory does not require the– 3 –\ndistribution of ∆ nto be Gaussian or to have a second-order moment.\nThe notion that the density difference is characterized by a L´ evy d istribution is a con-\nstraint on dynamical models for electron density fluctuations in the ISM. Consequently the\nquestion of how a L´ evy distribution can arise in electron density fluc tuations assumes con-\nsiderable importance in understanding the ISM.\nPrevious work has laid the groundwork for answering this question. It has been es-\ntablished that electron density fluctuations associated with inters tellar magnetic turbulence\nundergoasignificantchangeincharacternearthescale10 ρs, whereρsistheionsoundgyrora-\ndius (Terry et al. 2001). At larger scales, electron density is passiv ely advected by the turbu-\nlent flowofanMHDcascade mediatedby nonlinear shear Alfv´ enwave s (Goldreich & Sridhar\n1995). At smaller scales the electron density becomes compressive and the turbulent energy\nis carried into a cascade mediated by kinetic Alfv´ en waves (KAW) (Te rry et al. 2001). The\nKAW cascade brings electron density into equipartition with the magn etic field, allowing\nfor a significant increase in amplitude. The conversion to a KAW casca de has been ob-\nserved in numerical solutions of the gyrokinetic equations (Howes e t al. 2006), and is consis-\ntent with observations from solar wind turbulence (Harmon & Coles 2 005; Bale et al. 2005;\nLeamon et al. 1998). Importantly, it puts large amplitude electron d ensity fluctuations (and\nlarge amplitude density gradients) at the gyroradius scale ( ∼108−1010cm), a desirable set\nof conditions for pulsar scintillation (Boldyrev & Konigl 2006). It is th erefore appropriate to\nconsider whether large-amplitude non-Gaussian structures can a rise in KAW turbulence.\nThis question has been partially answered in a study of current filame nt formation in\ndecaying KAW turbulence (Terry & Smith 2007, 2008). In numerical solutions to a two-\nfield model with broadband Gaussian initial conditions large amplitude c urrent filaments\nspontaneously arose. Each filament was associated with a large-am plitude electron density\nstructure, circular in cross-section, that persisted in time. Thes e electron density structures\nwere not as localized as the corresponding current filaments, but w ere coherent and not\nmixed by surrounding turbulence. The observation of large amplitud e current filaments is\nsimilar to the large-amplitude vortex filaments found in decaying 2D hy drodynamic turbu-\nlence (McWilliams 1984). Counterparts of such structures in 3D are predicted to be the\ndominant component for higher order structure functions (She & Leveque 1994).\nTerry & Smith (2007) proposed that a nonlinear refractive magnet ic shear mechanism\npreventsthestructuresfrommixingwithturbulence. Radialshea rintheazimuthallydirected\nmagnetic field associated with each large-amplitude current filament decreases the radial\ncorrelation length of the turbulent eddies and enhances the decor relation rate. Eddies are\nunabletopersist longenoughtopenetratetheshearboundarylay er anddisrupt thestructure\ncore. The structure persistence mechanism allows large-amplitude fluctuations to persist for– 4 –\nmany eddy-turnover times. As the turbulent decays these struc tures eventually dominate the\nstatistics of the system. The spatial structure of the density fie ld associated with localized\ncircularly symmetric current filaments was shown from analytical th eory (Terry & Smith\n2007) to yield a L´ evy-distributed density gradient field. The kurto sis for the current field\nwas significantly larger than the Gaussian-valued kurtosis of 3, indic ating enhanced tails.\nThe electron density and magnetic field kurtosis values were not sign ificantly greater than 3.\nHowever, just as the current is non-Gaussian when the magnetic fi eld is not, it is expected\nthat numerical solutions should show non-Gaussian behavior for th e density gradient. In the\npresent paper, density gradientstatisticsaremeasuredandfou ndtobenon-Gaussian.Rather\nthan relying onkurtosis values alone, the probability density functio ns (PDFs) are computed\nfrom ensembles of numerical solutions, showing non-Gaussian PDFs for the density gradient\nfield.\nThe previous studies of filament generation in KAW turbulence leave s ignificant unan-\nswered questions relating to structure morphology and its effect o n scintillation. It is well\nestablished that MHD turbulence admits structures that are both filament-like and sheet-\nlike. Can sheet-like structures arise in KAW turbulence? If so, what are the conditions or\nparameters favoring one type of structure versus the other? I f sheet-like structures dominate\nin some circumstances, what are the statistics of the density grad ient? Can they be suffi-\nciently non-Gaussian to be compatible with pulsar scintillation scaling? I t is desirable to\nconsider such questions prior to calculation of rf wave scattering p roperties in the density\ngradient fields obtained from numerical solutions.\nInthispaper, weshowthatbothcurrent filaments andcurrent sh eet structures naturally\narise in numerical solutions of a decaying KAW turbulence model. Each has a structure of\nthe same type and at the same location in the electron density gradie nt. These structures\nbecome prevalent as the numerical solutions progress in time, and e ach is associated with\nhighlynon-GaussianPDFs. Moreover, weshowthatsmall-scalecurr entfilamentsandcurrent\nsheets, along with their associated density structures, are highly sensitive to the magnitude\nof resistive damping and diffusive damping of density fluctuations. Cu rrent filaments persist\nprovided that resistivity ηis small; similarly, electron density fluctuations and gradients\nare diminished by large diffusive damping in the electron continuity equa tion. The latter\nresults from collisions assuming density fluctuations are subject to a Fick’s law for diffusion.\nThe magnitude of the resistivity affects (1) whether current filame nts can become large in\namplitude, (2) their spatial scale, and (3) the preponderance of t hese filaments as compared\nto sheets. The magnitude of the diffusive damping parameter, µ, similarly influences the\namplitude of density gradients and, to a lesser degree, influences t he extent to which electron\ndensity structures are non-localized.– 5 –\nIn the ISM resistive and diffusive damping become important near res istive scales. How-\never, it iswell known that collisionless damping effects arealso presen t (Lysak & Lotko 1996;\nBale et al. 2005), and quite possibly dominate over collisional damping in larger scales near\nthe ion Larmor radius. The collisional damping in the present work is un derstood as a\nheuristic approach that facilitates analysis of the effects of differe nt damping regimes on\nthe statistics of electron density fluctuations. By varying the rat io of resistive and diffusive\ndamping we can, as suggested above, control the type of struct ure present in the turbulence.\nThis allows us to isolate and study the statistics associated with each type of structure. It\nalso allows us to assess and examine the type of environment conduc ive to formation of the\nstructure. We consider regimes with large and small damping parame ters, enabling us to\nexplore damping effects on structure formation across a range fr om inertial to dissipative.\nFuture work will address collisionless damping in greater detail.\n1.1. Background Considerations for Structure Formation\nThe coherent structures observed in numerical solutions of deca ying KAW turbulence,\nwhether elongatedsheets orlocalizedfilaments, aresimilar tostruc tures observedindecaying\nMHD turbulence, as in Kinney & McWilliams (1995). In that work, the flo w field initially\ngivesrisetosheet-like structures. Afterselective decayofthev elocity fieldenergy, thesystem\nevolves into a state with sheets and filaments. During the merger of like-signed filaments,\nlarge-amplitude sheets arise, limited to the region between the merg ing filaments. These\nshort-lived sheets exist in addition to the long-lived sheets not asso ciated with the merger\nof filaments. In the two-field KAW system, however, there is no flow ; the sheet and filament\ngeneration is due to a different mechanism, of which the filament gene ration has previously\nbeen discussed (Terry & Smith 2007).\nOther work (Biskamp & Welter 1989; Politano et al. 1989) observed t he spontaneous\ngeneration of current sheets and filaments in numerical solutions, with both Orszag-Tang\nvortex and randomized initial conditions. These 2D reduced MHD num erical solutions mod-\neled the evolution of magnetic flux and vorticity with collisional dissipat ion coefficients η,\nthe resistivity, and ν, the kinematic viscosity. The magnetic Prandtl number, ν/η, was set\nto unity. These systems are incompressible and not suitable for mod eling the KAW system\nwe consider here – they do however illustrate the ubiquity of curren t sheets and filaments,\nand serve as points of comparison. For Orszag-Tang-like initial con ditions with large-scale\nflux tubes smooth in profile, current sheets are preferred at the interfaces between tubes.\nTearing instabilities cangive risetofilamentary current structures that persist forlong times,\nbut the large-scale and smoothness of flux tubes do not give rise to strong current filaments– 6 –\nlocalized at the center of the tubes. To see this, consider a given flu x tube, and model it as\ncylindrically symmetric and monotonically decreasing in rwith characteristic radial extent\na,\nψ(r) =ψ0/parenleftBig\n1−(r\na)2/parenrightBig\n, (1)\nfor 0≤r≤a. The current is localized at the center with magnitude\nJ=−4ψ0\na2. (2)\nThus flux tubes with large radial extent ahave a corresponding small current filament\nat their center. Hence, initial conditions dominated by a few large-s cale flux tubes are\nnot expected to have large amplitude current filaments at the flux t ube centers, but favor\ncurrent sheet formation and filaments associated with tearing inst abilities in those sheet\nregions. At X points current sheet folding and filamentary structu res can arise (see, e.g.\nBiskamp & Welter 1989, Fig. 10), but these regions are small in area c ompared to the quies-\ncent flux-tube regions. Note that if, instead of Orszag-Tang-like initial conditions, the initial\nstate is random, one expects some regions with flux tubes that hav easmall, and therefore\na sizable current filament at the center.\nConsider now the effect of comparatively large or small η. In the case of large η, the\ncentral region of a flux tube is smoothed by the collisional damping, t hus having a strong\nsuppressive effect on the amplitude of the current filament associa ted with such a flux tube.\nLarge-amplitude current structures are localized to the interfac es between flux tubes. In\nthe process of mergers between like-signed filaments (and repulsio n between unlike-signed),\nlarge current sheets are generated at these interfaces, similar t o the large-amplitude sheets\ngenerated in MHD turbulence during mergers (Kinney & McWilliams 1995 ). For small η,\nrelatively little suppression of isolated current filaments should occu r; if these filaments are\nspatially separated owning to the buffer provided by their associate d flux tube, they can\nbe expected to survive a long time and only be disrupted upon the mer ger with another\nlarge-scale flux tube. Large η, then, allows current sheets to form at the boundaries between\nflux tubes while suppressing the spatially-separated current filame nts at flux tube centers.\nSmallηallows interface sheets and spatially separated filaments to exist.\nThese simple argumentssuggest thattheevolutionofthelarge-am plitudestructures and\ntheir interaction with turbulence is thus strongly influenced by the d amping parameters. As\nsuch, the magnitudes of the damping parameters are expected to affect the resultant pulsar\nscintillation scalings. The present paper considers the effect of var iations of these damping– 7 –\nparameters, ηandµ. In the KAW model, the (unnormalized) resistivity takes the form\nη=meνe/ne2and the density diffusion coefficient is µ=ρ2\neνe, wheremeis the electron mass,\nνeis the electron collision frequency, nis the electron density, eis the electron charge, and\nρe=vTe/ωce, withvTethe electron thermal velocity, and ωcethe electron gyrofrequency. The\nratioof these terms, c2η/4πµ= 2/β, whereβ= 8πnkT/B2istheratio ofplasma tomagnetic\npressures. When we vary this ratio, as we will do in the numerical solu tions presented here,\nwe have in mind that we are representing regions of different β. However, as a practical\nmatter in the numerical solutions, we must vary the damping parame ters independently of\nthevariationof β, sincethekinetic Alfv´ enwave dynamics require asmall βtopropagate. For\nthewarmionizedmedium, typicalparametersare Te= 8000 K,n= 0.08 cm−3,|B|= 1.4µG,\nδB= 5.0µG (Ferri` ere 2001). With these parameters, the plasma βformally ranges from\n0.05−1.2, spanning a range of plasma magnetization.\nWe present the results of numerical solutions of decaying KAW turb ulence to ascertain\nthe effect of different damping regimes on the statistics of the fields of interest, in particular\nthe electron density and electron density gradient. In the η≪µregime (using normalized\nparameters), previous work (Craddock et al. 1991) had large-am plitude current filaments\nthat were strongly localized with no discernible electron density stru ctures (µwas large to\npreserve numerical stability). This regime is unable to preserve den sity structures or density\ngradients. The numerical solutions presented here have η∼µandη≫µ; in each limit\nthe damping parameters are minimized so as to allow structure forma tion to occur, and are\nlarge enough to ensure numerical stability for the duration of each numerical solution. We\ninvestigate the statistics of both filaments and sheets in the conte xt of scintillation in the\nwarm ionized medium.\nThe paper is arranged as follows: section 2 gives an overview of the K AW model and\nnormalizations, its regime of validity, and its dispersion relation. Sect ion 3 discusses the\nnumerical method used and the field initializations. The negligible effect of initial cross-\ncorrelation between fields is discussed. Results for the two damping regimes are given in\nsection 4, where the type of structures that form, whether she ets-and-filaments or predomi-\nnantly sheets, are seen to be dependent on the values of ηandµ. PDFs from ensemble nu-\nmerical solutions are presented in section 5, illustrating the strong ly non-Gaussian statistics\nin the electron density gradient field for both the η∼µandη≫µregimes. This suggests\nthat non-Gaussian electron-density gradients are robust to var iation inη, as long as the\noverall damping in the continuity equation is not too large. Some discu ssions regarding the\nlimitations ofnumerical approximation forthis workandpossible enha ncements–particularly\na model that addresses driven KAW turbulence–are given in conclud ing remarks.– 8 –\n2. Kinetic Alfv´ en Wave Model\nThe kinetic Alfv´ en wave (KAW) model used in this paper is the same mo del used in the-\nories of pulsar scintillation through the ISM (Terry & Smith 2007, 200 8) and in earlier work\n(Craddock et al. 1991). It is a reduced, two-field, small-scale limit of a more general reduced\nthree-fieldMHDsystem(Hazeltine1993;Rahman & Wieland1983;Fer nandez & Terry1997)\nthat accounts for electron dynamics parallel to the magnetic field.\nThe 3-field model applies to largeand small-scale fluctuations ascomp ared toρs, the ion\ngyroradius evaluated at the electron temperature. In large-sca le strong turbulence magnetic\nand kinetic fluctuations are in equipartition, with electron density pa ssively advected. In\nthe limit of small spatial scales ( ≤10ρs) the roles of kinetic and internal fluctuations are\nreversed – magnetic fluctuations are in equipartition with density flu ctuations, and kinetic\nenergy experiences a go-it-alone cascade without participating in t he magnetic-internal en-\nergy interaction. The shear-Alfv´ en physics at large scale is suppla nted by kinetic-Alfv´ en\nphysics at small scale (Terry et al. 2001).\nIn the Boldyrev et al.theory, the length scales that dominate scintillation for pulsars\nwithDM >30 pc cm−3are small, around 108−1010cm. This motivates our focus on the\nsmall-scale regime ofthemoregeneral 3-fieldsystem. The dominant interactions arebetween\nmagnetic and internal fluctuations, via kinetic Alfv´ en waves. In th ese waves, electron density\ngradientsalongthemagneticfieldactonaninductiveelectricfieldinOh m’slaw. Theelectron\ncontinuity equation serves to close the system. The normalized equ ations are\n∂tψ=∇/bardbln+η0J−η2∇2J, (3)\n∂tn=−∇/bardblJ+µ0∇2n−µ2∇2∇2n, (4)\n∇/bardbl=∂z+∇ψ×z·∇, (5)\nJ=∇2ψ, (6)\nwithψ= (Cs/c)eAz/Te, the normalized parallel component of the vector potential and\nn= (Cs/VA)˜n/n0the normalized electron density. The normalized resistivity is η0=\n(c2/4πVAρs)ηsp, withηspthe Spitzer resistivity, given in the introduction. The normal-\nized diffusivity is µ=ρ2\neνe/ρsVA. The time and space normalizations are τA=ρs/VAand\nρs=Cs/Ωi. HereCs= (Te/mi)1/2is the ion acoustic velocity, VA=B/(4πmin0)1/2is\nthe Alfv´ en speed, and Ω i=eB/m icis the ion gyrofrequency. Electron density diffusion is\npresumed to follow Fick’s law; more detailed damping would necessarily c onsider kinetic ef-\nfects and cyclotron resonances. The η2andµ2terms (hyper-resistivity and hyper-diffusivity)\nare introduced to mitigate large-scale Fourier-mode damping by the linear diffusive terms.– 9 –\nThroughout the remainder of the paper, we drop the 2 subscript f romη2andµ2and refer\nto the hyper-dissipative terms as ηandµ.\nThree ideal invariants exist: total energy E=/integraltext\nd2x[(∇ψ)2+αn2]; fluxF=/integraltext\nd2x ψ2\nand cross-correlation Hc=/integraltext\nd2xnψ. Energy cascades to small scale (large k) while the flux\nand cross-correlation undergo an inverse cascade to large-scale (smallk) (Fernandez & Terry\n1997). The inverse cascades require the initialized spectrum to pea k atk0∝negationslash= 0 to allow for\nbuildup of magnetic flux at large-scales for later times.\nLinearizing the system yields a (dimensional) dispersion relation ω=VAkzk⊥ρs. The\nmodecombines perpendicular oscillationassociatedwitha finitegyror adiuswithfluctuations\nalong a mean field ( z-direction). The oscillating quantities are magnetic field and density,\nout of phase by π/2 radians.\nIn thelimit of strong mean field, quantities along themean field ( z-direction) equilibrate\nquickly, which allows ∂/∂z→0, orkz→0. Kinetic Alfv´ en waves still propagate, as long\nas there are a broad range of scales that are excited, as in fully dev eloped turbulence. As\nkz→0, all gradients are localized to the plane perpendicular to the mean fi eld. Presuming a\nlarge-scale fluctuation at characteristic wavenumber k0, smaller-scale fluctuations propagate\nlinearly along this larger-scale fluctuation so long as their character istic scaleksatisfies\nk≫k0. In this reduced, two dimensional system, the above dispersion re lation is modified\nto beω=VA(bk0·k/B)kρswhich is still Alfv´ enic but with respect to a perturbed large-scale\namplitude perpendicular to the mean field. Relaxing the scale separat ion criterion yields\nω∝k2for the general case.\n3. Numerical Solution Method\nWe evolve Eqs. (3) and (4) in a 2D periodic box, size [2 π]×[2π] on a mesh of resolution\n512×512. Theψandnscalarfields areevolved intheFourierdomain, withthenonlinearities\nadvanced pseudospectrally and with full 2 /3 dealiasing in each dimension (Orszag 1971).\nThe diffusive and resistive terms normally introduce stiffness into the equations; using an\nintegrating factor removes any stability constraints stemming fro m these terms. Following\nthe scheme outlined in Canuto et al. (1990), we start with the semi-d iscrete formulation of\nEqs. (3) and (4):\ndψk\ndt=−ηk4ψk+F/bracketleftbig\n∇/bardbln/bracketrightbig\n(7)– 10 –\ndnk\ndt=−µk4nk−F/bracketleftbig\n∇/bardblJ/bracketrightbig\n, (8)\nwhereF[·]denotesthediscrete Fouriertransform. Wedonotexplicitly expa ndthenonlinear\nterms as they will be integrated separately. The hyper-damping te rms (proportional to k4)\nare included above. Damping terms corresponding to the Laplacian o perator (proportional\ntok2) are not included in this section for clarity, but are trivial to incorpo rate. Equations 7\nand 8 can be put in the form\nd\ndt/bracketleftBig\neηk4tψk/bracketrightBig\n=eηk4tF/bracketleftbig\n∇/bardbln/bracketrightbig\n(9)\nd\ndt/bracketleftBig\neµk4tnk/bracketrightBig\n=−eµk4tF/bracketleftbig\n∇/bardblJ/bracketrightbig\n. (10)\nA second-order Runge-Kutta scheme for the ψkdifference equation is\nψm+1/2\nk=e−ηk4∆t/2/bracketleftbig\nψm\nk+∆t/2F/bracketleftbig\n∇/bardblnm/bracketrightbig/bracketrightbig\n(11)\nψm+1\nk=e−ηk4∆t/bracketleftBig\nψm+1/2\nk+∆tF/bracketleftbig\n∇/bardblnm+1/2/bracketrightbig/bracketrightBig\n(12)\nwith a similar form for the nkscheme.\n3.1. Initial conditions\nTheψkandnkfields are initialized such that the energy spectra are broad-band w ith a\npeak neark0∼6−10 and a power law spectrum for k > k 0. The falloff in kis predicted\nto bek−2for small-scale turbulence. Craddock et al. (1991) use k−3, between the current-\nsheet limit of k−4and the kinetic-Alfv´ en wave strong-turbulence limit of k−2. The numerical\nsolutions considered here have either k−2ork−3. The only qualitative difference between the\ntwo spectra is the scale at which structures initially form. The k−2spectra has more energy\nat smaller scales, leading to smaller characteristic structure size. A fter a few tens of Alfv´ en\ntimes these smaller-scale structures merge and the system resem bles the initial k−3spectra.\nThenkandψkphases can beeither cross-correlated oruncorrelated. By cros s-correlated\nwe mean that the phase angle for each Fourier component of the nkandψkfields are equal.\nIn general,– 11 –\nnk=|Ak|eiθ1, ψ k=|Bk|eiθ2, (13)\nwhere|Ak|and|Bk|are the Fourier component’s amplitude, set according to the spect rum\npower-law. For cross-correlated initial conditions, θ1=θ2for allkat the initial time.\nFor uncorrelated initialization, there is no phase relation between co rresponding Fourier\ncomponents of the nkandψkfields.\nCraddock et al. (1991) focused on the formation and longevity of c urrent filaments in\na turbulent KAW system. To preserve small-scale structure in the c urrent filaments, these\nnumerical solutions set η= 0andhad µ∼10−3, witharesolutionof128 ×128, corresponding\nto akmaxof 44. Large-amplitude density structures that would have arisen were damped\nto preserve numerical stability up to an advective instability time of a few hundred Alfv´ en\ntimes, for the parameter values therein. The numerical solutions p resented here explore\na range of parameter values for ηandµ. They make use of hyper-diffusivity and hyper-\nresistivity of appropriate strengths to preserve structures in n,BandJ. An advective\ninstability is excited after ∼102Alfv´ en times if resistive damping is negligible. The η= 0\nsolutions–not presented here due to their poor resolution of small- scale structures–see large-\namplitude current filaments arise, but they can be poorly resolved a t this grid spacing. With\nno resistivity, the finite number of Fourier modes cannot resolve ar bitrarily small structures\nwithout Gibbs phenomena resulting and distorting the current field.\nWehavefoundthroughexperiencethatsmallhyper-resistivityan dsmallhyper-diffusivity\npreserve large-amplitude density structures and their spatial co rrelation with the magnetic\nandcurrent structures, whilepreventing thedistortionresulting frompoorly-resolved current\nsheets and filaments. They allow the numerical solutions to run for a rbitrarily long times,\nand the effects of structure mergers become apparent. These o ccur on a longer timescale\nthan the slowest eddy turnover times. The results presented her e will consider two regimes\nof parameter values, the η≈µandη≫µregimes. The effect of cross-correlated and\nuncorrelated initial conditions will be addressed presently.\n4. Results\nIt is of interest to examine whether cross-correlated or uncorre lated initial conditions\naffect the long-term behavior of the system. Two representative numerical solutions are\npresented here that reveal the system’s tendency to form spat ially-correlated structures in\nelectron density and current regardless of initial phase correlatio ns. This study establishes\nthe robustness of density structure formation in KAW turbulence and lends confidence that– 12 –\nsuch structures should exist in the ISM under varying circumstanc es. The first numerical\nsolution has cross-correlated initial conditions between the nandψfields; the second, un-\ncorrelated. Damping parameters ηandµare equal and large enough to ensure numerical\nstability while preserving structures in density, current and magne tic fields. These examples\nalso serve to explore the intermediate η/µregime.\nThe energy vs. time history for both numerical solutions are given in Figs. 1 and 2.\nTotal energy is a monotonically decreasing function of time. The mag netic and internal\nenergies remain in overall equipartition throughout the numerical s olutions. Magnetic en-\nergy increases at the expense of internal energy and vice versa . This energy interchange is\nconsistent with KAW dynamics and overall energy conservation in th e absence of resistive\nor diffusive terms. The exchange is crucial in routinely producing larg e amplitude density\nfluctuations in this two-field model of nonlinearly interacting KAWs.\nThe total energy decay rates for the uncorrelated and correlat ed initial conditions in\nFigs. 1 and 2 differ, with the latter decaying more strongly than the f ormer. The damping\nparameters are identical for the two numerical solutions, and the decay-rate difference re-\nmains under varying randomization seeds. The magnitudes of the no nlinear terms during\nthe span of a numerical solution in Eqs. (3) and (4) for uncorrelate d initial conditions are\nconsistently larger than those of correlated initial conditions by a f actor of 5. This difference\nlasts until 2500 Alfv´ en times, after which the decay rates are rou ghly equal in magnitude.\nThe steeper energy decay during the run of numerical solutions wit h uncorrelated initial con-\nditions (Fig. 2) suggests that the enhancement of the uncorrelat ed nonlinearities transports\nenergy to higher k(smaller scale) more readily than the nonlinearities in the correlated c ase.\nRelatively more energy in higher kenhances the energy decay rate as the linear damping\nterms dissipate more energy from the system. The initial configura tion, whether correlated\nor uncorrelated, is seen to have an effect on the long-term energy evolution for these de-\ncaying numerical solutions. It will be shown below, however, that th e correlation does not\nsignificantly affect the statistics of the resulting fields.\nFor cross-correlated initial conditions, we expect there to be a st rong spatial relation\nbetween current, magnetic field and density structures through time. Figs. 3 and 4 show\nthenand|B|contours at various times. For the latest time contour, the spatia l structure\nalignment is evident. Further, in Fig. 5, the circular magnetic field str uctures (magnetic field\ndirection and intensity indicated by arrow overlays) align with the larg e-amplitude density\nfluctuations. The correlation is evident once one notices that ever y positive-valued circular\nnstructure corresponds to counterclockwise-oriented magnetic field, and vice versa . Fig. 5\nis at a normalized time of 5000 Alfv´ en times, defined in terms of the lar ge B0. The system\npreserves the spatial structure correlation indefinitely, even af ter structure mergers.– 13 –\nThe second representative numerical solution is one with uncorrela ted initial conditions.\nContourplotsofdensityand |B|aregiveninFigs.6and7, respectively. Itisnoteworthythat,\nsimilar to the cross-correlated initial conditions, spatially correlate d density and magnetic\nfield structures are discernible at the latest time contour.\nIn Fig. 8 the circular density structures correspond to circular ma gnetic structures.\nUnlike Fig. 5 the positive density structures may correspond to cloc kwise or counterclock-\nwise directed magnetic field structures. This serves to illustrate th at, although the initial\nconditions have no phase relation between fields, after many Alfv´ e n times circular density\nstructures spatially correlate with magnetic field structures and p ersist for later times.\nThe kurtosis excess as a function of time, defined as K(Ξ) =∝angb∇acketleftΞ4∝angb∇acket∇ight/σ4\nΞ−3, is shown\nin Figs. 9 and 10 for correlated and uncorrelated initial conditions, r espectively. Positive\nKindicates a greater fraction of the distribution is in the tails as compa red to a best-fit\nGaussian. These figures indicate that the non-Gaussian statistics for the fields of interest\nare independent of initial correlation in the fields. In particular, the density gradients, |∇n|,\nare significantly non-Gaussian as compared to the current. Becau se scintillation is tied to\ndensity gradients, this situation is expected to favor the scaling inf erred from pulsar signals.\nThe tendency of density structures to align with magnetic field stru ctures regardless of\ninitial conditions indicates that the initial conditions are representa tive of fully-developed\nturbulence. After a small number of Alfv´ en times the memory of th e initial state is removed\nas the KAW interaction sets up a consistent phase relation between the fluctuations in the\nmagnetic and density fields. Previous work (Terry & Smith 2007) pre sented a mechanism\nwhereby these spatially correlated structures can be preserved via shear in the periphery of\nthe structures. The above figures indicate that this mechanism is a t play even in cases where\nthe initial phase relations are uncorrelated.\nIn the damping regime presented above, circularly symmetric struc tures in density,\ncurrent and magnetic fields readily form and persist for many Alfv´ e n times, until disrupted\nby mergers with other structures of similar amplitude. It is possible t o define, for each\ncircular structure, an effective separatrix that distinguishes it fr om surrounding turbulence\nand large-amplitude “sheets” that exist between structures. [se e, e.g., the magnetic field\ncontours at later times in Fig. 7.] The density field has significant grad ients in both the\nregions surrounding the structure and within the structures the mselves. The ability to\nseparate these circular structures from the background sheet s and turbulence is determined\nby the magnitudes – relative and absolute – of the damping paramete rs. Larger damping\nvalueserodethesmall-spatial-scalestructurestoagreaterexte ntand, iflargeenough, disrupt\nthestructurepersistence mechanism that, forafixeddiameter, dependsonasufficiently large\namplitude current filament to generate a sufficiently large radially she ared magnetic field.– 14 –\nThe preceding results were for a damping regime where η/µ∼1, an intermediate\nregime. Numerical solutions with µ= 0 andηsmall explore the regime where η/µ→0.\nIn this regime, which is opposite the regime used in Craddock et al., circ ularly symmetric\ncurrent and magnetic structures are not as prevalent, rather, sheet-like structures dominate\nthe large amplitude fluctuations. Current and magnetic field gradien ts are strongly damped,\nand the characteristic length scales in these fields are larger.\nContours of density for a numerical solution with µ= 0 are shown in Fig. 11. Visual\ncomparison with contours for runs with smaller damping parameters (Fig. 6, where η=µ)\nindicate a preponderance of sheets in the µ= 0 case, at the expense of circularly-symmetric\nstructures asseen above. All damping isin η; anycurrent filament thatwouldotherwise form\nis unable to preserve its small-scale, large amplitude characteristics before being resistively\ndamped. Inspection of the current and |B|contours for the same numerical solution [Figs.\n12 and 13] reveal broader profiles and relatively few circular curre nt and magnetic field\nstructures with a well-defined separatrix as in the small ηcase. Since there is no diffusive\ndamping, gradients in electron density are able to persist, and elect ron density structures\ngenerally follow the same structures in the current and magnetic fie lds.\nKurtosis excess measurements for the µ= 0 numerical solutions yield mean values\nconsistent with the η=µnumerical solutions, as seen in Fig. 14. Magnetic field strength and\nelectron density statistics are predominantly Gaussian, with curre nt statistics and density\ngradient statistics each non-Gaussian. Perhaps not as remarkab le in this case, the density\ngradient kurtosis excess is again seen to be greater than the curr ent kurtosis excess – this\nis anticipated since the dominant damping of density gradients is turn ed off. With fewer\nfilamentary current structures, however, the mechanism propo sed in Terry & Smith (2007) is\nnot likely to be at play in this case, since few large-amplitude filamentar y current structures\nexist. Sheets, evident in the density gradients in Fig. 15 and in the cu rrent in Fig. 12 are the\ndominant large-amplitudestructures anddetermine theextent to whichthedensity gradients\nhave non-Gaussian statistics. The current and density sheets ar e well correlated spatially.\nThe largest sheets can extend through the entire domain, and evo lve on a longer timescale\nthan the turbulence. Sheets exist at the interface between large -scale flux tubes, and are\nregions of large magnetic shear, giving rise to reconnection events . Withηrelatively large,\nthe sheets evolve on timescales shorter than the structure pers istence timescale associated\nwith the long-lived flux tubes.\nSheets and filaments are the dominant large-amplitude, long timesca le structures that\narise in the KAW system. Filaments arise and persist as long as ηis small, with their\namplitude and statistical influence diminished as ηincreases. Sheets exist in both regimes,\nbecoming the sole large-scale structure in the large ηregime. Density gradients are consis-– 15 –\ntently non-Gaussian in both regimes as long as µis small, although the density structures\nare different in both regimes. Density gradient sheets arise in the lar geηregime and these\ndensity gradient sheets are large enough to yield non-Gaussian sta tistics.\n5. Ensemble Statistics and PDFs\nTo explicitly analyze the extent to which the decaying KAW system dev elops non-\nGaussian statistics, ensemble runs were performed for both the η/µ∼1 andη/µ≪1\nregimes, and PDFs of the fields were generated.\nFor theη/µ∼1 regime, 10 numerical solutions were evolved with identical paramet ers\nbut for different randomization seeds. In this case η=µand both damping parameters have\nminimal values to ensure numerical stability. The fields were initially pha se-uncorrelated.\nThe density gradient ensemble PDF for two times in the solution result s is shown in Fig. 16.\nDensity gradients are Gaussian distributed initially. Many Alfv´ en time s into the numerical\nsolution the statistics are non-Gaussian with long tails. These PDFs a re consistent with\nthe time histories of density gradient kurtosis excess as shown abo ve. The distribution tail\nextends beyond 15 standard deviations, almost 90 orders of magn itude above a Gaussian\nbest-fit distribution. Similar behavior is seen in the current PDFs – init ially Gaussian\ndistributed tending to strongly non-Gaussian statistics with long ta ils for later times. Fig. 17\nisthecurrent PDFatanadvancedtime into thenumerical solution. I t istobenotedthat the\ndensity gradient PDF has longer tails at higher amplitude than does th e current PDF. One\nwould expect these to be in rough agreement, since the underlying d ensity and magnetic\nfields have comparable PDFs that remain Gaussian distributed throu ghout the numerical\nsolution. The discrepancy between the density gradient and curre nt PDFs suggests a process\nthat enhances density derivatives above magnetic field derivatives . Future work is required\nto explore causes of this enhancement. This result is significant for pulsar scintillation, which\nis most sensitive to density gradients. Although interstellar turbule nce is magnetic in nature,\nthe KAW regime has the benefit of fluctuation equipartition between nandB. The density\ngradient, however, is more non-Gaussian than the magnetic compo nent, suggesting that this\ntype of turbulence is specially endowed to produce the type of scint illation scaling observed\nwith pulsar signals.\nEnsemble runs for the η/µ≪1 regime yield distributions similar to the η/µ∼1 regime\ninallfields. The ensemble PDFfortwo timesisshown inFig.18. Theinitial density gradient\nPDF is Gaussian distributed. For later times long tails are evident and c onsistent with the\nkurtosis excess measurements as presented above for the µ= 0 case. The density gradient\ndistribution has longer tails at higher amplitude than the current dist ribution; the overall– 16 –\ndistributions are similar to those for the η/µ∼1 regime, despite the absence of filamentary\nstructures and the presence of sheets. The strongly non-Gaus sian statistics are insensitive to\nthe damping regime, provided that the diffusion coefficient is small eno ugh to allow density\ngradients to persist.\n6. Discussion\nUsing the normalizations for Eqs. (3) and (4) and using B= 1.4µG,n= 0.08 cm−3and\nTe= 1 eV,ηnorm, the normalized Spitzer resistivity, is 2 .4×10−7andµnorm, the normalized\ncollisional diffusivity, is 1 .9×10−7. For a resolution of 5122, these damping values are unable\ntokeep thesystem numerically stable. Thethresholdforstabilityre quires thesimulation ηto\nbegreater than5 ×10−6, which is almost within anorder of magnitude of the ISMvalue. The\nnumerical solutions presented here, while motivated bythe pulsar s ignal width scalings, more\ngenerally characterize the current and density gradient PDFs whe n the damping parameters\narevaried. Wewouldexpectthedensitygradientstobenon-Gauss ianwhenusingparameters\nthat correspond to the ISM. Future work will address the pulsar w idth scaling using electron\ndensity fields from the numerical solution.\nThe non-Gaussian distributions presented here are strongly tied t o the fact that the\nsystem is decaying and that circular intermittent structures are p reserved from nonlinear\ninteraction. One can show that, in the KAW system, circularly symme tric structures (or\nfilaments) are force free in Eqs. (3) and (4), i.e., the nonlinearity is z ero. Once a large-\namplitude structure becomes sufficiently circularly symmetric and is a ble to preserve itself\nfrombackgroundturbulenceviatheshearmechanism, thatstruc tureisexpectedtopersiston\nlong timescales relative to the turbulence. Structure mergers will le ad to a time-asymptotic\nstate with two oppositely-signed current structures and no turb ulence. As structures merge,\nkurtosis excess increases until the system reaches a final two-fi lament state, which would\nhave a strongly non-Gaussian distribution and large kurtosis exces s.\nIf the system were driven, energy input at large scales would replen ish large-amplitude\nfluctuations. New structures would arise from large amplitude regio ns whenever the radial\nmagnetic field shear were large enough to preserve the structure from interaction with tur-\nbulence. One could define a structure-replenishing rate from the d riving terms that would\ndepend on the energy injection rate and scale of injection. The non -Gaussian measures\nfor a driven system would be characterized by a competition betwee n the creation of new\nstructures through the injection of energy at large scales and th e annihilation of structures\nby mergers or by erosion from continuously replenished small-scale t urbulence. If erosion\neffects dominate, the kurtosis excess is maintained at Gaussian valu es, diminishing the PDF– 17 –\ntails relative to a L´ evy distribution. If replenishing effects dominate , however, the enhance-\nment of the tails of the density gradient PDF may be observed in a driv en system as it is\nobserved in the present decaying system. We note that structur e function scaling in hy-\ndrodynamic turbulence is consistent with the replenishing effects be coming more dominant\nrelative to erosion effects as scales become smaller, i.e., the turbulen ce is more intermittent\nat smaller scales. The large range of scales in interstellar turbulence and the conversion of\nMHD fluctuations to kinetic Alfv´ en fluctuations at small scales both support the notion that\nthe structures of the decaying system are relevant to interstella r turbulence at the scales of\nKAW excitations. This scenario is consistent with arguments sugges ted by Harmon & Coles\n(2005). They propose a turbulent cascade in thesolar wind that inj ects energy into the KAW\nregime, counteracting Landau damping at scales near the ion Larmo r radius. By doing so\nthey can account for enhanced small-scale density fluctuations an d observed scintillation\neffects in interplanetary scintillation.\nWe also observe that, although the numerical solutions presented here are decaying in\ntime, the decay ratedecreases inabsolute valueforlater times (Fig s. 1 and2), approximating\na steady-state configuration. The kurtosis excess (Figs. 9 and 1 0) for the density gradient\nfield is statistically stationary after a brief startup period. Despite the decaying character of\nthe numerical solutions, they suggest that the density gradient fi eld would be non-Gaussian\nin the driven case.\nThe kurtosis excess – a measure of a field’s spatial intermittency – is itself intermittent\nin time. The large spikes in kurtosis excess correspond to rare even ts involving the merger\nof two large-amplitude structures, usually filaments. A large-amplit ude short-lived sheet\ngrows between the structures and persists throughout the mer ger, gaining amplitude in time\nuntil the point of merger. The kurtosis excess during this merger e vent is dominated by\nthe single large-amplitude sheet between the merging structures. This would likely be the\nregion of dominant scattering for scintillation, since a correspondin g large-amplitude density\ngradient structure exists in this region as well. The temporal interm ittency of kurtosis excess\nsuggests that these mergers are rare and hence, of low probabilit y. The heuristic picture of\nlong undeviated L´ evy flights punctuated by large angular deviation s could apply to these\nmerger sheets.\n7. Conclusions\nDecaying kineticAlfv´ enwave turbulenceisshowntoyieldnon-Gauss ianelectrondensity\ngradients, consistent with non-Gaussian distributed density grad ients inferred from pulsar\nwidth scaling with distance to source. With small resistivity, large-am plitude current fila-– 18 –\nments form spontaneously from Gaussian initial conditions, and the se filaments are spatially\ncorrelated with stable electron density structures. The electron density field, while Gaussian\nthroughout the numerical solution, has gradients that are stron gly non-Gaussian. Ensem-\nble statistics for current and density gradient fields confirm the ku rtosis measurements for\nindividual runs. Density gradient statistics, when compared to cur rent statistics, have more\nenhanced tails, even though both these fields are a single derivative away from electron den-\nsity and magnetic field, respectively, which are in equipartition and Ga ussian distributed\nthroughout the numerical solution.\nWhen all damping is placed in resistive diffusion ( η/µ→0 regime), filamentary struc-\ntures give way to sheet-like structures in current, magnetic, elec tron density and density\ngradient fields. Kurtosis measurements remain similar to those for t he smallηcase, and the\nfield PDFs also remain largely unchanged, despite the different large- amplitude structures\nat play.\nThe kind of structures that emerge, whether filaments or sheets , is a function of the\ndamping parameters. With ηandµminimal to preserve numerical stability and of com-\nparable value, the decaying KAW system tends to form filamentary c urrent structures with\nassociated larger-scale magnetic and density structures, all gen erally circularly symmetric\nand long-lived. Each filament is associated with a flux tube and can be w ell separated from\nthe surrounding turbulence. Sheets exist in this regime as well, and t hey are localized to the\ninterface between flux tubes. With ηsmall andµ= 0, the system is in a sheet-dominated\nregime. Both regimes have density gradients that are non-Gaussia n with large kurtosis.\nThe effects on pulsar signal scintillation in each regime have yet to be a scertained\ndirectly. TheconventionalpictureofaL´ evyflightisarandomwalkw ithstepsizesdistributed\naccording to a long-tailed distribution with no defined variance. This g ives rise to long,\nuninterrupted flightspunctuatedbylargescattering events. Th isisincontrasttoanormally-\ndistributed random walk with relatively uniform step sizes and small sc attering events. The\nintermittent filaments that arise in the small ηandµregime are suggestive of structures that\ncould scatter pulsar signals through large angles, however the ass ociated density structures\nare broadened in comparison to the current filament and would not g ive rise to as large a\nscattering event. Even broadened structures can yield L´ evy dis tributed density gradients\n(Terry & Smith 2007), but it is not clear how the L´ evy flight picture c an be applied to these\nbroad density gradient structures. In the µ= 0 regime, the large-aspect-ratio sheets may\nserve to provide the necessary scatterings through refraction and may map well onto the\nL´ evy flight model.\nAn alternative possibility, suggested by the temporal intermittency of the kurtosis (itself\na measure of a field’s spatialintermittency), is the encounter between the pulsar signal and– 19 –\na short-lived sheet that arises during the merger of two filamentar y structures. These sheets\nare limited in extent and have very large amplitudes. At their greates t magnitude they are\nthe dominant structure in the numerical solution. Their temporal in termittency distinguish\nthem from the long-lived sheets surrounding them. It is possible tha t a pulsar signal would\nundergo large scattering when interacting with a merger sheet. Th is scattering would be a\nrare event, suggestive of a scenario that would give rise to a L´ evy flight.\n8. Acknowledgments\nWe thankS. Boldyrev, S. Spangler andE. 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Lett., 72, 336\nSutton, J.M. 1971, MNRAS, 155, 51\nTerry, P.W., McKay, C., & Fernandez, E. 2001, Phys. Plasmas, 8, 27 07\nTerry, P.W., Smith, K.W. 2007, ApJ, 665, 402\nTerry, P.W., Smith, K.W. 2008, Phys. Plasmas, 15, 056502\nWilliamson, I.P. 1972, MNRAS, 157, 55\nWilliamson, I.P. 1973, MNRAS, 163, 345\nWilliamson, I.P. 1974, MNRAS, 166, 499\nThis preprint was prepared with the AAS L ATEX macros v5.2.– 21 –\n0.0 0.5 1.0 1.5 2.0\nTime (arb. units) 1e30.00.20.40.60.81.01.2Energy (arb. units)1e\u00002Energy vs. time -- correlated ICs\nTotal Energy\nMagnetic Energy\nInternal Energy\nFig. 1.— Energy vs. time for cross-correlated initial conditions. Tot al energy is monotoni-\ncallydecreasing withtime, andmagneticandinternal energies remain inroughequipartition.– 22 –\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nTime (arb. units) 1e30.00.20.40.60.81.01.2Energy (arb. units)1e\u00012Energy vs. time -- uncorrelated ICs\nTotal Energy\nMagnetic Energy\nInternal Energy\nFig. 2.— Energy vs. time for uncorrelated initial conditions. Total en ergy is monotonically\ndecreasing with time, and magnetic and internal energies remain in ro ugh equipartition.– 23 –\n0 1 2 3 4 5\n1e20123451e2 Time = 0\u00024.5\n\u00033.0\n\u00041.50.01.53.01e\u00054\n0 1 2 3 4 5\n1e20123451e2 Time = 5000\u00061.5\n\u00071.0\n\b0.50.00.51.01.52.01e\t4Density -- \n \u000b \f\nFig. 3.— Contours of nfor various times in a numerical solution with correlated initial\nconditions.\n0 1 2 3 4 5\n1e20123451e2 Time = 0\n0.00.51.01.52.02.53.03.54.04.51e \r4\n0 1 2 3 4 5\n1e20123451e2 Time = 5000\n0.00.20.40.60.81.01.21.41.61.81e \u000e4Magnetic Field -- \u000f \u0010 \u0011\nFig. 4.— Contours of |B|for various times in a numerical solution with correlated initial\nconditions.– 24 –\nFig. 5.— Contour plot of nwithBvectors overlaid. The positive, circularly-symmetric\ndensity structures correspond to counterclockwise-directed Bstructures; the opposite holds\nfor negative circularly-symmetric density structures. These spa tial correlations are to be\nexpected for correlated initial conditions.– 25 –\n0 1 2 3 4 5\n1e20123451e2 Time = 0\u00124.5\n\u00133.0\n\u00141.50.01.53.04.51e\u00154\n0 1 2 3 4 5\n1e20123451e2 Time = 5000\u00163.2\n\u00172.4\n\u00181.6\n\u00190.80.00.81.62.43.21e\u001a4Density -- \u001b \u001c \u001d uncorrelated\nFig. 6.— Contours of nfor various times in a numerical solution with uncorrelated initial\nconditions.\n0 1 2 3 4 5\n1e20123451e2 Time = 0\n0.51.01.52.02.53.03.54.04.51e \u001e4\n0 1 2 3 4 5\n1e20123451e2 Time = 5000\n0.30.60.91.21.51.82.12.42.71e \u001f4Magnetic Field -- ! \" uncorrelated\nFig. 7.— Contours of |B|for various times in a numerical solution with uncorrelated initial\nconditions.– 26 –\nFig. 8.— Contour plot of nwithBvectors overlaid for a numerical solution with initially\nuncorrelated initial conditions. The positive, circularly-symmetric d ensity structures corre-\nspond to magnetic field structures, although the sense (clockwise or counterclockwise) of the\nmagnetic field structure does not correlate with the sign of the den sity structures. Circled in\nblack are symmetric structures that display a high degree of spatia l correlation. The circle\ngives an approximate indication of the separatrix for the structur e.– 27 –\n0 1 2 3 4 5\nTime (arb. units) 1e3012345Kurtosis ExcessKurtosis excess vs. time -- correlated ICs\nb_x\ndendat\ncur\ndengrad_x\nFig. 9.— Kurtosis excess for a numerical solution with phase-correla ted initial conditions\nandη/µ= 1.– 28 –\n0 1 2 3 4 5 6 7\nTime (arb. units) 1e302468Kurtosis ExcessKurtosis excess vs. time -- uncorrelated ICs\nb_x\ndendat\ncur\ndengrad_x\nFig. 10.— Kurtosis excess for a numerical solution with phase-uncor related initial conditions\nandη/µ= 1.– 29 –\n0 1 2 3 4 5\n1e20123451e2 Time = 0#4.5\n$3.0\n%1.50.01.53.01e&4\n0 1 2 3 4 5\n1e20123451e2 Time = 5000'1.5\n(1.0\n)0.50.00.51.01.52.01e*4Density -- + , -\nFig. 11.— Electron density contour visualization with diffusive damping p arameterµ= 0\nfor various times.\n0 1 2 3 4 5\n1e20123451e2 Time = 0.6\n/4\n0202461e 13\n0 1 2 3 4 5\n1e20123451e2 Time = 500023.2\n32.4\n41.6\n50.80.00.81.62.41e 63Current -- 7 8 9\nFig. 12.— Current density contour visualization with diffusive damping p arameterµ= 0\nfor various times.– 30 –\n0 1 2 3 4 5\n1e20123451e2 Time = 0\n0.00.51.01.52.02.53.03.54.04.51e :4\n0 1 2 3 4 5\n1e20123451e2 Time = 5000\n0.00.20.40.60.81.01.21.41.61.81e ;4Magnetic Field -- < = >\nFig. 13.— Magnitude of magnetic field contour visualization with diffusive damping param-\neterµ= 0 for various times.– 31 –\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nTime (arb. units) 1e3\n?1012345678Kurtosis ExcessKurtosis excess vs. time -- zero-diffusivity\nb_x\ndendat\ncur\ndengrad_x\nFig. 14.— Kurtosis excess for a numerical solution with diffusive param eterµ= 0. Density\ngradient kurtosis remains greater than current kurtosis for the duration of the numerical\nsolution.– 32 –\n0 1 2 3 4 5\n1e20123451e2 Time = 0@6.0\nA4.5\nB3.0\nC1.50.01.53.04.51e D3\n0 1 2 3 4 5\n1e20123451e2 Time = 5000E3\nF2\nG1012341e H3Density Gradient -- I J K\nFig. 15.— Electron density gradient ( xdirection) contour visualization with diffusive damp-\ningµ= 0 for various times.– 33 –L0.8M0.6N0.4O0.2 0.0 0.2 0.4 0.6\nDensity gradient (dimensionless units) 1eP210-210-1100101102103Probability densityLog-PDF of Density Gradient with Gaussian Best-Fit\nden. gradient, t=0\nden. grad. gaussian, t=0\nden. gradient, t=5000\nden. grad. gaussian, t=5000\nFig. 16.— log-PDF of density gradients for an ensemble of numerical s olutions with η/µ= 1\natt= 0 andt= 5000. The density gradient field at t= 0 is Gaussian distributed, while for\nt= 5000 the gradients are enhanced in the tails, and deviate from a Ga ussian. A best-fit\nGaussian for each PDF is plotted for comparison.– 34 –Q1.0 R0.5 0.0 0.5\nCurrent (dimensionless units) 1eS210-310-210-1100101102Probability densityLog-PDF of Current with Gaussian Best-Fit\ncurrent, t=0\ncurrent gaussian, t=0\ncurrent, t=5000\ncurrent gaussian, t=5000\nFig. 17.— log-PDF of current for an ensemble of numerical solutions w ithη/µ= 1 att= 0\nandt= 5000. The current at t= 0 is Gaussian distributed. For t= 5000 the current is\nnon-Gaussian. Unlike the density gradient, the current is not enha nced in the tails of the\nPDF for later times relative to its initial Gaussian envelope.– 35 –T4U2 0 2 4\nDensity gradient (dimensionless units) 1eV210-310-210-1100101102Probability densityLog-PDF of Density Gradient with Gaussian Best-Fit, W=0\nden. gradient, t=0\nden. grad. gaussian, t=0\nden. gradient, t=5000\nden. grad. gaussian, t=5000\nFig. 18.— Log-PDF of density gradient for an ensemble of numerical s olutions with µ= 0\natt= 0 andt= 5000. The density gradient field at t= 0 is Gaussian distributed, while for\nt= 5000 the gradients are enhanced in the tails, and deviate from a Ga ussian. A best-fit\nGaussian for each PDF is plotted for comparison." }, { "title": "1306.1893v1.Observation_of_a_Berry_phase_anti_damping_spin_orbit_torque.pdf", "content": "Observation of a Berry phase anti-damping spin-orbit torque\nH. Kurebayashi,1, 2Jairo Sinova,3, 4D. Fang,1A. C. Irvine,1J. Wunderlich,4, 5\nV. Nov\u0013 ak,4R. P. Campion,6B. L. Gallagher,6E. K. Vehstedt,3, 4\nL. P. Z^ arbo,4K. V\u0013 yborn\u0013 y,4A. J. Ferguson,1and T. Jungwirth4, 6\n1Microelectronics Group, Cavendish Laboratory,\nUniversity of Cambridge, CB3 0HE, UK\n2PRESTO, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan\n3Department of Physics, Texas A&M University,\nCollege Station, Texas 77843-4242, USA\n4Institute of Physics ASCR, v.v.i., Cukrovarnick\u0013 a 10, 162 53 Praha 6, Czech Republic\n5Hitachi Cambridge Laboratory, Cambridge CB3 0HE, UK\n6School of Physics and Astronomy,\nUniversity of Nottingham, Nottingham NG7 2RD, UK\n(Dated: February 9, 2020)\n1arXiv:1306.1893v1 [cond-mat.mes-hall] 8 Jun 2013Recent observations of current-induced magnetization switching at\nferromagnet/normal-conductor interfaces1,2have important consequences for fu-\nture magnetic memory technology. In one interpretation, the switching origi-\nnates from carriers with spin-dependent scattering giving rise to a relativis-\ntic anti-damping spin-orbit torque (SOT)3{7in structures with broken space-\ninversion symmetry.1,1{3,3{7,7{9,13,15{20The alternative interpretation1,2,4{6,19{21\ncombines the relativistic spin Hall e\u000bect (SHE),22{28making the normal-\nconductor an injector of a spin-current, with the non-relativistic spin-transfer\ntorque (STT)29{32in the ferromagnet. Remarkably, the SHE in these ex-\nperiments originates from the Berry phase e\u000bect in the band structure of a\nclean crystal2,24,25,27,33and the anti-damping STT is also based on a disorder-\nindependent transfer of spin from carriers to magnetization. Here we report the\nobservation of an anti-damping SOT stemming from an analogous Berry phase\ne\u000bect to the SHE. The SOT alone can therefore induce magnetization dynamics\nbased on a scattering-independent principle. The ferromagnetic semiconductor\n(Ga,Mn)As we use has a broken space-inversion symmetry in the crystal.2,3,7,13\nThis allows us to consider a bare ferromagnetic \flm which eliminates by design\nany SHE related contribution to the spin torque. We provide an intuitive picture\nof the Berry phase origin of the anti-damping SOT and a microscopic modeling\nof measured data.\nIn the quasiclassical transport theory, the linear response of the carrier system to the\napplied electric \feld is described by the non-equilibrium distribution function of carrier\neigenstates which are considered to be unperturbed by the electric \feld. The form of the\nnon-equilibrium distribution function is obtained by accounting for the combined e\u000bects of\nthe carrier acceleration in the \feld and scattering. On the other hand, in the time-dependent\nquantum-mechanical perturbation theory the linear response is described by the equilibrium\ndistribution function and by the perturbation of carrier wavefunctions in the applied electric\n\feld. The latter framework was the basis of the intrinsic Berry phase mechanism introduced\nto explain the anomalous Hall e\u000bect in (Ga,Mn)As34and, subsequently, in a number of\nother ferromagnets.35Via the anomalous Hall e\u000bect, the Berry phase physics entered the\n\feld of the SHE in spin-orbit coupled paramagnets. Here the concept of a scattering-\nindependent origin brought the attention of a wide physical community to this relativistic\n2phenomenon, eventually turning the SHE into an important \feld of condensed matter physics\nand spintronics.28In our work we demonstrate that the SOT can also have the relativistic\nquantum-mechanical Berry phase origin.\nWe start by deriving the intuitive picture of our Berry phase anti-damping SOT based on\nthe Bloch equation description of the carrier spin dynamics. In (Ga,Mn)As, the combination\nof the broken inversion symmetry of the zinc-blende lattice and strain can produce spin-orbit\ncoupling terms in the Hamiltonian which are linear in momentum and have the Rashba\nsymmetry, HR=\u000b\n~(\u001bxpy\u0000\u001bypx), or the Dresselhaus symmetry, HD=\f\n~(\u001bxpx\u0000\u001bypy) (see\nFig. 1a).2,3,7,13Here\u001bare the Pauli spin matrices and \u000band\frepresent the strength of the\nRashba and Dresselhaus spin-orbit coupling, respectively. The interaction between carrier\nspins and magnetization is described by the exchange Hamiltonian term, Hex=J\u001b\u0001M.\nIn (Ga,Mn)As, Mcorresponds to the ferromagnetically ordered local moments on the Mn\nd-orbitals and Jis the antiferromagnetic carrier{local moment kinetic-exchange constant.36\nThe physical origin of our anti-damping SOT is best illustrated assuming for simplicity a 2D\nparabolic form of the spin-independent part of the total Hamiltonian, H=p2\n2m+HR(D)+Hex,\nand the limit of Hex\u001dHR(D). In equilibrium, the carrier spins are then approximately\naligned with the exchange \feld, independent of their momentum. The origin of the SOT\ncan be understood from solving the Bloch equations for carrier spins during the acceleration\nof the carriers in the applied electric \feld, i.e., between the scattering events. Let's de\fne\nx-direction as the direction of the applied electric \feld E. For\u0000MkE, the equilibrium\ne\u000bective magnetic \feld acting on the carrier spins, s=\u001b\n2, due to the exchange term is,\nBeq\neff\u0019(2JM; 0;0), in units of energy. During the acceleration in the applied electric \feld,\ndpx\ndt=eEx, and the e\u000bective magnetic \feld acquires a time-dependent y-component due to\nHRfor whichdBeff;y\ndt=2\u000b\n~dpx\ndt, as illustrated in Fig. 1b. For small tilts of the spins from\nequilibrium, the Bloch equationsds\ndt=1\n~(s\u0002Beff) yield,sx\u0019s,sy\u0019sBeff;y\nBeq\neff, and\nsz\u0019\u0000~s\n(Beq\neff)2dBeff;y\ndt=\u0000s\n2J2M2\u000beEx: (1)\nThe non-equilibrium spin orientation of the carries acquires a time and momentum indepen-\ndentszcomponent.\nAs illustrated in Figs. 1b,c, szdepends on the direction of the magnetization Mwith\nrespect to the applied electric \feld. It has a maximum for M(anti)parallel to Eand\n3vanishes for Mperpendicular to E. For a general direction of Mwe obtain,\nsz;M\u0019s\n2J2M2\u000beExcos\u0012M\u0000E: (2)\nThe total non-equilibrium spin polarization Szis obtained by integrating sz;Mover all oc-\ncupied states. The non-equilibrium spin polarization produces an out-of-plane \feld which\nexerts a torque on the in-plane magnetization. From Eq. (2) we obtain that this intrinsic\nSOT is anti-damping-like,\ndM\ndt=J\n~(M\u0002Sz^z)\u0018M\u0002([E\u0002^z]\u0002M): (3)\nFor the Rashba spin-orbit coupling, Eq. (3) applies to all directions of the applied electric\n\feld with respect to crystal axes. By replacing HRwithHDwe can follow the same argu-\nments and arrive at the corresponding expressions for the anti-damping SOT. In the case\nof the Dresselhaus spin-orbit coupling, the symmetry of the anti-damping SOT depends\non the direction of Ewith respect to crystal axes, as seen from Fig. 1a. For a particular\nelectric \feld direction one can interpolate the angle, \u0012M\u0000E, dependence of the Dresselhaus\nout of plane \feld by the relative phase of the Rashba and Dresselhaus polarization along E.\nIn the following table we summarise the angle dependence of the Rashba and Dresselhaus\ncontributions to sz;Mfor electric \felds along di\u000berent crystal directions.\nRashba:sz;M\u0018Dresselhaus: sz;M\u0018\nEk[100] cos\u0012M\u0000E sin\u0012M\u0000E\nEk[010] cos\u0012M\u0000E\u0000sin\u0012M\u0000E\nEk[110] cos\u0012M\u0000E cos\u0012M\u0000E\nEk[1\u000010] cos\u0012M\u0000E\u0000cos\u0012M\u0000E\nTo highlight the analogy between our anti-damping SOT and the Berry phase origin of\nthe SHE24,25we illustrate in Fig. 1d the solution of the Bloch equations in the absence of the\nexchange Hamiltonian term.25In this case Beq\neffdepends on the carrier momentum which\nimplies a momentum-dependent z-component of the non-equilibrium spin,\nsz;p\u0019s~2\n2\u000bp2\u000beExsin\u0012p: (4)\nClearly the same spin rotation mechanism which generates the spin accumulation in the case\nof our anti-damping SOT (Fig. 1b) is responsible for the scattering-independent spin-current\nin the SHE (Fig. 1d).\n4To complete the picture of the common origin between the microscopic physics of the\nBerry phase SHE and our anti-damping SOT we point out that equivalent expressions for\nthe SHE spin current and the SOT spin polarization can be obtained from the quantum-\ntransport Kubo formula. The expression for the out-of-plane non-equilibrium spin polariza-\ntion that generates our anti-damping SOT is given by\nSz=~\nVX\nk;a6=b(fk;a\u0000fk;b)Im[hk;ajszjk;bihk;bjeE\u0001vjk;ai]\n(Ek;a\u0000Ek;b)2; (5)\nwherea;bindicate the band indices. Here fkaare the Fermi-Dirac distribution functions\ncorresponding to band energies Eka. This expression is analogous to Eq. (9) in Ref. 25 for\nthe Berry phase intrinsic SHE.\nWe now discuss our low-temperature (6 K) experiments in which we identify the pres-\nence of the anti-damping SOT in our in-plane magnetized (Ga,Mn)As samples. We follow\nthe methodology of several previous experiments2,7and use current induced ferromagnetic\nresonance to investigate the magnitude and symmetries of the alternating \felds responsible\nfor resonantly driving the magnetisation. In our experiment, illustrated schematically in\nFig. 2a, a signal generator drives a microwave frequency current through a 4 \u0016m\u000240\u0016m\nmicro-bar patterned from a 18 nm thick (Ga,Mn)As epilayer with nominal 5% Mn-doping.\nA bias tee is used to measure the dc voltage across the sample, which is generated according\nto Ohm's law due to the product of the oscillating magneto-resistance (during magnetisation\nprecession) and the microwave current.37Solving the equation of motion for the magnetisa-\ntion (the LLG equation) for a small excitation \feld vector ( hx;hy;hz) exp [i!t] we \fnd dc\nvoltages containing symmetric ( VS) and anti-symmetric ( VA) Lorentzian functions, shown\nin Fig. 2b. As the saturated magnetization of the sample is rotated, using \u0012M\u0000Eto indicate\nthe angle from the current/bar direction, the in-plane and out-of plane components of the\nexcitation \feld are associated with VSandVAvia:\nVS/hzsin 2\u0012M\u0000E; (6)\nVA/\u0000hxsin\u0012M\u0000Esin 2\u0012M\u0000E+hycos\u0012M\u0000Esin 2\u0012M\u0000E: (7)\nIn this way we are able to determine, at a given magnetization orientation, the current\ninduced \feld vector. In Fig. 2c we show the angle dependence of VSandVAfor an in-\nplane rotation of the magnetization for a micro-bar patterned in the [100] crystal direction.\n5As described in the Supplementary information, the voltages VSandVAare related to the\nalternating excitation \feld, using the micro-magnetic parameters and AMR of the sample.\nThe in-plane \feld components, determined from VA, are well \ftted by a M-independent\ncurrent induced \feld vector ( \u00160hx;\u00160hy)=(-91,-15) \u0016T referenced to a current density of\n105Acm\u00002. SinceVSis non-zero, it is seen that there is also a signi\fcant hzcomponent\nof the current induced \feld. Furthermore, since VSis not simply described by sin 2 \u0012M\u0000E,\nhzdepends on the in-plane orientation of the magnetization. To analyse the symmetry of\nthe out-of-plane \feld we \ft the angle dependence of VS, \fnding for the [100] bar shown in\n(Fig. 2c) that \u00160hz= (13 + 95 sin \u0012M\u0000E+ 41 cos\u0012M\u0000E)\u0016T.\nWe measure 8 samples, 2 patterned in each crystal direction and plot in Fig. 3 the resulting\nsin\u0012M\u0000Eand cos\u0012M\u0000Ecoe\u000ecients of hz. The corresponding in-plane \felds are also shown:\nsince these are approximately magnetisation-independent they can be represented in Fig. 3\nby a single vector. In the [100] bar we found that the sin \u0012M\u0000Ecoe\u000ecient of hz, which\naccording to the theoretical model originates in the Dresselhaus spin-orbit term, is greater\nthan the cos \u0012M\u0000Ecoe\u000ecient related to the Rashba spin-orbit term (see Table 1). If we\nexamine the symmetries of hzin our sample set, we \fnd that they change in the manner\nexpected for samples with dominant Dresselhaus term; a trend that is in agreement with the\nin-plane \felds. The angle-dependence of hzmeasured throughout our samples indicates an\nanti-damping like SOT with the theoretically predicted symmetries. Since the magnitude of\nthe measured hzis comparable to the in-plane \felds (see Supplementary information for a\ndetailed comparison), the anti-damping and \feld-like SOTs are equally important for driving\nthe magnetisation dynamics in our experiment.\nTo model the measured anti-damping SOT, assuming its Berry phase intrinsic origin,\nwe start from the e\u000bective kinetic-exchange Hamiltonian describing (Ga,Mn)As:36H=\nHKL+Hstrain +Hex. HereHex=JpdcMnSMn^M\u0001s,HKLandHstrain refer to the strained\nKohn-Luttinger Hamiltonian for the hole systems of GaAs (see Supplementary information),\nsis the hole spin operator, SMn= 5=2,cMnis the Mn density, and Jpd= 55 meV nm3is\nthe kinetic-exchange coupling between the localized d-electrons and the valence band holes.\nThe Dresselhaus and Rashba symmetry parts of the strain Hamiltonian in the hole-picture\n6are given by\nHstrain =\u00003C4[sx(\u000fyy\u0000\u000fzz)kx+ c:p:] (8)\n\u00003C5[\u000fxy(kysx\u0000kxsy) + c:p:];\nwhereC4= 10 eV \u0017A and we take C5=C4.5,6These momentum-dependent Hstrain terms are\nessential for the generation of SOT because they break the space-inversion symmetry. The\nmomentum-dependent spin-orbit contribution to HKLdoes not produce directly a SOT but\nit does interfere with the linear in-plane momentum terms in Hstrain to reduce the magnitude\nof the SOT and introduce higher harmonics in the \u0012M\u0000Edependence of \u00160hz. We have also\nreplacedHKLwith a parabolic model with e\u000bective mass m\u0003= 0:5meand included the spin-\norbit coupling only through the Rashba and Dresselhaus-symmetry strain terms given by\nEq. (8). The expected cos \u0012M\u0000Eor sin\u0012M\u0000Esymmetry without higher harmonics follows.\nIn addition, a large increase of the amplitude of the e\u000bect is observed since the broken\ninversion symmetry spin-texture does not compete with the centro-symmetric one induced\nby the large spin-orbit coupled HKLterm. This indicates that for a system in which the\ndominant spin-orbit coupling is linear in momentum our Berry phase anti-damping SOT\nwill be largest.\nIn Fig. 4 we show calculations for our (Ga,Mn)As samples including the spin-orbit coupled\nHKLterm (full lines) term or replacing it with the parabolic model (dashed lines). The\nnon-equilibrium spin density induced by the Berry phase e\u000bect is obtained from the Kubo\nformula:3\nSz=~\n2\u0019VReX\nk;a6=bhk;ajszjk;bihk;bjeE\u0001vjk;ai[GA\nkaGR\nkb\u0000GR\nkaGR\nkb]; (9)\nwhere the Green's functions GR\nka(E)jE=EF\u0011GR\nka= 1=(EF\u0000Eka+i\u0000), with the property\nGA= (GR)\u0003.EFis the Fermi energy and \u0000 is the disorder induced spectral broadening,\ntaken in the simulations to be 25 meV. Note that in the disorder-free limit, Eq. (9) turns\ninto Eq. (5) introduced above. The relation between Szand the e\u000bective magnetic \feld\ngenerating the Berry phase SOT is given by \u00160hz=\u0000(Jpd=g\u0016 B)Sz, where\u0016Bis the Bohr\nmagneton, and g= 2 corresponds to the localized d-electrons in (Ga,Mn)As (for more details\non the modeling see Supplementary Information).\nResults of our calculations are compared in Fig. 4 with experimental dependencies of hzon\n\u0012M\u0000Emeasured in the 4 micro-bar directions. As expected, the parabolic model calculations\n7strongly overestimate the SOT \feld hz. On the other hand, including the competing centro-\nsymmetric HKLterm, which is present in the (Ga,Mn)As valence band, gives the correct\norder of magnitude of hzas compared to experiment. Moreover, by including the HKLterm\nwe can also explain the presence of higher harmonics in the \u0012M\u0000Edependencies seen in\nexperiment. This con\frms that the experimentally observed anti-damping SOT is of the\nBerry phase origin.\nTo conclude, we have predicted a Berry phase SOT phenomenon and experimentally\nidenti\fed the e\u000bect in (Ga,Mn)As which is a model ferromagnetic system with broken space\ninversion symmetry in the bulk crystal. Learning from the analogy with the intrinsic Berry\nphase anomalous Hall e\u000bect, \frst identi\fed in (Ga,Mn)As and subsequently observed in a\nnumber of ferromagnets, we infer that our Berry phase SOT is a generic phenomenon in\nspin-orbit coupled magnetic systems with broken space-inversion symmetry. In particular,\nthe Berry phase SOT might be present in ferromagnet/paramagnet bilayers with the broken\nstructural inversion symmetry. The resulting Rashba-like anti-damping SOT has the same\nbasic symmetry of its magnetization dependence as the earlier reported SHE-STT mech-\nanism. Therefore, two strong relativistic mechanisms of scattering-independent origin can\ncontribute in the current-induced magnetization switching in these technologically important\nmagnetic structures.\nMethods and Materials\nMaterials: The 18 nm thick (Ga 0:95,Mn 0:05)As epilayer was grown on a GaAs [001]\nsubstrate by molecular beam epitaxy, performed at a substrate temperature of 230 C. It\nwas subsequently annealed for 8 hours at 200 C. It has a Curie temperature of 132 K; a\nroom temperature conductivity of 387 \n\u00001cm\u00001which increases to 549 \n\u00001cm\u00001at 5 K;\nand has a carrier concentration at 5 K determined by high magnetic \feld Hall measurement\nof 1:1\u00021021cm\u00003.\nDevices: Two terminal microbars are patterned in di\u000berent crystal directions by electron\nbeam lithography to have dimensions of 4 \u000240\u0016m. These bars have a typical low temperature\nresistance of 10 k\n (data-table in supplementary information).\nExperimental procedure: A pulse modulated (at 789 Hz) microwave signal (at 11\nGHz) with a source power of (20 dBm) is transmitted down to cryogenic temperatures using\n8low-loss, low semi-rigid cables. The microwave signal is launched onto a printed circuit\nboard patterned with a coplanar waveguide, and then injected into the sample via a bond-\nwire. The recti\fcation voltage, generated during microwave precession, is separated from\nthe microwave circuit using a bias tee, ampli\fed with a voltage ampli\fer and then detected\nwith lock-in ampli\fer. All measurements are performed with the samples at 6 K.\nCalibration of microwave current: The resistance of a (Ga,Mn)As micro-bar depends\non temperature, and therefore on the Joule heating by an electrical current. First, the\nresistance change of the micro-bar due to Joule heating of a direct current is measured.\nThen, the resistance change is measured as a function of applied microwave power. We\nassume the same Joule heating (and therefore resistance change of the micro-bar) for the\nsame direct and rms microwave currents, enabling us to calibrate the unknown microwave\ncurrent against the known direct current.\nCorresponding author\nCorrespondence and requests for materials should be addressed to AJF.\n(ajf1006@cam.ac.uk).\nAcknowledgment\nWe acknowledge fruitful discussions with, and support from EU ERC Advanced Grant\nNo. 268066, EU grant FP7 215368 SemiSpinNet, from the Ministry of Education of the\nCzech Republic Grant No. LM2011026, from the Academy of Sciences of the Czech Republic\nNo. AV0Z10100521, Praemium Academiae, and from U.S. grants onr-n000141110780, NSF-\nDMR-1105512 and NSF TAMUS LSAMP BTD Award 1026774. AJF acknowledges support\nfrom a Hitachi research fellowship. 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Strain-induced valence-subband splitting in\nIII-V semiconductors. Physica B 46 , 6781 (1992).\n39Stefanowicz, W. et al. Magnetic anisotropy of epitaxial (Ga,Mn)As on (113)A GaAs. Phys.\nRev.B 81 , 155203 (2010).\n12'paeEt'Beffa'pE\n[100][110][010]ab\ncd[1-10][100][001]\n[100][001][001]M\n0=MMB\u0010~eqeff\nEM\nE[100][100]\nMB\u0010~eqeffeqeffBFIG. 1: Spin-orbit coupling and anti-damping SOT. a, Rashba (red) and Dresselhaus (blue)\nspin textures. b,For the case of a Rashba-like symmetry, the out-of-plane non-equilibrium carrier\nspin-density that generates the Berry phase anti-damping SOT has a maximum for E(anti)parallel\ntoM. In this con\fguration the equilibrium e\u000bective \feld Beq\neffand the additional \feld \u0001 Beff?M\ndue to the acceleration are perpendicular to each other causing all spins to tilt in the same out-\nof-plane direction. c,For the case of a Rashba-like symmetry, the out-of-plane non-equilibrium\ncarrier spin-density is zero for E?MsinceBeq\neffand \u0001Beffare parallel to each other. d,The\nanalogous physical phenomena for zero magnetization induces a tilt of the spin out of the plane\nthat has opposite sign for momenta pointing to the left or the right of the electric \feld, inducing\nin this way the intrinsic Berry phase SHE.25\n13a\nb\nc\n[100]\n0.2\n0.3\n0.4\n0.5\n-10\n-5\n0\n5\nVdc(µV)\nµ0H0(T)\nExp.\nS + A\nS\nA\n0\n90\n180\n270\n360\n-10\n-5\n0\n5\n10\nVdc(µV)\nS\nA\nθM-E(deg.)\n[001]\ny x\nθM-EFIG. 2: Spin-orbit FMR experiment. a, Schematic of the sample, measurement setup and\nmagnetisation precession. Microwave power goes through a bias-tee and into the (Ga,Mn)As\nmicro-bar which is placed inside a cryostat. The injected microwave current drives FMR that is\ndetected via a dc voltage Vdcacross the micro-bar. We de\fne \u0012M\u0000Eas an angle of the static\nmagnetisation direction determined by the external magnetic \feld, measured from the current\n\row direction. b,Typical spin-orbit FMR signal driven by an alternating current at 11 GHz and\nmeasured by Vdcas a function of external magnetic \feld. The data were \ftted by a combination of\nsymmetric and anti-symmetric Lorentzian functions. c,Symmetric and antisymmetric component\nofVdcas a function of \u0012M\u0000Efor current along the [100] direction.\n14[100]\n[010]\n[1-10][110]\nµ0hx\nµ0hyop\nj\n100 mT\nsin\ncos\nconst.\n-200\n-100\n0\n100\n200\nsin\ncos\nconst.\nsin\ncos\nconst.\nsin\ncos\nconst.\nµ0hz(µT)\nθM-E-dep.FIG. 3: In-plane and out-of-plane SOT \felds. In-plane spin-orbit \feld and coe\u000ecients of the\ncos\u0012M\u0000Eand sin\u0012M\u0000E\fts to the angle-dependence of out-of-plane SOT \feld for our sample set.\nFor the in-plane \felds, a single sample in each micro-bar direction is shown (corresponding to the\nsame samples that yield the blue out-of-plane data points). In the out-of-plane data, 2 samples are\nshown in each micro-bar direction. The symmetries expected for the anti-damping SOT, on the\nbasis of the theoretical model for the Dresselhaus term in the spin-orbit interaction, are shown by\nlight green shading. All data are normalised to a current density of 105Acm\u00002.\n1590 180 270-150-100-50050100150µ0 hz [ µT ][100]\nExperimentK-L modelParabolic model90 180 270-150-100-50050100150[010]\n0 90 180 270 360eM-J-150-100-50050100150µ0 hz [ µT ][1-10]90 180 270 360eM-J-150-100-50050100150[110]⇥1603.3⇥\nθM-E (deg.) θM-E (deg.) FIG. 4: Theoretical modeling of the measured angular dependencies of the SOT \felds.\nMicroscopic model calculation for the measured (Ga,Mn)As samples assuming Rashba ( \u000fxy=\n\u00000:15%) and Dresselhaus ( \u000fxx=\u00000:3%) strain. Solid blue lines correspond to the calculations\nwith the centro-symmetric HKLterm included in the (Ga,Mn)As Hamiltonian. Dashed blue lines\ncorrespond to replacing HKLwith the parabolic model. Both calculations are done with a disorder\nbroadening \u0000 = 25 meV. Black points are experimental data whose \ftting coe\u000ecients of the\ncos\u0012M\u0000Eand sin\u0012M\u0000E\frst harmonics correspond to blue points in Fig. 3.\n16Supplementary Information\n1. FMR linewidth analysis and sample parameters\nWe use the phenomenological Landau-Lifshitz-Gilbert (LLG) equation to describe the\nspin-orbit-induced magnetisation dynamics in our (Ga,Mn)As micro-bars:\n@M\n@t=\u0000\rM\u0002Htot+\u000b\nMs\u0012\nM\u0002@M\n@t\u0013\n\u0000\rM\u0002hso (10)\nHere,\rand\u000b, Msare the gyromagnetic ratio, the dimensionless Gilbert damping con-\nstant and the saturation magnetisation respectively. The \frst term describes precession\nof the magnetisation Maround the total static magnetic \feld Htot, which includes both\nmagneto-crystalline anisotropy \felds and the externally-applied \feld. Relaxation towards\nthe equilibrium direction is expressed by the second term. When Mis resonantly driven,\nin our case by the SO-torques included as the third term in the equation, it undergoes\nsteady-state precession around the H totdirection. We assume a small precession angle, such\nthat the magnetisation dynamics is within the linear excitation regime, hence we can write\nM= (Ms;mbei!t;mcei!t) within the right-hand coordinate system de\fned by the equilib-\nrium orientation of M(shown in Fig. 5). In this coordinate system hsocan be given by the\nfollowing, where \u0012M\u0000Eis the angle between Mand the current direction.\nhso=0\nBBB@hxcos\u0012M\u0000E+hysin\u0012M\u0000E\n\u0000hxsin\u0012M\u0000E+hycos\u0012M\u0000E\nhz1\nCCCAei!t; (11)\nFIG. 5: The co-ordinate systems used. These are either de\fned with respect to the current\ndirection (in the case of the spin orbit \feld) or with respect to the magnetisation (in the\nderivation of the recti\fcation voltage).\n17Solving the LLG equation to \frst order, the expression for m bcan be found as:\nmb=\u0000[i(!=\r)hz+ (H0+H1+i\u0001H)(\u0000hxsin\u0012M\u0000E+hycos\u0012M\u0000E)]Ms\n(!=\r)2\u0000(H0+H1+i\u0001H)(H0+H2+i\u0001H)(12)\nwhere \u0001H=\u000b!=\r andH1andH2contain magnetic anisotropy terms:\nH1=Ms\u0000H2?+H2kcos2\u0010\n'+\u0019\n4\u0011\n+1\n4H4k(3 + cos 4') (13)\nH2=H4kcos 4'\u0000H2ksin 2'; (14)\nH2?,H2kandH4krepresent the out-of-plane uniaxial, in-plane uniaxial and in-plane biaxial\nanisotropy respectively, and 'is the angle between the magnetisation vector Mand the\n[100] crystallographic axis. For in-plane equilibrium orientation of M, only the alternating\nin-plane angle (\u0018mb(t)/Ms) will lead to a recti\fcation voltage, and we can neglect the out-\nof-plane component of the precession. The magnetisation precession causes a time-varying\nresistance change originating in the anisotropic magnetoresistance (AMR): R(t) =R0\u0000\n\u0001Rcos2(\u0012M\u0000E+mb(t)=Ms). This, together with a microwave current at the same frequency,\nproduces a voltage, V(t) =Icos(!t)\u0001R(t), and we measured the dc component which is\ngiven byVdc= (I\u0001Rmb=2Ms) sin 2\u0012M\u0000E. Using Eq. (12) with the above approximation and\nfocusing on the real components, we can \fnd the dc component as:\nRefVdcg=Vsym\u0001H2\n(H0\u0000Hres)2+ \u0001H2+Vasy\u0001H(H0\u0000Hres)\n(H0\u0000Hres)2+ \u0001H2(15)\nVsym(\u0012M\u0000E) =I\u0001R!\n2\r\u0001H(2Hres+H1+H2)sin(2\u0012M\u0000E)hz (16)\nVasy(\u0012M\u0000E) =I\u0001R(Hres+H1)\n2\u0001H(2Hres+H1+H2)sin(2\u0012M\u0000E)(\u0000hxsin\u0012M\u0000E+hycos\u0012M\u0000E) (17)\nWe used these equations to quantify hx,hyandhzfrom the in-plane angle dependence\nofVdc. Each FMR trace was \frst \ft by a function with symmetric and anti-symmetric\nLorentzians and both components are analysed by Vsym(\u0012M\u0000E) andVasy(\u0012M\u0000E). In table I\nwe list experimental measurements of the magnetisation independent in-plane and magneti-\nsation dependent out-of-plane spin-orbit \felds for our set of 8 samples.\nIn table II, we give the uniaxial ( Hu) (along [1-10]) and cubic ( Hc) anisotropies; \u00160Me\u000b\nand the linewidth (at a frequency of 11 GHz) for each of our 8 samples, extracted from\nthe angle-dependent FMR measurements. In addition, we show the sample resistances and\nAMRs.\n18Sample 1 2 3 4 5 6 7 8\nDirection [100] [100] [010] [010] [110] [110] [1-10] [1-10]\n\u00160hx(\u0016T) -49 -91 132 96 2 2<1<1\n\u00160hy(\u0016T) -17 -15 -49 -30 127 120 -201 -145\n\u00160hz\u0000sin\u0012M\u0000E(\u0016T)51 95-122 -107 4 6 8 5\n\u00160hz\u0000cos\u0012M\u0000E(\u0016T)20 41 19 42 161 203 -127 -86\n\u00160hz- const. (\u0016T) -10 13 -23 25 27<127 2\nTABLE I: Amplitudes of the spin-orbit e\u000bective \felds for di\u000berent directions and symmetries.\nSample 1 2 3 4 5 6 7 8\nDirection [100] [100] [010] [010] [110] [110] [1-10] [1-10]\n\u00160Hc(mT) 59 66 61 65 62 62 60 58\n\u00160Hu(mT) 59 43 45 68 40 38 51 65\n\u00160Me\u000b(mT) 429 411 437 360 404 402 350 368\n\u00160\u0001H(mT) 7.1 7.4 8.2 6.8 9.4 8.8 7.5 6.9\nAMR (\n) 45 44 44 45 151 154 140 129\nR (k\n) 11.3 11.3 11.3 11.3 11.4 11.4 11.5 10.7\nTABLE II: Magnetic anisotropy and transport parameters in the studied devices.\n2. Theory of Intrinsic Spin-Orbit Torque\nThe dynamical interaction of the magnetization originating from localized moments aris-\ning from the d-electrons and the delocalized hole carriers in (Ga,Mn)As gives rise to an\ne\u000bective current-induced \feld \u000eH. The magnetization dynamics is then described by the\nLandau-Lifshitz-Gilbert equation\nd^M\ndt=\u0000\r^M\u0002(H+\u000eH) +\u000bd^M\ndt\u0002^M (18)\n19where His the external and internal equilibrium e\u000bective magnetic \feld, \u000bis the Gilbert\ndamping parameter and \r=ge=2m0is the gyromagnetic factor with ethe elementary charge\nandm0the electron mass. The current induced \feld is given by\n\u000eH=\u0000Jex\ng\u0016B\u000es; (19)\nwhere\u0016Bis the Bohr magneton, g= 2 corresponds to the localized d-electrons in (Ga,Mn)As,\nandJex= 55 meV nm3is the antiferromagnetic kinetic-exchange coupling between the\nlocalized d-electrons and the valence band holes, termed Jpd.\u000esis the current induced\nnon-equilibrium spin densities. We model the carriers in these systems are modeled by\na Hamiltonian with a kinetic exchange coupling term H=HGaAs +Hex, whereHex=\nJexcMnSMn^M\u0001s,HGaAs refers to the 4-band strained Kohn-Luttinger Hamiltonian for the\nhole systems of GaAs (see below), sis the 4\u00024 spin operator for the holes described by\nthe four-band Kohn-Luttinger model, SMn= 5=2, andcMncorresponds to the Mn local\nspin-density.\nThe current-induced spin density has two contributions, \u000es=\u000esext+\u000esint. The extrin-\nsic contribution, \u000esext,1,2arises from the non-equilibrium steady state distribution function\nof the carriers due to the interaction of the applied electric \feld and the spin-orbit cou-\npling(SOC) carriers, i.e.predominantly independent of the magnetization and therefore of\n\feld-like form. However, there is another contribution not discussed theoretically before\nwhich is the focus of our study. This contribution arises from the electric-\feld induced po-\nlarization of the spins as they accelerate between scattering events, i.e.of purely intrinsic\norigin arising from the band structure of the system, which has the form, \u000esint/^M\u0002a(E),\nwhere a(E) is an in-plane function linear in the electric \feld that depends on the symmetry\nof the SOC responsible for the e\u000bect, Rashba or Dresselhaus, as discussed in the main text.\nThis gives rise to an anti-damping torque, \u001canti\u0000damp/^M\u0002(^M\u0002a(E)) and it is, in the\ncase of (Ga,Mn)As, of the same order of magnitude as the extrinsic \feld-like SOT.\nThis current-induced non-equilibrium spin densities, \u000es, can be calculated by the linear\nKubo response theory:3\n\u000es=~\n2\u0019VReX\nk;a;b(\u001b)ab(eE\u0001v)ba[GA\nkaGR\nkb\u0000GR\nkaGR\nkb]; (20)\nwhere the Green's functions GR\nka(E)jE=EF\u0011GR\nka= 1=(EF\u0000Eka+i\u0000), with the property\nGA= (GR)\u0003. The carrier states are labeled by momentum k, band index a, andEFis the\n20Fermi energy. \u0000 = ~=2\u001cis the spectral broadening corresponding to a relaxation time \u001c.\nHere the matrix elements of an operator ^Care ( ^C)ab\u0011hkaj^Cjkbior (^C)a\u0011hkaj^Cjkai.\nThe intra-band contributions in the above expressions correspond to the component already\ndiscussed before which gives rise to the \feld-like torque,1{3and the inter-band contribution\nis the one that gives rise to the intrinsic anti-damping SOT in analogy to the intrinsic SHE.\nThe expression for \u000esintin the clean limit is given by\n\u000esint=~\nVX\nk;a6=bIm [(s)ab(eE\u0001v)ba]\n(Eka\u0000Ekb)2(fka\u0000fkb): (21)\nHerefkaare the Fermi-Dirac distribution functions corresponding to band energies Eka.\nIn the presence of disorder, as it is the case for (Ga,Mn)As, the resulting expression are\napproximated by\n\u000esint=\u000es(1)+\u000es(2)\n\u000es(1)=\u00001\nVX\nk;a6=b2Re [(\u001b)ab(eE\u0001v)ba]\n\u0002\u0000(Eka\u0000Ekb)\n[(Eka\u0000Ekb)2+ \u00002]2(fka\u0000fkb) (22)\n\u000es(2)=\u00001\nVX\nk;a6=b2Im [(\u001b)ab(eE\u0001v)ba]\n\u0002\u00002\u0000(Eka\u0000Ekb)2\n[(Eka\u0000Ekb)2+ \u00002]2fka:\nHere we have ignored small numerical corrections due to the GR\nkaGR\nkbterms which can be\nshown to formally vanish in a weak disorder situation and whose rapid oscillations can lead\nto numerical instabilities giving rise to systematic errors.\nThe hole-valence system is described by HGaAs =HKL+Hstrain, where the \frst term is\nthe Kohn-Luttinger Hamiltonian and the second contains the strain e\u000bects. The four-band\nKohn-Luttinger Hamiltonian in the hole-picture is\nHKL=~2k2\n2m0\u0012\n\r1+5\n2\r2\u0013\nI4\u0000~2\nm0\r3(k\u0001J)2(23)\n+~2\nm0(\r3\u0000\r2)\u0000\nk2\nxJ2\nx+k2\nyJ2\ny+k2\nzJ2\nz\u0001\n:\nHere, kis the momentum of the holes, m0is the electron mass, \r1= 6:98,\r2= 2:06, and\n\r3= 2:93 are the Luttinger parameters, I4is the 4\u00024 identity matrix and J= (Jx;Jy;Jz)\nare the 4\u00024 angular momentum matrices of the holes. Here the hole spin s=J=3, where\nsare the spin matrices for holes.4\n21The strain Hamiltonian in the hole-picture is\nHstrain =b\u0014\u0012\nJ2\nx\u0000J2\n3\u0013\n\u000fxx+ c:p:\u0015\n\u0000C4[Jx(\u000fyy\u0000\u000fzz)kx+ c:p:] (24)\n\u0000C5[\u000fxy(kyJx\u0000kxJy) + c:p:];\nwhere\u000fijis the strain tensor and b=\u00001:7 eV is the axial deformation potential. C4is\nthe magnitude of the momentum-dependent Dresselhaus-symmetric strain term and C5is\nthe magnitude of the Rashba-symmetric strain term. In our calculations, we use the value\nC4= 10 eV \u0017A calculated5,6from \frst principles for holes in (Ga,Mn)As and C5=C4. To the\nbest of our knowledge, there is no measurement or calculation for the C5term in (Ga,Mn)As.\nIn our calculations we set \r2=\r3within the spherical approximation and for the parabolic\napproximation we set \r2=\r3= 0 and take \r1= 2. The external electric \feld magnitude is\nset toE= 0:02 mV/nm (from the experimental values), the disorder broadening to \u0000 = 25\nmeV, and the strain to \u000fxx=\u000fyy=\u00001:1\u000fzz=\u00000:3% and\u000fxy=\u00000:15%. The \frst term\nof the strain Hamiltonian is momentum independent. The other two terms are momentum-\ndependent and they are essential for the generation of SOT because they break the space\ninversion symmetry. The second term has a Dresselhaus symmetry and the third has a\nRashba symmetry. As described in the experimental results, these symmetries are shared\nby the observed SOT. In this discussion we have neglected cubic Dresselhaus terms, allowed\nby the GaAs symmetry, since the experimentally observed SOTs vary linearly with strain.2,7\n1Manchon, A. & Zhang, S. Theory of spin torque due to spin-orbit coupling. Phys. Rev. B 79 ,\n094422 (2009).\n2Chernyshov, A. et al. Evidence for reversible control of magnetization in a ferromagnetic material\nby means of spin-orbit magnetic \feld. Nature Phys. 5, 656 (2009). arXiv:0812.3160.\n3Garate, I. & MacDonald, A. H. In\ruence of a transport current on magnetic anisotropy in\ngyrotropic ferromagnets. Phys. Rev. B 80 , 134403 (2010). arXiv:0905.3856.\n4Abolfath, M., Jungwirth, T., Brum, J. & MacDonald, A. H. Theory of magnetic anisotropy in\nIII1\u0000xMnxV ferromagnets. Phys. Rev. B 63 , 054418 (2001). arXiv:cond-mat/0006093.\n225Silver, M., Batty, W., Ghiti, A. & OReilly, E. P. Strain-induced valence-subband splitting in\nIII-V semiconductors. Physica B 46 , 6781 (1992).\n6Stefanowicz, W. et al. Magnetic anisotropy of epitaxial (Ga,Mn)As on (113)A GaAs. Phys. Rev.\nB 81 , 155203 (2010).\n7Fang, D. et al. Spin-orbit driven ferromagnetic resonance: A nanoscale magnetic characterisation\ntechnique. Nature Nanotech. 6, 413 (2011). arXiv:1012.2397.\n23" }, { "title": "1905.07296v2.Chiral_p_wave_superconductors_have_complex_coherence_and_magnetic_field_penetration_lengths.pdf", "content": "Chiral p-wave superconductors have complex coherence and magnetic \feld\npenetration lengths\nMartin Speight,1Thomas Winyard,1and Egor Babaev2\n1School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom\n2Department of Physics, KTH-Royal Institute of Technology, Stockholm, SE-10691 Sweden\nWe show that in superconductors that break time reversal symmetry and have anisotropy, such\nasp+ipmaterials, all order parameters and magnetic modes are mixed. Excitation of the gap \felds\nproduces an excitation of the magnetic \feld and vice versa. Correspondingly the long-range decay\nof the magnetic \feld and order parameter are in general given by the same exponent. Thus one\ncannot characterize p+ipsuperconductors by the usual coherence and magnetic \feld penetration\nlengths. Instead the system has normal modes that are associated with linear combinations of\nmagnetic \felds, moduli of and phases of the order parameter components. Each such normal mode\nhas its own decay length that plays the role of a hybridized coherence/magnetic \feld penetration\nlength. On a large part of the parameter space these exponents are complex. Therefore the system\nin general has damped oscillatory decay of the magnetic \feld accompanied by damped oscillatory\nvariation of the order parameter \felds.\nPACS numbers:\nINTRODUCTION\nSuperconducting states that spontaneously break time\nreversal symmetry (BTRS) are a subject of intense ex-\nperimental pursuit. Two types of BTRS state that at-\ntract particular interest are chiral p-wave superconduc-\ntors where the most intense discussions were focused on\nSr2RuO 4[1, 2], and s+isors+idsuperconducting\nstates, evidence for which was recently found in iron-\nbased superconductors [3, 4]. BTRS states are described\nby an order parameter that has at least two components,\nbecause they break at least U(1)\u0002Z2symmetry. Also,\nrather generically, there is anisotropy in such supercon-\nducting states. In this work we investigate the most basic\nproperty of that state: the magnetic \feld penetration and\ncoherence lengths.\nThe basic fundamental length scales of superconduc-\ntors were \frst discussed by Fritz and Heinz London [5]\nand Ginzburg and Landau [6] in an ordinary supercon-\nductor. This was done in the model for the simplest su-\nperconductor that breaks U(1) symmetry, described by\na single complex \feld j\tjneglecting crystal anisotropies.\nThe London magnetic \feld penetration length \u0015is the\npower in the exponential law of decay of the magnetic\n\feld:B=B0e\u0000r=\u0015. The coherence length \u0018is the scale\nassociated with the exponential law describing how the\nmodulusj\tjof the complex \feld, describing the order\nparameter, restores its ground state value\u0016j\tjaway from\na perturbation:j\tj(r)\u0019\u0016j\tj\u0000const e\u0000r=\u0018. The micro-\nscopic BCS theory of superconductivity related the mod-\nulus of the order parameter \feld j\tjto a superconducting\ngap \u0001 in the single-electron spectrum. The de\fnition of\nthe coherence length in the context of superconductiv-\nity has an extra factorp\n2 which we absorb for brevity in\nthe de\fnition of \u0018. Often the coherence length is assessed\nonly approximately, and it is important to remember thelimitations of these approximate de\fnitions. For example\nwhile in the simplest Ginzburg-Landau model coherence\nlength is often estimated via vortex core size or slope of\nthe order parameter near the centre of the vortex core,\nsuch estimates are known to fail even in the simplest\nmodels at low temperates [7]. Another indirect way to as-\nsess coherence length assumes its inverse proportionality\nto the gap function \u0001 in the BCS expression \u00180/1=\u0001.\nLikewise this expression has very limited validity. It can-\nnot serve as an estimate at strong coupling or in the\nmulti-component case. For example in the case of several\ngaps that would give unphysical divergence of coherence\nlength where a gap is closing (i.e. at the crossover from\ns++tos\u0006states where all coherence lengths should be\n\fnite because there is no symmetry breaking and no acci-\ndental degeneracies). Similarly that estimate would miss\nthe divergence of coherence length when a superconduc-\ntor transitions from ordinary to BTRS state i.e. stos+is\norstos+idstate, whose existence is dictated by sym-\nmetry. These examples shows that accurate coherence\nlength calculations are required while simple estimates\ncan be highly misleading. Calculations of coherence and\nmagnetic \feld penetration lengths have been made for\nisotropic multicomponent models for general interactions\nboth in phenomenological and microscopic models [8{12].\nThe multi-component nature of these systems strongly\na\u000bects only the coherence lengths, while the magnetic\n\feld penetration length is merely renormalized by inter-\ncomponent couplings. The situation was found to be\nvery di\u000berent in U(1) multiband superconductors if dif-\nferent bands have di\u000berent anisotropies. While usually\nthe magnetic (London) modes decouple from other nor-\nmal modes of the system, such as density and phase dif-\nference (Leggett) modes, having di\u000berent anisotropies in\ndi\u000berent bands results in a hybridization of the London\nmode with the phase di\u000berence mode [13{15]. For a sys-arXiv:1905.07296v2 [cond-mat.supr-con] 3 Dec 20192\ntem withNbands that means that magnetic \feld de-\ncay is described by several modes with di\u000berent expo-\nnents and there could be up N+ 1 such modes in the\nsystems considered in [13{15]. Furthermore the powers\nin the corresponding exponents under certain conditions\nare complex leading to a damped oscillatory decay of the\nmagnetic \feld.\nThat raises the question: what is the behaviour of the\nmagnetic \feld and what are the coherence lengths in p+ip\nsuperconductors, since such systems are inherently both\nmulticomponent and anisotropic? The important di\u000ber-\nence with the systems considered in [13{15], as discussed\nbelow, is the fact that such a superconducting state has\nspontaneously broken time reversal symmetry.\nThe standard Ginzburg-Landau model for a p+ipsu-\nperconductor can we written in dimensionless units as\nF=1\n2Q\u000b\f\nijDi \u000bDj \f+1\n2B2+Fp (1)\nwhere the greek indices enumerate components of the\norder parameter and latin indices stand for space di-\nrections. Summation over repeated indices is implied,\nDi=@i\u0000iAiis the covariant derivative with the gauge\n\feldAi, and the complex \felds\n \u000b=\u001a\u000bei\u0012\u000b\u000b= 1;2 (2)\nrepresent the di\u000berent superconducting components. We\nconsider here a quasi-two-dimensional system or a con-\n\fguration of a three dimensional system that is trans-\nlation invariant in the zdirection. The magnetic \feld\nB= (0;0;B) = (0;0;@1A2\u0000@2A1) is directed so that the\nspatial indices take only the values 1 ;2.Fprepresents the\npotential terms which, by gauge invariance, may depend\nonly on\u001a\u000band\u001212:=\u00121\u0000\u00122. For the standard p+ipsu-\nperconductor, the \u001212dependence enters only via a term\nof the form ( 1 \u0003\n2)2+c:c:, [16{18] that is,\nFp=V(\u001a1;\u001a2) +\u0011\n8\u001a2\n1\u001a2\n2cos 2\u001212 (3)\nwith\u0011 >0. Then the ground states (minima) of Fpare\ndegenerate occurring with \u001212=\u0006\u0019=2. The ground state\nof this system is not gauge equivalent to its complex con-\njugate. Hence the system exhibits broken time reversal\nsymmetry. Note that, although our focus below will be\non the example of p+ipsuperconductors, the model is\nvery general, also describing other BTRS states such as\ns+isands+idsuperconductors [19, 20]. Our results\nobtained below apply also to such states when anisotropy\nis present.\nThe anisotropy of the system enters through the pa-\nrametersQ\u000b\f\nij, which must satisfy Q\f\u000b\nji= (Q\u000b\f\nij)\u0003to en-\nsureFis real. Henceforth we assume, as is standard,\nthat allQ\u000b\f\nijare real.CALCULATION OF LENGTH SCALES\nThe spatial dependence of the \felds at equilibrium\nis governed by the Ginzburg-Landau (Euler-Lagrange)\nequations for the functional F=R\nR2F,\nQ\u000b\f\nijDiDj \f= 2@Fp\n@ \u000b; (4)\n@j(@jAi\u0000@iAj) =Ji; (5)\nwhere the total supercurrent is\nJi:= Im(Q\u000b\f\nij \u000bDj \f): (6)\nConsider the behaviour of the system a long distance\nfrom some defect (e.g. a vortex, domain wall or material\nboundary). Since the \felds are close to their ground state\nvalues, they should be well approximated by solutions of\nthe linearization of the Euler-Lagrange equations about\nthe ground state. That is, since the characteristic expo-\nnents, such as coherence lengths, de\fne the exponential\ndecay of a small perturbation of a \feld from its ground\nstate, in order to calculate them one expands \felds in\nthe Euler-Lagrange equations around their ground state\nvalues (see e.g. [21, 22]). For a conventional supercon-\nductor the coherence length is obtained by expanding in\nsmall deviations of the \feld modulus j j[23], but that\ncannot a priori be done for our system involving multiple\n\felds. Instead we should expand in small deviations in\nall degrees of freedom and see if there is a coupling be-\ntween the \felds arising at the lowest order. Because we\nare dealing with a superconductor we have a coupling to\nthe gauge \feld Aand some care must be taken in han-\ndling the gauge invariance of the system. Let us de\fne\nthe phase \feld\n\u0012\u0006:=1\n2(\u00121+\u00122): (7)\nNote that\u001a\u000b=j \u000bjand\u001212are gauge invariant, while\n\u0012\u0006andAiare not. The combination\npi:=Ai\u0000@i\u0012\u0006 (8)\nisgauge invariant, and our strategy is to reexpress the\nEuler-Lagrange equations in terms of \u001a\u000b,\u001212andpi. Let\nus denote the ground state values of \u001a\u000band\u001212byu\u000band\n\u00120respectively; for the p+ipmodel (3),\u00120=\u0006\u0019\n2, but it\nis instructive to leave it general, for the time being. Then\nsaying that the \felds are close to their ground state values\nmeans precisely that pi,\"\u000band\u0012\u0001are small, where\n\"\u000b:=\u001a\u000b\u0000u\u000b; \u0012 \u0001:=1\n2(\u001212\u0000\u00120): (9)\nIn particular, the small quantities \"\u000b;pi;\u0012\u0001should obey\nthe linearization of (5) about ( pi;\u001a\u000b;\u001212) = (0;u\u000b;\u00120).\nThe left hand side is exactly @j(@jpi\u0000@ipj) which is3\nalready of linear order, but we must compute the super-\ncurrentJito linear order. This is straightforward once\nwe recognize that Di \u000bis to linear order,\nDi 1= (@i\"1\u0000i(pi\u0000@i\u0012\u0001)u1)ei(\u0012\u0006+1\n2\u00120)+\u0001\u0001\u0001\nDi 2= (@i\"2\u0000i(pi+@i\u0012\u0001)u2)ei(\u0012\u0006\u00001\n2\u00120)+\u0001\u0001\u0001(10)\nso the linearization of (5) is\n@j(@jpi\u0000@ipj)\n=\u0000Q11\niju2\n1(pj\u0000@j\u0012\u0001)\u0000Q22\niju2\n2(pj+@j\u0012\u0001)\n\u0000u1u2cos\u00120fQ12\nij(pj+@j\u0012\u0001)\u0000Q21\nij(pj\u0000@j\u0012\u0001)g\n\u0000sin\u00120fQ12\niju1@j\"2\u0000Q21\niju2@j\"1g; (11)\nNote that the left hand side of this equation is pre-\ncisely the usual curl of the magnetic \feld ( pidi\u000bers\nfromAiby a gradient, so their curls coincide). The\nkey observation is that, unless \u00120= 0 or\u0019, that is, un-\nless the ground state is phase locked or antilocked (or\nQ12\nij\u00110) this PDE couples all the degrees of freedom\ntogether (through its \fnal term), so that they all decay\nto zero with the same dominant length scale. Any other\nvalue of\u00120(including\u0006\u0019\n2) corresponds to a ground state\n( 1; 2) = (u1ei1\n2\u00120;u2e\u0000i1\n2\u00120) which is not gauge equiv-\nalent to its complex conjugate, and hence breaks time\nreversal symmetry. Hence, the e\u000bects described below\nare generic when one has BTRS and spatial anisotropy.\nTo compute the length scales, we must linearize (5) in\n(pi;\"\u000b;\u0012\u0001) also. Henceforth, we specialize to the p+ip\ncase with potential (3), so that \u00120=\u0006\u0019\n2. Substituting\n(8),(9) into the Euler-Lagrange equations and discarding\nall terms nonlinear in small quantities yields\n\u0000\u0000\n@2\n1+@2\n2\u0001\npi+@i@jpj\u0000Lij@j\u0012\u0001+Kijpj\n\u0006Q12\nij(u1@j\"2\u0000u2@j\"1) = 0 (12)\n\u0006Q12\nij(u2@i@j\"1+u1@i@j\"2)\u0000Kij@i@j\u0012\u0001\n+Lij@ipj+ 2\u0011u2\n1u2\n2\u0012\u0001= 0 (13)\n\u0000Q11\nij@i@j\"1\u0006Q12\niju2@i(pj+@j\u0012\u0001) +H1\f\"\f= 0 (14)\n\u0000Q22\nij@i@j\"2\u0007Q12\niju1@i(pj\u0000@j\u0012\u0001) +H2\f\"\f= 0 (15)\nwhere we have de\fned the matrix coe\u000ecients,\nKij=Q11\niju2\n1+Q22\niju2\n2; (16)\nLij=Q11\niju2\n1\u0000Q22\niju2\n2; (17)\nand\nH\u000b\f=@2Fp\n@\u001a\u000b@\u001a\f\f\f\f\f(u1;u2;\u0006\u0019\n2)(18)\nis the Hessian of the potential Fpabout the ground state,\nwith respect to ( \u001a1;\u001a2).\nNote that in the case where there are no mixed gradi-\nent termsQ12= 0 the linearized equations decouple intoa pair for ( pi;\u0012\u0001) and a pair for ( \"1;\"2). That means\nthat small \ructuations in the density \felds do not cause\na perturbation of the phase di\u000berence and do not create\nmagnetic \feld, as is indeed the case in ordinary super-\nconductors, or in the class of anisotropic models studied\nin [13{15]. However we see that for anisotropic supercon-\nductors that break time reversal symmetry, such as p+ip\nsuperconductors, no such simpli\fcation takes place: all\nthe gauge invariant \felds \"\u000b;\u0012\u0001;piare coupled to one an-\nother, and when one changes all the others should change\ntoo. The implication of this is that systems like chiral\np+ipsuperconductors cannot be characterized by coher-\nence and magnetic \feld penetration lengths in the usual\nsense, but the decay length scale of a small perturbation\nof the order parameter \feld and magnetic \feld is in gen-\neral the same. Furthermore it implies that one cannot\nreliably use the London limit to calculate the magnetic\n\feld penetration length because density modes do not\nasymptotically decouple from magnetic modes. Below\nwe calculate these length scales.\nSince the equations are anisotropic, to extract the\nlength scales we must \frst select a direction (normal to\nthe domain wall or material boundary, or radial from the\nvortex core, depending on context), denoted by a unit\nvector n= (n1;n2), and then reduce the equations to or-\ndinary di\u000berential equations (ODEs) with n-dependent\ncoe\u000ecients, by imposing translation invariance orthogo-\nnal to n. So, we demand that\npi=a(X)n?\ni+b(X)ni\n\u0012\u0001=\u0012\u0001(X); \"\u000b=\"\u000b(X) (19)\nwhereX=nixiandn?= (\u0000n2;n1). Substituting (19)\ninto (12)-(15), one obtains a coupled set of \fve ODEs.\nThe two-vector valued ODE (12) implies a pair of scalar-\nvalued ODEs, obtained by taking its scalar product with\nnandn?. The ncomponent implies\nb=\u0000n\nn\u0001Kn\u0001\u0000\nKn?a\u0000Ln\u00120\n\u0001\u0006Q12n(u1\"0\n2\u0000u2\"0\n1)\u0001\n(20)\n(where0\u0011d=dX ), which can be used to eliminate b(X)\nfrom the other ODEs. We now have 4 coupled ODEs,\nforming a linear system, that describes the response of\nthe system to a small perturbation about its ground\nstate.\nA~ w00+B~ w0+C~ w= 0; (21)\nwhere~ w= (\"1;\"2;\u0012\u0001;a)TandA,B,Care certain con-\nstant 4\u00024 real matrices. It is important to note that A\nandCare symmetric, while Bis skew, and that all three\ndepend on the choice of direction n. Their exact form is\ngiven in Appendix A.\nRecall that (21) is the linearized system of \feld equa-\ntions describing how a system recovers from a perturba-\ntion in the n-direction under the assumption of trans-\nlation invariance orthogonal to n, for example, how the4\nsystem behaves near the boundary of a superconductor\nsubject to an external magnetic \feld. Its general solution\nis\n~ w(X) =8X\ni=1ci~ vie\u0000\u0016iX(22)\nwhere\u00161;\u00162;:::;\u0016 8, are the solutions of the degree 8\npolynomial equation\ndet\u0000\n\u00162A\u0000\u0016B+C\u0001\n= 0: (23)\nThe constants \u0016ishould be interpreted as \feld masses\nwhich set the length scale \u0015iof spatial decay of the as-\nsociated linear combination of \felds via\n\u0015i=1\n\u0016i: (24)\nThe quantities ~ v1;~ v2;:::;~ v 8, are the corresponding eigen-\nvectors (by eigenvector we mean a unit length vector sat-\nisfying (\u00162\niA\u0000\u0016iB+C)~ vi=~0), andc1;c2;:::;c 8are arbi-\ntrary constants, determined by boundary conditions and\nnonlinearities. Each exponential power is associated to a\nnormal mode, determined by ~ vi. In an ordinary super-\nconductor the normal mode associated with the coher-\nence length is the modulus of the order parameter, while\nthe magnetic \feld penetration length is attributed to a\nmassive vector \feld: the magnetic \feld. Instead, we see\nthat in the chiral p+ipsuperconductor the normal modes\nare associated with linear combinations of magnetic and\nmatter degrees of freedom.\nIndeed, the polynomial equation (23) has real coe\u000e-\ncients and is quartic in \u00162(sinceA;Care symmetric,\nwhileBis skew); hence, if \u0016is a solution, so are \u0000\u0016,\u0016\u0003\nand\u0000\u0016\u0003. This demonstrates that complex length scales\nare caused by mixing, as this is the only way for multi-\nple length scales to become linked and hence be complex\nconjugates of each other. Exactly half the eigenvalues,\nwhich we choose to label \u00161;:::;\u0016 4have positive real\npart, while the others have negative real part. We seek\nsolutions that decay to 0 as X!1 ; these are obtained\nby settingci= 0 fori\u00155 in equation (22).\nThe long-range behaviour of the \felds, in direction n,\nis governed by the dominant eigenvector ~ vi=~ v\u0003, de-\n\fned to be the eigenvector whose eigenvalue \u0016i=\u0016\u0003has\nsmallest positive real part (hence the longest length scale\n\u0015\u0003= 1=\u0016\u0003of spatial decay). Note that, in general, \u0016\u0003\nmay be complex , in which case the \felds at large Xare\nspatially oscillatory, behaving like\n(\"1;\"2;\u0012\u0001;a)\u0018c~ vr;\u0003e\u0000Re(\u0016\u0003)Xcos Im(\u0016\u0003)X; (X!1 )\n(25)\nwherecis some real constant and ~ vr;\u0003is the real part of\n~ v\u0003. The complex magnetic \feld penetration length im-\nplies oscilatory decay of the magnetic \feld as observed in\nanisotropic systems without BTRS [13{15]. Here we \fnd\nthat in ap+ipsuperconductor one cannot assume that aperturbation of the gap \felds will decay with real expo-\nnents: i.e. there are no real coherence lengths in general.\nNote that for dirty isotropic multiband superconductors,\nthe phase di\u000berence and density modes can be mixed\neven without breaking time reversal symmetry [12]. Our\n\fndings of the complete mixing of the order parameters\nand magnetic modes would apply also for that case.\nImportantly, as detailed below, in general one needs\nto retain contributions from the modes associated with\nshorter length scales.\nOf course, our analysis should reproduce the usual pic-\nture of separate real length scales (the coherence length\nand magnetic penetration depth) in the case of a spatially\nisotropic system, where Q\u000b\f\nij=\u000e\u000b\f\u000eij, and should hold\napproximately for a small perturbation of this. In the\nnear-isotropic regime, when Q11;Q22\u0019I2andQ12\u00190,\nthe coupling between \"\u000b;\u0012\u0001;ais weak, the spectrum is\nreal, and one of the eigenvectors, ~ v4say, is approximately\n(0;0;0;1), while the others, ~ v1;~ v2;~ v3, are approximately\nnormal to (0 ;0;0;1). We then recover the usual picture\nof separate length scales associated with the magnetic\n\feld,\u0015mag=\u00154, and the condensates, \u00151;2;3. Consider\nthe case where \u0015\u0003=\u0015mag, that is,~ v4is dominant. Al-\nthough all \felds do, strictly speaking, decay like (25)\n(with Im(\u0015\u0003) = 0) at very large X, the coe\u000ecients in\nfront of\"1;\"2;\u0012\u0001are very small, while the coe\u000ecient in\nfront ofais of order of unity ( ~ v\u0003=~ v4\u0019(0;0;0;1)) so\nat intermediate range contributions from the subdomi-\nnant eigenvectors are larger. This allows one to identify\napproximately \u0015magas a penetration depth and min f\u0015ig\nas a coherence length, and classify the system as type-\n2 (since\u0015mag is the largest length scale). Similar re-\nmarks apply if one of the condensate modes is domi-\nnant. This is consistent with the numerical solutions ob-\ntained earlier in such regimes [24]. One therefore may\napproximately call the exponents associated to matter-\n\feld-dominated modes coherence lengths and those as-\nsociated with magnetic-\feld dominated modes magnetic\n\feld penetration lengths. However this approximate pic-\nture disappears as one increases the anisotropy and mag-\nnetic and matter \feld couplings in (21) become signi\f-\ncant.\nIn summary, the long range behaviour of spatially de-\ncaying solutions of our system is (25) where \u0016\u0003= 1=\u0015\u0003\nis the solution of (23) with smallest positive real part.\nIn general, \u0016\u0003depends on n, the direction along which\nwe impose spatial decay, and may be complex, in which\ncase the decay of both magnetic and gap \felds is oscil-\nlatory. We have also shown that this coupling and the\noscillations in all four \felds is a direct result of BTRS\nand anisotropy.\nIn the next section we consider the implications of\nthese \fndings for the Meissner state of a p+ipsuper-\nconductor.5\nMEISSNER STATE IN A p+ipSYSTEM\nWe consider a simple p+ipmodel, such as the one\ndiscussed in the context of the debate of the nature of\nsuperconducting state in Sr 2RuO 4in [25]. This is of the\nform (1) with (after a trivial rescaling of \felds which is\nshown in detail in appendix D)\nQ11=\u00123 +\u00170\n0 1\u0000\u0017\u0013\n; Q22=\u00121\u0000\u00170\n0 3 +\u0017\u0013\n;\nQ12=\u00120 1\u0000\u0017\n1\u0000\u00170\u0013\nand potential,\nFp=V0\u001a\n1\u0000(\u001a2\n1+\u001a2\n2) +1\n8(3 +\u0017)\u0000\n\u001a2\n1+\u001a2\n2\u00012\n\u00001\n4(1 + 3\u0017)\u001a2\n1\u001a2\n2+1\n4(1\u0000\u0017)\u001a2\n1\u001a2\n2cos 2\u001212\u001b\n:(26)\nThe model contains two unknown parameters: \u00001<\u0017 <\n1 which measures the anisotropy of the Fermi surface,\nandV0, the overall strength of the potential (coinciding\nwithb=(\u0019\r2K2) in the notation of Ref. [25]). Its ground\nstates are ( \u001a\u000b;\u001212) = (1;\u0006\u0019=2), sou1=u2= 1.\nConsider a semi-in\fnite superconductor occupying the\nhalf-spaceX\u00150 (where, as before, X=n1x1+x2n2),\ndenoted \n, with the region X < 0 occupied by an insula-\ntor. Denote by @\n the boundary between these regions\n(whereX= 0). Note that n= (cos';sin') is an inward\npointing unit normal to this boundary. The system is\nsubjected to a uniform external magnetic \feld Hin the\nx3direction. Provided His not too strong, the system\nwill approach the ground state \u001a1=\u001a2= 1,\u001212=\u0019=2\n(say) in the bulk (as X!1 ). To \fnd the Meissner\nstate, we minimize the Gibbs free energy\nG=Z\n\nF\u0000HZ\n\nB+Z\n@\nFsurf(27)\nover all \felds in \n, assuming invariance under transla-\ntions normal to n. Here, we use the standard boundary\nconditions, advocated in [26], by including in the free\nenergy the surface term\nFsurf=\u001f1(\u001a2\n1+\u001a2\n2) +\u001f2(n2\n1\u0000n2\n2)(\u001a2\n1\u0000\u001a2\n2)\n+2\u001f3n1n2( \u0003\n1 2+ 1 \u0003\n2) (28)\nFor simplicity, we assume re\rection from the boundary\nisspecular , meaning that \u001f1=\u001f2=\u001f3=\u001f>0. Having\nimposed translation invariance, the problem reduces to a\none-dimensional variational problem on [0 ;1), with nat-\nural boundary conditions at 0, which can be solved by a\nstandard gradient-descent method. A more detailed dis-\ncussion of the boundary conditions is given in appendix\nB. There is a caveat here. It has been demonstrated\nrecently for s-wave superconductors, that boundary con-\nditions can be di\u000berent in superconductors from thosebased on the standard assumptions of Caroli-deGennes-\nMatricon type theory [27], which implies that the stan-\ndard theory of boundary conditions for p+ipshould also\nbe revised. However here we are interested not in the pre-\ncise \feld values at the boundary but rather in the laws\ngoverning their decay away from the boundary. There-\nfore the precise form of the boundary conditions is not\nvery important. In Appendix C we present results with\nthe extra boundary terms omitted entirely, giving the\nsame \feld decay behaviour.\nThe solutions depend on the unknown model parame-\nters\u0017;V0;\u001fas well as the applied \feld Hand the bound-\nary orientation angle '. We have run simulations for\n\u001f2f0;0:01;0:1;1;10g, \fnding no qualitative change in\nthe physics we are focussed on. For that reason we \fx\n\u001f= 1 for the remainder of this section and present a\nrepresentative sample of the other parameters. We have\nincluded a plot, \fgure 4 for \u001f= 0 in Appendix C for com-\nparison, to demonstrate that the oscillatory behaviour of\nthe \felds originates in complex coherence lengths, not\nfrom the boundary terms in eq. (28).\nForV0= 3,\u0017=\u00000:95,H= 0:3, the Meissner states\nwith boundary orientations '= 0 and'=\u0019=3 are pre-\nsented in \fgure 1. Both exhibit oscillatory tails and \feld\ninversion of both Band the condensates, consistent with\nexponential decay with a complex coherence and mag-\nnetic \feld penetration lengths. We shall return to this\nshortly. If the external \feld His increased further, the\ncondensates separate more until, for some ', one or other\nof the densities \u001a1or\u001a2hits zero. This produces a novel\nMeissner state depicted in \fgure 2 for H= 1 (well below\nthe lower critical \feld Hc1= 1:34) for several angles '.\nWe see that neither matter \feld component vanishes for\n'=\u0019=4, whereas\u001a1vanishes for '= 0, and\u001a2vanishes\nfor'=\u0019=2. As one condensate component goes to zero,\nthe other achieves a maximum exceeding its ground state\nvalue, producing a stripe (orthogonal to n) of depletion\nof one condensate and surfeit of the other.\nReturning to our main goal of testing the analysis of\nthe previous section, it is straightforward to compute,\nfor any given '(boundary orientation), \u0017(anisotropy\nparameter) and V0(potential energy scale in the GL en-\nergy) the dominant eigenvalue \u0016\u0003, and hence map out\nthe parameter set on which \u0016\u0003is complex. Figure 3\npresents pictures of the ( ';\u0017) parameter plane, for a se-\nquence of values of V0, coloured to show the parameter\ndomain where \u0016\u0003is complex. For V0small, the param-\neter domain of complex \u0016\u0003is small and con\fned to the\nedges wherej\u0017jis close to 1, but as V0increases, the\ndomain swells, eventually covering the whole parameter\nspace (when V0\u00194), predicting that the Meissner state\nshould be spatially oscillatory for all anisotropies \u0017and\nall boundary orientations 'ifV0is around this value.\nIncreasingV0still further, pockets of real \u0016\u0003return and\ngradually re\fll the whole parameter space for very high\nvalues ofV0. Turning to the parameter sets of \fgure 1,6\nFIG. 1: Superconductor-insulator boundary of a p+ipsuperconductor with V0= 3,\u0017=\u00000:95,\u001f= 1 and external \feld\nH= 0:3 for two di\u000berent boundary orientations: '= 0 (top set of four plots) and '=\u0019=3 (bottom set of four plots). The\nboundary is at X= 0, the plotted \felds are the condensate magnitudes \u001a1and\u001a2and the magnetic \feld strength B. The green\ndots mark points where the spatially oscillating \felds cross their ground state values and the blue dots mark local extrema.\nThe distances between these successive points are compared with the prediction of our linear analysis in the bottom right plot\nof each set.7\nFIG. 2: Superconductor-insulator boundary at high external \feld H= 1 and boundary orientations '= 0;\u0019\n4;\u0019\n3;\u0019\n2, showing\nstripe formation in the Meissner state: one condensate component goes to zero and the other achieves a maximum, producing\na stripe (orthogonal to n) of depletion of one condensate and surfeit of the other. Note that the \u001a1and\u001a2curves coincide in\nthe case'=\u0019\n4. The model parameters are as in Figure 1).\nFIG. 3: Plots of jIm(\u0016\u0003)j=jRe(\u0016\u0003)j, where\u0016\u0003=\u0015\u00001\n\u0003is\nthe leading mass scale (inverse length scale with smallest real\npart), in the ( ';\u0017) parameter space, for various values of\nV0. Here'is the orientation of the sample boundary, and\n\u0017,V0are parameters in the GL energy controlling the spatial\nanisotropy and the potential energy scale respectively. The\nblack regions indicate where \u0016\u0003is real and hence there will\nbe no oscillations of the magnetic \feld, or condensates, away\nfrom the sample boundary.\nV0= 3,\u0017=\u00000:95 and'= 0;\u0019=3, we \fnd in both cases\nthat\u0016\u0003is complex, consistent with the nonlinear numer-\nics (\u0016\u0003(0) = 0:689 + 0:548i,\u0016\u0003(\u0019=3) = 1:013 + 0:648i).\nThe oscillatory decay predicted by linear analysis pre-\ndicts that the zeros of B, and of\u001a\u000b\u0000u\u000bshould be\nequally spaced with period \u0019=Im(\u0016\u0003), as should succes-\nsive extrema of these functions. These gap widths can\neasily be extracted from the nonlinear numerics, and are\ndisplayed, for these parameter sets, alongside the linearprediction, in \fgure 1. The agreement is remarkable. Fi-\nnally, we have also chosen parameter sets for which \u0016\u0003\nis real, so that the linearization predicts non-oscillatory\ndecay. While the solutions still generically exhibit a sin-\ngle peak in each \feld, after this initial overshoot in the\nnonlinear regime, the \felds decay exponentially without\noscillation as the linearization predicts.\nCONCLUSIONS\nIn conclusion we have shown that the normal modes\nin an anisotropic superconductor that breaks time re-\nversal symmetry mix density and phase \felds with the\nmagnetic \feld. This precludes using the usual notion\nof coherence and magnetic \feld penetration lengths be-\ncause long range decay of matter and magnetic \felds is\ngiven by the same exponent. Additionally the fundamen-\ntal length scales, associated with the normal modes, that\nmix the order parameters and magnetic \feld, are in gen-\neral complex. We have also shown that this mode mixing\nrequires BTRS along with anisotropy and that mixing is\nrequired for complex length scales. While systems ex-\nist with oscillations in magnetic \feld and phase di\u000ber-\nence due to anisotropy driven mixing between these two\nmodes [13], all four \felds having complex length scales\ncan only happen in an anisotropic BTRS system. Calcu-\nlating numerically the Meissner e\u000bect in a chiral p+ip\nsuperconductor, we indeed \fnd that application of an\nexternal magnetic \feld is screened in an oscillatory way\nand produces damped oscillatory decay of the order pa-\nrameter \felds. For strong anisotropy the e\u000bect should be\ndetectable in muon spin relaxation experiments. Cutting\nsample boundaries under di\u000berent angles relative to crys-\ntal axes and measuring magnetic \feld inversion can allow\none to recover information about the order parameter.\nFinally we note that our analysis dictates that the e\u000bect\nis present for any inhomogeneous situation, including the\ndomain wall excitations in p+ipsuperconductors con-\nsidered in [25]; these should also exhibit oscillation and8\n\feld inversion. That this was not observed in [25] might\nbe an artifact of an overly restrictive ansatz. We plan to\nexamine this further in a separate publication.\nA: The coupling matrices\nHere we record the non-zero matrix elements of the\n4\u00024 matrices appearing in equation 21:\nA11=\u0000n\u0001Q11n+\u0000\nn\u0001Q12n\u00012\nn\u0001Knu2\n2; (29)\nA12=\u0000\u0000\nn\u0001Q12n\u00012\nn\u0001Knu1u2; (30)\nA13=\u0006u2n\u0001Q12n\u0012\n1 +n\u0001Ln\nn\u0001Kn\u0013\n; (31)\nA22=\u0000n\u0001Q22n+\u0000\nn\u0001Q12n\u00012\nn\u0001Knu2\n1; (32)\nA23=\u0006u1n\u0001Q12n\u0012\n1\u0000n\u0001Ln\nn\u0001Kn\u0013\n; (33)\nA33=(n\u0001Ln)2\nn\u0001Kn\u0000n\u0001Kn; (34)\nA44=\u00001 (35)\nB14=\u0006u2\u0012\nn\u0001Q12n?\u0000n\u0001Q12nn\u0001Kn?\nn\u0001Kn\u0013\n;(36)\nB24=\u0007u1\u0012\nn\u0001Q12n?\u0000n\u0001Q12nn\u0001Kn?\nn\u0001Kn\u0013\n;(37)\nB34=\u0012\nn\u0001Ln?\u0000n\u0001Kn?n\u0001Ln\nn\u0001Kn\u0013\n; (38)\nC\u000b\f=H\u000b\f;1\u0014\u000b;\f\u00142 (39)\nC33= 2\u0011u2\n1u2\n2; (40)\nC44=n?\u0001Kn?\u0000\u0000\nn\u0001Kn?\u00012\nn\u0001Kn: (41)\nRecall thatAij\u0011Aji,Bij\u0011\u0000BjiandCij=Cji.\nB: Boundary Conditions\nTo compute the Meissner state in the region \n numer-\nically we must minimize the Gibbs free energy\nG=Z\n\n(F\u0000HB)+Z\n@\nFsurf=:Z\n\nG+Z\n@\nFsurf(42)\namong all \felds de\fned on \n. It is convenient to include\na gauge-\fxing term1\n2(@iAi)2inF, and to denote the\ndynamical \felds collectively as \u001ea,a= 1;:::; 6 (consist-\ning of the real and imaginary parts of \u000b, andA1,A2).\nThen, under a variation \u000e\u001ea,Gvaries as\n\u000eG=Z\n\n\u0012@G\n@\u001ea\u0000@i\u0012@G\n@(@i\u001ea)\u0013\u0013\n\u000e\u001ea\n+Z\n@\n\u0012@Fsurf\n@\u001ea\u0000ni@G\n@(@i\u001ea)\u0013\n\u000e\u001ea; (43)where we have used the divergence theorem, and recalled\nthatnis an inward pointing normal to @\n. Demand-\ning that\u000eG= 0 for all variations requires both these\nintegrals vanish identically, and hence that \u001easatisfy the\nusual Euler-Lagrange equations in \n together with the\nboundary conditions\n@Fsurf\n@\u001ea\u0000ni@G\n@(@i\u001ea)= 0 (44)\non@\n. For the model studied here, this reduces to\nniQ1\f\nijDj \f= 2[(\u001f1+\u001f2(n2\n1\u0000n2\n2)) 1+ 2\u001f3n1n2 2];\nniQ2\f\nijDj \f= 2[(\u001f1\u0000\u001f2(n2\n1\u0000n2\n2)) 2+ 2\u001f3n1n2 1];\n@iAi= 0;\nB=H: (45)\nImposing the translationally invariant ansatz \u000b=\n \u000b(X),Ai=a(X)n?\ni+b(X)ni, whereX=nixi, this\nreduces further to\nn\u0001Q1\fn( 0\n\f(0) +ib(0) \f(0)) +in\u0001Q1\fn?a(0) \f(0)\n= 2[(\u001f1+\u001f2(n2\n1\u0000n2\n2)) 1(0) + 2\u001f3n1n2 2(0)];\nn\u0001Q2\fn( 0\n\f(0) +ib(0) \f(0)) +in\u0001Q2\fn?a(0) \f(0)\n= 2[(\u001f1\u0000\u001f2(n2\n1\u0000n2\n2)) 2(0) + 2\u001f3n1n2 1(0)];\nb0(0) = 0;\na0(0) =H: (46)\nThese are the boundary conditions we impose at X= 0.\nAtX=L, large (our e\u000bective in\fnity), we demand that\nb0=a0= 0, 1=u1and 2=iu2(the \felds are in their\nground state state).\nC:\u001f= 0results\nTo con\frm that the long-range decay behaviour holds\nfor various boundary conditions, we include here a plot\nin \fgure 4 for the parameters used in our results section\nbut with the boundary term removed, \u001f= 0.\nD: rescaling of \felds\nWe have made use of the form of the potential argued\nfor in [25], however we have made a few rescalings to\nrewrite the proposed model in a simpler fashion. The\nproposed model is,\nEb=Z\nR2n\nap(j\u0011xj2+j\u0011yj2) +b1(j\u0011xj2+j\u0011yj2)2\n+b2\n2(\u00112\nx\u00112\ny+\u00112\ny\u00112\nx) +b3j\u0011xj2j\u0011yj2\n+K1(jD1\u0011xj2+jD2\u0011yj2) +K2(jD1\u0011yj2+jD2\u0011xj2\n+D1\u0011xD2\u0011y+D2\u0011yD1\u0011x+D1\u0011yD2\u0011x+D2\u0011xD1\u0011y)\n+B2\n8\u0019\u001b\nd2xb: (47)9\nFIG. 4: The Meissner state at a superconductor-insulator interface in the model (26) with V0= 3,\u0017=\u00000:95,\u001f= 0 and\nexternal \feld H= 0:3 for two di\u000berent boundary orientations: '= 0 (top set of plots) and '=\u0019=3 (bottom set of plots).\nThe boundary is at X= 0, the plotted \felds are the condensate magnitudes \u001a1and\u001a2and the magnetic \feld strength B. The\ngreen dots mark points where the \felds cross their ground state values and the blue dots mark local extrema.10\nwhereDi=@i\u0000i\rAb\niand some of the parameters are\ncoupled such that,\nK1=K\n4(3 +\u0017)K2=K\n4(1\u0000\u0017)\nb1=b\n8(3 +\u0017)b2=b\n4(1\u0000\u0017)\nb3=\u0000b\n4(1 + 3\u0017): (48)\nWe will write our condensate \felds as,\n 1=\u0011x=\u0015; 2=\u00112=\u0015; \u0015 :=q\n\u0000ap=b; (49)\nand rescale our gauge \feld,\nAi=Ab\ni=\u0015A; \u0015A:=\u0015p\n4\u0019K: (50)\nFinally we can use a spatial rescaling,\nxi=xb\ni=\u0015x; \u0015x:= 1=\r\u0015A; (51)\nand then rescale the total energy to be,\nE=Eb=\u0015E; \u0015E:=K\u00152=2: (52)\nThis \fnally gives the form of the energy given in equation\n1 with potential,\nFp=V0\u001a\n1\u0000(\u001a2\n1+\u001a2\n2) +1\n8(3 +\u0017)\u0000\n\u001a2\n1+\u001a2\n2\u00012\n\u00001\n4(1 + 3\u0017)\u001a2\n1\u001a2\n2+1\n4(1\u0000\u0017)\u001a2\n1\u001a2\n2cos 2\u001212\u001b\n;(53)\nand anisotropy tensors,\nQ11=\u00123 +\u00170\n0 1\u0000\u0017\u0013\n; Q22=\u00121\u0000\u00170\n0 3 +\u0017\u0013\n;\nQ12=\u0012\n0 1\u0000\u0017\n1\u0000\u00170\u0013\n:\nWhere we have collected multiple parameters together,\nV0=b\n2\u0019\r2K2: (54)\nNote that without loss of generality we have reduced the\nnumber of parameters to two ( V0;\u0017). This leads to the\nvacua and hence asymptotic values being \u001212=\u0006\u0019=2\nas required for BTRS and \u001a1=\u001a2= 1 without loss of\ngenerality.\nAcknowledgements\nWe thank Mihail Silaev for collaboration and discus-\nsions. The work of MS and TW is supported by the\nUK Engineering and Physical Sciences Research Coun-\ncil through grant EP/P024688/1. EB is supported bythe Swedish Research Council Grants No. 642-2013-7837,\n2016-06122, 2018-03659 and G oran Gustafsson Founda-\ntion for Research in Natural Sciences and Medicine. This\nwork was performed in part at the Aspen Center for\nPhysics, which is supported by National Science Foun-\ndation grant PHY-1607611.\n[1] A. Mackenzie, npj Quantum Mater. 2, 40 (2017).\n[2] A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75,\n657 (2003), URL http://link.aps.org/doi/10.1103/\nRevModPhys.75.657 .\n[3] V. Grinenko, P. Materne, R. Sarkar, H. Luetkens,\nK. Kihou, C. H. Lee, S. Akhmadaliev, D. V. Efre-\nmov, S.-L. Drechsler, and H.-H. 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Babaev, Physica C: Super-\nconductivity and its Applications 533, 63 (2017).\n[20] V. Vadimov and M. Silaev, Physical Review B 98, 10450411\n(2018).\n[21] L. Landau and E. Lifshitz, Course of theoretical physics,\nPergamon International Library of Science, Technol-\nogy, Engineering and Social Studies, Oxford: Pergamon\nPress, 1980| c1980, 3rd rev. and enlarg. ed. (1980).\n[22] M. Plischke and B. Bergersen, Equilibrium statis-\ntical physics (Prentice Hall Englewood Cli\u000bs, N.J,\n1989), ISBN 978-981-256-048-3,978-981-256-155-8, URL\nhttp://www.worldscientific.com/worldscibooks/10.\n1142/5660 .\n[23] M. Tinkham, Introduction To Superconductivity\n(McGraw-Hill, 1995).[24] J. Garaud, E. Babaev, T. A. Bojesen, and A. Sudb\u001c,\nPhys. Rev. B 94, 104509 (2016), URL http://link.aps.\norg/doi/10.1103/PhysRevB.94.104509 .\n[25] A. Bouhon and M. Sigrist, New Journal of Physics 12,\n043031 (2010).\n[26] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63,\n239 (1991), URL https://link.aps.org/doi/10.1103/\nRevModPhys.63.239 .\n[27] A. Samoilenka and E. Babaev, arXiv e-prints\narXiv:1904.10942 (2019), 1904.10942." }, { "title": "0805.1320v2.Spin_dynamics_in__III_Mn_V_ferromagnetic_semiconductors__the_role_of_correlations.pdf", "content": "arXiv:0805.1320v2 [cond-mat.str-el] 25 Aug 2008Spin dynamics in (III,Mn)V ferromagnetic semiconductors: the role of correlations\nM. D. Kapetanakis and I. E. Perakis\nDepartment of Physics, University of Crete, and Institute o f Electronic Structure & Laser,\nFoundation for Research and Technology-Hellas, Heraklion , Crete, Greece\n(Dated: November 6, 2018)\nWe address the role of correlations between spin and charge d egrees of freedom on the dynamical\nproperties of ferromagnetic systems governed by the magnet ic exchange interaction between itiner-\nant and localized spins. For this we introduce a general theo ry that treats quantum fluctuations\nbeyond the Random Phase Approximation based on a correlatio n expansion of the Green’s function\nequations of motion. We calculate the spin susceptibility, spin–wave excitation spectrum, and mag-\nnetization precession damping. We find that correlations st rongly affect the magnitude and carrier\nconcentration dependence of the spin stiffness and magnetiz ation Gilbert damping.\nPACS numbers: 75.30.Ds, 75.50.Pp, 78.47.J-\nIntroduction— Semiconductors displaying carrier–\ninduced ferromagnetic order, such as Mn–doped III-V\nsemiconductors, manganites, chalcogenides, etc, have re-\nceived a lot of attention due to their combined magnetic\nand semiconducting properties [1, 2]. A strong response\nof their magnetic properties to carrier density tuning via\nlight, electrical gates, or current[3, 4, 5] canlead to novel\nspintronics applications [6] and multifunctional magnetic\ndevices combining information processing and storage on\na single chip. One of the challenges facing such magnetic\ndevices concerns the speed of the basic processing unit,\ndetermined by the dynamics of the collective spin.\nTwo key parameters characterize the spin dynam-\nics in ferromagnets: the spin stiffness, D, and the\nGilbert damping coefficient, α.Ddetermines the long–\nwavelength spin–wave excitation energies, ωQ∼DQ2,\nwhereQis the momentum, and other magnetic prop-\nerties.Dalso sets an upper limit to the ferromagnetic\ntransition temperature: Tc∝D[1]. So far, the Tcof\n(Ga,Mn)As has increased from ∼110 K [2] to ∼173 K\n[1, 7]. It is important for potential room temperature\nferromagnetism to consider the theoretical limits of Tc.\nTheGilbertcoefficient, α, characterizesthedampingof\nthe magnetization precession described by the Landau–\nLifshitz–Gilbert (LLG) equation [1, 8]. A microscopic\nexpression can be obtained by relating the spin suscepti-\nbility of the LLG equation to the Green’s function [9]\n≪A≫=−iθ(t)<[A(t),S−\nQ(0)]> (1)\nwithA=S+\n−Q,S+=Sx+iSy.∝angbracketleft···∝angbracketrightdenotes the\naverage over a grand canonical ensemble and SQ=\n1/√\nN/summationtext\njSje−iQRj, whereSjare spins localized at N\nrandomly distributed positions Rj. The microscopic ori-\ngin ofαisstill notfully understood[9]. Amean–fieldcal-\nculation of the magnetization damping due to the inter-\nplay between spin–spin interactions and carrier spin de-\nphasingwasdevelopedin Refs.[9, 10]. Themagnetization\ndynamics can be probed with, e.g., ferromagnetic res-\nonance [11] and ultrafast magneto–optical pump–probe\nspectroscopy experiments [5, 12, 13, 14]. The interpre-tation of such experiments requires a better theoretical\nunderstanding of dynamical magnetic properties.\nIn this Letter we discuss the effects of spin–charge cor-\nrelations, due to the p–d exchange coupling of local and\nitinerant spins, on the spin stiffness and Gilbert damp-\ningcoefficient. Wedescribequantumfluctuationsbeyond\nthe Random Phase Approximation (RPA) [15, 16] with a\ncorrelationexpansion[17]ofhigherGreen’sfunctionsand\na 1/S expansion of the spin self–energy. To O(1/S2), we\nobtain a strong enhancement, as compared to the RPA,\nof the spin stiffness and the magnetization damping and\na different dependence on carrier concentration.\nEquations of motion— The magnetic propertiescan be\ndescribedby the Hamiltonian [1] H=HMF+Hcorr, where\nthe mean field Hamiltonian HMF=/summationtext\nknεkna†\nknaknde-\nscribes valence holes created by a†\nkn, wherekis the mo-\nmentum, nis the band index, and εknthe band disper-\nsion in the presenceof the mean field created by the mag-\nnetic exchangeinteraction[16]. The Mn impurities act as\nacceptors, creating a hole Fermi sea with concentration\nch, and also provide S= 5/2 local spins.\nHcorr=βc/summationdisplay\nq∆Sz\nq∆sz\n−q+βc\n2/summationdisplay\nq(∆S+\nq∆s−\n−q+h.c.),(2)\nwhereβ∼50–150meV nm3in (III,Mn)V semiconductors\n[1] is the magnetic exchane interaction. cis the Mn spin\nconcentration and sq= 1/√\nN/summationtext\nnn′kσnn′a†\nk+qnakn′the\nhole spin operator. ∆ A=A− ∝angbracketleftA∝angbracketrightdescribes the quan-\ntum fluctuations of A. The ground state and thermo-\ndynamic properties of (III,Mn)V semiconductors in the\nmetallic regime ( ch∼1020cm−3) are described to first\napproximation by the mean field virtual crystal approxi-\nmation,HMF, justified for S→ ∞[1]. Most sensitive to\nthe quantum fluctuations induced by Hcorrare the dy-\nnamical properties. Refs.[9, 15] treated quantum effects\ntoO(1/S) (RPA). Here we study correlations that first\narise atO(1/S2). By choosing the z–axis parallel to the\nground state local spin S, we have S±= 0 and Sz=S.\nThe mean hole spin, s, is antiparallel to S,s±= 0 [1].2\nThe spin Green’s function is given by the equation\n∂t≪S+\n−Q≫=−2iSδ(t)+βc≪(s×S−Q)+≫\n−i∆≪s+\n−Q≫+βc\nN×\n/summationdisplay\nkpnn′≪(σnn′×∆Sp−k−Q)+∆[a†\nknapn′]≫,(3)\nwhere ∆ = βcSis the mean field spin–flip energy gap\nands= 1/N/summationtext\nknσnnfknis the ground state hole spin.\nfkn=∝angbracketlefta†\nknakn∝angbracketrightis the hole population. The first line on\nthe right hand side (rhs) describes the mean field pre-\ncession of the Mn spin around the mean hole spin. The\nsecond line on the rhs describes the RPA coupling to the\nitinerant hole spin [10], while the last line is due to the\ncorrelations. The hole spin dynamics is described by\n(i∂t−εkn′+εk−Qn)≪a†\nk−Q↑ak↓≫\n=βc\n2√\nN/bracketleftbigg\n(fk−Qn−fkn′)≪S+\n−Q≫\n+/summationdisplay\nqm≪(σn′m·∆Sq)∆[a†\nk−Qnak+qm]≫\n−/summationdisplay\nqm≪(σmn·∆Sq)∆[a†\nk−Q−qmakn′]≫/bracketrightbigg\n.(4)\nThe firstterm on the rhsgivesthe RPAcontribution[10],\nwhile the last two terms describe correlations.\nThe correlation contributions to Eqs.(3) and (4) are\ndetermined by the dynamics of the interactions be-\ntween a carrier excitation and a local spin fluctuation.\nThis dynamics is described by the Green’s functions\n≪∆Sp−k−Q∆[a†\nknapn′]≫, whose equations of motion\ncouple to higher Green’s functions, ≪Sa†aa†a≫and\n≪SSa†a≫, describingdynamicsof threeelementaryex-\ncitations. To truncate the infinite hierarchy, we apply a\ncorrelation expansion [17] and decompose ≪Sa†aa†a≫\ninto all possible products of the form ∝angbracketlefta†aa†a∝angbracketright ≪S≫,\n∝angbracketleftS∝angbracketright∝angbracketlefta†a∝angbracketright ≪a†a≫,∝angbracketlefta†a∝angbracketright ≪∆S∆[a†a]≫, and∝angbracketleftS∝angbracketright ≪\na†aa†a≫c, where≪a†aa†a≫cis obtained after sub-\ntracting all uncorrelated contributions, ∝angbracketlefta†a∝angbracketright ≪a†a≫,\nfrom≪a†aa†a≫(we include all permutations of mo-\nmentum and band indices) [18]. Similarly, we decompose\n≪SSa†a≫into products of the form ∝angbracketleftSS∝angbracketright ≪a†a≫,\n∝angbracketleftS∝angbracketright∝angbracketlefta†a∝angbracketright ≪S≫,∝angbracketleftS∝angbracketright ≪∆S∆[a†a]≫, and∝angbracketlefta†a∝angbracketright ≪\n∆S∆S≫. This corresponds to decomposing all opera-\ntorsAinto average and quantum fluctuation parts and\nneglecting products of three fluctuations. We thus de-\nscribe all correlations between any twospin and charge\nexcitations and neglect correlations among threeor more\nelementary excitations (which contribute to O(1/S3))\n[18]. In the case of ferromagnetic β, as in the mangan-\nites, we recover the variational results of Ref.[19] and\nthus obtain very good agreement with exact diagonaliza-\ntionresultswhilereproducingexactlysolvablelimits (one\nelectron, half filling, and atomic limits, see Refs.[18, 19]).When treating correlations in the realistic (III,Mn)V\nsystem, the numerical solution of the above closed sys-\ntem of equations of motion is complicated by the cou-\npling of many momenta and bands and by unsettled is-\nsues regarding the role on the dynamical and magnetic\nanisotropy properties of impurity bands, strain, localized\nstates, and sp–d hybridization [1, 20, 21, 22, 23]. In the\nsimpler RPA case, which neglects inelastic effects, a six–\nband effective mass approximation [16] revealed an order\nof magnitude enhancement of D. The single–band RPA\nmodel [15] also predicts maximum Dat very small hole\nconcentrations, while in the six–band model Dincreases\nand then saturates with hole doping. Here we illustrate\nthe main qualitative features due to ubiquitous corre-\nlations important in different ferromagnets [19, 24] by\nadopting the single–band Hamiltonian [15]. We then dis-\ncuss the role of the multi–band structure of (III,Mn)V\nsemiconductors by using a heavy and light hole band\nmodel.\nIn the case of two bands of spin– ↑and spin– ↓states\n[15], we obtain by Fourier transformation\n≪S+\n−Q≫ω=−2S\nω+δ+ΣRPA(Q,ω)+Σcorr(Q,ω),(5)\nwhereδ=βcsgives the energy splitting of the local spin\nlevels. Σ RPAis the RPA self energy [15, 16].\nΣcorr=βc\n2N/summationdisplay\nkp/bracketleftBigg\n(Gpk↑+Fpk)ω+εk−εk+Q\nω+εk−εk+Q+∆+iΓ\n−(Gpk↓−Fpk)ω+εp−Q−εp\nω+εp−Q−εp+∆+iΓ/bracketrightBigg\n(6)\nis the correlated contribution, where\nGσ=≪S+∆[a†\nσaσ]≫\n≪S+≫, F=≪∆Sza†\n↑a↓≫\n≪S+≫.(7)\nΓ∼10-100meV is the hole spin dephasing rate [25]. Sim-\nilar to Ref.[10] and the Lindblad method calculation of\nRef.[14], we describe such elastic effects by substituting\nthe spin–flip excitation energy∆ by ∆+ iΓ. We obtained\nGandFbysolvingthecorrespondingequationstolowest\norder in 1/S, with βSkept constant, which gives Σcorrto\nO(1/S2). More details will be presented elsewhere [18].\nResults— Firstwestudythe spinstiffness D=DRPA+\nDcorr\n++Dcorr\n−. The RPA contribution DRPAreproduces\nRef.[15]. The correlated cotributions Dcorr\n+>0 and3\n0 0.1 0.2 0.3 0.4 0.5\np00.020.040.060.08D/D0 D\nDRPA\nDRPA+D(-)\n0 0.2 0.4\np00.020.040.06\n50 100 150\nβc (meV)00.010.02D/D0\n50 100 150\nβc (meV)00.020.040.06a) βc =70meV b) βc =150meV\nc) p =0.1 d) p =0.5\nFIG. 1: (Color online) Spin stiffness Das function of hole\ndoping and interaction strength for the single–band model.\nc= 1nm−3, Γ=0,D0=/planckover2pi12/2mhh,mhh= 0.5me.\nDcorr\n−<0 were obtained to O(1/S2) from Eq.(6) [18]:\nDcorr\n−=−/planckover2pi12\n2mhS2N2/summationdisplay\nkp/bracketleftBigg\nfk↓(1−fp↓)εp(ˆp·ˆQ)2\nεp−εk\n+fk↑(1−fp↑)εk(ˆk·ˆQ)2\nεp−εk/bracketrightBigg\n, (8)\nDcorr\n+=/planckover2pi12\n2mhS2N2/summationdisplay\nkpfk↓(1−fp↑)×\n/bracketleftBig\nεk(ˆk·ˆQ)2+εp(ˆp·ˆQ)2/bracketrightBig\n×\n/bracketleftbigg2\nεp−εk+1\nεp−εk+∆−∆\n(εp−εk)2/bracketrightbigg\n,(9)\nwhereˆQ,ˆk, andˆ pdenote the unit vectors.\nFor ferromagnetic interaction, as in the manganites\n[19, 24], the Mn and carrier spins align in parallel. The\nHartree–Fock is then the state of maximum spin and\nan exact eigenstate of the many–body Hamiltonian (Na-\ngaoka state). For anti–ferromagnetic β, as in (III,Mn)V\nsemiconductors, the ground state carrier spin is anti–\nparallel to the Mn spin and can increase via the scat-\ntering of a spin– ↓hole to an empty spin– ↑state (which\ndecreases Szby 1). Such quantum fluctuations give rise\ntoDcorr\n+, Eq.(9), which vanishes for fk↓= 0.Dcorr\n−comes\nfrom magnon scattering accompanied by the creation of\naFermi seapair. In the caseofaspin– ↑Fermi sea, Eq.(8)\nrecovers the results of Refs.[19, 24].\nWe evaluated Eqs.(8) and (9) for zero temperature\nafter introducing an upper energy cutoff corresponding\nto the Debye momentum, k3\nD= 6π2c, that ensures the\ncorrect number of magnetic ion degrees of freedom [15].0 0.1 0.2 0.3 0.4 0.5\np00.20.4D/D0\n0 0.1 0.2 0.3 0.4 0.5\np00.20.4\n0 0.1 0.2 0.3\nεF (eV)00.010.02D/D0\n0 0.1 0.2 0.3 0.4 0.5\nεF (eV)00.020.04a) βc =70meV b) βc =150meV\nc) βc =70meV d) βc =150meV\nFIG. 2: (Color online) Spin stiffness Dfor the parameters of\nFig. 1. (a)and(b): two–bandmodel, (c)and(d): dependence\non the Fermi energy within the single–band model.\nFigs. 1(a) and (b) show the dependence of Don hole\ndoping, characterized by p=ch/c, for two couplings β,\nwhile Figs. 1(c) and (d) show its dependence on βfor\ntwo dopings p. Figure 1 also compares our full result, D,\nwithDRPAandDRPA+Dcorr\n−. It is clear that the cor-\nrelations beyond RPA have a pronounced effect on the\nspin stiffness, and therefore on Tc∝D[1, 7] and other\nmagnetic properties. Similar to the manganites [19, 24],\nDcorr\n−<0 destabilizes the ferromagneticphase. However,\nDcorr\n+stronglyenhances Das comparedto DRPA[15] and\nalso changes its p–dependence.\nThe ferromagnetic order and Tcvalues observed in\n(III,Mn)V semiconductors cannot be explained with the\nsingle–band RPA approximation [15], which predicts a\nsmallDthat decreases with increasing p. Figure 1\nshows that the correlations change these RPA results in\na profound way. Even within the single–band model,\nthe correlations strongly enhance Dand change its p–\ndependence: Dnow increases with p. Within the RPA,\nsuch behavior can be obtained only by including multiple\nvalence bands [16]. As discussed e.g. in Refs.[1, 7], the\nmain bandstructure effects can be understood by con-\nsidering two bands of heavy ( mhh=0.5me) and light (\nmlh=0.086me) holes. Dis dominated and enhanced by\nthe more dispersive light hole band. On the other hand,\nthe heavily populated heavy hole states dominate the\nstatic properties and EF. By adopting such a two–band\nmodel, we obtain the results of Figs. 2(a) and (b). The\nmain difference from Fig. 1 is the order of magnitude en-\nhancement of all contributions, due to mlh/mhh= 0.17.\nImportantly,thedifferencesbetween DandDRPAremain\nstrong. Regarding the upper limit of Tcdue to collective\neffects, we note from Ref.[7] that is is proportional to D\nand the mean field Mn spin. We thus expect an enhance-\nment, as compared to the RPA result, comparable to the4\n0 0.5 100.020.04αα\nαRPA\n0 0.5 100.020.04\n0 0.5 1\np00.020.04α\n0 0.5 1\np00.020.04a) βc =70meV b) βc =100meV\nc) βc =120meV d) βc =150meV\nFIG. 3: (Color online) Gilbert damping as function of hole\ndoping for different interactions β.c= 1nm−3,Γ = 20meV.\ndifference between DandDRPA.\nThe dopingdependence of Dmainlycomesfromits de-\npendence on EF, shown in Figs. 2(c) and (d), which dif-\nfers strongly from the RPA result. Even though the two\nband model captures these differences, it fails to describe\naccuratelythe dependence of EFonp, determined by the\nsuccessive population of multiple anisotropicbands. Fur-\nthermore, thespin–orbitinteractionreducesthe holespin\nmatrix elements [22]. For example, |σ+\nnn′|2is maximum\nwhen the bandstates arealsospin eigenstates. The spin–\norbit interaction mixes the spin– ↑and spin– ↓states and\nreduces|σ+\nnn′|2. From Eq.(3) we see that the deviations\nfromthe meanfield resultaredetermined bythe coupling\nto the Green’s functions ≪σ+\nnn′∆[a†\nnan′]≫(RPA),≪\n∆Szσ+\nnn′∆[a†\nnan′]≫(correctiontoRPAdueto Szfluctu-\nationsleadingto Dcorr\n+>0), and≪∆S+σz\nnn′∆[a†\nnan′]≫\n(magnon–Fermi sea pair scattering leading to Dcorr\n−<0).\nBoth the RPA and the correlation contribution arising\nfrom ∆Szare proportional to σ+\nnn′. Our main result, i.e.\ntherelativeimportance of the correlation as compared to\nthe RPA contribution, should thus also hold in the real-\nistic system. The full solution will be pursued elsewhere.\nWe now turn to the Gilbert damping coefficient, α=\n2S/ω×Im≪S+\n0≫−1atω→0 [9]. We obtain to\nO(1/S2) thatα=αRPA+αcorr, where αRPArecovers\nthe mean–field result of Refs [9, 10] and predicts a linear\ndependence on the hole doping p, while\nαcorr=∆2\n2N2S2/summationdisplay\nkpIm/bracketleftBigg\nfk↓(1−fp↑)\n∆+iΓ×\n/parenleftbigg1\nεp−εk−δ+1\nεp−εk+∆+iΓ/parenrightbigg/bracketrightBigg\n(10)\narises from the carrier spin–flip quantum fluctuations.Fig.(3) compares αwith the RPA result as function of\np. The correlations enhance αand lead to a nonlinear\ndependence on p, which suggests the possibility of con-\ntrolling the magnetization relaxation by tuning the hole\ndensity. A nonlinear dependence of αon photoexcitation\nintensity was reported in Ref.[13] (see also Refs.[12, 21]).\nWe conclude that spin–charge correlations play an im-\nportant role on the dynamical properties of ferromag-\nnetic semiconductors. For quantitative statements, they\nmust be addressed together with the bandstructure ef-\nfects particular to the individual systems. The correla-\ntions studied here should play an important role in the\nultrafast magnetization dynamics observed with pump–\nprobe magneto–optical spectroscopy [12, 13, 14, 21, 22].\nThis work was supported by the EU STREP program\nHYSWITCH.\n[1] T. Jungwirth et al., Rev. Mod. Phys. 78, 2006.\n[2] H. Ohno, Science 281, 951 (1998).\n[3] S. Koshihara et al., Phys. Rev. Lett. 78, 4617 (1997).\n[4] H. Ohno et al., Nature 408, 944 (2000).\n[5] J. Wang et al., Phys. Rev. Lett. 98, 217401 (2007).\n[6] S. A. Wolf et al., Science 294, 1488 (2001).\n[7] T. K. Jungwirth et al., Phys. Rev. B 72, 165204 (2005).\n[8] L. D. Landau, E. M. Lifshitz, and L. P. Pitaeviski, Sta-\ntistical Physics, Part 2 (Pergamon, Oxford, 1980).\n[9] J. Sinova et. al., Phys. Rev. B69, 085209 (2004); Y.\nTserkovnyak, G.A.Fiete, andB. I.Halperin, Appl.Phys.\nLett.84, 25 (2004).\n[10] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y, Phys. Sta t.\nSol.23, 501 (1967).\n[11] S. T. B. Goennenwein et al., Appl. Phys. Lett. 82, 730\n(2003).\n[12] J. Wang et al., J. Phys: Cond. Matt. 18, R501 (2006).\n[13] J. Qi et al., Appl. Phys. Lett. 91, 112506 (2007).\n[14] J. Chovan, E. G. Kavousanaki, and I. E. Perakis, Phys.\nRev. Lett. 96, 057402 (2006); J. Chovan and I. E. Per-\nakis, Phys. Rev. B 77, 085321 (2008).\n[15] J. K¨ onig, H–H Lin and A. H. MacDonald, Phys. Rev.\nLett.84, 5628, (2000); M. Berciu and R. N. Bhatt, Phys.\nRev. B66, 085207 (2002).\n[16] J. K¨ onig, T. Jungwirth, and A. H. MacDonald, Phys.\nRev. B64, 184423 (2001).\n[17] J. Fricke, Ann. Phys. 252, 479 (1996).\n[18] M. D. Kapetanakis and I. E. Perakis, arXiv:0806.0938v1 .\n[19] M. D. Kapetanakis, A. Manousaki, and I. E. Perakis,\nPhys. Rev. B 73, 174424 (2006); M. D. Kapetanakis and\nI. E. Perakis, Phys. Rev. B 75, 140401(R) (2007).\n[20] K. S. Burch et. al., Phys. Rev. Lett. 97, 087208 (2006).\n[21] J. Wang et. al., arXiv:0804.3456; K. S. Burch at. al.,\nPhys. Rev. B 70, 205208 (2004).\n[22] L. Cywi´ nski and L. J. Sham, Phys. Rev. B 76, 045205\n(2007).\n[23] X. Liu et. al., Phys. Rev. B 71, 035307 (2005); K.\nHamaya et. al., Phys. Rev. B 74, 045201 (2006).\n[24] D. I. Golosov, Phys. Rev. Lett. 84, 3974 (2000); N.\nShannon and A. V. Chubukov, Phys. Rev. B 65, 104418\n(2002).5\n[25] T. Jungwirth et. al., Appl. Phys. Lett. 81, 4029 (2002)." }, { "title": "2204.09101v2.Role_of_shape_anisotropy_on_thermal_gradient_driven_domain_wall_dynamics_in_magnetic_nanowires.pdf", "content": "Role of shape anisotropy on thermal gradient-driven domain wall\ndynamics in magnetic nanowires\nM. T. Islama,<, M. A. S. Akandaa, F. Yesmina, M. A. J. Pikulband J. M. T. Islama\naPhysics Discipline, Khulna University, Khulna 9208, Bangladesh\nbDepartment of Physics, Colorado State University, Fort Collins, Colorado 80523, USA\nARTICLE INFO\nKeywords :\nDomain wall dynamics\nThermal gradient\nsLLG equation\nShape anisotropyABSTRACT\nWeinvestigatethemagneticdomainwall(DW)dynamicsinuniaxial _biaxialnanowiresunderather-\nmal gradient (TG). The findings reveal that the DW propagates toward the hotter region in both\nnanowires. Themainphysicsofsuchobservationsisthemagnonicangularmomentumtransfertothe\nDW. The hard (shape) anisotropy exists in biaxial nanowire, which contributes an additional torque,\nhence DW speed is larger than that in uniaxial nanowire. With lower damping, the DW velocity is\nsmaller and DW velocity increases with damping which is opposite to usual expectation. To explain\nthis, it is predicted that there is a probability to form the standing spin-waves (which do not carry\nnet energy/momentum) together with travelling spin-waves if the propagation length of thermally-\ngenerated spin-waves is larger than the nanowire length. For larger-damping, DW decreases with\ndamping since the magnon propagation length decreases. Therefore, the above findings might be\nuseful in realizing the spintronic (racetrack memory) devices.\n1. Introduction\nEfficientmanipulationofdomainwall(DW)inmagnetic\nnanostructures has drawn much attention because of its po-\ntentialapplicationsinspintronicdevicessuchasindatastor-\nage devices [1, 2] and logic operations [3, 4]. Several con-\ntrollingparameters,suchasmagneticfields,microwaves,and\nspin-polarizedcurrents,havebeenreportedintherecentdecade\ntodriveDWinmagneticnanostructures. Buttheseparame-\nters suffer from certain limitations in applications. Particu-\nlarly,underexternalmagneticfield,astaticDWcannotexist\ninahomogeneousmagneticnanowireandhenceapropaga-\ntion of DW is obtained due to the dissipation of energy for\nGilbert damping. The DW velocity is proportional to the\nrateofenergydissipation[5,6]. However,themagneticfield\nis unable to drive a series of DWs synchronously [7–9]. On\ntheotherhand,anspinpolarizedcurrentcandriveaDW,or\na series of DWs in the same direction with the mechanism\nof spin transfer torque [10–13]. Besides, a high critical cur-\nrent density is required to obtain a useful DW speed, which\ncauses a Joule heating problem. [9, 14, 15]. To overcome\nsuch challenges, spin-wave-spin-current generated by ther-\nmalgradient(TG)emergesasanalternativedrivingforcefor\nthe DW motion [16, 17]. Therefore, the study of TG-driven\nDW motion is significant not only for spintronic device ap-\nplicationsbutalsofortheunderstandingoftheinterplaybe-\ntween spin-wave and magnetic DW [18, 19].\nPresently,toexplainthephysicalpictureoftheTGdriven\nDWdynamics,twotheoriesofdifferentoriginsareavailable,\nnamely,microscopic(magnonictheory)theory[20–23]and\nmacroscopic (thermodynamic theory) theory [24–26]. The\nmicroscopic theory predicts that more magnons are gener-\n1>3\nSNS 170×88nm22620.6-<0.84\nBi-2212 4×2.5µm2125∼40755.6\naatH≃0.5Oe\nThermal fluctuations lead to premature switching and\nretrapping. This means that fluctuations tend to de-\ncrease the switching current ISwith respect to Ic0, but\nincrease the retrapping current, IR, with respect to IR0.\nTherefore, thermal fluctuations help in returning the\nJJ from RtoSstate. One should also keep in mind\nthat since retrapping critically depends on damping and\ndamping depends on bias, retrapping can become promi-\nnent at the switching current even for junctions which\nare underdamped at zero bias.\nIII. SAMPLES\nAnalysis of dissipation effects on phase dynamics re-\nquiresjunctionswithwelldefinedandcontrolleddamping\nparameters. Ideally such junctions should be RCSJ-type\nwith bias independent R; and have Rmuch smaller than\nthehighfrequencyimpedanceofthecircuitry Z0∼100Ω.\nPrevious studies of fluctuation phenomena in JJ’s were\nperformed predominantly on underdamped, Q0≫1, SIS\ntunnel junctions, which are not described by the sim-\nple RCSJ model with constant, Q0, both due to strongly\nnon-linear IVC’s and considerable shunting by the high\nfrequency impedance [9]. This ambiguity does not ex-\nists for SNS junctions, which are well described by RCSJ\nmodel with constant R, typically much smaller than Z0.\nAlthough the quality factor of SNS junctions is ex-\npected to be constant, verification of this as well as ex-\nact evaluation of Q0is non trivial. Parameters Ic0and\nCstill need to be independently defined. Furthermore,\nEq.(1) is valid only for junctions with sinusoidal CPR.\nThe CPR in SNS junctions deviates from sinusoidal [22],\nwhich changes the plasma frequency ωp0and thus affects\nthe effective Centering Eq.(1). The effective Cis no\nlonger equal to the real, explicitly measurable junction\ncapacitance. Therefore, quantitative analysis of dissipa-\ntion effects requires a possibility of tuning the qualitySi/Si0 CuNiNb \n0 20 40 0100 200 I ( µΑ )\nV ( µV) 66,4mK \n 138mK \n 302mK \n 580mK \n 855mK \n 1,67K \n 2,31K \n0.0 0.2 0.4 0.6 0.8 1.0 \n0 50 100 150 200 IrIc\nRN=0.24 Ohm \nV ( µV) I (mA) \n T=1.2 K \n 2.34 K \n 3.32 K \n 3.65 K \n 4.2 K b) c) a) \nNb/CuNi (70/50 nm) \n#1a Nb/CuNi (25/50 nm) #2a \nFIG. 2: (Color online). a) Sketch of the planar Nb-CuNi-\nNb junction. The junctions were made by cutting Nb/CuNi\nbilayers by FIB. Panels b) and c) show I−Vcharacteristics\nat different Tfor junctions made from 70/50 nm and 25/50\nnm thick Nb/CuNi bilayers, respectively. The IVC’s of both\njunctions are RCSJ-like and exhibit hysteresis at low T.\nfactor by at least as many independent parameters as\nthe amount of unknown variables in Eq.(1).\nInthisworkwefocusontheanalysisofphasedynamics\nin low ohmic SNS-type junctions with moderate damp-\ning 1/lessorsimilarQ0<10. Emphasis was made on the ability\nto control and tune the damping parameter of the junc-\ntions. Below we describe five different ways used for tun-\ning and verification of the quality factor of our junctions:\ntogether with conventional ways of tuning Ic0by apply-\ning the magnetic field or changing temperature, we were\nalso able to tune Ic0by applying gate voltage, adding a\nferromagnetic material into the junction barrier and by\ntuningCby in-situ capacitive shunting.\nThe parameters of the studied junctions are summa-\nrized in Table 1.\nPlanar SFS (Nb-CuNi-Nb) junctions\nPlanar Nb-CuNi-Nb junctions were made by cutting a\nsmall Nb/Cu 0.47Ni0.53bilayer bridge by a Focused Ion\nBeam (FIB) [23] . A sketch of the SFS junction is\nshown in Fig.2 a). We made junctions from two types\nof Nb/CuNi bilayers with either 70 or 25 nm thick Nb\nlayers. ThethicknessoftheCuNilayerwasalways50nm.\nIn-plane dimensions of the JJ’s, presented here, were the\nsame. Details of sample fabrication and characterization\ncan be found elsewhere [24].\nFig. 2 b) and c) show the IVC’s at different Tfor\nplanar SFS junctions made from 70/50 nm and 25/50\nnm Nb/CuNi bilayers, respectively. From Fig. 2 it is\nseen that the IVC’s are consistent with the RCSJ model5\n0.0 0.5 1.0 1.5 2.0050100150\n0.0 0.2 0.4 0.6 0.8 1.00.00.5\n#2b#2a\nIRIS\n I (µA)\nT (K)Nb-CuNi-Nb\n 1 - IR /IS\nPower at IR (nW)\nFIG. 3: (Color online). Temperature dependence of the\nswitchingandretrappingcurrentsfor twoSFSjunctionsmad e\nfrom thesame 25/50 nmNb/CuNibilayer at H= 0. Itis seen\nthatthehysteresis IS> IRexistsatlow Tfor bothJJ’s. Solid\nand dashed lines represent IR(T) calculated within the RCSJ\nand self-heating models, respectively. The inset shows the\nsize of the hysteresis 1 −IR/ISat the lowest Tas a function\nof the dissipated power at I=IR. Note that the hysteresis\nis not proportional to the power at retrapping.\nwith constant R. SinceR≪Z0∼100Ω, shunting by the\ncircuitry impedance is negligible also at high frequencies.\nThecriticalcurrentdependsverystronglyonthedepth\nof the FIB cut. By varying the depth of the cut we\nwereabletofabricateJJ’swiththreeordersofmagnitude\ndifference in Ic0[24]. To the contrary, the resistance of\njunctions remained almost unchanged R∼0.25Ω as seen\nfromIVC’sin Fig. 2. Sincethe in-planegeometriesofthe\njunctions are the same, Cand the thermal conductances\nof the junctions are also similar. Therefore, by changing\nthe depth of the FIB cut we could vary in a wide range\ntheQ0of the junctions by solely affecting Ic0and leaving\nall other parameters intact.\nFig. 3 shows T−dependencies of switching and retrap-\npingcurrentsfortwoJJ’sonthesamechip. Development\nof the hysteresis, IS> IR, withTis seen. The JJ’s had\ngood uniformity of the critical current, as follows from\nthe clear Fraunhofer modulation of the critical current\nas a function of magnetic field, shown in Fig. 4.\nNano-sculptured SNS (Nb-Pt-Nb) junctions\nNano-scale SNS junctions were made from Nb-Pt-Nb\ntrilayers [25] by three-dimensional FIB sculpturing. The\nthicknesses of bottom and top Nb layers were 225 and\n350 nm, respectively. The thickness of Pt was 30 nm. A\nsketch of the junction is shown in the inset in Fig. 5.\nNano-fabrication was required both for increasing Rand\ndecreasing Ic0to easily measurable values.\nThe main panel in Fig. 5 shows a set of IVC’s at\nT= 3.2Kfor a Nb-Pt-Nb JJ (170 ×88nm2) at differ--2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0050100150\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.01.01.52.0a)Planar SFS \nNb/CuNi #2a\n(25/50 nm) \nT=37 mK\n \n I (µA) IS\n IR\n IR0(RCSJ)\nIc0/IR0(RCSJ)\nIS/IR(Heat) b) IS / IR\nH (Oe)\nFIG. 4: (Color online). a) Magnetic field dependence of the\nswitching andretrappingcurrentsfor the planar SFSjuncti on\n#2a atT= 37mK. The solid line is the calculated IR0(H)\nwithin the RCSJ model for Q0(H= 0) = 2 .55. b)IS/IRvs\nHfor the same junction. The solid line is the Ic0/IR0within\nthe RCSJ model for Q0(H= 0) = 2 .55. The dashed line\nrepresents Ic0/IRcalculated for the case when the hysteresis\nis caused solely by self-heating (see sec.IV).\nent magnetic fields along the long side of the JJ. Strong\nmodulation of the critical current is seen. The IVC’s\nare well described by the RCSJ model with constant\nR≃0.6Ω≪Z0.\nSome JJ’s were anodized to remove possible shorts\ncaused by redeposition of Nb during FIB etching, and to\nfurther decrease the junction area. The absence of shorts\nand the uniformity of junctions was confirmed by clear\nFraunhofer modulation of Ic(H), shown in Fig. 6. De-\ntailsofthe junction fabricationandcharacterizationwere\ndescribed in Ref.[26] and will be published elsewhere.\nS-2DEG-S junctions\nS-2DEG-Sjunctionswithplanargeometrywereformed\nby two Nb electrodes connected via the 2DEG (InAs)\n[27]. Properties ( Ic0andR) of the JJ’s depend on the\nwidth of Nb-electrodes (either 10, or 40 µm), the length\nofthe 2DEG(either400or500nm) andthe transparency\nofthe contactbetweenthe Nb and the 2DEG[29]. Anar-\nrowgateelectrodewasmadeontopofthe2DEG,forming\na Josephson field-effect transistor [28]. This provides a\nunique opportunity to tune properties of the JJ’s by ap-6\nFIG. 5: (Color online). I−Vcharacteristics at different H\nfor a nano-sculptured SNS junctions at T= 3.2K. IVC’s\nare RCSJ-like and exhibit hysteresis at low H. Inset shows a\nsketch of the junction.\nplying a gate voltage, Vg. Details of sample fabrication\nand characterization can be found elsewhere [28, 29].\nFig. 7 shows a set of IVC’s at T= 30mKfor different\nVg. It demonstrates that the critical current is increased\nat positive Vgand strongly suppressed at small negative\nVg. Note that the resistance of the junction starts to\nincrease at substantially larger negative Vg/lessorsimilar−1V, at\nwhich the critical current is already strongly suppressed.\nTherefore the IVC’s at Vg>−1Vare reasonably well\ndescribed by the RCSJ model with a constant R.\nThe majority of the switching current measurements\nwere performed on the junction (#2 b) with a wide\n(40µm)andshort(400 nm)2DEGandgoodtransparency\nof the Nb/2DEG interface. For 40 µmwide junctions,\nshunting by high frequency impedance was insignificant\nbecause of the small junction resistance R≃7.5−10Ω.\nJunctions with narrower 2DEG (10 µm) had proportion-\nally smaller Ic0and larger R∼30−40Ω. All JJ’s studied\nherehadauniformcriticalcurrentdistribution, asjudged\nfrom periodic Fraunhofer modulations Ic(H), see Fig. 8.\nFig. 9 shows T−dependencies of ISandIRfor the\nsame JJ at two magnetic fields, marked by circles in Fig.\n8. TheT−dependence of ISandIRfor S-2DEG-S JJ’s\nis similar to that of planar SFS JJ’s, see Fig. 3. In both\ncases the IRisT−independent in a wide T−range.\nBi-2212 intrinsic Josephson junctions\nIntrinsic Josepson junctions (IJJ’s) are naturally\nformed between adjacent Cu-O layers in strongly\nanisotropic HTSC single crystals [30]. IJJ’s behave as\nSIS-type junctions [31, 32] with high Q0, inspite of the\nd-wave symmetry of the order parameter in HTSC. This-2000 -1000 0 1000 2000050100150200250\n-1000 0 10001,001,051,10a) IR(RCSJ)\nQ0(H=0)\n =1.07IS\nIR\n I (µA)\nH (Oe)\nRCSJ\nQ0(H=0)\n =1.07Self-\nheatingNb-Pt-Nb\n T=3.2K b)\n IS / IR\nH (Oe)\nFIG. 6: (Color online). a) Magnetic field modulation of the\nswitching and retrapping currents for the same Nb-Pt-Nb\njunction as in Fig. 5. b) IS/IRvsH. The lines represent\ncalculations within the RCSJ model for Q0(H= 0) = 1 .07\n(solid), and in case when the hysteresis is solely caused by\nself-heating (dashed), see sec. IV. It is seen that the capac -\nitive hysteresis within the RCSJ model disappears abruptly\nat certain Ic(H), while the self-heating hysteresis decreases\ngradually with Ic.\nwas confirmed by observation of geometric Fiske reso-\nnances [33, 34] and energy level resolution in the MQT\nexperiments on Bi-2212 IJJ [35] and YBa 2Cu3O7−δbi-\nepitaxial c-axis (IJJ-like) JJ’s [20].\nIJJ’s were made by micro/nano-patterning of small\nmesa structure on top of Bi-2212 single crystals. Here\nwe present data for an optimally doped Bi-2212 single\ncrystal with Tc≃94.5K. Mesas were cut in two parts by\nFIB to allow true four-probe measurements. Details of\nmesa fabrication can be found elsewhere [36]. Properties\nof our IJJ’s were described in detail before [31, 32]. Fig.\n10 shows IVC’s at different Tfor a Bi-2212 mesa. IVC’s\nofIJJ’sarenon-linearandexhibit stronghysteresisbelow\nTc. However, at elevated temperatures, 70 < T < T c, the\nIVC’s are almost linear in a small voltage range [2, 31].\nEach mesa contains several stacked IJJ’s. Therefore,\nIVC’s exhibit a multi-branch structure due to one-by-one\nswitching of stacked IJJ’s from StoRstate.\nFig. 11 shows the T−dependence of the most probable\nswitching current, ISmax, the retrapping current IR, and\nthe fluctuation-free critical current Ic0for a single IJJ\nfrom the same Bi-2212 sample as in Fig. 10 [37]. The\nIc0was obtained from the analysis of switching statis-7\n0 50 100 150 20001234Nb-2DEG-Nb #1a\n T=30mK\n Vg (V) = +0.2, +0.02 , \n0, -0.03 , -0.1, -0.2, -0.4,\n-0.8, -1.4, -1.7, -1.8.\n I (µA)\nV (µV)\nFIG. 7: (Color online). The IVC’s of a S-2DEG-S junction\nfor different gate voltages at T= 30mK.\n-15 -10 -5 0 5 10 15-40-2002040\nIR0 (RCSJ)\nQ0(H=0)\n =2.35T=30 mKS-2DEG-S #2b\n width 40 µm, \n length 400 nm\n I (µA)\nB (µT)IS\nIR\nFIG. 8: (Color online). Magnetic field modulation of switch-\ning and retrapping currents for the S-2DEG-S junction #2 b\natT= 30mK. The solid line represents a simulation within\nthe RCSJ model for Q0(H= 0) = 2 .35.\ntics [36]. The Ic0(T) follows the Ambeokar-Baratoff de-\npendence typical for conventional SIS JJ’s [36]. The IR\nisT−independent at low Tand increases with Tup to\n∼85K. Such behavior is also typical for conventional\nSIS JJ’s and is attributed to strong T−dependence of\nthe low bias quasiparticle resistance, which determines\nthe effective dissipation for the retrapping process [38].\nIV. THE ORIGIN OF HYSTERESIS\nFigs. 2-11 show that the IVC’s of all the four types of\nJJ’sstudied hereexhibit ahysteresisatlow T. According\nto the RCSJ model the hysteresis is related to damping0.0 0.5 1.0 1.5 2.0102030\n0.2 0.4 0.6 0.80.80.91.0\nH = 3.66 µTH = 0\nIRIS\nS-2DEG-S #2b\n I (µA)\nT (K)Ic0(µA)=37,\n 29, 25, 21.\n Ic0 / Ic0(0)\nT (K)\nFIG. 9: (Color online). T−dependence of the switching and\nretrapping currents for the S-2DEG-S junction #2 bat mag-\nnetic fields marked by two of the circles in Fig. 8. The inset\nshowsT−dependencies of normalized fluctuation-free critical\ncurrents Ic0(T)/Ic0(T= 0) at four Hmarked in Fig. 8.\nand appears in underdamped JJ’s with Q0>0.84.\nFor the case of Bi-2212 IJJ’s the hysteresis can be un-\nambiguously attributed to a large capacitance caused by\natomic scale separation between electrodes. From mea-\nsurements of Fiske step voltages [33] the specific capac-\nitance of our IJJ’s was estimated as C∼68.5fF/µm2.\nSubstituting typical parameters of IJJ’s [32], the criti-\ncal current density Jc(4.2K)≃103A/cm2; the large bias\nc−axis tunnel resistivity ρc≃30 Ωcmand the stack-\ning periodicity s≃1.5nm, we obtain Q0(4.2K)≃20.\nThis value will become up to two orders of magnitude\nlarger if we use the low bias quasiparticle resistivity at\nT= 4.2Kinstead of ρc. In any case, IJJ’s are strongly\nunderdamped, Q0≫1, atT≪Tcand also remain un-\nderdamped in practically the whole T−rangeT < T c, as\nseen from Fig. 11. It has been demonstrated that the\nhysteresis IS/IRin Bi-2212 IJJ’s agrees well with the\ncalculated Q0using the specific capacitance of IJJ’s [36].\nOn the other hand, explanation of the hysteresis in\nSNS-type JJ’sis less straightforward. Typically SNS JJ’s\nare strongly overdamped, Q0≪1, and the hysteresis is\ncaused by either self-heating [39], non-equilibrium effects\n[40], orfrequencydependent damping[9], ratherthan the\njunctions capacitance.\nSelf-heating\nIt is known that self-heating can cause hysteresis in\nsuperconducting weak links with negligible C[39]. In\nthis case the ”retrapping” current simply represents IS\nat the elevated temperature due to power dissipation at\nthe resistive branch of the IVC:8\nFIG. 10: (Color online). Four-probe I−Vcharacteristics\nof a Bi-2212 mesa at different temperatures. It is seen that\nthe multi-branch structure and the hysteresis persists up t o\n∼3KbelowTc.\nIR=IS(T0+∆T(IR))≃IS(T0)+dIS\ndT∆T(IR)).(16)\nThe temperature rise is given by ∆ T(IR) =PRRth≃\nRRthI2\nR, whereRthis the thermal resistance of the junc-\ntion and PRis the power dissipation at IR. Thus, Eq.\n(16) forIRbecomes:\nIR≃IS(T0)[1−αI2\nR], (17)\nwhereα=−(dIS/dT)RRth/IS(T0). The solution of\nthis quadratic equation yields:\nIR≃/radicalbig\n1+4αI2\nS−1\n2αIS. (18)\nThe dashed lines in Figs. 4 b) and 6 b) show fits\nto experimental IR(H) in the self-heating model, Eq.\n(18). The solid lines in the same figures represent fits\nwithin the RCSJ model, Eqs. (1,10). IR(H) for both\nself-heating and RCSJ models were obtained using a\nsingle fitting parameter ( RthandQ0(H= 0), respec-\ntively), which are unambiguously determined from hys-\nteresisIS/IRatH= 0. The self-heating and RCSJ\nmodels provides almost equally good fits to the IR(H)\nmodulation for the SFS junction in Fig. 4. Similarly,\nT−dependencesof IRwithin thetwoscenariosarepracti-\ncally indistinguishable for this sample, as seen from com-\nparison of dashed and solid lines in Fig. 3.\nFortheSNSJJ,theRCSJmodelprovidesabetterfitto\ntheexperimentaldatathanthesimpleself-heatingmodel,\nsee Fig. 6b). In experiment and in the RCSJ model the\nhysteresis disappears abruptly at certain H, while self-\nheating is always present and leads to smooth variation0 10 20 30 40 50 60 70 80 90 100020406080100120140\nBi-2212Ic0\nISmax\nIRI (µA)\nT (K)\nFIG. 11: (Color online). Temperature dependence of\nthe most probable switching ISmax, retrapping IRand\nfluctuation-free critical Ic0currents for the same Bi-2212 IJJ.\nof the hysteresis near the minima. On the other hand, a\nbetterfitcanbeobtainedifweallowsome T−dependence\nofRth.\nA clue to the origin of hysteresis can be obtained from\ncomparison of IVC’s of JJ’s with identical geometry but\ndifferent IS. As described in the previous section, for\nplanar SFS JJ’s, a minor variation of the FIB-cut depth\nchangesIc0byseveralordersofmagnitudewithoutaffect-\ning other characteristics of JJ’s ( C,RandRth), as seen\nfrom Figs. 2 and 3. The inset in Fig. 3 shows the values\n1−IR/ISatT≃30mKfor three SFS JJ’s with similar\ngeometry on the same chip. Within the self-heating sce-\nnario, Eq.(16), 1 −IR/ISwould be proportional to the\npower at retrapping, PR, as shown by the dashed line in\nthe inset to Fig. 3. This seems to be true for JJ’s with\nlargeIc0andPR. However, provided that the hysteresis\nin JJ’swith larger Ic0iscaused byself-heatingand Rthof\nall JJ’s are the same, there should be no hysteresis due\nto self-heating for the JJ with the smallest Ic0in Fig.\n3. This is indicated by the green (lower) dashed line in\nthe main panel of Fig.3, which representsthe self-heating\nIR(T) for the JJ with small IS, calculated from Eq.(18)\nusingthe parameter αobtainedfrom the fit IR(T) forthe\nJJwith larger IS, shownby the black(upper) dashedline\nin Fig. 3.\nFurthermore, experimental IR(T) for our junctions are\nalmostT−independent at low T. This is in contrast to\nthe prediction of the self-heating model, Eq.(18): IR∝\nI1/2\nS, see the black (upper) dashed line in Fig. 3. Note\nthat the flatter IR(T) dependence can not be explained\nby the flatter Ic0(T) in comparison to IS(T), because\nwithin the self-heating model IRis correlated with IS,\nnotIc0. Exactly the same behavior was observed for S-\n2DEG-S junctions, see Fig. 9.\nThus, we conclude that self-heating does not satisfac-\ntory explain the hysteresis in SFS junctions, although\nit is probably responsible for a considerable part of the\nhysteresis in JJ’s with larger Is.\nFor Bi-2212 IJJ’s, self-heating was measured directly9\n[41] for the same mesa. The Rthof the mesa ranged from\n∼70K/mWatT= 4.2Kto∼10K/mWat 80K. Since\nthe dissipated power at I < I Sat the first branch in\nthe IVC never exceeded a few µW, self-heating can be\nexcluded as the origin of hysteresis in IJJ’s.\nCapacitance\nIfthehysteresiswereduetofinitejunctioncapacitance,\nthen magnetic field modulation of IR0(H) should be a\nunique function of Ic0(H), given by Eqs. (9,10) with field\ndependent Q0(H) =Q0(H= 0)[Ic0(H)/Ic0(H= 0)]1/2,\nasfollows fromEq.(1). The corresponding IR0(H) curves\ncalculated within the RCSJ model are shown by solid\nlines in Figs. 4, 6, and 8. In all cases the agreement\nwith the experimental data is remarkable, considering\nthat there is only one fitting parameter Q0(H= 0) for\neach curve (indicated in the figures).\nNext we estimate the capacitances that would be re-\nquired for reaching Q0= 1:C[Q0(H= 0) = 1] ∼35pF\nfor SFS #2a (Fig.4), ∼4pFfor SNS (Fig.6) and ∼0.2pF\nfor S-2DEG-S#2b (Fig.8) JJ’s, respectively. Thosemust\nbe comparedwith the expectedgeometrical Cofthe JJ’s.\nThe overlap capacitance of the SNS junction, Fig.6,\nis small, ∼a fewfF, due to small area of the JJ ( ∼\n0.015µm2). The stray capacitance was estimated to be\nof the same order of magnitude. Therefore, the total\nCof this junction is insufficient for observation of the\nhysteresis within the simple RCSJ model.\nThe total (stray) capacitance of wide S-2DEG-S JJ’s\nwith the gate electrode is estimated to be ∼0.1−0.2pF.\nThisCcan cause a substantial hysteresis in the junc-\ntion #2bwith large Ic0and may be just sufficient for a\ntiny hysteresis in the other junctions with smaller Ic0.\nThis conclusion is also supported by observation of un-\nderdamped phase dynamics in those junctions, as will be\ndiscussed below.\nWe argued above that the hysteresis in SFS JJ’s can\nnot be caused solely by self-heating. But how could the\nhugeC∼35pFappearinthoseplanarJJ’swiththestray\ncapacitance in the range of few fF? To understand this\nwe should consider the specific junction geometry, shown\nin Fig. 2 a). The JJ’s are made of Nb/CuNi bilayers.\nThe CuNi-layer may act as a ground plane for the JJ\nand may create the large overlap capacitance, provided\nthere is a certain barrier for electron transport between\nthe layers. The transparency of Nb/Cu interfaces, made\nin the same setup, was previously estimated to be ∼0.4\n[42, 43, 44]. The interface transparency between Nb and\nCuNi is expected to be even smaller due to appearance\nof excess interface resistance between normal metals and\nspin-polarized ferromagnets [45]. For typical values of\nthe overlap capacitance C∼20−40fF/µm2, the re-\nquiredC∼35pFcan originate from the bilayer within\njust∼30−40µmradius from the JJ. An unambiguous0 100 200 0246810 \nT=30mK, V g=0 #3a (10 µm) S-2DEG-S \n#3b (40 µm) \nC-shunted \nUnshunted \n I ( µA) \nV ( µV) Nb Nb Nb Nb \n2DEG 2DEG Al Al 22OO33Al Al \nFIG. 12: IVC’s of two S-2DEG-S junctions on the same chip\natT= 30mK,H= 0, before and after in-situ C−shunting.\nInset shows a sketch of the C−shunted junction.\nconfirmation of the presence of the large Cin our SFS\nJJ’s follows also from observation of the underdamped\nphase dynamics, as reported below (Fig. 16).\nIn-situ capacitive shunting\nThe frustratingly similar behavior of IR(T,H) within\nself-heating and RCSJ models hinders discrimination be-\ntweenself-heatingandcapacitiveoriginsofthehysteresis.\nToclarifythe originofhysteresis,wefabricatedanin-situ\nshuntcapacitor,consistingof300 µmwideAl 2O3/Aldou-\nble layer deposited right on top of S-2DEG-S JJ’s. The\nsketch of the C−shunted junction is shown in Fig. 12.\nThe IVC’s for two JJ’s from the same chip before and\nafterC−shunting are shown in Fig. 12. The length of\n2DEG in those junctions was 500 nm, which results in\nconsiderably smaller Ic0than for the JJ#2 bwith 400 nm\nlong 2DEG, Fig.8. It is seen that the hysteresis for both\njunctions increased considerably, while Rwas little af-\nfected by C−shunting [46]. The total capacitance of the\nshunt was ∼5pF, much larger than the initial capaci-\ntance of unshunted junctions C∼0.1−0.2pF.\nIt should be emphasized that introduction of the\nC−shunt improve thermal conductance from the junc-\ntions. Indeed, since the sample was placed in vacuum,\ntheC−shunt double layeron top ofthe 2DEG acts as the\ntop heat spreading layer [47, 48], creating an additional\nheat sinking channel. Thus, the self-heating hysteresis\nmust decrease in the C−shunted JJ. Therefore, increase\nof the hysteresis in C−shunted JJ’s unambiguously indi-\ncates that the hysteresis is indeed caused by the junction\ncapacitance, rather than self-heating.\nTo summarize this section, the hysteresis in the IVC’s\nof IJJ’s are caused solely by the junction capacitance.\nTheC−shunted S-2DEG-S JJ’s are underdamped with10\npredominantly capacitive hysteresis. For most SNS-type\njunctions the hysteresis is considerably affected by self-\nheating. However, planar SFS and S-2DEG-S junctions\nwith short (400 nm) 2DEG are underdamped, Q0/greaterorsimilar1,\nat lowT. Therefore, a substantial part of hysteresis in\nthose junctions has a capacitive origin. This conclusion\nis confirmed below by observation of the underdamped\nphase dynamics in those junctions.\nV. COLLAPSE OF THERMAL ACTIVATION\nMeasurements of switching and retrapping current\nstatistics were made in a carefully shielded dilution re-\nfrigerator(sampleinvacuum)inashieldedroomenviron-\nment. Measurements in the temperature range1.2-100K\nweremeasuredinaHe4cryostat(sampleinliquidorgas).\nSwitching and retrapping currents were measured using\na standard sample-and-hold technique. All histograms\nwere made for 10240 switching events.\nCollapse in Bi-2212 intrinsic Josephson junctions\nA sporadic switching of simultaneously biased stacked\nIJJ’sin the Bi-2212mesa resultsin ratherchaoticswitch-\ningbetweenquasiparticlebranchesintheIVC,seeFig.10.\nThis makes the analysisof switching statistics quite com-\nplicated. In addition, strong electromagnetic coupling\nof atomic scale stacked IJJ’s in the mesa leads to ap-\npearance of metastable fluxon states [49, 52] and re-\nsults in multiple valued critical current [49, 51]. It has\nbeen reported that switching histograms of IJJ’s can be\nvery broad and contain multiple maxima [49, 50, 51],\nconsistent with frustration caused by the presence of\nmetastable states in long, strongly coupled stacked JJ’s\n[49, 51].\nIn order to avoid the metastable states, we studied\nswitching statistics of a single IJJ [36, 37]. In the stud-\nied mesa, one of the junctions occasionally had slightly\nsmallerIc0(by∼20%) than the rest of the IJJ’s. Thus,\nwewereableto achievestableswitchingofthis singleIJJ,\nwhile the rest of the IJJ’s remained in the S−state [36].\nFig. 13showsswitchinghistogramsforthesingleIJJat\ndifferent T. The black solid lines represent simulated his-\ntograms for conventional TA escape, Eqs.(2,13), at given\nT,andforcorrespondingjunctionparametersandexperi-\nmental sweepingrates. Detailed analysisof the switching\nhistograms can be found elsewhere [36]. The switching\nhistograms are perfectly described by the TA theory up\ntoT∗∼75K. However, at higher Tthe histograms sud-\ndenly become narrower. This is clearly seen from Fig.\n13 b) which represents the effective escape temperature\nTescobtained from the fit of experimental escape rate\nby the TA expression Eq.(2), with T=Tescbeing the\nfitting parameter. For conventional TA, Tesc=T, as20 40 60 80 100 120020406080100120\n0 20 40 60 80020406080a)\nBi-2212T=81K\nT=71.3K\nT=57.6K\nT=46.2K\nT=32.5K\nT=18.8K\nT=4.2KCounts\nI (µA)b)Tesc (K)\nT (K)\nFIG. 13: (Color online). a) Switching histograms of a sin-\ngle IJJ at different T. The solid lines represent theoretical\nresults for thermal activation at a given T. Note that the ex-\nperimental histograms initially become wider with increas ing\nT, but suddenly become narrower and change the shape at\nT∗between 71 Kand 81K. b) The effective escape temper-\nature vs. Textracted from fitting the switching histograms\nto the TA theory. The solid line represents the prediction of\nconventional TA theory, Tesc=T. A sudden collapse of Tesc\nis seen at T∗≃75K.\nshown by the solid line in Fig. 13 b). A sudden collapse\nofTescatT∗≃75Kis clearly seen from Fig. 13 b). We\nalso note that the histograms become progressively more\nsymmetric at T > T∗.\nCollapse in S-2DEG-S junctions\nS-2DEG-S junctions provide a unique opportunity to\ntuneIc0,EJ0andQ0by applying the gate voltage Vg, see\nFig. 7. Fig. 14 a) shows switching current histograms\natT= 37mKfor the S-2DEG-S #3 b’(identical to #3 b)\nat different Vg. Panel b) shows that the most probable\nswitching current, ISmax, decreases monotonously with\nincreasing negative Vg. Panel c) shows the width at half-\nheight, ∆ I, versus ISmax. It is seen that initially his-\ntograms are getting wider with increasing negative Vg,\ndue to the increase of TA with decreasing EJ0/T. How-\never, at Vg<−0.35Va sudden change occurs and ∆ I\nstarts to rapidly collapse.\nFig. 15 a) shows Tescvs.Tfor the S-2DEG-S #2 bat\nH= 0,2.32,3.05,and 3.66µT(marked in Fig.8). In all\ncases we can distinguish three T−regions:\n(i) The MQT regime . At low T,Tescis independent\nofT. BoththeTA-MQTcrossovertemperature, atwhich\nsaturation of Tesc(T) occurs with decreasing T, and the\nvalue of Tesc(T→0) decrease with H, which leaves no\ndoubts that we observe the MQT state [7, 9, 10, 11]. Fig.\n15b)showsthedependence ofthe TescatT= 20mKasa\nfunction of Ic0(H).Ic0for eachHwas extrapolated from11\n3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.00200400600800\n3 4 5 6 7 820304050\n-0.8 -0.4 0.034567a) S-2DEG-S #3b’\nT=37mK\n-0.47V-0.71V\n-0.59V\n-0.35V\n-0.23V\n-0.1VVg= -0.79V\n Counts\nI(µA)\nc)\nVg=0V-0.79V-0.35V\n ΔI (nA)\nISmax(µA)b)\n ISmax (µA)\nVg (V)\nFIG. 14: (Color online). a) Switching histograms of S-2DEG-\nS #3b’ at T= 37mKfor different gate voltages Vg. It is seen\nthat the height (inversely proportional to the width) of the\nhistograms first decreases, but then start to increase with\nincreasing negative Vg. b) Dependence of the most probable\nswitching current ISmaxon the gate voltage. c) The width\nof the histograms ∆ Ivs.ISmax. A sudden collapse of ∆ I\noccurs at Vg<−0.35V.\nthe switching histograms. The inset in Fig. 9 demon-\nstrates that Ic0(T)/Ic0(T→0) for different Hcollapse\ninto one curve, confirming the accuracy of determination\nofIc0(T,H). The solid line in Fig. 15 b) shows the fit\nto the TA-MQT crossover temperature, Eq.(6), taking\nQ0(H= 0) = 2 .35, following from the value of hysteresis\nin the IVC’s, see Fig. 8. Obviously, the MQT calcu-\nlations are consistent with the previous conclusion that\nthis JJ is underdamped at low Tand that the hysteresis\nis predominantly caused by the junction capacitance.\nWe tried to perform quantum level spectroscopy [20]\nin the MQT state for this JJ, but couldn’t observe any\ninterlevel transitions. This implies that the level width\n∼/planckover2pi1/(RC) is of the order of level spacing /planckover2pi1ωp, which in\nturn is caused by a relatively low Q0for this JJ. There-\nfore, Fig. 15 confirms the theoretical prediction [7, 8]\nthat the MQT occurs even in the absence of well defined\nquantum levelsinslightly underdampedand overdamped\nJJ’s,Q0/lessorsimilar1. Both the absolute value and the character-\nistic parabolic shape of Tesc(T) in the MQT state are in\ngood agreement with theoretical predictions, as shown in\nFig. 15 c). The solid lines in Fig.15 c) show simulated\nTescvsTin the MQT state, calculated from Eq.(5) using\nthe experimental conditions for the data in Fig. 15 a).\nIt is seen that both the absolute value and the shape of\nsimulated Tesc(T) agree with the experimental data.0 200 400 600 8000200400600800\n0 10 20 30 40050100150200250300\n0 200 4000200400a)S-2DEG-S #2b I\nc0=37\nµA I\nc0=29\nµA I\nc0=25\nµA I\nc0=21\nµA\n Tesc (mK)\nT (mK)\n TMQT [Eq.(6)]\nQ0(H=0)=2.35T=20mK\nb)\n Tesc (mK)\nIc0 (µA)c)Ic0(µA)=\n 37,\n 29,\n 25,\n 21.MQT \nT (mK)\nFIG. 15: (Color online). a) Escape temperature vs. Tfor\nthe S-2DEG-S JJ #2 bat four magnetic fields (marked by cir-\ncles in Fig. 8). Three T−regions can be distinguished: the\nMQT at low T, the TA at intermediate T, and the collapse\nregion at T > T∗. b)Tescin the MQT state at T= 20mK\nas a function of the fluctuation free Ic0, suppressed by apply-\ning magnetic field. Solid line represents the fit to Eq.(6) for\nQ0(H= 0) = 2 .35. c) Results of MQT simulations, Eq.(5),\nfor the experimental conditions in panel a).\n(ii)Thermal activation regime . Atintermediate T,\n∆Iincreases in agreement with TA calculations, Tesc=\nTshown by dashed lines in Fig. 15 a). Slightly larger\ninclination of the experimental Tesc(T) may be due to\nnon-sinusoidal CPR in this SNS-type JJ [22].\n(iii) Collapse of thermal activation . At higher T,\nthe width of histograms start to rapidly collapse, leading\ntoa downturnof Tesc(T). The magnetic field dependence\nfrom Fig. 15 a) reveals that the collapse temperature T∗\ndecreases quite rapidly with Ic0,i.e., the collapse occurs\nat lower Tin junctions with smaller Q0.\nCollapse in planar SFS junctions\nFig. 16showsswitchingcurrentstatisticsatdifferent T\nfor the planar Nb-CuNi-Nb junction #2 aatH≃0.5Oe.\nThe scales of both axes were kept constant for all his-\ntograms to facilitate direct comparison of the histograms\nat different T. It is seen that the collapse of TA occurs\natT∗≃200mK.12\n39.0 39.2 0200 400 \n47.2 47.4 51.0 54.2 57.2 \n59.8 0200 400 \n61.6 61.8 63.0 63.6 63.8 63.8 64.0 T =500 mK Counts Counts 400 mK SFS #2a, H~0.5 Oe \n350 mK a) \n300 mK \nI ( µA) 250 mK \n200 mK \n 150 mK 100 mK \n 50 mK \nI ( µA) T=25 mK \n0 100 200 300 400 500 40 45 50 55 60 65 70 \nb) \nPlanar Nb-CuNi-Nb \n junction \n ΔI (nA) \nT(mK) \nFIG. 16: (Color online). a) Switching histograms of the pla-\nnar SFS junction #2 aat different T. The histograms become\nwider with increasing Tup toT= 200mK(lower row) but\nthen start to shrink at T⋝250mK(upper row). b) The\nwidth of histograms ∆ Ivs.T. The ∆ I(T) follows the TA\nbehavior up to T∗= 200mKbut collapses at higher T.\nThe shape of switching histograms\nFig. 17 shows switching histograms of the same single\nIJJ as in Fig.13 just before and after the collapse. It\ndemonstrates that not only the width, but also the shape\nof the histogram changes upon the collapse. At T < T∗\nthe histogramshavethe characteristicasymmetricshape,\nperfectly consistent with the TA theory [36], as shown by\nthe black dashed line (coincides with the blue solid line)\nin Fig. 17 a). However, at T > T∗histograms become\nnarrower and loose the characteristic asymmetric shape,\nas seen from Fig. 17 b). Such a tendency was observed\nfor all JJ’s, as can be seen from Figs. 13,14,16. As will\nbe discussed in sec. VI below, the transformation of the\nhistogram shape at T > T∗is caused by the interference\nof switching and retrapping processes.\nEffect of C−shunting\nFig.18shows TescvsTobtainedfromswitchingcurrent\nstatistics, forthesameS-2DEG-SJJ’sasinFig.12,before\nand after capacitive shunting. Apparently, C−shunting\nqualitatively changed the phase dynamics of the junc-\ntions, even though it had a minor effect on Ic0andR.25 30 35 40020\n01\n14 16 18 20 22 24040\n01Bi-2212 single IJJ\nQ0=5.6 a)T =71.3 K < T *\nTesc =70.5 K\n Counts\nPS(81K)Q0=4.9b)T = 81 K > T *\nTesc= 23.1 KProbability of not being retrapped\n Counts\nI (µA)\nFIG. 17: (Color online). Switching histograms of a single IJ J\na) below and b) above the collapse temperature T∗≃75K.\nSymbolsrepresent experimental data, dashed lines - TAprob -\nability density of S→Rswitching, and dashed-dotted lines -\nprobabilities of not being retrapped PnR, Eq.(15). Solid lines\nshow the conditional probability density PSRof switching\nwithout being retrapped. It is seen that close to the collaps e\ntemperature, the retrapping process becomes significant an d\neffectively ”cuts-off” thermal activation at small bias. Not e\nthat both the width and the shape of the histograms change\natT∗. Data from Ref. [2].\nHowever, C−shunting strongly affected the quality fac-\ntor of the JJ’s. From Fig. 18 it is seen that the switching\nstatistics of the underdamped C−shunted JJ #3b is well\ndescribed by the TA theory, for which Tesc=T. To the\ncontrary, for the unshunted junction, which is just at the\nedge of being overdamped, Q0≃1, theTescdecreases\nwith increasing Talmost in the whole T−range.\nFailure of the thermal activation theory in\nmoderately damped junctions\nThe observed collapse can not be caused by fre-\nquency dependent damping due to shunting by circuitry\nimpedance [9]. Indeed, we observed the collapse in pla-\nnar SFS junctions with R <1Ω, for which such shunt-\ning plays no role. Neither can it be due to T−depen-\ndence of the TA prefactor atin Eq.(4). Indeed, damp-\ning changes only gradually through T∗and enters only\ninto the (logarithmic) prefactor atof the TA escape rate,\nEq.(2). Gradual variations of at(T) do not cause any\ndramatic variation of the TA escape rate. Moreover, in\nall calculations presented here we did take into account\ntheQ(T) dependence of the TA prefactor at, so that Tesc\nmustbydefinition beequivalentto Tforthe conventional13\n0 100 200 300 400 500 600 7000100200300400500600700\nC-shunted #3bS-2DEG-S \n #3a \nC-shunted\nUnshunted \n #3b \n Tesc (mK)\nT (mK)\nFIG. 18: (Color online). Escape temperature vs Tfor the\nS-2DEG-S junctions on chip #3 before and after in-situ\nC−shunting. It is seen that the conventional TA behavior,\nTesc=T, is restored upon increasing Q0afterC−shunting.\nTA and the drastic drop in TescatT > T∗, can not be\nexplained within a simple TA scenario. Therefore, the\nobserved collapse of switching current fluctuations with\nincreasing Trepresents a dramatic failure of the classi-\ncal TA theory, which was supposed to be valid even for\noverdamped JJ’s [4, 7].\nVI. DISCUSSION\nFrom Figs. 13-16 it is clear that those very different\nJJ’s exhibit the same paradoxical collapse of switching\ncurrent fluctuations with increasing T. A very similar\ncollapse was observed also in moderately damped SIS\ntype Al-AlO x-Al [1] and Nb-AlO x-Nb [3] junctions and\nSQUID’s. Therefore, the collapse of TA must be a gen-\neral property of all moderately damped JJ’s.\nThe dramatic effect of C−shunting on the collapse T∗\nclearly shows that damping has a crucial significance for\nthe observed phenomenon. From the experimental data\npresented above it is also clear that T∗decreases with\nincreasing damping and that for overdamped junctions\nT∗→0, see the curve for unshunted JJ #3 bin Fig. 18.\nThe data also shows that the T∗is close to the temper-\nature at which the hysteresis in IVC’s vanishes, compare\nFigs. 3 and 16, 9 and 15 a), 11 and 13, which implies\nthat retrapping processes may become important in the\nvicinity of the collapse state.\nInfluence of retrapping on the switching statistics of\nmoderately damped junctions\nThe paradoxical collapse of thermal fluctuations and\nthe corresponding failure of the conventional TA theory\nin moderately damped JJ’s can be explained by the in-fluence of retrapping processes on the switching current\nstatistics [1, 2, 3]. Indeed, in moderately damped junc-\ntionsIc0andIR0are close to each other. As discussed\nin sec. II, increasing Ttends to decrease ISand increase\nIR. Therefore, at sufficiently high T, both switching and\nretrapping events may become possible at the same bias.\nIf so, the criterion for measuring the switching event has\nto be reformulated:\nThe probability of switching from the Sto theRstate\nis a conditional probability of switching and not being re-\ntrapped back, during the time of experiment .\nPSR(I) =PS(I)PnR(I). (19)\nHerePSRis the probability density of measuring the\nswitching event, PSis the probability density of switch-\ning, Eq.(13), and PnRis the probability of not being re-\ntrapped Eq.(15).\nDashed-dotted lines in Fig. 17 a,b) show PnR, cal-\nculated for experimental parameters typical for Bi-2212\nIJJ’s. The corresponding quality factors are indicated in\nthe figures. From Fig. 17 a) it is seen that at T < T∗\nthePnR= 1 in the region where PS>0, therefore re-\ntrappingis insignificant. However, at T > T∗, retrapping\nbecomes significant at small currents. The resulting con-\nditional probability density of measuring the switching\ncurrent, PSR, Eq.(19), normalized by the total number\nof switching events, is shown by the solid line in Fig. 17\nb). This explains very well both the reduced width and\nthe almost symmetric shape of the measured histogram.\nThe collapse temperature\nFig. 19 a) shows the bias dependence of switching\n∆USand retrapping ∆ URbarriers. As was noted in\nRef.[9], there is always a current IR0< Ie< Ic0at\nwhich∆US(Ie) = ∆UR(Ie), sothatswitchingandretrap-\nping processbecome equally probable. However, this will\nhave an influence on the switching current statistics only\nin the case when the probability density of switching at\nthis current is considerable. For the case Q0= 3, shown\nin Fig. 19 a), Ie/Ic0≃0.7. However, at low Tthe major\npart of switching events will occur at ISmax≃Ic0> Ie.\nSuch situation is seen in Fig. 17 a) for T < T∗: the\nprobability of not being retrapped PnR≃1 at the most\nprobable switching current ISmaxand according to the\ncriterium, Eq.(19), retrapping has no influence on the\nswitching statistics.\nFigs. 19 b and c) represent numerical simulations in\nwhich we intentionally disregarded T−dependencies of\nIc0= 20µA,Q0andEJ0for simplicity of analysis. Pa-\nrameters were chosen similar to that for the S-2DEG-\nS #2batH= 3.66µT, see Fig. 9. The two consid-\nered cases correspond to the estimated capacitance of14\n0.0 0.5 1.0 1.5 2.00501001502000.0 0.5 1.0 1.5 2.0141618200.0 0.2 0.4 0.6 0.8 1.00123\nQ0=1.373Q0=1.942c)\nΔIΔISΔIR\nQ0=1.942\nQ0=1.373\nT (K)\n ΔI (nA)Eq.(24)\nb)ISmax\nIRmax\nEq.(23)\n \nQ0=1.373\nQ0=1.942I (µA)Ic0Q0=5 a)\nQ0=3\nIR0ΔURΔUS\n ΔU / EJ0\nI / Ic0\nFIG. 19: (Color online). a) Escape and retrapping barrier\nheights as a function of bias current for Q0= 3 and 5.\nb) Numerical simulations of the T-dependencies of the most\nprobable switching ISmaxand retrapping IRmaxcurrents for\ntwo values of Q0. Simulations were made for T−independent\nIc0= 20µA(dotted line) and parameters typical for the S-\n2DEG-S #2 bjunction. c) The simulated width of switching\n∆ISand retrapping ∆ IRhistograms disregarding the mutual\ninfluence of switching and retrapping processes. ∆ Iis the\nresulting width of histograms taking into account switchin g\nand retrapping. From Figs. b) and c) it is seen that the\ncollapse of ∆ Ioccurs at the condition ISmax≃IRmax.\nthe junction #2 b(Q0= 1.373) and twice the capaci-\ntance (Q0= 1.942), respectively. From Fig. 19 b) it\nis seen that the most probable switching current ISmax\ndecreases, while the most probable retrapping current\nIRmaxincreases with Tas a result of thermal fluctua-\ntions. The width of both switching ∆ ISand retrappling\n∆IRhistogramscontinuouslyincrease with Tfor conven-\ntional TA, as seen from Fig. 19 c). As expected, switch-\ning histograms are unaffected by the small variation of\nQ0, so that both ISmaxand ∆IScoincide for the two\nvalues of Q0. To the contrary, retrapping histograms are\nstrongly affected by Q0: theIRmax(T) dependence be-\ncomes weaker and ∆ IRsmaller with increasing Q0. This\nis caused by the increase of the retrappping barrier, ∆ UR\nwithQ0as seen from Fig. 19 a).SinceISmaxdecreases, while IRmaxincreases with T,\nswitching and retrappinghistogramsinevitably will over-\nlap at a certain temperature T∗. Fig. 19 b and c) clearly\ndemonstrates that the collapse of thermal fluctuations of\nthe measuredswitchingcurrent∆ Ioccursat T∗andthat\ntheT∗itself strongly depends on Q0. From the simula-\ntions presented in Fig. 19 it is clear that the collapse\nis not caused by a crossover from underdamped to over-\ndamped state since Q0wasT−independent in this case.\nThe collapse temperature can be estimated from the\nsystem of equations:\nΓTA(ISmax)≃(dI/dt)/Ic0, (20)\nΓR(T∗,ISmax) = ΓTA(T∗,ISmax). (21)\nEq.(20) states that the JJ switches into the R-state\nduring the time of the experiment. From Eqs.(2,20) it\nfollows that:\n∆US(ISmax)\nkBT≃ln/bracketleftbiggatωpIc0\n2π(dI/dt)/bracketrightbigg\n≡Y.(22)\nIn the measurements presented here Y≃24, as seen\nfrom Fig. 20 a). The parameter Yis weakly (logarithmi-\ncally) dependent on experimental parameters and, there-\nfore, has approximately the same value in different stud-\nies of switching statistics of JJ’s. From Eqs.(3,22) we\nobtain the value of the most probable switching current\nISmax(disregarding retrapping):\nISmax/Ic0≃1−[T/TJ]2/3, (23)\nwhereTJ= (4√\n2EJ0)/(3YkB). This dependence is\nshown by the dashed line in Fig. 19 b) and agrees with\nnumerical simulations (squares).\nSimilarly, the most probable retrapping current IRmax\n(disregarding switching) is obtained from Eqs.(7,21,22):\nIRmax≃IR0+Ic0/radicalBigg\n2kBT(Y+X)\nEJ0Q2\n0,(24)\nwhereX=ln[2π(ISmax−IR0)\natIc0(1−(ISmax/Ic0)2)1/4/radicalBig\nEJ0\n2πkBT] is the log-\narithm of the ratio of prefactors of TA rettraping and\nswitching rates, Eqs.(2,7). The factor Xis only weakly\ndependent on experimental parameters and in the first\napproximation can be considered constant (or even ne-\nglected). For the case of S-2DEG-S #2 b,X≃3. Red\ndashed lines in Fig. 19 b) represent IRmax(T) calculated\nfrom Eq.(24) with Y= 24 and X= 3, which perfectly\nreproduce the simulated IRmax(T) for both Q0values.\nKnowing ISmax(T)andIRmax(T), wecaneasilyobtain\nT∗from the condition (cf. Figs. 19 b and c):15\n1 10 1000.00.51.01.5\n0.000.020.040.06\n0 200 400 600 80005101520\n unshunted \nS-2DEG-S #3b\nSFS #2a\nS-2DEG-S#2b C-shunted \nS-2DEG-S #3aBi-2212 \nT * / Ic0 (K/µA)\nQ0 (at T * )b)kBT * / EJ0at T = T * \na)\nS-2DEG-S #2b\n ΔUS [ISmax ] (K)\nT (mK)\nFIG. 20: (Color online). a) The height of the escape barrier\nat the most probable switching current as a function of T(at\nT=T∗): symbols represent experimental data for S-2DEG-S\n#2bfrom Fig. 15 a), the solid line corresponds to Eq.(22). b)\nThe normalized collapse temperature vs. the quality factor :\ndashed line represents numerical solution of Eq.(25), symb ols\nrepresent experimental data for different JJ’s.\nIRmax(T∗) =ISmax(T∗). (25)\nA simple analytic estimation of T∗can be obtained\nby observing that ISmax(T) is almost linear in a wide\nT−range, as seen from Figs. 19 b) and 11. In this case\nEq.(23) can be approximated as\nISmax/I∗\nc0≃1−βT, (26)\nwhereβ= 2/(3T2/3\nJT1/3\n0−T0),I∗\nc0=Ic0[1−\n(T0/TJ)2/3/3] andT0is some characteristic tempera-\nture∼TJ. Substituting Eq.(26) into Eq. (25), taking\na simple approximation for IR0, Eq.(7), and neglecting\nT−dependence of Ic0we obtain a quadratic equation for\nT∗, which yields:\nT∗≃kB(Y+X)\n2β2EJ0Q2\n0/bracketleftBigg/radicalBigg\n1+/parenleftbiggI∗\nc0\nIc0−4\nπQ0/parenrightbigg2Q2\n0βEJ0\nkB(Y+X)−1/bracketrightBigg2\n.\n(27)\nFrom Eq.(27) it follows that kBT∗/EJ0strongly de-\npends on Q0, but is independent of EJ0because Tap-\npear in Eqs. (20,24) only in combination T/EJ0(in the\ncase of Eq. (27) because β∼1/EJ0).\nThe dashed line in Fig. 20 b) represents the numeri-\ncally simulated T∗normalized by EJ0andIc0, (left and\nright axes, respectively) as a function of the quality fac-\ntorQ0. It was obtained by numerical solution of Eq.(25),\nwithout simplifications used for derivation of Eq.(27). It\nis seen that for overdamped JJ’s, Q0<0.84,T∗/Ic0→0.\nTheT∗/Ic0continuouslygrowswithincreasing Q0>0.84and saturates at ∼2K/µAfor strongly underdamped\nJJ’sQ0≫1.\nThesymbolsinFig.20b)representexperimentalvalues\nofT∗/Ic0for the JJ’s studied in this work. The experi-\nmentaldataagreeswellwiththeproposedtheory(dashed\nline). The simulated values of T∗/Ic0are also consistent\nwith experimental data for underdamped SIS-type JJ’s\nAl-AlO x-Al,T∗/Ic0≃1−3.3K/µA[1], and Nb-AlO x-\nNb,T∗/Ic0≃1K/µA[3] (with a reservation that those\nmeasurements were done on SQUID’s, which may have\ndifferent activation energies than single JJ’s, Eqs.(3,8).\nTheT∗/Ic0dependence, shown in Fig. 20 b), is al-\nmost universal and explains the paradoxical collapse of\nthermal fluctuations, reported in Refs.[1, 2, 3], as well\nas the new data presented here. For example, recovery\nof conventional TA-switching in C−shunted S-2DEG-S\njunctions, see Fig. ??, is caused by the increase of the\nQ0, which according to Fig. 20 b) result in larger T∗for\nthe same Ic0. Similarly, the decrease of Q0due to sup-\npression of Ic0causes the collapse of TA vs. Vgin Fig.\n14 and the decrease of T∗withHin Fig. 15.\nFromFig. 20b)itisseenthatinoverdampedjunctions\nT∗= 0, implying that retrapping is crucially affecting\nswitching statistics at any T. In this case Tesc(T) and\n∆I(T) decrease at all T. We observed such behavior\nfor S-2DEG-S junctions with small Ic0, and consequently\nsmallQ0. The tendency of decreasing T∗withIc0is\napparent from Fig 15 a).\nTherefore, observationofthe collapse, i.e., amaximum\nofTesc(T) atT∗>0, is the most unambiguousindication\nof underdamped, Q0>0.84, state in the studied JJ’s.\nThe estimation of Q0from the value of T∗confirms our\nassessment of junction capacitances, made in sec. IV.\nPhase dynamics in the collapsed state\nThe insight into the phase dynamics at T > T∗can\nbe obtained from Fig. 20 a), in which the dependence of\n∆US(I=ISmax) vs.Tis shown for the case of Fig. 15\na). The dashed line corresponds to ∆ US(ISmax)/kBT=\n24.3≃Yobtained from simulations presented in Fig. 19\nand demonstrates excellent agreement with the experi-\nment. The large value of ∆ US/kBTimplies that the JJ\ncanescapefromthe StotheRstateonlyafewtimesdur-\ning the time of the experiment. Therefore, the collapse\nis not due to transition into the phase-diffusion state,\nwhich may also lead to reduction of ∆ I[14]. Indeed,\nphase diffusion requires repeated escape and retrapping,\nwhich is only possible for ∆ US/kBT∼1 [9, 53]. Careful\nmeasurements of supercurrent branches in the IVC’s at\nT/greaterorsimilarT∗did not reveal any dc-voltage down to ∼10nV\nfor S-2DEG-S JJ’s and ∼1µVfor IJJ’s. Furthermore,\nthe IVC’s remain hysteretic at T > T∗, which is incom-\npatible with the phase diffusion within the RCSJ model\n[9]. As can be seen from Fig. 10, the phase diffusion16\nin IJJ’s appears only at T >90K, meaning that all the\ncollapse of TA shown in Fig. 13 b) at 75 K < T < 85K\noccursbeforeentering into the phase diffusion state.\nTherefore, in the collapse state the junction makes a\nfewveryshortexcursionsfromthe StotheRstateduring\nthe current sweep, before it eventually switches into the\nRstate. However, the number of excursions and the\ntotal excursion time is so small that it does not lead to\na measurable dc-voltage in the JJ. The occurrence of the\ncorresponding phase dynamic state, prior to the phase-\ndiffusion, has been observed by numerical modelling and\ndiscussed in Ref. [9].\nCONCLUSIONS\nWe have analyzed the influence of damping on\nthe switching current statistics of moderately damped\nJosephson junctions, employing a variety of methods for\naccurate tuning of the damping parameter. A paradox-\nical collapse of switching current fluctuations with in-\ncreasing temperature was observed in various types of\nJosephson junctions [1, 2, 3], including low- TcSNS, SFS,\nS-2DEG-S, SIS, and high- Tcintrinsic Josephson junc-\ntions. The unusual phenomenon was explained by an in-\nterplay of two conflicting consequences of thermal fluctu-\nations, whichon the one hand assistin prematureswitch-\ning to the R−state and on the other hand help in retrap-\nping back into the S−state. In this case the probability\nof measuring a switching event becomes a conditional\nprobability of switching and not being retrapped during\nthe time of the experiment. Numerical calculations has\nshown that this model provides a quantitative explana-\ntion of both the value of the collapse temperature T∗,\nand the unusual shape of switching histograms in the\ncollapsed state. Based on the theoretical analysis, we\nconclude that the collapse represents a very general phe-\nnomenon, which must occur in any underdamped JJ at\nsufficiently high T.\nThe collapse of switching current fluctuations in\nJosephsonjunctions representsan exceptionfromthe law\nof increasing of thermal fluctuations with temperature.\nIn the studied case, the ”failure” of this general law of\nnature, iscausedbycoexistenceoftwocounteractingpro-\ncesses (switching and retrapping). It should be empha-\nsized that fluctuations for each of the two processesalone\nfollowthelawandenhancewith Tinaconventionalman-\nner. It is, however, remarkable and unusual that fluctu-\nations may cancel each other and lead to reduction of\nthermal fluctuations of a physically measurable quantity.\nFinally, we note that the reduced width of switching\nhistograms in the collapsed state of moderately damped\nJJ’s may be advantageous for single-shot read-out of su-\nperconducting qubits, which requires accurate discrimi-\nnation of two close current states.ACKNOWLEDGMENTS\nWe are grateful to S.Intiso, E.H¨ urfeld, H.Frederiksen,\nI.Zogaj, A.Yurgens, V.A.Oboznov and V.V.Ryazanov\nfor assistance with sample fabrications and/or measure-\nments; to T.Akazaki and H.Takayanagi for providing S-\n2DEG-S samples; and to R.Gross for lending the sample-\nand-hold equipment.\n[1] J.M.Kivioja, T.E.Nieminen, J.Claudon, O.Buisson,\nF.W.J.Hekking, and J.P.Pekola, Phys.Rev.Lett. 94,\n247002 (2005)\n[2] V.M.Krasnov, T.Bauch, S.Intiso, E.H¨ urfeld, T.Akazak i,\nH.Takayanagi, and P.Delsing, Phys.Rev.Lett 95, 157002\n(2005)\n[3] J.M¨ annik, S.Li, W.Qiu, W.Chen, V.Patel, S.Han, and\nJ.E.Lukens, Phys.Rev.B 71, 220509(R) (2005)\n[4] P.H¨ anggi, P.Talkner, and M.Borkovec, Rev.Mod.Phys.\n62, 251 (1990)\n[5] A.O.Caldeira and A.J.Leggett , Phys.Rev.Lett. 46211\n(1981)\n[6] E.Ben-Jacob, D.J.Bergman, B.J.Matkowsky, and\nZ.Schuss, Phys.Rev.A 262805 (1982)\n[7] H.Grabert, P.Olschowski and U.Weiss, Phys.Rev.B 36,\n1931 (1987)\n[8] J.M.Martinis and H.Grabert, Phys.Rev.B 38, 2371 (1988)\n[9] R.L.Kautz and J.M.Martinis, Phys.Rev.B 42, 9903\n(1990)\n[10] S.Washburn,R.A.Webb, R.F.Voss and S.M.Faris,\nPhys.Rev.Lett. 542712 (1985)\n[11] J.M.Martinis, M.H.Devoret, and J.Clarke, Phys.Rev.B\n35, 4682 (1987)\n[12] P.Silvestrini, S.Pagano, R.Cristiano, O.Liengme and\nK.E.Gray, Phys.Rev.Lett 60, 844 (1988)\n[13] E.Turlot, D.Esteve, C.Urbina, J.M.Martinis,\nM.H.Devoret, S.Linkwitz, andH.Grabert, Phys.Rev.Lett.\n621788 (1989)\n[14] D.Vion, M.Gotz, P.Joyez, D.Esteve, M.H.Devoret,\nPhys.Rev.Lett. 773435 (1996).\n[15] M.G.Castellano, et al., J.Appl.Phys. 80, 2922 (1996);\nibid.,86, 6405 (1999).\n[16] M.Schlosshauer Rev.Mod.Phys. 761267 (2004).\n[17] R.McDermott, et al., Science 307, 1299 (2005); A.J.\nBerkley, et al., ibid.3001548 (2003).\n[18] I.Chiorescu, et al., Nature431159 (2004).\n[19] D.Vion, et al., Science296886 (2002); Yu.A.Pashkin, et\nal.,Nature421823 (2003);\n[20] T.Bauch, T.Lindstr¨ om, F.Tafuri, G.Rotoli, P.Delsin g,\nT.Claeson, and F.Lombardi, Science 311, 57 (2006)\n[21] K.K.Likharev, ”Dynamics of Josephson junctions and\ncircuits” (Gordon & Breach 1991)\n[22] A.A.Golubov, M.Yu.Kupriyanov and E.Ilichev, Rev.\nMod. 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Rev. B 71,\n174502 (2005)\n[30] R.Kleiner and P.M¨ uller, Phys.Rev.B 49, 1327 (1994)\n[31] V.M.Krasnov, A.Yurgens, D.Winkler, P.Delsing,\nand T.Claeson, Phys.Rev.Lett. 84, 5860 (2000);\nV.M.Krasnov, A.E.Kovalev, A.Yurgens, and D.Winkler\nibid.86, 2657 (2001)\n[32] V.M.Krasnov, Phys.Rev.B. 65, 140504(R) (2002)\n[33] V.M.Krasnov, N.Mros, A.Yurgens, and D.Winkler,\nPhys.Rev.B 59, 8463 (1999);\n[34] A.Irie, Y.Hirari, and G.Oya, Appl.Phys.Lett. 72, 2159\n(1998); Yu.I.Latyshev, A.E.Koshelev, V.N.Pavlenko,\nM.B.Gaifullin, T.Yamashita, and Y.Matsuda, Physica\nC 367, 365 (2002); S.M.Kim, et.al., Phys.Rev.B 72,\n140504(R) (2005).\n[35] K.Inomata, et al., Phys.Rev.Lett. 95, 107005\n(2005); X.Y.Jin, J.Lisenfeld, Y.Koval, A.Lukashenko,\nA.V.Ustinov and P.M¨ uller, ibid.96, 177003 (2006)\n[36] V.M. Krasnov, T.Bauch and P.Delsing, Phys.Rev.B 72,\n012512 (2005).\n[37] The studied Bi-2212 structure had a triple-mesa config-\nuration, containing two contact mesas on top of a com-\nmon pedestal. This allowed four-probe measurement of\nIJJ’s in the pedestal mesa, shown in Fig. 10. The switch-\ning statistics in Figs. 11, 13 and 17 was obtained from\na stable switching of one of the IJJ’s from the contact\nmesa. Since contact mesas had approximately two timessmaller area than the pedestal mesa, IJJ’s in the con-\ntact mesa had approximately two times smaller critical\ncurrent, as seen from comparison of Figs. 10 and 11.\nMore details about triple mesa structures can be found\nin V.M.Krasnov, Phys.Rev.Lett. 97, 257003 (2006).\n[38] A.T.Johnson, C.J.Lobb, and M.Tinkham,\nPhys.Rev.Lett. 65, 1263 (1990).\n[39] T.A. Fulton and L.N.Dunkleberger, J.Appl.Phys.\n45, 2283 (1974); W.J.Skocpol’, M.R.Beasley, and\nM.Tinkham, ibid 45, 4054 (1974).\n[40] Y. 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Phys. 89, 5578 (2001); ibid., 93, 1329 (2003).\n[49] V.M.Krasnov, V.A.Oboznov, V.V.Ryazanov, N.Mros,\nA.Yurgens, and D.Winkler, Phys. Rev. B 61, 766 (2000).\n[50] P.A.Warburton, et al., J.Appl.Phys. 95, 4941 (2004)\n[51] N.Mros, V.M.Krasnov, A.Yurgens, D.Winkler, and\nT.Claeson, Phys.Rev.B 57 R8135 (1998)\n[52] V.M.Krasnov and D.Winkler, Phys.Rev.B 56, 9106\n(1997)\n[53] A.Franz, et al., Phys.Rev.B 69, 014506 (2004)." }, { "title": "2111.00586v1.Thermally_induced_all_optical_ferromagnetic_resonance_in_thin_YIG_films.pdf", "content": "1 \n Thermally induced all-optical ferromagnetic resonance in thin YIG films \nE. Schmoranzerová1*, J. Kimák1, R. Schlitz3, S.T. B. Goennenwein3,6, D. Kriegner2,3, H. Reichlová2,3, Z. Šobáň2, \nG. Jakob5, E.-J. Guo5, M. Kläui5, M. Münzenberg4, P. Němec1, T. Ostatnický1 \n1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic \n2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic \n3Technical University Dresden, 01062 Dresden, Germany \n4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany \n5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany \n6 Department of Physics, University of Konstanz, 78457 Konstanz, Germany \n \nLaser-induced magnetization dynamics is one of the key methods of modern opto-spintronics which aims \nat increasing the spintronic device speed1,2. Various mechanisms of interaction of ultrashort laser pulses \nwith magnetization have been studied, including ultrafast spin-transfer3, ultrafast demagnetization4, \noptical spin transfer and spin orbit torques 5,6,7 , or laser-induced phase transitions8,9. All these effects can \nset the magnetic system out of equilibrium, which can result in precession of magnetization. Laser-induced \nmagnetization precession is an important research field of its own as it enables investigating various \nexcitation mechanisms and their ultimate timescales2. Importantly, it also represents an all-optical analogy \nof a ferromagnetic resonance (FMR) experiment, providing valuable information about the fundamental \nparameters of magnetic materials such as their spin stiffness, magnetic anisotropy or Gilbert damping10. \nThe “all-optical FMR” (AO-FMR) is a local and non-invasive method, with spatial resolution given by the \nlaser spot size, which can be focused to the size of few micrometers. This makes it particularly favourable \nfor investigating model spintronic devices. \nMagnetization precession has been induced in various classes of materials including ferromagnetic \nmetals11, semiconductors10, 12, or even in materials with a more complex spin structure, such as non-\ncollinear antiferromagnets13. Ferrimagnetic insulators, with Yttrium Iron Garnet (YIG, Y 3Fe5O12) as the \nprime representative14, are of particular importance for spintronic applications owing to their high spin \npumping efficiency15 and the lowest known Gilbert damping16. However, inducing magnetization dynamics \nin ferrimagnetic garnets using optical methods is quite challenging, as it requires large photon energies 2 \n (bandgap of YIG is Eg ≈ 2.8 eV)17. This spectral region is rather difficult to access with most common ultrafast \nlaser systems, which are usually suited for near-infrared wavelengths. Therefore, methods based mostly \non non-thermal effects, such as inverse Faraday18,19 and Cotton-Mouton effect20 or photoinduced magnetic \nanisotropy21, 22 have been used to trigger the magnetization precession in YIG so far. For these phenomena \nto occur, large laser fluences of tens of mJ/cm2 are required23. In contrast, laser fluences for a thermal \nexcitation of magnetization precession usually do not exceed tens of J/cm2 (Refs. 12, 21, 13). Using the \nlow fluence excitation regime allows for the determination of quasi-equilibrium material parameters, not \ninfluenced by strong laser pulses. In magnetic garnets, an artificial engineering of the magnetic anisotropy \nvia the inclusion of bismuth was necessary to achieve thermally-induced magnetization precession21. \nIn this paper, we show that magnetization precession can be induced thermally by femtosecond laser \npulses in a thin film of pure YIG only by adding a metallic capping layer. The laser pulses locally heat the \nsystem, which sets the magnetization out of equilibrium due to the temperature dependence of its \nmagnetocrystalline anisotropy. This way we generate a Kittel (n = 0, homogeneous precession) FMR mode, \nwith a precession frequency corresponding to the quasi-equilibrium magnetic anisotropy of the thin YIG \nfilm10. We thus prove that the AO-FMR method is applicable for determining micromagnetic parameters \nof thin YIG films. Using the AO-FMR technique we revealed that at low temperature the Kittel mode \ndamping is significantly faster than at room-temperature, in accord with previous FMR experiments24,25. \nOur experiments were performed on a 50 nm thick layer of pure YIG grown by pulsed-laser deposition on \na gadolinium-gallium-garnet (GGG) (111)-oriented substrate. One part of the film was covered by 8 nm of \nAu capping layer, the other part by Pt capping, both being prepared by ion-beam sputtering. Part of the \nsample was left uncapped as a reference. X-ray diffraction confirmed the excellent crystal quality of the \nYIG film with a very low level of growth-induced strain, as described in detail in Ref. 26. The magnetic \nproperties were further characterized using SQUID magnetometry and ferromagnetic resonance \nexperiments, showing the in-plane orientation of magnetization (see Supplementary Material, Part 1 and \nFigs. S1 and S2). The deduced low-temperature (20 K) saturation magnetization µ0Ms 180 mT is in \nagreement with results published on qualitatively similar samples27 again confirming a good quality of the \nstudied YIG film. Magnetic anisotropy of the system at 20 K was established from an independent \nmagneto-optical experiment (Ref. 28), the corresponding anisotropy constants for cubic anisotropy of the \nfirst and second order are Kc1 = 4680 J/m3 and Kc2 = 223 J/m3, while the overall uniaxial out-of-plane \nanisotropy is vanishingly small. 3 \n Laser-induced dynamics was studied in a time-resolved magneto-optical experiment in transmission \ngeometry, as schematically shown in Fig. 1(a). An output of a Ti:Sapphire oscillator generating 200 fs laser \npulses was divided into a strong pump beam, with fluences tuned between 70 and 280 µJ/cm2, and a 20-\ntimes weaker probe beam. The beams were focused on a 30 m spot on the sample, which was placed in \na cryostat and kept at cryogenic temperatures (typically 20 K). An external magnetic field (up to 550 mT) \ngenerated by an electromagnet was applied in y direction (see Fig. 1). The wavelength of pump pulses (800 \nnm) was set well below the absorption edge of the YIG layer, as indicated in the transmission spectrum of \nthe sample in Fig. 1(b). The wavelength of probe pulses (400 nm) was tuned to match the maximum of the \nmagneto-optical response of bulk YIG [see inset in Fig. 1(b) and Ref. 29]. \nThe detected time-resolved magnetooptical (TRMO) signal corresponding to the rotation of polarization \nplane of the probe beam Δβ, was measured as a function of the time delay Δt between pump and probe \npulses. In Fig. 1(c), we show an example of TRMO signals observed in uncapped YIG and two YIG/metal \nheterostructures. Clearly, in the presence of the metallic capping layer an oscillatory TRMO signal is \nobserved, whose amplitude depends on the capping metal used. Frequency and damping of the \noscillations, on the other hand, remain virtually unaffected by the type of the capping layer, while no \noscillations are observed in the uncapped YIG sample. \nThe TRMO signals can be phenomenologically described by a damped harmonic function after removing a \nslowly varying background (see Supplementary Material, Part 2 and Fig. S3),12 \n∆𝛽(Δ𝑡)=𝐴cos(2𝜋𝑓𝛥𝑡+𝜑)exp(−𝛥𝑡 𝜏⁄ ), (1) \nwhere A is the amplitude of precession, f its frequency, φ the phase and τ the damping time. The fits are \nshown in Fig. 1(c) as solid lines. \nIn order to demonstrate that the TRMO signals result from (laser-induced) magnetization dynamics, we \nvaried the external magnetic field Hext and extracted the particular precession parameters by fitting the \ndetected signals by Eq. (1). As depicted in Fig. 2(a), the experimentally observed dependence of the \nprecession frequency on the applied field is in excellent agreement with the solution of Landau-Lifshitz-\nGilbert (LLG) equation, using the free energy of a [111] oriented cubic crystal [see Supplementary, section \n5, Eq. (S5) and Ref. 28]. This correspondence with the LLG model proves that our oscillatory signals reflect \nindeed the precession of magnetization in uniform (Kittel) mode in YIG. We stress that the precession \nfrequency is inherent to the YIG layer and does not depend on the type of the capping layer. 4 \n The detection of the uniform Kittel mode can be further confirmed by comparing the frequency of the \noscillatory TRMO signal with the frequency of resonance modes observed in a conventional, microwave-\ndriven ferromagnetic resonance (MW-FMR) experiment. The MW-FMR experiment was performed in the \nin-plane ( H = 0°) and out-of-plane ( H = 90°) geometry of the external field. We measured the TRMO signals \nin YIG/Au sample in a range of magnetic field angles H and modelled the angular dependency of f by LLG \nequation with the same parameters that were used in Fig. 2(a). The output of the model is presented in \nFig. 2(b), together with precession frequencies obtained from TRMO and FMR experiments. The MW-FMR \ndata fit well to the overall trend, confirming the presence of uniform magnetization precession [Ref. 30] \nTo find the exact physical mechanism that triggers laser-induced magnetization precession in our \nYIG/metal bilayers, we measured the TRMO signals at different sample temperatures T. For comparison \nwe calculated also the dependence of f on the first order cubic anisotropy constant Kc1 from the LLG \nequation, which is shown in the inset of Fig. 2(c). This graph reveals that f should be directly proportional \nto Kc1 in the studied range of temperatures . In Fig 2(c) we plot f as a function of T, together with the \ntemperature dependence of Kc1 (T) obtained from Ref. 28 and Ref. 32. Clearly, both Kc1 and f show a similar \ntrend in temperature. Considering also the temperature dependence of the precession amplitude [see \nFig. S5 (a) and Section 4 of Supplementary Material], we identify the pump pulse-induced heating and \nconsequent modification of the magnetocrystalline anisotropy constant Kc1 as the dominant mechanism \ndriving laser-induced magnetization precession. \nIn order to estimate the pump-induced increase in quasi-equilibrium temperature of the sample, we first \nfit the temperature dependence of the parameter Kc1 reported in literature by a second order polynomial \n[Fig. 2(c)]. Owing to the linear relation between f and Kc1 and the known temperature dependence of f, \nthe measured dependence of f on pump fluence I can be converted to the intensity dependence of the \ntemperature increase T(I), which is shown Fig. 2(d). As expected, higher fluence leads to more \npronounced heating, which results in a decrease of the precession frequency. Note that for the highest \nintensity of 300 J/cm2, the sample temperature can increase by almost 80 K. \nNature of the observed laser-induced magnetization precession was further investigated by comparing \nsamples with different capping layers. In Fig. 3(a) we show the amplitude A of the oscillatory signal in the \nYIG/Pt and YIG/Au layers as a function of I. The difference between the samples is apparent both in the \nabsolute amplitude of the precession and in its increase with I, the YIG/Pt showing stronger precession. \nFurthermore, precession damping is stronger in YIG/Au than in YIG/Pt, as apparent from Fig 3 (b) where \neffective Gilbert damping parameter eff is presented as a function of Hext. These values of eff were 5 \n obtained by fitting the TRMO data by the LLG equation, as described in the Supplementary Material \n(Section 5). Despite the relatively large fitting error, we can still see that YIG/Pt shows slightly lower \n0.020, while the YIG/Au has 0.025. To understand these differences, we modeled the propagation of \nlaser-induced heat in GGG/YIG/Pt and GGG/YIG/Au multilayers by using the heat equation (see \nSupplementary Material, Section 7). In Fig. 3(c), T is presented as a function of time delay t after pump \nexcitation for selected depths from the sample surface. In Fig. 3(d), the same calculations are presented \nfor variable depths and fixed t. The model clearly demonstrates that a significantly higher T can be \nexpected in the Pt-capped layer simply due to its smaller reflection coefficient as compared to Au-capping \n(see Supplementary Material, Section 7). This in turn leads to a higher amplitude of the laser-induced \nmagnetization precession in YIG/Pt compared to the YIG/Au, as apparent in Fig. 3(a). \nAccording to our model, an extreme increase in temperature is induced in the first few picoseconds after \nexcitation, which acts as a trigger of magnetization precession. After approximately 10 ps, precession takes \nplace in quasi-equilibrium conditions. The system returns to equilibrium on a timescale of nanoseconds, \nwhich shows also in the TRMO signals as the slowly varying background (Fig. S3). The precession frequency \nwe detect reflects the quasi-equilibrium state of the system. Therefore, the temperature increase T \ndeduced from the TRMO signal can be compared with our model for large time delays after the excitation \n(t 10 ps). In YIG/Au sample, the experimental values of T = (25 10) K for excitation intensity of 150 \nJ/cm2 [see Fig. 2(d)], while the model gives us T 5K [Fig. 3 (c)]. Clearly, the values match in the order \nof magnitude but there is a factor of 5 difference. This difference results from the boundary conditions \nof the model that assumes ideal heat transfer between the sample and the holder, which is experimentally \nrealized using a silver glue with less than perfect performance at cryogenic conditions. \nFrom Fig. 3(d) it also follows that large thermal gradients are generated across the 50 nm layer. This could \nlead to significant inhomogeneity in magnetic properties of the layer, that would increase the damping \nparameter by an extrinsic term. In our TRMO measurements, is indeed very large for a typical YIG \nsample (TRMO 2-2.5 x10-2) and exceeds the value obtained from room-temperature MW-FMR by almost \nan order of magnitude ( FMR 1x10-3, see Supplementary Material, Section 1b). As the modeled thermal \ngradient alone cannot account for such a large change in Gilbert damping (see Supplementary Material, \nSection 6), we attribute this increase in Gilbert damping to the difference in the ambient temperatures. \nLarge change of Gilbert damping (by a factor of 30) between room and cryogenic (20 K) temperature has \nrecently been reported on a seemingly high quality YIG thin film24. It was explained in terms of the presence \nof rare earth or Fe2+ impurities that are activated at cryogenic temperatures. It is likely that the same 6 \n process occurs in our sample. Even though other mechanisms related to the optical excitation can also \ncontribute to the increase in TRMO (see Supplementary Material, Section 6), the all-optical and standard \nFMR generated Kittel modes correspond very well [see Fig. 2(b)]. Furthermore, also the observed sample-\ndependent Gilbert damping is consistent with this explanation. The YIG/Pt sample is heated to higher \ntemperature by the pump laser pulse [Fig. 3(c), (d)] than the YIG/Au sample, which according to Ref. 24 \ncorresponds to a lower Gilbert damping. It is worth noting that damping parameter can be increased also \nby spin-pumping from YIG to the metallic layer. However, this effect is expected to be significantly higher \nwhen Pt is used as a capping, which does not agree with our observations. \nIn conclusion, we demonstrated the feasibility of the all-optical ferromagnetic resonance method in 50-\nnm thin films of plain YIG. Magnetization precession can be triggered by laser-induced heating of a metallic \ncapping layer deposited on top of the YIG film. The consequent change of sample temperature modifies \nits magnetocrystalline anisotropy, which sets the system out of equilibrium and initiates the magnetization \nprecession. Based on the field dependence of precession frequency, we identify the induced magnetization \ndynamics as the fundamental (Kittel) FMR mode, which is virtually independent of the type of capping and \nreflects the quasi-equilibrium magnetic anisotropy. The Gilbert damping parameter is influenced by line-\nbroadening mechanism due to low-temperature activation of impurities, which is an important aspect to \nbe taken into account for low-temperature spintronic device applications. \nRegarding the efficiency of the optical magnetization precession trigger, it was found that the type of \ncapping layer strongly influences the precession amplitude. The precession in YIG/Pt attained almost twice \nthe amplitude of that in YIG/Au under the same conditions. This indicates that a suitable choice of capping \nlayer should be considered in an optimization of this local non-invasive magnetometric method. \n \nAcknowledgments: \nThis work was supported in part by the INTER-COST grant no. LTC20026 and by the EU FET Open RIA \ngrant no. 766566. We also acknowledge CzechNanoLab project LM2018110 funded by MEYS CR for the \nfinancial support of the measurements at LNSM Research Infrastructure and the German Research \nFoundation (DFG SFB TRR173 Spin+X projects A01 and B02 #268565370). \n \n 7 \n LITERATURE \n[1] A. Hirohata et al., JMMM 509, 16671 (2020) \n[2] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010) \n[3] F. Siegries et al., Nature 571, 240–244 (2019) \n[4] E. Beaurepaire et al., Phys. Rev. Lett. 76, 4250 (1996). \n[5] P. Nemec et al., Nature Physics 8, 411-415 (2012) \n[6] G.M. Choi et al., Nat. Comm. 8, 15085 (2017) \n[7] N. Tesařová et al., Nat. Phot. 7, 492-498 (2013) \n[8] A. Kimel et al., Nature 429, 850–853 (2004). \n[9] Y.G. Choi and G.M. Choi, Appl. Phys. Lett. 119, 022404 (2021) \n[10] P. Němec et al., Nature Communications 4, 1422 (2013) \n[11] V. N. Kats et al., PRB 93, 214422 (2016) \n[12] Y. Hashimoto, S. Kobayashi, and H. 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Mat. 3, 034403 (2019) \n[28] E. Schmoranzerova et al., ArXiv XXX (2021) \n[29] E. Lišková Jakubisova et al., Appl. Phys. Lett. 108, 082403 (2016) \n[30] We note that the FMR data were obtained at room temperature while the TRMO experiment was \nperformed at 20K. However, as apparent from Fig. 2(c), the precession frequency varies by less than \n10% between 20 K and 300 K, which is well below the experimental error of (H). This justifies \ncomparison of the precession frequencies obtained from the TRMO experiment with the FMR data. \n[31] M. Haider et al., J. Appl. Phys. 117, 17D119 (2015) \n[32] N. Beaulieu et al., IEEE Magnetics Letters 9, 3706005 (2018) \n \n \n \nFIGURES \n \n \n9 \n Fig. 1: (a) Schematic illustration of the pump&probe experimental setup, where Eprobe is the probe beam linear \npolarization orientation which is rotated by an angle after transmission through the sample with respect to the \norientation E’probe. An external magnetic field H ext is applied at an angle H. (b) Absorption spectrum of the studied \nYIG sample, where OD stands for the optical density defined as minus the decadic logarithm of sample \ntransmittance. The red arrow indicates the wavelength of the pump beam PUMP = 800 nm. Inset: Spectrum of Kerr \nrotation K of bulk YIG crystal29. The blue arrow shows the wavelength of the probe beam PROBE = 400 nm. (c) \nTypical time-resolved magneto-optical signals of a plain 50 nm YIG film (black dots), YIG /Pt (green dots) and \nYIG/Au bilayer (blue dots) at 20 K and 0Hext = 100 mT, applied at an angle H = 40°. Lines indicate fits by Eq. (1). \nThe data were offset for clarity. \n \n \n \nFig. 2: (a) Frequency f of magnetization precession as a function of magnetic field applied at an angle H = 40°, for \nYIG/Pt (blue dots) and YIG/Au (green triangles) at T = 20 K and I = 150 J/cm2. The line is calculated from LLG equation \n(Eq. S3) with the free energy given by (Eq.S5) (b) Field-angle dependence of f in YIG/Au sample for 0Hext = 300 mT \n(blue dots), compared to a model by LLG model (line) and to frequencies measured by MW-FMR (red stars)32. (c) \nTemperature dependence of f in YIG/Au sample (black points), where 0Hext = 300 mT was applied at H = 40°. The \ntemperature dependence of cubic anisotropy constant Kc1 was obtained from Ref. 28 (red dots) and Ref. 32 (red star, \nT = 20 K). The data were fitted by an inverse polynomial dependence 𝐾ଵ(𝑇)= ଵ\n(ା்ା்మ), with parameters: a = 0.18 \nm2/kJ; b= 9 x 10-4 m2/kJ.K; c = 9 x 10-6m2/kJ.K2. Inset: Dependence f(Kc1) obtained from the LLG equation. (d) f as a \nfunction of pump pulse fluence I, from which the increase of sample temperature T for the used pump fluences was \nevaluated using the f(T) dependence. \n \n10 \n \nFig. 3: Comparison of magnetization precession in YIG/Pt and YIG/Au samples. (a) Precession amplitude A as a \nfunction of pump fluence I (dots) with the corresponding linear fits 𝐴 = 𝑠∙𝐼. The parameter s Pt = (1.05 0.09)x10-2 \nrad.cm2/J in the YIG/Pt, and s Au = (0.50.1)x10-2 rad.cm2/J in YIG/Au. These dependencies were measured for \n0Hext = 300 mT and T 0 = 20 K. In YIG/Pt sample the as-measured data obtained for H = 40° are shown. In the YIG/Au \nsample, the A(I) dependence was originally measured for H = 21° and recalculated to H = 40° according to the \nmeasured angular dependence, as described in detail in Supplementary Material, Section 3. (b) Gilbert damping eff \nfor Hext applied at an angle H = 40°. The values of eff result from fitting the TRMO signals to LLG equation; I = 140 \nJ/cm2. (c) and (d) Increase in lattice temperature as a function of time delay between pump and probe pulses for \nselected depths from the sample surface (c) and as a function of depth for fixed time delays (d). I = 140 J/cm2, T0 = \n20 K. The heat capacities and conductivities of individual layers are provided in the Supplementary Material, Section \n7. \n \n \n \n11 \n Thermally induced all-optical ferromagnetic resonance in thin YIG films: \nSupplementary Material \nE. Schmoranzerová1*, J. Kimák1, R. Schlitz3 , S.T. B. Goennenwein3, D. Kriegner2,3, H. Reichlová2,3, , Z. \nŠobáň2, G. Jakob5, E.-J. Guo5, M. Kläui5, M. Münzenberg4, P. Němec1 , T. Ostatnický1 \n1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic \n2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic \n3Technical University Dresden, 01062 Dresden, Germany \n4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany \n5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany \n \n6 Department of Physics, University of Konstanz, 78464 Konstanz, Germany \n \n \n1. Magnetic characterization \n \nA. SQUID magnetometry \nA superconducting quantum device magnetometer (SQUID) was used to characterize the magnetic \nproperties of the thin YIG film at several sample temperatures. The magnetic hysteresis loops, detected \nwith magnetic field applied in [2-1-1] crystallographic direction of the YIG layer, are shown in Fig. S1. As \nexpected [t26], the saturation magnetization increases at low temperatures, which is accompanied by a \nslight increase in coercive field. At room temperature, the effective saturation magnetization is estimated \nto be Ms = 95 kA/m. This value is in good agreement with the effective magnetization Meff obtained from \nthe ferromagnetic resonance (FMR) measurement (see Section 1b), which indicates only a weak out-of-\nplane magnetic anisotropy [s1]. However, as discussed in detail in Ref. 26, the Ms from our SQUID \nmeasurement is burdened by a relatively large error. Therefore, mere comparison of SQUID and FMR \nexperiment is not sufficient to evaluate the size of the out-of-plane magnetic anisotropy. An additional \nexperiment such as static magneto-optical measurement [28] is needed in order to get more precise \nestimation of the out-of plane magnetic anisotropy. \n \nB. FMR measurement \nThe SQUID magnetometry was complemented by so-called broad band ferromagnetic resonance \nmeasurements using a co-planar waveguide to apply electromagnetic radiation of a variable frequency f \n=/2 to the sample. The measurement was performed at room temperature and further details on the \nmethod can be found in Ref. s2. An exemplary set of spectra showing the normalized microwave \ntransmission | S21|norm obtained at different external fields magnitudes applied in the sample plane, is \nshown in Fig. S2(a). The set of Lorentzian-shape resonances can be fitted by the equation: 12 \n |𝑆ଶଵ|୬୭୰୫=ቀഘ\nమቁమ\nቀഘ\nమഏିഘబ\nమഏቁାቀഘ\nమቁమ+𝑦 (S1) \nWhere f0 =0/2 is the FMR resonance frequency, /2 is the half width half maximum line width, B the \namplitude of the FMR line and y0 a frequency independent offset. From an automated fitting of the set of \nlines obtained at different Hext, we extract the magnetic field dependence of the resonance frequency \n0/2 (Hext) [Fig. S2(b)] and linewidth (Hext) [Fig. S2(c)]. Clearly, the resonance frequencies correspond \nto the fundamental (Kittel) mode, and can correspondingly be fitted by the Kittel formula [s3]: \nఠబ\nଶగ=ఊ\nଶగඥ𝜇𝐻ୣ୶୲(𝜇𝐻ୣ୶୲+𝜇𝑀ୣ) (S2) \nWhere Meff is the effective saturation magnetization that includes the out-of-plane anisotropy term, and \nis gyromagnetic ratio. From this fit, it is possible to evaluate Meff , Kittel = 94.9 kA/m \nFrom the linewidth dependence (Hext)=2 + 0 we can extract both the inhomogeneous line \nbroadening and the Gilbert damping parameter, as shown in Fig. S2(c) [s2]. In our experiment, the \ninhomogeneous linewidth broadening is 0 = 55.8 MHz, and the Gilbert damping parameter = 0.001. \nBoth values are on a higher side compared e.g. with YIG prepared by liquid phase epitaxy [s8] but in good \nagreement with typical YIG thin films similar to our layers, which were prepared by pulsed laser deposition \n[27]. This again confirms the good quality of the studied thin YIG films. \n \n2. Processing of time-resolved magneto-optical data \nIn order to extract the parameters describing the precession of magnetization correctly from the time-\nresolved magneto-optical (TRMO) signals, it is first necessary to remove the slowly varying background on \nwhich the oscillatory signals are superimposed. For this purpose, we fitted the measured data by the \nsecond-order polynomial. The fitted curve was then subtracted from the measured signals, as \ndemonstrated in Fig. (S3). \nFrom the physical point of view, the background can be attributed to a slow return of magnetization to its \nequilibrium state after the pump beam induced heating, which can take place on the timescale of tens of \nnanoseconds [10]. Since both saturation magnetization Ms and magnetocrystalline anisotropy Kc are \ntemperature-dependent, their temporal variation can in principle contribute to the background signal. \nHowever, as explained later in Section 4, the variation of Ms is very weak at cryogenic temperatures. The \nheat-induced modification of Kc, and the resulting change of the magnetization quasi-equilibrium \norientation, is, therefore, more probable origin of the slowly varying background, which is detected in the \nMO experiment by the Cotton-Mouton effect [28]. \n3. Angular dependence of precession amplitude \nIn order to mutually compare the values of precession amplitudes measured in YIG/Pt and YIG/Au samples \nat different angles of the external magnetic field H, it is necessary to correct their values for the value of \nH. The following procedure was used to correct the data presented in Fig. 3 of the main text . \n 13 \n First, we measured in detail angular dependence of the precession amplitude in the YIG/Au layer, which is \npresented in Fig. S4. Amplitude of the oscillatory signal detected in our experiment does not depend solely \non the amplitude of the magnetization precession but also on the size of the magneto-optical (MO) effect. \nIn our experimental setup, the change of H was achieved by tilting the sample relative to the position of \nelectromagnet poles [see Fig. 1(a)]. The MO response, however, varies also with the angle of incidence \nwhich is modified simultaneously with a change of H [see Fig. 1(a)] . Therefore, it is not straightforward to \ndescribe the A(H) analytically. Instead, we fitted the measured dependence A(H) by a rational function in \na form of y = 1/(A+Bx2), which is the lowest order polynomial function that can describe the signal properly. \nFrom the fit we derived a correction factor of 1.7 by which the amplitudes A measured at H =21° has to \nbe multiplied to correspond to that measured at H =40°. This factor was then used to recalculate the \nintensity dependence of the precession amplitude A(I) in YIG/Au measured at H =21° to the A(I) at H \n=40°, which could be directly compared to the A(I) dependence detected at YIG/Pt for H =40° - see Fig. \nS4(b). \n \n4. Temperature dependence of precession amplitude \n \nIn order to further investigate the origin of the laser-induced magnetization precession, the amplitude of \nthe oscillatory MO signal was measured as a function of the sample temperature in YIG/Pt sample, see Fig. \nS5(a). In Fig. S5 (b), we show temperature dependence of saturation magnetization Ms, as obtained from \nRef. 32 \nThe only parameter changed within this experiment was the sample temperature. It is reasonable to \nexpect that the size of the magneto-optical effect is not strongly temperature dependent in the studied \ntemperature range between 20 and 50 K (see Ref. 28) Therefore, the dependence A(T) presented in Fig S5 \ncorresponds directly to the temperature dependence of magnetization precession amplitude. By \ncomparing the Ms(T) and A(T) data, it is immediately apparent that the laser-induced heating would not \nmodify Ms enough to account for the large change of the magnetization precession amplitude with the \nsample temperature. Even assuming the most extreme laser-induced temperature increase T 80 K \nshown in Fig. 3(c), the laser-induced Ms variation would be less than 5%, while the precession amplitude \nchanges by more than 50% between 20 and 50 K. In contrast, the magnetocrystalline anisotropy Kc1 \nchanges drastically even in this relatively narrow temperature range [see Fig. 2(c)]. Consequently, the \nchange of Kc1, which leads to a significant change of the position of quasi-equilibrium magnetization \norientation in the studied sample (see Section 5) provides a more plausible explanation for the origin of \nthe laser-induced magnetization precession in the YIG/metal layer. \n \n5. LLG equation model \nThe data were modelled by numerical solution of the Landau-Lifshitz-Gilbert (LLG) equation, as defined in \n[s9]: \nௗ𝑴(௧)\nௗ௧= −𝜇𝛾ൣ𝑴(𝑡)×𝑯𝒆𝒇𝒇(𝑡)൧+ఈ\nெೞቂ𝑴(𝑡)×ௗ𝑴(௧)\nௗ௧ቃ, (S3) 14 \n where is the gyromagnetic ratio, is the Gilbert damping constant, and MS is saturated \nmagnetization.The effective magnetic field Heff is given by: \n𝑯𝒆𝒇𝒇(𝑡)=డி\nడ𝑴 (S4) \nwhere F is energy density functional that contains contributions from the external magnetic field Hext, \ndemagnetizing field and the magnetic anisotropy of the sample. We consider the form of F including first- \nand second-order cubic terms as defined in Ref. [t24]. The polar angle is measured with respect to the \ncrystallographic axis [111] and the azimuthal angle = 0 corresponds to the direction [21ത1ത], with an \nappropriate index referring to the magnetization position (index M) or the direction of the external \nmagnetic field (index H). The resulting functional takes the form (in the SI units): \n \n𝐹= −𝜇𝐻𝑀[sin𝜃ெsin𝜃ு+cos𝜃ெcos𝜃ுcos(𝜑ு−𝜑ெ)]+ቀଵ\nଶ𝜇𝑀ଶ−𝐾uቁsinଶ𝜃ெ \n +𝐾c1\n12ൣ7cosସ𝜃ெ−8cosଶ𝜃ெ+4−4√2cosଷ𝜃ெsin𝜃ெcos3𝜑ெ൧ \n +c2\nଵ଼ൣ−24cos𝜃ெ+45cosସ𝜃ெ−24cosଶ𝜃ெ+4−2√2cosଷ𝜃ெsin𝜃ெ(5cosଶ𝜃ெ−\n2)cos3𝜑ெ+cos𝜃ெcos6𝜑ெ൧ , (S5) \n \nwhere 0 is the vacuum permeability and we consider the following values of constants: magnetization M \n= 174 kA/m, first-order cubic anisotropy constant Kc1=4.68 kJ/m3, second-order cubic anisotropy constant \nKc2 = 222 J/m3 [t24]. \nFor modelling the dependence of precession frequency on the external magnetic field Hext [Fig. 2 (a)] and \non the angle H [Fig. 2 (a)], we assumed that in a steady state magnetization direction is parallel to Hext, i.e. \nM = H, and M = H. This is surely fulfilled for large enough magnitude of Hext. Since the coercive field is \nvery small, we can assume the procedure to be correct. Further correspondence to experimental data \nEvaluation of the Gilbert damping factor from the as-measured magneto-optical oscillatory data was done \nby fitting signals by a theoretical curve calculated by solving numerically LLG equation [Eq. (S3)]. We \nconsidered the magnetization free energy density in a form of Eq. (S5) using magnitude and direction of \nthe external magnetic field from the experiment. The electron g-factor was set to 2.0 and then the Gilbert \nfactor and five parameters of the fourth-order polynomial to remove the background MO signal were the \nfitting parameters. The resulting dependence of fitted effective Gilbert factors αeff on external magnetic \nfield is displayed in Fig. 3(b) in the main text, from which the field-independent Gilbert factor α can be \nevaluated. \n \n6. Comparison of Gilbert damping parameter from MW-FMR and TRMO experiments \nThe Gilbert damping from the room-temperature FMR measurement on the YIG film 1∙10-3 and the \nresults from fits of the low-temperature pump&probe data 2∙10-2, differ by an order of magnitude. As \ndetailed in the main text, we attribute this difference to the different sample temperatures in the AO-FMR \nand MW-FMR measurements. However, one might also argue that the increased damping in the optical \nexperiments is caused either by a spatial inhomogeneity of the magnetization oscillations or it is the result \nof the perturbation of the YIG surface. 15 \n In the former case, we expect that the spatial inhomogeneity of the temperature distribution [see Fig. \n3(d)] causes the magnetization to oscillate in a form of a superposition of harmonic waves with well-\ndefined in-plane wavevectors. Considering the dispersion of the allowed oscillatory modes [s4] and \nincluding the relevant value of the exchange stiffness [s5], we revealed that neither the inhomogeneity \ndue to the finite cross section of the excitation laser beam nor the temperature gradient perpendicular to \nthe sample surface can cause such a strong decrease of the Gilbert damping factor that is observed \nexperimentally. Here, we provide an estimate on which time scales the mode dispersion influences the \ndecay of the signal if the exchange stiffness is taken into account. Following [s5], the mode dispersion is \ndescribed by the additive exchange field in the form: \n𝜇𝐻ex=𝐷ቈ𝜋ଶ\n𝑑ଶ𝑛ଶ+𝑘∥ଶ , \nwhere D ≈ 5∙10-17 T.m2 is the exchange stiffness, n is the order of the confined magnon mode, d is the YIG \nlayer thickness and k‖ is the in-plane magnon wave vector. We consider here only the n = 0 case since this \nis the only visible harmonic mode observed in the experimental MO data, as proven by the numerical \nfitting. Note that the frequency shift ∆𝜔/2𝜋=|𝛾|𝜇𝐻ex/2𝜋, where the symbol γ stands for the electron \ngyromagnetic ratio, of the n = 1 mode would be 5.5 GHz, which would be then clearly distinguishable from \nthe basic n = 0 mode in the lowest external magnetic fields. The in-plane wave vector k‖ can be calculated \nfrom the FWHM (full width at half maximum) width of the laser spot on the sample L, which is about 30 \nµm in our case, that leads to the order of magnitude k‖ ≈ (2π/L) ≈ 105 m-1. The frequency increase due to \nthe finite laser spot size can be estimated as ∆𝜔/2𝜋=|𝛾|𝐷𝑘∥ଶ/2𝜋≈14 kHz. Inverse of this value ( 0.1 \nms) determines the typical time scale at which the magnon dynamics is influenced by their dispersion due \nto the finite laser spot size, which is clearly out of the range of the experimental time scale. \nThe presence of a metallic layer on the top of the YIG sample surface can result into two significant \ndamping processes. First, the magnetization oscillations (and thus oscillations of the macroscopic magnetic \nfield) are coupled to electromagnetic modes which penetrate the surrounding material and can be \neventually radiative for small magnon wave vectors. Penetration into conductive material in turn causes \nenergy dissipation through finite conductivity of such material. We checked the magnitudes of the \nadditional damping caused by the radiative field and energy dissipation in a thin metallic layer and we \nfound that these processes exist but the additional energy loss cannot explain the observed magnitude of \nthe Gilbert damping parameter. The second possible explanation of the increased precession damping due \nto the presence of the metal/YIG surface may be that there is an additional perturbation to otherwise \nhomogeneous sample due to some inhomogeneity through surface roughness or spatially inhomogeneous \nlocal spin pinning. Since both the surface roughness and spin pinning can depend on the composition of \nthe capping layer, it can also cause a minor difference in the resulting damping factor, as observed in Fig. \n3(b). \nOverall, we attribute the experimentally observed difference in Gilbert damping measured by FMR and \npump&probe techniques to the difference in ambient temperatures that were used in these experiments, \nwhich is in accord with the results of Ref. t22. \n \n7. Heat propagation in YIG/Pt and YIG/Au 16 \n Heat propagation in our sample structures was modelled in terms of the heat equation: \nడ்\nడ௧=ఒ\n∆𝑇 , (S4) \nwhere T is the local temperature, λ is the local thermal conductivity, c is the heat capacity and the symbol \nΔ denotes the Laplace operator. The spatio-temporal temperature distribution in the studied sample has \nbeen calculated by a direct integration of Eq. (S4) in a time domain, assuming excitation of the metallic \nlayer by an ultrashort optical pulse [with a temporal duration of 100 fs (FWHM)]. We have taken the whole \nstructure profile of vacuum/metal/YIG/GGG into consideration, assuming that the GGG substrate had a \nperfect heat contact with the cold finger of the cryostat, which has been held on a constant temperature. \nThe respective heat conductivities ( λ) and heat capacities ( c) were set to the following values. Au: λ = 5 \nW/m K [s6], c = 1.3∙104 J/cm3, Pt: λ = 10 W/m K [s7], c = 1.2∙104 J/cm3, YIG: λ = 60 W/m K, c = 6.7∙103 J/cm3, \nGGG: λ = 300 W/m K, c = 2.1∙104 J/m3. \nTo evaluate the initial heat transfer from the optical pulses to the capping metallic layer, we considered \nthe proper geometry of our experiment, i.e. a 8 nm thick metallic layer deposited on the YIG sample, the \nincidence angle of the laser beam of 45 degrees and its p-polarization. We then used optical constants of \ngold and platinum in order to calculate transmission and reflection coefficients of a nanometer-thick \nmetallic layers by means of the transfer matrix method. From those, we estimated the efficiency of power \nconversion from the optical field to heat to be 3% for gold and 6.5% for platinum. The total amount of heat \ndensity was then calculated by multiplication of the pump pulse energy density and the above-mentioned \nefficiency. \nThe data shown in Fig. 3(c)-(d) were then extracted from the full spatio-temporal temperature distribution. \nClearly, the temperature increase in the YIG/Pt sample is approximately twice larger than that of the \nYIG/Au sample as a consequence of twice larger efficiency of the light-heat energy conversion in favour of \nplatinum. Correspondingly, also the amplitudes of the MO oscillations in Fig. 3(a) reveal the ratio 2:1. \n \n \nLITERATURE \n[s1] B. Bhoi et al.: J. Appl.Phys. 123, 203902 (2018) \n[s2] H. Maier-Flaig et al,.PRB 95, 214423 (2017) \n[s3] Ch. Kittel: “Introduction to solid state physics (8th ed.)”. New Jersey: Wiley. (2013). \n \n[s4] D. D. Stancil, A. Prabhakar: Spin waves – theory and applications (Springer, New York, 2009). \n[s5] S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. Hillebrands, \nand A. Conca, J. Phys. D: Appl. Phys. 48, 015001 (2014). \n[s6] G. K. White, Proc. Phys. Soc. A 66, 559 (1953). \n[s7] X. Zhang, H. Xie, M. Fujii, H Ago, K. Takahashi, T. Ikuta, T. Shimizu, Appl. Phys. Lett. 86, 171912 (2005) 17 \n \n[s8] C. Dubs et al., Phys. Rev. Materials 4, 024416 (2020) \n[s9] J. Miltat, G. Albuquerque, and A. Thiaville, An introduction to micromagnetics in the dynamic regime, \nin Spin dynamics in confined magnetic structures I, edited by B. Hillebrands and K. Ounadjela, Springer, \nBerlin, 2002, vol. 83 of Topics in applied physics. \n \nFIGURES \n \n \n \nFig. S1: Magnetic hysteresis loops measured by SQUID magnetometry with magnetic field Hext applied in \ndirection [2-1-1] at several sample temperatures. The saturation magnetization obtained from SQUID \nmagnetometry measurement at room temperature is roughly Ms = 95 kA/m , assuming a YIG layer thickness \nof 50 nm. \n \n18 \n \n \nFig. S2: (a) Ferromagnetic resonance spectra measured at room temperature for several different external \nmagnetic field magnitudes µ0Hext from 0 to 540 mT applied in the sample plane. Resonance peaks were \nfitted by Eq. (S1) and the obtained resonance frequencies and linewidths are plotted as points in panels \n(b) and (c), respectively. The lines correspond to fit by Kittel formula [Eq. (S2)], which enables to evaluate \neffective magnetization Meff = 94.9 kA/m and Gilbert damping parameter of = 0.001. \n \n19 \n Fig. S3: Removal of slowly varying background from time-resolved magneto-optical signals. The red dots \ncorrespond to as-measured signals, line indicates the polynomial background that is subtracted from the \nraw signals. Black dots show the signal after background subtraction, black line representing the fit by Eq. \n(1) of the main text. The data were taken at external field of 0Hext = 300 mT, temperature 20 K and pump \nfluence I = 140 J/cm2.. \n \n \nFig. S4: (a) Dependence of the amplitude A of oscillatory magneto-optical signal on the sample tilt \n(different field angles of magnetic field H ) measured in YIG/Au sample . 0Hext = 300 mT, temperature T \n= 20 K and pump fluence I = 150 J/cm2.. (b) Pump intensity dependence of A measured for YIG/Au \nsample at H =21° (red points), the same dependence recalculated to correspond to H =40 (blue \npoints) where A(I) was measured for YIG/Pt sample (green points); T = 20 K. \n \n \nFig. S5: (a) Temperature dependence of amplitude of the time-resolved magneto-optical signals measured \nfor external field 0Hext = 300 mT applied at an angle H = 30°. (b) Temperature dependence of saturation \nmagnetization Ms obtained from Ref. 32. \n \n" }, { "title": "2309.03116v1.Strong_magnon_magnon_coupling_in_an_ultralow_damping_all_magnetic_insulator_heterostructure.pdf", "content": "1 \n Strong magnon -magnon coupling in an ultralow damping all-magnetic -insulator heterostructure \nJiacheng Liu*1,5, Yuzan Xiong*2, Jingming Liang*3, Xuezhao Wu1,5, Chen Liu4, Shun Kong Cheung1,5 \nZheyu Ren1,5, Ruizi Liu1,5, Andrew Christy2, Zehan Chen1,5,6, Ferris Prima Nugraha1,5, Xi-Xiang Zhang4, \nChi Wah Leung3, Wei Zhang#2, Qiming Shao#1,5,6 \n1Department of Electronic and Computer Engineering, The Hong Kong University of Science and \nTechnology, Hong Kong SAR \n2Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC \n27599, USA \n3Department of Applied Physics, The Hong Kong Polytechnic University, Hong Kong SAR \n4 Physical Science and Engineering Division (PSE), King Abdullah Universi ty of Science and Technology \n(KAUST), Thuwal 23955 –6900, Saudi Arabia \n5IAS Center for Quantum Technologies, The Hong Kong University of Science and Technology, Hong \nKong SAR \n6Department of Physics, The Hong Kong University of Science and Technology, Hong K ong SAR \n* Equal contributions # Corresponding emails: zhwei@unc.edu ; eeqshao@ust.hk \n \nMagnetic insulators such as y ttrium iron garnets (YIGs) are of paramount importance for spin-wave \nor magnonic devices as their ultralow damping enables ultra low power dissipation that is free of \nJoule heating , exotic magnon quantum state, and coherent coupling to other wave excitations. \nMagne tic insulator heterostructures bestow superior structural and magnetic properties and house \nimmense design space thanks to the strong and engineerable exchange interaction between individual \nlayers. To full y unleash their potential, realizing low damping and strong exchange coupling \nsimultaneously is critical , which often requires high quality interface . Here, we show that such a \ndemand is realized in an all-insulator thulium iron garnet (TmIG) /YIG bilayer system. The ultralow \ndissipation rates in both YIG and TmIG , along with their significant spin-spin interaction at the \ninterface, enable strong and coherent magnon -magnon coupling with a benchmarking cooperativity \nvalue larger than the conventional ferromagnetic metal -based heterostructures . The coupling \nstrength can be tuned by varying the magnetic insulator layer thickness and magnon mode s, which \nis consistent with analytical calculations and micromagnetic simulatio ns. Our results demonstrate \nTmIG/YIG as a novel platform for investigating hybrid magnonic phenomena and open \nopportunities in magnon devices comprising all-insulator heterostructures. \n \nSpin-wave (or magnonic ) devices utilize magnon spin degree of freedom to process information, which can \noccur in magnetic insulators free from any charge current , and therefore, are promising contenders for \nultralow -power functional circuits 1–4. Magnetic garnets such as yttrium iron garnet ( Y3Fe5O12, YIG) have \nan ultralow damping factor , and they have enabled long magnon spin transmission 5, efficient magnon spin \ncurrent generation 6, and magnon logic circuits 2,3. Another type of magnetic garnet, thulium iron garnet \n(Tm 3Fe5O12, TmIG), has been engineered to a binary memory with a robust perpendicular magnetic \nanisotropy 7,8. Besides, TmIG thin films can exhibit topological magnetic skyrmion phase 9,10, promising \nfor future magnetic insulator -based racetrack memory devices. In addition to these promising practical 2 \n applications, magnetic insulators are well-known for hosting novel quantum phases such as Bose –Einstein \ncondensate11, spin superfluidity12, and topological magnonic insulators13. \nMagnetic heterostructures can provi de more functionalities and richer properties because exchange \ninteractions between different layers provide another control knot 14,15. While ferromagnetic metal -based \nheterostructures have been extensively studied and applied in commercial devices such as magneto -resistive \nrandom -access memor y 15, magnetic insulator -based heterostructures are still on the horizon yet already \nshowcased a few promises, including strong interfacial coupling s 16–20, magnon valve effects21–23, control \nof magnon transport in the magnetic insulator layer using another magnetic layer24,25, magnonic crystal 26, \ncoherent magnon -magnon coupling 27–30, and topological spin textures 10. Magnetic insulator \nheterostructures are also theoretically predicted to host exotic quantum phase such as magnon flat band 31. \nHowever, to date, coherent magnon -magnon coupling has only been studied in hybrid systems consisting \nof a low damping YIG and another ferromagnetic metal 27–30. The d emonstration of low damping and strong \ncoherent coupling in purely magneti c insulator bilayers is lacking. \nIn this work, we demonstrate ultralow damping and strong magnon -magnon coupling in a TmIG/YIG \nheterostructure. We characterize the structural and magnetic properties of our TmIG/YIG heterostructures \non gadolinium gallium garnet (Gd 3Ga5O12, GGG) using high-angle annular dark -field scanning \ntransmission electron micro scopy (HAADF -STEM ), X-ray diffraction (XRD), and vibrating sample \nmagnetometry (VSM) . Then, we investigate the magnetic dynamics in these bilayers by using a broadband \nferromagnetic resonance (FMR) technique . We observe a strong coupling between the Kittel mode of YIG \nand perpendicular standing spin wave (PSSW) mode of TmIG. By matching t he experimental FMR spectra \nwith analytical calculations and micromagnetic simulations, we obtain the exchange coupling strength at \nthe interface, which is dependent on the magnetic insulator layer thickness and coupling mode. Finally , we \nbenchmark the di ssipation rates and cooperativity in our samples against these in ferromagnetic metal -based \nheterostructures. \nWe prepare our TmIG/YIG on GGG substrates using pulsed laser deposition (see Methods). Atomic images \nfrom HAADF -STEM show a single crystallinity and perfect interfaces at the YIG/GGG and TmIG/YIG \nboundaries (Fig. 1a). Elemental mapping (Fig. 1b) proves there is no inter diffusion between different layers. \nFig. 1c presents the high -resolution XRD spectra of TmIG/GGG, YIG/GGG, and TmIG/YIG/GGG bilayer \nfilms measured with the scattering vector normal to the <001> oriented cubic substrate. Along the sharp \n<004> peaks from the GGG substrate, the XRD spectra shows Laue oscillations, indicating a smooth \nsurface and interface. We also measured the magnetic hysteresis loops for YIG, TmIG, and TmIG/YIG \nsamples to quantify their saturation magnetizations (see Supplementary Note 1). In pr inciple, e xchange \ncoupling strength (J) between different layers can be estimated from major and minor hysteresis loops 15. \nWe can estimate the interfa cial J at the CoFeB(50 nm)/ TmIG(350 nm) interface is −0.031 mJ/m2, \nindicating an antiferromagnetic exchange coupling (see Supplementary Note 1). However, YIG and TmIG \nhave very similar coercive fields, preventing us from obtaining the coupling strength directly from the \nhysteresis loop measurements. \nWe measure the magnetization dynamics in TmIG(200 nm)/ YIG(200 nm) bilayers using a field modulated \nFMR technique (see Methods). We mount the sample on a coplanar waveguide and apply a microwave \ncurre nt that generates radiofrequency magnetic fields (Fig. 2a) . The absorption coefficient exhibits a peak \nwhen the FMR conditions for YIG and TmIG are met (Fig. 2b) . We experimentally extract the resonance \nfrequency at a specific field by fitting the frequenc y scan at the field using Lorentz functions (Fig. 3a). In \naddition to regular FMR peaks, we also observe anti -crossing at specific field-frequency points, which are \nsignatures of exchange interaction -driven coupling of Kittel mode in YIG and PSSW modes in TmIG. To 3 \n identify the underlying magnon modes responsible for the coupling, we list the formula of generalized \nexcited spin wave modes in two layers (𝜔𝑖\n2𝜋 𝑜𝑟 𝑓𝑖): \n𝜔𝑖\n2𝜋=𝑓𝑖=𝛾𝑖\n2𝜋√(𝜇0𝐻ext+2𝐴ex,𝑖\n𝑀𝑠,𝑖 𝑘����2)(𝜇0𝐻ext+2𝐴ex,𝑖\n𝑀𝑠,𝑖 𝑘𝑖2+𝜇0𝑀𝑠,𝑖), (1) \nwhere i=YIG or TmIG, 𝛾𝑖\n2𝜋=(𝑔eff,i/2)×28 GHz /T is the gyromagnetic ratio, 𝜇0 is the permeability, 𝐻ext \nis the external field, 𝑀𝑠 is the effective magnetization, 𝐴ex is the exchange stiffness, and 𝑘 is the \nwavevector of the excited spin wave. Note that if there is no exchange interaction between YIG and TmIG, \n𝑘=𝑛𝜋\n𝑑, where n is an integer and 𝑑 is the thickness of the magnetic insulator . By fitting the Kittel mode \nwith n=0, we get 𝑔eff,YIG=2 (𝜇0𝑀𝑠,𝑌𝐼𝐺=0.25 T) and 𝑔eff,TmIG =1.56 (𝜇0𝑀𝑠,𝑇𝑚𝐼𝐺 =0.24 T) for YIG \nand TmIG , respectively, which are consistent with the previous report 34. Then, by assuming zero exchange \ninteraction and matching 𝜔YIG=𝜔TmIG from Eq. (1), we can understand the first (second) anti -crossing \nshown in Fig. 2b is from the coupling between n=0 mode in YIG and n=1 (n=2) mode in TmIG. A schematic \nof n=0 mode in YIG and n=1 mode in TmIG is shown in Fig. 2a. In addition, we determine t he exchange \nstiffness of the TmIG to be 2.69 pJ/m, which is consistent with the previous report 35. When there is an \nexchange interaction between YIG and TmIG, we expect an anti -crossing gap , whic h can be described by \nthe minimum frequency separation of 2g. However, with only Eq. (1) the relation between the exchange \ninteraction and the g value cannot be uniquely determined . \nTo fully understand the exchange coupling -driven magnon -magnon coupling, we perform the \ncomprehensive numerical analysis and micromagnetic simulations (see Method s). We consider the \nboundary conditions at the interface and two surface s of the TmIG/YIG bilayers and arrive at the formula \n(see Supplementary Note 2) : \n2𝐴ex,YIG\n𝑀s,YIG𝑘YIGtan(𝑘YIG𝑑YIG)∙2𝐴ex,TmIG\n𝑀s,TmIG𝑘TmIG tan(𝑘TmIG 𝑑TmIG )=\n2𝐽\n𝜇0(𝑀𝑠,YIG+𝑀𝑠,TmIG )[2𝐴ex,YIG\n𝑀s,YIG𝑘YIGtan(𝑘YIG𝑑YIG)+2𝐴ex,TmIG\n𝑀s,TmIG𝑘TmIG tan(𝑘TmIG 𝑑TmIG )], (2) \nwhere J is the interfacial exchange coupling strength. By solving 𝜔YIG=𝜔TmIG from Eq. (1) and Eq. (2) \ntogether , we can get a set of ( 𝑘YIG, 𝑘TmIG ) values that correspond to different modes. In the presence of \nexchange interaction, 𝑘 will not be precisely equal to 𝑛𝜋\n𝑑 anymore . As a result, the degeneracy is lifted at \nthe crossing point and we have two frequencies corresponding to two ( 𝑘YIG, 𝑘TmIG ) values . By employing \n𝐽=−0.057 mJ/m2 (see Supplementary Table 1) , we have obtained high consistency between the \nexperimental and calculated spectra of field-frequency points in the entire range (Fig. 3a). The negative \nsign suggests an antiferromagnetic exchange coupling between TmIG and YIG. The strength is also \ncomparable with the ferromagnetic metal/YIG bilayers 27. We have also carried the FMR measurement on \nthe reference TmIG(350 nm)/CoFeB(50 nm) sample (see Supplementary Note 4). We get 𝐽=\n−0.032 𝑚J/m2, which is close to the result from the VSM loop measurements. This consistency suggests \nthat we can reliably extract 𝐽 values of TmIG/YIG samples from the FMR measurement. \nWe further study the thickness and mode dependence of anti -crossing gap (2 g). We extract the g value from \nthe frequency scan , for example, g = 85 MHz for the TmIG(200 nm)/YIG(200 nm) bilayer ( Fig. 3b). We \nfind the gap reduces as the layer thickness increases (Fig. 3c). To understand this, we derive the approximate \nsolution (see Supplementary Note 3) : 4 \n 𝑔≈𝛾𝑌𝐼𝐺𝛾𝑇𝑚𝐼𝐺\n4𝜋2𝐽\n(𝑀𝑠,𝑌𝐼𝐺+𝑀𝑠,𝑇𝑚𝐼𝐺 )∙√(2𝜇0𝐻res+𝜇0𝑀𝑠,𝑌𝐼𝐺)(2𝜇0𝐻res+𝜇0𝑀𝑠,𝑇𝑚𝐼𝐺 )\n𝑓𝑟𝑒𝑠∙1\n√𝑑𝑌𝐼𝐺𝑑𝑇𝑚𝐼𝐺, (3) \nwhere 𝜔res and 𝐻res are the resonance frequency and field in the gap center, respectively. The calculated \nresults show the same trend as in the experiment s (Fig. 3c). Also, Eq. (3) allows us to analyze the g valu e \nfor coupling of the YIG Kittel mode to different TmIG PSSW modes. We compare the experimental and \ncalculated g values for the coupling of n=0 mode in YIG and n=2 mode in TmIG in Fig. 3c, where we \nconfirm that the higher mode coupling results in a lower g in our case. \nFinally, t o evaluate the coupling cooperativity in TmIG/YIG bilayers , we have determine d the individual \ndissipation rates. We first get Gilbert damping factors for YIG and TmIG from field scans at different \nfrequencies when they are not coupled (see Supplementary Note 4). The extracted damping factors are \nplotted in Fig. 3d, where we find a damping factor as low as 4.91 (± 0.79) × 10-4 in the 350 nm -thick TmIG. \nWe also extract the dissipation rates for YIG and TmIG from f requency scans at different fields when they \nare not coupled (see Supplementary Note 4). As an example, 𝜅𝑌𝐼𝐺=10 𝑀𝐻𝑧 and 𝜅𝑇𝑚𝐼𝐺 =29.5 𝑀𝐻𝑧 for \nthe TmIG(200 nm)/YIG(200 nm) bilayer. Therefore, 𝑔>𝜅𝑌𝐼𝐺,𝜅𝑇𝑚𝐼𝐺 , and 𝐶=𝑔2\n𝜅𝑌𝐼𝐺𝜅𝑇𝑚𝐼𝐺=24.5, \nconcluding a strong coupling in the bilayer. In Fig. 4, we summarize the dissipation rates and cooperativity \nfor TmIG - and ferromagnetic metal -based heterostructures that show magnon -magnon coupling. The TmIG \nhas a very low dissipation rate compared to ferromagnetic metals, which is consistent with the ultralow \nGilbert damping . \nIn summary, we demonstrate ultralow damping and dissipation rates in the TmIG and achieve strong \nmagnon -magnon coupling and high cooperativity in the TmIG/YIG bilayers. The combined experimental \nand theoretical analyses allow us to determine the interfacial exchange coupling strength s in our all-\ninsulator bilayers . The all-magnetic -insulator bilayers allow us to achieve ultralow damping insulati ng \nsynthetic antiferromagnets, magnonic crystals, and other artificial structures to realize energy -efficient spin \nwave devices. Besides, the strong coupling between two distinct magnetic insulators with ultralow damping \nallows to explore the novel quantum phases, such as topological magnon insulators and magnon flatband . \n 5 \n Figures and Captions \n \nFigure 1. Structural characterizations of YIG /TmIG heterostructures on GGG substrates. a, High-\nangle annular dark -field scanning t ransmission electron micro scopy (HAADF -STEM ) image for the \nYIG/TmIG heterostructure on GGG substrates . The YIG/GGG and YIG/TmIG interfaces are denoted by \nthe green and red lines in partial enlarged figure, respectively. Pt is used as a capping layer to pre vent \ndamage when preparing TEM samples. The inset of a demonstrates that two 1/8 of unit cells corresponding \nto YIG and TmIG at the interface, which have a garnet -type structures (𝐶)3[𝐴]2(𝐷)3𝑂12. b, Energy \ndispersive x -ray (EDX) spectra of different elements in the TmIG/YIG/GGG heterostructure. The images \nwere taken along the <100> direction of the GGG substrate, and the distribution of elements is marked in \nthe figure with color, respectively. c, High-resolution of X-ray di ffraction spectr a for the YIG(100 \nnm)/GGG, TmIG(100 nm)/GGG, and TmIG(100 nm)/YIG(100 nm)/GGG samples . \n \n26 27 28 29 30 31\n2q (degrees) TmIG(100nm)XRD Intensity (arb.units) YIG(100 nm) TmIG(100 nm)/YIG(100 nm)\n¨\n·\n§¨ GGG\n· TmIG\n§ YIG\nTmIG\nYIG\nGGG\nGGGYIGYIGTmIG\nPt(a) (c)\n(b)\nYIGTmIG\nTm\nFe\nY\nMd\nMd\nMaMa Mc\nO2-\nC-site Y3+\nA-site Fe3+D-site Fe3+C-siteTm3+\nHAADF Pt Tm Y O Fe Ga Gd\n50 nm 50 nm 50 nm 50 nm 50 nm 50 nm 50 nm 50 nm6 \n \nFigure 2. Schematic diagram of the spin waves in the heterostructure and the measured resonance \nspectra . a, Schematic illustration of the measurement set -up, where ℎ𝑟𝑓 and 𝐻𝑒𝑥𝑡 stand for the microwave \nmagnetic field and external static magnetic field, respectively. Spin -wave spectra are obtained by placing \nthe sample face -down on a coplanar waveguide (CPW ). The inset depicts the Kittel uniform spin wave \nmode in the YIG and the perpendicular standing spin wave (PSSW) mode in the TmIG . b, Experimentally \ncolor -coded spin -wave absorption spectra of the YI G(200 nm)/TmI G(200 nm) for the first three resonance \nmodes of TmIG (n=0, 1, 2) and the uniform mode of YIG (n=0). \n \ny\nxzHext\n(a) (b)\n0 200 400 6001.02.03.04.0\nm0Hr (Oe)w/2p (GHz)-3E-4 -2E-4 -3E-5 9E-5 2E-4\n2g=\n0.06GHz\n2g=\n0.16GHzAmplitude (V)\n0 20 40 601.02.03.04.0\nm0Hext (mT)w/2p (GHz)-3E-4 -2E-4 -3E-5 9E-5 2E-47 \n \nFigure 3. Observation of s trong magnon -magnon coupling and ultralow damping in YIG/TmIG \nbilayers . a, Resonant absorption peaks of the two hybrid modes as a function of external magnetic field \nwith the YIG(200 nm)/TmI G(200 nm) bilayer. Solid curves show the numerical theory method fitting as \nhybrid modes. Data points are extracted from experimental data by reading out the minimum of each \nresonant peak from Fig. 2(b). b, Spin wave spectra at minimum resonance separation ( 𝜇0𝐻𝑒𝑥𝑡=10 mT) \nin magnetic insulator bilayers with the YIG(200 nm)/TmI G(200 nm) bilayer . c, Coupling strength g \nbetween TmIG (n=1,2) mode and YIG (n=0) mode as a function of the YIG thickness. Red circles are \nexperimental results and black squares are from the oretical calculations . Red dots: Experiments. d, \nThickness dependence of Gilbert damping factors of YIG and TmIG in the YIG/TmIG bilayers . \n \n0 20 40 601.02.03.04.0\n1.0 1.2 1.4 1.6 1.8-15-10-505\n012345\n0.040.080.120.160.20w/2p (GHz)\nm0Hext (mT)dots: Experiment\ndash line: Fitting\nAmplitude (V)\n0.030.060.090.120.150.18\ndTmIG/dYIG (nm)100/100 140/140 200/200TmIG (n=2) couple YIG (n=0)TmIG (n=1) couple YIG (n=0)Frequency (GHz)2g m0Hext:\n10 mT200nm/200nm YIG/TmIG\nsignal ´105\n1.56-1.40=2g\nk1=0.032/2\nk2=0.04821 /2\nC=16.59 Experiment\n Lorentz Fitsignal ´10-5\n0.17 GHzaYIG\nThickness (nm) TmIG\n YIG´10-3\n012345\naTmIG´10-3\n100 140 200 350g (GHz) Experiment\n Calculation(a) (b)\n(c) (d)8 \n \nFigure 4. Summary of dissipation rates in TmIG and ferromagnetic metals versus cooperativities in \nTmIG - and ferromagnetic metal -based heterostructures. Star and square symbols are dissipation rates \nfor TmIG and ferromagnetic metals, respectively. 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Ferroma gnetic resonance linewidth in metallic thin films: Comparison of \nmeasurement methods. J. Appl. Phys. 99, 093909 (2006). \n34. Crossley, S. et al. Ferromagnetic resonance of perpendicularly magnetized Tm3Fe5O12/Pt \nheterostructures. Appl. Phys. Lett. 115, (2019). \n35. Rosenberg, E. R. et al. Magnetic Properties and Growth‐Induced Anisotropy in Yttrium Thulium \nIron Garnet Thin Films. Adv. Electron. Mater. 7, 2100452 (2021). \n36. Adhikari, K., Sahoo, S., Mondal, A. K., Otani, Y. & Barman, A. Large nonlinear fer romagnetic \nresonance shift and strong magnon -magnon coupling in N i80 F e20 nanocross array. Phys. Rev. B \n101, 054406 (2020). \n37. Adhikari, K., Choudhury, S., Barman, S., Otani, Y. & Barman, A. Observation of magnon –\nmagnon coupling with high cooperativity in Ni 80 Fe 20 cross -shaped nanoring array. \nNanotechnology 32, 395706 (2021). \n38. Shiota, Y., Taniguchi, T., Ishibashi, M., Moriyama, T. & Ono, T. Tunable Magnon -Magnon 11 \n Coupling Mediated by Dynamic Dipolar Interaction in Synthetic Antiferromagnets. Phys. R ev. \nLett. 125, 017203 (2020). \n39. Wang, H. et al. Hybridized propagating spin waves in a CoFeB/IrMn bilayer. Phys. Rev. B 106, \n064410 (2022). \n40. Hayashi, D. et al. Observation of mode splitting by magnon –magnon coupling in synthetic \nantiferromagnets. Appl . Phys. Express 16, 053004 (2023). \n \n 12 \n Methods \nMaterial growth \nHigh -quality epitaxial YIG (lattice constant 12.376 Å), TmIG (lattice constant 12.324 Å), and YIG/TmIG \nthin films were deposited on <100> -oriented GGG (lattice constant 12.383 Å) single -crystal substrates \nusing pulsed laser deposition (PLD). Prior to deposition, the substrates were cleaned with acetone, alcohol, \nand deionized water, followed by annealing in air at 1000° C for 6 hours. The films were deposited at 710 °C \nin an oxygen atmosphere of 100 mTorr, with a base pressure of better than 2×10−6 𝑚𝑇𝑜𝑟𝑟 , using a 248 \nnm KrF excimer laser with a repetition rate of 10 Hz. In -situ post -anneali ng was carried out at the deposition \ntemperature for 10 minutes in 10 Torr oxygen ambient, followed by natural cooling to room temperature. \n \nMaterial characterizations \nThe cross -sectional (TmIG/YIG/GGG) TEM lamella with a thickness of ~70 nm was fabricated by the \nfocused ion beam (FIB) technique in a Thermofisher Helios G4 UX dual beam system. The atomic \nresolution high -angle annular dark -field scanning transmission electron microscopy (HAADF -STEM) \nimages viewed along <100> orientation and the EDS data were obtained using FEI Titan Themis G2 TEM \nwith a probe corrector at an acceleration voltage of 300 kV. The C2 aperture is 70 µ m and the camera length \nis 115 mm, corresponding to a probe convergence semi -angle of 23.9 mrad. The image filtering and EDS \nmapping analysis were processed using the Velox software. \nFilm thickness was measured using Surface Profiler (Bruker DektakXT). The microstructure of the samples \nwas analyzed using X -ray diffractometry (SmartLab, Rigaku Co.) with Cu Ka1 radiation (λ = 1.5406 Å). \nMagnetic properties were assessed with vibrating sample magnetometer s (Physical Property Measurement \nSystem , Quantum Design , or Lakeshore ). \n \nResonance studies \nThe magnetization dynamics measurement was performed using the field -modulation FMR technique at \nroom temperature. During the measurement, the GGG/YIG/TmIG sample was mounted in the flip -chip \nconfiguration (TmIG side facing down) on top of the signal -line of a coplanar waveguide for broadband \nmicrowave excitation. An external bias field, H, was applied in -plane perpendicular to the rf field of the \nCPW. We used a modulation frequency of Ω/2π = 81.57 Hz (supplied by a lock -in amplifier and provided \nby a pair of modulation coil) and a modulation field of about 1.1 Oe. The microwave signal was delivered \nfrom a signal generator (0 - 5 dBm) to one port of the board. The field -modulated FMR signal was measured \nfrom the other port by the lock -in amplifier in the form of a dc voltage, V, by using a sensitive rf diode. We \nswept the bias field, H, and at each incremental frequency f, to construct the V[f, H] dispersion contour \nplots. \n \nMicromagnetic simulations \nOur finite -element model implements coupled LLG equation with antiferromagnetic interfacial exchange \ninteraction in magnetic insulator heterost ructure in order to simulate the strong magnon -magnon coupling \nin frequency -domain. The technical details are provided in Supplementary Note 2. 13 \n \nAcknowledgements \nThe authors appreciate insightful discussions with S. S. Kim, Y. Tserkovnyak , Z. Zhang , A. Com stock, D. \nSun, Y. Li . The sample fabrication, structural characterization, and data analysis at HKUST were supported \nby National Key R&D Program of China (Grants No.2021YFA1401500). The magnetic dynamics \nmeasurement and analysis were supported by the U.S. National Science Foundation under Grant No. ECCS -\n2246254. The thin film deposition work in PolyU through the GRF grant 15302320 . The authors also \nacknowledge support from RGC General Research Fund (Grant No. 16303322), State Key Laboratory of \nAdvanced Displays and Optoelectronics Technologies (HKUST), and Guangdong -Hong Kong -Macao Joint \nLaboratory for Intelligent Micro -Nano Optoelectronic Technology (Grant No. 2020B1212030010). \n \nData availability \nThe data that support the plots within this paper and other findings of this study are available at XXX. \n(Authors’ note: the data will be uploaded after the acceptance of this manuscript.) \n \nAuthor contributions \nW. Z. and Q. S. conceived the idea. J. L. did the VSM, XRD, partial FMR measurements, and data analysis \nwith help from X. W., S. K. C., Z. R., R. L., Z. C., F. P. N. , and S. K. Kim . Y. X. and W.Z. did FMR \nmeasurements with help from A. C. J. L. and D. C.W. L. grew the samples. C. L. and X.X. Z. did the TEM. \nJ. L. and Q. S. and W. Z. wrote the manuscript with help from other co-authors. \n \nCompeting interests \nThe authors de clare that they have no competing financial interests. \n \n \n \n \n1 \n Supplementary Information \nJiacheng Liu, et al. \n2 \n Supplementary Note 1. Hysteresis loops by vibrating sample magnetometer (VSM) \nmeasurement for Tm IG, YIG , and TmIG/Y IG samples and TmIG, CoFeB, and \nTmIG /CoFeB samples \nIn principle , due to the existence of antiferromagnetic coupling at the heterostructure interface , \nwe could also measure the major and minor hysteresis loop s through VSM to estimate the \ninterfac ial coupling energy (J). We prepared 2 series of samples to do a comparison , [TmIG( 100 \nnm), YIG(100 nm), TmIG(100 nm)/YIG (100 nm)) and TmIG (350 nm), CoFeB(50 nm), \nTmIG(350 nm)/CoFeB(50 nm)]. Due to the extremely similar coercive fields of YIG and TmIG , \nwe cannot get the J directly from hysteresis loop data in Supplementary Fig. 1. Then , we made \nanother series of samples ( TmIG, CoFeB, and TmIG/CoFeB ) for comparison. Measuring the \nhysteresis loops of TmIG(350 nm) /CoFeB(50 nm) and individual layer s in Sup plementary Fig. \n2a, we could clearly observe that the process of magnetization flipping (black solid line) from \nTmIG(350 nm)/CoFeB(50 nm), which is caused by the different coercivity between TmIG \n(light blue solid line) and Co FeB (red solid line). We noticed that no sharp switching (like \narrow points A to B to C) of the CoFeB layer is visible but a smooth increase (like arrow points \nA to C) of the measured magnetic moment until the bilayer magnetization is saturated. This \ncould be explained by a direct interfacial exchange coupling between TmIG and CoFeB \nmagnetizations 321-3. Process (1 -5) in Supplementary Figs. 2b -c show a possible magnetization \nflipping process in an exchange coupled heterostructure at an exte rnal magnetic field . \nSubsequently, to quantify the interfacial exchange coupling field and energy , we measured the \nminor hysteresis loops of TmIG(350 nm)/CoFeB(50 nm) in Supplementary Fig. 3a . In an ideal \nstate, (2) and (3) state is similar, but the difference is that the moment of (3) is about to flip the \nspin of the Tm IG layer, and the moment of (2) has not yet reached the flip condition. The -\nminor loop measured from the experiment is a C -A-B-C loop (Supplementary Fig. 3b) , \nindicating the existence of antiferromagnetic interfacial coupling . If there is no interfac ial \ncoupling, the loop will be like E -A-D-E, because it only depends on its coercivity of TmIG , \nthat is, the forward and reverse min or loops should basically coincide (just like a single TmIG \n3 \n hysteresis loop). The shift from the minor hysteresis loops is precisely because of the interfac ial \nexchange coupling that the two layers are in an antiferromagnetic relationship such that the \nextra 𝐻𝑒𝑥 want s to make spins parallel, which is why the flip ping (3) will be performed faster \nby (𝐻𝑒𝑥−𝐻𝑒𝑥𝑡). So, we can estimate the interfacial exchange coupling J value from the minor \nloops: 𝐽=𝜇0𝐻𝑒𝑥×∆𝑀𝑠\n𝑉×𝑡=−0.394×10−3(𝑇)∙1.975×105(𝐴/𝑚)∙400×10−9(𝑚)=\n−0.03113 𝑚𝐽/𝑚2, where 𝐻𝑒𝑥 is exchange field induced by the interfacial coupling, ∆𝑀𝑠 is \nthe value of magnetization reduction in the shad ow region (C-B-D-E loop) for additional \nexchange filed , 𝑉=(350𝑛𝑚+50𝑛𝑚)∙5𝑚𝑚∙5𝑚𝑚 is volume of the heterostructure, t is \nthickness of the sample. After the analysis of the interfacial exchange coupling by the m inor \nloops, we also extract the interfacial coupling energy of the TmIG (350nm)/CoFeB (50nm) \nsample by ferromagnetic resonance ( FMR ) measurement , which is described in detail in \nSupplementary Note 3. \n \nSupplementary Note 2. Numerical analysis and micromagnetic simulation for the strong \nmagnon -magnon coupling induced by the interface exchange interaction \nNumerical analysis \nTo solve the eigenfrequency of the anti -crossing curve under the strong magnon -magnon \ncoupling (Fig. 3a) , we conduct numerical analysis and micromagnetic simulations on the \nbilayer heterostructure following the method in ref erence s 4,5. We assume that magnetic \ninsulator layer 1 (MI1) describes index 1, and magnetic insulator layer 2 (MI2) describes index \n2. The interface of the MI1 and MI2 is at z=0 (Supplementary Fig. 4a). \nDispersion relation : for the magnetic system, Landau -Lifshitz -Gilbert ( LLG) equation [Eq. \n(S1)] describe s the intrinsic precess ion of the spin in the materials : \n𝜕𝑴𝑖̇⃗⃗⃗⃗⃗ \n𝜕𝑡=−𝜇0𝛾𝑖𝑴𝑖̇⃗⃗⃗⃗⃗ ×𝑯𝑒𝑓𝑓,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ +𝛼𝑖\n𝑀𝑠,𝑖(𝑴𝑖̇⃗⃗⃗⃗⃗ ×𝜕𝑴𝒊̇⃗⃗⃗⃗⃗ \n𝜕𝑡), (S1) \n4 \n where whole magnetic insulator bilayers subjected to the static magnetic field 𝑯𝒆𝒙𝒕⃗⃗⃗⃗⃗⃗⃗⃗ in the x \ndirection and the dynamic magnetic field 𝒉𝒓𝒇⃗⃗⃗⃗⃗⃗ generated by the coplanar waveguide (CPW) in \nthe y direction. Since the dynamic magnetic field induced by the microwave is so small \n|𝒉𝒓𝒇⃗⃗⃗⃗⃗⃗ |≪|𝑯𝒆𝒙𝒕⃗⃗⃗⃗⃗⃗⃗⃗ | that precessing amplitude of spin moment can be divided into static and \ndynamic parts: \n𝑴𝑖̇⃗⃗⃗⃗⃗ =𝑀𝑠,𝑖(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ ),𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ =[1\n0\n0],𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ =𝛿𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ 𝑒𝑗𝜔𝑡=[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖],(S2) \nwhere 𝒎0⃗⃗⃗⃗⃗⃗ is normalization constant modulo 1, which is the static expression of spin moment . \n𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ is micro -perturbations induced by microwave fields, and its amplitude is much smaller \nthan |𝒎0⃗⃗⃗⃗⃗⃗ |. 𝑯𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗ include s an external magnetic statistic magnetic field 𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ [Eq. (S3)] in x \ndirection, dynamic magnetic field 𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ , |𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ |≪|𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ |,𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ ⊥𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ , and demagnetization \nfield 𝑯𝑑𝑒,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗⃗ [Eq. (S4)]. In particular, the sample is considered as a thin film, so 𝑁𝑥=𝑁𝑦=\n0,𝑁𝑧=1. \n𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ =[𝐻𝑥\n0\n0],𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ ∥𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ , (S3) \n𝑯𝑑𝑒,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗⃗ =−𝒩∙𝑴𝑖̇⃗⃗⃗⃗⃗ =−𝑀𝑠,𝑖[𝑁𝑥00\n0𝑁𝑦0\n00𝑁𝑧](𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ )=−𝜇0𝑀𝑠,𝑖[0\n0\n𝛿𝑚𝑧,𝑖],(S4) \nSubstitute 𝑴𝑖̇⃗⃗⃗⃗⃗ and 𝑯𝑒𝑓𝑓,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ into Eq. (S1): \n𝑀𝑠,𝑖𝜕(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ )\n𝑑𝑡=−𝜇0𝛾𝑖𝑀𝑠,𝑖(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ )×(𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ +𝑯𝑑𝑒⃗⃗⃗⃗⃗⃗⃗ +𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ ) \n+𝛼𝑖\n𝑀𝑠,𝑖𝑀𝑠,𝑖(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ )×𝑀𝑠,𝑖𝜕(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ )\n𝜕𝑡, (S5) \nwhere 𝜕𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗⃗ \n𝜕𝑡=0, and 𝜕𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ \n𝑑𝑡=𝑗𝜔𝛿𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ 𝑒𝑗𝜔𝑡=𝑗𝜔𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ . In Eq. (S5) , 𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ ×𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ =0 is due to \n5 \n the magnetization and the static component of the external magnetic fiel d are parallel to each \nother , 𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ ×𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ =0 is due to the second -order epsilon, and 𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ ×𝜕𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ \n𝜕𝑡=𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ ×\n𝑗𝜔𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ =0. So, Eq. (S5) could be simplified to: \n𝑗𝜔𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ =−𝜇0𝛾(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ ×𝑯𝑑𝑒⃗⃗⃗⃗⃗⃗⃗ +𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ ×𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ ×𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ )+𝑗𝜔𝛼(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ ×𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ ),(S6) \nThe matrix form of Eq. (S6) is: \n𝑗𝜔[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖]=−𝜇0𝛾𝑖[[1\n0\n0]×[0\n0\n−𝑀𝑠,𝑖𝛿𝑚𝑧,𝑖]+[1\n0\n0]×[ℎ𝑥\nℎ𝑦\nℎ𝑧]+[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖]×[𝐻𝑥\n0\n0]]\n+𝑗𝜔𝛼𝑖([1\n0\n0]×[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖]), (S7) \n𝑗𝜔\n𝜇0𝛾𝑖[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖]=−[[0\n𝑀𝑠,𝑖𝛿𝑚𝑧,𝑖\n0]+[0∙��𝑥 \n−ℎ𝑧\nℎ𝑦]+[0\n𝐻𝑥𝛿𝑚𝑧\n−𝐻𝑥𝛿𝑚𝑦]]+𝑗𝜔𝛼𝑖\n𝜇0𝛾𝑖[0\n−𝛿𝑚𝑧\n𝛿𝑚𝑦],(S8) \nSimplify Eq. (S8) into 𝒉⃗⃗ =𝜒−1𝛿𝒎⃗⃗⃗ : \n[0∙ℎ𝑥\nℎ𝑦\nℎ𝑧]=\n[ 𝑗𝜔\n𝜇0𝛾0 0\n0(𝐻𝑥+𝑗𝜔𝛼\n𝜇0𝛾) −𝑗𝜔\n𝜇0𝛾\n0 𝑗𝜔\n𝜇0𝛾(𝑀𝑠+𝐻𝑥−𝑗𝜔𝛼\n𝜇0𝛾)] \n[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖], (S9) \nWhen the FMR condition is met , the system will reach maximum resonance, that means the \nsystem, ℎ⃗ =0,𝜒−1𝑚⃗⃗ =0 has a non -zero solution [Eq. (S10)]: \n|𝜒−1|=0, (S10) \n(𝑀𝑠,𝑖+𝐻𝑥−𝑗𝜔𝛼𝑖\n𝜇0𝛾𝑖)(𝐻𝑥+𝑗𝜔𝛼𝑖\n𝜇0𝛾𝑖)−𝜔2\n𝜇02𝛾𝑖2=0, (S11) \n(1−𝛼𝑖2)𝜔2−𝑗𝜇0𝛾𝑀𝑠,𝑖𝛼𝑖𝜔−𝜇02𝛾𝑖2𝐻𝑥(𝐻𝑥+𝑀𝑠,𝑖)=0, (S12) \n6 \n 𝜔𝑐𝑜𝑚𝑝𝑙𝑒𝑥=𝑗𝜇0𝛾𝑖𝑀𝑠,𝑖𝛼𝑖\n2(1−𝛼𝑖2)+\n1\n2(1−𝛼𝑖2)√𝜇02𝛾𝑖2𝑀𝑠,𝑖2𝛼𝑖2+4(1−𝛼𝑖2)[𝜇02𝛾𝑖2𝐻𝑥,𝑖(𝐻𝑥,𝑖+𝑀𝑠,𝑖)], (S13) \nWhen 𝛼≪1, the Eq. (S13) will be simplified to : \n𝜔𝑖=𝛾𝑖√𝜇0𝐻𝑥(𝜇0𝐻𝑥+𝜇0𝑀𝑠,𝑖), (S14) \nEq. (S14) is Kittel mode by solving the linearized LLG equation with wave vector 𝑘=0. If \nconsider the exchange field 𝑯𝑒𝑥⃗⃗⃗⃗⃗⃗⃗ =2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖𝑘𝑖2, the Eq. (S14) will be revised to: \n𝜔𝑐𝑜𝑚𝑝𝑙𝑒𝑥=𝑗𝜇0𝛾𝑖𝑀𝑠,𝑖𝛼𝑖\n2(1−𝛼𝑖2)+ \n1\n2(1−𝛼𝑖2)√𝜇02𝛾𝑖2𝑀𝑠,𝑖2𝛼𝑖2+4(1−𝛼𝑖2)𝛾𝑖2[ (𝜇0𝐻𝑥+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 𝑘𝑖2)(𝐻𝑥,𝑖+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 𝑘𝑖2+𝑀𝑠,𝑖)] (S15) \nWhen 𝛼≪1, the Eq. (S15) will be simplified to : \n𝜔𝑖=𝛾𝑖√(𝜇0𝐻𝑥+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 𝑘𝑖2)(𝜇0𝐻𝑥+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 𝑘𝑖2+𝜇0𝑀𝑠,𝑖), (S16) \nwhere 𝑘𝑖 is the wave vector of the perpendicular standing spin wave (PSSW) , and the \npropagation of the spin wave is in the z direction . \n𝜔1−𝜔2=0, (S17) \nFor the magnetic insulator bilayer, Eq. (S15) with 𝛼 and Eq. (S16) without 𝛼 describe the \ndispersion relation within each layer. Eq. (S17) means that the necessary condition for the \nbilayer spin waves to exist simultane ously in the same external condition ( same ℎ𝑟𝑓 and \n𝐻𝑒𝑥𝑡) in the two layers . \nBoundary condition s—free boundary condition and interface exchang e boundary \ncondition : Now, we can focus on the boundary conditions in the magnetic bilayers. For the \nmagnetic insulator bilayer system, free boundary condition should be applied to the top (𝑧=\n𝑑1) [Eq. (S18)] and the bottom (𝑧=−𝑑2) [Eq. (S19)] of the bilayer system, which al so \nmean no pining at the surfaces: \n7 \n 𝜕𝛿𝑚𝑧,1\n𝜕𝑧|\n𝑧=𝑑1=0, (S18) \n𝜕𝛿𝑚𝑧,2\n𝜕𝑧|\n𝑧=−𝑑2=0, (S19) \nConsidering that the dynamic magnetization in x and y directions is uniform, there is a wave \nvector (𝑘⊥) along the thickness direction. The spatial distribution of the PSSW can be expressed \nas: \n𝛿𝑚𝑧,𝑖=𝛿𝑚0,𝑖cos(𝑘𝑖𝑧+𝜙𝑖), (S20) \nwhere 𝜙𝑖 is the phase of the spin wave . Substitute Eq . (20) into Eq . (S18, S19), we can obtain: \n−𝑘1𝑑1+𝑛1𝜋=𝜙1, (S21) \n 𝑘2𝑑2+𝑛2𝜋=𝜙2, (S22) \nInterface exchange energy J describes the number and strength of exchange bonds between \nmagnetic insulator layer 1 and layer 2. The effect of exchange coupling energy 𝐽 (𝑚𝐽/𝑚2) at \nthe interface (z = 0) will be shown by the interface boundary conditions generated by the \ncombination of the Huffman boundary conditions 6-8. The conservation of magnetic energy flow \nat the interface leads to: \n2𝐴𝑒𝑥,1\n𝑀𝑠,1𝜕𝛿𝑚𝑧,1\n𝜕𝑧+2𝐽\n(𝑀𝑠,1+𝑀𝑠,2) (𝛿𝑚𝑧,2−𝛿𝑚𝑧,1)|\n𝑧=0=0, (S23) \n−2𝐴𝑒𝑥,2\n𝑀𝑠,2𝜕𝛿𝑚𝑧,2\n𝜕𝑧+2𝐽\n(𝑀𝑠,1+𝑀𝑠,2) (𝛿𝑚𝑧,1−𝛿𝑚𝑧,2)|\n𝑧=0=0, (S24) \nThe matrix form of Eqs. (S23, S24) is as follows : \n[0\n0]=[−2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)cos𝜙1−2𝐴𝑒𝑥,1\n𝑀𝑠,1𝑘1𝑠𝑖𝑛𝜙12𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑐𝑜𝑠𝜙2\n2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑐𝑜𝑠𝜙1 −2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)cos𝜙2+2𝐴𝑒𝑥,2\n𝑀𝑠,2𝑘2𝑠𝑖𝑛𝜙2][𝛿𝑚0,1\n𝛿𝑚0,2] (S25) \nThe resonant condition requires that the determinant of the coeffic ient matrix of Eq. S25 \nvanishes: \n8 \n [2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)cos𝜙1+2𝐴𝑒𝑥,1\n𝑀𝑠,1𝑘1𝑠𝑖𝑛𝜙1][2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)cos𝜙2−2𝐴𝑒𝑥,2\n𝑀𝑠,2𝑘2𝑠𝑖𝑛𝜙2]\n=[2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑐𝑜𝑠𝜙2][2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑐𝑜𝑠𝜙1], (S26) \nEq. (S26) is the relationship under the interface -exchanged boundary conditions, combined \nwith the free boundary conditions of Eqs. (S21, S22) , we can obtain: \n[2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)−2𝐴𝑒𝑥,1\n𝑀𝑠,1𝑘1𝑡𝑎𝑛(𝑘1𝑑1)][2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)−2𝐴𝑒𝑥,2\n𝑀𝑠,2𝑘2𝑡𝑎𝑛(𝑘2𝑑2)]\n=[2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)][2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)], (S27) \n2𝐴𝑒𝑥,1\n𝑀𝑠,1k1tan(𝑘1𝑑1)∙2𝐴𝑒𝑥,2\n𝑀𝑠,2k2tan(𝑘2𝑑2)\n=2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)[2𝐴𝑒𝑥,1\n𝑀𝑠,1k1tan(𝑘1𝑑1)+2𝐴𝑒𝑥,2\n𝑀𝑠,2k2tan(𝑘2𝑑2)], (S28) \nTo get the eigenfrequency, we can s olve the transcendental equations of the di spersion \ncondition [Eq. (S17)] and boundary condition [Eq. (S28)] for each external magnetic field 𝐻𝑒𝑥𝑡 \nby traversal . \nAlthough 𝑓1(𝑘1,𝑘2,𝐻𝑒𝑥𝑡)=0 [Eq. (S17)] and 𝑓2(𝑘1,𝑘2,𝐻𝑒𝑥𝑡)=0 [Eq. (S28)] cannot be \nexplicitly functionalized , we can solve them numeri cally by handling them as implicit functions. \nSupplementary Fig. 4b shows the frame work of the numerical analysis using MATLAB \nsoftware. Next, we will solve the eigenfrequency in two cases (𝐽=0 𝑚𝐽/𝑚2,𝐽≠0 𝑚𝐽/𝑚2): \nWhen 𝐽=0 𝑚𝐽/𝑚2, transcendental Eq. (S28) will degenerate to: \n𝑘1𝑘2tan(𝑘1𝑑1)tan(𝑘2𝑑2)=0, (S29) \n𝑘1=𝑛𝜋/𝑑1 or 𝑘2=𝑛𝜋/𝑑2 is the solution of Eq. (S29), which means spin wave in magnetic \ninsulator layer 1 and layer 2 is quantized as 𝑛𝜋/𝑑 and not influenced with each othe r. \nSubstitute Eq. (S29) into Eq. (S16) (assuming that 𝛼≪1): \n9 \n 𝜔𝑖=𝛾𝑖√(𝜇0𝐻𝑥+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 (𝑛𝜋\n𝑑)2\n)(𝜇0𝐻𝑥+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 (𝑛𝜋\n𝑑)2\n+𝜇0𝑀𝑠,𝑖), (S30) \nThe resonant peaks in Supplementary Fig. 5a shows the FMR and PSSW modes of TmIG and \nYIG FMR mode as a function of magnetic field when 𝐽=0 𝑚𝐽/𝑚2. We also show the process \nof solving 𝑘1,𝑘2 by graphical method in Supplementary Figs. 5b-c. Without inter facial \nexchange energy, the hybrid modes will be degenerate to the uniform mode and PSSW modes, \nwhich also means that spin -wave vector is quantized with 𝑛𝜋/𝑑. \nWhen 𝐽≠0=−0.0573𝑚𝐽/𝑚2 , we can clearly see the anti -crossing curve in the \nSupplementary Fig. 6a. A set of degenerate solutions of the shaded region in the Supplementary \nFig. 6a are composed of a set of intersection points of Figs. 6 b-c. It also needs to be noted that \nthe evanescent wave emerges in the YIG layer when the frequency of the PSSW in TmIG layer \nis lower than the uniform mode in YIG layer ( 𝑘1 is pure imaginary part). \nMicromagnetic simulation s based on finite -element model \nIn addition to proposing a relatively complete theory, we show the strong magnon -magnon \ncoupling regime by solving the LLG equation in the frequency domain base d on COMSOL's \nmicromagnetic simulation module 9. We apply the frequency -domain LLG equations of Eq. \n(S31 , S32) to the two magnetic thin films respectively : \n−𝑖𝜔𝛿𝒎𝒊=−𝛾𝑖𝒎𝒊×(𝑯𝒆𝒇𝒇+𝑯𝒆𝒙)−𝑖𝜔𝛼𝑖𝒎𝒊×𝛿𝒎𝒊 (S31) \n−𝑖𝜔𝛿𝒎𝒋=−𝛾𝑗𝒎𝒋×(𝑯𝒆𝒇𝒇+𝑯𝒆𝒙)−𝑖𝜔𝛼𝑗𝒎𝒋×𝛿𝒎𝒋 (S32) \nwhere all parameters are consistent with the theory calc ulations. 𝐻𝑒𝑥 is the effective filed-like \ntorque induced by the interfacial exchange energy, which is defined as: \n𝐻𝑒𝑥=2𝐽\n𝑀𝑖+𝑀𝑗𝛿(𝑧)𝒎𝒋,∫𝛿(𝑧)=1+∞\n−∞ (S33) \nwhere 2𝐽/(𝑀𝑖+𝑀𝑗) means that t he average value of the interfac ial exchange energy for \ndifferent magnetization materials on both sides of the interface . 𝛿(𝑧) is impulse function, \n10 \n which shows that the interaction only occur s at the interface of the TmIG/YIG heterostructure. \nThe simulation parameters to be set are shown in the following Supplementary Table S1. \nTo demonstrate the coupling strength between YIG and TmIG, we perform full -frequency full -\nmagnetic field simulations . We characterize the strength of resonance through the response of \n𝛿𝑚𝑧 to magnetic field and mi crowave field . Through the simulation results shown in the \nSupplementary Fig. 7a, we can clearly see that TmIG (n=1,2) is coupled with YIG (n=0) to \nform anti -cross ing at 105 Oe and at 470 Oe. Supplementary Fig. 7b shows the spectral \nresonance curve of the spin wave at 105 Oe. Then, w e can see the internal dynamics of spin \nwaves through simulation . To perform the resonance direction of the spin wave on the interface \nof the bilayers, we show the 𝛿𝑚𝑧 distribution o f the normalized intensity in the z direction for \nthe two hybrid modes A and B (marked) in Supplementary Fig. 8. \n \nSupplementary Note 3. Thickness dependence of interfacial exchange coupling energy \nand coupling strength for TmIG/YIG samples \nThe sign judgment of interfacial exchange coupling energy \nJudgment 1: we obtained the interfacial coupling energy (𝐽<0 𝑚𝐽/𝑚2) by fitting the \nexperimental data with the theory of Supplementary Note 2. The values of the J are summarized \nin the Supplementary Table S2. \nJudgment 2: We plot the spin -wave absorption spectra for TmIG(100 nm)/YIG(100 nm) in \ncomparison with YIG(100 nm)/GGG (grey dashed line) in Supplementary Fig. 9. The lower \nresonant magnetic field (𝐻𝑟𝑒𝑠) or higher resonant frequency (𝜔𝑟𝑒𝑠) means that for the \nTmIG/YIG sample, the interfacial exchange field applied on YIG is opposite to the external \nmagnetic field, which supports the antiferromagnetic coupling nature concluded in the main \ntext. In the meantime, Eq. (S38) shows that 𝛿𝑘1(𝑌𝐼𝐺)2 has the same sign as J, which means if \n𝐽<0 𝑚𝐽/𝑚2 , 𝛿𝑘1,𝑌𝐼𝐺2<0 . According to the Eq. (S16), we can know the u nder the same \n11 \n external magnetic field, the resonant frequency required by TmIG/YIG sample is smaller than \nthat of YIG/GGG, which is consisten t with our results (Supplementary Fig.9). We co uld also \nroughly estimate the coupling strength (𝐽≈𝐻𝑠ℎ𝑖𝑓𝑡×𝑀𝑠1+𝑀𝑠2\n2×𝑑1=−0.0042 𝑇×\n133.3 𝑘𝐴/𝑚×100𝑛𝑚=−0.0598 𝑚𝐽/𝑚2) by observing the shift [Eq. (S38)] in the FMR \ncurve between the TmIG/YIG/GGG heterostructure and YIG/GGG single layer . \nThickness dependence of interfacial coupling energy and coupling strength for \nTmIG/YIG samples \nInterfacial coupling energy: Supplementary Figs. 10a-c show the colormap of the spin-wave \nabsorption spectra for the PSSW modes of TmIG and the uniform mode of YIG measured for \nTmIG(100 nm)/YIG(100 nm), TmIG(140 nm)/YIG(140 nm), and TmIG(200 nm)/YIG(200 nm) \nheterostructures . We fit the experimental data (Supplementary Figs. 10e -f) through numerical \nanalysis in Supplementary Note 2 . The results are summarized in the Supplementary Table 2. \nCoupling strength g : We could also extract the magnon -magnon coupling strength, which is \ndefined as the half of the minimal peak to peak frequency spacing in the anti -crossing induced \nby the interface exchange energy10,11. When the coupling strength is 0 (𝐽=0 𝑚𝐽/𝑚2), the \nhybrid modes will be decoupled. The resonant magnetic field (𝐻𝑒𝑥𝑡) and resonant frequency \n(𝑓) where the minimum resonant separation is located should intersect , which means that \n𝑘1=0,𝑘2=𝑛𝜋/𝑑2 . At the existence of the interfacial exchange coupling energy (𝐽≠\n0 𝑚𝐽/𝑚2), we can obtain the perturbative solution. we make a difference to Eq. (S16) to get \nan expression of the coupling strength g (𝛿𝑓\n2): \n2fres𝛿𝑓=(𝛾\n2𝜋)2\n∙[2(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠)2𝐴𝑒𝑥\n𝑀𝑠𝑘𝛿𝑘+(2𝐴𝑒𝑥\n𝑀𝑠)24𝑘3𝑑𝑘],(S34) \nFor Eq. (S28), we can let 2𝐴𝑒𝑥,1\n𝑀𝑠,1k1tan(𝑘1𝑑1) be A and 2𝐴𝑒𝑥,2\n𝑀𝑠,2k2tan(𝑘2𝑑2) be B. So, the Eq. \n(S28) can be simplified: \n1−2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)(1\n𝐴+1\n𝐵)=0, (S35) \n12 \n We now consider 𝑘1=0+𝛿𝑘1,𝑘2=𝑛𝜋/𝑑2+𝛿𝑘2,|𝛿𝑘1|≪1 ,|𝛿𝑘2|≪1 as the \nperturbation solution corresponding to the minimum resonance separation of YIG's FMR \nmode and TmIG PSSW mode (n) . Eq. (S35) will yield layer 1 -dominated and layer 2 -\ndominated reson ance. For layer 1 -dominated resonance, we have: \n2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝐴≈1 and 2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝐵≪1, (𝑆36) \n2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)≈2𝐴𝑒𝑥,1\n𝑀𝑠,1𝑘1𝑡𝑎𝑛(𝑘1𝑑1)=2𝐴𝑒𝑥,1\n𝑀𝑠,1𝑘1𝛿𝑘1𝑑1=2𝐴𝑒𝑥,1\n𝑀𝑠,1𝛿𝑘12𝑑1,(S37) \n𝛿𝑘12=2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑀𝑠,1\n2𝐴𝑒𝑥,11\n𝑑1, (S38) \nFor 𝑘1=𝛿𝑘1, (2𝜇0𝐻+𝜇0𝑀𝑠)2𝐴𝑒𝑥\n𝑀𝑠2𝛿k1≫(2𝐴𝑒𝑥\n𝑀𝑠)2\n4𝛿𝑘13 \n𝛿𝑓1≈(𝛾1\n2𝜋)2\n[2(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,1)2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝑑1]1\n2𝑓𝑟𝑒𝑠, (S39) \nFor layer 2 -dominated resonance, we have: \n2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝐵≈1 and 2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝐴≪1, (S40) \n𝛿𝑘2=2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑀𝑠,2\n2𝐴𝑒𝑥,21\n𝑛𝜋/𝑑21\n𝑑2, (S41) \nFor 𝑘2=𝑛𝜋\n𝑑2+𝛿𝑘2 , (2𝜇0𝐻+𝜇0𝑀𝑠)~10−1,2𝐴𝑒𝑥\n𝑀𝑠~10−18,𝑘2~𝑛𝜋\n𝑑2~107 , so we can get that \n(2𝜇0𝐻+𝜇0𝑀𝑠)2𝐴𝑒𝑥\n𝑀𝑠2𝑘2~10−12≫(2𝐴𝑒𝑥\n𝑀𝑠)2\n4𝑘3~10−15. \n𝛿𝑓2≈(𝛾2\n2𝜋)2\n[2(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,2)2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝑑2]1\n2𝑓𝑟𝑒𝑠, (S42) \n𝛿𝑓2=(𝛾1\n2𝜋)2\n(𝛾2\n2𝜋)2\n(2𝐽\n(𝑀𝑠,1+𝑀𝑠,2))2[(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,1)(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,2)]\n𝑓𝑟𝑒𝑠21\n𝑑1𝑑2,(S43) \n𝑔=𝛿𝑓\n2=��1\n2𝜋𝛾2\n2𝜋𝐽\n(𝑀𝑠,1+𝑀𝑠,2)√(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,1)(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,2)\n𝑓𝑟𝑒𝑠1\n√𝑑1𝑑2,(S44) \n13 \n Where 𝑔 is the coupling strength, 𝐻𝑟𝑒𝑠,𝑓𝑟𝑒𝑠 are the resonant magnetic field and frequency at \nthe minimum resonance separation , 𝑑1,𝑑2 is the thickness of the YIG layer (FMR mode) and \nTmIG layer (PSSW mode). The g value extracted in the experiment (Supplementary Figs. 11a -\nc) and calcu lation from Eq. (S4 4) are shown as Fig. 3c. \n \nSupplementary Note 4. The extraction of Gilbert d amping , dissipation rate, and \ncooperativity from the FMR measurements \nGilbert damping 𝜶: High coupling strength 𝑔 and extremely low Gilbert damping 𝛼 are \nkey factors to realize long -distance information transmission that is free of Joule heating. \nGilbert damping refers to an intrinsic feature of magnetic substances which dictates the speed \nat which angular momentum is transferred to the crystal lattice , which is the key parameter that \ndetermines the spin -wave relaxation 13,14 . To demonstrate the superiority of low damping \nMI/MI bilayer system of this work , we focus on the line widths variation with the frequency of \nYIG/ TmIG heterostructure samples. The linewidth versus frequency experiment data with \nerror bar for different thicknesses were extracted by fitting Lorentz equations in Supplementary \nFig. 12. In strong coupling region, we can clearly observe that the line width value of the pink \ncircle is significantly higher than that of the light blue circle, which is suggested that a coherent \ndamping -like torque which acts along or against the intrinsic damping torque depending on the \nphase difference of the coupled dynamics of YIG and TmIG 4; Away from the strong coupling \nregion, we use ∆𝐻=∆𝐻0+4𝜋𝛼\n𝛾𝑓 for linear characterization, and the result is indicated by \nthe dashed line in the Supplementa ry Fig. 12 . The fitting effective damping constants are \nsummarized in the Supplementary Table 2. \nDissipation rate: We extract the dissipation rate (half width at half maximum ) from frequency \nscans at different fields when they are away from the coupling region (Supplementary Fig. 13) . \nWe fit the FMR mode of YIG and TmIG in Supplementary Fig. 13 through the Lorentz peak \nto get the dissipation rate. \n14 \n Cooperativity 𝑪: The strong c oupling implies coherent dynamics between the magnon and \nthe magnon. In our case, all samples’ 𝑔>𝜅1,𝑔>𝜅2 mean that strong coupling is formed \n11,12,15. Through Lorentz peak fitting, we found that the dissipation rate will decrease as the \nthickness increases, which is consistent with the damping trend we calculated. Due to the \nmagnetic insulator heterostructure, the dissipation rate s in TmIG/YIG bilayers are particularly \nlow compared w ith ferromagnetic metal -based heterostructures so that we can get a largest \ncooperativity 𝐶=(𝑔/𝜅1)(𝑔/𝜅2)=24.5 in the TmIG(200 nm)/YIG(200 nm) case. The \nresults are summarized in the Supplementary Table 2 . As the thickness increases, the \ncooperativity increases significantly, mainly because the decrease rate of the dissipation rates \nis greater than that of coupling strength. The cooperativity is summarized in the Supplementary \nTable S2. \nAntiferromagnetic interfacial exchange coupling energy for the TmIG(350 \nnm)/CoFeB(50 nm) sample \nWe also verify antiferromagnetic interfacial exchange coupling energy of the MI/FM \nTmIG(350 nm)/CoFeB(50 nm) by FMR modulation . We use theory ( Supplementary Note 2) \nto analyze and fit the color -coded experiment data (Supplementary Figs. 14a -b) and get the \nTmIG/CoFeB interfacial coupling energy 𝐽=−0.0321 𝑚𝐽/𝑚2 , showing that the FMR \nmethod is consistent with the minor hysteresis loop method (𝐽=−0.0311 𝑚𝐽/𝑚2). \n \n15 \n Parameters Symbol Value \nMesh 𝑁 20×20×50 \nMicrowave field ℎ𝑟𝑓 0.001𝑇 \nInterfacial exchange energy 𝐽 −0.0537 𝑚𝐽/𝑚2 \nYIG/TmIG gyromagnetic ratio 𝛾𝑌𝐼𝐺/𝛾𝑇𝑚𝐼𝐺 28/21.81 𝐺𝐻𝑧/𝑇 \nYIG/TmIG stiffness constant 𝐴𝑒𝑥,𝑌𝐼𝐺/𝐴𝑒𝑥,𝑇𝑚𝐼𝐺 3.08/2.67 𝑝𝐽/𝑚 \nYIG/TmIG Gilbert damping 𝛼𝑌𝐼𝐺/𝛼𝑇𝑚𝐼𝐺 6.66×10−4/1.17×10−3 \nFrequency range (𝑓𝑠𝑡𝑎𝑟𝑡,∆𝑓,𝑓𝑒𝑛𝑑) (1,0.01,4) 𝐺𝐻𝑧 \nMagnetic field range (𝐻𝑠𝑡𝑎𝑟𝑡,∆𝐻,𝐻𝑒𝑛𝑑) (30,5,700) 𝑂𝑒 \nSupplementary Table S1. | Parameters used to do the micromagnetic simulation with \nTmIG 200nm/ YIG 200nm bilayer in Supplementary Figs. 7-8. \n16 \n Materials 𝜅𝑌𝐼𝐺 (𝛼𝑌𝐼𝐺) 𝜅𝑇𝑚𝐼𝐺 (𝛼𝑇𝑚𝐼𝐺) \nor 𝜅𝐹𝑀 (𝛼𝐹𝑀) g C J (mJ/m2) \nYIG(100 nm)/TmIG \n(100 nm) (this work) 0.058 GHz \n(2.98e -3 ± 1.4e-4) 0.046 GHz \n(4.4e -3 ± 3e -4) 0.155 GHz 8.76 -0.0618 \nYIG(140 nm)/TmIG \n(140 nm) (this work) 0.0185 GHz \n(7.78e -4 ± 1.7e -4) 0.054 GHz \n(1.69e -3 ± 2e -4) 0.105 GHz 11.04 -0.072 0 \nYIG(200 nm)/TmIG \n(200 nm) (this work) 0.01 GHz \n(6.64e-4 ± 4.2e-5) 0.0295 GHz \n(1.07e -3 ± 1e -4) 0.085 GHz 24.49 -0.0573 \nTmIG(350 nm)/CoFeB \n(50 nm) (this work) -- 𝜅𝑇𝑚𝐼𝐺 : 0.0255 GHz \n(4.91e -4±8e-5) \nor \nκFM: 0.232 GHz \n(3e-3±3.9e-5) 0.263GHz 11.69 -0.0321 \nYIG(20 nm)/Ni(20 nm)29 * 0.06GHz 0.63 GHz 0.12 GHz 0.38 -- \nYIG(20 nm)/Co(30 nm)29 * 0.06GHz 0.5 GHz 0.79 GHz 21 -- \nYIG(100 nm)/ Ni80Fe20 \n(9 nm)27 0.106GHz \n(2.3e -4) 0.192 GHz \n(1.75e -3) 0.35 GHz 6 -0.060 ± \n0.011 \nYIG(1 µm)/Co(50 nm) 28 (7.2e -4 ± 3e -5) (7.7e -3 ± 1e -4) -- -- -- \nNi80Fe20(20 nm)34 * -- 0.66 GHz -- 0.6 -- \nNi80Fe20(20 nm)35 * -- 0.31 GHz -- 2.25 -- \nIrMn(10 nm)/CoFeB \n(80 nm)36 -- 0.26 GHz -- 2 -- \nCoFeB(15 nm)/Ru(0.6 nm) \n/CoFeB(15 nm )37 * -- 0.31 GHz -- 5.39 -- \nCoFeB(15 nm)/Ru(0.6 nm) \n/CoFeB(15 nm)38 * -- 0.23 GHz -- 8.4±1.3 -- \nSupplementary Table 2. Summary of dissipation rates, damping factors, coupling strengths, \ncooperativities, and exchange interaction strengths in the studied bilayers and some reported \nYIG/ferromagnetic metal (FM) bilayers that show magnon -magnon coupling. All reference \nnumbers are consistent with the main text . * In -plane confinement introduces extra dynamical \ndipolar coupling. \n17 \n \nSupplementary Figure 1 | Hysteresis loop s for YIG (100 nm), TmIG(100 nm) and Y IG \n(100nm) /TmIG (100nm) samples. Since YIG and TmIG have approximately equal coercive \nfields (~1.5Oe), we could not obtain the interfacial coupling energy through the dynamic \nprocess of the switching of the two layers in the hysteresis loop. \n-50 -40 -30 -20 -10 0 10 20 30 40 50-0.6-0.4-0.20.00.20.40.6M (memu)\nField (Oe)In-plane\n YIG (100nm)\n TmIG (100nm)\n YIG (100nm)/TmIG (100nm) \n18 \n \nSupplementary Figure 2 | Hysteresis loop s for TmIG (350 nm), CoFeB(50 nm), and TmIG \n(350 nm)/CoFeB(50 nm) samples . a, The individual magnetic thin film layers TmIG (light \nblue solid line) and Co FeB (red solid line) exhibit different coercivity, which could be directly \nobserved the process of magnetization flipping between the two coercivity (black solid line). \nFurthermore, due to the existence of interfacial exchange coupling (J), no sharp switching (like \nthe arrow point A to B to C) of the CoFeB layer is visible but a smooth increase (the arrow \npoint A to C) of the measured magnetic moment until the bilayer magnetization is saturated. \nb, Enlarged detail of the TmIG(350 nm)/CoFeB(50 nm) heterostructure hysteresis loop. c, \nProcess (1 -5) shows a possible magnetization flipping in an exchange coupled heterostructure \nat an external magnetic field in b. \nHext(3) Hext(1)TmIG CoFeBHext(4) Hext(5) Hext(2)\n-40 0 40 80-1.00.01.0\n-44 -22 0 22 44M/Ms\nField (Oe) CoFeB (50nm)\n TmIG (350nm)\n TmIG (350nm)/CoFeB (50nm)\nIn-plane\nTmIG (350nm) /CoFeB (50nm)\n-40 0 40 80-1.00.01.0\n-44 -22 0 22 44M/Ms\nField (Oe) CoFeB (50nm)\n TmIG (350nm)\n TmIG (350nm)/CoFeB (50nm)\nIn-plane\nTmIG (350nm) /CoFeB (50nm)(1)\n(3)\n(4)\n(5)AB\nC(2)\n(a) \n(c) \n(b) \n19 \n \nSupplementary Figure 3 | Hysteresis loop and minor loops for TmIG (350 nm)/CoFeB (50 \nnm) sample . a, Minor loops for TmIG(350 nm)/CoFeB(50 nm) sample . The non -coincidence \nof +minor loop and -minor loop indicates the existence of interface coupling. b, The process \nof analyzing interfacial exchange coupling using minor loop s. The result of the experiment \nmeasurement is a C -A-B-C loop, indicating the existence of antiferromagnetic interfacial \ncoupling . However, if there is no interfac ial exchange coupling, the loop will be like E -A-D-E, \nbecause it only depends on its coercivity of TmIG , that is, the forward and reverse minor loops \nshould basically coincide (just like individual TmIG Hysteresis loop). We can also get the \n-30 -20 -10 0 10 20 30-3.5-2.5-1.5-0.50.51.52.53.5\n Hysteresis Loop TmIG (350nm)/CoFeB (50nm)\n + Minor LoopM (memue)\nField (Oe) - Minor Loop\nIn-planeABC\nDE\nHex=3.94OeΔMs=1.975memue\n-30 -20 -10 0 10 20 30-3.5-2.5-1.5-0.50.51.52.53.5\n Hysteresis Loop TmIG (350nm)/CoFeB (50nm)\n + Minor LoopM (memue)\nField (Oe) - Minor Loop\nIn-plane(2)(1)\n(3)M (memu ) M (memu )\n(a) \n(b) \n20 \n similar results from the + minor loop with the similar method. \n21 \n \n \nSupplementary Figure 4 | Schematic of the magnetic insulator heterostructure with \nthickness of 𝒅𝟏 and 𝒅𝟐 , respectively. a, For the convenience of calculation, we set the \ninterface of bilayer as the interface of z=0. The direction of the static external magnetic field \n(𝐻𝑒𝑥𝑡) is set as the x -axis, and the dynamic magnetic field (ℎ𝑟𝑓) generated by CPW is \nperpendicular to the static magnetic field on the y -axis, where 𝐻𝑒𝑥𝑡≫ℎ𝑟𝑓 . b, Theoretical \ncalculation framework from MATLAB. We traverse the value of 𝐻𝑒𝑥𝑡 through the loop cycle \nand solve Eqs. (S17, S28) under each external magnetic field value to obtain the 𝑘1and 𝑘2, \nthen substitute them into the dispersion relationship to obtain the eigen frequency. \nStartStart Start Loop though StartSolve \nStartCalculate equation \nS16 to get Trune parameter?\nrangeStartStartYES\nNOTune parameter?\nHext range\n(a) \n(b) \n22 \n \nSupplementary Figure 5 | Numerical solution for the eigenfrequency when 𝑱=𝟎 𝒎𝑱/𝒎𝟐. \na, Frequencies of FMR and PSSW modes of TmIG (200 nm) and FMR mode of YIG (200 nm) \nas a function of H when 𝐽=0 𝑚𝐽/𝑚2. b-c, Numerical solutions 𝑘1,𝑘2 obtained by solving \ndispersion condition and boundary condition without interfacial exchange energy. When 𝐽=\n0 𝑚𝐽/𝑚2, we could obviously see that the crossing points are degenerate solution in a from \nthe shaded region, which indicates that 𝑘1,𝑘2 is qu antized solution with 𝑛𝜋/𝑑1 and 𝑛𝜋/𝑑2 \nrespectively . \n0 100 200 300 4000.00.51.01.52.02.53.0\n0.0 0.5 1.0 1.50123\n0 1 2 30.51.01.52.02.5Frequency (GHz)\nH (Oe) TmIG kittel \n YIG kittel \n TmIG n=1 \n TmIG n=2k2 (units of p/d2)\nk1 (units of i p/d1) Eq S28\n Eq S17\n·\n★TmIG n=0YIG n=0J=0 mJ/m2\nk2 (units of p/d2)\nk1 (units of i p/d1) Eq S28\n Eq S17\nTmIG n=1TmIG n=2J=0 mJ/m2\n◆▲\n(a) \n(b) \n (c) \n23 \n \nSupplementary Figure 6 | Numerical solution for the eigenfrequency when 𝑱=\n−𝟎.𝟎𝟓𝟕𝟑𝒎𝑱/𝒎𝟐 . a, Anti-crossing coupling f requencies of hybrid modes of TmIG (200 \nnm)/ YIG(200 nm) as a function of H when 𝐽=−0.0573 𝑚𝐽/𝑚2. b-c, Numerical solutions \n𝑘1,𝑘2 obtained by solving dispersion condition and boundary condition without interfacial \nexchange energy. With the interfacial exchange energy, we could see that the spin -wave vector \nsolution is no longer the integer of 𝑛𝜋/𝑑. It is noted that the evanescent wave emerges in the \nYIG layer when the frequency of the PSSW in TmIG layer is lower than the uniform mode in \nYIG layer ( 𝑘1 is pure imaginary part). \n \n \n ( ) \n b \n b \n \n ( ) \n \n \n ( ) \n \n \n \n(a) \n(b) \n (c) \n24 \n \n \nSupplementary Figure 7 | Full micromagnetic simulation based on frequency domain. a, \nColor -coded spin -wave spectra with frequency and magnetic field with the TmIG (200 nm) / \nYIG(200 nm) heterostructure. we can clearly see that TmIG(n=1,2) is coupled with YIG (n=0) \nto form anti -cross ing at 105 Oe and at 470 Oe. Then, w e use the value of change in 𝛿𝑚𝑧 as \nthe peak response. b, The spin spectrum curve at the minimum resonance separation of the first \nanti-crossing (H=105 Oe) . \n \n1234\n200 400 600Frequency (GHz)\nMagnetic Field (Oe)0.0 0.80\n0.011T 1.32GHz\n0.011T 1.56GHz\n1 2 3 40.00.30.71.0\n1234\n200 400 600Frequency (GHz)\nMagnetic Field (Oe)0.0 0.80\n0.011T 1.32GHz\n0.011T 1.56GHz\nAmplitude (Norm.)\nFrequency (GHz)Hext=105 Oe\nGraph19\n0.011T 1.56GHz2g=0.22GHz\n(a) \n(b) \n25 \n \nSupplementary Figure 8 | The two hybrid eigenmodes at 105 Oe . The 𝛿𝑚𝑧 distribution of \nthe normalized intensity in the z direction for the two hybrid modes at 1.33 GHz and 1.55 GHz \nrespectively. \n \n0 100 200200 300 400\n200 300 400 0100 200\ndmz (norm.)\nz (nm)\ndmz (norm.)A: 1.33GHz B: 1.55GHz \n26 \n \n \nSupplementary Figure 9 | The spin -wave absorption spectra for TmIG(100 nm)/YIG(100 \nnm) in comparison with YIG(100 nm)/GGG (grey dashed line) . Lower resonant magnetic \nfield 𝐻𝑟𝑒𝑠, or higher resonant frequency 𝜔𝑟𝑒𝑠, is observed for YIG( 100 nm)/GGG both before \nand after the avoided crossing. This shows that for the TmIG/YIG sample, the interfacial \nexchange field applied on YIG is opposite to the external magnetic field, which supports the \nantiferromagnetic coupling nature concluded in the main text. \n0 400 800 12002.03.04.05.0\n0 200 400 6001.02.03.04.0\n0 200 400 6001.02.03.04.00.080.120.16\n0.00 0.02 0.04 0.0624\n04008001200\n2.0 3.0 4.0 5.0Hext (Oe)\nw/2p (GHz)-2.8E-5 2.8E-5\n(a) (b) (c)\n(e) (f) (g)0 200 400 6001.02.03.04.0\nHext (Oe)w/2p (GHz)-1.1E-4 6.7E-5\n0 200 400 6001.02.03.04.0\nHext (Oe)w/2p (GHz)-2.8E-4 2.2E-4w/2p (GHz)\nHext (Oe)YIG(100nm)/TmIG(100nm) YIG(140nm)/TmIG(140nm) YIG(200nm)/TmIG(200nm)V (f,H) V (f,H) V (f,H)\n原始文件在 ‘画图全’中的folder1\nw/2p (GHz)\nHext (Oe)dw/2p (GHz)\nHext (Oe)\ng (GHz) Experiment\n Calculation0.08 0.10 0.12tYIG\n100 140 200 F1F1\n YIG single \n27 \n \nSupplementary Figure 10 | Magnetic insulator heterostructure thickness dependence of \nthe strong magnon -magnon coupling . a-c, Experimentally color -coded spin -wave absorption \nspectra with the YIG (100nm) /TmIG (100nm) (a), YIG (140nm) /TmIG (140nm) (b), YIG \n(200nm) /TmIG (200nm) (c). Inset magnification was performed at each anti -crossing region . \ne-g, Resonant absorption peaks of the two hybrid modes as a function of external magnetic \nfield with YIG (100nm) /TmIG (100nm) (e), YIG (140nm) /TmIG (140nm) (f), YIG \n(200nm) /TmIG (200nm) (g) bilayer . Solid curves show the numerical theory method fitting as \nhybri d modes using Supplementary note2 . Data points are extracted from experimental data by \nfitting the line shapes to two independent derivative Lorentzian functions from (a-c). The \nincrease of effective magnetization in YIG and TmIG and the stiffening of YIG and TmIG \nresonance frequency is due to the increase of different magnetic insulator thickness. \n0 400 800 12002.03.04.05.0\n0 200 400 6001.02.03.04.0\n0 200 400 6001.02.03.04.00.080.120.16\n0 400 800 12002.03.04.05.0\nHext (Oe)w/2p (GHz)-2.8E-5 2.8E-5\n(a) (b) (c)\n(e) (f) (g)0 200 400 6001.02.03.04.0\nHext (Oe)w/2p (GHz)-1.1E-4 6.7E-5\n0 200 400 6001.02.03.04.0\nHext (Oe)w/2p (GHz)-2.8E-4 2.2E-4w/2p (GHz)\nHext (Oe)YIG(100nm)/TmIG(100nm) YIG(140nm)/TmIG(140nm) YIG(200nm)/TmIG(200nm)V (f,H) V (f,H) V (f,H)\n原始文件在 ‘画图全’中的folder1\nw/2p (GHz)\nHext (Oe)dw/2p (GHz)\nHext (Oe)2g=\n0.31GHz2g=\n0.17GHz2g=\n0.06GHz\n2g=\n0.21GHz\ng (GHz) Experiment\n Calculation\n0.080.100.12\ntYIG100 140 200 \n28 \n \nSupplementary Figure 11 | Magnetic insulator heterostructure thickness dependence of \nthe coupling strength . a-c, Experimentally spin-wave spectra at the minimum resonance \nseparation with frequency with the TmIG (100 nm)/ YIG(100 nm) (a), TmIG (140 nm)/ YIG(140 \nnm) (b), TmIG (200 nm)/ YIG(200 nm) (c). At the same time, the red line is fitted by the Lorentz \npeak function to coupling strength . \n3.0 3.5 4.0-10-50\n2.5 3.0 3.5 4.0-505\n1.0 1.2 1.4 1.6-10-505Amptitude (V)\nFrequency (GHz) Experiment\n Lorentz Fitsignal ´106\n2g\nHext:\n500.3 Oe140nm/140nm YIG/TmIG\n Experiment\n Lorentz Fit\n1.56-1.40=2g\nk1=0.032/2\nk2=0.04821 /2\nC=16.59Amptitude (V)\nFrequency (GHz)signal ´106\n2g\nHext:\n574.5 Oe100nm/100nm YIG/TmIG\n Experiment\n Lorentz Fit\n3.49-3.18=2g\nk1=0.1888/2\nk2=0.0473/2\nC=10.763.51-3.30=2g\nk1=0.0673/2\nk2=0.0521/2\nC=12.577\nAmptitude (V)\nFrequency (GHz)2gHext:\n97.7 Oe200nm/200nm YIG/TmIG signal ´105\n(a) \n (b) \n (c) \n29 \n \nSupplementary Figure 12 | Thickness dependence of the Gilbert damping . a-c, \nExperimentally line-width with frequency with the TmIG (100 nm)/ YIG(100 nm) (a), \nTmIG (140 nm)/ YIG(140 nm) (b), TmIG (200 nm)/ YIG(200 nm) (c). Circle points with the \nerror bars represent linewidth extracted from experimental data by fitting the line shapes to \nthree independent derivative Lorentzian functions from (a-c). Dashed lines are linear fits away \nfrom the strong coupling region through equation 5 . The fitting effective damping constant s \nare summarized in the Supplementary Table S2. \n1.0 1.5 2.0 2.5 3.0 3.5 4.001020\n2.0 2.5 3.0 3.5 4.0 4.50204060\n1.0 1.5 2.0 2.5 3.0 3.5 4.005101520\n Hybrid 1\n Hybrid 2H (Oe)\nFrequency(GHz)YIG FMR\nTmIG FMR\n F\n Linear Fit of Sheet1 D\n Linear Fit of Sheet1 B\na\nYIG： 7.87e-4+-1.66e-4\nTmIG： 1.699e-3+-1.45e-4\nslope tmig FMR: 9.79046+- 0.83577\nslope YIG FMR: 3.53367+-0.74594140nm/140nm YIG/TmIG\n Fit\n FitStrong \nCoupling \nEquation y = a + b*x\nPlot B\nWeight Instrumental (=1/ei^2)\nIntercept 11.78418 ± 0.14368\nSlope 1.558 ± --\nResidual Sum of Squares 12.79439\nPearson's r 0.91465\nR-Square (COD) 0.65215\nAdj. R-Square 0.65215H (Oe)\nFrequency(GHz) Fit\n Fit\na\nYIG： 2.98e-3+-1.35e-4\nTmIG： 4.4e-3+-2.59e-4\nslope tmig FMR: 25.4005+- 1.4966\nslope YIG FMR: 13.82892+-0.60861100nm/100nm YIG/TmIG\nStrong \nCoupling Equation y = a + b*x\nPlot B\nWeight Instrumental (=1/ei^2)\nIntercept 6.75777 ± 0.03877\nSlope 2.2009 ± --\nResidual Sum of Squares 0.21698\nPearson's r 0.99552\nR-Square (COD) 0.97839\nAdj. R-Square 0.97839\nHybrid 1\nHybrid 2\nH (Oe)\nFrequency(GHz)YIG FMR\nTmIG FMR\n Hybrid 4\n Hybrid 3\n Linear Fit of Sheet1 D\n Linear Fit of Sheet1 B\na\nYIG： 6.66e-4+-0.42e-4\nTmIG： 1.07e-3+-1.16e-4\nslope tmig FMR: 6.17513+- 0.6707\nslope Hybrid 1: 2.98248+-0.19075200nm/200nm YIG/TmIG\n Fit\n FitStrong \nCoupling 2\ncell://[Book28]FitLinear6!No cell://[Book28]FitLinear\nPlot [Book28]FitLinear6!Par\n[Book28]FitLinear6!Notes. [Book28]FitLinear6!Not\nIntercept [Book28]FitLinear6!Par\nSlope [Book28]FitLinear6!Par\n[Book28]FitLinear6!RegStat [Book28]FitLinear6!Reg\n[Book28]FitLinear6!RegStat [Book28]FitLinear6!Reg\n[Book28]FitLinear6!RegStat [Book28]FitLinear6!Reg\n[Book28]FitLinear6!RegStat [Book28]FitLinear6!Reg\n原始文件在‘耦合系数'中的folder4Hybrid 1\nHybrid 2\n(a) \n (b) \n (c) \n30 \n \nSupplementary Figure 13 | Thickness dependence of the dissipation rate . a-c, Frequency \nsweep curve s away from the coupling region . Dissipation rate is obtained by Lorentzian peak \nfunction f itting. Red solid line presents YIG FMR, blue solid line presents TmIG FMR. \n2.0 2.5 3.0 3.5024681012 Amplitude (V)\nFrequency (GHz) 0.03068\n Lorentz Fit of Sheet1 AP\"0.03068\"\n 0.03216\n Lorentz Fit of Sheet1 AR\"0.03216\"\n 0.0344\n Lorentz Fit of Sheet1 AU\"0.0344\"\n 0.03588\n Lorentz Fit of Sheet1 AW\"0.03588\"\n 0.04927\n Lorentz Fit of Sheet1 BO\"0.04927\"\n 0.05001\n Lorentz Fit of Sheet1 BP\"0.05001\"\n 0.0515\n Lorentz Fit of Sheet1 BR\"0.0515\"\n 0.05076\n Lorentz Fit of Sheet1 BQ\"0.05076\"TmIG FMR peaks\nYIG FMR peaks30.0 mT32.1 mT34.4 mT35.8 mT49。2 mT50.0 mT50.7 mT51.0 mTTmIG（100nm）/YIG(100 nm)Signal´105\n1.0 1.5 2.0 2.5-40481216202428 Amplitude (V)\nFrequency (GHz)10.1 mT12.1 mT14.2 mT16.3 mT18.3 mT20.0 mTTmIG FMR \npeaksYIG FMR \npeaksYIG(140 nm)/TmIG(140 nm)Signal´105\n1.5 2.0 2.5 3.0-8-4048121620 Amplitude (V)\nFrequency (GHz)221Oe242Oe262Oe280Oe300Oe321Oe338Oe\nTmIG FMR \npeaks\nYIG FMR \npeaksSignal´105TmIG(200 nm)/YIG(200 nm)\n(a) \n (b) \n (c) \n31 \n \n \nSupplementary Figure 14 | Magnetic field dependency of resonant peaks for \nCoFeB 50nm/TmIG 350nm heterostructure. a, Experimentally color -coded spin -wave absorption \nspectra of the Co FeB50nm/TmIG 350nm for the first seven resonance modes of TmIG (n=0 -6) and \nthe uniform mode of CoFeB (n=0). b, Fitting of antiferromagnetic interfacial coupling energy \nby theoretical inversion. \n0 50 100 150 200 2501234frequency (Hz)\n-3E-4-2E-4-1E-4-7E-54E-68E-52E-42E-43E-4\nField (Oe)\n0 50 100 150 200 2501234frequency (Hz)\n-3E-4-2E-4-1E-4-7E-54E-68E-52E-42E-43E-4\nField (Oe) Theory fitting\n(a) \n(b) \n32 \n 1. Magnetic heterostructures: advances and perspectives in spinstructures and spintransport. (Springer \nVerlag, 2008). doi:10.1088/0031 -9112/23/4/020. \n2. Livesey, K. L., Crew, D. C. & Stamps, R. L. Spin wave valve in an exchange spring bilayer. Phys. Rev. \nB 73, 184432 (2006). \n3. Klingler, S. et al. Spin-Torque Excitation of Perpendicular Standing Spin Waves in Coupled YIG / Co \nHeterostructures. Phys. Rev. Lett. 120, 127201 (2018). \n4. Li, Y . et al. Coherent Spin Pumping in a Strongly Coupled Magnon -Magnon Hybrid System. Phys. Rev. \nLett. 124, 117202 (2020). \n5. Zhang, Z., Yang, H., Wang, Z., Cao , Y . & Yan, P. Strong coupling of quantized spin waves in \nferromagnetic bilayers. Phys. Rev. B 103, 104420 (2021). \n6. Hoffmann, F., Stankoff, A. & Pascard, H. Evidence for an Exchange Coupling at the Interface between \nTwo Ferromagnetic Films. Journal of Appl ied Physics 41, 1022 –1023 (1970). \n7. Hoffmann, F. Dynamic Pinning Induced by Nickel Layers on Permalloy Films. phys. stat. sol. (b) 41, \n807–813 (1970) \n8. Cochran, J. F. & Heinrich, B. Boundary conditions for exchange -coupled magnetic slabs. Phys. Rev. B \n45, 13096 –13099 (1992). \n9. Zhang, J., Yu, W., Chen, X. & Xiao, J. A frequency -domain micromagnetic simulation module based \non COMSOL Multiphysics. AIP Advance s 13, 055108 (2023). \n10. Zhang, X., Zou, C. -L., Jiang, L. & Tang, H. X. Strongly Coupled Magnons and Cavity Microwave \nPhotons. Phys. Rev. Lett. 113, 156401 (2014). \n11. Chen, J. et al. Strong Interlayer Magnon -Magnon Coupling in Magnetic Metal -Insulator Hybrid \nNanostructures. Phys. Rev. Lett. 120, 217202 (2018). \n12. Chen, J. et al. Excitation of unidirectional exchange spin waves by a nanoscale magnetic grating. Phys. \nRev. B 100, 104427 (2019). \n13. Khodadadi, B. et al. Conductivitylike Gilbert Damping due to Intraband Scattering in Epitaxial Iron. \nPhys. Rev. Lett. 124, 157201 (2020). \n14. Li, Y . et al. Giant Anisotropy of Gilbert Damping in Epitaxial CoFe Films. Phys. Rev. Lett. 122, 117203 \n(2019). \n15. Xiong, Y . et al. Probi ng magnon –magnon coupling in exchange coupled Y 3Fe5O12/Permalloy bilayers \nwith magneto -optical effects. Sci Rep 10, 12548 (2020). \n " }, { "title": "2403.08478v1.Thermal_Hall_effect_incorporating_magnon_damping_in_localized_spin_systems.pdf", "content": "Thermal Hall e ffect incorporating magnon damping in localized spin systems\nShinnosuke Koyama1and Joji Nasu1\n1Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan\n(Dated: March 14, 2024)\nWe propose a theory for thermal Hall transport mediated by magnons to address the impact of their damping\nresulting from magnon-magnon interactions in insulating magnets. This phenomenon is anticipated to be par-\nticularly significant in systems characterized by strong quantum fluctuations, exemplified by spin-1 /2 systems.\nEmploying a nonlinear flavor-wave theory, we analyze a general model for localized electron systems and de-\nvelop a formulation for thermal conductivity based on a perturbation theory, utilizing bosonic Green’s functions\nwith a nonzero self-energy. We derive the expression of the thermal Hall conductivity incorporating magnon\ndamping. To demonstrate the applicability of the obtained representation, we adopt it to two S=1/2 quantum\nspin models on a honeycomb lattice. In calculations for these systems, we make use of the self-consistent imag-\ninary Dyson equation approach at finite temperatures for evaluating the magnon damping rate. In both systems,\nthe thermal Hall conductivity is diminished due to the introduction of magnon damping over a wide temperature\nrange. This e ffect arises due to the smearing of magnon spectra with nonzero Berry curvatures. We also discuss\nthe relation to the damping of chiral edge modes of magnons. Our formulation can be applied to various local-\nized electron systems as we begin with a general Hamiltonian for these systems. Our findings shed light on a\nnew aspect of topological magnonics emergent from many-body e ffects and will stimulate further investigations\non the impact of magnon damping on topological phenomena.\nI. INTRODUCTION\nIn condensed matter physics, the concept of topology pro-\nvides profound insights into exotic electronic structures and\nthe phenomena that arise from them. The topology of bands,\nformed by Bloch electrons, is characterized by a topologi-\ncal invariant defined for each band. The presence of a non-\nzero topological invariant predicts the emergence of gapless\nedge modes and the quantization of Hall conductivity in in-\nsulating states [1, 2]. This concept has been extended to in-\nclude systems composed not only of fermionic particles, such\nas electrons, but also those with bosonic quasiparticles like\nphonons [3–10] and magnons [11–19]. In bosonic systems,\nit is also possible to introduce the Berry curvature and topo-\nlogical invariants, such as the Chern number, for each band,\nsimilar to fermionic systems. Because bosonic quasiparti-\ncles are charge-neutral, the Hall e ffect does not manifest in\nthese systems. Instead, these quasiparticles can carry heat,\nimplying the potential for the emergence of a thermal Hall\neffect in systems with topologically nontrivial band struc-\ntures [3–6, 11–14]. For instance, in localized electron sys-\ntems with spin degrees of freedom, it has been proposed that\nanisotropic spin interactions, such as Dzyaloshinskii-Moriya\n(DM) interactions [20] and Kitaev couplings [21], can induce\na nonzero Chern number in magnon bands [16, 19, 22–36].\nSuch magnons, termed topological magnons, have recently\ngarnered significant attention [13, 14]. In fact, the thermal\nHall e ffect originating from magnons has been observed in\nmaterials with pyrochlore [37–39], honeycomb [40, 41], and\nkagome structures [42, 43].\nThus far, the topological properties of magnons, elemen-\ntary excitations arising from a magnetic order in localized\nspin systems, have been conducted within the framework of\nlinear spin-wave theory. This approximation enables the rep-\nresentation of the thermal Hall coe fficient through the Berry\ncurvature of magnon bands, incorporating a contribution be-\nyond the conventional Kubo formula. This contribution isknown as the heat magnetization arising from the orbital mo-\ntion of magnons [12, 15, 17, 44]. The formalism based on the\nfree-magnon picture has successfully explained experimental\nresults, such as the magnetic-field dependence of a thermal\nHall coe fficient in Lu 2V2O7[37]. Furthermore, this approach\nhas been expanded to include calculations of other topologi-\ncal phenomena, including the spin Nernst e ffect and nonlin-\near responses [19, 22, 27, 45–47]. However, the spin-wave\ntheory beyond the linear approximation indicates the pres-\nence of magnon-magnon interactions due to quantum fluctu-\nations [48]. These interactions become notably significant in\nsystems with short spin lengths and when a large number of\nmagnons are thermally excited. Such magnon-magnon inter-\nactions could play a crucial role in topological thermal trans-\nport phenomena, as magnons occupy bands with finite Berry\ncurvature only at finite temperatures. Furthermore, the fact\nthat magnons are bosons emphasizes the significance of their\ninteractions. Quasiparticles that describe elementary excita-\ntions as bosons do not obey the conservation law of particle\nnumbers, owing to their zero chemical potential. This leads\nto magnon-magnon interactions that do not conserve particle\nnumbers, resulting in the decay of high-energy magnons even\nat low temperatures [23, 34, 49–70]. It has been suggested\nthat this e ffect also influences the topological properties of\nmagnons [23, 34, 35, 59, 61, 64, 71], particularly for the\ndamping of the chiral edge modes [68, 69], potentially sup-\npressing the thermal Hall e ffect. Therefore, to evaluate the\nthermal Hall conductivity, it is essential to consider the e ffects\nof magnon-magnon interactions properly. This contrasts with\nthat in electronic systems at low temperatures, where chiral\nedge modes remain robust against such interactions [72].\nRecent experimental results suggest that such magnon-\nmagnon interactions significantly influence the thermal Hall\neffect. For instance, it has been reported that the measured\nvalues of thermal Hall conductivity in the layered material\nCr2Ge2Te6with a honeycomb structure exhibiting ferromag-\nnetic order are considerably lower than those obtained by the-arXiv:2403.08478v1 [cond-mat.str-el] 13 Mar 20242\noretical calculations under the linear spin-wave theory [73].\nFurthermore, in the Shastry-Sutherland model, the elemen-\ntary excitations termed triplons, which are bosonic quasipar-\nticles similar to magnons, have been theoretically predicted\nto contribute to the thermal Hall conductivity within the free-\nparticle approximation [74, 75]. However, experiments have\nnot detected a thermal Hall e ffect in the candidate material\nSrCu 2(BO 3)2[76]. These studies imply that the discrepan-\ncies between theory and experiment may be attributed to ne-\nglecting interaction e ffects between magnons in the theoreti-\ncal calculations. Nevertheless, formulating a framework for\nthe thermal Hall e ffect extending beyond the free-magnon ap-\nproximation remains challenging, partly due to the complex-\nity of considering the contribution of heat magnetization from\nmagnons to the thermal Hall conductivity, in addition to the\ncalculations from the Kubo formula.\nIn this paper, we formulate the thermal Hall conductivity\nin the presence of magnon damping in localized electron sys-\ntems to elucidate the e ffect of magnon-magnon interactions\non the thermal Hall e ffect. Beginning with a general Hamil-\ntonian for localized electron models, we adopt a mean-field\n(MF) approximation assuming a long-range order and intro-\nduce magnons as elementary excitations through the Holstein-\nPrimako fftransformation. This transformation not only pro-\nduces a bilinear term of bosonic operators but also brings\nabout additional terms responsible for magnon-magnon scat-\ntering. We introduce the magnon Green’s function, treating\nthe former as an unperturbed term and the latter as a pertur-\nbation term. We consider the e ffect of magnon damping as\nthe imaginary part of the self-energy. By assuming this part\nto be nonzero, we derive the expression for the thermal Hall\nconductivity incorporating magnon damping. We apply this\nframework to the Kitaev model under a magnetic field and an\nS=1/2 spin model with Heisenberg and DM interactions,\ncalculating the temperature dependence of thermal Hall con-\nductivity in the presence of magnon damping. The results\nreveal that the value of thermal Hall conductivity is signifi-\ncantly suppressed when magnons in bands with large Berry\ncurvature decay strongly. Our findings suggest that magnon\ndamping plays a crucial role in thermal Hall conductivity in a\nwide temperature range.\nThis paper is organized as follows. In Sec. II, we introduce\nthe model Hamiltonian and review the calculation method.\nThe MF approximation and Holstein-Primako fftransforma-\ntion are presented in Secs. II A and II B, respectively, to in-\ntroduce magnons as bosonic quasiparticules. Based on the\nHolstein-Primako fftransformation, the spin-wave Hamilto-\nnian is obtained in Sec. II C. In Sec. II D, we show the details\nof the perturbation theory based on the bosonic Green’s func-\ntion for magnons. Analytical properties of the Green’s func-\ntion are also discussed in this section. In Sec. III, we formu-\nlate the thermal conductivity using the bosonic Green’s func-\ntion. We briefly present the general theory of thermal trans-\nport in Sec. III A. The representation of thermal conductivity\nis shown in Sec. III B. In Sec. III C, we present the expres-\nsion of thermal Hall conductivity where the imaginary part\nof the self-energy is taken into account. Section III D shows\nthe fundamental properties of thermal Hall conductivity in thepresence of the magnon damping. In Sec. IV, we show the\ncalculation results of the thermal Hall conductivity using our\nframework in the following two quantum spin models: the Ki-\ntaev model under a magnetic field (Sec. IV A) and an S=1/2\nspin model with Heisenberg and DM interactions (Sec. IV B).\nFinally, Sec. V is devoted to summary and discussion.\nII. MODEL AND METHOD\nA. Mean-field theory\nIn this section, we briefly review the MF approximation.\nWe start from a general localized electron model, which is\nrepresented by\nH=1\n2X\ni,jX\nαβJαβ\ni jOα\niOβ\nj−X\niX\nαhα\niOα\ni, (1)\nwhereOα\niis theαcomponent of the local operator defined\nat site i, and Jαβ\ni jrepresents the exchange matrix between the\noperatorsOα\niandOβ\nj. We consider Nlocal states for each\nsite. The last term of Eq. (1) is the one-body term with the\nlocal field hα\ni. In the MF theory, Eq. (1) is divided to\nH=HMF+H′, (2)\nwhere the first term represents the MF Hamiltonian, which is\ngiven by\nHMF=X\niHMF\ni+const. (3)\nThe local MF Hamiltonian HMF\niat site iis represented as\nHMF\ni=X\nαMX\nl′Nu/MX\nj∈l′X\nβJαβ\ni j⟨Oβ⟩l′−hα\niOα\ni, (4)\nwhere NuandMare the number of unit cells and sublattices,\nrespectively. A sublattice here refers to the set of sites where\nthe same MF is assumed. The expectation value ⟨Oα⟩l=\n⟨0;i|Oα|0;i⟩is introduced for the ground state |0;i⟩of the lo-\ncal HamiltonianHMF\niwith site ibelonging to sublattice l. This\nground state is obtained by diagonalizing HMF\ni. Moreover, we\nobtain the m-th excited states|m;i⟩form=1,2,···,N−1\nofHMF\niin a similar manner.\nB. Generalized Holstein-Primako fftransformation\nIn this section, we rewrite the original Hamiltonian using\nbosons to describe the elementary excitations from the MF\nground state. We expand the local operator using the eigen-\nstates of the local MF Hamiltonian at site iin sublattice las\nOα\ni=N−1X\nm,m′=0Xmm′\niOα\nmm′;l, (5)3\nwhere Xmm′\ni≡|m;i⟩⟨m′;i|andOα\nmm′;l=⟨m;i|Oα|m′;i⟩, which\ndepends only on the sublattice index lto which site ibelongs.\nXmm′\niis represented by bosons using the generalized Holstein-\nPrimako fftransformation, which is known as a flavor-wave\ntheory [28, 68, 77–82]. We introduce N−1 bosonic operators\na†\nmi(ami) with m=1,2,···,N−1 for each site. For m≥1,\nX0m\niandXm0\niare given by\nXm0\ni=a†\nmiS−N−1X\nn=1a†\nniani1/2\n,X0m\ni=\u0010\nXm0\ni\u0011†.(6)\nHere,Sis introduced as\nS=X00\ni+N−1X\nn=1a†\nniani, (7)\nand it should be unity because ofPN−1\nm=0Xmm\ni=1. For 1≤\nm,m′,Xmm′\niis given by\nXmm′\ni=a†\nmiam′i. (8)\nWhen the number of bosons is small enough, one can expand\nthe square root in Eq. (6) with respect to 1 /S[28, 68, 77, 78,\n82, 83]. Using the expression, His represented by the bosons\nand expanded for 1 /Sas\nH=S \nH0+1√\nSH3+1\nSH4+O(S−3/2)!\n+const.,(9)\nwhereH0is the bilinear term consisting of bosonic operators\nandH3andH4are the terms composed of the three and four\nbosonic operators. The explicit expression for H0is shown\nthe next section. Note that, while a†\nmiandamido not appear\nalone because of the stable condition of the MF solution, other\nodd-order terms are allowed to appear in the Hamiltonian [56].\nC. Flavor-wave theory\nIn the bosonic representation in Eq. (9), H0is regarded as\na noninteracting Hamiltonian. This is written as\nH0=MX\nlNuX\nuN−1X\nm=1∆El\nma†\nm(l,u)am(l,u)\n+MX\nll′NuX\nuu′X\nαβN−1X\nmm′=1Jαβ\ni j\n2\n×\u0010\nOα\nm0;la†\nm(l,u)+H.c.\u0011\u0010\nOβ\nm′0;l′a†\nm′(l′,u′)+H.c.\u0011\n,(10)\nwhere ∆El\nmis the energy di fference between the excited state\nand ground state of the local MF Hamiltonian at site ibelong-\ning to sublattice l. The site label iis expressed by the two\nindices ( l,u) with unit cell uand sublattice l. We also intro-\nduce s=(l,m) as the composite index of sublattice land local\nexcited state mwith N=M(N−1) being the number thatscan take. Note that Nis the number of branches for the\ncollective modes [82]. Then, H0is represented as\nH0=1\n2X\nuu′ss′\"\u0010\nM11\nuu′\u0011\nss′a†\nusau′s′+\u0010\nM12\nuu′\u0011\nss′a†\nusa†\nu′s′\n+\u0010\nM21\nuu′\u0011\nss′ausau′s′+\u0010\nM22\nuu′\u0011\nss′ausa†\nu′s′#\n,(11)\nwhereM11\nuu′,M12\nuu′,M21\nuu′, andM22\nuu′are the N×Nmatrices\nsatisfying the following relations [82]:\nM11\nuu′=\u0010\nM11\nu′u\u0011†=\u0010\nM22\nu′u\u0011T=\u0010\nM22\nuu′\u0011∗, (12)\nM12\nuu′=\u0010\nM12\nu′u\u0011T=\u0010\nM21\nu′u\u0011†=\u0010\nM21\nuu′\u0011∗. (13)\nWe also introduce the 2 N-dimensional vector A†\nu, which is\ngiven by\nA†\nu=\u0010\na†\nu,1a†\nu,2···a†\nu,Nau,1au,2···au,N\u0011\n. (14)\nThen,H0is rewritten as follows:\nH0=1\n2NuX\nuu′A†\nuMuu′Au′. (15)\nWe introduce the 2 N×2NHermitian matrix Muu′, which is\ngiven by\nMuu′=M11\nuu′M12\nuu′\n\u0010\nM12\nuu′\u0011∗\u0010\nM11\nu′u\u0011T. (16)\nNote thatMuu′depends only on the relative positions of unit\ncells uandu′. By introducing the Fourier transformation of\nau,swith respect to u, the Hamiltonian H0is formally written\nas\nH0=1\n2X\nkA†\nkMkAk, (17)\nwhereMkis a 2 N×2NHermitian matrix. The sum of kis\ntaken in the first Brillouin zone. The 2 N-dimensional vector\nA†\nkis given by\nA†\nk=\u0010\na†\nk,1a†\nk,2···a†\nk,Na−k,1a−k,2···a−k,N\u0011\n, (18)\nwhere ak,sis the Fourier transformation of ami, which is rep-\nresented by\nak,s=r\n1\nNuX\nuau,se−ik·ri. (19)\nHere, we replace the index ( mi) inamito (u,s), and riis the\nposition of site ibelonging to sublattice lin unit cell u. By\nintroducing the representative position of unit cell ˜ ru,riis\nrepresented as ri=˜ru+˜δs, where ˜δsfors=1,2,···,Nis a4\nrelative vector from ˜ rutori. Note that ˜δsdepends only on the\nsublattice index lins=(l,m).Mkis given as\n(Mk)ss′=X\nuexph\n−ik·( ˜ru+˜δs−˜ru′−˜δs′)i\n(Muu′)ss′.\n(20)\nWe diagonalizeMkby applying the Bogoliubov transfor-\nmation asEk=T†\nkMkTk, where Tkis a paraunitary ma-\ntrix, which sastisfies the relation Tkσ3T†\nk=T†\nkσ3Tk=σ3\nwith the paraunit matrix σ3≡\u00101N×N 0\n0−1N×N\u0011\n, where 1N×Nis\ntheN×Nunit matrix.Ekis the diagonal matrix given by\nEk=diag{εk,1,εk,2,···,εk,N,ε−k,1,ε−k,2,···,ε−k,N}[84].\nUsing this transformation, we rewrite the Hamiltonian as the\nfollowing diagonalized form:\nH0=1\n2X\nkB†\nkEkBk, (21)\nHere, we introduce the set of bosonic operators Bk=T−1\nkAk,\nwhich is given by\nB†\nk=\u0010\nb†\nk,1b†\nk,2···b†\nk,Nb−k,1b−k,2···b−k,N\u0011\n. (22)\nNote thatBksatisfies the following commutation relation:\nh\nBk,η,B†\nk,η′i\n=σ3,ηδη,η′. (23)\nWhileH0in Eq. (21) is written as a free-boson Hamiltonian,\nhigher order terms such as H3andH4in Eq. (9) describe in-\nteractions between bosons.\nD. Perturbation theory using Green’s functions\nIn this section, we introduce the method addressing higher-\norder terms describing the interactions between bosons in-\ntroduced in Sec. II B. Here, the bosonic representation of\nthe Hamiltonian in Eq. (9) is split into two terms: H/S=\nH0+Hint, whereHintis the interactions between bosons, as\nshown in Sec. II B. The higher-order contributions Hintare\nincorporated by using the perturbation theory, where H0is re-\ngarded as an unperturbed term [50, 53, 54]. The perturbation\nterm is given byHint=H3/√\nS+H4/S+O(S−3/2).\nTo perform the perturbation expansion systematically, we\nemploy the Green’s function approach [85]. We define the\ntemperature Green’s function for the Bogoliubov bosons Bk,η\nas\nGk,ηη′(τ)=−\nTτBk,η(τ)B†\nk,η′\u000b, (24)\nwhere Tτis the time-ordering operator in imaginary time τ,\nand⟨·⟩stands for the thermal average. The Fourier represen-\ntation for imaginary time is introduced as\nGk,ηη′(iωn)=Zβ\n0dτeiωnτGk,ηη′(τ), (25)whereωn=2nπ/βis the Matsubara frequency with nbeing\ninteger, kBis the Boltzmann constant, and Tis the tempera-\nture.\nThe bare Green’s function is given by\nG(0)\nk(iωn)=[iωnσ3−Ek]−1, (26)\nwhich is a 2 N×2Nmatrix. Moreover, the temperature Green’s\nfunction can be expanded as\nGk(τ)=G(0)\nk(τ)+Zβ\n0dτ1D\nTτHint(τ1)Bk(τ)B†\nkE\n0\n−1\n2!Zβ\n0dτ1Zβ\n0dτ2D\nTτHint(τ1)Hint(τ2)Bk(τ)B†\nkE\n0\n+···, (27)\nwhere⟨·⟩ 0represents the thermal average for the unperturbed\nHamiltonianH0. We also introduce the self-energy as\nΣk(iωn)≡h\nG(0)\nk(iωn)i−1−h\nGk(iωn)i−1. (28)\nThe Green’s function is written as Gk(iωn)=h\niωnσ3−Ek−\nΣk(iωn)i−1. The retarded and advanced self-energies, ΣR\nk(ω)\nandΣA\nk(ω), are calculated by performing the analytic contin-\nuation. By using the self-energy, the retarded, and advanced\nGreen’s functions can be written as [85]\nGR\nk(ω)=h\n(ω+i0+)σ3−Ek−ΣR\nk(ω)i−1, (29)\nGA\nk(ω)=h\n(ω−i0+)σ3−Ek−ΣA\nk(ω)i−1. (30)\nNote that the temperature Green’s function satisfies the con-\nditionsGk,ηη′(iωn)=G−k,η′+N,η+N(−iωn) andGk,η,η′+N(iωn)=\nG−k,η′,η+N(−iωn) for 0≤η,η′≤N, which are obtained from\nEq. (24). From them, the following relations hold for the re-\ntarded, and advanced Green’s functions with 0 ≤η,η′≤N:\nGR\nk,ηη′(ω)=GA\n−k,η′+N,η+N(−ω), (31)\nGR\nk,η,η′+N(ω)=GA\n−k,η′,η+N(−ω). (32)\nSimilarly, the temperature, retarded, and advanced self ener-\ngies satisfy the following relations for 0 ≤η,η′≤N:\nΣk,ηη′(iωn)= Σ−k,η′+N,η+N(−iωn) (33)\nΣk,η,η′+N(iωn)= Σ−k,η′,η+N(−iωn), (34)\nand\nΣR\nk,ηη′(ω)= ΣA\n−k,η′+N,η+N(−ω) (35)\nΣR\nk,η,η′+N(ω)= ΣA\n−k,η′,η+N(−ω). (36)\nFinally, we introduce the spectral function as follows:\nρk,η(ω)=1\n2πZ∞\n−∞dteiωtDh\nBk,η(t),B†\nk,ηiE\n. (37)5\nFrom Eq. (23), the sum rule is given by\nZ∞\n−∞ρk(ω)dω=σ3. (38)\nFurthermore, the spectral function is connected to the diago-\nnal component of the retarded Green’s function by the follow-\ning relation:\nρk,η(ω)=−1\nπImGR\nk,ηη(ω). (39)\nMeanwhile, the spectral function satisfies the condition\nsgn(ω)ρk,η(ω)≥0, which is obtained from its Lehmann repre-\nsentation. Thus, the following relation holds for the imaginary\npart of the retarded Green’s function:\nsgn(ω)ImGR\nk,ηη(ω)≤0. (40)\nIII. FORMALISM FOR THERMAL TRANSPORT\nIn this section, we formulate thermal conductivity using the\nbosonic Green’s function introduced in the previous section.\nA. Introduction to thermal conductivity\nFirst, we briefly review a general theory for thermal trans-\nport based on Ref. [86–89]. Thermal responses can be mi-\ncroscopically evaluated as a response against a virtually in-\ntroduced gravitational field, which is the mechanical counter-\npart of the temperature gradient applied to the system [90].\nHere, we consider the local Hamiltonian h(ri) involving site\ni, which satisfiesH=P\nih(ri). From this local Hamilto-\nnian, the external field is introduced by replacing Hwith\nHχ=P\ni\u00021+χ(ri)\u0003h(ri), whereχ(r) is the gravitational field\napplied to the system. Since we consider a bosonic system\nwith zero chemical potential, the thermal current operator is\nequivalent to the energy current, which is given by [85]\nJQ=1\nV∂PE\n∂t=i\nVℏ\u0002H,PE\u0003, (41)\nwhere PEis the energy polarization operator defined as\nPE=X\nirih(ri). (42)\nIn the presence of the gravitational field, the local Hamiltonian\nis replaced to h(ri)→hχ(ri)=\u00021+χ(ri)\u0003h(ri) [17, 90], and\nthermal current is also changed to JQ;χ, which is written as\nJQ;χ=i\nVℏh\nHχ,Pχ\nEi\n, (43)\nwhere Pχ\nE=P\niri\u00021+χ(ri)\u0003h(ri)=P\nirihχ(ri).\nHere, we introduce the thermal conductivity κλλ′, whereλ(=\nx,y,z) is the component of Cartesian coordinate, as\nJtr\nλ=κλλ′(−∇λ′T). (44)Here,∇Tis the temperature gradient, and Jtris the thermal\ntransport response [86–89]. Note that Jtrmust vanish in the\nabsence of the thermal gradient. As mentioned at the begin-\nning of this section, the thermal conductivity can be evaluated\nas the response against the gradient of the gravitational field\ninstead of the temperature gradient as follows [90]:\nJtr\nλ=Lλλ′(−∇λ′χ(r)), (45)\nwhere Lλλ′=κλλ′Tis the thermal transport coe fficient with T\nbeing the temperature in equilibrium. The thermal transport\nresponse Jtris not equivalent to ⟨JQ;χ⟩∇χin the first order of\n∇χ, where⟨·⟩∇χis the expectation value in the presence of the\ngravitational-field gradient. To enforce the condition of Jtr=\n0 for∇χ=0, one needs to subtract the contribution from a\nheat magnetization from ⟨JQ;χ⟩∇χ(see Appendix A) [86–89].\nFinally, the thermal transport coe fficient is written as\nLλλ′=Sλλ′+X\nλ′′ελλ′λ′′2MQ\nλ′′\nV, (46)\nwhere Sλλ′is the contribution obtained by the well-known\nKubo formula [91], MQis the heat magnetization originat-\ning from thermal carriers [12, 89], Vis the volume of the\nsystem, and ελλ′λ′′is the Levi-Civita symbol. Note that the\nsecond term in Eq. (46) does not contribute to the symmetric\ncomponents of the thermal transport coe fficient but plays cru-\ncial role in the antisymmetric components, namely the thermal\nHall conductivity.\nThe first term of Eq. (46) is evaluated from\nSλλ′=−lim\nΩ→0PR\nλλ′(Ω)−PR\nλλ′(Ω)\niΩ, (47)\nwhere PR\nλλ′(Ω) is the retarded correlation function between\nthermal currents, which is calculated from the imaginary-\ntime correlation function Pλλ′(iΩ) via analytic continuation:\nPR\nλλ′(Ω)=Pλλ′(iΩ→ℏΩ +i0+). Here, Pλλ′(iΩ) is given by\nPλλ′(iΩ)=−1\nVZβ\n0dτeiΩτ⟨TτJQ\nλ(τ)JQ\nλ′⟩, (48)\nwhereβ=1/kBTand the Heisenberg representation of an\noperatorOis defined asO(τ)=eτHOe−τH. On the other hand,\nthe heat magnetization MQis evaluated from the following\nrelations:\n2MQ+β∂MQ\n∂β=1\nβ∂\n∂β\u0010\nβ2MQ\u0011\n=˜MQ, (49)\nwhere ˜MQis given by\n˜MQ\nλ=−β\n2iX\nλ′λ′′ελλ′λ′′∂\n∂qλ′′D\nh−q;jQ\nq,λ′E\f\f\f\fq→0. (50)\nThe di fferential equation in Eq. (49) is solved under the\nboundary condition: lim β→∞β∂MQ\n∂β=0, namely, 2 MQ=\n˜MQat zero temperature limit [89]. Here, we introduce the6\nFourier transforms of the local Hamiltonian h(ri) and the ther-\nmal current density defined by JQ=P\nijQ(ri) as\nhq=X\nih(ri)e−iq·ri,jQ\nq=X\nijQ(ri)e−iq·ri. (51)\nAdditionally,D\nh−q;jQ\nqE\nstands for the canonical correlation\nbetween them, which is defined by\nD\nh−q;jQ\nqE\n=1\nβZβ\n0dτD\nh−q(τ)jQ\nqE\n. (52)\nTo evaluate heat magnetization from Eq. (49), the follow-\ning scaling relation must be imposed for the thermal current\ndensity [87, 89]:\njQ;χ(ri)=\u00021+χ(ri)\u00032jQ(ri). (53)\nIn the following calculations, the local Hamiltonian and cur-\nrent density are introduced from the bilinear bosonic Hamilto-\nnian given in Eq. (15), for simplicity. Then, the local Hamil-\ntonian is represented as\nhq=1\n2X\nkA†\nkMk+Mk+q\n2Ak+q, (54)\nand the thermal current density satisfying Eq. (53) is writtenas\njQ\nq=1\n4X\nkA†\nk\u0010\nvkσ3Mk+q+Mkσ3vk+q\u0011\nAk+q\n−1\n16X\nλX\nkℏqλA†\nk\u0010\nvkσ3vk+q,λ−vk,λσ3vk+q\u0011\nAk+q,\n(55)\nwhere vq=1\nℏ∂Mq\n∂q. The derivation of the above representation\nis given in Appendix B. Note that the total thermal current JQ\nis given by JQ=jQ\nq\f\f\fq→0, where the second term in Eq. (55)\ndoes not contribute to JQ, and thereby, JQis written as\nJQ=1\n4X\nkA†\nk\u0010\nvkσ3Mk+Mkσ3vk\u0011\nAk. (56)\nB. Green’s function representation of thermal conductivity\nIn this section, we show the representations of Sλλ′and\n˜MQusing the Green’s functions introduced in Eqs. (29) and\n(30), where we neglect vertex corrections (see Appendix C).\nFor simplicity, we omit their o ff-diagonal components with\nrespect toηand only consider diagonal components of the\nretarded and advanced Green’s functions, which are written\nasGR\nk,η(ω) and GA\nk,η(ω), respectively. Under this assumption,\nSλλ′and ˜MQ\nλare represented as\nSλλ′=−iℏ\n8V2NX\nη,η′=1X\nk\u0010\nT†\nkvk,λTk\u0011\nηη′\u0010\nT†\nkvk,λ′Tk\u0011\nη′η\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u00112\n×PZ∞\n−∞dω\nπg(βω)Imh\nGR\nk,η(ω)i∂GR\nk,η′(ω)\n∂ω−∂GA\nk,η(ω)\n∂ωImh\nGR\nk,η′(ω)i, (57)\nand\n˜MQ\nλ=−1\n16iX\nλ′λ′′ελλ′λ′′∂\n∂qλ′′2NX\nη,η′=1X\nkh\nT†\nk\u0010\nMk+Mk−q\u0011\nTk−qi\nηη′\n×\"\nT†\nk−qvk−q,λσ3Mk+Mk−qσ3vk,λ−X\nλ′′′ℏqλ′′′vk−q,λσ3vk,λ′′′−vk−q,λ′′′σ3vk,λ\n4Tk#\nη′η\n×PZ∞\n−∞dω\nπg(βω)(\nImh\nGR\nk,η(ω)i\nGR\nk−q,η′(ω)+GA\nk,η(ω)Imh\nGR\nk−q,η′(ω)i)\f\f\f\f\f\fq→0, (58)\nrespectively. Here, g(x)=(ex−1)−1is the Bose distribu-\ntion function with zero chemical potential, and PR\nstands for\nthe principal value integral. The details of the derivations for\nEqs. (57) and (58) are given in Appendices C 1 and C 2, re-\nspectively.C. Thermal Hall conductivity with approximate Green’s\nfunction\nHere, we rewrite Sλλ′and ˜MQ\nλgiven in Eqs. (57) and (58),\nrespectively, as more convenient expressions. In the present\nstudy, we focus on e ffects of magnon damping on the thermal7\nHall e ffect, and hence, we take into account the imaginary part\nof the self-energy in Eq. (29) and (30) and neglect its real part.\nIn the previous section, we only consider the diagonal part\nof the Green’s functions. This simplification corresponds to\nomitting the o ff-diagonal components of the self-energy, and\nhence, we here only consider the imaginary part of the diag-\nonal components of the self-energy. We define the imaginary\npart of the retarded self-energy as\nΓk,η(ω)=−ImΣR\nk,η(ω), (59)\nwhich corresponds to the damping rate of the magnon with\nmomentum ℏkand branch η. Using the above approximation\nand the damping rate Γk,η(ω), we represent the retarded and\nadvanced Green’s functions as\nGR\nk,η(ω)≃1\n(ω+i0+)σ3,η−Ek,η+iΓk,η(ω), (60)\nGA\nk,η(ω)≃1\n(ω−i0+)σ3,η−Ek,η−iΓk,η(ω), (61)\nwhere we use the relation Im ΣR\nk,η(ω)=−ImΣA\nk,η(ω), which\nis obtained from the Lehmann representation of the Green’s\nfunctions [85].Here, we discuss the analytical properties of Γk,η(ω). From\nEq. (40), the retarded self-energy satisfies sgn( ω)ImΣR\nk,η(ω)≤\n0. This leads to the following conditions:\nsgn(ω)Γk,η(ω)≥0. (62)\nAdditionally, from Eq. (35), Γk,η(ω) also satisfies the follow-\ning relation for η=1,···,N:\nΓk,η(ω)=−Γ−k,η+N(−ω). (63)\nHereafter, we assume that Γk(ω) varies slowly enough as\na function of k,ω, and T. We neglect the di fferential coef-\nficients with respect to these variables and focus only on the\nantisymmetric part of the thermal conductivity matrix κλλ′to\ndiscuss the thermal Hall e ffect. Within the assumption, the ω\nderivative of the retarded and advanced Green’s functions are\nrepresented as ∂GR\nk,η/∂ω≃−1/[ωσ 3,η−Ek,η+iΓk,η(ω)]2and\n∂GA\nk,η/∂ω≃−1/[ωσ 3,η−Ek,η−iΓk,η(ω)]2, respectively. Us-\ning the representations of Green’s functions in Eqs. (60) and\n(61) and the above approximations, the antisymmetric part of\nSλλ′and 2 MQ\nλ/Vare calculated as\nSa\nλλ′≃1\n4ℏVX\nλ′′ελλ′λ′′NX\nη=12NX\nη′=1X\nk˜Ωλ′′\nk,ηη′\u0010\nεk,η+σ3,η′Ek,η′\u00112\u0010\nεk,η−σ3,η′Ek,η′\u00112\n×RePZ∞\n−∞dωρk,η(ω)2g(βω)+1\n\u0010\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)\u00112(64)\nand\n2MQ\nλ\nV≃−1\n2β2ℏVNX\nη=12NX\nη′=1X\nk˜Ωλ\nk,ηη′\u0010\nεk,η+σ3,η′Ek,η′\u00112\u0010\nεk,η−σ3,η′Ek,η′\u0011\n×Re\"\nPZ∞\n−∞dωρk,η(ω)\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)#Zβ\n0˜βh\n2g(˜βω)+1i\nd˜β\n−1\n4β2ℏVNX\nη=12NX\nη′=1X\nk2˜Ωλ\nk,ηη′εk,η\u0014\u0010\nεk,η+σ3,η′Ek,η′\u00112−4ε2\nk,η\u0015\nPZ∞\n−∞dωρk,η(ω)Zβ\n0˜β∂g\n∂ωd˜β, (65)\nwhere Sa\nλλ′=(Sλλ′−Sλ′λ)/2 and ˜Ωλ\nk,ηη′are defined as\n˜Ωλ\nk,ηη′=−iℏ2\n2X\nλ′λ′′ελλ′λ′′σ3,ησ3,η′\u0010\nT†\nkvk,λ′Tk\u0011\nηη′\u0010\nT†\nkvk,λ′′Tk\u0011\nη′η\u0010\nσ3,ηEk,η−σ3,η′Ek,η′\u00112\n(66)\nThe detailed derivations of Eqs. (64) and (65) are given in Ap-\npendixes D 1 and D 2. From Eq. (46), we find that the thermalHall conductivity κH\nλλ′=(κλλ′−κλ′λ)/2 is evaluated by\nκH\nλλ′=La\nλλ′\nT0=Sa\nλλ′+X\nλ′′ελλ′λ′′2MQ\nλ′′\nV, (67)\nwhere La\nλλ′=(Lλλ′−Lλ′λ)/2.\nTo further proceed the calculations of the thermal Hall\nconductivity, we presume that ρk,η(ω) forω→0 is su ffi-\nciently smaller than the maximum of ρk,η(ω). This assump-\ntion is justified when the damping rate of magnons is small\nenough in the vicinity of the zero energy. Previous studies8\nhave suggested that this situation is realized at low tempera-\ntures [53, 54, 64, 68]. Then, the contributions around the pole\nof the Green’s function, ω≃εk,η, are dominant in the evalu-\nations of Eq. (64) and (65) for η=1,2,···,N. Moreover, we\nhypothesize that Γk,η(ω) is small and incorporate this contri-\nbution up to the first order. Based on the above assumptions,\nwe obtain the representation of the thermal Hall conductivity\nas\nκH\nλλ′≃−k2\nBT\nℏVX\nλ′′NX\nη=1X\nkελλ′λ′′Ωλ′′\nk,η\n×Z∞\n−∞dωρk,η(ω)c2(g(βω)). (68)\nThe derivation of the above expression is given in Ap-\npendix D 3. Here, we introduce c2(x) as\nc2(x)=Zx\n0 \nln1+t\nt!2\ndt (69)\nandΩk,ηis the Berry curvature given by [15, 17]\nΩλ\nk,η=iX\nλ′λ′′ελλ′λ′′σ3∂T†\nk\n∂kλ′σ3∂Tk\n∂kλ′′\nηη. (70)\nThis Berry curvature satisfies the sum rulePN\nη=1P\nkΩλ\nk,η=\n0 for the positive-energy branches [15, 17] and is related to\n˜Ωλ\nk,ηη′in Eq. (66) as Ωλ\nk,η=−2P2N\nη′(,η)˜Ωλ\nk,ηη′[see Eq. (D14)].\nHere, we briefly discuss the noninteracting limit with\nΓk,η(ω)=0. In this limit, ρk,η(ω)→δ(ω−εk,η) forω≥0,\nand hence,κH\nxyis written as\nκH;free\nλλ′=−k2\nBT\nℏVX\nλ′′NX\nη=1X\nkελλ′λ′′Ωλ′′\nk,ηc2(g(βεk,η)).(71)\nThis result coincides with the previous studies on free magnon\nsystems [12, 15, 17].\nIn numerical calculations, it is considerably di fficult to eval-\nuate theωdependence of Γk,η(ω). To reduce the calcula-\ntion cost, we omit the ωdependence from Γk,η(ω) as˜Γk,ηfor\nη=1,2,···,N. The damping rate ˜Γk,ηcan be obtained by nu-\nmerical calculations such as on-shell approximation as ˜Γk,η=\nΓk,η(εk,η) [49, 56] or o ff-shell methods [53, 56, 57, 60, 68].\nFrom Eq. (62), we find that ˜Γk,η≥0 becauseεk,ηis positive.\nAs an approximate form of Γk,η(ω) to satisfy these conditions\nand Eq. (62), we introduce the following expression:\nΓk,η(ω)≃˜Γk,ηθ(ω) (η≤N), (72)\nwhereθ(ω) is the step function. In this approximation, the\nretarded Green’s function is simplified as\nGR\nk,η(ω)≃1\nω+i0+−εk,η+i˜Γk,ηθ(ω)(η≤N). (73)\nThis expression obviously satisfies Eq. (40). Furthermore,\nwithin this approximation, ρk,η(ω) is written as ρk,η(ω)≃˜ρk,η(ω)=Lk,η(ω)θ(ω), whereLk,η(ω) is the Lorentz func-\ntion given by\nLk,η(ω)=˜Γk,η/π\n(ω−εk,η)2+˜Γ2\nk,η(η≤N). (74)\nNote that Ik,η=R∞\n−∞˜ρk,η(ω) does not coincide with unity due\nto the simplification given in Eq. (72). This quantity is writ-\nten as Ik,η=1\n2+1\nπarctanεk,η\n˜Γk,η, indicating that Ik,η≃1 for\n˜Γk,η≪εk,η. Thus sum rule for ˜ ρk,η(ω) is approximately satis-\nfied as long as ˜ ρk,η(ω) forω→0 is su fficiently small, which\nis assumed before. Finally, we obtain the thermal Hall con-\nductivity incorporating the magnon damping ˜Γk,ηas\nκH\nλλ′≃−k2\nBT\nℏVX\nλ′′NX\nηX\nkελλ′λ′′Ωλ′′\nk,η\n×Z∞\n−∞dω˜ρk,η(ω)c2(g(βω)). (75)\nIn the following, we focus on κH\nxyin two-dimensional systems\nstacked along the zdirection, where the interlayer distance is\nassumed to be unity.\nD. Fundamental properties of thermal Hall conductivity\nincorporating magnon damping\nIn the previous section, we have formulated the thermal\nHall conductivity incorporating magnon damping as Eq. (75).\nIn this section, we examine the fundamental properties by in-\ntroducing a simple two-band magnon model ( N=2), where\nthe magnon dispersions and the corresponding damping rates\nare given by εk,ηand˜Γk,ηwithη=1,2, respectively. Here, we\nomit the kdependence of the magnon energy and the damp-\ning rate and introduce the parameters αandγasα=εk,1/εk,2\nandγ=˜Γk,1/εk,1=˜Γk,2/εk,2, respectively. We assume that\nεk,1< εk,2, namely,α < 1. Furthermore, the Chern numbers\nCz\nηof the two magnon branches are set to be Cz\n1=−Cz\n2=1\nwhereCz\nη=1\n2πR\nBZdkxdkyΩλ\nk,η. The thermal Hall conductivity\nin this simple model is written as\nκH\nxy≃−k2\nBT\n2πℏZ∞\n−∞dω\u0002˜ρ1(ω)−˜ρ2(ω)\u0003c2(g(βω)). (76)\nNote that the temperature dependence of κH\nxy/Tcomes from\nc2(g(βω)), and c2(g(x)) is a monotonically decreasing function\nofxfromπ2/3 atx=0 to 0 at x→∞ .\nFigure 1 shows the temperature dependence of κH\nxy/Tin\nthe simple two-band model for several values of α. As an\noverall behavior regardless of α, we find that introducing\nthe magnon damping enhances the absolute value of κH\nxyin\nthe low-temperature region and suppresses it in the high-\ntemperature region. The impact of the magnon damping on\nthe thermal Hall conductivity is understood as follows. At\nlow temperatures, the ωdependence of c2(g(βω)) predom-\ninantly enhances the contribution of the low-energy part of\nthe integral in Eq. (76) to the thermal Hall conductivity. This9\n0.0\n−0.1\n−0.2\n−0.3\n−0.4κH\nxy/T(a)α=0.1\n0.0\n−0.1\n−0.2κH\nxy/T(b)α=0.5\n0.00 0.25 0.50\nT/ε20.0\n−0.1κH\nxy/T(c)α=0.8\nγ=0.0001\nγ=0.2\nγ=0.4\nγ=0.6\nγ=0.8\nγ=1.0\nFIG. 1. Temperature dependence of the thermal Hall conductivity\nin the two-band magnon model introduced in Sec. III D for several\nvalues of the energy ratio α=εk,1/εk,2.\nis amplified by magnon damping, which increases the lower-\nenergy spectral weight, thereby enhancing κH\nxy/T. On the other\nhand, at higher temperatures, the ωdependence of c2(g(βω))\nbecomes less pronounced. Here, the magnon damping leads\nto a broadening of the spectrum. This broadening facilitates\nan cancellation e ffect between contributions from the bands\npossessing the opposite Chern numbers, resulting in a reduc-\ntion of the absolute value of κH\nxy/T. This e ffect is particularly\npronounced when αis large, as shown in Fig. 1(c), because\nof the proximity of the two branches. In contrast, at a smaller\nα, the temperature range over which κH\nxy/Tis enhanced by γ\nbecomes more restricted, as shown in Fig. 1(a).\nIV . APPLICATION TO LOCALIZED SYSTEMS\nIn this section, we apply our theory for the thermal Hall\nconductivity formulated above to the two localized spin-1 /2\nmodels on a honeycomb lattice: the Kitaev model under mag-\nnetic fields and the Heisenberg-DM model. We evaluate the\ndamping rate ˜Γk,ηfor each magnon branch ηusing the self-\nFIG. 2. Schematic picture of the honeycomb lattice on which the\nKitaev model is defined. The red, blue, and green lines represent\ntheX,Y, and Zbonds, respectively. The inset shows the relation\nbetween the coordinate of the spin space ( SX,SY,SZ) and that of the\nreal space ( x,y,z).\nconsistent imaginary Dyson equation (iDE) approach at finite\ntemperatures developed in Ref. [68]. In this method, we con-\nsider contributions up to O(1/S) corrections from the bilinear\ntermH0in Eq. (9). To this end, we deal with H3/√\nSup to\nsecond-order perturbations and H4/Sup to first-order pertur-\nbations [53, 64]. We calculate κH\nxybased on Eq.(75), where the\nhoneycomb lattice is defined on the xyplane.\nA. Kitaev model under magnetic field\nFirst, we address thermal transport in the Kitaev model on\na honeycomb lattice under an external magnetic field, whose\nHamiltonian is given as follows:\nH=2KX\n⟨i j⟩ΛSΛ\niSΛ\nj−X\nih·Si, (77)\nwhere Kandhare the strengths of the Kitaev interaction and\nmagnetic field, respectively, and SΛ\ni(Λ=X,Y,Z) is theΛ\ncomponent of an S=1/2 spin defined at site ion the hon-\neycomb lattice, whose bonds are classified into three types:\nX,Y, and Zbonds, as shown in Fig. 2. The Λbond connect-\ning between sites iandjare denoted as⟨i j⟩Λ. In this system,\nthe [111] axis of the spin space is taken to be parallel to the\nzdirection in the real space for the correspondence to real\nmaterials with spin-orbit coupling. The other axes are deter-\nmined as presented in the inset of Fig. 2. Here, we consider\nthe ferromagnetic Kitaev model with K<0. under a ma-\ngentic field along the [111] axis in the spin space, which cor-\nresponds to the out-of-plane zdirection. We assume a forced\nferromagnetic state along the external-field direction as a clas-\nsical ground state. Within the linear spin-wave approximation,10\nΓ KM Γ0.01.02.03.0ω/|K|(a)h/|K|=0.1\nΓ KM Γ0.01.02.03.0(b)h/|K|=0.3\nΓ KM Γ0.01.02.03.0ω/|K|(c)h/|K|=0.5\nΓ KM Γ0.01.02.03.0(d)h/|K|=0.7\n-1.00.01.0\nΩk,η−1 0 1−0.50.00.5\nΓ KM\nFIG. 3. Dispersion relations of magnons from a spin polarized state\nin the Kitaev model with (a) h/|K|=0.1, (b) 0.3, (c) 0.5, and (d) 0.7.\nThe line color indicates the Berry curvature Ωk,η. The inset in (a)\nshows the first Brillouin zone of the honeycomb lattice. The disper-\nsion relations are plotted along the red dashed lines in this inset.\ngapped two magnon modes appear in the presence of the mag-\nnetic fields, and they exhibit nonzero nonzero Chern numbers\nwith±1 [23, 24], leading to nonzero thermal Hall conductiv-\nity. Note that the Hamiltonian does not commute with the total\nspin operatorP\ni˜SZ\ni, where ˜SZ\ni=(SX\ni+SY\ni+SZ\ni)/√\n3 is the\nspin component along the field direction. This suggests the\nappearance of magnon scattering processes without the parti-\ncle number conservation, and thereby, the magnon damping\nshould arise even in lower-order corrections for 1 /Sin the\nHolstein-Primako fftheory [56, 68].\nBefore showing the results for the thermal Hall conductiv-\nity, we briefly comment on the magnon band structure under\nseveral magnetic fields. Figures 3(a)–3(d) present the disper-\nsion relations of magnons for h/|K|=0.1, 0.3, 0.5, and 0.7,\nrespectively [23]. There are two magnon branches in this sys-\ntem, and our analysis has confirmed that the Chern numbers\nof low-energy and high-energy branches are +1 and−1, re-\nspectively. Across all parameters, the absolute value of the\nBerry curvature around the K point takes a large value. We\nobserve that with increasing the magnetic field, the magnon\ndispersion shifts to the high-energy side and the gap between\nthe two bands becomes narrow. Despite the Chern number of\nthe low-energy branch being +1, the Berry curvature at this\nbranch around the Γpoint takes a small negative value, partic-\nularly for h/|K|=0.1 [23].\nHere, we present the temperature dependence of the ther-\nmal Hall conductivity κH\nxycalculated with the magnon damp-\ning in Fig. 4. For comparison, we also provide results obtained\nunder the free-magnon approximation based on Eq. (71).\nFirst, we review the results in the free-magnon system [23].\nThe thermal Hall conductivity takes a negative value in the\ntemperature range except for extremely low temperatures, and\nκH\nxy/Tasymptotically approaches zero at high temperatures.\nThis phenomenon can be attributed to the overall structure\nof Berry curvature associated with magnon bands and func-\ntional form of c2(g(βεk,η)); the low-energy band with the pos-\n0.0 0.25 0.5−0.20−0.100.000.100.20κH\nxy/T(a) h/|K|=0.1 (free)\nh/|K|=0.3 (free)h/|K|=0.1 (interacting)\nh/|K|=0.3 (interacting)\n0.00 0.25 0.50\nT/|K|−0.20−0.100.000.100.20κH\nxy/T(b) h/|K|=0.5 (free)\nh/|K|=0.7 (free)h/|K|=0.5 (interacting)\nh/|K|=0.7 (interacting)FIG. 4. Temperature dependence of thermal Hall conductivity di-\nvided by temperature in the Kitaev model for (a) h/|K|=0.1 and 0.3\nand (b) h/|K|=0.5 and 0.7. The dashed-dotted lines represent the\nresults for the free magnon system within the linear spin-wave ap-\nproximation. On the other hand, the solid lines represent the results\nfor the systems with magnon-magnon interactions calculated based\non the iDE approach.\nitive Chern number largely contributes to the thermal Hall ef-\nfect compared to the high-energy band because c2(g(βεk,η))\nrapidly decreases with increasing εk,ηat low temperatures.\nThis feature results in the negative value of κH\nxybecause of the\nnegative sign in Eq. (71). At high temperatures, c2(g(βεk,η))\nis almost independent on εk,η, and thereby, contributions from\ntwo bands with opposite Chern numbers to the thermal Hall\nconductivity chancel out each other. On the other hand, in the\nlow-temperature region, the thermal Hall conductivity turns\nto be positive at h/|K|=0.1, as shown in Fig. 4(a) [23]. As\nmentioned before, the sign change of κH\nxyis ascribed to the neg-\native Berry curvature of the low-energy branch in the vicinity\nof the Γpoint. We also find that the absolute value of κH\nxy/T\ndecreases with increasing the external magnetic field, as seen\nin Fig. 4(b). This trend can be comprehended through the in-\ncrease of the excitation energy and the narrowing of the gap\nenergy between the two branches in Fig. 3.\nNext, we discuss the temperature dependence of the ther-\nmal Hall conductivity incorporating magnon damping. We\nfind that the thermal Hall conductivity is strongly suppressed11\nΓ K M Γ0.01.02.03.0ω/|K|(a)h/|K|=0.1,T/|K|=0\nΓ K M Γ0123ω/|K|(b)h/|K|=0.1,T/|K|=0.25\nΓ K M Γ0123ω/|K|(c)h/|K|=0.1,T/|K|=0.5\nΓ K M Γ0.01.02.03.0ω/|K|(d)h/|K|=0.3,T/|K|=0\nΓ K M Γ0123ω/|K|(e)h/|K|=0.3,T/|K|=0.25\nΓ K M Γ0123ω/|K|(f)h/|K|=0.3,T/|K|=0.5\nΓ K M Γ0.01.02.03.0ω/|K|(g)h/|K|=0.5,T/|K|=0\nΓ K M Γ0123ω/|K|(h)h/|K|=0.5,T/|K|=0.25\nΓ K M Γ0123ω/|K|(i)h/|K|=0.5,T/|K|=0.5\nΓ K M Γ0.01.02.03.0ω/|K|(j)h/|K|=0.7,T/|K|=0\nΓ K M Γ0123ω/|K|(k)h/|K|=0.7,T/|K|=0.25\nΓ K M Γ0123ω/|K|(l)h/|K|=0.7,T/|K|=0.5\n012345\n˜ρk,η(ω)\nFIG. 5. (a)–(c) Color map of the spectral function ˜ ρk,η(ω) calculated by the iDE approach in the Kitaev model at (a) T/|K|=0, (b) 0.25, (c)\n0.5 for h/|K|=0.1. The cyan dashed-dotted lines stand for the dispersion relations obtained by the linear spin-wave approximation. (d)–(f),\n(g)–(i), (j)–(l) correspond to the results at h/|K|=0.3, 0.5, and 0.7, respectively. The spectral functions are plotted along the red dashed lines\nin the inset of Fig. 3(a).\nby the magnon damping, especially in the systems under low\nmagnetic fields. The sign change in κH\nxy, predicted by calcu-\nlations using the free-magnon approximation, vanishes at low\ntemperatures when h/|K|=0.1. This is understood from the\napproximated spectral function ˜ ρk,η(ω). From Eq. (75), the\nthermal Hall conductivity depends on this quantity with the\ndamping rate ˜Γk,ηin addition to the Berry curvature. Fig-\nures 5(a)–5(c) show the spectral function ˜ ρk,η(ω) on the k-\nωplane for h/|K|=0.1. At T=0, the lower-energy\nmode survives around the Γpoint, which is responsible for\nthe sign change of the thermal Hall conductivity in the free-\nmagnon approximation. On the other hand, the two magnon\nmodes near the K-M path are strongly damped. Note that theBerry curvature in the lower-energy mode around the K point\ntakes a large positive value in the free-magnon approximation\n[Fig. 3(a)]. The broadened spectrum contributes to the spec-\ntral weight near zero energy, which results in a negative ther-\nmal Hall conductivity. Therefore, the thermal Hall conductiv-\nity remains negative even at low temperatures. With increas-\ning temperature, the smearing becomes more pronounced,\nwhich causes the strong suppression of κH\nxy/Tat higher tem-\nperatures, as discussed in Sec. III D.\nAs the external magnetic field increases, the di fference be-\ntweenκH\nxywith and without magnon-magnon interactions di-\nminishes. This behavior arises from the reduction of magnon\ndamping due to the application of the magnetic field, as de-12\nqk+qqk-qkkkk\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¯V⇤¯V¯V¯V⇤(a) (b)\nFIG. 6. Schematic picture of the honeycomb lattice on which the\nHeisenberg-DM model is defined. The yellow arrows represent the\nvectors connecting next-nearest sites clockwise. The inset shows the\ndefinition of θ, which is an angle between the direction of the DM\nvector and ferromagnetic spin moment.\npicted in Fig. 5. The magnon damping at low tempera-\ntures is primarily attributed to a decay process in which a\nmagnon splits into two magnons [68]. Applying the magnetic\nfield decreases the overlap between the magnon branch and\nthe two-magnon continuum, thereby suppressing magnon-\nmagnon scatterings. Similar phenomena have been observed\nin the damping of a chiral magnon edge mode in the Kitaev\nmodel on a ribbon-shaped cluster, where the damping rate de-\ncreases monotonically with increasing a magnetic field [68].\nThese results suggest a close relationship between the decay\nof the chiral edge mode and the impact of magnon-magnon\ninteractions on thermal Hall conductivity.\nWe also find that a sign change occurs when h/|K|=0.5\nand 0.7, as shown in Fig. 4(b). This phenomenon can be at-\ntributed the di fference between the magnon damping for the\ntwo branches. As shown in Fig. 5(j), even at zero tempera-\nture, magnons within the high-energy branch undergo signif-\nicant damping, primarily due to the magnon decay process\nsplit into two magnons [68]. As discussed in Sec. III D, pro-\nnounced magnon damping amplifies the impact of the Berry\ncurvature in the corresponding branch on thermal Hall con-\nductivity. Consequently, the sign change to a positive value\nofκH\nxyat low temperatures, depicted in Fig.4(b), results from\nthe magnon damping in the high-energy branch with negative\nBerry curvatures, as demonstrated in Fig.3(d).\nB. Heisenberg-DM model\nIn this section, we examine the thermal Hall response in an\nS=1/2 quantum spin model with Heisenberg and DM inter-\nactions on a honeycomb lattice, which is represented as [69],\nH=JX\n⟨i j⟩Si·Sj+X\n⟨⟨i j⟩⟩Di j·\u0010\nSi×Sj\u0011\n, (78)\nΓ KM Γ0.01.02.03.0ω/|J|(a)θ=45◦\nΓ KM Γ0.01.02.03.0(b)θ=80◦\n-1.00.01.0\nΩk,ηFIG. 7. Dispersion relations of magnons in the Heisenberg-DM\nmodel from a ferromagnetic state with (a) θ=45◦and (b) 80◦. The\nline color indicates the Berry curvature Ωk,η. The dispersion relations\nare plotted along the red dashed lines in the inset of Fig. 3(a).\nwhere JandDi jare the exchange constant of the Heisenberg\ninteraction between nearest-neighbor sites ⟨i j⟩and the DM\nvector for the bond ⟨⟨i j⟩⟩connecting next nearest-neighbor\nsites iandj, respectively. In this model, an S=1/2 spin at\nsiteiis represented by Si=(Sx\ni,Sy\ni,Sz\ni). Unlike the Kitaev\nmodel, the axes of the spin space are aligned with those in\nreal space (see Fig. 6). We assume that the Heisenberg inter-\naction is ferromagnetic ( J<0), and the DM vector is parallel\nto the zaxis,Di j=(0,0,±D), where the plus (minus) sign\nis assigned when the vector connecting from site itojis ori-\nented clockwise (anticlockwise) in a hexagon plaquette of the\nhoneycomb lattice (see Fig. 6). In the present calculations,\nwe choose D=0.3|J|. In this case, the classical ground state\nis a fully polarized ferromagnetic state and is degenerate for\nthe direction of the spin polarization [69]. Here, we introduce\nthe parameter θto denote a tilting angle of the spin moment\nfrom the zaxis to the xaxis (see the inset of Fig. 6). Note\nthat, atθ=0, the total spin operatorP\niSz\nicommutes with the\nHamiltonian, preventing the emergence of magnon-magnon\ninteractions without particle-number conservation in the non-\nlinear spin-wave theory [59].\nFigures 7(a) and 7(b) illustrate the dispersion relations of\nmagnons at θ=45◦andθ=80◦, respectively. In both cases,\nthe dispersion relations exhibit two branches, with the abso-\nlute value of the Berry curvature increasing significantly near\nthe K-M path, while approaching zero in the vicinity of the\nΓpoint. We have confirmed that the Chern numbers of low-\nenergy and high-energy branches are +1 and−1, respectively.\nWe also find that the gap between the two magnon bands nar-\nrows with increasing θ.\nFigure 8 displays the temperature dependence of κH\nxy/Tboth\nwith and without magnon-magnon interactions for θ=45◦\nand 80◦. Given that the Chern number for the lower-energy\nbranch is positive, κH\nxyexhibits negative values across a broad\ntemperature range. We find that the absolute value of κH\nxyfor\nθ=80◦is smaller than that for θ=45◦. This phenomenon\nis understood from the reduction of the magnon gap between\nthe two bands with increasing θ. As discussed in Sec. III D,\ndecreasing the magnon gap leads to suppressing the thermal\nHall conductivity.\nNext, we discuss the e ffect of the magnon-magnon inter-\naction on the thermal Hall conductivity. As shown in Fig. 8,\nthis e ffect slightly enhances the absolute value of κH\nxyat low13\n0.00 0.25 0.50\nT/|J|−0.2−0.10.00.10.2κH\nxy/Tθ=45◦(free)\nθ=80◦(free)θ=45◦(interacting)\nθ=80◦(interacting)\nFIG. 8. Temperature dependence of thermal Hall conductivity di-\nvided by temperature in the Heisenberg-DM model from a ferromag-\nnetic state with θ=45◦and 80◦. The dashed-dotted lines represent\nthe results for the free magnon system within the linear spin-wave ap-\nproximation. On the other hand, the solid lines represent the results\nfor the systems with magnon-magnon interactions calculated based\non the iDE approach.\ntemperatures but suppresses it at higher temperatures. This\nbehavior is expected from the simplified model with magnon\ndamping introduced in Sec. III D. To examine the contribu-\ntion of magnon-magnon interactions, we calculate the spec-\ntral function of magnons calculated by the iDE approach. The\ncolor map of the spectral function is presented in Fig. 9 at\nseveral temperatures. In both cases with θ=45◦and 80◦,\nmagnon damping occurs around the K point. Given that\nmagnon bands near this point possess large values of the Berry\ncurvature, it is found that magnon damping significantly influ-\nences the thermal Hall conductivity. It has been demonstrated\nthat a chiral edge mode is strongly damped by incorporating\nmagnon-magnon interactions in a cluster with open bound-\naries [69]. Our findings suggest that the damping of the chi-\nral edge mode corresponds to the significant contribution of\nmagnon-magnon interactions to the thermal Hall conductivity\nfound in the present study, which is similar to the case of the\nKitaev model introduced in the previous section.\nV . SUMMARY AND DISCUSSION\nIn summary, we have derived the expression for the ther-\nmal Hall conductivity incorporating the magnon damping in\nlocalized electron systems based on nonlinear spin-wave the-\nory, where magnons are introduced as elementary excitations\nfrom a magnetic order. We have formulated the thermal re-\nsponse by accounting for both the Kubo formula and heat\nmagnetization based on Green’s functions of magnons. The\neffect of the magnon damping is introduced as the imaginary\npart of the self-energy, which gives rise to the broadening\nof the magnon spectrum. The thermal Hall conductivity ob-\ntained in the present study reproduces the previous result of\nfree-magnon systems in the zero limit of the magnon damp-ing. Based on the expression of the thermal Hall conductivity,\nwe first discussed the impact of magnon damping in a sim-\nple magnon model with nonzero Chern numbers. We have\nfound that the magnon damping slightly enhances the ther-\nmal Hall conductivity at very low temperatures due to the in-\ncrease of the low-energy spectral weight of magnons result-\ning from the spectrum broadening. Meanwhile, the thermal\nHall conductivity is suppressed by the magnon damping at\nhigher temperatures by the cancellation of contributions from\nhigher-energy magnon branches with Berry curvatures taking\nopposite values, which is also caused by the broadening of the\nmagnon spectrum. We have also applied the present theory to\ntwo localized spin models on a honeycomb lattice: the Kitaev\nmodel under magnetic fields and the ferromagnetic Heisen-\nberg model with Dzyaloshinskii-Moriya interactions. In these\nmodels, the imaginary part of the self-energy, which arises\nfrom the magnon-magnon interactions beyond the linear spin-\nwave theory, has been evaluated by the self-consistent imagi-\nnary Dyson equation approach at finite temperatures. We have\nclarified that magnon damping substantially a ffects the tem-\nperature dependence of the thermal Hall conductivity in both\nsystems. In particular, in the presence of significant quan-\ntum fluctuations, the low-energy magnon branches largely de-\ncay, and the absolute value of the thermal Hall conductivity is\nstrongly reduced from the value obtained in the free-magnon\nsystem. We have found that such a substantial change of the\nthermal Hall conductivity occurs when a chiral edge mode is\nlargely damped, suggesting the presence of bulk-edge corre-\nspondence even in the presence of magnon-magnon interac-\ntions.\nSince our study begins with a general form of localized\nelectron systems with multiple local degrees of freedom, the\npresent results are easily applied to other models, such as lo-\ncalized systems with multipole interactions including spin-\norbital systems, spin dimer systems typified by the Shastry-\nSutherland model, skyrmion crystals, and localized electron\nsystems coupled with lattice vibrations. In this study, we have\nfocused on elucidating the impact of magnon damping by con-\nsidering only the imaginary part of the self-energy on the ther-\nmal Hall conductivity. On the other hand, this study has yet\nto incorporate the real part of the self-energy, which shifts the\nmagnon energy, and the e ffects of vertex correction. These\ncontributions will be addressed in future work. Additionally,\nformulating other thermal responses, exemplified by the spin\nNernst e ffect, remains challenging for future research.\nACKNOWLEDGMENTS\nThe authors thank S. Murakami, M. Zhitomirsky, A. Shi-\ntade, and A. Ono for fruitful discussions. Parts of the nu-\nmerical calculations were performed in the supercomput-\ning systems in ISSP, the University of Tokyo. This work\nwas supported by Grant-in-Aid for Scientific Research from\nJSPS, KAKENHI Grant No. JP19K03742, JP20H00122,\nJP22H01175, JP23H01129, and JP23H04865, and by JST,\nthe establishment of university fellowships towards the cre-\nation of science technology innovation, Grant Number JP-14\nΓ K M Γ0.01.02.03.0ω/|J|(a)θ=45◦,T/|J|=0\nΓ K M Γ0123ω(b)θ=45◦,T/|J|=0.25\nΓ K M Γ0123ω(c)θ=45◦,T/|J|=0.5\nΓ K M Γ0.01.02.03.0ω/|J|(d)θ=80◦,T/|J|=0\nΓ K M Γ0123ω(e)θ=80◦,T/|J|=0.25\nΓ K M Γ0123ω(f)θ=80◦,T/|J|=0.5\n012345\n˜ρk,η(ω)\nFIG. 9. (a)–(c) Color map of the spectral function ˜ ρk,η(ω) calculated by the iDE approach in the Heisenberg-DM model at (a) T/|J|=0, (b)\nT/|J|=0.25, (c) T/|J|=0.5 forθ=45◦. The cyan dashed-dotted lines stand for the dispersion relations obtained by the linear spin-wave\napproximation. (d)–(f) Corresponding results for θ=80◦. The spectral functions are plotted along the red dashed lines in the inset of Fig. 3(a).\nMJFS2102.\nAppendix A: Definition of heat magnetization\nWhen we calculate a response for a statistical force (e.g.,\nthermal gradient) applied to a system, we must pay attention\nto the impact of such a gradient on the rotational motion in-\ntrinsic to the wave packet of carriers [88]. In the present pa-\nper, we focus on magnon systems with zero chemical poten-\ntial, and hence, we omit the e ffect of the chemical-potential\ngradient and focus on the thermal transport. In the thermal\nHall e ffect, the contribution of heat magnetization originating\nfrom the rotational motion appears in addition to that evalu-\nated from the Kubo formula [89]. When the system is in equi-\nlibrium without thermal gradients, the thermal current density\nsatisfies∇·D\njQ(r)E\n=0. In this case, the rotational motion of\ncarriers only contributes to the thermal current density. Thus,\nwe define the heat magnetization density mQ(r) by [89],\nD\njQ(r)E\n=∇×mQ(r). (A1)\nIn a similar manner, the macroscopic transport thremal current\nJtrin the presence of the gravitational-field gradient ∇χis\nintroduced as\nJtr=1\nVZ\ndr\u0014D\njQ;χ(r)E\n∇χ−∇× mQ;χ(r)\u0015\n, (A2)\nwhere⟨·⟩∇χandmQ;χ(r) represent the expectation value and\nheat magnetization density in the presence of ∇χ. Up to the\nfirst order of∇χ, the first term of the above equation is writtenas\n1\nVZ\ndrD\njQ;χ(r)E\n∇χ≃D\nJQE\n∇χ+1\nVZ\ndrD\njQ;χ(r)E\n,(A3)\nwhere JQ=1\nVR\ndrjQ(r). The first term can be evaluated by\nthe Kubo formula as\nD\nJQE\n∇χ≃X\nλ′Sλλ′(−∇λ′χ). (A4)\nOn the other hand,D\njQ;χ(r)E\nin the second term is calculated\nas\nD\njQ;χ(r)E\n≃∇× mQ;χ(r)−2mQ(r)×∇χ, (A5)\nup to the first order of χ. From these expressions, we obtain\nEqs. (45) and (46). Here, the total heat magnetization is intro-\nduced as MQ=R\nmQ(r)dr. This quantity is evaluated from\nEqs. (49) and (50), which are derived from Eq. (A1) [89].\nAppendix B: Thermal current operator and scaling low\n1. Thermal current operator\nIn this section, we derive the representation of the thermal\ncurrent density operator given in Eq. (55) from the bilinear\nbosonic Hamiltonian H0in Eq. (15). We start from the fol-\nlowing equation of continuity in a continuum limit:\n∂hχ(r)\n∂t=−i\nℏ\u0002hχ(r),Hχ\u0003=−∇·jQ;χ(r). (B1)15\nwhere hχ(r),Hχ, andjQ;χ(r) are the local Hamiltonian, total\nHamiltonian, and thermal current density in the presence of\nthe gravitational field χ(r), which are defined as follows. In\nthe continuum limit, the bosonic Hamiltonian H0without the\ngravitational field is represented as [17]\nH0=1\n2X\nδZ\ndrA†(r)MδA(r+δ)\n=1\n2Z\ndrA†(r)ˆM0A(r) (B2)\nwhere ˆM0is given by\nˆM0=X\nδMδeiˆp·δ/ℏ, (B3)\nHere,Mδis a 2 N×2NHermitian matrix depending on δ\nowing to the translational symmetry and represented as\nMδ=M11\nδM12\nδ \u0010\nM12\nδ\u0011∗\u0010\nM11\n−δ\u0011T. (B4)\nThis corresponds to Eq. (16) for lattice systems. Since ˆM0\nis a Hermitian matrix, Mδsatisfies the relation M†\nδ=M−δ.\nWe also introduce A(r) as a set of the 2 Nbosonic operators,\nwhich is given by\nAs(r)=as(r) ( s=1,···,N)\na†\ns−N(r) (s=N+1,···,2N)(B5)where as(r) and a†\ns(r) are annihilation and creation operators\nsatisfying the commutation relations such ash\na†\ns(r),as′(r′)i\n=\nδss′δ(r−r′). Moreover, the operator ˆ pin Eq. (B3) is defined\nas a generator of translation for the bosonic operator As(r),\nwhich satisfies the relation eiˆp·δ/ℏAs(r)=As(r+δ). From\nEq. (B2), the local Hamiltonian for H0can be written as\nh(r)=1\n2A†(r)ˆM0A(r). (B6)\nIn a similar manner, the local Hamiltonian hχ(r) in the\npresence of the gravitational field, which satisfies Hχ= R\ndrhχ(r), is represented as,\nhχ(r)=h(r)+1\n2[χ(r)h(r)+h(r)χ(r)]\n≃1\n2˜A†(r)ˆM0˜A(r), (B7)\nwhere ˜A(r)=h\n1+χ(r)\n2i\nA(r). Note that we apply sym-\nmetrization for χ(r) and h(r) because these do not commute\ndue to the operator ˆ pinh(r) in the continuum limit [17]. Us-\ning the local Hamiltonian, we introduce the energy polariza-\ntionPχ\nEas\nPχ\nE=1\n2Z\ndr\u0002rhχ(r)+hχ(r)r\u0003. (B8)\nFrom Eq. (41), we calculate the thermal current as follows:\nJQ;χ=i\nVℏ\u0002Hχ,PE\u0003=1\n4VZ\ndr˜A†(r)ˆvσ3\"\n1+χ(r)\n2#2\nˆM0+ˆM0\"\n1+χ(r)\n2#2\nσ3ˆv˜A(r), (B9)\nwhere we use the following relations (Mδ)m,n=(M−δ)n+N,m+Nand(Mδ)m,n+N=(M−δ)m+N,n, corresponding to Eqs. (12) and\n(13), andAm(r)=A†\nm+N(r) andA†\nm(r)=Am+N(r) for n,m=1,2,···,N. The velocity ˆ vis given by\nˆv=−i\nℏh\nr,ˆM0i\n=i\nℏX\nδδMδeiˆp·δ/ℏ. (B10)\nThus, the thermal current densities in the absence and presence of χare expressed by\njQ(r)=1\n4A†(r)\u0010\nˆvσ3ˆM0+ˆM0σ3ˆv\u0011\nA(r), (B11)\njQ;χ(r)=1\n4A†(r)\"\n1+χ(r)\n2#ˆvσ3\"\n1+χ(r)\n2#2\nˆM0+ˆM0\"\n1+χ(r)\n2#2\nσ3ˆv\"\n1+χ(r)\n2#\nA(r), (B12)\nrespectively. Using the relation χ(r)=r·∇χ(r) with∇χbeing constant, we expand Eq. (B12) with respect to ∇χas\njQ;χ(r)=jQ(r)+jQ\n∇χ(r). (B13)\nHere,jQ\n∇χ(r) is the term proportional to ∇χinjQ;χ, which is represented as\njQ\n∇χ(r)=−iℏ\n8X\nλ(∇λχ)A†(r)(ˆvσ3ˆvλ−ˆvλσ3ˆv)A(r)\n+1\n8X\nλ(∇λχ)h\nA†(r)(rλˆvσ3+3 ˆvσ3rλ)ˆM0A(r)+A†(r)ˆM0(3rλσ3ˆv+σ3ˆvrλ)A(r)i\n. (B14)16\n2. Scaling law for thermal current operator\nTo evaluate the heat magnetization MQ\nz, we impose the scaling relation given in Eq. (53) for the current density operator.\nSimilar to Eq. (B7), the scaling relation is applied as the following symmetric form in the continuum limit:\njQ;χ(r)=1\n2n\u00021+χ(r)\u00032jQ(r)+jQ(r)\u00021+χ(r)\u00032o\n(B15)\nFrom Eq. (B1), the equation of continuity is invariant under the gauge transformation inherent in the thermal current density as\nfollows:\njQ→jQ+∇×f(r), (B16)\nwhere f(r) is an arbitrary vector function. Using the redundant degrees of freedom, we determine the expression of thermal\ncurrent density so as to satisfy the scaling relation given in Eq. (B15). Here, by applying r·∇χ=χ(r) to the second term of\nEq. (B14), we can rewrite jQ;χ(r) up to the first order of χas\njQ;χ(r)=\u00021+χ(r)\u00032jQ(r)+jQ(r)\u00021+χ(r)\u00032\n2−\u00021+χ(r)\u00032[∇×Λ(r)]+[∇×Λ(r)]\u00021+χ(r)\u00032\n2\n+∇×\u00021+χ(r)\u00032Λ(r)+Λ(r)\u00021+χ(r)\u00032\n2, (B17)\nwhere Λ(r) is given by\nΛ(r)=ℏ\n16iA†(r)(ˆv×σ3ˆv)A(r). (B18)\nBy redefining jQ;χ(r)−∇×n\u00021+χ(r)\u00032Λ(r)+Λ(r)\u00021+χ(r)\u00032o\n/2 and jQ(r)−∇× Λ(r) asjQ;χ(r) andjQ(r) respectively\nin Eq. (B17), we find that the new thermal current operators satisfy the scaling relation given in Eq. (B15). Thus, the thermal\ncurrent density is written as\njQ(r)=1\n4A†(r)\u0010\nˆvσ3ˆM0+ˆM0σ3ˆv\u0011\nA(r)−ℏ\n16iX\nλ∇λh\nA†(r)(ˆvσ3ˆvλ−ˆvλσ3ˆv)A(r)i\n. (B19)\nFinally, we obtain Eq. (55) by introducing the Fourier transformations of Mδ,jQ(r), andA(r) as,\nMq=X\nδMδeiq·δ,jQ\nq=Z\ndrjQ(r)e−iq·r,Aq=Z\ndrA(r)e−iq·r. (B20)\nAppendix C: Expression of transport coe fficient\n1. Expression of Sxy\nIn this section, we present the detailed derivation of Sλλ′given in Eq. (57). The current-current correlation in Eq. (48) is\nwritten by the sum of four products of the bosonic operators AandA†by using the expression of JQin Eq. (56). We apply the\nfollowing decouplings to them, which corresponds to neglecting vertex corrections:\nPλλ′(iΩ)≃−1\n16VZβ\n0dτeiΩτ2NX\ns1s2s3s4=1X\nkk′\u0000Xk,λ\u0001\ns1s2\u0000Xk′,λ′\u0001\ns3s4\"D\nTτA†\nk,s1(τ)Ak′,s4ED\nTτAk,s2(τ)A†\nk′,s3E\n+D\nTτA†\nk,s1(τ)A†\nk′,s3ED\nTτAk,s2(τ)Ak′,s4E#\n, (C1)\nwhere we introduce Xk,λ=vk,xσ3Mk+Mkσ3vk,λ. The bosonic operators AandA†are written by using the Bogoliubov\nbosons given in Eq. (22) as\nAk,s=NX\nη=1\u0010\nTk\u0011\nsηbk,η+NX\nη=1\u0010\nTk\u0011\ns,η+Nb†\n−k,η, (C2)\nA†\nk,s=NX\nη=1\u0010\nT†\nk\u0011\nηsb†\nk,η+NX\nη=1\u0010\nT†\nk\u0011\nη+N,sb−k,η. (C3)17\nBy neglecting the o ff-diagonal part of the temperature Green’s function for ηin Eq. (24), we obtain the following form:\nPλλ′(iΩ)≃−1\n8VZβ\n02NX\nη,η′=1X\nk\u0010\nT†\nkXλ\nkTk\u0011\nηη′\u0010\nT†\nkXλ′\nkTk\u0011\nη′ηGk,η(−τ)Gk,η′(τ)\n=−kBT\n8V∞X\nn2NX\nη,η′=1X\nk\u0010\nT†\nkXx\nkTk\u0011\nηη′\u0010\nT†\nkXy\nkTk\u0011\nη′ηGk,η(iωn−iΩ)Gk,η′(iωn) (C4)\nThe Matsubara sum can be taken by performing the integrals along the three contours shown in Fig. 10 on the complex plane [92].\nCarrying out the analytic continuation for the Matsubara frequency, we obtain the retarded correlation function between thermal\ncurrents as\nPR\nλλ′(Ω)≃−1\n8V2NX\nη,η′=1X\nk\u0010\nT†\nkXk,λTk\u0011\nηη′\u0010\nT†\nkXk,λ′Tk\u0011\nη′η\n×PZ∞\n−∞dω\n2πig(βω)h\nGR\nk,η(ω)GR\nk,η′(ω+ℏΩ)−GA\nk,η(ω)GR\nk,η′(ω+ℏΩ)+GA\nk,η(ω−ℏΩ)GR\nk,η′(ω)−GA\nk,η(ω−ℏΩ)GA\nk,η′(ω)i\n=−1\n8V2NX\nη,η′=1X\nk\u0010\nT†\nkvk,λTk\u0011\nηη′\u0010\nT†\nkvk,λ′Tk\u0011\nη′η\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u00112\n×PZ∞\n−∞dω\n2πig(βω)h\nGR\nk,η(ω)GR\nk,η′(ω+ℏΩ)−GA\nk,η(ω)GR\nk,η′(ω+ℏΩ)+GA\nk,η(ω−ℏΩ)GR\nk,η′(ω)−GA\nk,η(ω−ℏΩ)GA\nk,η′(ω)i\n,\n(C5)\nwhere we use the relation\u0010\nT†\nkXk,λTk\u0011\nηη′=\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u0011\u0010\nT†\nkvk,λTk\u0011\nηη′, which is calculated from T†\nkMkσ3=Ekσ3T†\nk.\nFinally, substituting the above equation to Eq. (47), we obtain Eq. (57).\niΩ\nRe zIm z\nFIG. 10. Paths of the contour integrals for F(z)=g(βz)Gk,η(z−iΩ)Gk,η′(z). The horizontal dashed line represents Im z= Ω.18\n2. Expression of ˜MQ\nz\nNext, we derive the expression of ˜MQ\nzin Eq. (58). By carrying out calculations similar to the procedure obtaining Pλλ′(iΩ) for\n⟨h−q;jq,λ⟩with Eqs. (54) and (55), we obtain the following result:\nD\nh−q;jq,λE\n≃1\n4β2NX\nη,η′=1X\nk \nT†\nkMk+Mk−q\n2Tk−q!\nηη′\u0010\nT†\nk−qYk−q,k,λTk\u0011\nη′η\n×PZ∞\n−∞dω\nπg(βω)(\nImh\nGR\nk,η(ω)i\nGR\nk−q,η′(ω)+GA\nk,η(ω)Imh\nGR\nk−q,η′(ω)i)\n, (C6)\nwhere Yk−q,k,λis defined as\nYk−q,k,λ=\u0010\nvk−q,λσ3Mk+Mk−qσ3vk,λ\u0011\n−1\n4X\nλ′ℏqλ′\u0010\nvk−q,λσ3vk,λ′−vk−q,λ′σ3vk,λ\u0011\n. (C7)\nSubstituting this expression to Eq. (50), we obtain Eq. (58).\nAppendix D: Evaluation of thermal Hall conductivity\n1. Calculation of Sλλ′\nIn this section, we derive Eq. (64). By substituting Eqs. (60) and (61) to Eq. (57), Sλλ′is written as\nSλλ′≃−iℏ\n8V2NX\nη,η′=1X\nkσ3,ησ3,η′\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u00112\u0010\nT†\nkvk,λTk\u0011\nηη′\u0010\nT†\nkvk,λ′Tk\u0011\nη′η\n×PZ∞\n−∞dωσ3,ηρk,η(ω)g(βω)\n\u0010\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)\u00112−σ3,η′ρk,η′(ω)g(βω)\n\u0010\nω−σ3,ηEk,η−iσ3,ηΓk,η(ω)\u00112\n=ℏ\n4V2NX\nη,η′=1X\nkσ3,ησ3,η′\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u00112Im\u0010\nT†\nkvk,λTk\u0011\nηη′\u0010\nT†\nkvk,λ′Tk\u0011\nη′ηPZ∞\n−∞dωσ3,ηρk,η(ω)g(βω)\n\u0010\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)\u00112.\n(D1)\nHere, we introduce the symmetric and antisymmetic parts of the above expression with respect to ( λ,λ′) as Ss\nλλ′andSa\nλλ′,\nrespectively, which are represented as\nSs\nλλ′=ℏ\n4V2NX\nη,η′=1X\nkσ3,ησ3,η′\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u00112\n×Re\u0014\u0010\nT†\nkvk,λTk\u0011\nηη′\u0010\nT†\nkvk,λ′Tk\u0011\nη′η\u0015\nImPZ∞\n−∞dωσ3,ηρk,η(ω)g(βω)\n\u0010\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)\u00112, (D2)\nSa\nλλ′=ℏ\n4V2NX\nη,η′=1X\nkσ3,ησ3,η′\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u00112\n×Im\u0014\u0010\nT†\nkvk,λTk\u0011\nηη′\u0010\nT†\nkvk,λ′Tk\u0011\nη′η\u0015\nRePZ∞\n−∞dωσ3,ηρk,η(ω)g(βω)\n\u0010\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)\u00112. (D3)19\nAs we calculate thermal Hall conductivity, we focus on the antisymmetric part Sa\nλλ′. Using ˜Ωλ\nk,ηη′introduced in Eq. (66), we\nrewrite Eq. (D3) as\nSa\nλλ′=1\n4ℏVX\nλ′′ελλ′λ′′2NX\nη,η′=1X\nk˜Ωλ′′\nk,ηη′\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u00112\u0010\nσ3,ηEk,η−σ3,η′Ek,η′\u00112\n×RePZ∞\n−∞dωσ3,ηρk,η(ω)g(βω)\n\u0010\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)\u00112\n=1\n4ℏVX\nλ′′ελλ′λ′′NX\nη=12NX\nη′=1X\nk˜Ωλ′′\nk,ηη′\u0010\nεk,η+σ3,η′Ek,η′\u00112\u0010\nεk,η−σ3,η′Ek,η′\u00112\n×RePZ∞\n−∞dωρk,η(ω)2g(βω)+1\n\u0010\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)\u00112, (D4)\nwhich is the same as Eq. (64). Here, we use ρ−k,η+N(−ω)=−ρk,η(ω) and ˜Ωλ\n−k,η+N,η′+N=−˜Ωλ\nk,ηη′forη,η′=1,···,Nand the\nrelation for η,η′as follows:\n\u0010\nT†\nkvk,λTk\u0011\nηη′=1\nℏ\u0010\nσ3,η′Ek,η′−σ3,ηEk,η\u0011 \nT†\nkσ3∂Tk\n∂kλ!\nηη′. (D5)\n2. Calculation of MQ\nλ\nIn this section, we derive the expression of MQ\nλin Eq. (65). Similar to the previous section, we substitute the Green’s functions\ngiven in Eqs. (60) and (61) to Eq. (58), and thereby, we obtain the following form:\n˜MQ\nλ≃1\n16iX\nλ′λ′′ελλ′λ′′∂\n∂qλ′′2NX\nη,η′=1X\nkσ3,ησ3,η′h\nT†\nk\u0010\nMk+Mk−q\u0011\nTk−qi\nηη′\u0010\nTk−qYk−q,k,xTk\u0011\nη′η\n×PZ∞\n−∞dω\"σ3,ηρk,η(ω)g(βω)\nω−σ3,η′Ek−q,η′+iσ3,η′Γk−q,η′(ω)+σ3,η′ρk−q,η′(ω)g(βω)\nω−σ3,ηEk,η−iσ3,ηΓk,η(ω)#\f\f\f\f\f\fq→0. (D6)\nHere, we divide the above expression into two parts for η,η′andη=η′, which are defined as ˜MQ;inter\nλand ˜MQ;intra\nλ, respectively.\nFirst, we focus on ˜MQ;inter\nλ. This is calculated as\n˜MQ;inter\nλ=1\n16iX\nλ′λ′′ελλ′λ′′∂\n∂qλ′′2NX\nη,η′X\nkσ3,ησ3,η′h\nT†\nk\u0010\nMk+Mk−q\u0011\nTk−qi\nηη′\u0010\nTk−qYk−q,k,xTk\u0011\nη′η\n×PZ∞\n−∞dω\"σ3,ηρk,η(ω)g(βω)\nω−σ3,η′Ek−q,η′+iσ3,η′Γk−q,η′(ω)+σ3,η′ρk−q,η′(ω)g(βω)\nω−σ3,ηEk,η−iσ3,ηΓk,η(ω)#\f\f\f\f\f\fq→0\n≃−1\n16iX\nλ′λ′′ελλ′λ′′2NX\nη,η′X\nkσ3,ησ3,η′\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u00112 \nT†\nkσ3∂Tk\n∂kλ′′!\nηη′\u0010\nT†\nkvk,λ′Tk\u0011\nη′η\n×PZ∞\n−∞dω\"σ3,ηρk,η(ω)g(βω)\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)+σ3,η′ρk,η′(ω)g(βω)\nω−σ3,ηEk,η−iσ3,ηΓk,η(ω)#\n=−1\n4ℏ2NX\nη,η′=1X\nk˜Ωλ\nk,ηη′\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u00112\u0010\nσ3,ηEk,η−σ3,η′Ek,η′\u0011\nRe\"\nPZ∞\n−∞dωσ3,ηρk,η(ω)g(βω)\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)#\n,\n(D7)\nwhere we neglect the first derivative of Γk,η(ω) with respect to kand use the relations\u0010\nT†\nkMkTk\u0011\nηη′=0 and\u0010\nT†\nkvk,λTk\u0011\nηη′=\n1\nℏ\u0010\nσ3,η′Ek,η′−σ3,ηEk,η\u0011\u0012\nT†\nkσ3∂Tk\n∂kλ\u0013\nηη′forη,η′.20\nNext, we calculate ˜MQ;intra\nλfor the case with η=η′as follows:\n˜MQ;intra\nλ=1\n16iX\nλ′λ′′ελλ′λ′′∂\n∂qλ′′2NX\nη=1X\nkh\nT†\nk\u0010\nMk+Mk−q\u0011\nTk−qi\nηη\u0010\nTk−qYk−q,k,λ′Tk\u0011\nηη\n×PZ∞\n−∞dω\"σ3,ηρk,η(ω)g(βω)\nω−σ3,ηEk−q,η+iσ3,ηΓk−q,η(ω)+σ3,ηρk−q,η(ω)g(βω)\nω−σ3,ηEk,η−iσ3,ηΓk,η(ω)#\f\f\f\f\f\fq→0\n=−1\n4X\nλ′λ′′ελλ′λ′′2NX\nη=1X\nkE2\nk,ηIm \nT†\nkσ3∂Tk\n∂kλ′′!\nηη\u0010\nT†\nkvk,λ′Tk\u0011\nηηRe\"\nPZ∞\n−∞dω2σ3,ηρk,η(ω)g(βω)\nω−σ3,ηEk,η+iσ3,ηΓk,η(ω)#\n−1\n8X\nλ′λ′′ελλ′λ′′2NX\nη=1X\nkEk,ηIm∂T†\nk\n∂kλ′′\u0010\nMkσ3vk,λ′+vk,λ′σ3Mk\u0011\nTk\nηηRe\"\nPZ∞\n−∞dω2σ3,ηρk,η(ω)g(βω)\nω−σ3,ηEk,η+iσ3,ηΓk,η(ω)#\n−ℏ\n16X\nλ′λ′′ελλ′λ′′2NX\nη=1X\nkEk,ηIm\u0014\u0010\nTkvk,λ′′σ3vk,λ′Tk\u0011\nηη\u0015\nRe\"\nPZ∞\n−∞dω2σ3,ηρk,η(ω)g(βω)\nω−σ3,ηEk,η+iσ3,ηΓk,η(ω)#\n. (D8)\nUsing the relation σ3=T†\nkσ3Tk=Tkσ3T†\nkand Eq. (D5), we rewrite the above form as\n˜MQ;intra\nλ=−1\n16ℏX\nλ′λ′′ελλ′λ′′2NX\nη,η′=1σ3,ηEk,ηImσ3∂T†\nk\n∂kλ′σ3Tk\nηη′\u0014\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u00112−4σ3,ηE2\nk,η\u0015 \nσ3T†\nkσ3∂Tk\n∂kλ′′!\nη′η\n×Re\"\nPZ∞\n−∞dω2σ3,ηρk,η(ω)g(βω)\nω−σ3,ηEk,η+iσ3,ηΓk,η(ω)#\n. (D9)\nBy neglecting the ω-derivative of Γk,η(ω), theω-integral in the above equation is approximated as\nRe\"\nPZ∞\n−∞dω2σ3,ηρk,η(ω)g(βω)\nω−σ3,ηEk,η+iσ3,ηΓk,η(ω)#\n≃−PZ∞\n−∞dωσ 3,η∂ρk,η\n∂ωg(βω)=PZ∞\n−∞dωσ 3,ηρk,η(ω)∂g\n∂ω. (D10)\nFinally, ˜MQ;intra\nλis represented as\n˜MQ;intra\nλ≃−1\n8ℏ2NX\nη,η′=1X\nk˜Ωλ\nk,ηη′σ3,ηEk,η\u0014\u0010\nσ3,ηEk,η+σ3,η′Ek,η′\u00112−4\u0010\nσ3,ηEk,η\u00112\u0015\nPZ∞\n−∞dωσ 3,ηρk,η(ω)∂g\n∂ω, (D11)\nwhere the temperature derivative of Γk,η(ω) is neglected. From Eqs.(D7) and (D11), we obtain ˜MQ\nλ, and MQ\nλin Eq. (65) is\nderived by solving the di fferential equation in Eq. (49) under the boundary condition lim β→∞β∂MQ\n∂β=0.\n3. Calculation of κH\nλλ′\nIn this section, we derive the expression of the thermal Hall conductivity in Eq. (68). First, we divide Eq (65) into to parts as\nMQ\nλ≃MQ;inter\nλ+MQ;intra\nλ, which are defined by\n2MQ;inter\nλ\nV=−1\n2β2ℏVNX\nη=12NX\nη′=1X\nk˜Ωλ\nk,ηη′\u0010\nεk,η+σ3,η′Ek,η′\u00112\u0010\nεk,η−σ3,η′Ek,η′\u0011\n×Re\"\nPZ∞\n−∞dωρk,η(ω)\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)#Zβ\n0˜βh\n2g(˜βω)+1i\nd˜β\n−1\n2β2ℏVNX\nη=12NX\nη′=1X\nk˜Ωλ\nk,ηη′\u0010\nεk,η+σ3,η′Ek,η′\u00112PZ∞\n−∞dωρk,η(ω)εk,η\nωZβ\n0˜βω∂g\n∂ωd˜β, (D12)\nand\n2MQ;intra\nλ\nV=−1\nβ2ℏVNX\nη=1X\nkΩλ\nk,ηε3\nk,ηPZ∞\n−∞dωρk,η(ω)Zβ\n0˜β∂g\n∂ωd˜β, (D13)21\nwhere we use the following relation for the Berry curvature given in Eq. (70) [28]:\nΩλ\nk,η=−X\nλ′λ′′ελλ′λ′′Imσ3∂T†\nk\n∂kλ′σ3∂Tk\n∂kλ′′\nηη=−ℏ2X\nλ′λ′′ελλ′λ′′2NX\nη′(,η)σ3,ησ3,η′Im\u0014\u0010\nT†\nkvk,λ′Tk\u0011\nηη′\u0010\nT†\nkvk,λ′′Tk\u0011\nη′η\u0015\n\u0010\nσ3,ηEk,η−σ3,η′Ek,η′\u00112=−22NX\nη��(,η)˜Ωλ\nk,ηη′.\n(D14)\nWe carry out the temperature integrals in Eqs. (D12) and (D13) using the following relations:\nZβ\n0˜βg(˜βω)d˜β=1\n2β2g(βω)−1\n2ω2˜c2(g(βω)),Zβ\n0˜βω∂g\n∂ωd˜β=1\nω2˜c2(g(βω)). (D15)\nThen, the thermal transport coe fficients, La;inter\nλλ′=Sa\nλλ′+P\nλ′′ελλ′λ′′2MQ;inter\nλ′′\nVandLa;intra\nλλ′=P\nλ′′ελλ′λ′′2MQ;intra\nλ′′\nV, where La\nλλ′is written\nasLa\nλλ′≃La;inter\nλλ′+La;intra\nλλ′, are calculated as\nLa;inter\nλλ′=−1\n2β2ℏVX\nλ′′ελλ′λ′′NX\nη=12NX\nη′=1X\nk˜Ωλ′′\nk,ηη′\u0010\nεk,η+σ3,η′Ek,η′\u00112\u0010\nεk,η−σ3,η′Ek,η′\u0011\n×Re(\nPZ∞\n−∞dωρk,η(ω)\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)\"\n1−εk,η−σ3,η′Ek,η′\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)#)(\nβ2g(βω)+β2\n2−1\nω2˜c2(g(βω)))\n−1\n2β2ℏVX\nλ′′ελλ′λ′′NX\nη=12NX\nη′=1X\nk˜Ωλ′′\nk,ηη′\u0010\nεk,η+σ3,η′Ek,η′\u00112\n×RePZ∞\n−∞dωρk,η(ω)\nω2εk,η\nω−\u0010\nεk,η−σ3,η′Ek,η′\u00112\n\u0010\nω−σ3,η′Ek,η′+iσ3,η′Γk,η′(ω)\u00112˜c2(g(βω)), (D16)\nand\nLa;intra\nλλ′=−1\nβ2ℏVX\nλ′′ελλ′λ′′NX\nη=1X\nkΩλ′′\nk,ηPZ∞\n−∞dωε3\nk,η\nω3ρk,η(ω)˜c2(g(βω)), (D17)\nrespectively.\nAs discussed in Sec. III C, we assume that the damping rate of magnons is small enough in the vicinity of the zero energy. This\nimplies that the contribution at ω≃εk,η, corresponding to the peak of ρk,η(ω), are dominant in the ωintegrals of Eqs. (D16) and\n(D17). Thus, we approximate ωappearing explicitly in these equations to ω≃εk,η. Furthermore, we incorporate contributions\nup to the first order of Γk,η(ω) into the thermal Hall conductivity. Since the spectral function given in Eq. (39) does not contain\nzeroth order contributions of Γk,η(ω), we apply the following approximation [ εk,η−σ3,η′Ek,η′+iσ3,η′Γk,η′]−x≃[εk,η−σ3,η′Ek,η′]−x\nforx=1,2 in Eqs. (D16) and (D17). Using the above approximations, we find La;inter\nλλ′≃0 and the thermal Hall conductivity is\nrepresented as\nκH\nλλ′≃La;intra\nλλ′\nT≃−k2\nBT\nℏVX\nλ′′NX\nη=1X\nkελλ′λ′′Ωλ′′\nk,ηZ∞\n−∞dωρk,η(ω)\"\nc2(g(βω))−π2\n3#\n. (D18)\nUsing the sum rule for the spectral function given in Eq. 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Goossens1\n1Centre for Plasma Astrophysics, Department of Mathematics , Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3 001 Leuven,\nBelgium\n2Departament de F´ ısica, Universitat de les Illes Balears, E -07122, Palma de Mallorca, Spain and\n3School of Computing, Engineering and Information Sciences , Northumbria University, Newcastle Upon Tyne, NE1 8ST, Eng land, UK\nDraft version August 29, 2018\nABSTRACT\nIt has been shown that resonant absorption is a robust physical m echanism to explain the observed\ndamping of magnetohydrodynamic (MHD) kink waves in the solar atmo sphere due to naturally oc-\ncurring plasma inhomogeneity in the direction transverse to the dire ction of the magnetic field. Theo-\nretical studies of this damping mechanism were greatly inspired by th e first observations of post-flare\nstanding kink modes in coronal loops using the Transition Region And Coronal E xplorer (TRACE).\nMore recently, these studies have been extended to explain the at tenuation of propagating coronal\nkink waves observed by the Coronal Multi-Channel Polarimeter (Co MP). In the present study, for the\nfirst time we investigate the properties of propagating kink waves in solar waveguides including the\neffects of bothlongitudinal and transverse plasma inhomogeneity. Importantly, it is found that the\nwavelength is only dependent on the longitudinal stratification and t he amplitude is simply a product\nof the two effects. In light of these results the advancement of so lar atmospheric magnetoseismology\nby exploiting high spatial/temporal resolution observations of prop agating kink waves in magnetic\nwaveguides to determine the length scales of the plasma inhomogene ity along and transverse to the\ndirection of the magnetic field is discussed.\nSubject headings: Sun: oscillations — Sun: corona — Sun: atmosphere — Magnetohydro dynamics\n(MHD) — Waves\n1.INTRODUCTION\nResonant absorption, caused by plasma inhomogene-\nity in the direction transverse to the magnetic field, has\nproved to be the most likely candidate for explaining the\nobserved attenuation of magnetohydrodynamic (MHD)\nkink waves in the solar atmosphere (see Goossens et al.\n2006; Goossens 2008, for a review about this damp-\ning mechanism). E.g., resonant absorption is a feasi-\nble explanation for the damping of kink MHD waves\nin both coronal loops (Goossens et al. 2002) and in the\nfine structure of solar prominences (see, e.g., Oliver\n2009; Arregui & Ballester 2010, and references therein).\nMost of theoretical studies of resonant absorption as-\nsume that the inhomogeneity is in the radial direction\nexclusively, and only a few works (e.g., Andries et al.\n2005; Arregui et al. 2005; Dymova & Ruderman 2006;\nAndries et al. 2009a; Soler et al. 2010) have studied the\nproperties of resonantly damped standing kink waves\nwhen plasma inhomogeneity is also present in the di-\nrection of the magnetic field. These studies have been\nrestricted to standing waves, so the amplitude of the\nstanding wave is damped in time as a result of the reso-\nnance. Nevertheless, there are observational indications\nof spatial damping of propagating waves along differ-\nent waveguides in the solar atmosphere. Some clear\nexamples have been reported by Tomczyk et al. (2007)\nand Tomczyk & McIntosh (2009) in coronal loops us-\ning Coronal Multi-Channel Polarimeter (CoMP) data.\nOther observational evidence of propagating kink waves\nin wave guides of the solar atmosphere are, e.g.,\nroberto.soler@wis.kuleuven.beDe Pontieu et al. (2007); He et al. (2009a,b) in spicules,\nOkamoto et al. (2007) in the fine structure of promi-\nnences, and Lin et al. (2007, 2009) in filament threads.\nThe theoretical modeling of the spatial damping of trav-\neling kink waves due to resonant absorption has been\ncarried out by Terradas et al. (2010) and Verth et al.\n(2010) analytically, and by Pascoe et al. (2010, 2011)\nnumerically. All these works assumed a homogeneous\ndensity in the longitudinal direction. Terradas et al.\n(2010) showed that the damping length by resonant\nabsorption is inversely proportional to the wave fre-\nquency. It was predicted that high-frequency waves be-\ncomedamped inlengthscalessmallerthanlow-frequency\nwaves. Verth et al. (2010) showed that this theory\nis consistent with the CoMP observations reported by\nTomczyk et al. (2007) and Tomczyk & McIntosh (2009).\nThis means that in solar waveguides resonant absorption\nacts as a natural low-pass filter.\nSome recent investigations have extended the analyti-\ncal work of Terradas et al. (2010) by adding effects not\nconsidered in their original paper. Soler et al. (2011a)\ntook into account the effect of partial ionization in the\nsingle-fluid approximation, so kink waves are damped\nby both resonant absorption and ion-neutral collisions in\ntheir configuration. Soler et al. (2011a) concluded that\nin thin tubes the effect of resonant absorption domi-\nnates, recovering the result of Terradas et al. (2010) in\nthe fully ionized case that the damping length is in-\nversely proportional to the frequency. Therefore, the re-\nsult of Soler et al. (2011a) allows the theory developed\nby Terradas et al. (2010) to be applied to kink waves in\npartially ionized plasmas as, e.g., chromospheric spicules2\nand solar prominences. Alternately, Soler et al. (2011b)\nhave shown that in the presence of flow the damping\nlength remains inversely proportional to the frequency,\nbut the factor of proportionality is different for forward\nand backward propagating waves to the flow direction\nand depends on the characteristics of the flow. However,\nnone of these previous works investigated the effect of\nlongitudinal stratification.\nThe analysis of kink MHD waves in longitudinally ho-\nmogeneous models make some problems more tractable\nfrom an analytical point of view. Realistic models of\nsolar waveguides should include longitudinal stratifica-\ntion. E.g., coronal loops can extend out into the atmo-\nsphere up to heights of the order of several density scale\nheights. This means that the differences in the density\nat their footpoints and apex can be significant. There\nis evidence of such stratification in coronal loops both\nfrom emission measure analysis (e.g., Aschwanden et al.\n1999) and coronal magnetoseismology(see Andries et al.\n2009b, forreviewofthistopic). Regardingspicules, there\nhavebeen spectroscopicline intensity studies to showthe\nplasma density is longitudinally strongly stratified (e.g.,\nBeckers 1968; Makita 2003) and more recently by mag-\nnetoseismology (Verth et al. 2011). It is also clear from\nHαobservations (e.g., Lin et al. 2008) that prominence\nthreads too exhibit longitudinal density inhomogeneity.\nForthese reasons,the main aim ofthe presentpaper is to\ninvestigate the effect of longitudinal stratification on the\npropertiesofresonantlydamped propagatingkinkwaves.\nThis is an interesting problem from a theoretical, as well\nasobservationalpointofview. Thereasonisthataradial\nvariation of density causes damping due to resonant ab-\nsorption while a longitudinal variation might cause an\nincrease of the amplitude. An increase of the ampli-\ntude due to stratification along the magnetic field direc-\ntion has been obtained by, e.g., De Moortel et al. (1999)\nfor phase mixed Alfv´ en waves and De Moortel & Hood\n(2004) for propagating slow modes.\nIn this paper, we investigate the effect of longitudinal\ndensity variation on propagating kink waves by using\nboth analytical and numerical approaches. Analytical\ntheory is used to the largest possible extent. To do so,\nweapplystandardapproximationsusuallyadoptedinthe\npreceding literature for the investigationof kink wavesin\nmagnetic tubes. The approximations used in the analyt-\nical part of this paper are briefly discussed and justified\nin the next two paragraphs. In addition to the analyti-\ncal theory, the problem is solved numerically beyond the\nlimitations of the analytical approximations. The com-\nparison between the analytical results and those from\nthe full numerical simulations will enable us to check the\nvalidity of the analytical theory.\nOur waveguide model is a straight cylindrical mag-\nnetic flux tube, inhomogeneous in both the radial and\nlongitudinal directions, and embedded in a magnetized\nenvironment. To investigate propagating kink waves an-\nalytically we adopt the following approximations. We\nuse the thin tube (TT) approximation, so that the wave-\nlength,λ, is much larger than the radius of the tube, R.\nThis condition is easily fulfilled for propagating waves\nobserved in the solar atmosphere (see the various refer-\nences given above). The variation of density in the radial\ndirection is confined to a thin transitional layer, and we\nadopt the thin boundary (TB) approximation (see de-tails in, e.g., Hollweg & Yang 1988; Sakurai et al. 1991;\nGoossens et al. 1992; Goossens 2008). The TB approxi-\nmation has been proved to be very accurate even when\nradialinhomogeneityisnot restrictedtoa thin layer(see,\ne.g., Van Doorsselaere et al. 2004; Andries et al. 2005;\nArregui et al. 2005). Regarding inhomogeneity in the\nlongitudinal direction, we assume that the kink mode\nwavelength is shorter than the longitudinal inhomogene-\nity length scale, so that the WKB approximation can be\napplied. In the numerical part of this paper, the prob-\nlem is solved in the general case where large variations\nof density in both the radial and longitudinal directions\nare considered and for arbitrary wavelength.\nCurvature is neglected in the model. The effect of\ncurvature was studied by, e.g., Van Doorsselaere et al.\n(2005) and Terradas et al. (2006), and it has been re-\nviewed by Van Doorsselaere et al. (2009). Analytically,\nVan Doorsselaere et al.(2005)showedthatcurvaturehas\nno first-order effect on the frequency and the damping of\nkink modes in curved cylindrical models. The result that\ncurvaturehasonlyaminoreffectonkinkMHDwaveswas\nalso numerically confirmed by Van Doorsselaere et al.\n(2005) and Terradas et al. (2006). Wave damping due\nto leakage is not considered. Leaky waves, i.e., waves\ndamped by MHD radiation, were first studied in mag-\nnetic flux tubes by Spruit (1982), whose conclusion was\nthat leakage is only important for wavelengths of the\norder of the tube radius or smaller. For thin tubes\nleakage is unimportant. This result was confirmed by\nGoossens & Hollweg (1993) and Goossens et al. (2009).\nFinally, the β= 0 approximation is also adopted, where\nβis the ratio of gas pressure to magnetic pressure.\nThe assumption of β= 0 implies that waves do not\nhave motions along the equilibrium magnetic field di-\nrection. It has been explained by, e.g., Spruit (1982) and\nGoossens et al. (2009) that the longitudinal component\nof the velocity is proportional to the transverse compo-\nnent, with a factor of proportionality that depends of β\nand (R/λ)2. In the low- βcase and for thin tubes the lon-\ngitudinal component of the velocity is much smaller than\nthe transverse component. In magnetic structures of the\nsolar atmosphere β≪1. This means that the longitudi-\nnal component of the velocity can be neglected and the\nβ= 0 approximation can be safely adopted. For the ef-\nfect of plasma βon the resonant damping of kink waves,\nSoler et al. (2009) (see also Goossens et al. 2009) showed\nthat the contribution of the slow continuum damping,\npresent when β∝negationslash= 0, is negligible compared to the Alfv´ en\ncontinuum damping.\nThis paper is organized as follows. The description of\nthe model configuration and basic equations are given\nin Section 2. Then, we obtain general expressions for\nthe amplitude and wavelengthof propagatingkink waves\nin longitudinally and transversely inhomogeneous mag-\nnetic flux tubes in Section 3. Later, we use these gen-\neral expressions to study resonantly damped propagat-\ning kink waves in stratified solar waveguides and com-\npare the analytical predictions with full numerical time-\ndependent simulations in Section 4. The implications\nfor the method of solar atmospheric magnetoseismology\nare given in Section 5. Finally, Section 6 contains the\ndiscussion of the results and our conclusions.KINK WAVES IN STRATIFIED SOLAR WAVEGUIDES 3\n2.MODEL AND BASIC EQUATIONS\nIn theβ= 0 approximation, the basic equations for\nthe discussion of linear ideal MHD waves are,\nρ∂v\n∂t=1\nµ(∇×b)×B, (1)\n∂b\n∂t=∇×(v×B), (2)\nwhereρis the plasma density, Bis the equilibrium mag-\nnetic field, vis the velocity perturbation, bis the mag-\nnetic field perturbation, and µis the magnetic permit-\ntivity. We use linear theory in our investigation be-\ncause the Alfv´ en velocity in the solar corona is of the\norder of 1000 km s−1, while the peak-to-peak veloc-\nity amplitude of the propagating waves observed using\nCoMP is only about 1 km s−1(see Tomczyk et al. 2007;\nTomczyk & McIntosh 2009). We study MHD waves in\na static background, so the equilibrium flow is absent\nfrom our discussion. The effect of flow on resonantly\ndamped propagating kink waves has been recently stud-\nied by Soler et al. (2011b).\nThe equilibrium configuration is a straight cylindri-\ncal magnetic flux tube of average radius Rembedded\nin a magnetized plasma environment. For convenience,\nwe use cylindrical coordinates, namely r,ϕ, andzfor\nthe radial, azimuthal, and longitudinal coordinates, re-\nspectively. The axis of the cylinder is set along the\nz-direction. In the following expressions, subscripts i\nand e refer to the internal and external plasmas, respec-\ntively. We denote by ρi(z) andρe(z) the internal and\nexternal densities, respectively. Both of these quantities\nare functions of z. There is a nonuniform transitional\nlayer in the transverse direction that continuously con-\nnects the internal density to the external density. The\nthickness of the layer is l, and extends in the interval\nR−l/2≤r≤R+l/2. The equilibrium magnetic field\nis straight and homogeneous, B=Bˆez, withBcon-\nstant. As ϕis an ignorable coordinate in our model and\nwe consider waves propagating along the tube with a\nfixed frequency, we write all perturbations proportional\nto exp(imϕ−iωt), where mis the azimuthal wavenum-\nberandωisthewave(angular)frequency. Thefrequency\nfis related to ωbyω= 2πf. Kink modes, i.e., the\nonly wave modes that can displace the magnetic cylinder\naxis and so produce transverse motions of the whole flux\ntube (see, e.g., Edwin & Roberts 1983; Goossens et al.\n2009), are described by m= 1. Due to the presence\nof a transverse inhomogeneous transitional layer, wave\nmodes with m∝negationslash= 0 are spatially damped by resonant\nabsorption. In the position within the transverse inho-\nmogeneous transitional layer where the wave frequency\ncoincides with the local Alfv´ en frequency, i.e., the reso-\nnance position, the wave character changes and becomes\nmore Alfv´ enic as the wavepropagates. As a result of this\nprocess, transverse motions of the flux tube are damped\nwhile azimuthal motions within the transitional layer are\namplified.\nWe use the TT approximation, i.e., λ≫R, where\nλis the wavelength along the tube, and apply the for-\nmalism of Dymova & Ruderman (2005, 2006). The ef-\nfect of the resonant damping is included here by intro-\nducing jump conditions for perturbations at the reso-\nnance position along with the TB approximation(see de-tails in, e.g., Hollweg & Yang 1988; Sakurai et al. 1991;\nGoossens et al. 1992). The combination of both TT and\nTB approximationsis usually called the TTTB approach\n(see the recent review by Goossens 2008). Then, fol-\nlowing the scaling arguments of Dymova & Ruderman\n(2005,2006), thetotalpressureperturbation P=Bbz/µ,\nwithbzthe longitudinal component of the magnetic field\nperturbation, evaluated at both sides of the transverse\ntransitional layer can be cast as\nPi≈/parenleftBigr\nR/parenrightBigm\nAi(z),atr≈R−l\n2,(3)\nPe≈/parenleftbiggR\nr/parenrightbiggm\nAe(z),atr≈R+l\n2,(4)\nwithAi(z) andAe(z) arbitrary functions of z. In the TB\napproximation, the condition of Pcontinuity across the\nresonant layer (e.g., Sakurai et al. 1991; Goossens et al.\n1992, 1995) implies Pi=Peatr=R, which means that\nAi(z) =Ae(z) =A(z).\nOn the other hand, from Equations (1) and (2) we\nobtain the following relation\n∂2vr\n∂t2−v2\nA(z)∂2vr\n∂z2=−1\nρ(z)∂2P\n∂r∂t, (5)\nwherevris the radial velocity perturbation, and v2\nA(z) =\nB2\nµρ(z)is the Alfv´ en velocity squared. After using Equa-\ntions (3) and (4) in Equation (5) we arrive at the two\nfollowing expressions,\nv2\nAi(z)+∂2vri\n∂z2+ω2vri=−iω\nρi(z)m\nRA(z),atr≈R−l\n2,(6)\nv2\nAe(z)+∂2vre\n∂z2+ω2vre=iω\nρe(z)m\nRA(z),atr≈R+l\n2,(7)\nwherevriandvreare the radial velocity perturbations\natr≈R−l/2 andr≈R+l/2, respectively. Now,\nwe take into account that in the TB approximation the\njump of vracross the resonance layer, namely [ vr], is\nassumed to be the same as the jump of vracross the\nwhole transverse inhomogeneous layer, i.e., vre−vri=\n[vr]. For a straight and constant magnetic field, [ vr] is\ngiven by (see, e.g., Sakurai et al. 1991; Goossens et al.\n1992, 1995; Tirry & Goossens 1996; Andries et al. 2005;\nDymova & Ruderman 2006),\n[vr] =−πm2/R2\nω|∂rρ|P,atr≈R, (8)\nwhere|∂rρ|is the radial derivative of the transverse den-\nsity profile at the resonance position, which is assumed\nto take place at r≈R. This is a reasonable assumption\nin the TTTB approximation. Subsequently, we combine\nEquations (6) and (7) to eliminate function A(z) and use\nthe jump condition (Equation (8)) to arrive at a single\nequation for vriand [vr]. Forthwith, we denote vriasvr\nto simplify the notation and the resultant expression is\n∂2vr\n∂z2+ω2\nv2\nk(z)vr=−1\n2/parenleftbigg∂2[vr]\n∂z2+ω2\nv2\nAe(z)[vr]/parenrightbigg\n,(9)\nwhere\nv2\nk(z) =2B2\nµ(ρi(z)+ρe(z)), (10)4\nis the squared of the kink velocity, i.e, the phase velocity\nof the propagating kink wave. For future use, it is in-\nstructive to note that the kink velocity vkis a function\nofz. Finally, we use Equation (6) along with the result\nthatPis constant across the resonant layer to relate [ vr]\nwithvr, obtaining\n[vr]≈ −iπm\nRρi(z)\n|∂rρ|/parenleftbiggv2\nAi(z)\nω2∂2vr\n∂z2+vr/parenrightbigg\n,(11)\nwhichweputinEquation(9)togetanequationinvolving\nvronly, namely\nC4∂4vr\n∂z4+C3∂3vr\n∂z3+C2∂2vr\n∂z2+C1∂vr\n∂z+C0vr= 0,(12)\nwith\nC4=−iπ\n2m\nR1\nω2ρi(z)\n|∂rρ|v2\nAi(z),\nC3=−iπm\nRρi(z)\nω2∂\n∂z/parenleftbigg1\n|∂rρ|/parenrightbigg\nv2\nAi(z),\nC2=1−iπ\n2m\nR/bracketleftbiggρi(z)\nω2∂2\n∂z2/parenleftbigg1\n|∂rρ|/parenrightbigg\nv2\nAi(z)+ρi(z)+ρe(z)\n|∂rρ|/bracketrightbigg\n,\nC1=−iπm\nR∂\n∂z/parenleftbiggρi(z)\n|∂rρ|/parenrightbigg\n,\nC0=ω2\nv2\nk(z)−iπ\n2m\nR/bracketleftbigg∂2\n∂z2/parenleftbiggρi(z)\n|∂rρ|/parenrightbigg\n+ω2\nv2\nAe(z)ρi(z)\n|∂rρ|/bracketrightbigg\n.(13)\nEquation (12) is the main equation of this investigation.\nNote that the imaginary parts of the coefficient functions\nC0–C4are all proportional to m.\n2.1.Case Without Longitudinal Stratification\nAs a limiting case of Equation (12), we study wave\npropagation in a magnetic tube which is uniform in the\nlongitudinal direction and try to recover the results of\nTerradas et al. (2010, hereafter TGV). In order to do so,\nwe takeρiandρeconstants and neglect their derivatives\nin the longitudinal direction. Since the background is in-\ndependent of zwe can Fourier-analyze in the z-direction\nand write vrproportional to exp( ikzz), withkzthe lon-\ngitudinal wavenumber. Equation (12) becomes\nC4k4\nz−C2k2\nz+C0= 0. (14)\nSince we are investigating spatial damping, we write\nkz=kzR+ikzI, approximate k2\nzR≈ω2/v2\nk, and assume\nkzI≪kzR(weak damping), sowe neglect terms with k2\nzI.\nFinally the following relation is obtained,\nkzI\nkzR=π\n8m\nR1\n|∂rρ|(ρi−ρe)2\nρi+ρe, (15)\nwhich coincides with Equation (10) of TGV. Therefore,\nwerecoverthe resultsofTGVforpropagationanddamp-\ning of kink waves in longitudinally homogeneous tubes.\nThe reader is referred to TGV for a complete analysis of\nthis case. It is worth noting that Equation (15) also\nagrees with Equation (20) of Soler et al. (2011a), ob-\ntained for the spatial damping ofpropagatingkink waves\ninpartiallyionizedthreadsofsolarprominences,iftheef-\nfect of damping by ion-neutral collisions is removed from\ntheir expression.3.GENERAL FORMULATION\n3.1.No Resonant Damping\nIn the present and next Sections, we investigate the\neffects of longitudinal density stratification. First, we\nstudy the casewhen resonantdampingis absent toassess\nthe effect of density stratification on the amplitude of\nthe propagating kink wave. Damping will be considered\nlater. Then, Equation (12) reduces to\n∂2vr\n∂z2+ω2\nv2\nk(z)vr= 0. (16)\nNote that Equation (16) is independent of the azimuthal\nwavenumber, m. In the absence of damping, the waves\nare degenerate with respect to mas long as m∝negationslash= 0.\nTo find a solution to Equation (16), we use the\nWentzel-Kramers-Brillouin (WKB) approximation (see,\ne.g., Bender & Orszag 1978, for details about the\nmethod). We define the dimensionless longitudinal co-\nordinate as,\nζ=z\nΛ, (17)\nwhere Λ is the longitudinal inhomogeneity scale height,\nandexpressthe solution ofEquation(16) in the following\nform,\nvr(ζ)≈Q(ζ)exp[iΛK(ζ)], (18)\nwithQ(ζ) andK(ζ) functions to be determined. The\nvalidity of the WKB approximation, and so of Equa-\ntion (18), is restricted to large values of Λ, i.e., weak\ninhomogeneity, such as λ/Λ≪1. We combine Equa-\ntions (16) and (18), and separate the different terms ac-\ncording to their order with respect to Λ. As Λ is large,\nthe dominant terms are those with the highest order in\nΛ. From the leading term, i.e., that with O/parenleftbig\nΛ2/parenrightbig\n, we\nobtain the following expression,\n˜k2\nz(ζ) =ω2\nv2\nk(ζ), (19)\nwhere˜kz(ζ) has been defined as\n˜kz(ζ)≡dK(ζ)\ndζ. (20)\nWe see that ˜kz(ζ) plays the role of the longitudinal\nwavenumber, which in our case is a function of ζthrough\nthe dependence ofthe kinkvelocity on ζ(Equation(10)).\nTherefore, the wavelength, λ(ζ), is\nλ(ζ) =2π\n˜kzR(ζ)=2π\nωvk(ζ) =τ vk(ζ),(21)\nwithτthe wave period. We obtain that λ(ζ) changes in\nζdue to stratification. Therefore, function K(ζ) is\nK(ζ) =ω/integraldisplayζ\n01\nvk(ξ)dξ, (22)\nwhere we have assumed K(0) = 0.\nOn the other hand, from the contribution of O/parenleftbig\nΛ1/parenrightbig\nin\nthe WKB expansion of Equation (16), we get the follow-\ning expression\n∂Q(ζ)\n∂ζ+1\n2˜kz(ζ)∂˜kz(ζ)\n∂ζQ(ζ) = 0,(23)KINK WAVES IN STRATIFIED SOLAR WAVEGUIDES 5\nwhich can be easily integrated to\nQ(ζ) =Q(0)/parenleftbiggvk(ζ)\nvk(0)/parenrightbigg1/2\n. (24)\nEquation (24) tells us how the amplitude changes with\nheightduetodensitystratification. Densitystratification\nmodifies both the amplitude and the wavelength of the\nkink wave as it propagates along the magnetic tube. If\ndensity decreases with height, and hence the kink veloc-\nity increases with height, Equations (24) and (21) show\nthat both the wavelength and the amplitude increase.\nThe complete expression for vrin the WKB approxi-\nmation using the original variable zis\nvr(z)≈vr(0)/parenleftbiggvk(z)\nvk(0)/parenrightbigg1/2\nexp/parenleftbigg\niω/integraldisplayz\n01\nvk(˜z)d˜z/parenrightbigg\n.(25)\n3.2.Effect of Resonant Damping\nHereafter, weinclude the effect ofdampingby resonant\nabsorption and we try to solve the full Equation (12). As\nacompletelygeneralsolutiontoEquation(12)isverydif-\nficult to obtain, we restrict ourselves to the case in which\nthe internal and external densities are proportional. We\ndefine the density contrast, χ, as\nχ=ρi(z)\nρe(z). (26)\nWe assume that parameter χis a constant independent\nofz. In addition, we write |∂rρ|as\n|∂rρ|=Fπ2\n4ρi(z)−ρe(z)\nl=Fπ2\n4χ−1\nχρi(z)\nl,(27)\nwithFa factor that depends on the form of the trans-\nverse density profile. For example, F= 4/π2for a linear\nprofile (Goossens et al. 2002) and F= 2/πfor a sinu-\nsoidal profile (Ruderman & Roberts 2002). Then, coeffi-\ncientsC0–C4in Equations (13) simplify to\nC4=−i2\nπl\nRm\nFχ\nχ−11\nω2v2\nAi(z),\nC3=−i4\nπl\nRm\nFχ\nχ−11\nω2∂v2\nAi(z)\n∂z,\nC2=1+i2\nπl\nRm\nF/bracketleftbiggχ\nχ−11\nω2∂2v2\nAi(z)\n∂z2−χ+1\nχ−1/bracketrightbigg\n,\nC1=0,\nC0=ω2\nv2\nk(z)−i2\nπl\nRm\nF1\nχ−1ω2\nv2\nAi(z). (28)\nAs before, we use the WKB approximation to solve\nEquation (12) with the coefficients given in Equa-\ntions (28). Then, we re-introduce the dimensionless lon-\ngitudinal coordinate ζ=z/Λ (Equation (17)) and write\nvr(ζ) in the form given in Equation (18). Following the\nsame process as in the case without resonant damping,\nwe separate the different terms of Equation (12) accord-\ning to their order with respect to Λ. From the leading\ncontribution, i.e., that with O/parenleftbig\nΛ4/parenrightbig\n, we obtain the fol-\nlowing expression,\n˜k2\nz(ζ)−ω2\nv2\nk(ζ)+iπ\n2m\nRµ\nB2ρi(ζ)ρe(ζ)\n|∂rρ|×/parenleftBig\nω2−˜k2\nz(ζ)v2\nAi(ζ)/parenrightBig/parenleftBig\nω2−˜k2\nz(ζ)v2\nAe(ζ)/parenrightBig\nω2= 0.(29)\nwhere˜kz(ζ) is defined in Equation (20). For conve-\nnience, we have used Equation (27) to introduce |∂rρ|\nagain, in order to illustrate that Equation (29) is for-\nmally equivalent to Equation (6) of TGV if the longi-\ntudinal dependence of the density is added to their ex-\npression. We follow the analytical process described in\nTGV and write ˜kz(ζ) =˜kzR(ζ) +i˜kzI(ζ). The effect of\nresonant absorption is to introduce an imaginary part\nof˜kz(ζ), while we can reasonably assume that ˜kzR(ζ)\nis the same as in the case without resonant damping,\ni.e.,˜kzR(ζ)≈ω/vk(ζ) according to Equation (19). Next,\nwe assume weak damping, i.e., ˜kzI(ζ)≪˜kzR(ζ), so we\nneglect terms with ˜k2\nzI(ζ) in Equation (29). Finally, a\nrelation between ˜kzR(ζ) and˜kzI(ζ) is derived, namely\n˜kzI(ζ)\n˜kzR(ζ)=π\n8m\nR1\n|∂rρ|(ρi(ζ)−ρe(ζ))2\nρi(ζ)+ρe(ζ),(30)\nwhich coincides with our Equation (15) and with Equa-\ntion (10) of TGV if the dependence of the densities in\nthe longitudinal direction is explicitly included. Taking\ninto account the definitions of Equations (26) and (27),\nwe rewrite Equation (30) as\n˜kzI(ζ)\n˜kzR(ζ)=1\n2πm\nFl\nRχ−1\nχ+1, (31)\nand we clearly see that the right-hand side of Equa-\ntion (31) is independent of ζ. We use Equation (21)\nand define the damping length by resonant absorption as\nLD(ζ) = 1/˜kzI(ζ), so we can obtain from Equation (31)\nthat\nLD(ζ)\nλ(ζ)=F\nmR\nlχ+1\nχ−1. (32)\nAgain, note that the right-hand side of Equation (32)\nis independent of ζ. Finally, we consider the expression\nforλ(ζ) of Equation (21) to give the expression for the\ndamping length as\nLD(ζ) = 2πF\nmR\nlχ+1\nχ−11\nωvk(ζ), (33)\nwhich is equivalent to Equation (22) of TGV. The damp-\ning length is inversely proportional to both ωandm. By\nusing Equations (21) and (33), we compute the real and\nimaginary parts of K(ζ) =KR(ζ)+iKI(ζ), namely\nKR(ζ)=/integraldisplayζ\n02π\nλ(ξ)dξ=ω/integraldisplayζ\n01\nvk(ξ)dξ,(34)\nKI(ζ)=/integraldisplayζ\n01\nLD(ξ)dξ\n=1\n2πm\nFl\nRχ−1\nχ+1ω/integraldisplayζ\n01\nvk(ξ)dξ,(35)\nwhere again we assumed KR(0) =KI(0) = 0.\nOn the other hand, we get an equation for Q(ζ) from\nthe contribution of O/parenleftbig\nΛ3/parenrightbig\nin the WKB expansion of6\nEquation (12), namely\n∂Q(ζ)\n∂ζ+1\n2˜kzR(ζ)∂˜kzR(ζ)\n∂ζQ(ζ) = 0,(36)\nwhich coincides with Equation (23). Therefore, the solu-\ntion to Equation (36) is given by Equation (24), mean-\ning that the change of the amplitude in ζdue to density\nstratification is the same as in the case without resonant\ndamping.\nFinally, using Equations (24), (34) and (35) in Equa-\ntion (18), and returning to the original variable z, we\nobtain\nvr(z)≈ A(z)exp/parenleftbigg\niω/integraldisplayz\n01\nvk(˜z)d˜z/parenrightbigg\n,(37)\nwithA(z) the amplitude as a function of zgiven by\nA(z)=vr(0)/parenleftbiggvk(z)\nvk(0)/parenrightbigg1/2\n×exp/parenleftbigg\n−1\n2πm\nFl\nRχ−1\nχ+1ω/integraldisplayz\n01\nvk(˜z)d˜z/parenrightbigg\n.(38)\nEquation (38) shows that there are two effects that de-\ntermine the amplitude of the propagating kink wave, i.e,\nlongitudinal density stratification and resonant absorp-\ntion. While the effect of longitudinal stratification de-\npends on whether the density increases or decrease with\nz, the role of resonant absorption is always to damp the\nwave and so to decrease its amplitude. It is clear from\nEquation (38) that resonant absorption acts as a natural\nfilter for waves with high frequencies and large m.\n4.APPLICATION TO PROPAGATING KINK\nWAVES IN STRATIFIED CORONAL LOOPS\n4.1.Analytical Expressions\nHere, we apply the general theory described in the last\nSection to the case of driven kink waves in a stratified\nflux tube. For a particular application we model a coro-\nnal loop being driven at one footpoint. Assuming the\nlongitudinal stratification of the coronal loop is symmet-\nric about its apex, we focus our analysis on the loop half\nthat contains the footpoint driver. We take the plasma\nto be more dense at the loop’s footpoints to mimic a\ngravitationally stratified solar atmosphere and assume\nan exponential zdependence in density, namely\nρi(z)=ρi0exp/parenleftBig\n−z\nΛ/parenrightBig\n, (39)\nρe(z)=ρe0exp/parenleftBig\n−z\nΛ/parenrightBig\n, (40)\nwhereρi0andρe0are the internal and external densities,\nrespectively, at the footpoints of loop, i.e., at z= 0, and\nΛ is the scale height defined as\nΛ =L/2\nln(ρi0/ρapex), (41)\nwithLthe total length of the loop and ρapexthe internal\ndensity at the apex, i.e., at z=L/2. As we assume the\nsame stratification within the tube and in the corona,\nχ=ρi(z)/ρe(z) =ρi0/ρe0is a constant. According toEquations (39) and (40), the square of the kink velocity\nis\nv2\nk(z) =v2\nk0exp/parenleftBigz\nΛ/parenrightBig\n, (42)\nwherev2\nk0= 2B2/[µ(ρi0+ρe0)] is the square of the\nkink velocity at z= 0. As in our model the mag-\nnetic field is straight and constant, the density scale\nheight and the kink velocity scale height are propor-\ntional. CoMP observations of coronal propagating waves\n(Tomczyk & McIntosh 2009) show that the kink velocity\nis relatively constant with height so that the kink veloc-\nity scale height is much longer that the typical coronal\nscale height. The effect of flux tube expansion might be\nresponsible for keeping the kink velocity relatively con-\nstant. The effect of flux tube expansion is discussed in\nSection 5.\n4.1.1.Case Without Resonant Damping\nFirst, we analyze the case without resonant damp-\ning. The kink mode propagation is governed by Equa-\ntion(16). Theexpressionfor v2\nk(z)giveninEquation(42)\nenable us to rewrite Equation (16) as a Bessel Equation\noforder0. Then, it ispossibletoobtainanexactsolution\nof Equation (16) as\nvr(z)=A1H(1)\n0/bracketleftbigg\n2Λω\nvk0exp/parenleftBig\n−z\n2Λ/parenrightBig/bracketrightbigg\n+A2H(2)\n0/bracketleftbigg\n2Λω\nvk0exp/parenleftBig\n−z\n2Λ/parenrightBig/bracketrightbigg\n,(43)\nwithH(1)\n0andH(2)\n0the Hankel functions of the first and\nsecondkindoforder0,and A1andA2complexconstants.\nThe real part of Equation (43) contains the solution with\nphysical meaning. The functions H(1)\n0andH(2)\n0repre-\nsent waves propagating towards the positive and nega-\ntivez-directions, respectively. By setting A2= 0, we\nrestrict ourselves to propagation towards the positive z-\ndirection. In addition, we chose A1so thatℜ(vr) = 1\nandℜ/parenleftbig∂vr\n∂z/parenrightbig\n= 0 atz= 0, hence\nA1=Y1/parenleftBig\n2Λω\nvk0/parenrightBig\n+iJ1/parenleftBig\n2Λω\nvk0/parenrightBig\nJ0/parenleftBig\n2Λω\nvk0/parenrightBig\nY1/parenleftBig\n2Λω\nvk0/parenrightBig\n−J1/parenleftBig\n2Λω\nvk0/parenrightBig\nY0/parenleftBig\n2Λω\nvk0/parenrightBig,\n(44)\nwithJandYthe Bessel functions of the first and second\nkind, respectively.\nApart from the exact expression of vrgiven in Equa-\ntion (43), we also can obtain an expression in the WKB\napproximation, which is instructive to understand the\nwave properties. Then, by using Equation (25) with the\nkink velocity given in Equation (42), we get velocity\nvr(z)≈vr(0)exp/parenleftBigz\n4Λ/parenrightBig\nexp/braceleftbigg\ni2Λω\nvk0/bracketleftBig\n1−exp/parenleftBig\n−z\n2Λ/parenrightBig/bracketrightBig/bracerightbigg\n,\n(45)\nand according to Equation (21) the wavelength is\nλ(z) =2π\nωvk0exp/parenleftBigz\n2Λ/parenrightBig\n. (46)\nWe note that Equation (45) is consistent with the con-\ndition of validity of the WKB approximation, since itKINK WAVES IN STRATIFIED SOLAR WAVEGUIDES 7\ncoincides with the expression obtained with the asymp-\ntotic expansion of H(1)\n0for large arguments, i.e., large Λ,\nis used in Equation(43) (see, e.g., Abramowitz & Stegun\n1972). We see that density stratification increases both\namplitude and wavelength of the kink wave as it propa-\ngates along the loop from the footpoint to the apex. In\nthe limit of no longitudinal stratification, i.e., Λ → ∞,\nEquations (45) and (46) simply reduce to\nvr(z)≈vr(0)exp/parenleftbigg\niω\nvk0z/parenrightbigg\n, (47)\nλ(z)=λ=2π\nωvk0. (48)\nwhich correspondsto the propagatingkink wavein a lon-\ngitudinally homogeneous tube with constant kink veloc-\nityvk0. Both the amplitude and the wavelength are con-\nstants in this case.\nFigure 1. Kink wave radial velocity perturbation (solid line), as\na function of z/Lin a longitudinally stratified tube. In this case,\nthe transverse transitional layer is absent so there is no re sonant\ndamping. The symbols correspond to the WKB approximation\n(Equation (25)). The dashed lines outline the envelope exp( z/4Λ).\nResults for χ= 3,ρi0/ρapex= 10, and ωL/vk0= 100.\nWe comparein Figure 1the exact vr(z) givenby Equa-\ntion (43) with the expressionin the WKB approximation\n(Equation (45)) for a particular set of parameters. We\nsee that there is a very good agreement between both\nEquations. The effect of longitudinal stratification on\nboth the wavelength and the amplitude is clearly seen in\nFigure 1.\n4.1.2.Case with Resonant Damping\nHere, we include the effect of damping by resonant\nabsorption. As an exact solution of Equation (12) is very\ndifficult to obtain, we use the expressions derived in the\nWKB approximation. First ofall, the expressionfor λ(z)\nis the same as given in Equation (46) in the case without\nresonantdamping. Nowwecomputethedampinglength,\nLD(z), due to resonant absorption using Equation (33)\nwith the kink velocity given in Equation (42). We obtain\nLD(z) = 2πF\nmR\nlχ+1\nχ−11\nωvk0exp/parenleftBigz\n2Λ/parenrightBig\n.(49)\nWe note again that the ratio LD(z)/λ(z) is independent\nofz.\nFinally, using Equations (37) and (38) we obtain the\nexpression for vr(z) andA(z), namelyFigure 2. Kink wave radial velocity perturbation (solid line), as\na function of z/Lin a longitudinally stratified tube. The different\nlines correspond to l/R= 0 (solid), l/R= 0.1 (dotted), and l/R=\n0.2 (dashed). In all cases, χ= 3,ρi0/ρapex= 10, and ωL/vk0=\n100.\nvr(z)≈ A(z)exp/braceleftbigg\ni2Λω\nvk0/bracketleftBig\n1−exp/parenleftBig\n−z\n2Λ/parenrightBig/bracketrightBig/bracerightbigg\n,(50)\nwithA(z) the amplitude as a function of zgiven by\nA(z)=vr(0)exp/parenleftBigz\n4Λ/parenrightBig\n×exp/braceleftbigg\n−1\nπm\nFl\nRχ−1\nχ+1Λω\nvk0/bracketleftBig\n1−exp/parenleftBig\n−z\n2Λ/parenrightBig/bracketrightBig/bracerightbigg\n.(51)\nIn Equation (51) we see that there are two competing\neffects that determine the amplitude of the propagating\nkinkwave. Onthe onehand, densitystratificationcauses\nthe amplitude to increasewith height; on the other hand,\nresonant absorption damps the kink wave, so its effect\nis to decrease the amplitude. Whereas the increase of\nthe amplitude due to longitudinal stratification is inde-\npendent of ω, the damping by resonant absorption does\ndepend on ω. Thus, we can anticipate that for a par-\nticular, critical frequency, ωcrit, the amplitude may be a\nconstant independent of z. However, due to the func-\ntional dependence of Equation (51) on z, we clearly see\nthat the amplitude cannot remain constant for all z. De-\nspite this fact, it is still possible to give an expression for\nthe critical frequency for which the amplitude is a ap-\nproximately constant for small zonly. To do so, we ap-\nproximate 1 −exp[−z/(2Λ)]≈z/(2Λ) in Equation (51).\nThen, we obtain\nωcrit≈π\n2F\nmR\nlχ+1\nχ−1vk0\nΛ. (52)\nForω/lessorsimilarωcritlongitudinal stratification is the dominant\neffect and the amplitude increase with z. On the con-\ntrary, for ω/greaterorsimilarωcritdamping by resonant absorption is\nmoreefficient andthe amplitude decreasesin z. We must\nbear in mind that Equation (52) is only valid for small\nz, i.e., close to the footpoint of the loop, but gives us\nsome qualitative information about the behavior of the\namplitude depending on the value of the frequency.\nIn Figure 2 we plot vr(z) for fixed frequency and lon-\ngitudinal inhomogeneity scale height, but for different\nvalues of l/R. In the case l/R= 0 there is no reso-\nnant damping. As l/Rincreases, the wave amplitude8\nFigure 3. Kink wave amplitude as a function of z/Lin a longitudinally stratified tube with a transverse transi tional layer. (a) Results\nforl/R= 0, 0.1, 0.2, and, 0.4 with ωL/vk0= 100 and ρi0/ρapex= 10. (b) Results for ωL/vk0= 50, 100, 200, and 400 with l/R= 0.1 and\nρi0/ρapex= 10. (c) Results for ρi0/ρapex= 2, 5, 10, and 20 with ωL/vk0= 100 and l/R= 0.1. In all cases χ= 3.\ndecreases as a result of resonant absorption. However,\nwe see that the wavelength is independent of the value\nofl/R. This means that, in the TTTB approximation,\nresonantabsorptiondoesnotaffect thevalueofthewave-\nlength, which is exclusively determined by the longitu-\ndinal inhomogeneity scale height. This is an important\nresult from the observational point of view, because the\nwavelength carries information about longitudinal strat-\nification only (Equation (21)), while the amplitude is\ninfluenced by both longitudinal and transverse density\nprofiles (Equation (38)).\nFigure 3 displays the kink wave amplitude computed\nfrom Equation(51) for different combinationsof parame-\nters. In Figure 3(a) we determine the effect of l/Rwhen\nthe remaining quantities are kept constant, while Fig-\nure 3(b) and (c) show the effect of ωandρi0/ρapex, re-\nspectively. Both l/Randωhave a similar effect on the\namplitude as both of them control the efficiency of the\nresonant damping in front of the increase of the ampli-\ntude due to stratification. When both l/Randωget\nlarger, the effect of resonant damping becomes stronger.\nOn the contrary, ρi0/ρapexcontrols the role of longi-\ntudinal stratification through Λ (Equation (41)). As\nρi0/ρapexgrows (Λ decreases), the increase in amplitude\ndue to stratification becomes more important. For the\nset of parameters used in Figure 3(b), the approximate\ncritical frequency for a constant amplitude according to\nEquation (52) is ωcritL/vk0≈100. We see that, as ex-\npected, for the critical frequency, amplitude is constant\nfor small z, but then increases slightly with larger z. We\nstress again that, for fixed frequency, the amplitude isdetermined by the combined effect of longitudinal strat-\nification and transverse inhomogeneity, while the wave-\nlength is only affected by longitudinal stratification.\n4.2.Time-dependent Numerical Simulations\nHere the aim is to show how the analytical results de-\nrived in the previous sections for the eigenmode problem\nare related to the time-dependent problem. In addition,\nwe go beyond the TT and TB approximations used in\nthe derivation of the analytical expressions, allowing us\nfurther insight. To study the problem of linear propagat-\ning waves, Equations (1) and (2) are numerically solved\nin cylindrical coordinates. Since we are only interested\nin the kink mode we assume an azimuthal dependence of\nthe form exp( imϕ) withm= 1, and the equations are\nintegrated in the radial and longitudinal coordinates ( r\nandzin our case, respectively). A driver at the foot-\npoint of the loop ( z= 0), with a single frequency ω, is\nintroduced to excite propagating waves along the tube.\nThe spatial form of the driver in the radial direction is\nchosen, to simplify things, to be the eigenmode of a loop\nwith no resonant layer. This enables us to mainly excite\ntheresonantlydampedmodes, whilethedirectexcitation\nof Alfv´ en modes is very weak. Thus, the driver at z= 0\nis implemented through the periodic variation in time\nand the radial dependence of the kink mode eigenfunc-\ntions. The simulations are done with the code MoLMHD\n(see Bona et al. 2010, for further details about the nu-\nmerical method) in a uniform grid of 1000 ×100 points\nin ther-zplane, and in the range 0 < r/R < 10 and\n0< z/R < 100. For convenience, in the code time isKINK WAVES IN STRATIFIED SOLAR WAVEGUIDES 9\nexpressed in units of τA=R/vAi0, withvAi0the internal\nAlfv´ en velocity at z= 0, so that the dimensionless fre-\nquency is ωτA. Note that in the previous sections, the\ndimensionless frequency has been written as ωτk, with\nτk=L/vk0. The relation between the two time scales is\nτA=τkR/L/radicalbig\n2χ/(1+χ).\nHereafter, we concentrate on the analysis of the two\nmain variables, vrandvϕ, i.e., the radial and azimuthal\nvelocity perturbations, respectively. Figure 4 shows the\ntwo-dimensional spatial distribution of the two velocity\ncomponents, vrandvϕ, at a given time and for a partic-\nular set of parameters. The driver excites propagating\nwaves that move upwards along the tube with a charac-\nteristic wavelength and amplitude. The amplitude inside\nthe tube decreases with height for both vrandvϕ, while\nthevϕcomponent showsa significantincreasein zwithin\nthe inhomogeneous layer (see the work by Pascoe et al.\n2010, for a longitudinally homogeneous loop). In Fig-\nure 5, we perform two cuts along the z-direction of vrat\nr= 0 and vϕatr=R. The radial velocity perturbation\nat the center of the loop decreases its amplitude in zdue\nto the combined effect of longitudinal stratification and\nresonant absorption. On the contrary, the amplitude of\nvϕatr=Rgrows in zbecause of the conversion from\nradial to azimuthal motions that takes place in the in-\nhomogeneous layer due to the Alfv´ en resonance. As a\nconsequence of this process more energy is concentrated\nin the layer in the form of azimuthal motions as the wave\npropagates upwards along the loop.\nFirst, we perform some test cases in order to compare\nthe full numerical solutions with the analytical expres-\nsions in the TTTB and WKB approximation. We start\nwith an example that satisfies both the TTTB and WKB\napproximations. In this example the frequency of the\ndriver is ωτA= 0.36. The transitional layer thickness is\nset tol/R= 0.1, i.e., we consider a a thin layer, and the\nstratification scale height is Λ /R= 200. Waves with a\nwavelength much longer that the tube radius and much\nshorter than the stratification scale height are excited.\nTherefore, we are in the TTTB and WKB regimes. In\nFigure 6 we plot the z-dependence of vratr= 0. The\nradial velocity perturbation shows an almost constant\namplitude in z, while the wavelength increases slightly\nwith height. In this particular example, the effect of lon-\ngitudinal stratification and resonant absorption on the\namplitude cancel each other. The numerical curve is\nin quite good agreement with the analytical expression\ngiven by Equation (50). Thus, this simulation shows\nthat the assumptions made to derive Equation (50) are\nwell justified when R≪λ≪Λ. The almost negligible\ndifferences between the numerical and analytical results\nbecome smaller as we take smaller values of Rand larger\nvalues of Λ.\nNext, we show how the results are modified when we\ndepart from the valid regimes of the TTTB and WKB\napproximations. In a second example (see Figure 7) the\ndriver frequency is ωτA= 0.72 and the rest of parame-\nters are the same as in the previous case, but now the\nexcited kink wavelength is almost twice smaller than be-\nfore and the assumption of TT may be compromised.\nThe comparison of the numerical result and analytical\nTTTB expression indicates that for large zthe analyt-\nical formula overestimates both amplitude of oscillationand wavelength. In our third example, we return to the\ncase with ωτA= 0.36 but we use a larger value of the\ntransitional layer thickness, l/R= 0.4. Then, we are\nnot in the TB regime. The results in this case (see Fig-\nure 8) point out that the analytical expressions predict a\nstronger damping of amplitude and a larger increase of\nwavelength with zin comparison with our numerical re-\nsults. In our last case, we test the WKB approximation\nby considering that the wavelength of the excited kink\nmode is not much shorter than the stratification scale\nheight. First we take in Figure 9 the same parameters\nas in Figure 6 but with Λ /R= 50 and ωτA= 0.36. Now\nthe wavelength and the stratification scale height are of\nthe same order, approximately, and we are not in the\nWKB regime strictly. Next in Figure 10 we use exactly\nthe same parameters but ωτA= 0.18, so that the wave-\nlength is longer than the stratification scale height and\nthe condition of applicability of the WKB approximation\nis violated. However, both Figures 9 and 10 show that\nthe full numerical solutions and the WKB results are in\nremarkable good agreement.\nThe numerical simulations performed in this Section\ntell us that the analytical expressions derived in the\nTTTBandWKBapproximationsareveryaccuratewhen\nkinkwavelengthisintherange R≪λ≪Λandl/R≪1.\nWe also obtain a very good agreement when the condi-\ntionλ≪Λ is relaxed. For thick layers, i.e., outside the\nTB approximation, the analytical expression for the am-\nplitude gives a slightly stronger damping than in the full\nnumericalcase,althoughthewavelengthremainsapprox-\nimately correct. If instead we depart from the TT ap-\nproximation, the analytical approximation of amplitude\nand wavelength givesslightly largervalues in comparison\nwith the numerical results. However, as can be seen in\nFigures 7–10, the differences between the numerical sim-\nulations and the analytical predictions are remarkably\nsmall even in the worst case scenarios studied here.\n5.IMPLICATIONS FOR SOLAR ATMOSPHERIC\nMAGNETOSEISMOLOGY\nThe theoretical results of this paper show that deter-\nmining the longitudinal and transverse inhomogeneity\nlength scales of solar waveguides from observations of\npropagating kink waves is not a trivial matter. As was\nshown in Section 4.1.2, if we can observationally iden-\ntify the wavelength (or phase speed) as a function of z,\nwe know this is independent of the transverse inhomo-\ngeneity. Therefore, if we can simultaneously estimate the\nwavelength (or phase speed) and amplitude of the kink\nwave as a function of zwe will have enough information\nto quantify the possible contribution of resonant absorp-\ntion on attenuating the observed amplitude. However,\nobservationally this presents difficult challenges. In the\nWKB approximation, for a flux tube with density strat-\nification, constant magnetic field and no resonant layer,\nEquation (25) shows that the velocity amplitude is re-\nlated to the phase speed by\nvr(z)∝/radicalbig\nvk(z). (53)\nHence if this relationship does not hold for an observed\nwave, the discrepancy could be caused by damping due\nto the presence of a resonant layer, i.e., transverse in-\nhomogeneity. At present a possible avenue to for this10\nFigure 4. Snapshot of the velocity perturbations (a) vrand (b) vϕat a given time (indicated on top of the panels). The driver is located\natz= 0 and its frequency is ωτA= 0.36. In this plot the width of the transverse transitional lay er (see dashed lines) is l/R= 0.2,χ= 3,\nand Λ/R= 200. Note the different spatial scale in the randzaxes.\nFigure 5. Velocity perturbations vr(solid line) at r= 0 and vϕ\n(dashed line) at r=Ras functions of the position along the loop\nat a given time ( t/τA= 103.4). These results correspond to the\nsimulation shown in Figure 4.\nFigure 6. Simulated radial velocity perturbation (solid line) and\nanalytical curve (dotted line)at r= 0 as functions of position along\nthe loop at t/τA= 103.4. In this plot l/R= 0.1, Λ/R= 200 and\nωτA= 0.36.Figure 7. Simulated radial velocity perturbation (solid line) and\nanalytical curve (dotted line)at r= 0 as functions of position along\nthe loop at t/τA= 103.4. In this plot l/R= 0.1, Λ/R= 200 and\nωτA= 0.72.\nFigure 8. Simulated radial velocity perturbation (solid line) and\nanalytical curve (dotted line)at r= 0 as functions of position along\nthe loop at t/τA= 103.4. In this plot l/R= 0.4, Λ/R= 200, and\nωτA= 0.36.KINK WAVES IN STRATIFIED SOLAR WAVEGUIDES 11\nFigure 9. Simulated radial velocity perturbation (solid line) and\nanalytical curve (dotted line)at r= 0 as functions of position along\nthe loop at t/τA= 70.1. In this plot l/R= 0.1, Λ/R= 50, and\nωτA= 0.36.\nFigure 10. Same as Figure 9 but for ωτA= 0.18.\ntype of magnetoseismology is by using the high spa-\ntial/temporal resolution Solar Optical Telescope (SOT)\nonboard Hinode. Using the Ca II broadband filter of\nSOT chromospheric kink waves propagating in spicules\ncan be analyzed. As was shown by He et al. (2009b), it\nis possible to track both the propagating kink wave am-\nplitudesandphasespeedsfromthephotosphereuptothe\nlower corona. He et al. (2009a) have also shown time\nsnapshots of these propagating waves that nicely show\nthe relationship between wavelength and amplitude with\nheight. Unfortunately, since there are only a few wave-\nlengths detected up to the visible apex of spicules it is\ndifficult to detect a clear trend in the variation of wave-\nlength with height. Therefore a more practical approach\nof estimating vk(z) is by tracking the phase travel time\nof the kink wave as was done by He et al. (2009b).\nRegarding CoMP data, Verth et al. (2010) attempted\nto estimate the damping length of kink waves propagat-\ning along coronal loops, assuming the plasma was lon-\ngitudinally homogeneous. To interpret CoMP data, as-\nsuming there is longitudinal variation in plasma density,\nas stated previously, we need both information about ve-\nlocity amplitude and phase speed to see if Equation (53)\nholds. Certainly, Tomczyk & McIntosh (2009) showed\nthat it was possible to estimate vk(z) from their time-\ndistance plots (see Figure 5 in Tomczyk & McIntosh\n2009). However, since the velocity amplitudes were small\n(only a few km−1s) it will be more of a challenge to es-\ntimate the longitudinal variation of this quantity.We must also mention the physical limitation of using\na model flux tube with constant magnetic field. It was\nshown by Ruderman et al. (2008) that if the magnetic\nfield is varying longitudinally (still with density strati-\nfication), in the thin tube approximation Equation (53)\nbecomes\nvr(z)∝R(z)/radicalbig\nvk(z), (54)\nwhereR(z) is the flux tube radius. Indeed, Equation\n(54) was used by Verth et al. (2011) as the basis for in-\nterpreting propagating kink waves in a spicules to deter-\nminechromosphericfluxtubeexpansion,sinceboth vr(z)\nandvk(z) were estimated from observation. It can be\nseen from Equation (54) that if there is a resonant layer\npresent, assuming ideal MHD will cause the expansion of\nflux tube with height to be underestimated. Of course,\nEquation (54) can also be applied to CoMP observations\nofpropagatingkinkwavestoestimatethevariationofthe\nmagnetic field along coronal loops since in the thin tube\nflux conservation is given by B(z)∝1/R2(z). Magneto-\nseismology of post-flare standing kink waves in coronal\nloops, allowing for both longitudinal variation in mag-\nnetic field strength and plasma density was first done by\nVerth et al. (2008), therefore it would be interesting to\nsee if propagating coronal kink waves could be exploited\nfor a similar purpose.\n6.DISCUSSION AND CONCLUSIONS\nIn this paper we have investigated the propagation of\nresonantlydampedkinkMHDwavesinbothtransversely\nandlongitudinallyinhomogeneoussolarwaveguides. The\npresentworkextends the previousinvestigationby TGV,\nwhere longitudinal inhomogeneity was not taken into ac-\ncount. By using the WKB method, we have derived gen-\neral expressions for amplitude and wavelength of kink\nwaves in the TT and TB approximations. Variation of\nwavelengthalongthe magneticfluxtube onlydepends on\nlongitudinal stratification. However, wave amplitude is\naffectedbybothtransverseandlongitudinalinhomogene-\nity. The kink mode is damped by resonant absorption in\nthe Alfv´ en continuum due to transverse inhomogeneity\nand the damping length changes along the waveguide\nsince it is dependent on the longitudinal density profile.\nHowever, it is important to note that the ratio of damp-\ning length to wavelength along the magnetic flux tube\nis constant. It is obvious that the effect of longitudinal\ninhomogeneity can both amplify or attenuate the wave,\ndepending on whether the density decreases or increases\ntowards the direction of wave propagation, respectively.\nIn the case of radial and longitudinal inhomogeneity we\nhave shown that the variation of kink mode amplitude\nalong magnetic tubes is determined by the combined ef-\nfect of both these mechanisms.\nWe have performed an application of the theory to the\ncaseofdriven kink wavesin stratified coronalloops prop-\nagating upward from the loop footpoints. In this model\nthe density decreases exponentially along the flux tube,\nand so resonant absorption and longitudinal stratifica-\ntion have opposite effects on the kink wave amplitude.\nTheefficiencyofresonantabsorptionasadampingmech-\nanism depends on the frequency. For frequencies larger\nthan an approximate critical value, the net effect is the\ndecrease of the amplitude in z. The opposite effect,\ni.e., increase of the amplitude, takes place for frequencies12\nsmallerthanthe approximatecriticalone. Theseanalyti-\ncalresults havebeen checkedusingfull numericalsimula-\ntions beyond the TTTB and WKB approximations. The\nanalytical expressions and the full numerical solution of\ndriven waves are in good agreement even when the re-\nquirements of the TTTB and WKB approximations are\nnot strictly fulfilled.\nThe analytical expressions for the wavelength and the\namplitude derived in this paper have direct implica-\ntions for solar magnetoseismology of, e.g., coronal loops,\nspicules, and prominence threads, allowing the equilib-\nrium model to have realistic variation in plasma den-\nsity in both the longitudinal and transverse direction to\nthe magnetic field. This is an important step forward\nfor exploiting the observations of ubiquitous propagat-\ning kink wavesto probe the plasmafine structure of solar\natmosphere by implementing magnetoseismologicaltech-\nniques.\nRS acknowledges support from a postdoctoral fellow-\nship within the EU Researchand Training Network“SO-\nLAIRE” (MTRN-CT-2006-035484). JT acknowledges\nsupport from the Spanish Ministerio de Educaci´ on y\nCiencia through a Ram´ on y Cajal grantand funding pro-\nvided underprojectsAYA2006-07637andFEDER funds.\nGV and MG acknowledge support from K.U. Leuven via\nGOA/2009-009.\nREFERENCES\nAbramowitz, M., & Stegun, I. A. 1972, Handbook of\nMathematical Functions, Dover Publications\nAndries, J., Goossens, M., Hollweg, J. V., Arregui, I., & Van\nDoorsselaere, T. 2005, A&A, 430, 1109\nAndries, J., Arregui, I., & Goossens, M. 2009a, A&A, 497, 265\nAndries, J., van Doorsselaere, T., Roberts, B., Verth, G.,\nVerwichte, E. & Erd´ elyi, R. 2009b, Space Sci. 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B. 2008, ApJ, 687, L45\nVerth, G., Goossens, M. & He, J.-S. 2011, ApJ, in press\nVerth, G., Terradas, J., & Goossens, M. 2010, ApJ, 718, L102" }, { "title": "1708.00757v1.Spatially_Localized_Particle_Energization_by_Landau_Damping_in_Current_Sheets_Produced_by_Strong_Alfven_Wave_Collisions.pdf", "content": "arXiv:1708.00757v1 [physics.plasm-ph] 21 Jul 2017Under consideration for publication in J. Plasma Phys. 1\nSpatially Localized Particle Energization by\nLandau Damping in Current Sheets\nProduced by Strong Alfv´ en Wave Collisions\nGREGORY G. HOWES1†, ANDREW J. MCCUBBIN1,\nand KRISTOPHER G. KLEIN2\n1Department of Physics and Astronomy, University of Iowa, Io wa City, IA 52242, USA\n2Department of Climate and Space Sciences and Engineering, U niversity of Michigan, Ann\nArbor, MI 48109, USA\n(Received ?; revised ?; accepted ?.)\nUnderstanding the removalof energy from turbulent fluctuation s in a magnetized plasma\nand the consequent energization of the constituent plasma partic les is a major goal\nof heliophysics and astrophysics. Previous work has shown that no nlinear interactions\namongcounterpropagatingAlfvenwaves—orAlfvenwavecollisions— arethefundamental\nbuilding block of astrophysical plasma turbulence and naturally gene rate current sheets\nin the strongly nonlinear limit. A nonlinear gyrokinetic simulation of a str ong Alfven\nwave collision is used to examine the damping of the electromagnetic flu ctuations and\nthe associated energization of particles that occurs in self-consis tently generated current\nsheets.Asimplemodelexplainstheflowofenergyduetothecollisionle ssdampingandthe\nassociated particle energization, as well as the subsequent therm alization of the particle\nenergy by collisions. The net particle energization by the parallel elec tric field is shown\nto be spatially intermittent, and the nonlinear evolution is essential in enabling that\nspatial non-uniformity. Using the recently developed field-particle correlation technique,\nwe show that particles resonant with the Alfven waves in the simulatio n dominate the\nenergy transfer, demonstrating conclusively that Landau dampin g plays a key role in\nthe spatially intermittent damping of the electromagnetic fluctuatio ns and consequent\nenergization of the particles in this strongly nonlinear simulation.\n1. Introduction\nThe space and astrophysical plasmas that fill the heliosphere, and other more remote\nastrophysical environments, are found generally to be both magn etized and turbulent.\nUnderstanding the removalof energy from turbulent fluctuation s in a magnetized plasma\nand the consequent energization of the constituent plasma partic les is a major goal of\nheliophysics and astrophysics. Although plasma heating and particle energization are\ngoverned by microscopic processes typically occurring at kinetic len gth scales in the\nplasma, these important energy transport mechanisms can have a significant impact on\nthe macroscopic evolution of the systems. For example, the diffuse plasma of the solar\ncorona is found to be nearly three ordersof magnitude hotter tha n the solarphotosphere.\nThe dissipation of turbulent fluctuations, through a physical mech anism that is poorly\nunderstood at present, is believed to be responsible for this drama tic heating of the coro-\nnal plasma. This very high coronal temperature leads to the super sonic solar wind that\npervades the entire heliosphere (Parker 1958), so the kinetic plas ma physics governing\n†Email address for correspondence: gregory-howes@uiowa.e du2 G. G. Howes, A. J. McCubbin, and K. G. Klein\nthe heating of the coronal plasma at small scales indeed impacts the global structure of\nthe heliosphere.\nThe low density and high temperature conditions of the plasma in many astrophysical\nsystems lead to a mean free path for collisions among the constituen t charged particles\nthat is often much longer than the length scales of the turbulent flu ctuations. Under such\nweaklycollisionalplasmaconditions, the dynamics ofthe turbulence a nd its dissipationis\ngoverned by kinetic plasma physics. Unlike in the more well-known case of fluid systems\n(which corresponds to the strongly collisional regime), in weakly collis ional plasmas, the\ndissipation of turbulent energy into plasma heat is inherently a two-s tep process (Howes\n2017). First, energy is removed from the turbulent electromagne tic fluctuations through\ncollisionless interactions between the fields and particles, transfer ring that energy to\nnon-thermal fluctuations in the particle velocity distribution funct ions, a process that is\nreversible.Subsequently, arbitrarilyweakcollisionscan smooth out the small fluctuations\nin velocity space, leading to entropy increase and irreversible heatin g of the plasma\n(Howeset al.2006; Howes 2008; Schekochihin et al.2009). In this two-step process, the\nremoval of energy from turbulent fluctuations and the subseque nt conversion of that\nenergy into plasma heat may even occur at different locations (Nava rroet al.2016).\nIn fluid simulations of plasma turbulence using the magnetohydrodyn amic (MHD)\napproximation—a strongly collisional limit of the large-scale dynamics ( relative to the\ncharacteristic kinetic plasma length scales)—the nonlinear evolution leads to the devel-\nopment of intermittent current sheets (Matthaeus & Montgomer y1980; Meneguzzi et al.\n1981). Furthermore, it has been found that the dissipation of tur bulent energy is largely\nconcentrated in these intermittent current sheets (Uritsky et al.2010; Osman et al.2011;\nZhdankin et al.2013). Numerous studies have recently sought evidence for the s patial lo-\ncalizationofplasmaheatingbythedissipationofturbulenceincurren tsheetsthroughsta-\ntistical analysesofsolarwind observations(Osman et al.2011; Borovsky & Denton 2011;\nOsmanet al.2012; Perri et al.2012; Wang et al.2013; Wu et al.2013; Osman et al.\n2014)andnumericalsimulations(Wan et al.2012;Karimabadi et al.2013;TenBarge & Howes\n2013; Wu et al.2013; Zhdankin et al.2013).\nThe mechanisms of the spatially localized dissipation found in MHD simulat ions are\nresistive (Ohmic) heating and viscous heating (Zhdankin et al.2013; Brandenburg 2014;\nZhdankin et al.2015). But, resistivity and viscosity arise from microscopic collisions in\nthe strongly collisional (or small mean free path) limit, a limit that is not applicable to\nthe dynamics of dissipation in many space and astrophysicalenviron ments (Howes 2017).\nUnder the weakly collisional conditions appropriate for most space a nd astrophysical\nplasmas, which physical mechanisms are responsible for the damping of the turbulent\nfluctuations and the consequent energization of the plasma partic les remains an open\nquestion. Our aim here is to identify the mechanisms governing the da mping of the\nturbulent fluctuations and the particle energization using a kinetic s imulation code that\nfollows the three-dimensional evolution of a weakly collisional plasma in which current\nsheets develop self-consistently.\nEarly research on incompressible MHD turbulence in the 1960s (Iros hnikov 1963;\nKraichnan 1965) emphasized the wave-like nature of turbulent plas ma motions, sug-\ngesting that nonlinear interactions between counterpropagating Alfv´ en waves—orsimply\nAlfv´ en wave collisions —mediate the turbulent cascade of energy from large to small\nscales. In fact, the physics of the nonlinear interactions among Alf v´ en waves provides the\nfoundation for modern scaling theories of plasma turbulence that e xplain the anisotropic\nnature of the turbulent cascade (Goldreich & Sridhar 1995) and th e dynamic alignment\nof velocity and magnetic field fluctuations (Boldyrev 2006).\nFollowing a number of previous investigationsof weak incompressible M HD turbulenceParticle Energization in Alfv´ enic Current Sheets 3\n(Sridhar & Goldreich 1994; Ng & Bhattacharjee 1996; Galtier et al.2000), the nonlin-\near energy transfer in Alfv´ en wave collisions in the weakly nonlinear lim it has been\nsolved analytically (Howes & Nielson 2013), confirmed numerically with g yrokinetic nu-\nmerical simulations (Nielson et al.2013), and verified experimentally in the laboratory\n(Howeset al.2012, 2013; Drake et al.2013), establishing Alfv´ en wave collisions as the\nfundamental building block of astrophysical plasma turbulence. Mo re recent research\nhas found that Alfv´ en wave collisions in the strongly nonlinear limit nat urally generate\ncurrent sheets (Howes 2016), providing a first-principles explana tion for the ubiquitous\ndevelopment of intermittent current sheets in plasma turbulence. This self-consistent\ngeneration of current sheets is found to persist even in the more r ealistic case of strong\ncollisions between localized Alfv´ en wave packets (Verniero et al.2017).\nHere we explore the damping of the electromagnetic fluctuations an d the associated\nenergization of particles that occurs in current sheets that are g enerated self-consistently\nby strong Alfv´ en wave collisions. Previous work using a simulation of k inetic Alfv´ en wave\nturbulence has shown that, although enhanced plasma heating rat es are well correlated\nwith thepresenceofcurrentsheets,therateofheatingasafun ctionofwavenumberiswell\npredicted assuming that linear Landau damping is entirely responsible for the removal\nof energy from the turbulence (TenBarge & Howes 2013). This res ult suggests that the\nphysical mechanism governing the removal of energy from turbule nt fluctuations, even\nin spatially intermittent current sheets, is Landau damping. Using no nlinear gyrokinetic\nsimulations of strong Alfv´ en wave collisions, we aim to answer two que stions:\n(i) Isthedissipationassociatedwithcurrentsheetsthataregene ratedbystrongAlfven\nwave collisions spatially intermittent?\n(ii) What is the physical mechanism governing the removal of energy from the turbu-\nlence and the consequent energization of the particles?\n2. Simulation\nSimilar to a previous study showing the development of current shee ts in strong Alfv´ en\nwave collisions (Howes 2016), we employ the Astrophysical Gyrokine tics code AstroGK\n(Numata et al.2010) to perform a gyrokinetic simulation of the nonlinear interactio n\nbetween two counterpropagating Alfv´ en waves in the strongly no nlinear limit. AstroGK\nevolves the perturbed gyroaveraged distribution function hs(x,y,z,λ,ε ) for each species\ns, the scalar potential ϕ, the parallel vector potential A/bardbl, and the parallel magnetic field\nperturbation δB/bardblaccording to the gyrokinetic equation and the gyroaveraged Maxw ell’s\nequations (Frieman & Chen 1982; Howes et al.2006). Velocity space coordinates are λ=\nv2\n⊥/v2andε=v2/2. The domain is a periodic box of size L2\n⊥×L/bardbl, elongated along the\nstraight, uniform mean magnetic field B0=B0ˆz, where all quantities may be rescaled\nto any parallel dimension satisfying L/bardbl/L⊥≫1. Uniform Maxwellian equilibria for ions\n(protons)andelectronsarechosen,withareducedmassratio mi/me= 36suchthat,even\nwith the modest spatial resolution of this simulation, the collisionless d amping by ions\nand electrons is sufficiently strong within the resolved range of lengt h scales to terminate\nthe nonlinear transfer of energy to small scales. Spatial dimension s (x,y) perpendicular\nto the mean field are treated pseudospectrally; an upwind finite-diff erence scheme is used\nin the parallel direction, z. Collisions employ a fully conservative, linearized collision\noperator with energy diffusion and pitch-angle scattering (Abel et al.2008; Barnes et al.\n2009).\nTo set up the simulation of an Alfv´ en wave collision, following Nielson et al.(2013), we\ninitialize two perpendicularly polarized, counterpropagating plane Alf v´ en waves, z+=\nz+cos(k⊥x−k/bardblz−ω0t)ˆyandz−=z−cos(k⊥y+k/bardblz−ω0t)ˆx, where ω0=k/bardblvA,4 G. G. Howes, A. J. McCubbin, and K. G. Klein\nk⊥= 2π/L⊥, andk/bardbl= 2π/L/bardbl. Herez±=u±δB//radicalbig\n4π(nimi+neme) are the El-\nsasser fields (Elsasser 1950) which represent Alfv´ en waves that propagate up or down the\nmean magnetic field at the Alfv´ en velocity vA=B0//radicalbig\n4π(nimi+neme) in the MHD\nlimit,k⊥ρi≪1. We specify a balanced collision with equal counterpropagating wav e\namplitudes, z+=z−, such that the nonlinearity parameter is χ=k⊥z±/(k/bardblvA) = 1,\nrelevant to the regime of strong turbulence (Goldreich & Sridhar 19 95). To study the\nnonlinear evolution in the limit k⊥ρi≪1, we choose a perpendicular simulation domain\nsizeL⊥= 8πρiwith simulation resolution ( nx,ny,nz,nλ,nε,ns) = (64,64,32,128,32,2).\nThe fully resolved perpendicular range in this dealiased pseudospect ral method covers\n0.25/lessorequalslantk⊥ρi/lessorequalslant5.25, or 0.042/lessorequalslantk⊥ρe/lessorequalslant0.875 given the chosen mass ratio mi/me= 36\nand temperature ratio Ti/Te= 1. Here the ion thermal Larmor radius is ρi=vti/Ωi, the\nion thermal velocity is v2\nti= 2Ti/mi, the ion cyclotron frequency is Ω i=qiB0/(mic), and\nthe temperature is given in energy units. The plasma parameters of the simulation are\nβi= 1 and Ti/Te= 1, typical of near-Earth solar wind conditions. The linearized Lan-\ndau collision operator (Abel et al.2008; Barnes et al.2009) is employed with collisional\ncoefficients νi=νe= 6×10−4k/bardblvA, yielding weakly collisional dynamics with νs/ω≪1.\nTo prepare the simulation, the two initial Alfv´ en wave modes are evo lved linearly for\nfive periods with enhanced collision frequencies νi=νe= 0.01k/bardblvAto eliminate any\ntransient behavior arising from the initialization that does not agree with the proper-\nties of the Alfv´ en wave mode (Nielson et al.2013). The simulation is then restarted\nwith the nonlinear terms enabled, beginning the nonlinear evolution of the strong Alfv´ en\nwave collision. Note that the two Alfv´ en waves are already overlapp ing at the beginning\nof this simulation before the nonlinear evolution begins, an idealized ca se which facili-\ntates the comparison to an asymptotic analytical solution in the wea kly nonlinear limit\n(Howes & Nielson 2013; Howes 2016). The nonlinear evolution of deve lopment of current\nsheets is found to persist in the more realisticcase of collisionsbetwe en two initially sepa-\nratedAlfv´ enwavepacketsoffinite parallelextent (Verniero et al.2017; Verniero & Howes\n2017).\nFor the plasma parameters of this gyrokinetic simulation, we solve th e linear colli-\nsionless gyrokinetic dispersion relation (Howes et al.2006) for the Alfv´ en/kinetic Alfv´ en\nwavemode to determine the linear frequency and collisionlessdamping rate for this mode\nas a function of perpendicular wavenumber. Note that the collisionle ss damping of this\nmode is due to the Landau resonances with the ions and electrons. I n the upper panel\nof Fig. 1 is plotted the normalized real frequency ω/k/bardblvAvs. the normalized perpendic-\nular wavenumber k⊥ρi. In the lower panel is plotted the total collisionless damping rate\nnormalized to the wave frequency γ/ω(solid black), as well as the separate contribu-\ntions to this linear collisionless damping rate from the ions (red dotted ) and electrons\n(blue dashed). These gyrokinetic results have been verified by com parison with the so-\nlutions of the full Vlasov-Maxwell linear dispersion relation using the P LUME solver\n(Klein & Howes 2015). Since gyrokinetic theory resolves the Landau resonances but not\nthecyclotronresonances,thisagreementbetweenthegyrokine ticandtheVlasov-Maxwell\nresults confirms that the linear collisionless damping is due to the Land au resonance.\nFig. 1 shows that the collisional damping by the ions (red dotted) has a relatively\nbroad peak over the range 0 .5/lessorsimilark⊥ρi/lessorsimilar2.0. The range of resonant parallel phase ve-\nlocitiesω/k/bardblassociated with this broad peak in damping, normalized in terms of the\nion thermal velocity, is 1 .0/lessorsimilarω/k/bardblvti/lessorsimilar1.5. Therefore, if Landau damping with the\nions is active, the energy transfer should be dominated by resonan t ions with parallel\nvelocities in the range 1 .0/lessorsimilarv/bardbl/vti/lessorsimilar1.5. The collisionless damping by the electrons,\non the other hand, increases monotonically with perpendicular wave number, becoming\nsufficiently strong with γe/ω/greaterorsimilar0.1 atk⊥ρi/greaterorsimilar1.2. From this point, up to the maximumParticle Energization in Alfv´ enic Current Sheets 5\nFigure 1. (a) The normalized frequency ω/k/bardblvAand (b) total collisionless damping rate γtot/ω\n(black solid) vs. k⊥ρifor Alfv´ en and kinetic Alfv´ en waves with mi/me= 36 from the linear\ncollisionless gyrokinetic dispersion relation, includin g the separate contributions to the linear\ncollisionless damping rate from the ions γi/ω(red dotted) and the electrons γe/ω(blue dashed).\nSquares indicate values computed from linear runs of AstroGK. Solid vertical lines indicate the\nlimits of the fully resolved perpendicular scales of the non linear simulation at k⊥ρi= 0.25 and\nk⊥ρi= 5.25. The vertical dashed line indicates the highest k⊥ρivalue,k⊥ρi= 5.25√\n2≃7.42,\nof the modes in the corner of Fourier space.\nfully resolved perpendicular scale of k⊥ρi= 5.25, the range of resonant parallel phase\nvelocities ω/k/bardblin terms of the electron thermal velocity is 0 .17/lessorsimilarω/k/bardblvte/lessorsimilar0.6. There-\nfore, if the collisionless energy transfer from the turbulent electr omagnetic fields to the\nplasma particles is governed by a Landau resonant mechanism, we wo uld expect to see\nthe transfer of energy localized at parallel velocities within this rang e of resonant values.\n3. Evolution of Energy\nUnder weakly collisional plasma conditions typical of many heliospheric and astro-\nphysical plasmas, the removal of energy from turbulent fluctuat ions and the eventual\nconversion of that energy into plasma heat, unlike in the more familiar fluid limit, is a\ntwo-step process (Howes 2017). Specifically, the turbulent fluct uations are first damped\nthrough reversible, collisionless interactions between the electrom agnetic fields and the\nplasma particles, leading to energization of the particles. This non-t hermal energization\nof the particlevelocity distributions is subsequently thermalized by a rbitrarilyweakcolli-\nsions, thereby accomplishingthe ultimate conversionof the turbule nt energyinto particle\nheat. An analysis of the flow of energy in this Alfv´ en wave collision simu lation illustrates\nthese two distinct steps of the turbulent dissipation.\nInagyrokineticsystem,the total fluctuatingenergy δW(Howeset al.2006;Brizard & Hahm6 G. G. Howes, A. J. McCubbin, and K. G. Klein\n2007; Schekochihin et al.2009) is given by †\nδW=/integraldisplay\nd3r/bracketleftBigg\n|δB|2+|δE|2\n8π+/summationdisplay\ns/integraldisplay\nd3vT0s��f2\ns\n2F0s/bracketrightBigg\n, (3.1)\nwhere the index sindicates the plasma species and T0sis the temperature of each species’\nMaxwellian equilibrium. The left term represents the electromagnetic energy and the\nright term represents that microscopic fluctuating kinetic energy of the particles of each\nplasma species s. Note that the elimination of the parallel nonlinearity in the stan-\ndard form of gyrokinetic theory means that the appropriate cons erved quadratic quan-\ntity in gyrokinetics is the Kruskal-Obermann energy, E(δf)\ns≡/integraltext\nd3r/integraltext\nd3vT0sδf2\ns/2F0s\n(Kruskal & Oberman 1958; Morrison 1994), in contrast to the usu al kinetic theory defi-\nnition of microscopic kinetic energy,/integraltext\nd3r/integraltext\nd3v(msv2/2)fs. Note also that δWincludes\nneither the equilibrium thermal energy,/integraltext\nd3r3\n2n0sT0s=/integraltext\nd3r/integraltext\nd3v1\n2msv2F0s, nor the\nequilibrium magnetic field energy,/integraltext\nd3rB2\n0/8π. Thus, the terms of δWin (3.1) represent\nthe perturbed electromagnetic field energies and the microscopic k inetic energy of the\ndeviations from the Maxwellian velocity distribution for each species.\nA moreintuitive formofthe total fluctuating energy δWcanbe obtainedby separating\nout the kinetic energy of the bulk motion of the plasma species from t he non-thermal\nenergy in the distribution function that is not associated with bulk flo ws (Liet al.2016),\nδW=/integraldisplay\nd3r/bracketleftBigg\n|δB|2+|δE|2\n8π+/summationdisplay\ns/parenleftbigg1\n2n0sms|δus|2+3\n2δPs/parenrightbigg/bracketrightBigg\n(3.2)\nwheren0sistheequilibriumdensity, msismass,and δusisthefluctuatingbulkflowveloc-\nity. The non-thermal energy in the distribution function (not including the bulk kinetic\nenergy) is defined by E(nt)\ns≡/integraltext\nd3r3\n2δPs≡/integraltext\nd3r(/integraltext\nd3vT0sδf2\ns/2F0s−1\n2n0sms|δus|2)\n(TenBarge et al.2014). The turbulent energy is defined as the sum of the electromag-\nnetic field and the bulk flow kinetic energies (Howes 2015; Li et al.2016),E(turb)≡/integraltext\nd3r[(|δB|2+|δE|2)/8π+/summationtext\ns1\n2n0sms|δus|2]. Therefore the total fluctuating energy\nis simply the sum of the turbulent energy and species non-thermal e nergies, δW=\nE(turb)+E(nt)\ni+E(nt)\ne.\n3.1.Evolution of Turbulent and Non-Thermal Energies\nIn Fig. 2, we plot the evolution of these three different contribution s to the total fluc-\ntuating energy normalized to the total initial fluctuating energy δW0≡δW(t= 0). In\nFig. 2(a), we plot the total fluctuating energy δW/δW 0(black), the turbulent energy\nE(turb)/δW0(purple), the ion non-thermal energy E(nt)\ni/δW0(red), and the electron\nnon-thermal energy E(nt)\ne/δW0(blue). Note that collisions in AstroGK, as well as in real\nplasma systems, convert non-thermal to thermal energy, repr esenting irreversible plasma\nheating with an associated increase of entropy. The energy lost fr omδWby collisions\nis tracked by AstroGK and represents thermal heating of the plasma species, but this\nenergy is not fed back into the code to evolve the equilibrium thermal temperature, T0s\n(Howeset al.2006; Numata et al.2010; Li et al.2016). The evolution in Fig. 2(a) makes\nclear that, over 7.5 periods of the initial Alfv´ en waves, more than 6 0% of the initial\nfluctuating energy in the simulation is lost to collisional heating.\nIn Fig. 2(b), we plot the different components that contribute to t he turbulent energy\n†Note that in the gyrokinetic approximation, the electric fie ld energy is relativistically small\nrelative to the magnetic field energy (Howes et al.2006).Particle Energization in Alfv´ enic Current Sheets 7\nE(turb). In orderofdecreasingmagnitude, these contributionsare the p erpendicular mag-\nnetic energy EB⊥(green dashed), perpendicular ion kinetic energy Eui,⊥(red dashed),\nperpendicular electron kinetic energy Eue,⊥(blue dashed), parallel magnetic energy EB/bardbl\n(green dotted), parallel ion kinetic energy Eui,/bardbl(red dotted), and parallel electron ki-\nnetic energy Eue,/bardbl(blue dotted). The turbulent energyis dominated bythe perpendic ular\nmagnetic energy and perpendicular ion kinetic energy. This is expect ed for Alfv´ enic fluc-\ntuations at k⊥ρi≪1: transversemotion of the plasma dominated by ion kinetic energyis\nfirst arrestedby magnetictension, followedby the accelerationof the plasmaback toward\nthe equilibrium point by magnetic tension, thereby leading to the oscilla tory transfer of\nenergy back and forth between perpendicular magnetic energy an d perpendicular ion\nkinetic energy, as evident in Fig. 2(b). Note that this energy is integ rated over the entire\nsimulation domain, so neither of these energies is expected to drop t o zero, as would\noccur for the energy density at a single point in space as an Alfv´ en w ave passes through\nthat point. In the MHD limit k⊥ρi≪1, Alfv´ enic fluctuations also have very little par-\nallel motion, u/bardbl≪u⊥and a very small parallel magnetic field fluctuation, δB/bardbl≪δB⊥.\nFinally, the electron kinetic energies are down from the respective io n kinetic energies\napproximately by a factor of the mass ratio, me/mi= 1/36, so electrons make a sub-\ndominant contribution to the turbulent energy. Finally, note that a lthough the volume\nintegrated energy of each component of E(turb)shows oscillations with the period T0,\ntheir sum varies smoothly in time, suggesting that this definition of tu rbulent energy is\nphysically well motivated.\n3.2.Evolution of Collisional Heating\nIn Fig. 3, we present the evolution of the collisional heating rate per unit volume of\nionsQi(red) and electrons Qe(blue) as well as the total collisional heating rate Qtot=\nQi+Qe(black) for this nonlinear Alfv´ en wave collision simulation (thick lines). The\nheating rates are normalized by a characteristic heating rate per u nit volume, Q0=\n(n0iT0ivti/L/bardbl)(π/8)(L⊥/L/bardbl)2. The total fluctuating energy δWin Fig. 2(a) diminishes\nin time due tothermalizationbycollisions.Thiscollisionalenergylossfro mδWis tracked\ninAstroGK by this collisional heating rate, enabling energy conservation to be m easured\nin the simulation.\nNote that the rapid initial rise in the collisional damping rate for the ele ctronsQeat\nt/T0/lessorsimilar0.5 in Fig. 3 is due to the fact that the linear initialization uses higher collisio n\ncoefficients, νs= 0.01k/bardblvA, than the subsequent nonlinear evolution, νs= 6×10−4k/bardblvA.\nWhen the collisional coefficients are reduced, smaller velocity scale st ructures in the ve-\nlocity distribution must develop (through the kinetic evolution) befo re the collisional\nheating is able to effectively thermalize the non-thermal energy con tained in those fluc-\ntuations.\nAlso plotted in Fig. 3 is the evolution of the collisional heating rates in a lin ear simula-\ntion (thin lines), where the simulation is started from the same initial c onditions but the\nnonlinear terms are turned off. In this linear simulation, there is no no nlinear transfer\nof energy to other Fourier modes—meaning that there is no nonlinea r turbulent cascade\nof energy to small scales—so the evolution of the energy is solely due to linear Landau\ndamping of the initial Alfv´ en waves and the subsequent collisional th ermalization of the\nfluctuations in the velocity distribution functions that were genera ted by this linear Lan-\ndau damping. It is important to note that the nonlinear evolution eve ntually leads to a\nhigher collisional heating rate, presumably through the nonlinear tr ansfer of energy to\nsmaller scale fluctuations that have higher collisionless damping rates than the initial\nAlfv´ en waves, although we do not directly analyze that nonlinear ca scade of energy in\nthis study.8 G. G. Howes, A. J. McCubbin, and K. G. Klein\nFigure 2. (a) Evolution of the normalized energy E/δW 0as a function of time t/T0for the total\nfluctuatingenergy δW(black), theturbulentenergy E(turb)(purple),the ion non-thermalenergy\nE(nt)\ni(red) and the electron non-thermal energy E(nt)\ne(blue). (b) Evolution of the different\ncomponents of the turbulent energy E(turb)(purple), dominated by the perpendicular magnetic\nfield energy EB⊥(green dashed) and the perpendicular ion bulk flow kinetic en ergyEi,u⊥\n(red dashed), with successively smaller contributions by t he perpendicular electron bulk kinetic\nenergyEe,u⊥(blue dashed), the parallel magnetic field energy EB/bardbl(green dotted), the parallel\nion bulk flow kinetic energy Ei,u/bardbl(red dotted), and the parallel electron bulk flow kinetic ene rgy\nEe,u/bardbl(blue dotted).\n3.3.Model of Energy Flow\nA physical interpretation of the two-step energy flow in this stron g Alfv´ en wave collision\nsimulation is illustrated by the diagram in Fig. 4. The energy of turbulen t fluctuations\nE(turb), consisting of the sum of the electromagnetic field fluctuations and the kinetic en-\nergy of the bulk flows (first velocity moment) of each plasma species (Howes 2015, 2017),\ncan be removed by collisionless interactions ˙E(fp)\nsbetween the electromagnetic fields andParticle Energization in Alfv´ enic Current Sheets 9\nFigure 3. Ion collisional heating rate Qi/Q0(red), electron collisional heating rate (blue) and\ntotal collisional heating rate Qtot=Qi+Qe(black) and as a function of time t/T0for the\nnonlinear simulation (thick lines). Also plotted (thin lin es) is the linear evolution from the same\ninitial conditions.\nCollisional E(fp)\ni\nE(fp)\neField−Particle\nInteractionsE(nt)\ni\nE(nt)\neE(turb)Turbulent\nEnergyQi\nQeIon Non−Thermal\nElectron Non−\nThermal EnergyEnergyCollisionalIon Heating\nElectron Heating\nFigure 4. Diagram of the energy flow in weakly collisional turbulent pl asmas, showing that\ninteractions between the electromagnetic fields and plasma particles ˙E(fp)\nscan reversibly trans-\nfer energy between the turbulent energy E(turb)and the non-thermal energy in the velocity\ndistribution function of each species E(nt)\ns. Collisional heating Qsthen can irreversibly convert\nthis non-thermal energy, represented by fluctuations in vel ocity space of each species, into heat\nof each plasma species s. This is the two-step process of reversible particle energi zation and\nsubsequent irreversible thermalization of that particle e nergy.\nthe plasma particles. This energy is converted to non-thermal ene rgy of the ions and\nelectrons, E(nt)\ns. This non-thermal energy is represented by fluctuations in the pa rticle\nvelocity distribution functions that have no associated bulk flow (fir st moment), and\ntherefore do not contribute to the turbulent motions. A key prop erty of this collisionless\nenergy transfer ˙E(fp)\nsis that it is reversible (two-headed arrows in Fig. 4), representing\nthe electromagnetic work done on the particles by the fields, which c an be positive or\nnegative.\nThe non-thermal energy E(nt)\nsis contained in fluctuations in velocity-space of the\nparticle velocity distribution functions for each species, δfs(v). If these fluctuations reach\nsufficiently small scales in velocity space, arbitrarily weak collisions can smooth out\nthose fluctuations, thermalizing their energy and thereby realizing irreversible plasma\nheating, Qs. The kinetic equation for each species governstwo mechanisms tha t facilitate10 G. G. Howes, A. J. McCubbin, and K. G. Klein\nthe transfer of energy to ever smaller scales in velocity space: linea r phase mixing and\nnonlinear phase mixing.\nThe first mechanism is linear phase mixing governedby the ballistic term in the kinetic\nequation, which couples spatial variations with velocity-space fluct uations and can lead\nto the transfer of energy to small scales in velocity space. †In linear Landau damping, for\nexample, the energy of a damped wave is first transferred collisionle ssly into non-thermal\nvelocity space fluctuations, which subsequently phase mix linearly to small enough scales\nin velocity space that weak collisions can irreversibly convert the non -thermal energy\ninto plasma heat. Boltzmann’s Htheorem proves that the entropy increase associated\nwith irreversible plasma heating is ultimately collisional (Howes et al.2006).\nIn addition to this linear phase-mixing process, at perpendicular spa tial scales com-\nparable to the particle thermal Larmor radii, k⊥ρs/greaterorsimilar1, a nonlinear phase-mixing pro-\ncess (Dorland & Hammett 1993), also known as the entropy cascad e (Schekochihin et al.\n2009; Tatsuno et al.2009; Plunk et al.2010; Plunk & Tatsuno 2011; Kawamori 2013),\ncan be very effective at transferring energy to ever smaller scales in velocity space. Ul-\ntimately, when the non-thermal particle energy in the velocity distr ibution functions\nδfs(v) has reached sufficiently small scales in velocity, due to some combina tion of linear\nand nonlinear phase mixing, collisions may thermalize that particle ener gy, completing\nthe final step in the conversion of turbulent energy into plasma hea t. InAstroGK, this\ncollisional heating removes energy from fluctuating energy in the pla sma,δW.\nIt is worthwhile to contrast this two-step mechanism in weakly collision al plasmas—\ncollisionless particle energization followed by collisional thermalization— with the more\nfamiliar picture of turbulent dissipation in the fluid (strongly collisional) limit. A di-\nmensionless measure of the collisionality is the ratio of the thermal co llision rate to the\nfrequency of typical fluctuations in the plasma, ν/ω. In the strongly collisional limit,\nν/ω≫1, collisions can directly remove energy from both the bulk plasma flow s through\nviscosity and the plasma currents through resistivity. Because bo th viscosity and resistiv-\nity are collisional, entropy increases through these mechanisms, an d the energy from the\nturbulent electromagnetic field and plasma flow fluctuations is immedia tely thermalized\nto plasma heat. Thus, the dissipation of turbulence in the strongly c ollisional, fluid limit\nis a single-step process. Consider the example of resistive MHD, whe re Ohm’s Law gives\nthe electric field in terms of the plasma fluid velocity, magnetic field, an d current density,\nE+U/c×B=ηj(Spitzer 1962; Kulsrud 1983). The work done by the electric field is\nj·E=−j·(U/c×B)+ηj2, where second term is the non-negative Ohmic heating due to\nresistive dissipation of the current, showing that the resistivity lea ds directly to plasma\nheating.\nThe strong Alfv´ en wave collision simulation presented here has ν/ω∼6×10−4≪\n1, firmly in the weakly collisional limit. Unlike in the MHD Ohm’s Law above, wh ere\nthe current density jand electric field Edue to the resistive term are in phase, and\nthereby yield a zero or positive change in energy, in the weakly collision al case the\ncurrent density jand electric field Eneed not be in phase, enabling the work done by\ncollisionless interactions between the fields and particles to give ener gy to or take energy\nfrom the particles. In fact, if the current and electric field are exa ctly 90 degrees out of\nphase, there is zero net energy transfer between fields and part icles over one complete\noscillation, corresponding to undamped wave motion. The bottom line , a point that\n†It has been recently suggested that, under particular condi tions in a turbulent plasma\nof sufficiently low collisionality, a turbulent anti-phase- mixing process can prevent velocity\nspace fluctuations from reaching sufficiently small-scales t o enable thermalization by collisions\n(Schekochihin et al.2016; Parker et al.2016).Particle Energization in Alfv´ enic Current Sheets 11\ncannot be overstated, is that in a weakly collisional plasma, the elect romagnetic work\nj·Edoes not correspond to irreversible plasma heating, but rather to reversible work\ndone on the particles by the fields, or vice versa.\nDeveloping a detailed understanding of particle energization and plas ma heating in\nheliospheric plasmas is grand challenge problem in heliophysics, and this simple model\nof the energy flow provides important constraints to focus effort s in that endeavor. Note\nthat the final step of the process in Fig. 4, the thermalization of th e particle energy, is\nfundamentally collisional, independent of what mechanism (which we ha ve not specified\nhere) removed energy from the turbulent fluctuations initially. The key question in un-\nderstanding particle energization and plasma heating in heliospheric p lasmas is therefore\nto understand the first step: what collisionless and reversible mech anism is responsible\nfor the removal of energy from the turbulent fluctuations and co nversion of that energy\ninto non-thermal energy of the plasma species?\n3.4.Rate of Energy Transfer\nNow we use the strong Alfv´ en wave collision simulation presented her e to analyze the\nchannels of energy transfer shown in Fig. 4. For each species, the rate of change of non-\nthermal energy is given by\n˙E(nt)\ns=˙E(fp)\ns−Qs, (3.3)\nwhere the irreversible collisional heating Qs/greaterorequalslant0 but the reversible collisionless field-\nparticle energy transfer ˙E(fp)\nscan be either positive or negative. In addition, the rate\nof change of turbulent energy must be the sum of the collisionless fie ld-particle energy\ntransfer for each species,\n−˙E(turb)=˙E(fp)\ni+˙E(fp)\ne. (3.4)\nNote that we have not specified the physical mechanism governing t he field-particle\nenergy transfer, but we are simply showing that the transfers of energy indeed follow the\ndiagram in Fig. 4. The rate of field-particle energy transfer presen ted below is calculated\nfrom (3.3) as the difference between the rate of change of non-th ermal particle energy\nand the collisional heating rate for each species.\nIn Fig. 5, we present the terms of these energy transfer relation s for the (a) ions and\n(b) electrons, as well as (c) the balance between the loss of turbu lent energy and the\nfield-particle energy transfer to each species. A few very interes ting aspects of Fig. 5\nare worth highlighting. First, although the change of turbulent ene rgyE(turb)and non-\nthermal energies E(nt)\nsin Fig. 2 appears to be smooth, the time derivative, which gives\nthe rate of change, indeed varies rapidly, including a significant fluct uation with period\nT0/2.\nSecond, in Fig. 5(b), the energy transferred into electron non-t hermal energy at the\nrate˙E(fp)\ne(solid) is very quickly thermalized by collisions into electron heat (dash ed);\nthe time lag between these two curves is ∆ t= 0.6T0(not shown), suggesting that non-\nthermal energy transferred into the electron velocity distributio n is rapidly transferred\nby phase mixing to sufficiently small velocity-space scales to be therm alized by the weak\ncollisions. For the ions in Fig. 5(a), on the other hand, the time lag bet ween the en-\nergy transferred into non-thermal ion energy ˙E(fp)\niand the thermalization of that ion\nenergy is approximately ∆ t= 3.6T0, a factor of/radicalbig\nmi/me= 6 longer, suggesting that\nthe phase-mixing occurs more slowly for ions by the ratio of the elect ron-to-ion ther-\nmal velocity. Note also that the collisionless field-particle energy tra nsfer to ions indeed\nbecomes negative at a few points in time, as allowed for a reversible pr ocess.\nFurthermore, note that the magnitudes of ˙E(fp)\niand˙E(fp)\neare fairly similar, as ex-12 G. G. Howes, A. J. McCubbin, and K. G. Klein\nFigure 5. The rate of energy transfer by field-particle interactions ˙E(fp)\ns(solid), the rate of\nchange of non-thermal energy ˙E(nt)\ns(dotted), and the collisional heating rate Qs(dashed) for\n(a) ions (red) and (b) electrons (blue). (c) The energy balan ce between the loss of turbulent\nenergy−˙E(turb)(purple solid) and the summed transfer of energy to both ions and electrons,\n˙E(fp)\ni+˙E(fp)\ne(black dashed).\npected because the linear damping rates, shown in Fig. 1, are fairly s imilar for ions and\nelectrons, γi≃γe, over the range of spatial scales k⊥ρi<1 that contain most of the\nenergy in the simulation. Finally, in Fig. 5(c), we see that the energy lo st by the turbu-\nlence−˙E(turb)(purple solid) is indeed balanced by the sum of the field-particle energ y\ntransfer to ions and electrons (black dashed).\n3.5.Evolution of the Total Energy Budget\nPlots of the total energy budget as a function of time in the simulatio n nicely summarize\nthe flow of energy in the simulation. First, we account for the energ y lost from δWinParticle Energization in Alfv´ enic Current Sheets 13\ntheAstroGK simulation to collisional plasma heating by accumulating the thermalized\nenergy in each species over time, E(coll)\ns(t) =/integraltextt\n0dt′QS(t′).\nIn Fig. 6(a), we plot the evolution of the energy budget over the co urse of the sim-\nulation, showing that turbulent energy E(turb), which dominates at the beginning of\nthe simulation, is largely converted to ion heat E(coll)\niand electron heat E(coll)\neby the\nend of the simulation, with a smaller fraction of the lost turbulent ene rgy persisting as\nnon-thermal ion energy E(nt)\niand electron energy E(nt)\ne. Also indicated in Fig. 6(a) is\nthe evolution of the total fluctuating energy δW(thick black line), showing that 60% of\nthis energy has been lost to plasma heating over 7.5 periods of the init ial Alfv´ en waves.\nAnother interesting point is that, although electrons are heated t wice as much as ions,\nthe non-thermal electron energy content of the simulation always remains very small.\nThis point is consistent with the idea, introduced in §3.4 above, that non-thermal energy\ntransferred into the electron velocity distribution function by collis ionless damping of the\nturbulence is very rapidly thermalized into electron heat. This analys is of the evolution\nof the total energy budget shows that energy is conserved to wit hin 0.1% over the course\nof the simulation.\nOne can alternatively divide the contributions to the energy budget in terms of (3.1),\nas shown in Fig. 6(b), showing the perpendicular magnetic field energ yEB⊥(green),\nthe parallel magnetic field energy EB/bardbl(cyan), the total fluctuating ion kinetic energy\nE(δf)\ni(red), and the total fluctuating electron kinetic energy E(δf)\ne(blue). Note that,\nas anticipated from the contributions to the turbulent energy in Fig . 2(b), the turbulent\nenergyin Fig.6(a)is largelycomposedofperpendicular magneticene rgyEB⊥andkinetic\nenergy of the perpendicular ion bulk flows Ei,u⊥. The wiggly boundary between EB⊥and\nEi,u⊥is a consequence of the Alfv´ enic fluctuations, and their nonlinear in teractions, in\nthe simulation.\nOne final point is that, although one may choose to decompose the d ifferent contribu-\ntions to the energy using (3.1) in Fig. 6(b), by organizing the energie s instead according\nto the turbulent energy E(turb)=/integraltext\nd3r[(|δB|2+|δE|2)/8π+/summationtext\ns1\n2n0sms|δus|2] and the\nspecies non-thermal energies E(nt)\ns, the interpretation of the energy flow is much more\nphysically motivated, as illustrated by Fig. 4. By simply plotting E(δf)\nias a function\nof time, one does not see the important split between the large frac tion of the total\nfluctuating ion kinetic energy E(δf)\nithat is associated with turbulent fluctuations and\nthe remainder that corresponds to non-thermal energy not ass ociated with turbulent\nfluctuations.\n4. Development of Current Sheets and Intermittent Particle\nEnergization\nIn the limit of strong nonlinearity, χ∼1—corresponding to the important case of crit-\nically balanced, strong MHD turbulence (Goldreich & Sridhar 1995)—r ecent work has\nshown that Alfv´ en wave collisions self-consistently develop intermit tent current sheets\n(Howes 2016). This finding may indeed explain the ubiquitous current sheets found to\ndevelop in simulations of plasma turbulence (Wan et al.2012; Karimabadi et al.2013;\nTenBarge & Howes 2013; Wu et al.2013; Zhdankin et al.2013) and inferred from space-\ncraftobservationsofthesolarwind(Osman et al.2011;Borovsky & Denton2011;Osman et al.\n2012; Perri et al.2012; Wang et al.2013; Wu et al.2013; Osman et al.2014). Yet how\nthis self-consistentdevelopmentofcurrentsheetsinfluences th e physicalmechanismsthat\nremoveenergy from plasma turbulence remains unanswered. We sh ow in this section that14 G. G. Howes, A. J. McCubbin, and K. G. Klein\nFigure 6. (a) The energy budget of the simulation vs. time, showing the turbulent energy\nE(turb), non-thermal ion energy E(nt)\ni, non-thermal electron energy E(nt)\ne, ion heat E(coll)\niand\nelectron heat E(coll)\ne. (b) The same energy budget decomposed according to (3.1), s howing the\nperpendicular magnetic field energy EB⊥, parallel magnetic field energy EB/bardbl(cyan, not labeled,\nappearing between EB⊥andE(δf)\ni,), total fluctuating ion kinetic energy E(δf)\ni, total fluctuating\nelectron kinetic energy E(δf)\ne, ion heat E(coll)\niand electron heat E(coll)\ne. The total fluctuating\nenergyδWis shown in both panels (thick black line).Particle Energization in Alfv´ enic Current Sheets 15\n0 5 10 15 20 250510152025\n-2.5-2-1.5-1-0.500.511.522.5\n0 5 10 15 20 250510152025\n-2.5-2-1.5-1-0.500.511.522.5\n0 5 10 15 20 250510152025\n-2.5-2-1.5-1-0.500.511.522.5\n0 5 10 15 20 250510152025\n-2.5-2-1.5-1-0.500.511.522.5\nFigure 7. Plots of parallel current j/bardbl/j0(colorbar) and contours of the parallel vector potential\nA/bardbl(contours, positive black, negative white) at times t/T0= (a) 1.38, (b) 1.75, (c) 1.86, and\n(d) 2.03.\nthe simulationreportedhereindeed developsintermittent current ssheets (intermittent in\nboth time and space), and in section §5 we employ the field-particle correlationtechnique\nto examine the physical mechanism that removes energy from the t urbulent fluctuations.\nIn Fig. 7 we plot the current density parallel to the mean magnetic fie ldj/bardbl/j0(col-\norbar) and contours of parallel vector potential A/bardbl(positive black, negative white) in\nthe plane z/L/bardbl=−0.25, where the simulation domain spans −L/bardbl/2/lessorequalslantz/lessorequalslantL/bardbl/2 and\nj0=n0qivtiL⊥/L/bardbl. We plot evolution of the current in this plane at four different times\nin the evolution of the strong Alfv´ en wave collision, t/T0= (a) 1.38, (b) 1.62, (c) 1.86,\nand (d) 2.10. Here T0= 2π/ωis the period of the initial Alfv´ en waves, where the gyroki-\nnetic linear dispersion relation gives ω/k/bardblvA= 0.995 andγ/k/bardblvA=−6.10×10−3. These\nplots show the presence of intermittent, elongated sheets of loca lized current density.\nOver a single initial Alfv´ en wave period T0, two current sheets form at slightly differ-\nent times, become thinner and more intense, and then disappear. O ne of these current\nsheets appears in the upper right quadrant of the plane z/L/bardbl=−0.25, and the other in\nthe lower left quadrant, as shown in Fig. 7. During this time, their cro ss sections in the\nplane plotted in Fig. 7 moves slowly across the quadrant of the domain in which each\nappears (but these intermittent current sheets do not cross th e entire domain, as would\nbe expected from a strictly linear fluctuation). The generalpictur e of current sheet devel-\nopment and evolution in a strong Alfv´ en wave collision is described in mo re quantitative\ndetail by Howes (2016); although the parameters of this simulation are slightly different,\nthe evolution of the current sheets is qualitatively similar here.16 G. G. Howes, A. J. McCubbin, and K. G. Klein\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC−20−15−10−505101520\nj/bardblE/bardbl/Q0(a)\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC\n−30−20−100102030\nj/bardblE/bardbl/Q0(b)\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC\n−18−12−6061218\nj/bardblE/bardbl/Q0(c)\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC−4−2024\nj/bardblE/bardbl/Q0(d)\nFigure 8. Plots of j/bardblE/bardbl(colorbar) and contours of the parallel vector potential A/bardbl(contours,\npositive solid, negative dashed) at times t/T0= (a) 1.75, (b) 1.86, and (c) 2.03, as well as (d)\n/angbracketleftj/bardblE/bardbl/angbracketrightτ, the rate of electromagnetic work per unit volume averaged o ver approximately one full\nwave period, τ= 0.992T0, centered at time t/T0= 1.86.\n4.1.Spatial Distribution of Parallel Electromagnetic Work, j/bardblE/bardbl\nAs shown in §3, over the full time of the simulation, 7 .5T0, 60% of the fluctuating energy\nδWof the initial Alfv´ en waves is removed from the fluctuations in the pla sma. Fig. 3\nshows that this energy is ultimately irreversibly converted into elect ron and ion heat\nthrough the weak but finite collisionality in the plasma. As the model of energy flow\nillustrated in Fig. 4 shows, this energy is initially removed from the turb ulent electro-\nmagnetic fluctuations (Howes 2015, 2017) through collisionless inte ractions between the\nelectromagneticfields and the individual plasma particles. In a kinetic plasma, the rateof\nelectromagnetic work done on the particles by the fields is given by dW/dt=/integraltext\nd3r j·E\n(Howeset al.2017; Klein 2017). Therefore, plotting the rate of electromagnet ic work\nj·Eas a function of position provides useful insights into the particle en ergization in the\nplasma.\nAs shown in Appendix B, in this simulation the dominant electromagnetic work is\ndone by the component of the electric field parallel to the magnetic fi eld,E/bardbl, so in Fig. 8\nwe plot the instantaneous value of dimensionless rate of work per un it volume j/bardblE/bardbl/Q0\nas a function of position in the plane z/L/bardbl=−0.25 at three different times during the\nsimulation t/T0=(a)1.75,(b) 1.86,and(c)2.03.Notethat thevalueof j/bardblE/bardblisphysically\ninterpreted as the rate of transfer of spatial energy density be tween the parallel electric\nfieldE/bardbland the plasma ions and electrons. Since this electromagnetic work is reversible,\nits value can be positive or negative, where positive means work done on the particles by\nthe field, and negative means work done on the field by the particles.Particle Energization in Alfv´ enic Current Sheets 17\nAs emphasized in Howes et al.(2017), the instantaneous energy transfer between fields\nand particles has two components: (i) an oscillating energy transfer back and forth be-\ntween fields and particles that is typical of undamped linear wave mot ion in a kinetic\nplasma, and (ii) a secular energy transfer that represents the energy lost from the elec-\ntromagnetic fluctuations to the plasma particles through collisionles s damping. To de-\ntermine the particle energization, it is the secular energy transfer that is of interest, but\nthe challenge is that the oscillating energy transfer often has a muc h larger amplitude\nthan the secular energy transfer. However, if a time-average is t aken over a suitably cho-\nsen averaging interval, the oscillating energy transfer will largely ca ncel out, exposing\nthe smaller secular energy transfer that is sought. In this strong Alfv´ en wave collision\nsimulation, the linear period T0of the initial Alfv´ en waves is an appropriate choice for\nthis time-averaging, and we plot in Fig. 8(d) the time-average of /angb∇acketleftj/bardblE/bardbl/angb∇acket∇ightτover an interval\nτ= 0.992T0centered at time t/T0= 1.86.\nThe plots shown in Fig. 8 convey a number of valuable insights into the p article\nenergization in this simulation. First, the plots in Fig. 8(a)-(c) show c learly that the\ninstantaneous rate of energy transfer is both spatially and tempo rally intermittent, with\nthe energy transfer localized in sheet-like structures reminiscent of the current sheets\nplotted in Fig. 7. An example of the temporal variation is illustrated by observing the\nchanges in the instantaneous energy transfer rate at point A mar ked on each plot. At\n(a)t/T0= 1.75, the plasma is energized by E/bardbl, but later at (b) t/T0= 1.86 the plasma\nis losing energy to E/bardbl, and finally at (c) t/T0= 2.03 there is very little energy transfer\neither direction. Averaged over one period, Fig. 8(d) shows that t he plasma gains energy\nfrom the parallel electric field at point A. Curiously, the instantaneo us energy transfer\nfrom fields to particles at t/T0= 1.86 in Fig. 8(b) is negative at point A, but the single-\nperiod average, centered at that same time t/T0= 1.86 in Fig. 8(d) shows a positive\ntransfer of energy to the particles at the same position. This plot s tresses the importance\nof appropriate time-averaging to properly understand the net pa rticle energization in a\nturbulent plasma.\nSecond, the net plasma energization over one period in Fig. 8(d) is als o spatially in-\ntermittent, with plasma energization at point A, a net loss of energy at point B, and\nlittle energy change at point C. It is also worthwhile pointing out that t he magnitude of\nthe time-averaged energy transfer is smaller in magnitude than the instantaneous energy\ntransfer, as expected if some fraction of this energy transfer is oscillatoryand largelycan-\ncels out when averagedoverone period T0. Together,the four panels demonstratethe key\npoint that that the particle energization is spatially non-uniform, bo th instantaneously\nas well as when averaged over one period T0of the initial Alfv´ en waves.\nThird, something that cannot be appreciated by the single time slice in Fig. 8(d), is\nthe surprising result that the single-period-averaged plasma ener gization has very little\ntemporal variation as the center of the time-averagewindow is adv anced over one period.\nIn fact, one observes only a very slow evolution of this particle ener gization pattern over\na number of periods, probably due to the long-term evolution and ac cumulating loss of\nfluctuating energy δWover the course of the simulation.\nA final point is that the plasma energization—the sum of the energy t ransfer to both\nions and electrons—has a net positive value when integrated over th e entire simulation\ndomain, as demonstrated by the sum ˙E(fp)\ni+˙E(fp)\neplotted in Fig. 5(c). Therefore, al-\nthough there is a loss of plasma energy in some regions of the domain, the net effect is\nthat plasma species gain energy at the expense of the turbulent ele ctromagnetic field and\nbulk plasma flow fluctuations, as depicted in the energy flow diagram in Fig. 4.\nFurther insight into the effect of the nonlinear evolution on the resu lting plasma en-18 G. G. Howes, A. J. McCubbin, and K. G. Klein\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC−4−2024\nj/bardblE/bardbl/Q0(a)\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC−0.14−0.070.000.070.14\nj/bardblE/bardbl/Q0(b)\nFigure 9. Comparison of time-averaged /angbracketleftj/bardblE/bardbl/angbracketrightτover an interval τ= 0.992T0centered at time\nt/T0= 1.86 for both (a) a nonlinear run and (b) a linear run, starting f rom identical initial\nconditions, showing a much more spatially intermittent dis tribution of plasma energization in\nthe nonlinear case.\nergization can be gained by comparing a linear simulation starting from identical initial\nconditions. In the linear case, no energy is transferred to other F ourier modes, and all of\nthe particle energization is due to linear collisionless damping via the Lan dau resonance.\nIn Fig. 9, weplot the time-averaged /angb∇acketleftj/bardblE/bardbl/angb∇acket∇ightτoveraninterval τ= 0.992T0centeredat time\nt/T0= 1.86 for both (a) the nonlinear run and (b) the linear run. This figure d irectly\ndemonstrates the striking fact that the spatial non-uniformity o f particle energization\narises due to the nonlinear transfer of energy to other Fourier mo des. This is fully con-\nsistent with the picture of current sheet generation by construc tive interference among\nthe initial Alfv´ en wave modes and the nonlinearly generated fluctua tions (Howes 2016).\nIn§5, the field-particle correlation technique will be used to identify the nature of the\ncollisionless energy transfer that yields this spatially non-uniform pa rticle energization.\nThe rate ofplasma energizationis the sum of the ratesof ion and elec tron energization,\nj/bardblE/bardbl=j/bardbliE/bardbl+j/bardbleE/bardbl, and, in Appendix B, we plot in Figs. 16 and 17 the separate ion\nand electron energization contributing to Fig. 8. In this simulation, t he single-period\naveraged particle energization in the plane z/L/bardbl=−0.25 shown in Fig. 8 yield about\ntwice the energy transfer to electrons relative to the ions at t/T0= 1.86.\n5. Analysis of Energy Transfer Mechanism Using Field-Particle\nCorrelations\nThe rates of total energy transfer as a function of time between the turbulent fluctu-\nations and the ions and electrons, plotted in Fig. 5, give the desired in formation about\nthe net collisionless particle energization over the entire simulation do main. But this\nsimple approach cannot be applied to the analysis of spacecraft mea surements to under-\nstand heating in heliospheric plasmas, because spacecraft measur e the particle velocity\ndistributions and electromagnetic fields at only one or a few points in s pace, so it is not\npossible to integrate the plasma heating over the entire plasma volum e. In addition, such\nan energy flow analysis alone, such as that given in the diagram in Fig. 4 , tells us nothing\nof the mechanism leading to the particle energization.\nThe electromagnetic work, j·E, can be computed with single point measurements,pro-\nviding moreinsight intothe natureoftheparticle energizationmecha nismandthe spatial\ndistribution of energy transfer than an energy analysis alone, but the newly developed\nfield-particlecorrelationtechnique(Klein & Howes2016;Howes et al.2017), whichyieldsParticle Energization in Alfv´ enic Current Sheets 19\nthe distribution of the energy transfer as a function of particle ve locity, gives far greater\ndetail about the nature of the energy transfer mechanism. This t echnique requires only\na single-point time series of both field and velocity distribution functio n measurements,\nwhich can be obtained using modern spacecraft instrumentation.\nThe field-particle correlation technique has been successfully applie d to examine the\nelectron energization due to the damping of electrostatic fluctuat ions in a 1D-1V Vlasov-\nPoisson plasma (Klein & Howes 2016; Howes et al.2017), to determine the transfer of\nfree energy in kinetic instabilities from unstable particle velocity distr ibutions to electro-\nstatic fluctuations (Klein 2017), and to explore the particle energiz ation caused by the\ncollisionless damping of strong, broadband, gyrokinetic plasma turb ulence (Klein et al.\n2017). Here we apply the technique to discover the nature of the p hysical mechanism re-\nsponsible for the spatially intermittent transfer of energy from th e turbulent fluctuations\nto the non-thermal energy of the ions and electrons in the plasma.\nSpecifically, since we know from Fig. 15 in Appendix B that the net ener gy transfer is\ndominated by the parallel electric field, we will evaluate the correlatio n of the ion and\nelectron fluctuations with the parallel electric field. The correlation ofthe parallel electric\nfield,E/bardbl, with a species sis defined by\nCE/bardbl,s(v,t,τ) =C/parenleftBigg\n−qsv2\n/bardbl\n2∂fs(r0,v,t)\n∂v/bardbl,E/bardbl(r0,t)/parenrightBigg\n. (5.1)\nThis unnormalized correlation is taken over an appropriately chosen correlation interval\nτto suppress the signal of the oscillatory energy transfer relative to the secular energy\ntransfer (Howes et al.2017). Defining the phase-space energy density byws(r,v,t) =\nmsv2fs(r,v,t)/2, this unnormalized correlation yields the phase-space energy tra nsfer\nrate between the parallel electric field E/bardbland species sgiven by the Lorentz term in\nthe Vlasov equation (Howes 2017; Klein et al.2017). A key aspect of this novel analysis\nmethod is that it retains the dependence of the energy transfer o n velocity space. Note\nthat integratingthiscorrelationovervelocityspacesimplyyieldsthe parallelcontribution\nto the electromagnetic work, j/bardblE/bardbl=/integraltext\ndvCE/bardbl(v,t,τ) (Howes et al.2017; Klein et al.\n2017).\nFor the application of this technique to data from our gyrokinetic sim ulation using\nAstroGK, we note that the gyrokinetic distribution function hs(x,y,z,v ⊥,v/bardbl) is related\nto the total distribution function fsvia (Howes et al.2006)\nfs(r,v,t) =F0s(v)/parenleftbigg\n1−qsφ(r,t)\nT0s/parenrightbigg\n+hs(r,v/bardbl,v⊥,t). (5.2)\nAs a technical step, we transform from the gyrokinetic distributio n function hsto the\ncomplementary perturbed distribution function\ngs(r,v/bardbl,v⊥) =hs(r,v/bardbl,v⊥)−qsF0s\nT0s/angbracketleftbigg\nφ−v⊥·A⊥\nc/angbracketrightbigg\nRs, (5.3)\nwhere/angb∇acketleft.../angb∇acket∇ightis the gyroaveraging operator (Schekochihin et al.2009). The complemen-\ntary distribution function gsdescribes perturbations to the background distribution in\nthe frame of reference moving with the transverse oscillations of a n Alfv´ en wave. Field-\nparticle correlations calculated using hsorfsyield qualitatively and quantitatively sim-\nilar results to those computed with gs(Kleinet al.2017).\nBelow, we present the correlations between the complementary pe rturbed distribution20 G. G. Howes, A. J. McCubbin, and K. G. Klein\nfunction and the parallel electric field E/bardblat a single-point r0\nCE/bardbl,s(v/bardbl,v⊥,t,τ) =C/parenleftBigg\n−qsv2\n/bardbl\n2∂gs(r0,v/bardbl,v⊥,t)\n∂v/bardbl,E/bardbl(r0,t)/parenrightBigg\n. (5.4)\nTo explore the particle energization over time, we can also integrate over the perpendic-\nular velocity v⊥to obtain a reduced parallel correlation\nCE/bardbl,s(v/bardbl,t,τ) =/integraldisplay\nv⊥dv⊥CE/bardbl,s(v/bardbl,v⊥,t,τ). (5.5)\nThis reduced parallel correlation CE/bardbl,s(v/bardbl,t,τ) can be plotted as a function of ( v/bardbl,t) to\nillustrate the time evolution particle energization using a timestack plo t of the energy\ntransfer as a function of the parallel velocity of the particles.\n5.1.Timestack Plots of Field-Particle Correlations\nHere we present the results of the field-particle technique applied a t three points in the\nsimulation domain, labeled A, B, and C in Fig. 8. From Fig. 8(d), we see th at, averaged\nover one period τ/T0= 0.992 centered at t/T0= 1.86, there is a net gain of energy by the\nplasma at point A, a net loss of energy by the plasma at point B, and litt le net change in\nthe plasma energy at point C. Note that the reduced parallel corre lationCE/bardbl,s(v/bardbl,t,τ)\nin the plots presented in this section is normalized by the energy tran sfer rate per unit\nvolume per unit velocity, Q0/vti.\nIn Fig. 10(b), we present a timestack plot of the reduced parallel fi eld-particle corre-\nlation for the ions CE/bardbl,i(v/bardbl,t,τ) at position A with a correlation interval τ/T0= 0.992,\nshowing the distribution of the energy transfer to the ions as a fun ction of the parallel\nvelocity v/bardbl/vtivs. normalized time t/T0. Vertical solid and dashed black lines indicate\nthe limits of resonant parallel phase velocities from Fig. 1, 1 .0/lessorsimilar|ω/k/bardblvti|/lessorsimilar1.5 for ions;\nthere are both positive and negative ranges of parallel phase veloc ities, corresponding to\nAlfv´ en waves traveling up or down the mean magnetic field. Also plott ed in Fig. 10(a) is\nthe velocity-space integrated correlation, ∂wi/∂t=/integraltext\ndv/bardblCE/bardbli(v/bardbl,t,τ), equivalent to the\nparallel ion contribution of the electromagnetic work j/bardbliE/bardblat position A, showing a net\nenergization of the ions over the course of the simulation.\nThedistributionoftheenergytransferasafunctionof v/bardbl/vtiinFig.10(b)isthe velocity\nspace signature of the energy transfer mechanism. The localization of the energy t ransfer\nin the marked range of resonant parallel phase velocities for kinetic Alfv´ en waves clearly\nindicates that the energy transfer is resonant. The specific distr ibution of this energy\ntransfer, with a transfer of energy from E/bardblto the ions (red) at |v/bardbl/vti|>|vres/vti|and a\nloss of energy from the ions (blue) at |v/bardbl/vti|<|vres/vti|is the characteristicsignature of\nthe Landau damping of kinetic Alfv´ en waves(Howes 2017; Klein et al.2017). The change\nof sign in the energy transfer occurs at the resonant phase veloc ity for the collisionlessly\ndamped wave. Here the change of sign occurs at v/bardbl/vti=ω/k/bardblvti≃vA/vti=±1,\nindicating that larger scale Alfv´ en waves with k⊥ρi≪1, which have a parallel phase\nvelocityω/k/bardbl=vA, appear to dominate the energy transfer at point A. This is consist ent\nwith the fact the energy in the electromagnetic and plasma bulk flow fl uctuations in this\nsimulation is dominated by the low k⊥ρi≪1 modes. This novel field-particle correlation\nanalysis shows definitively the key result that ion Landau damping con tributes to the\nenergization of ions at position A in this strong Alfv´ en wave collision sim ulation.\nIn Fig. 10(d), we plot the the reduced parallel field-particlecorrela tionfor the electrons\nCE/bardbl,e(v/bardbl,t,τ) at position A with the same correlation interval τ/T0= 0.992, where\nverticalsolid and dashed blacklines indicate the rangeofresonantp arallelvelocities fromParticle Energization in Alfv´ enic Current Sheets 21\nCE||,ix10-2\n(b)\n-4-3-2-101234\nv||/vti-6-4-20246\n01234567\n-3-2-10123t/T0\n10-2 x ∂wi / ∂tCE||,ix10-2\n(b) (a)\nCE||,ex10-2\n(d)\n-4-3-2-101234\nv||/vte-9-6-30369\n01234567\n-1-0.500.51t/T0\n10-1 x ∂we / ∂tCE||,ex10-2\n(d) (c)\nFigure 10. Timestack plots of ion and electron energization at positio n A for a correlation\nintervalτ/T0= 0.992. (a) Velocity-space integrated correlation, giving th e rate of ion energiza-\ntion∂wi/∂tdue to ion interactions with E/bardbl. (b) The reduced parallel field-particle correlation\nfor the ions CE/bardbl,i(v/bardbl,t,τ). (c) Velocity-space integrated correlation, giving the r ate of electron\nenergization ∂we/∂tdue to electron interactions with E/bardbl.(d) The reduced parallel field-particle\ncorrelation for the electrons CE/bardbl,e(v/bardbl,t,τ). Vertical solid black indicate resonant velocities for\na parallel phase velocity at the Alfv´ en speed ω/(k/bardblvts) =vA/vts.Vertical dashed lines indi-\ncate the highest parallel phase velocities for modes with si gnificant collisionless damping in the\nsimulation.\nFig. 1, 0.17/lessorsimilarω/k/bardblvte/lessorsimilar0.6 for electrons. Also plotted in Fig. 10(c) is the the velocity-\nspace integrated correlation, ∂we/∂t=/integraltext\ndv/bardblCE/bardble(v/bardbl,t,τ), equivalent to the parallel\nelectron contribution of the electromagnetic work j/bardbleE/bardbl, showing a net energization of\nthe electrons at position A.22 G. G. Howes, A. J. McCubbin, and K. G. Klein\nThe velocity space signature of the electron energization in Fig. 10( d) also shows the\ntypical characteristics of electron Landau damping, with a slight diff erence from the ion\ncase. Because kinetic Alfv´ en waves are dispersive, with a parallel p hase velocity that in-\ncreasesfor k⊥ρi/greaterorsimilar1 givenapproximatelyby ω=k/bardblvA/radicalbig\n1+(k⊥ρi)2/[βi+2/(1+Te/Ti)]\n(Howeset al.2014), the parallel resonant velocity will increase for kinetic Alfv´ e n waves\nwith larger k⊥ρi. The velocity space signature of linear Landau damping typically show s\nthe change of sign of the energy transfer at the resonant velocit y. In Fig. 10(d), that\nchange of sign for 1 /lessorequalslantt/T0/lessorequalslant3 occurs at a resonant velocity slightly larger than vA/vte\n(vertical black line), suggesting that the kinetic Alfv´ en wave involv ed in the electron\nLandau damping has a value of k⊥ρi/greaterorsimilar1 leading to a higher resonant parallel velocity.\nDespite this minor detail, the energy transfer still shows that the e lectron energization is\nmediatedbyresonantelectrons,withavelocityspacesignaturety picalofelectronLandau\ndamping (Howes 2017). Therefore, this analysis definitively yields a s econd key result,\nthat electron Landau damping contributes to the energization of e lectrons at position A\nin this strong Alfv´ en wave collision simulation.\nWe can also investigate the regions in the simulation domain where the p lasma loses\nenergy to the parallel electric field at point B, plotted in Fig. 11. The ( a) velocity-\nintegrated ion energization ∂wi/∂tdue toE/bardbland (c) velocity-integrated electron ener-\ngization ∂we/∂tdue toE/bardblboth show that the net energy transfer to ions and electrons\nat point B is negative. Here, as in Fig. 10, we see that the energy tra nsfer for both ions\nin panel (b) and electrons in panel (d) is dominated by particles with v elocities that fall\nwithin the range of parallel velocities expected to be resonant with t he parallel phase ve-\nlocity of Alfv´ en waves, demonstrating directly that energy trans fer between the particles\nand the parallel electric field E/bardblis governed by Landau resonant interactions.\nAt point C in the simulation domain, there is very little net energy trans fer from the\nparallel electric field to the plasma particles. The same field-particle c orrelation analysis\nat point C, presented in Fig. 12, shows that the velocity-integrate d (a) ion energiza-\ntion∂wi/∂tand (c) electron energization ∂we/∂tdue toE/bardblyields a very small positive\ntransfer of energy to the particles over the first couple of period sT0, with an ampli-\ntude about an order of magnitude smaller than the energy transfe r at points A and B.\nThe reduced parallel field-particle correlation CE/bardbl,s(v/bardbl,t,τ) for (b) the ions and (d) the\nelectrons shows that the majority of this very small amount of ene rgy transfer is still\ndominated by resonant particles.\nBut there is a very significant difference between the reduced para llel field-particle\ncorrelation CE/bardbl,sat point C for both ions and electrons compared to the same correla tion\nat points A and B: the pattern of energy transfer at point C is domin antly odd in v/bardbl,\nwhereas the patterns at points A and B are dominantly even in v/bardbl. When integrated\nover the parallel velocity to obtain the net change of energy of a sp ecies, an odd pattern\nlargelycancelsout,whereasanevenpatterndoesnot.Therefor e,thereislittlenetparticle\nenergization at point C, even though individual particles do gain and lo se energy through\nresonant interactions with E/bardbl. Particles with v/bardbl>0 gain nearly the same amount of\nenergy as that lost by particles with v/bardbl<0, yielding little net change of particle energy.\nThe important point that the field-particle correlation analysis here demonstrates is\nthat collisionless interactions of the Landau resonance between E/bardbland the ions and\nelectrons contribute to the spatially intermittent pattern of time- averaged particle en-\nergization, shown in Fig. 8(d). This result disproves by counterexa mple the commonly\nstated belief that Landau damping can only lead to spatially uniform pa rticle energiza-\ntion. Rather, we see clearly here that collisionless damping via the Lan dau resonance can\nindeed be responsible for spatially localized particle energization, as p reviously suggested\nin the literature (TenBarge & Howes 2013; Howes 2015, 2016). Fur thermore, the nonlin-Particle Energization in Alfv´ enic Current Sheets 23\nCE||,ix10-2\n(b)\n-4-3-2-101234\nv||/vti-8-6-4-202468\n01234567\n-5-3-1135t/T0\n10-2 x ∂wi / ∂tCE||,ix10-2\n(b) (a)\nCE||,ex10-1\n(d)\n-4-3-2-101234\nv||/vte-3-2-10123\n01234567\n-6-4-20246t/T0\n10-2 x ∂we / ∂tCE||,ex10-1\n(d) (c)\nFigure 11. Plots of the same field-particle correlation analysis as Fig . 10, but taken at point\nB.\near energy transfer by collisionless damping via the Landau resonan ce is not inhibited\nby the strong nonlinear interactions that play an important role in th is strong Alfv´ en\nwave collision simulation. Indeed, nonlinear gyrokinetic simulations of s trong, broadband\nplasma turbulence have indeed shown that the collisionless transfer of energy between\nfields and ions is dominated by particles approximately in Landau reson ance with the\nparallel phase velocity of Alfv´ enic fluctuations (Klein et al.2017).\n5.2.Particle Energization in Gyrotropic Velocity Space\nFinally, we can examine the distribution of particle energization in gyro tropic velocity\nspace (v/bardbl,v⊥) (Howes 2017) using the field-particle correlation CE/bardbl,s(v/bardbl,v⊥,t,τ). Al-\nthough plots of this analysis are limited to the correlation centered a t just a single point24 G. G. Howes, A. J. McCubbin, and K. G. Klein\nCE||,ix10-2\n(b)\n-4-3-2-101234\nv||/vti-2-1012\n01234567\n-3-2-10123t/T0\n10-3 x ∂wi / ∂tCE||,ix10-2\n(b) (a)\nCE||,ex10-2\n(d)\n-4-3-2-101234\nv||/vte-3-2-10123\n01234567\n-1-0.500.51t/T0\n10-2 x ∂we / ∂tCE||,ex10-2\n(d) (c)\nFigure 12. Plots of the same field-particle correlation analysis as Fig . 10, but taken at point\nC.\nin time, by not integrating over perpendicular velocity v⊥one obtains complete infor-\nmation about which particles in gyrotropic velocity space ( v/bardbl,v⊥) participate in the\ncollisionless transfer of energy. Note that the parallel correlation in gyrotropic velcotiy\nspace,CE/bardbl,s(v/bardbl,v⊥,t,τ) in the plots presented in this section is normalized by the energy\ntransfer rate per unit volume per unit velocity squared, Q0/v2\nti.\nIn Fig. 13, we plot CE/bardbl,s(v/bardbl,v⊥,t,τ) for the same correlation interval τ/T0= 0.992\ncentered at time t/T0= 2.10: (a) ion and (b) electron energization at point A, (c) ion\nand (d) electron energization at point B, and (e) ion and (f) electro n energization at\npoint C. As before, vertical solid and dashed black lines denote the r ange of resonant\nparallel velocities for Alfv´ en waves. Three important points can be inferred from theseParticle Energization in Alfv´ enic Current Sheets 25v⊥/vt,iCEx10-1\n(a)\n-4-3-2-101234\nv||/vt,i01234\n-9-6-30369\nv⊥/vt,e CE||,ex10-1\n(b)\n-4-3-2-101234\nv||/vt,e01234\n-5-3-1135v⊥/vt,iCE||,ix10-0\n(c)\n-4-3-2-101234\nv||/vt,i01234\n-3-2-10123\nv⊥/vt,e CE||,ex10-0\n(d)\n-4-3-2-101234\nv||/vt,e01234\n-2-1012v⊥/vt,iCE||,ix10-2\n(e)\n-4-3-2-101234\nv||/vt,i01234\n-5-3-1135\nv⊥/vt,e CE||,ex10-2\n(f)\n-4-3-2-101234\nv||/vt,e01234\n-2-1012\nFigure 13. Plots of the field-particle correlation CE/bardbl(v/bardbl,v⊥,t,τ) on gyrotropic velocity space\n(v/bardbl,v⊥) for a correlation interval τ/T0= 0.992 centered at time t/T0= 2.10: (a) ion and (b)\nelectron energization at point A, (c) ion and (d) electron en ergization at point B, and (e) ion and\n(f) electron energization at point C. Vertical solid lines d enote the resonant parallel velocities\nfor a parallel phase velocity at the Alfv´ en speed ω/(k/bardblvts) =vA/vts.\ngyrotropicvelocityspaceplots.First,otherthan asteadydecre aseofthe amplitude ofthe\nsignal with increasing v⊥—as expected because the equilibrium Maxwellian distribution\ndrops off exponentially as exp( −v2/v2\nts), so the amplitude of fluctuations δf(v) would be\nexpected to have a similar drop off in amplitude—the energy transfer shows very little\nvariation with v⊥. The variation in the energy transfer is organized almost entirely by v/bardbl,\nas expected for a Landau resonant energy transfer process. S econd, this energy transfer\nis dominated by particles with parallel velocities resonant with Alfv´ en ic fluctuations,\nv/bardbl≃vA, demonstrating that the energy transfer is governed by the Lan dau resonance.\nThird, the odd or even character in v/bardblat the different points A, B, and C seen in the\ntimestack plots is also clearly apparent here in these gyrotropic velo city space plots.\nSummarizing, the gyrotropic velocity space ( v/bardbl,v⊥) plots in Fig. 13 demonstrate how\nthe field-particle correlation technique maximizes the use of the full particle velocity26 G. G. Howes, A. J. McCubbin, and K. G. Klein\ndistribution function information, enabling the physical mechanism r esponsible for the\nremoval of energy from turbulent fluctuations and consequent p article energization to\nbe identified definitively. In this case, the velocity-space signature of the field-particle\ncorrelation is unmistakably that of Landau damping of a kinetic Alfv´ e n wave (Howes\n2017), proving that Landau damping indeed plays a role in the spatially intermittent\nremoval of the energy of electromagnetic and bulk plasma flow fluct uations, even in the\npresence of strong nonlinearity.\n6. Conclusion\nUsing a nonlinear gyrokinetic simulation of a strong Alfv´ en wave collisio n, we examine\nhere the damping of the electromagnetic fluctuations and the asso ciated energization\nof particles that occurs in current sheets that are generated se lf-consistently during the\nnonlinear evolution.\nThe flow of energy due to the collisionless damping and the associated particle ener-\ngization, as well as the subsequent thermalization of the particle en ergy by collisions,\nprovides an important framework for interpreting the nonlinear dy namics and dissipa-\ntion. Fig. 4 presents a simple model of the energy flow from turbulen t energy to plasma\nheat in the simulation, with the following two key stages: (i) the turbu lent fluctuation\nenergy is removed by collisionless field-particle interactions, transf erring that energy re-\nversibly into non-thermal energy of the plasma species; and (ii) the non-thermal energy,\nrepresented by fluctuations in the particle velocity distribution fun ctions, is driven to\nsufficiently small velocity-space scales that weak collisions can therm alize that energy, ir-\nreversibly heating the plasma species. In the strong Alfv´ en wave c ollision simulated here,\nthis two-step processes ultimately leads to more than 60% of the or iginal fluctuation\nenergy being dissipated collisionally as thermal ion and electron energ y.\nIt has long been appreciated that the nonlinear evolution of plasma t urbulence leads to\nthedevelopmentofintermittentcurrentsheets(Matthaeus & Mo ntgomery1980;Meneguzzi et al.\n1981), and a recent study has shown that strong Alfv´ en wave co llisions—nonlinear inter-\nactions between counterpropagatingAlfv´ enwaves—self-consis tentlydevelopintermittent\ncurrentsheets through the constructiveinterferenceofthe o riginalAlfv´ en wavesand non-\nlinearly generated fluctuations (Howes 2016). MHD turbulence simu lations have shown\nthat the dissipation of turbulent energy is largely concentrated in t hese intermittent\ncurrent sheets (Uritsky et al.2010; Osman et al.2011; Zhdankin et al.2013), so a nat-\nural question is whether the collisionless damping of current sheets generated by strong\nAlfv´ en wave collisions leads to such spatially intermittent particle ene rgization. Plotting\nthe spatial distribution of the electromagnetic work done by the pa rallel electric field\nE/bardbl, shown in Fig. 8(a)–(c), shows that the instantaneous particle en ergization is indeed\nspatially intermittent with a sheet-like morphology.\nA key point, however, is that much of this reversible electromagnet ic work leads to an\noscillatory transfer of energy to and from the particles associate d with undamped wave\nmotion. Only by averagingovera suitable time interval, in this case an a veraginginterval\nthat is approximately a single wave period τ≃T0, can we determine the secular particle\nenergization /angb∇acketleftj/bardblE/bardbl/angb∇acket∇ightτassociated with the net removal of energy from the turbulent fluc tu-\nations.Fig. 8(d) showsthat the secularparticleenergization /angb∇acketleftj/bardblE/bardbl/angb∇acket∇ightτin this strongAlfv´ en\nwave collision indeed remains spatially intermittent, although less localiz ed than the in-\nstantaneous rates of energy transfer in Fig. 8(a)–(c). The bot tom line is that the current\nsheets arising in strong Alfv´ en wave collisions indeed generate spat ially localized particle\nenergization, consistent with that found in simulations of plasma tur bulence (Wan et al.\n2012; Karimabadi et al.2013; TenBarge & Howes 2013; Wu et al.2013; Zhdankin et al.Particle Energization in Alfv´ enic Current Sheets 27\n2013) and inferred from spacecraft observations of the solar win d (Osman et al.2011,\n2012; Perri et al.2012; Wang et al.2013; Wu et al.2013; Osman et al.2014).\nThe next obvious question is what is the physical mechanism governin g the removal of\nenergy from the turbulence and the consequent spatially intermitt ent energization of the\nparticles?Usingtherecentlydevelopedfield-particlecorrelationte chnique(Klein & Howes\n2016; Howes et al.2017), weexamine howthe energytransfertoions and electronsb y the\nparallel electric field E/bardblis distributed in velocity space. In other words, which particles\nin velocity space receive the energy transferred collisionlessly from the electromagnetic\nfields? The results, exemplified by Fig. 10, show that the particles th at are resonant with\nthe parallel velocity of the Alfv´ en waves in the simulation dominate th e energy transfer,\ndemonstrating conclusively that Landau damping plays a role in the da mping of the\nelectromagnetic fluctuations and consequent energization of the particles in this strongly\nnonlinear simulation.\nBasedontheplane-wavedecompositiontypicallyusedtoderivelinear Landaudamping\nanalytically, one may naively expect that Landau damping leads to spa tially uniform\nenergization. Together, the results presented here definitively s how instead that Landau\ndamping can indeed lead to spatially intermittent particle energization . The comparison\nto a strictly linear simulation from identical initial conditions, present ed in Fig. 9, shows\nthat the nonlinear energy transfer to other Fourier modes is esse ntial for the localization\nofthe particle energization.This is consistent with the model forcu rrent sheet generation\nin Alfv´ en wave collisions in which nonlinearly generated modes constru ctively interfere\nwith theinitialAlfv´ enwavestocreatespatiallyintermittentcurren tsheets;linearLandau\ndamping of each of these modes, which occurs spatially locally, leads t o the intermittent\nspatial pattern of the energization, as previously suggested (Ho wes 2015, 2016).\nOur result here, that Landau damping is effective even in a plasma whe re strong non-\nlinear interactions are playing an important role, also addresses the important question\nof whether Landau damping is effective in a strongly turbulent plasma (Plunk 2013;\nSchekochihin et al.2016). Our results here complement a recent field-particle correla tion\nanalysis of gyrokinetic turbulence simulations showing that Landau d amping indeed per-\nsists as an effective physical mechanism for ion energization in broad band, strong plasma\nturbulence (Klein et al.2017).\nWe emphasize here that we have not shown that Landau damping is th eonlydamping\nmechanism, but we have provided conclusive evidence that Landau d amping does play\na role in the collisionless damping of turbulence in intermittent current sheets that arise\nfrom strong Alfv´ en wave collisions. As mentioned in Appendix B, tran sit-time damping\n(Barnes 1966; Quataert 1998)—which does work on particles via th eir magnetic mo-\nment through the magnetic mirror force arising from fluctuations in the magnetic field\nmagnitude—is another effective physical mechanism for collisionless d amping and ener-\ngization of particles via the Landau resonance in gyrokinetics. Here we have focused only\non the contribution to the particle energization by the parallel elect ric fieldE/bardbl; future\nwork will address the additional contribution by the magnetic mirror force arising from\n∇/bardbl|B|.\nThe comparisonof the time evolution ofthe field-particle energytra nsfer˙E(fp)\nsand the\ncollisional heating Qsfor each species in Fig. 5 also raises important questions about the\nrelative rates of linear and nonlinear phase-mixing processes that e nable the non-thermal\nenergy, represented by fluctuations in the particle velocity distrib ution function, to reach\nsufficiently small velocity-space scales to be thermalized by arbitrar ily weak collisions.\nThese questions, and more, about the flow of energy in weakly collisio nal heliospheric28 G. G. Howes, A. J. McCubbin, and K. G. Klein\nFigure 14. Comparison of the results from the linear collisionless gyr okinetic dispersion relation\nfor the reduced mass ratio mi/me= 36 (thick) and a realistic proton-to-electron mass ratio\nmi/me= 1836 (thin): (a) normalized frequency ω/k/bardblvAand (b) total collisionless damping rate\nγtot/ω(black solid), ion collisionless damping rate γi/ω(red dotted), and electron collisionless\ndamping rate γe/ω(blue dashed). Solid and dashed vertical lines are the same a s in Fig. 1.\nplasmas, lie at the forefront of kinetic heliophysics (Howes 2017), a nd will drive research\nefforts by the next generation of space plasma physicists.\nThis work was supported by NSF CAREER Award AGS-1054061, NASA HSR grant\nNNX16AM23G, DOE grant DE-SC0014599, and the University of Iow a Mathematical\nand Physical Sciences Funding Program. This work used the Extrem e Science and Engi-\nneering Discovery Environment (XSEDE), which is supported by Nat ional Science Foun-\ndation grant number ACI-1053575, through NSF XSEDE Award PHY 090084.\nAppendix A. Collisionless Damping Rate as a Function of Mass Ratio\nHere we present in Fig. 14 a comparison of the linear physics of the Alf v´ en/kinetic\nAlfv´ en wave mode for the reduced mass ratio used here mi/me= 36 (thick lines) to\nthat for a realistic proton-to-electron mass ratio mi/me= 1836 (thin lines). Specifically,\nfor a plasma with βi= 1 and Ti/Te= 1, we solve the linear collisionless gyrokinetic\ndispersionrelation(Howes et al.2006) for the (a) normalizedwavefrequency ω/k/bardblvAand\n(b) total collisionless damping rate γtot/ω(black solid) as a function of the perpendicular\nwavenumber k⊥ρi. In addition, in panel (b) we also show the separate contributions o f\nthe ions (red dotted) and electrons (blue dashed) to the collisionles s damping rate.\nThe comparison shows that the linear wave frequency ω/k/bardblvAbegins to differ only\nslightly between the two cases at k⊥ρi/greaterorsimilar5. Note that the fully resolved perpendicu-\nlar range of the dealiased pseudospectral method for the strong Alfv´ en wave collision\nsimulation covers 0 .25/lessorequalslantk⊥ρi/lessorequalslant5.25, denoted by the two vertical solid black lines;\nmodes in the corner of ( kx,ky) Fourier space represent perpendicular wavenumbers out\ntok⊥ρi= 5.25√\n2≃7.42, denoted by the vertical dashed black line. Therefore, there\nis very little difference in the linear wave frequency over the perpend icular range of the\nsimulation between mi/me= 36 and mi/me= 1836.\nThe noticeable difference between the two cases arises in the electr on collisionlessParticle Energization in Alfv´ enic Current Sheets 29\ndamping rate γe/ωin Fig. 14(b). The ion damping is slightly smaller for the realistic\nmass ratio relative to the reduced mass ratio, but the electron dam ping drops by nearly\nanorderofmagnitude.Otherthantheamplitudechanges,howeve r,the individualspecies\ndamping rates γs/ωon a log-log plot have the same the functional form, only a different\nrelative weighting. The reduced mass ratio case has nearly a factor of ten larger rela-\ntive contribution to the collisionless damping than the realistic mass ra tio. Note that\nsignificant collisionless damping of a wave occurs when γ/ω/greaterorsimilar0.1 (marked by a horizon-\ntal dashed line), so total collisionless damping is relatively weak over t he perpendicular\nrange of the simulation for a realistic mass ratio mi/me= 1836, whereas the damping is\nvery strong with γtot/ω∼1.0 at the smallest perpendicular scales for the reduced mass\nratiomi/me= 36. This enables collisionless damping to remove energy from the tur bu-\nlent fluctuations completely over the resolve range of scales, avoid ing any problematic\nbottlenecks at the smallest scales.\nNote that reducing the mass ratio to values mi/me<32 (not shown) leads to a\nsignificant qualitative change from the behavior of the collisionless da mping shown here.\nAt these very low mass ratio values, the ion collisionless damping does n ot drop off at\nk⊥ρi≫1, potentially leading to qualitatively incorrect results about the rela tive ion and\nelectron damping (Klein et al.2017).\nAppendix B. Particle Energization by Component and Species\nIntheVlasov-Maxwellsystemofequations,therateofchangeof particleenergydensity\nat a given position in space is given by the rate of electromagnetic wor k,j·E(Kleinet al.\n2017), confirming the familiar concept the only the electric field can c hange the energy of\ncharged particles. In the low-frequency limit of kinetic plasma theor y, one may average\ntheVlasov-Maxwellequationsoverthegyrophase θincylindricalvelocityspace( v/bardbl,v⊥,θ)\nto obtain the reduced system of gyrokinetics (Frieman & Chen 1982 ; Howeset al.2006).\nThe benefit of this procedure is the reduction of velocity space to t wo dimensions ( v/bardbl,v⊥)\nat the expense of discarding the physics at cyclotron frequencies and higher; effectively,\nthe cyclotron resonances and fast magnetosonic waves are elimina ted, while retaining\nfinite Larmor radius effects and collisionless damping via the Landau re sonance.\nIn addition to this elimination of the cyclotron resonances,a compon ent of the perpen-\ndicular electromagnetic work, j⊥·E⊥is alternatively expressed in terms of the magnetic\nmirror force, Fmir=−µ∇/bardbl|B|, where the magnetic moment of a particle is given by\nµs=msv2\n⊥/2|B|. In the anisotropic limit k/bardbl≪k⊥of the gyrokinetic approximation,\nthe change in the magnetic field magnitude is dominated by the variatio n in the parallel\ncomponent of the field, δ|B|=δB/bardbl+O(|δB|2). For electromagnetic waves with a fluc-\ntuation in the magnetic field strength, the magnetic mirror force −µ∇/bardblδB/bardblacting on\nthe magnetic moment µof the particle gyromotion, leads to collisionless damping of the\nwave via the Landau resonance, a process denoted by the term tr ansit-time damping, or\nalternatively called Barnes damping (Barnes 1966; Quataert 1998) .\nThe bottom line is in gyrokinetics there are two separate mechanisms that can lead\nto resonant collisionless particle energization: Landau damping media ted by the parallel\nelectric field E/bardbland transit-time damping mediated by gradients in the parallel magne tic\nfield perturbation ∇/bardblδB/bardbl. In this paper, we focus solely on the particle energization by\nLandau damping through the parallel electric field E/bardbl, leaving a detailed analysis of\ntransit-time damping to future work.\nAlthough gyrokinetics eliminates cyclotron resonant heating, it still describes the elec-\ntromagnetic work done by all three components of j·E. The primary focus of this paper\nis resonant heating by Landau damping through the parallel electric fieldE/bardbl, so plots in30 G. G. Howes, A. J. McCubbin, and K. G. Klein\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC −0.6−0.30.00.30.6\njxEx/Q0(a)\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC −0.6−0.30.00.30.6\njyEy/Q0(b)\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC−303\nj/bardblE/bardbl/Q0(c)\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC−303\nj·E/Q0(d)\nFigure 15. Plots of the different components of the electromagnetic wor k (a)/angbracketleftjxEx/angbracketrightτ, (b)\n/angbracketleftjyEy/angbracketrightτ, and (c) /angbracketleftj/bardblE/bardbl/angbracketrightτ, as well as the total work (d) /angbracketleftj·E/angbracketrightτaveraged over a single wave\nperiodτ= 0.992T0centered at time t/T0= 1.86.\nthe body of paper focus only on the parallel contribution j/bardblE/bardbl. In Fig. 15, we present\nhere for completeness the three components of the electromagn etic work (a) /angb∇acketleftjxEx/angb∇acket∇ightτ, (b)\n/angb∇acketleftjyEy/angb∇acket∇ightτ, and (c) /angb∇acketleftj/bardblE/bardbl/angb∇acket∇ightτaveraged over single wave period τ= 0.992T0centered at time\nt/T0= 1.86. The components jxExandjyEydominantly represent the energy transfer\nbetween fields and particles associated with undamped wave motion, for example repre-\nsenting magnetic tension as the restoring force for the Alfv´ en wa ve. Therefore, jxExand\njyEyrepresent oscillatory energy transfer, and averaged over one w ave period there is\nvery little net energy change. By comparison, the single-wave perio d averaged /angb∇acketleftj/bardblE/bardbl/angb∇acket∇ightτ\nrepresentsthesecularenergytransferassociatedwith collisionle ssdamping,ismuchlarger\nthan that for jxExorjyEy. Note that (d) the total single-wave period averaged total\nelectromagnetic work /angb∇acketleftj·E/angb∇acket∇ightτis dominated by the parallel component /angb∇acketleftj/bardblE/bardbl/angb∇acket∇ightτ. This com-\nparison motivates our focus in the body of this paper on the parallel contribution to the\nelectromagnetic work, j/bardblE/bardbl.\nWe also plot separately the parallel electromagnetic work on the ions j/bardbl,iE/bardblin Fig. 16\nand on the electrons j/bardbl,eE/bardblin Fig. 17. The spatial patterns of the instantaneous rate\nof work at t/T0= (a) 1.75, (b) 1.86, and (c) 2.03, as well as (d) the /angb∇acketleftj/bardbl,sE/bardbl/angb∇acket∇ightτaveraged\nover one full wave period τ= 0.992T0centered at time t/T0= 1.86, are similar for both\nspecies, but the rate ofelectron energization in this plane is about t wice the magnitude of\nthat for the ions. Since the ion and electronlinear damping ratesare similar for k⊥ρi/lessorsimilar1,\nthis may suggest significant energy removal at higher k⊥ρi>1 where electrons are\nexpected toreceiveagreatershareofthe removedturbulenten ergy.AfutureexaminationParticle Energization in Alfv´ enic Current Sheets 31\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC\n−6−3036\nj/bardbl,iE/bardbl/Q0(a)\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC −6−3036\nj/bardbl,iE/bardbl/Q0(b)\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC−303\nj/bardbl,iE/bardbl/Q0(c)\n0 5 10 15 20 25\nx/ρi0510152025y/ρi\nAB\nC\n−1.8−1.2−0.60.00.61.21.8\nj/bardbl,iE/bardbl/Q0(d)\nFigure 16. Plots of the instantaneous rate of parallel electromagneti c work on the ions j/bardbl,iE/bardbl\nand contours of the parallel vector potential A/bardbl(contours, positive solid, negative dashed) at\ntimest/T0= (a) 1.75, (b) 1.86, and (c) 2.03, as well as (d) /angbracketleftj/bardbl,iE/bardbl/angbracketrightτaveraged over one full wave\nperiodτ= 0.992T0centered at time t/T0= 1.86.\nof the ion and electron energization will investigate the typical lengt h scale at which\nparticles are energized in more detail.\nREFERENCES\nAbel, I. G., Barnes, M., Cowley, S. C., Dorland, W. & Schekoch ihin, A. A. 2008\nLinearized model Fokker-Planck collision operators for gy rokinetic simulations. I. Theory.\nPhys. Plasmas 15(12), 122509.\nBarnes, A. 1966 Collisionless Damping of Hydromagnetic Waves. Phys. Fluids 9, 1483–1495.\nBarnes, M., Abel, I. G., Dorland, W., Ernst, D. R., Hammett, G . W., Ricci, P.,\nRogers, B. N., Schekochihin, A. 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J.\n771, 124." }, { "title": "0807.5009v1.Scattering_Theory_of_Gilbert_Damping.pdf", "content": "arXiv:0807.5009v1 [cond-mat.mes-hall] 31 Jul 2008Scattering Theory of Gilbert Damping\nArne Brataas,1,∗Yaroslav Tserkovnyak,2and Gerrit E. W. Bauer3\n1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n2Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\nThe magnetization dynamics of a single domain ferromagnet i n contact with a thermal bath\nis studied by scattering theory. We recover the Landau-Lift shitz-Gilbert equation and express the\neffective fields and Gilbert damping tensor in terms of the sca ttering matrix. Dissipation of magnetic\nenergy equals energy current pumped out of the system by the t ime-dependent magnetization, with\nseparable spin-relaxation induced bulk and spin-pumping g enerated interface contributions. In\nlinear response, our scattering theory for the Gilbert damp ing tensor is equivalent with the Kubo\nformalism.\nMagnetization relaxation is a collective many-body\nphenomenon that remains intriguing despite decades of\ntheoretical and experimental investigations. It is im-\nportant in topics of current interest since it determines\nthe magnetization dynamics and noise in magnetic mem-\nory devices and state-of-the-art magnetoelectronic ex-\nperiments on current-induced magnetization dynamics\n[1]. Magnetization relaxation is often described in terms\nof a damping torque in the phenomenological Landau-\nLifshitz-Gilbert (LLG) equation\n1\nγdM\ndτ=−M×Heff+M×/bracketleftBigg˜G(M)\nγ2M2sdM\ndτ/bracketrightBigg\n, (1)\nwhereMis the magnetization vector, γ=gµB//planckover2pi1is the\ngyromagnetic ratio in terms of the gfactor and the Bohr\nmagnetonµB, andMs=|M|is the saturation magneti-\nzation. Usually, the Gilbert damping ˜G(M) is assumed\nto be a scalar and isotropic parameter, but in general it\nis a symmetric 3 ×3 tensor. The LLG equation has been\nderived microscopically [2] and successfully describes the\nmeasured response of ferromagnetic bulk materials and\nthin films in terms of a few material-specific parameters\nthatareaccessibletoferromagnetic-resonance(FMR) ex-\nperiments [3]. We focus in the following on small fer-\nromagnets in which the spatial degrees of freedom are\nfrozen out (macrospin model). Gilbert damping pre-\ndicts a striclylinear dependence ofFMR linewidts on fre-\nquency. This distinguishes it from inhomogenous broad-\nening associated with dephasing of the global precession,\nwhich typically induces a weaker frequency dependence\nas well as a zero-frequency contribution.\nThe effective magnetic field Heff=−∂F/∂Mis the\nderivative of the free energy Fof the magnetic system\nin an external magnetic field Hext, including the classi-\ncal magnetic dipolar field Hd. When the ferromagnet is\npart of an open system as in Fig. 1, −∂F/∂Mcan be\nexpressed in terms of a scattering S-matrix, quite anal-\nogous to the interlayer exchange coupling between ferro-\nmagnetic layers [4]. The scattering matrix is defined in\nthe space of the transport channels that connect a scat-\ntering region (the sample) to thermodynamic (left andleft\nreservoirF N Nright\nreservoir\nFIG. 1: Schematic picture of a ferromagnet (F) in contact\nwith a thermal bath via metallic normal metal leads (N).\nright) reservoirs by electric contacts that are modeled by\nideal leads. Scattering matrices also contain information\nto describe giant magnetoresistance, spin pumping and\nspin battery, and current-induced magnetization dynam-\nics in layered normal-metal (N) |ferromagnet (F) systems\n[4, 5, 6].\nIn the following we demonstrate that scattering the-\nory can be also used to compute the Gilbert damping\ntensor˜G(M).The energy loss rate of the scattering re-\ngion can be described in terms of the time-dependent\nS-matrix. Here, we generalize the theory of adiabatic\nquantum pumping to describe dissipation in a metallic\nferromagnet. Our idea is to evaluate the energy pump-\ningoutoftheferromagnetandtorelatethistotheenergy\nloss of the LLG equation. We find that the Gilbert phe-\nnomenology is valid beyond the linear response regime of\nsmall magnetization amplitudes. The only approxima-\ntion that is necessary to derive Eq. (1) including ˜G(M)\nis the (adiabatic) assumption that the frequency ωof the\nmagnetization dynamics is slow compared to the relevant\ninternal energy scales set by the exchange splitting ∆.\nThe LLG phenomenology works so well because /planckover2pi1ω≪∆\nsafely holds for most ferromagnets.\nGilbert damping in transition-metal ferromagnets is\ngenerally believed to stem from spin-orbit interaction in\ncombinationwith impurityscatteringthattransfersmag-\nnetic energy to itinerant quasiparticles [3]. The subse-\nquent drainage of the energy out of the electronic sys-\ntem,e.g.by inelastic scattering via phonons, is believed\nto be a fast process that does not limit the overall damp-\ning. Our key assumption is adiabaticiy, meaning that\nthe precession frequency goes to zero before letting the\nsample size become large. The magnetization dynam-\nics then heats up the entire magnetic system by a tiny2\namount that escapes via the contacts. The leakage heat\ncurrent then equals the total dissipation rate. For suf-\nficiently large samples, bulk heat production is insensi-\ntive to the contact details and can be identified as an\nadditive contribution to the total heat current that es-\ncapes via the contacts. The chemical potential is set\nby the reservoirs, which means that (in the absence of\nan intentional bias) the sample is then always very close\nto equilibrium. The S-matrix expanded to linear order\nin the magnetization dynamics and the Kubo linear re-\nsponse formalisms should give identical results, which we\nwill explicitly demonstrate. The role of the infinitesi-\nmal inelastic scattering that guarantees causality in the\nKubo approach is in the scattering approach taken over\nby the coupling to the reservoirs. Since the electron-\nphonon relaxation is not expected to directly impede the\noverall rate of magnetic energy dissipation, we do not\nneed to explicitly include it in our treatment. The en-\nergy flow supported by the leads, thus, appears in our\nmodel to be carried entirely by electrons irrespective of\nwhethertheenergyisactuallycarriedbyphonons, incase\nthe electrons relax by inelastic scattering before reaching\nthe leads. So we are able to compute the magnetization\ndamping, but not, e.g., how the sample heats up by it .\nAccording to Eq. (1), the time derivative of the energy\nreads\n˙E=Heff·dM/dτ= (1/γ2)˙ m/bracketleftBig\n˜G(m)˙ m/bracketrightBig\n,(2)\nin terms of the magnetization direction unit vector m=\nM/Msand˙ m=dm/dτ. We now develop the scatter-\ning theory for a ferromagnet connected to two reservoirs\nby normal metal leads as shown in Fig. 1. The total\nenergy pumping into both leads I(pump)\nEat low tempera-\ntures reads [11, 12]\nI(pump)\nE= (/planckover2pi1/4π)Tr˙S˙S†, (3)\nwhere˙S=dS/dτandSis the S-matrix at the Fermi\nenergy:\nS(m) =/parenleftbiggr t′\nt r′/parenrightbigg\n. (4)\nrandt(r′andt′) are the reflection and transmissionma-\ntrices spanned by the transport channels and spin states\nfor an incoming wave from the left (right). The gener-\nalization to finite temperatures is possible but requires\nknowledge of the energy dependence of the S-matrix\naround the Fermi energy [12]. The S-matrix changes\nparametrically with the time-dependent variation of the\nmagnetization S(τ) =S(m(τ)). We obtain the Gilbert\ndamping tensor in terms of the S-matrix by equating the\nenergy pumping by the magnetic system (3) with the en-\nergy loss expression (2), ˙E=I(pump)\nE. Consequently\nGij(m) =γ2/planckover2pi1\n4πRe/braceleftbigg\nTr/bracketleftbigg∂S\n∂mi∂S†\n∂mj/bracketrightbigg/bracerightbigg\n,(5)which is our main result.\nThe remainder of our paper serves three purposes. We\nshow that (i) the S-matrix formalism expanded to linear\nresponseis equivalentto Kubolinearresponseformalism,\ndemonstrate that (ii) energy pumping reduces to inter-\nface spin pumping in the absence ofspin relaxationin the\nscattering region, and (iii) use a simple 2-band toy model\nwith spin-flip scattering to explicitly show that we can\nidentify both the disorder and interface (spin-pumping)\nmagnetization damping as additive contributions to the\nGilbert damping.\nAnalogous to the Fisher-Lee relation between Kubo\nconductivity and the Landauer formula [15] we will now\nprove that the Gilbert damping in terms of S-matrix (5)\nis consistent with the conventionalderivation of the mag-\nnetization damping by the linear response formalism. To\nthis end we chose a generic mean-field Hamiltonian that\ndepends on the magnetization direction m:ˆH=ˆH(m)\ndescribes the system in Fig. 1. ˆHcan describe realistic\nband structures as computed by density-functional the-\nory including exchange-correlation effects and spin-orbit\ncouplingaswell normaland spin-orbitinduced scattering\noff impurities. The energy dissipation is ˙E=/angb∇acketleftdˆH/dτ/angb∇acket∇ight,\nwhere/angb∇acketleft.../angb∇acket∇ightdenotes the expectation value for the non-\nequilibriumstate. Inlinearresponse,weexpandthemag-\nnetization direction m(t) around the equilibrium magne-\ntization direction m0,\nm(τ)=m0+u(τ). (6)\nThe Hamiltonian can be linearized as ˆH=ˆHst+\nui(τ)∂iˆH, where ˆHst≡ˆH(m0) is the static Hamilto-\nnian and∂iˆH≡∂uiˆH(m0), where summation over re-\npeated indices i=x,y,zis implied. To lowest order\n˙E= ˙ui(τ)/angb∇acketleft∂iˆH/angb∇acket∇ight, where\n/angb∇acketleft∂iˆH/angb∇acket∇ight=/angb∇acketleft∂iˆH/angb∇acket∇ight0+/integraldisplay∞\n−∞dτ′χij(τ−τ′)uj(τ′).(7)\n/angb∇acketleft.../angb∇acket∇ight0denotes equilibrium expectation value and the re-\ntarded correlation function is\nχij(τ−τ′) =−i\n/planckover2pi1θ(τ−τ′)/angbracketleftBig\n[∂iˆH(τ),∂jˆH(τ′)]/angbracketrightBig\n0(8)\nin the interaction picture for the time evolution. In order\nto arrive at the adiabatic (Gilbert) damping the magne-\ntization dynamics has to be sufficiently slow such that\nuj(τ)≈uj(t) + (τ−t) ˙uj(t). Since m2= 1 and hence\n˙ m·m= 0 [7]\n˙E=i∂ωχij(ω→0)˙ui˙uj, (9)\nwhereχij(ω) =/integraltext∞\n−∞dτχij(τ)exp(iωτ). Next, we use\nthe scattering states as the basis for expressing the\ncorrelation function (8). The Hamiltonian consists of\na free-electron part and a scattering potential: ˆH=\nˆH0+ˆV(m). We denote the unperturbed eigenstates of3\nthe free-electron Hamiltonian ˆH0=−/planckover2pi12∇2/2mat en-\nergyǫby|ϕs,q(ǫ)/angb∇acket∇ight, wheres=l,rdenotes propagation\ndirection and qtransverse quantum number. The po-\ntentialˆV(m) scatters the particles between these free-\nelectron states. The outgoing (+) and incoming wave\n(-) eigenstates |ψ(±)\ns,q(ǫ)/angb∇acket∇ightof the static Hamiltonian ˆHst\nfulfill the completeness conditions /angb∇acketleftψ(±)\ns,q(ǫ)|ψ(±)\ns′,q′(ǫ′)/angb∇acket∇ight=\nδs,s′δq,q′δ(ǫ−ǫ′) [10]. These wave functions can be ex-\npressed as |ψ(±)\ns(ǫ)/angb∇acket∇ight= [1 +ˆG(±)\nstˆVst]|ϕs(ǫ)/angb∇acket∇ight, where the\nstatic retarded (+) and advanced (-) Green functions are\nˆG(±)\nst(ǫ) = (ǫ±iη−ˆHst)−1andηis a positive infinites-\nimal. By expanding χij(ω) in the basis of the outgo-\ning wave functions |ψ(+)\ns/angb∇acket∇ight, the low-temperature linear re-\nsponse leads to the followingenergydissipation (9) in the\nadiabatic limit\n˙E=−π/planckover2pi1˙ui˙uj/angbracketleftBig\nψ(+)\ns,q|∂iˆH|ψ(+)\ns′,q′/angbracketrightBig/angbracketleftBig\nψ(+)\ns′,q′|∂jˆH|ψ(+)\ns,q/angbracketrightBig\n,\n(10)\nwith wave functions evaluated at the Fermi energy ǫF.\nIn order to compare the linear response result, Eq.\n(10), withthat ofthe scatteringtheory, Eq. (5), weintro-\nduce the T-matrix ˆTasˆS(ǫ;m) = 1−2πiˆT(ǫ;m), where\nˆT=ˆV[1 +ˆG(+)ˆT] in terms of the full Green function\nˆG(+)(ǫ,m) = [ǫ+iη−ˆH(m)]−1. Although the adiabatic\nenergy pumping (5) is valid for any magnitude of slow\nmagnetization dynamics, in order to make connection to\nthe linear-response formalism we should consider small\nmagnetization changes to the equilibrium values as de-\nscribed by Eq. (6). We then find\n∂τˆT=/bracketleftBig\n1+ˆVstˆG(+)\nst/bracketrightBig\n˙ui∂iˆH/bracketleftBig\n1+ˆG(+)\nstˆVst/bracketrightBig\n.(11)\ninto Eq. (5) and using the completeness of the scattering\nstates, we recover Eq. (10).\nOur S-matrix approach generalizes the theory of (non-\nlocal) spin pumping and enhanced Gilbert damping in\nthin ferromagnets [5]: by conservation of the total an-\ngular momentum the spin current pumped into the\nsurrounding conductors implies an additional damping\ntorque that enhances the bulk Gilbert damping. Spin\npumping is an N |F interfacial effect that becomes impor-\ntant in thin ferromagnetic films [14]. In the absence of\nspin relaxation in the scattering region, the S-matrix can\nbe decomposed as S(m) =S↑(1+ˆσ·m)/2+S↓(1−ˆσ·\nm)/2, where ˆσis a vector of Pauli matrices. In this case,\nTr(∂τS)(∂τS)†=Ar˙ m2, whereAr= Tr[1−ReS↑S†\n↓]\nand the trace is over the orbital degrees of freedom only.\nWe recover the diagonal and isotropic Gilbert damping\ntensor:Gij=δijGderived earlier [5], where\nG=γMsα=(gµB)2\n4π/planckover2pi1Ar. (12)\nFinally, we illustrate by a model calculation that\nwe can obtain magnetization damping by both spin-\nrelaxationandinterfacespin-pumpingfromtheS-matrix.We consider a thin film ferromagnet in the two-band\nStoner model embedded in a free-electron metal\nˆH=−/planckover2pi12\n2m∇2+δ(x)ˆV(ρ), (13)\nwhere the in-plane coordinate of the ferromagnet is ρ\nand the normal coordinate is x.The spin-dependent po-\ntentialˆV(ρ) consists of the mean-field exchange interac-\ntion oriented along the magnetization direction mand\nmagnetic disorder in the form of magnetic impurities Si\nˆV(ρ) =νˆσ·m+/summationdisplay\niζiˆσ·Siδ(ρ−ρi),(14)\nwhich are randomly oriented and distributed in the film\natx= 0. Impurities in combination with spin-orbit cou-\npling will give similar contributions as magnetic impuri-\nties to Gilbert damping. Our derivation of the S-matrix\nclosely follows Ref. [8]. The 2-component spinor wave\nfunction can be written as Ψ( x,ρ) =/summationtext\nk/bardblck/bardbl(x)Φk/bardbl(ρ),\nwhere the transverse wave function is Φ k/bardbl(ρ) = exp(ik/bardbl·\nρ)/√\nAfor the cross-sectional area A. The effective one-\ndimensional equation for the longitudinal part of the\nwave function is then\n/bracketleftbiggd2\ndx2+k2\n⊥/bracketrightbigg\nck/bardbl(x) =/summationdisplay\nk′\n/bardbl˜Γk/bardbl,k′\n/bardblck/bardbl(0)δ(x),(15)\nwhere the matrix elements are defined by ˜Γk/bardbl,k′\n/bardbl=\n(2m//planckover2pi12)/integraltext\ndρΦ∗\nk/bardbl(ρ)ˆV(ρ)Φk′\n/bardbl(ρ)and the longitudinal\nwave vector k⊥is defined by k2\n⊥= 2mǫF//planckover2pi12−k2\n/bardbl. For\nan incoming electron from the left, the longitudinal wave\nfunction is\nck/bardbls=χs√k⊥/braceleftBigg\neik⊥xδk/bardbls,k′\n/bardbls′+e−ik⊥xrk/bardbls,k′\n/bardbls′,x<0\neik⊥xtk/bardbls,k′\n/bardbls′,x>0,\n(16)\nwheres=↑,↓andχ↑= (1,0)†andχ↓= (0,1)†. Inver-\nsion symmetry dictates that t′=tandr=r′. Continu-\nity of the wave function requires 1+ r=t. The energy\npumping (3) then simplifies to I(pump)\nE=/planckover2pi1Tr/parenleftbig˙t˙t†/parenrightbig\n/π.\nFlux continuity gives t= (1 +iˆΓ)−1, whereˆΓk/bardbls,k′\n/bardbls′=\nχ†\nsˆΓk/bardbls,k′\n/bardbls′χs′(4k⊥k⊥)−1/2.\nIn the absence of spin-flip scattering, the transmis-\nsion coefficient is diagonal in the transverse momentum:\nt(0)\nk/bardbl= [1−iη⊥σ·m]/(1+η2\n⊥), whereη⊥=mν/(/planckover2pi12k⊥).\nThe nonlocal (spin-pumping) Gilbert damping is then\nisotropic,Gij(m) =δijG′,\nG′=2ν2/planckover2pi1\nπ/summationdisplay\nk/bardblη2\n⊥\n(1+η2\n⊥)2. (17)\nIt can be shown that G′is a function of the ratio be-\ntween the exchange splitting versus the Fermi wave vec-\ntor,ηF=mν/(/planckover2pi12kF).G′vanishes in the limits ηF≪14\n(nonmagnetic systems) and ηF≫1 (strong ferromag-\nnet).\nWe include weak spin-flip scattering by expanding the\ntransmission coefficient tto second order in the spin-\norbit interaction, t≈/bracketleftbigg\n1+t0iˆΓsf−/parenleftBig\nt0iˆΓsf/parenrightBig2/bracketrightbigg\nt0, which\ninserted into Eq. (5) leads to an in general anisotropic\nGilbert damping. Ensemble averaging over all ran-\ndom spin configurations and positions after considerable\nbut straightforward algebra leads to the isotropic result\nGij(m) =δijG\nG=G(int)+G′(18)\nwhereG′is defined in Eq. (17). The “bulk” contribution\nto the damping is caused by the spin-relaxation due to\nthe magnetic disorder\nG(int)=NsS2ζ2ξ, (19)\nwhereNsis the number of magnetic impurities, Sis the\nimpurity spin, ζis the average strength of the magnetic\nimpurity scattering, and ξ=ξ(ηF) is a complicated ex-\npression that vanishes when ηFis either very small or\nvery large. Eq. (18) proves that Eq. (5) incorporates the\n“bulk” contribution to the Gilbert damping, which grows\nwith the number of spin-flip scatterers, in addition to in-\nterface damping. We could have derived G(int)[Eq. (19)]\nas well by the Kubo formula for the Gilbert damping.\nThe Gilbert damping has been computed before based\non the Kubo formalism based on first-principles elec-\ntronic band structures [9]. However, the ab initio appeal\nis somewhat reduced by additional approximations such\nas the relaxation time approximation and the neglect of\ndisorder vertex corrections. An advantage of the scatter-\ningtheoryofGilbertdampingisitssuitabilityformodern\nab initio techniques of spin transport that do not suffer\nfrom these drawbacks [16]. When extended to include\nspin-orbit coupling and magnetic disorder the Gilbert\ndamping can be obtained without additional costs ac-\ncording to Eq. (5). Bulk and interface contributions can\nbe readily separated by inspection of the sample thick-\nness dependence of the Gilbert damping.\nPhononsareimportantforthe understandingofdamp-\ning at elevated temperatures, which we do not explic-\nitly discuss. They can be included by a temperature-\ndependent relaxation time [9] or, in our case, structural\ndisorder. A microscopic treatment of phonon excitations\nrequires extension of the formalism to inelastic scatter-\ning, which is beyond the scope of the present paper.\nIn conclusion, we hope that our alternative formal-\nism of Gilbert damping will stimulate ab initio electronic\nstructure calculations as a function of material and dis-\norder. By comparison with FMR studies on thin ferro-\nmagnetic films this should lead to a better understanding\nof dissipation in magnetic systems.This work was supported in part by the Re-\nsearch Council of Norway, Grants Nos. 158518/143 and\n158547/431, and EC Contract IST-033749 “DynaMax.”\n∗Electronic address: Arne.Brataas@ntnu.no\n[1] For a review, see M. D. Stiles and J. Miltat, Top. Appl.\nPhys.101, 225 (2006) , and references therein.\n[2] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y, Phys. Sta-\ntus Solidi 23, 501 (1967); V. Kambersky, Can. J. Phys.\n48, 2906 (1970); V. Korenman, and R. E. Prange, Phys.\nRev. B6, 2769 (1972); V. S. 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Lett 99, 246603\n(2007).\n[15] D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981).\n[16] M. Zwierzycki et al., Phys. Stat. Sol. B 245, 623 (2008)" }, { "title": "1507.07724v1.Spatial_damping_of_propagating_sausage_waves_in_coronal_cylinders.pdf", "content": "arXiv:1507.07724v1 [astro-ph.SR] 28 Jul 2015Astronomy&Astrophysics manuscriptno.ver1.0 c∝circlecopyrtESO2018\nJanuary6,2018\nSpatialdamping of propagating sausagewavesincoronal\ncylinders\nMing-Zhe Guo1,2, Shao-Xia Chen1, BoLi1,Li-Dong Xia1,and Hui Yu1\n1Shandong Provincial Key Laboratory of Optical Astronomy an d Solar-Terrestrial Environment, and Institute of Space Sc iences,\nShandong UniversityWeihai, Weihai264209, China\ne-mail:bbl@sdu.edu.cn\n2CAS Key Laboratory of Geospace Environment, University of S cience & Technology of China, Chinese Academy of Sciences,\nHefei230026, China\nABSTRACT\nContext. Sausage modes are important in coronal seismology. Spatial ly damped propagating sausage waves were recently observed\ninthe solar atmosphere.\nAims.Weexamine how waveleakage influences thespatialdamping of sausage wavespropagating alongcoronal structuresmodele d\nbya cylindrical density enhancement embedded ina uniform m agnetic field.\nMethods. Workingintheframeworkofcoldmagnetohydrodynamics, wes olvethedispersionrelation(DR)governingsausagewaves\nforcomplex-valuedlongitudinalwavenumber katgivenrealangularfrequencies ω.Forvalidationpurposes,wealsoprovideanalytical\napproximations to the DR in the low-frequency limit and inth e vicinity ofωc, the criticalangular frequency separating trapped from\nleakywaves.\nResults. In contrast to the standing case, propagating sausage waves are allowed forωmuch lower thanωc. However, while able\nto direct their energy upwards, these low-frequency waves a re subject to substantial spatial attenuation. The spatial damping length\nshows little dependence on the density contrast between the cylinder and its surroundings, and depends only weakly on fr equency.\nThis spatial damping length is of the order of the cylinder ra dius forω/lessorsimilar1.5vAi/a, whereaandvAiare the cylinder radius and the\nAlfvénspeed inthe cylinder, respectively.\nConclusions. If a coronal cylinder is perturbed by symmetric boundary dri vers (e.g., granular motions) with a broadband spectrum,\nwave leakage efficientlyfiltersout the low-frequency components.\nKey words. magnetohydrodynamics (MHD)–Sun:corona – Sun:magnetic fie lds–waves\n1. Introduction\nConsiderable progress has been made in coronal seis-\nmology thanks to the abundantly identified waves and\noscillations in the structured solar atmosphere (for a re-\ncent review, see De Moortel& Nakariakov 2012, and\nalso Ballester et al. 2007, Nakariakov&Erdélyi 2009,\nErdélyi& Goossens 2011 for three recent topical issues).\nEqually important is a detailed theoretical understanding of the\ncollective wave modessupportedby magnetizedcylinders(e .g.,\nRoberts 2000). While kink waves (with azimuthal wavenumber\nm=1) have attracted much attention since their measurements\nwithTRACE(Nakariakovet al.1999;Aschwandenet al.1999),\nsausage waves prove important in interpreting second-scal e\nquasi-periodic pulsations (QPPs) in the lightcurves of sol ar\nflares (see Nakariakov&Melnikov 2009, for a recent review).\nTheir importance is strengthened given their recent detec-\ntionin both the chromosphere (Mortonet al. 2012) and the\nphotosphere(Dorotovi ˇcet al. 2014;Grantet al.2015).\nStanding sausage modes are well understood. For in-\nstance, two distinct regimes are known to exist, depending\non the longitudinal wavenumber k(Nakariakov&Verwichte\n2005). The trapped regime results when kexceeds some\ncritical value kc, whereas the leaky regime arises when the\nopposite is true. Both eigen-mode analyses (Kopylovaetal.\n2007; VasheghaniFarahaniet al. 2014) and numerical sim-ulations from an initial-value-problem perspective (e.g. ,\nNakariakovetal. 2012) indicate that the period Pof sausage\nmodes increases smoothly with decreasing kuntil reaching\nsomeP0in the thin-cylinder limit ( ka→0 withabeing\nthe cylinder radius). Likewise, being identically infinite in\nthe trapped regime, the attenuation time τdecreases with\ndecreasing kbefore saturating at τ0whenka=0. Furthermore,\nP0is determined primarily by a/vAi, wherevAiis the Alfvén\nspeed in the cylinder (Robertset al. 1984), with the detaile d\ntransverse density distribution playing an important role\n(Nakariakovetal. 2012, also Chen etal. 2015) . This is why\nsecond-scale QPPs are attributed to standing sausage modes ,\nsincea/vAiis of the order of seconds for typical coronal\nstructures. On the other hand, the ratio τ0/P0is basically\nproportionalto the density contrast (e.g., Kopylovaetal. 2007),\nmeaning that high-quality sausage modes are associated wit h\ncoronal structures with densities considerably exceeding their\nsurroundings.\nInterestingly, the dispersive behavior of trapped modes is\nimportant also in understanding impulsively generated sau sage\nwaves (Robertset al. 1984). When measured at a distance from\nthe source, the signals from these propagating waves posses s\nthreephases:periodic,quasi-periodic,anddecay.Thefre quency\ndependence of the longitudinal group speed vgris crucial in\nthis context. In particular, whether the quasi-periodic ph ase\nexists depends on the existence of a local minimum in vgr,\nArticlenumber, page 1of 6A&Aproofs: manuscript no. ver1.0\nwhich in turn depends on the density profile transverse to the\nstructure(Edwin&Roberts1988;Nakariakov&Roberts1995) .\nThisanalyticalexpectation,extensivelyreproducedinnu merical\nsimulations (e.g., Murawski&Roberts 1993; Selwa et al. 200 4;\nNakariakovet al. 2004),well explainsthe timesignatureso f the\nwave trains discovered with the Solar Eclipse Coronal Imagi ng\nSystem (Williamsetal. 2001, 2002; Katsiyanniset al. 2003) as\nwell asthose measuredwithSDO /AIA (Yuan etal.2013) .\nWe intend to examine the spatial damping of leaky\nsausage waves propagating along coronal cylinders in re-\nsponse to photospheric motions due to, say, granular con-\nvection (Berghmanset al. 1996). One reason for conduct-\ning this study is that, besides the observations showing\nthat propagating sausage waves abound in the chromo-\nsphere (Mortonet al. 2012), a recent study clearly demon-\nstrates the spatial damping of sausage waves propagating fr om\nthe photosphere to the transition region in a pore (Granteta l.\n2015). Another motivation is connected to the intensive int er-\nest(Terradaset al.2010;Hoodetal.2013;Pascoeetal.2013 )in\nemploying resonant absorption to understand the spatial da mp-\ning of propagating kink waves measured with the Coronal\nMulti-Channel Polarimeter (CoMP) instrument (Tomczyketa l.\n2007; Tomczyk&McIntosh 2009). A leading mechanism\nfor interpreting the temporal damping of standing kink\nmodes (Ruderman& Roberts 2002; Aschwandenetal. 2003,\nand references therein), resonant absorption is found to at ten-\nuate propagatingkinkwaveswith a spatial lengthinversely pro-\nportionalto wave frequency(Terradaset al. 2010). If attri buting\nthe generation of these kink waves to broadband photospheri c\nperturbations, one expects that resonant absorption essen tially\nfilters out the high-frequency components. One then natural ly\nasks:Whatroledoeswaveleakageplayinattenuatingpropag at-\ningsausagewaves?Andwhatwill bethefrequencydependence\noftheassociateddampinglength?\nThismanuscriptisstructuredasfollows.Wepresentthenec -\nessary equations, the dispersion relation (DR) in particul ar, in\nSect.2, and then present our numerical solutions to the DR in\nSect.3 where we also derive a couple of analytical approxima -\ntions to the DR for validation purposes. Finally, a summary i s\ngiveninSect.4.\n2. ProblemFormulation\nWe consider sausage waves propagating in a structured coron a\nmodeled by a plasma cylinder with radius aembedded in a\nuniform magnetic field B=Bˆz, where a cylindrical coordi-\nnate system ( r,θ,z) is adopted. The cylinder is directed along\nthez-direction. A piece-wise constant (top-hat) density profil e\nis adopted,with the densities inside and external to the cyl inder\nbeingρiandρe, respectively (ρe<ρi). The Alfvén speeds, vAi\nandvAe, follow fromthe definition vA=/radicalbig\nB2/4πρ. Appropriate\nfor the solar corona, zero- β, ideal MHD equations are adopted.\nInsuchacase,sausagewavesdonotperturbthe z-componentof\nthe plasma velocity. Let δvrdenote the radial velocity perturba-\ntion,andδbr,δbzdenotethe radialandlongitudinalcomponents\nof the perturbed magnetic field δb, respectively. The perturbed\nmagnetic pressure, or equivalently total pressure in the ze ro-β\ncase,is thenδptot=Bδbz/4π.\nLetusFourier-decomposeanyperturbedvalue δf(r,z;t) as\nδf(r,z;t)=Re/braceleftBig˜f(r)exp[−i(ωt−kz)]/bracerightBig\n. (1)Withthedefinition\nµ2\ni=ω2\nv2\nAi−k2, µ2\ne=ω2\nv2\nAe−k2(−π\n20 because Re(ω/µe)>0, see\nEq.(14)).Indeed,numericalstudiesstartingwithinitial standing\nwaves suggest that the temporal damping after a transient st age\nmatches exactly the attenuation rate given by the eigen-mod e\nanalysis(Terradaset al.2007).Forpropagatingwaves,one finds\nfrom Fig. 2a that once again Im (µe)<0 and Re(ω/µe)>0.\nHencesimilartothestandingcase,aneigen-modeanalysisd oes\nnot permit an investigation into the energetics of propagat ing\nwaves. However, if a coronal structure is perturbed with a ha r-\nmonicboundarydriver,oneexpectstoseethattheapparents pa-\ntial damping after some transient phase will be given by the\nattenuation length 1 /kIobtained from this eigen-mode analy-\nsis. And this attenuation is once again associated with the o ut-\nwardly directed∝angbracketleftFr∝angbracketright. We note that this expectationcan be read-\nily numerically tested, in much the same way that the expecte d\nspatial damping of kink waves due to resonant absorption was\ntested(Pascoeetal. 2013).\nArticlenumber, page 4of 6Ming-Zhe Guoet al.: Spatial damping of propagating sausage waves incoronal cylinders\n4. Summary\nThisstudyismotivatedbytheapparentlackofadedicatedst udy\nontherolethatwaveleakageplaysinspatiallyattenuating prop-\nagatingsausagewavessupportedbydensity-enhancedcylin ders\nin the corona. To this end, we worked in the framework of cold\nmagnetohydrodynamics(MHD),andnumericallysolvedthedi s-\npersion relation (DR, eq. (7)) for complex-valued longitud inal\nwavenumbers k=kR+ikIat given real angular frequencies ω.\nTovalidateournumericalresults,wealsoprovidedtheanal ytical\napproximationstothefullDRinthelow-frequencylimit ω→0\nand in the neighborhood of ωc, the critical angular frequency\nseparatingtrappedfromleakywaves.Oursolutionsindicat ethat\nwhile sausage waves can propagate for ω<ω cand can direct\ntheir energyupwards, they su ffer substantial spatial attenuation.\nThe attenuation length (1 /kI) is of the order of the cylinder ra-\ndiusaand shows little dependence on frequency or the density\ncontrast between a coronal structure and its surroundings f or\nω/lessorsimilar1.5vAi/a,wherevAiistheAlfvénspeedinthecylinder.This\nmeans that when a coronal cylinder is subject to boundary per -\nturbations with a broadband spectrum (e.g., granular motio ns),\nwave leakage removesthe low-frequencycomponentsrather e f-\nficiently. A comparison with the solutions to the DR for stand -\ning waves (real k, complexω) indicates that a close relation-\nship between propagatingand standing waves exists only whe n\nthe waves are trapped or weakly damped. In addition, while an\neigen-modeanalysisdoesnotallowaninvestigationintoth een-\nergeticsofpropagatingwaves,theattenuationlengthisex pected\nto play an essential role in numericalsimulationswherecor onal\nstructuresareperturbedbyharmonicboundarydrivers.\nAcknowledgements. This research is supported by the 973 program\n2012CB825601, National Natural Science Foundation of Chin a (41174154,\n41274176, 41274178, and 41474149), the Provincial Natural Science Foun-\ndation of Shandong via Grant JQ201212, and also by a special f und of Key\nLaboratory of Chinese Academy of Sciences.\nReferences\nAschwanden, M. J., Fletcher, L., Schrijver, C. 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T., et al. 2 002, MNRAS, 336,\n747\nWilliams, D.R.,Phillips, K.J.H.,Rudawy, P.,etal. 2001, M NRAS,326, 428\nYuan,D.,Shen, Y.,Liu, Y.,etal. 2013, A&A,554, A144\nAppendixA: Aderivationof theaveragedenergy\nfluxdensities\nThissectionoffersaderivationoftheaveragedenergyfluxdensi-\ntiesgiveninEqs.(8),(9)and(12),followingaproceduresi milar\nto that adopted by Li &Li (2007, page 1225). Let us first con-\nsider standing waves for which the axial wavenumber kis real,\nbut the angular frequency ωis allowed to be complex-valued\n(ω=ωR+iωI).Evaluatinga perturbation δf(r,z;t) withEq.(1)\natgivenvaluesof[ r,t] yieldsthat\nδf(r,z;t)=Re/braceleftBig˘fexp(ikz)/bracerightBig\n, (A1)\nwhere\n˘f=˜f(r)exp(−iωt)=˜f(r)exp(−iωRt)eωIt. (A2)\nWith another perturbation δg(r,z;t) in the same form, one finds\nthattheproductδfδgaveragedoverawavelength λ=2π/kis\n∝angbracketleftδfδg∝angbracketright(r,t)\n≡1\nλ/integraldisplayλ\n0δf(r,z;t)δg(r,z;t)dz\n=1\nλ/integraldisplayλ\n0Re/braceleftBig˘fexp(ikz)/bracerightBig\nRe/braceleftbig˘gexp(ikz)/bracerightbigdz\n=1\nλ/integraldisplayλ\n0/bracketleftBig˘fRcos(kz)−˘fIsin(kz)/bracketrightBig/bracketleftbig˘gRcos(kz)−˘gIsin(kz)/bracketrightbigdz\n=1\nλ/integraldisplayλ\n0/bracketleftBig˘fR˘gRcos2(kz)+˘fI˘gIsin2(kz)\n−/parenleftBig˘fR˘gI+˘fI˘gR/parenrightBig\ncos(kz)sin(kz)/bracketrightBig\ndz\n=1\n2/parenleftBig˘fR˘gR+˘fI˘gI/parenrightBig\n=1\n2Re/parenleftBig˘f˘g∗/parenrightBig\n=1\n2Re/parenleftBig˘f∗˘g/parenrightBig\n, (A3)\nwhere we have used the shorthand notations ˘fR=Re˘fand\n˘fI=Im˘f. By noting that ˘fis expressible in terms of ˜fthrough\nEq.(A2), onefindsthat\n∝angbracketleftδfδg∝angbracketright(r,t)=1\n2Re/bracketleftBig˜f(r)˜g∗(r)/bracketrightBig\ne2ωIt=1\n2Re/bracketleftBig˜f∗(r)˜g(r)/bracketrightBig\ne2ωIt.(A4)\nArticlenumber, page 5of 6A&Aproofs: manuscript no. ver1.0\nNow the energyflux density averagedovera wavelengthcan be\nevaluatedwithitsdefinition,Eq.(6),the resultsbeing\n∝angbracketleftFr∝angbracketright=1\n2Re/parenleftbig˜ptot˜v∗\nr/parenrightbige2ωIt=1\n2Re/parenleftbig˜p∗\ntot˜vr/parenrightbige2ωIt, (A5)\n∝angbracketleftFz∝angbracketright=1\n2Re/parenleftbigg\n−B\n4π˜br˜v∗\nr/parenrightbigg\ne2ωIt. (A6)\nFurthermore,bynotingthat Eq.(5)allows ˜ vrtobeexpressedby\n˜vr=−iω\nµ24π˜p′\ntot\nB2, (A7)\nonefindsthat\n∝angbracketleftFr∝angbracketright=2π\nB2Re/bracketleftBigg\n˜p∗\ntot/parenleftBigg\n−iω\nµ2˜p′\ntot/parenrightBigg/bracketrightBigg\ne2ωIt,\n=2π\nB2Re/bracketleftBigg\n˜p∗\ntot˜ptot/parenleftBigg\n−iω\nµ2˜p′\ntot\n˜ptot/parenrightBigg/bracketrightBigg\ne2ωIt,\n=2π|˜ptot|2\nB2Re/parenleftBigg\n−iω\nµ2˜p′\ntot\n˜ptot/parenrightBigg\ne2ωIt. (A8)\nThis expression is valid both in and outside the cylinder. Wh en\napplied to the external medium, it results in the first expres sion\ninEq.(12).\nNow consider propagating waves for which ωis real,\nwhereaskis allowed to be complex-valued ( k=kR+ikI). In\nthiscaseevaluatingaperturbation δf(r,z;t)withEq.(1)atgiven\nvaluesof[ r,z]yieldsthat\nδf(r,z;t)=Re/braceleftBig˘fexp(−iωt)/bracerightBig\n, (A9)\nwhere\n˘f=˜f(r)exp(ikz)=˜f(r)exp(ikRz)e−kIz. (A10)\nTo evaluate the product δfδgaveraged over a wave period T=\n2π/ω, one can follow the same procedure as in Eq. (A3) by re-\nplacingkzwith−ωt, theresult being\n∝angbracketleftδfδg∝angbracketright(r,z)\n≡1\nT/integraldisplayT\n0δf(r,z;t)δg(r,z;t)dt\n=1\n2Re/parenleftBig˘f˘g∗/parenrightBig\n=1\n2Re/parenleftBig˘f∗˘g/parenrightBig\n. (A11)\nWith theaidofEq.(A10) whichrelates ˘fto˜f, onefindsthat\n∝angbracketleftδfδg∝angbracketright(r,z)=1\n2Re/bracketleftBig˜f(r)˜g∗(r)/bracketrightBig\ne−2kIz\n=1\n2Re/bracketleftBig˜f∗(r)˜g(r)/bracketrightBig\ne−2kIz. (A12)\nTheenergyfluxdensityaveragedoveraperiodfollowsfromth e\ndefinition, Eq. (6), and the results are givenby Eqs. (8) and ( 9).\nThe second expression in Eq. (12), appropriate for propagat ing\nwaves,canbederivedinviewofEq.(A7).\nArticlenumber, page 6of 6" }, { "title": "1804.07080v2.Damping_of_magnetization_dynamics_by_phonon_pumping.pdf", "content": "Damping of magnetization dynamics by phonon pumping\nSimon Streib,1Hedyeh Keshtgar,2and Gerrit E. W. Bauer1, 3\n1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n2Institute for Advanced Studies in Basic Science, 45195 Zanjan, Iran\n3Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan\n(Dated: July 11, 2018)\nWe theoretically investigate pumping of phonons by the dynamics of a magnetic film into a non-\nmagnetic contact. The enhanced damping due to the loss of energy and angular momentum shows\ninterferencepatternsasafunctionofresonancefrequencyandmagneticfilmthicknessthatcannotbe\ndescribed by viscous (“Gilbert”) damping. The phonon pumping depends on magnetization direction\nas well as geometrical and material parameters and is observable, e.g., in thin films of yttrium iron\ngarnet on a thick dielectric substrate.\nThe dynamics of ferromagnetic heterostructures is at\nthe root of devices for information and communication\ntechnologies [1–5]. When a normal metal contact is at-\ntached to a ferromagnet, the magnetization dynamics\ndrives a spin current through the interface. This effect\nis known as spin pumping and can strongly enhance the\n(Gilbert) viscous damping in ultra-thin magnetic films\n[6–8]. Spin pumping and its (Onsager) reciprocal, the\nspin transfer torque [9, 10], are crucial in spintronics, as\nthey allow electric control and detection of magnetiza-\ntion dynamics. When a magnet is connected to a non-\nmagnetic insulator instead of a metal, angular momen-\ntum cannot leave the magnet in the form of electronic or\nmagnonic spin currents, but they can do so in the form\nof phonons. Half a century ago it was reported [11, 12]\nand explained [13–16] that magnetization dynamics can\ngenerate phonons by magnetostriction. More recently,\nthe inverse effect of magnetization dynamics excited by\nsurface acoustic waves (SAWs) has been studied [17–20]\nand found to generate spin currents in proximity normal\nmetals [21, 22]. The emission and detection of SAWs was\ncombined in one and the same device [23, 24], and adia-\nbatic transformation between magnons and phonons was\nobserved in inhomogeneous magnetic fields [25]. The an-\ngular momentum of phonons [26, 27] has recently come\ninto focus again in the context of the Einstein-de Haas\neffect [28] and spin-phonon interactions in general [29].\nThe interpretation of the phonon angular momentum in\ntermsoforbitalandspincontributions[29]hasbeenchal-\nlenged [30], a discussion that bears similarities with the\ninterpretation of the photon angular momentum [31]. In\nour opinion this distinction is rather semantic since not\nrequired to arrive at concrete results. A recent quantum\ntheory of the dynamics of a magnetic impurity [32] pre-\ndicts a broadening of the electron spin resonance and a\nrenormalized g-factor by coupling to an elastic contin-\nuum via the spin-orbit interaction, which appears to be\nrelated to the enhanced damping and effective gyromag-\nnetic ratio discussed here.\nA phonon current generated by magnetization dynam-\nics generates damping by carrying away angular momen-\ntum and energy from the ferromagnet. While the phonon\nphonon sinkzmagnet\nnon-magnet0\nphononsmHFigure 1. Magnetic film (shaded) with magnetization mat-\ntached to a semi-infinite elastic material, which serves as an\nideal phonon sink.\ncontribution to the bulk Gilbert damping has been stud-\nied theoretically [33–38], the damping enhancement by\ninterfaces to non-magnetic substrates or overlayers has\nto our knowledge not been addressed before. Here we\npresent a theory of the coupled lattice and magnetiza-\ntion dynamics of a ferromagnetic film attached to a half-\ninfinite non-magnet, which serves as an ideal phonon\nsink. We predict, for instance, significantly enhanced\ndamping when an yttrium iron garnet (YIG) film is\ngrown on a thick gadolinium gallium garnet (GGG) sub-\nstrate.\nWe consider an easy-axis magnetic film with static ex-\nternal magnetic field and equilibrium magnetization ei-\nther normal (see Fig. 1) or parallel to the plane. The\nmagnet is connected to a semi-infinite elastic material.\nMagnetization and lattice are coupled by the magne-\ntocrystalline anisotropy and the magnetoelastic interac-\ntion, giving rise to coupled field equations of motion in\nthe magnet [39–42]. By matching these with the lattice\ndynamics in the non-magnet by proper boundary con-\nditions, we predict the dynamics of the heterostructure\nas a function of geometrical and constitutive parameters.\nWe find that magnetization dynamics induced, e.g., by\nferromagnetic resonance (FMR) excites the lattice in the\nattachednon-magnet. Inanalogywiththeelectroniccase\nwecallthiseffect“phononpumping” thataffectsthemag-\nnetization dynamics. We consider only equilibrium mag-\nnetizations that are normal or parallel to the interface,\nin which the pumped phonons are pure shear waves that\ncarry angular momentum. We note that for general mag-arXiv:1804.07080v2 [cond-mat.mes-hall] 16 Jul 20182\nnetization directions both shear and pressure waves are\nemitted, however.\nWe consider a magnetic film (metallic or insulating)\nthat extends from z=\u0000dtoz= 0. It is subject to suffi-\nciently high magnetic fields H0such that magnetization\nis uniform, i.e. M(r) =M:For in-plane magnetizations,\nH0> Ms, where the magnetization Msgoverns the de-\nmagnetizing field [43]. The energy of the magnet|non-\nmagnet bilayer can be written\nE=ET+Eel+EZ+ED+E0\nK+Eme;(1)\nwhich are integrals over the energy densities \"X(r). The\ndifferent contributions are explained in the following.\nThe kinetic energy density of the elastic motion reads\n\"T(r) =(\n1\n2\u001a_u2(r); z> 0\n1\n2~\u001a_u2(r);\u0000d 0\n1\n2~\u0015(P\n\u000bX\u000b\u000b(r))2+ ~\u0016P\n\u000b\fX2\n\u000b\f(r);\u0000d 0\n~\u0016\n2\u0000\nu02\nx(z) +u02\ny(z)\u0001\n;\u0000d0. The\nmagnetoelastic energy derived above then simplifies to\nEz\nme=(B?\u0000K1)A\nMsX\n\u000b=x;yM\u000b[u\u000b(0)\u0000u\u000b(\u0000d)];(19)\nwhichresultsinsurfaceshearforces F\u0006(0) =\u0000F\u0006(\u0000d) =\n\u0000(B?\u0000K1)Am\u0006, withF\u0006=Fx\u0006iFy. These forces\ngenerate a stress or transverse momentum current in the\nzdirection (see Supplemental Material)\nj\u0006(z) =\u0000\u0016(z)u0\n\u0006(z); (20)\nwith\u0016(z) =\u0016forz >0and\u0016(z) = ~\u0016for\u0000d < z < 0,\nandu\u0006=ux\u0006iuy, which is related to the transverse mo-\nmentump\u0006(z) =\u001a( _ux(z)\u0006i_uy(z))by Newton’s equa-\ntion:\n_p\u0006(z) =\u0000@\n@zj\u0006(z): (21)\nThe boundary conditions require momentum conserva-\ntion and elastic continuity at the interfaces,\nj\u0006(\u0000d) = (B?\u0000K1)m\u0006;(22)\nj\u0006(0+)\u0000j\u0006(0\u0000) =\u0000(B?\u0000K1)m\u0006;(23)\nu\u0006(0+) =u\u0006(0\u0000): (24)\nWe treat the magnetoelastic coupling as a small pertur-\nbation and therefore we approximate the magnetization\nm\u0006entering the above boundary conditions as indepen-\ndent of the lattice displacement u\u0006. The loss of angular\nmomentum (see Supplemental Material) affects the mag-\nnetization dynamics in the LLG equation in the form of a\ntorque, which we derive from the magnetoelastic energy\n(19),\n_m\u0006jme=\u0006i!c\nd[u\u0006(0)\u0000u\u0006(\u0000d)]\n=\u0006i!cRe(v)m\u0006\u0007!cIm(v)m\u0006;(25)where!c=\r(B?\u0000K1)=Ms(for YIG:!c= 8:76\u0002\n1011s\u00001) andv= [u\u0006(0)\u0000u\u0006(\u0000d)]=(dm\u0006). We can\ndistinguish an effective field\nHme=!c\n\r\u00160Re(v)ez; (26)\nand a damping coefficient\n\u000b(?)\nme=\u0000!c\n!Imv: (27)\nThe latter can be compared with the Gilbert damping\nconstant\u000bthat enters the linearized equation of motion\nas\n_m\u0006j\u000b=\u0006i\u000b_m\u0006=\u0006\u000b!m\u0006: (28)\nWith the ansatz\nu\u0006(z;t) =(\nC\u0006eikz\u0000i!t; z> 0\nD\u0006ei~kz\u0000i!t+E\u0006e\u0000i~kz\u0000i!t;\u0000d z0), the time\nderivative of the transverse momentum P\u0006=Px\u0006iPy\nreads\n_P\u0006=\u001aZ\nVd3ru\u0006(z;t)\n=\u0016A\u0002\nu0\n\u0006(z1;t)\u0000u0\n\u0006(z0;t)\u0003\n:(S11)\nThe change of momentum can be interpreted as a trans-\nverse momentum current density j\u0006(z0) =\u0000\u0016u0\n\u0006(z0)\nflowing into the magnet at z0and a current j\u0006(z1) =\n\u0000\u0016u0\n\u0006(z1)flowing out at z1. The momentum current\nis related to the transverse momentum density p\u0006(z) =\n\u001a_u\u0006(z)by\n_p\u0006(z) =\u0000@\n@zj\u0006(z); (S12)\nwhich confirms that\nj\u0006(z;t) =\u0000\u0016u0\n\u0006(z;t): (S13)\nThe instantaneous conservation of transverse momentum\nisaboundaryconditionsattheinterface. Itstimeaverage\nhj\u0006i= 0, but the associated angular momentum along z\nis finite, as shown above.\nIII. SANDWICHED MAGNET\nWhen a non-magnetic material is attached at both\nsides of the magnet and elastic waves leave the magnet\natz= 0andz=\u0000d, the boundary condition are\nj\u0006(\u0000d\u0000)\u0000j\u0006(\u0000d+) = (B?\u0000K1)m\u0006;(S14)\nj\u0006(0+)\u0000j\u0006(0\u0000) =\u0000(B?\u0000K1)m\u0006;(S15)\nu\u0006(0+) =u\u0006(0\u0000); (S16)\nu\u0006(\u0000d+) =u\u0006(\u0000d\u0000); (S17)2\nwithd\u0006=d\u00060+. Since the Hamiltonian is piece-wise\nconstant\nu\u0006(z;t) =8\n><\n>:C\u0006eikz\u0000i!t; z> 0\nD\u0006ei~kz\u0000i!t+E\u0006e\u0000i~kz\u0000i!t;\u0000d0. The quasi-1D chains of Ba 3Ir4O10zigzag\nwith a relative angle \u00120as in Fig. 1a. Nonzero J2and\u00120\nare both required in order to produce a observe a four-3\nspinon hump for photon polarizations transverse to the\nchain axis.\nFIG. 2. Temperature dependence of Raman spec-\ntra in the bbphoton polarization for both compounds.\nThe Ba 4Ir3O10broadened phonons and broad 4-spinon\nhump at 10 K (shaded) persist up to 170 K, but\nare absent in the magnetically-ordered sister compound\n((Ba 0:98Sr0:02)4Ir3O10) at all temperatures below its N\u0013 eel\ntemperature of 130 K.\nResult of theoretical computation.\nWith this minimal model, we compute the inelastic\nRaman scattering spectrum using a mean \feld theory of\nfree 1D fermionic spinons [18] with Hamiltonian H0=P\nk\u000fk^cy\nk^ck, where\u000fk=\u0000tcosk=\u0000\u0019\n2Je\u000bcosk. Here,\nwe introduce an energy scale Je\u000bwhich is a function of\nHamiltonian parameters ( J1;J2) scaled so that, in the\nJ2!0 limit, it reproduces the exact Bethe ansatz result\nof\u000fk=\u0000\u0019\n2J1cos(k) [28]. Wavevectors kare in units of\ninverse bond length projected onto the chain axis.\nThe mean \feld four-spinon Raman susceptibility\n\u001f00\nR\u0017;R\u0017(!) is plotted in Fig. 3 for \u0017= 1;2 at various\ntemperatures. Here, R\u0017is one of two equivalent choices\nof mean \feld Raman operators for the scattering spec-\ntrum, and\u001f00\nR\u0017;R\u0017is the imaginary part of its associated\ndynamical susceptibility.\nThe low temperature hump feature agrees with exper-\nFIG. 3. Spinon Raman response computed within two\nmean \feld choices R\u0017=1;2, with J(\u0017)\ne\u000b=kB= 75 K at T=\n10;50;90;130 and 170 K (same vertical scale). The di\u000ber-\nence between R1andR2at higher temperatures quanti\fes\nthe self-consistency breakdown of the Raman mean \feld the-\nory.\niment (Fig. 1b); at higher temperatures ( kBT=J e\u000b>1)\nthe central frequency of the feature is lower, and the sus-\nceptibilities obtained from R1andR2become di\u000berent.\nThis di\u000berence quanti\fes the self-consistency breakdown\nof the theory at high temperatures.\nThe agreement between the mean \feld susceptibility\nand experiment at low temperature allows one to extract\nJe\u000b=kB= 75 K. The J1-J2mean \feld self-consistency\nequation derived in Ref. [18] relates Je\u000bto a di\u000berence\nbetweenJ1;J2inH0:Je\u000b\u00191:042J1\u00000:8106J2, and the\nminus sign in this expression (arising from the sublattice-\nrotated Jordan-Wigner transformation into spinons) al-\nlows for a given Je\u000bto arise from substantially larger\nJ1;J2. For example, taking Je\u000b=kB= 75 K as a reason-\nable energy scale for Ba 4Ir3O10, mean \feld self consis-\ntency permits J1=kB=J2=kB= 324 K. This scale for\nJ1;J2is consistent with Curie-Weiss measurements.\nAt higher temperatures the spinons are incoherent but\nnevertheless remain as the magnetic excitations. This is\nconsistent with the broad damping feature persisting as\ntemperature increases (Fig. 2). Now consider the sister\nmagnetic material with Sr replacement. Where this mag-\nnet exhibits magnetic order, no broad peak is observed,4\nbut aboveTNa broad peak is seen which appears sim-\nilar to the pure case at high temperature. This inter-\npretation suggests an intriguing possibility for the sis-\nter sample: the presence of the peak above TNsuggests\nthat its magnetic transition could be considered as an\ninstability of an incoherent parent quantum liquid state,\n\\existing\" (incoherently) at the high temperatures above\nTN; strictly speaking this high temperature state is just\na paramagnet, but here it evidently shows an unusually\ndense spectrum of strong short-ranged spin \ructuations,\nwhich could be interpreted as high-temperature relics of\nspinons.\nAs to using Raman to characterize the observed\nspinons, the model's agreement suggests they could be\nconsistently modeled as 1D spinons within a low temper-\nature e\u000bective theory, though we expect the spinons of\nany type of 2D quantum spin liquid phase to produce\na similar four-spinon continuum hump. One intriguing\npossibility for a 2D quantum liquid phase with spinons\ncloser to our model is the 2D \\Bose-Luttinger Liquid\"\nphase, a 2D bosonic generalization of 1D Luttinger Liq-\nuids recently introduced in Ref. [7]; its relevance would\nbe resolved by observing appropriate singularities away\nfrom the Brillouin zone center.\nPhonon linewidth broadening through spin-\nphonon coupling.\nPhonon broadening due to spins that show short\nranged correlation without long ranged magnetic order,\ncombined with spin-orbit interaction [29], is well known\ne.g. in Sr 2IrO4at higher temperatures [30]. The same ef-\nfect is here seen in Sr-substituted (Ba 0:98Sr0:02)4Ir3O10,\nagain only at higher temperatures. Most interestingly,\nthese broad phonons persist down to the lowest temper-\natures in pure Ba 4Ir3O10(Fig. 2) which serves as fur-\nther con\frmation of the magnetic quantum liquid. This\nobservation is very striking because increased disorder,\nsuch as introduced by Sr substitution, typically makes\nthe phonons broader, not narrower. The low tempera-\nture narrowing of the phonon peaks upon Sr substitution,\nand indeed the broadening in pure Ba 4Ir3O10down to\nthe lowest temperature, provide additional evidence for\na quantum liquid with spinon excitations in Ba 4Ir3O10.\nBeyond the particular quantum liquid state here, this\nbehavior shows that strong coupling between magnetic\ndegrees of freedom and phonons when the spin-orbit in-\nteraction is strong exists at the lowest temperatures, not\njust aboveTNas previously observed in Sr 2IrO4. Inter-\nestingly here the broadened phonon peaks are symmetric\nand do not show any signi\fcant Fano lineshape. Thus the\nbroad hump does not have a measurable interaction with\nphonons; instead, it appears that phonons are broadened\nby magnetic \ructuations that are not Raman active.\nPhonon peak assignments can be made based on corre-\nlating the phonon energies with particular features of the\ncrystal structure. Ba is heavy and most loosely bonded\natom, so we assign the peak at low energy of 50 cm\u00001to the Ba vibration. This peak is sharp in both sam-\nples. Substitution has a radical impact on peaks at\nhigher energies, which correspond to various rigid mo-\ntions of the IrO 6octahedra. These modes in the mag-\nnetically ordered sample narrow dramatically on cooling\neven though the Ba site is not a\u000bected. On the other\nhand, modes in the frustrated sample remain broad down\nto the lowest temperatures. This behavior dramatically\nillustrates that disorder of the pseudospins in the quan-\ntum liquid phase dominates phonon damping for a large\nsubset of phonons. This result naturally explains cou-\npling of the pseudospins to the phonons.\nThermal conductivity and future outlook for\nthermoelectricity.\nThe preceding discussion on phonon linewidth broad-\nening by spinon-phonon scattering suggests that the sup-\npressed phonon lifetimes should also be re\rected in a re-\nduced phonon contribution to thermal conductivity. In-\ndeed the thermal conductivity shows this type of be-\nhavior, with a surprisingly small value and an increase,\nwithin a window of low temperatures, upon Sr substitu-\ntion [16]. That this observed thermal conductivity behav-\nior can be mostly ascribed to phonon lifetimes is shown\nby the weak dependence of the thermal conductivity on\napplied magnetic \felds within the quantum spin liquid\nstate.\nSuch a quantum liquid with distinct phonon damp-\ning also presents a new direction for studies of ther-\nmoelectrics, in which poor phonon thermal conductivity\nis essential. Attempts to control thermal conductivity\nare focused on engineering structures that produce \rat\nphonon bands [31]. Alternatively, phonon lifetime can be\nreduced via damping by interactions with other phonons,\ndefects, or electrons [32]. But this type of phonon damp-\ning necessarily accompanies an unwanted consequence\nof reduced electrical conductivity. It is also established\nthat spin-phonon coupling via magnetostriction can re-\nduce the phonon lifetime, thus suppress phonon thermal\nconductivity [33]. Here we demonstrate that spin-orbit\ninteraction, rather than magnetostriction, can be an ef-\n\fcient driver of phonon damping which extends possible\nthermoelectricity to a new class of materials: magneti-\ncally non-ordered 4d and 5d transition metal materials\nwith strong spin-orbit interaction. When these exhibit\nstrongly correlated conducting phases, rather than Mott\ninsulators, the spin sector of the conducting electrons\nmay still damp phonons in analogy to the spinon-phonon\ncoupling mechanism in the present Mott insulator case.\nWe thank Martin Mourigal, Zhigang Jiang, and\nMichael Pustilnik for helpful conversations. I.K. ac-\nknowledges the Aspen Center for Physics where part of\nthis work was performed, which is supported by National\nScience Foundation grant PHY-1607611. Work at the\nUniversity of Colorado was funded by the U.S. Depart-\nment of Energy, O\u000ece of Basic Energy Sciences, O\u000ece\nof Science, under Contract No. DE-SC0006939 and by5\nNational Science Foundation grant DMR-1903888.\nSupplemental Material\nDetails of theoretical computation.\nTo derive the Raman response, we proceed as usual\n[22] by relating the inelastic Raman scattering spectrum\nI(!) to the dynamical susceptibility of the relevant mean\n\feld operator R:I(!) =1\n2\u0019R1\n\u00001dt ei!thR(t)R(0)i. The\nscattering spectrum Iis related to the dynamical sus-\nceptibility \u001f00by the \ructuation dissipation theorem:\nI(!) =\u001f00(!)=(1\u0000e\u0000!=T). Here,!is the energy of the\nincident or scattered photon, and the operator Ris the\nLoudon-Fleury photon-induced superexchange operator\n[34]\nR=X\nr1;r2(^ ei\u0001^ r12)(^ es\u0001^ r12)A(r12)Sr1\u0001Sr2 (2)\nFor brevity, we will refer to Ras a Raman operator. In\n(2),^ r1;2is the unit vector pointing from lattice site r2\nto lattice site r1, and ^ ei(respectively ^ es) is the polar-\nization vector of the incoming (outgoing) photon. The\nfactorA(r12) is di\u000ecult to compute, but it is known\nthat the ratio of Aon di\u000berent bonds is on the or-\nder of the spin-exchange couplings on the bonds (e.g.\nA(rj;j+2)=A(rj;j+1) isO(J2=J1)) [22]. Given a bare\nHamiltonian Hfor a system, it is clear from the de\f-nition ofI(!) and (2) that two Raman operators R;R0\nyield the same spectrum if there exists some real constant\nCsuch thatR0=R\u0000CH. If such aCexists, we will say\nthatRandR0arespectrally equivalent . Having reviewed\nsome preliminary results on inelastic Raman scattering,\nwe may consider the particular case of an isolated 1D\nmagnetic chain.\nUsing the coordinate system of Fig. 1, a transverse bb\npolarization corresponds to \u0012i=\u0012s=\u0000\u0019=2. In this case\nthe Raman operator is R=R1/sin2\u00120P\njSj\u0001Sj+1. We\nnote, however, that the freedom due to spectral equiva-\nlence also allows one to choose R=R2/P\njSj\u0001Sj+2. In\nthe limit of a straight chain ( \u00120!0), the Raman opera-\ntor vanishes (up to spectral equivalence), and no spinons\nare seen in the spectrum. Alternatively, in the limit of\nJ2!0, the Raman operator has neither the form of R1\nnorR2. In this case, the resultant susceptibility does not\nexhibit a broad hump feature [22]. Hence, both second\nneighbor interactions and a zig-zagged chain are su\u000ecient\nconditions to observe spinons in the scattering spectrum\nfor photon polarizations transverse ( bb) to the chain axis.\nWithin the minimal J1-J2model (1), we compute the\ninelastic scattering spectrum using a mean \feld theory of\nfree 1D spinons [18]. Taking H0=P\nk\u000fk^cy\nk^ckas the min-\nimal Hamiltonian and considering bbpolarization, one\nmakes a choice of the Raman operator up to spectral\nequivalence. That is, one chooses R=R\u0017(\u0017= 1;2).\nThe inelastic scattering spectrum is then given by (3)\nI(\u0017)(!)/Z\u0019\n\u0000\u0019dkZ\u0019\n\u0000\u0019dqX\nk0h(\u0017)(k;k0;q)[h(\u0017)(k;k0;q)\u0000h(\u0017)(k;k0;k0\u0000k\u0000q)]p\n(2tsin(q=2))2+ (\u000fk+q\u0000\u000fk\u0000!)2\u0002f(\u000fk)(1\u0000f(\u000fk+q))f(\u000fk0)(1\u0000f(\u000fk0\u0000q))\n(3)\nwherefis the Fermi function, h(1)(k;k0;q) = cos(q) and\nh(2)(k;k0;q) = cos(2q)\u0000cos(2k+q)\u0000cos(2k0\u0000q), and\nthe sum over k0is taken for all k02[\u0000\u0019;\u0019] which satisfy\n2tsin(q=2) sin(k0\u0000q=2) =\u000fk+q\u0000\u000fk\u0000!.\n[1] X. G. Wen, Mean-\feld theory of spin-liquid states with\n\fnite energy gap and topological orders, Phys. Rev. B\n44, 2664 (1991).\n[2] A. Kitaev, Anyons in an exactly solved model and be-\nyond, Annals of Physics 321, 2 (2006).\n[3] L. Savary and L. Balents, Quantum spin liquids: a re-\nview, IOP Publishing 80, 016502 (2016).\n[4] Y. Zhou, K. Kanoda, and T.-K. 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Rev. 166, 514 (1968)." }, { "title": "0804.3732v1.Ion_acoustic_waves_in_the_plasma_with_the_power_law_q_distribution_in_nonextensive_statistics.pdf", "content": "arXiv:0804.3732 \n \n \nIon acoustic w aves in the plasma w ith the pow er-law -distribution \nin nonextensive statistics q\n \nLiu Liyan, Du Jiulin \nDepartment of Physics, School of Science, Tianjin University , Tianjin 300072, China \n \nAbstract : We investig ate the d ispersion relati on and Landau dam ping of ion acoustic \nwaves in the collis ionless m agnetic-f ield-free plasm a if it is des cribed by th e \nnonextensive -distribu tions of Tsallis sta tistics. W e show that the increa sed \nnumbers of supertherm al particles and low velocity p article s can explain th e \nstreng thened and weak ened m odes of La ndau dam ping, respectively , with the \n-distribu tion. When the ion tem perature is equal to the electron tem peratu re, th e \nweakly dam ped waves are found to be the distributions with sm all values of . q\nq\nq\n \nPACS num ber(s): 52.35.Fp; 05.20.-y ; 05.90.+m \n \nKeywords: Ion acoustic waves; Landau dam ping; Tsallis statistics \n \n1．Introduction \nThe so-called ion acou stic waves are th e low-frequency longitudinal plasm a \ndensity oscillations. In the os cillations, electrons and ions are propagating in the pha se \nspace [1]. The ion acou stic waves were pr edicted first by T onks and Langum ir based \non the fluid dyna mics in 1929 [2]. The firs t experim ental observation for the waves \nwas reported in 1933 [3]. It is known that there are two models for the ion acoustic \nwaves [4]: one is the continuum models, in which the plasm a is treated as a fluid and \nso the fluid dynam ics is used for its theore tical studies; the othe r one is based on the \nkinetic equations in statistical theory , wh ere the distribution f unctions are used to \ndescribe the properties of the ion acoustic waves. As is well-known, Maxwellian \ndistribution in Boltzm ann-Gibbs (B-G) statis tics is believed valid unive rsally f or the \nmacroscopic er godic eq uilibrium system s, but for the sys tems with the long-rang e \ninteractions, such as plasm a and gravit ational system s, where the non-equilibrium \nstationary s tates ex ist, M axwellian distribution m ight be in adequate for the description \nof the system s. For example, the experim entally m easured phase velo city of the io n \nacoustic waves was 70% higher than th e theoretical value derived under the \npresupposition that the plasm a is described by Maxwellian distribution [5]. In the \nexperim ent for m easuring the ion acoustic wa ves, the ener gy distribution of electrons \nmay be actually not the Maxwellian one and hence we are hard to determ ine the valid \nelectron temperature [6]. In fact, the non-Maxwellian ve locity distributions for \nelectrons in plasm a were already m easured in th e experim ent where the tem perature \n 1gradient was steep [7]. And the non-Maxwelli an veloc ity d istributions for ions were \nalso reported in the studies of the earth’ plasm a sheet, the solar wind, and elsewhere \nthe long-range interacting system s containing pl entiful supertherm al particles, i.e., the \nparticles with the speeds f aster than the therm al speed [ 8, 9]. \nIn recent y ears there has been an increasing focus on a new statistical approach, \nnonextensive statistics (or T sallis statisti cs). It is described by a nonextensive \nparam eter q. For , it gives power -law distributi on functions and only when the \nparam eter Maxwellian distribution is recove red [10]. It is thought to be a \nuseful generalization of B- G statistics and to be appr opriate for the st atistical \ndescription of the long-rang interaction systems, charac terizing the non-equilibrium \nstationary state [1 1]. Ref. [12] shown that , for electrostatic plane-wave propagation in \nplasm as, Tsallis form alism presents a good fit to the experim ental data, while th e \nstandard Maxwellian distribution on ly provides a crude d escription. By re stricting th e \nvalue of -param eter from the experim ental data one can attain a good agreem ent \nbetween the theory and the experim ent [13]. We will show here that the plas ma \ndescribed by -distribu tion with 1q≠\n1q→\nqq\n1q< contains plentiful supply of supertherm al \nparticle s. Furtherm ore, the flexibility prov ided by th e nonextens ive param eter \nenables one to obtain a good agreem ent between the theory and experim ent. q\nThe paper is or ganized as follows. In S ec. II, we study the dispersion relations \nand Landau dam ping in the new st atistics with the power -law q-distr ibutions. In Se c. \nIII, th e Lan dau dam ping is discuss ed under th e different constraints on the values of \n. Sum mary and conclusion are given in Sec. IV . q\n2. The generaliz ed dispersion r elations and Landau damping \nIn Maxwellian description in B-G stat istics, the form ulae of the dis persion \nrelation and the ratio of Landa u damping to the frequency of ion acoustic waves in the \nequilibrium plasm a [1] are respectively \n2\n2\n22 21\n12Be r\nTi\nDe iKTvkk mω\nλ=+3+ ( 1) \n32 32\n22 222 133 exp81 1 2ie e e i\nrD e e i i DTm T TT\nkT m T kγπ\nωλ λe =− + + − − ++ ( 2) \nwhere rω is the frequency of ion acoustic wave, is the wave num ber, kDeλ is the \nDebye length of electrons, is the B oltzmann constant, BK γ is the Landau dam ping, \n is the thermal speed of ions. and T are the temperatu re of electrons and ions, \nrespec tively, and are the m ass of an ion and an electron, respectively . Here \nwe will s tudy the ion acoustic wa ves in the -distr ibution descrip tions in T sallis \nstatistics. Eq. (1) and ( 2) will be generalized in T sallis statistics, whic h may descr ibe \ntheir behaviors when the plasm a is in the non-equilibrium stationary state. TiveTi\nimem\nq\nFirst let us rem ind some basic facts about Tsallis statistics. In T sallis statistics, \nthe entropy has the form [10] of \n 2()33\n1q\nqBf fd xdv\nSKq−\n=−∫, (3) \nwhere f is the -distributions, and the param eter different from unity specifies \nthe degree of nonextensivity . The B-G entropy is recovered in the lim it . The \nbasic property of T sallis entropy is th e nonadditivity or nonextensivity for . For \nexam ple, for two system s A and B, the rule of com position [10] reads q q\n1q→\n1q≠\n() ()()()()() 1qq q q SA B SA SB qSA SB += ++−q . (4) \nThe q-equilibrium distribu tion f unction takes the power -law form. For \none-dim ensional case, it is given [14] by \n()() 112\n0\n0 211q\nq\nT TnA vf qv vπ−=− − ( 5) \n1\n110\n11\n12qqAq q\nqΓ−=− <≤Γ−−1, and 11\n12 1112 1\n1qq qAq\nqΓ+− + q = −≥Γ−, \nwhere is the par ticle num ber density , is the temperature, m is the m ass and \n is the the rmal speed, 0n T\nTv 2T B v KT=\n)m. It is worthy to notice th at for there \nis a the rmal cutof f on the m aximum value allowed f or the velocity o f the partic les, \nnamely 1q>\n(1T vv qα< −. As expected, Maxwellian dist ribution is obt ained in the \nlimit . Many class ical q uestions in B-G statis tics have been reconsid ered in th e \nfram ework of T sallis s tatistics. Among them , let us write the generalized Boltzm ann \nequation [15], 1→q\n()v qff fC ftm∂+⋅∇+⋅∇=∂Fv , ( 6) \nwhere is the q- collisio nal integ ral term , and is the extern al force. It has \nbeen dem onstrated that if the tim e dependent solu tions of the generalized \nequation (6) will evolve irrev ersibly towards the q-equilibrium distribution (5) and C()qCf F\n0q>\nq \nwill vanish. \n As we know , spacecraft measurem ents of plasm a velocity distributions, both in \nthe solar w ind and in the plan etary m agnetospheres and m agnetosheaths, have \nrevealed that non-Maxw ellian distributions are quite comm on. In m any situations the \ndistribution have a “suprather mal” power-law tail a t high en ergies. This h as been well \nmodeled by the so-called к-distribution [16], which, now we have known [17], is \nactually equivalent to the q-distribution Eq.( 5). In other words, E q.(5) and its lead ing \nresults can be directly applicab le to the above physical situ ations. In fact, in addition \nto the solar wind and the plan etary plasm a, the sun’ s interi or plasm a is the physical \nsituation where Eq.(5) can be applic able to f or the sta tistical description of it being the \nnonequilibrium stationary-state [18]. \n 3For a m agnetic -field-f ree plasm a which sli ghtly departs from equilibrium , we use \nEq.(6) and let the distribution function be 0 1 f f fααα=+ , where the le tter α in the \nsubscript of f denotes particle species ( α=i, e; for ion and for electron), i e0fα \ncorresponds to the one-dim ensional power -law -equilibrium distribution (5), and q\n1fα is the corresponding perturbation about th e distribution (5). As one knows, the \ndynam ical behavior of the plasm a is govern ed by a com bination of the generalized \nBoltzm ann equation and Poisson equation [1]. Here we assum e the plasm a \ntemperature to be high enough so that the q-collis ional term in the generalize d \nBoltzm ann equation (6) is negligible. Neglect ing high-order term s in the expansion of \nthe distribution function and linearizing the equation, one finds \n1\n1 0fQftmαα\nα\nα∂+⋅∇+⋅∇=∂1v vE0fα ( 7) \nand Poisson equation \n 1\n01Qf dαα\nαε∇⋅=∑∫ 1E v , ( 8) \nwhere is the electric field produ ced by the perturbation and 1E Qα is char ge of the \nparticle. \nWe consider the d irection of wave vector to be along k x-axis, and let . \nMaking Fourier transform ation for xvu=\nx and Laplace transformation for in Eq. (7 ), \nwe have t\n0\n11 ()Qfiku f Emuαα\nα\nαω 0∂−+ =∂. ( 9) \nCom bined with Eq,(8), the following dispersion equation is obtained, \n2\n0\n2ˆ\n1p fudukk uαα\nαω\nω∂∂+−∑∫0=, ( 10) \nwhere 2\n00 p nQ mα αα α ω= ε is the naturally oscillating frequency of the plasm a，0ˆfα \nis the norm alized distribution function,00ˆ\n0 f fnααα= . Inserting the power -law \ndistribution 0fα, Eq.(5), in Eq. (10), one readily gets \n()2211102q\nDqZkαα\nα αξξλ++ +∑=, ( 11) \nwhere 2D Tvp αα α λ ω= is the Debye length, αξ is a dim ensionless parameter , \ndefined as the ratio of the phase velocity , vφ kω= , to th e therm al speed Tvα, nam ely, \nTvvαφαξ= . (qZ)αξ is the generalized plasm a disp ersion function in the context of \nTsallis statistics, \n()()() 21211qq\nq\nqqx AZ dxxα\nαξξ π−−+∞\n−∞−− =−∫. ( 12) \nIn the lim it , it is reduced to the standard f orm in B-G statistics [ 19], 1q→\n 4()21 exp( )xZ dxxα\nαξξπ+∞\n−∞−=−∫. ( 13) \nObviously , there exists a singularity ， xαξ= ，in the integ rand of the dis persio n \nfunction (12). Here we follow the line deve loped by Landau to deal with the singular \npoint [20, 21]. The frequency is expressed in the com plex form: riq ωωγ=+ , where \nqγ is the gene ralized Lan dau dam ping that is re lated to q. For the weakly dam ped \nmodes, the real part of dispersion function (12) is the Cauc hy princip al value, while \nthe im aginary part is equal to half the residue of the integrand at the singularity , e.g. \n()()()\n()() 221221 11\nPr 1 1qq\nqqq\nqqqx AZ dxiAxα α\nαξπξ π−−+∞−−\n−∞−− = + − −∫qξ− . ( 14) \nThen Eq. (1 1) can be written as \n()()\n()() 221221\n2211 111P r 1 12qq\nqqq\nq\nDqx A qdx iA qkxαα\nα ααξξ πλξ π−−+∞−−\n−∞−− + ++ + −− −∑ ∫0αξ =\niTe. \n(15) \nIn a p lasma, if the electron tem perature is much higher than the ion tem perature, \n, and the ion m ass is much h eavier than the electron’ s, , the phase \nspeed eT\u0000im m\u0000\nvφ i eBmTK /= is m uch m ore than the the rmal speed o f ion s \nTivi imT/=BK but m uch less than that of electrons Teve eBmTK /= , \n. From the definition of th e dim ensionless param eter, Ti Te vv vφ\u0000\u0000Tvvαφαξ= , \nwe get 1eξ\u0000 and 1iξ\u0000. Making series expansion for th e integrand in Eq. (15) an d \nintegrating it, we obtain \n()() ()2 2221 2122\n22 2 2 22 2211 31 1 11 1123 1qq qqpi q q Ti\nee i i\nDe De DiiA iA kv qqqkq k kωπ πξξ ξ ξλω ωλ λ−− −− + +− + + −− + −− −0=\nq \n(16) \nInserting riωωγ=+ in Eq. (16) and m aking the real pa rt of left side to be zero, we \nfind \n ()2 22\n22 2 211 3\n23 1pi Ti\nDe r rkv q\nkqω\nλω ω+11 0 + −+−= , ( 17) \nThus we obtain the generalized dispersion relation, \n2\n2\n2\n221\n1 31\n2Be r\nTi\ni\nDeKTvq kmkω\nλ= ++ −+3\nq. ( 18) \nAs expected , in the lim it , Eq. (18) r educes to Eq . (1), the f amiliar resu lt in B-G \nstatistics. If the electro n tem peratu re is close to the ion te mperature in the lim iting \ncase of lon g-waveleng th 1q→\nDek 1λ\u0000, from Eq. (18), the gene ralized phase ve locity \nbecom es， ()()() 23iv 1q 1 31rT vk q qφω== ++−. Then the constraint 13 q> is \nimposed in order to ke ep the ph ase velocity positive. The tr aditional ph ase ve locity in \n 5the fram ework of B-G statistics, 2Ti vφ= v, is re covere d in the lim it . It is \neasy to ve rify that the genera lized phase ve locity is f aster than the trad itional phase \nvelocity when q1q→\n1< and is slower than the traditional one when . If the \nelectron temperature is m uch higher th an the ion’ s, in the lim it case of \nlong-wavelength, the generalized ph ase velo city will be m uch higher than the io n \ntherm al speed. 1q>\n)12\nq iπξ\n\n\n() 21\n3\n31qq−− 32\n32\n222 161111 83 1\n22q ie e e i\nq\nre i i\nDeTm T TTAqqq qT m T qkkγπ\nωλλ22\nDe =− + + −− + ++ −− ++ \n1→\nqA\n1≥An interesting feature of ion acoustic wa ves in plasm as is that they should be \ndamped even in the cas e of no collisions among particles [ 20], and the collisionless \ndamping is explained as the interactions between the wave and particles moving with \nthe speed close to the p hase velocity [22]. Let the im aginary part of Eq. (16) to b e \nzero, we obtain the generalized Landau dam ping as \n()()()(221 232\n211 11qq qqDi e\nqr e i\nDe iAq qλξγω ξ ξλξ−− −− =− −− +−− . ( 19) \nUsing the dispersion relation (18), we find \n(). (20) \nIn the lim it , Eq. (20) re duces to Eq. (2) in th e fram ework of B-G statis tics. In \nthe above equation, the second term in th e brace represen ts the contribution of th e \nions, which plays a main role, and th e first te rm in the b race is related to the ele ctrons, \nwhich is negligible f or the f act that the electron m ass is m uch lighter than the ion’ s. q\n3. The Landau damping and -parameter q\nIn this section, we will explore the extended Landau dam ping under dif ferent \nconstraints on the value of -param eter. Consid ering that Landau dam ping is related \nto the particle v elocity distribu tions [1], we f irst inves tigate the prop erties of the \npower law -distributions. In Fig. 1 (a), the cu rves of the d istributio n q\nq\n0 Ti f vfπ=% n\nq\nq as a f unction of the ratio of the particle ve locity to th e thermal \nspeed with the cons traint are plotted, and in Fig. 1 (b), the restric tion is \nmodified to . The valu es of the -param eter in Fig. 1 (a) are 1 (dashed line), \nwhich is jus t the Maxwe llian dis tribution, 0.6 (dotted line), and 0.2 (solid line). F rom \nFig. 1 (a), we can see th at in the cas e of the power law distribut ions, comparing with \nthe Maxwellian dis tribution, there are m ore supertherm al par ticles, i.e. p article s with \nthe speed faster than th e therm al speed, when the value of is sm all. In Fig. 1 (b), \nthe va lues of are 1.8 (dashed line), 1.4 (solid lin e), and 1.0 (dotted line). There is a \ncutof f in th e tail when , nam ely, 1≤q\n1>q\nq\nq ()1T vv qα< −\nq, and the maxim um value \nallowed f or the velo city of particles gets lower as increases. Thus we know the \n-distribu tions with the c onstra int are suitable for the des cription of sy stem s \ncontaining a lar ge num ber of low speed particles. q 1>q\nIn Fig. 2, we plot the curves of ()q Tikvγ as a function of the ion dim ensionless \n 6param eter ivvφ Tiξ= with som e selec ted va lues of the nonextensive param eter . \nThe selected values are 1.4 (dash-dotted line) , 1.0 (dashed line), 0.6 (dotted line), and \n0.4 (solid line). It show s that when th e phas e velocity is lo w or, equiv alently , q\niξ is \nsmall, the da mping is he avier f or higher value s of , but in the regions that the pha se \nvelocity b eing m uch faster than ion therm al speed, th e waves are m ore dam ped for \nlower value s of . Now we discuss the reason fo r this phenom enon. As one know s \nLandau damping stem s from the resonant interactions between the wave and particles \nmoving with the speed close to the p hase veloc ity, as a result, the dam ping condition s \nis determ ined by the number of resonant pa rticles. On the other hand, we know that \nthe num ber of the supertherm al particle s in the system obeying the power law \ndistributions increases as the values of decrease as in Fig. 1 (a). Therefore when \nthe phase velocity is m uch higher than the ion therm al speed there are more resonant \nsupertherm al ions in the sy stem s with lower valu es of . Consequ ently the \ninteractions between the wave and ions ar e stronger and the waves are more dam ped \nfor lower values of . The power law distributions (5) with the constraint \ndescrib es the system composed of a lar ge number of low velocity particles, hen ce \nheavy dam ping m odes are form ed in the region of low phase velocity . q\nq\nq\nq\nq 1>q\nq rγ\nei\n1q\nq\nrπThe ratio of the dam ping to the frequency , ω, as a function of the ratio of \nthe electron temperature to ion temperature in the lim iting case of l ong-waveleng th \n1Dekλ\u0000\nq is shown in Fig. 3. The selected values of are 1.4 (dash-dotted line), 1.0 \n(dashed line), 0.6 (dotted line), and 0.4 (solid line). It is obvious that for a fixed value \nof the damping is s mall when the electr on tem perature is m uch higher than ion \ntemperature, and it reaches the m aximum value when q\n=1 TT . First, we pay \nattention to the m aximum da mping presente d in the case of equal electron and ion \ntemperature. By Eq. (18), we can see that in such a case the generalized phas e \nvelocity of ion acoustic wave is low when , meanwhile for the case of th e power \nlaw distributions with the sam e value of , there are a lar ge number of low velocity \nparticle s. As result, heav ily damped modes appear . When 1q>\nq<, the generalized phase \nvelocity is h igher th an the trad itiona l phase velocity in B-G statistics an d it inc reases \nas the value of decreases. From Fig. 1 (a), we see that the proportion of particles \nwhose speed is m ore than two tim es of the therm al speed is sm all, so when the phas e \nvelocity is high, the number of par ticles moving with the speed close to the phas e \nvelocity is sm all. This w ill lead to the ex isten ce of weak dam ping m odes. If the \ncriterion used for a weak da mping m ode is that the im aginary part of ω shall be less \nthan the real part divided by 2π, i.e. 12ω< , in Fig. 3 weak dam ping modes are \nseen when takes the values 0.4 and 0.6. As for the increas ed dam ping for sm aller \nvalues of in the region of the electron tem perature being much higher than ion \ntemperature, it can be understood as a contribution by the increased num ber of \nsupertherm al particles in the tails close to the phase velocity . q\nqγ\n4. Summary and Conclusion \nIn th is paper we d iscuss the ion acoustic waves in plasm as with th e power la w \n 7q\nq-distribu tions in T sallis statistics, an d we obtain the extend ed dispers ion rela tions \nand Landau dam ping. Unlike the description of Maxwellian distribut ion that m ost of \nthe pa rticles cente r around the therm al spe ed, the power law distri butions characterize \nthe sys tems contain ing plentiful sup ply of s upertherm al particles with th e constrain t \n or includ ing a lar ge number of low velocity pa rticles by th e restric tion . \nThus the Landau dam ping which relies on the pa rticle v elocity dis tributions is re lated \nto the v alue of -param eter. Weak damping m odes are introdu ced for the power law \ndistributions with sm all values of in the case of equal electron and ion temperature, \nwhile for Maxwellian system only heavily damped modes exis t. W hen the phase \nvelocity of ion acoustic wave is m uch faster than the ion therm al speed, such as in the \ncase of th e electron temperat ure being m uch higher than ion’s, the extended Landau \ndamping be comes stronger for lower values of . Moreover , when the ph ase velo city \nis low , more heavily damped m odes are found for lar ger values of . All of these \nresults are explained by the increased num ber of supertherm al particle s or lo w \nvelocity particles contained in the plasm a with the power law -distributions. 1< 1q>\nq\nq\nq\nq\nq\nFinally , it is should be em phasized that the physical state described by the \n-distribu tion in T sallis statis tics is not the therm odynam ic equilib rium . The \nnonextensive param eter was prov ed to relate to the temperatu re gradient and th e \npotential en ergy of the system by the form ula q\nq\n() 10BKT qQαϕ ∇+− ∇=. Thus, the \ndeviation of from unity qualifies the degree of the inhom ogeneity of tem perature \nor the deviation from the equilibrium [11]. Therefore, the p roperties of ion acou stic \nwaves derived here are actually of th e ones of plasm as in non-equilibrium \nstationary -state. Furth ermore our re sults sugg est tha t Tsallis statistics is suitab le for \nthe system s being the nonequilibrium stationary-s tate with inhom ogeneous \ntemperature and containing plentiful supply of the supertherm al or low velocity \nparticles. q\n \nAcknow ledgements \n \nWe would like to thank the National Natu ral Science Foundation of China under the \ngrant No.10675088. \n \nRefer ences \n \n[1] Li Ding and Chen Y inhua et al, Plasm a Physics, Higher education press, Beijing, \n2006；Xu Jialuan and Jin Sha ngxian, Plasma Physics, Nuclear Ener gy Press, \nBeijing, 1981. \n[2] L. Tonks and I. Langm uir, Phy. Rev. 33(1929)195. \n[3] R.W . Revans, Phy . Rev. 44(1933)798. \n[4] Burton D. Fried and Roy W. Gould, The Phys. Fluids 4(1961)139. \n[5] A.Y. Wong, N.D' Angelo, and R. W. Motley , Phy. Rev. Lett. 9(1962)415. \n[6] I. Alexeff and R.V . Neidigh, Phy . Rev. 129(1963)516. \n[7] E. T. Sarris, S. M. Krimigis, A. T. Y. Lui, K. L. Ackerson, L. A. Frank, and D. J. \nWilliam s, Geophys. Res. Lett. 8(1981)349. D. J. William s, D. G . Mitchell, and S. P . \n 8Christon, Geophys. Res. Lett. 15(1988)303. \n[8] J.T . Gosling, J.R. Asbridge, S.J. Ba me, W.C. Feldm an, R.D. Zwickl, G . Paschm ann, \nN. Sckopke, and R. J. Hynds, J. Geophys. Res. [Space Phys.] 86(1981)547. \n[9] J.M. Liu, J.S. De Groot, J.P . Matte, T.W. Johnston, R.P . Drake, Phys. Rev . Lett. 72 \n(1994)2717. \n[10] C. Tsallis, J. S tat. Phys. 52(1988)479. \n[11] J.L. Du, Europhys.Lett. 67(2004)893; J.L. Du, Phy. Lett. A 329(2004)262. \n[12] J.A. S. Lim a, R. Silva, J. Santos, Phys. Rev. E 61(2000)3260. \n[13] R. Silva, J.S. Alcaniz, J.A. S. Lim a, Physica A 356(2005)509. \n[14] R. Silva, Jr ., A. R. Plastino, and J. A. S. Lim a, Phys. Lett. A 249(1998)401. \n[15] J.A.S. Lim a, R. Silva, A.R. Plastino, Phys. Rev . Lett. 86(2001)2938. \n[16] S.R.Cranm er, Astrophys. J. 508(1998)925. \n[17] M.P.Leubner, Astrophys. Space Sci. 282(2002)573; Astrophys. J. 604(2004)469; \nL N Guo, J L Du and Z P Liu, Phys. Lett. A 367(2007)431 . \n[18] J.L. Du, Europhys.Lett. 75(2006)861. \n[19] B. D. Fried, M. Gell-Mann, J.D. Jacks on, an d H.W . Wyld, J. Nuclear Ener gy: Part \nC 1, 190，1960. \n[20] L. D. Landau, J. Phys. USSR 10(1946 )25. \n[21] N.A. Krall and A .W. Trivelpiece, Prin ciple of Plasma Physics, McGraw-Hill, \nKogakusha, 1973. \n[22] D. Bohm, E. Gross, Phys. Rev . 75 ( 1949) 1851; D. Bohm, E. Gross, Phys. Rev . \n75(1949)1864. \n \n 9" }, { "title": "2202.06154v1.Generalization_of_the_Landau_Lifshitz_Gilbert_equation_by_multi_body_contributions_to_Gilbert_damping_for_non_collinear_magnets.pdf", "content": "Generalization of the Landau-Lifshitz-Gilbert equation by multi-body contributions to\nGilbert damping for non-collinear magnets\nSascha Brinker,1Manuel dos Santos Dias,2, 1,\u0003and Samir Lounis1, 2,y\n1Peter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich & JARA, 52425 J ulich, Germany\n2Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg, Germany\n(Dated: February 15, 2022)\nWe propose a systematic and sequential expansion of the Landau-Lifshitz-Gilbert equation utilizing\nthe dependence of the Gilbert damping tensor on the angle between magnetic moments, which arises\nfrom multi-body scattering processes. The tensor consists of a damping-like term and a correction\nto the gyromagnetic ratio. Based on electronic structure theory, both terms are shown to depend\non e.g. the scalar, anisotropic, vector-chiral and scalar-chiral products of magnetic moments: ei\u0001ej,\n(nij\u0001ei)(nij\u0001ej),nij\u0001(ei\u0002ej), (ei\u0001ej)2,ei\u0001(ej\u0002ek)..., where some terms are subjected to the\nspin-orbit \feld nijin \frst and second order. We explore the magnitude of the di\u000berent contributions\nusing both the Alexander-Anderson model and time-dependent density functional theory in magnetic\nadatoms and dimers deposited on Au(111) surface.arXiv:2202.06154v1 [cond-mat.mtrl-sci] 12 Feb 20222\nI. INTRODUCTION\nIn the last decades non-collinear magnetic textures have been at the forefront in the \feld of spintronics due to the\npromising applications and perspectives tied to them1,2. Highly non-collinear particle-like topological swirls, like\nskyrmions3,4and hop\fons5, but also domain walls6can potentially be utilized in data storage and processing devices\nwith superior properties compared to conventional devices. Any manipulation, writing and nucleation of these various\nmagnetic states involve magnetization dynamical processes, which are crucial to understand for the design of future\nspintronic devices.\nIn this context, the Landau-Lifshitz-Gilbert (LLG) model7,8is widely used to describe spin dynamics of materials\nranging from 3-dimensional bulk magnets down to the 0-dimensional case of single atoms, see e.g. Refs.9{12. The\nLLG model has two important ingredients: (i) the Gilbert damping being in general a tensorial quantity13, which can\noriginate from the presence of spin-orbit coupling (SOC)14and/or from spin currents pumped into a reservoir15,16;\n(ii) the e\u000bective magnetic \feld acting on a given magnetic moment and rising from internal and external interactions.\nOften a generalized Heisenberg model, including magnetic anisotropies and magnetic exchange interactions, is utilized\nto explore the ground state and magnetization dynamics characterizing a material of interest. Instead of the con-\nventional bilinear form, the magnetic interactions can eventually be of higher-order type, see e.g.17{23. Similarly to\nmagnetic interactions, the Gilbert damping, as we demonstrate in this paper, can host higher-order non-local contri-\nbutions. Previously, signatures of giant anisotropic damping were found24, while chiral damping and renormalization\nof the gyromagnetic ratio were revealed through measurements executed on chiral domain wall creep motion24{28.\nMost \frst-principles studies of the Gilbert damping were either focusing on collinear systems or were case-by-case\nstudies on speci\fc non-collinear structures lacking a general understanding of the fundamental behaviour of the Gilbert\ndamping as function of the non-collinear state of the system. In this paper, we discuss the Gilbert damping tensor\nand its dependencies on the alignment of spin moments as they occur in arbitrary non-collinear state. Utilizing linear\nresponse theory, we extract the dynamical magnetic susceptibility and identify the Gilbert damping tensor pertaining\nto the generalized LLG equation that we map to that obtained from electronic structure models such as the single\norbital Alexander-Anderson model29or time-dependent density functional theory applied to realistic systems10,30,31.\nApplying systematic perturbative expansions, we \fnd the allowed dependencies of the Gilbert damping tensor on the\ndirection of the magnetic moments. We identify terms that are a\u000bected by SOC in \frst and second order. We generalize\nthe LLG equation by a simple form where the Gilbert damping tensor is amended with terms proportional to scalar,\nanisotropic, vector-chiral and scalar-chiral products of magnetic moments, i.e. terms like ei\u0001ej, (nij\u0001ei)(nij\u0001ej),\nnij\u0001(ei\u0002ej), (ei\u0001ej)2,ei\u0001(ej\u0002ek)..., where we use unit vectors, ei=mi=jmij, to describe the directional dependence\nof the damping parameters and nijrepresents the spin-orbit \feld.\nThe knowledge gained from the Alexander-Anderson model is applied to realistic systems obtained from \frst-principles\ncalculations. As prototypical test system we use 3 dtransition metal adatoms and dimers deposited on the Au(111)\nsurface. Besides the intra-site contribution to the Gilbert damping, we also shed light on the inter-site contribution,\nusually referred to as the non-local contribution.\nII. MAPPING THE GILBERT DAMPING FROM THE DYNAMICAL MAGNETIC SUSCEPTIBILITY\nHere we extract the dynamical transverse magnetic response of a magnetic moment from both the Landau-Lifshitz-\nGilbert model and electronic structure theory in order to identify the Gilbert damping tensor Gij10,11,32,33. In linear\nresponse theory, the response of the magnetization mat siteito a transverse magnetic \feld bapplied at sites jand\noscillating at frequency !reads\nm\u000b\ni(!) =X\nj\f\u001f\u000b\f\nij(!)b\f\nj(!); (1)\nwith the magnetic susceptibility \u001f\u000b\f\nij(!) and\u000b;\fare thex;ycoordinates de\fned in the local spin frame of reference\npertaining to sites iandj.\nIn a general form13the LLG equation is given by\ndmi\ndt=\u0000\rmi\u00020\n@Be\u000b\ni+X\njGij\u0001dmj\ndt1\nA; (2)3\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)°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\n°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)\nabc\nFIG. 1. Illustration of the Landau-Lifshitz-Gilbert model and local density of states within the Alexander-Anderson model.\n(a) A magnetic moment (red arrow) precesses in the the presence of an external \feld. The blue arrow indicates the direction\nof a damping term, while the green arrow shows the direction of the precession term. (b) Density of states for di\u000berent\nmagnetizations in the range from 0 :2 to 0:8. Density of states of dimers described within the Alexander-Anderson model for\ndi\u000berent magnetizations in the range from 0 :2 to 0:8. Shown is the ferromagnetic reference state. The magnetizations are\nself-consistently constrained using a longitudinal magnetic \feld, which is shown in the inset. Model parameters: U= 1:0 eV,\nEd= 1:0 eV; t= 0:2 eV;\u0000 = 0:2 eV; 'R= 0 °.\nwhere\r= 2 is the gyromagnetic ratio, Be\u000b\ni=\u0000dHspin=dmiis the e\u000bective magnetic \feld containing the contributions\nfrom an external magnetic \feld Bext\ni, as well as internal magnetic \felds originating from the interaction of the\nmoment with its surrounding. In an atomistic spin model described by e.g. the generalized Heisenberg hamiltonian,\nHspin=P\nimiKimi+1\n2P\nijmiJijmj, containing the on-site magnetic anisotropy Kiand the exchange tensor Jij,\nthe e\u000bective \feld is given by Be\u000b\ni=Bext\ni\u0000Kimi\u0000P\njJijmj(green arrow in Fig. 1a). The Gilbert damping tensor\ncan be separated into two contributions { a damping-like term, which is the symmetric part of the tensor, S, (blue\narrow in Fig. 1a), and a precession-like term A, which is the anti-symmetric part of the tensor. In Appendix A we\nshow how the antisymmetric intra-site part of the tensor contributes to a renormalization of the gyromagnetic ratio.\nTo extract the magnetic susceptibility, we express the magnetic moments in their respective local spin frame of\nreferences and use Rotation matrices that ensure rotation from local to globa spin frame of reference (see Appendix B).\nThe magnetic moment is assumed to be perturbed around its equilibrium value Mi,mloc\ni=Miez\ni+mx\niex\ni+my\niey\ni,\nwhere e\u000b\niis the unit vector in direction \u000bin the local frame of site i. Using the ground-state condition of vanishing\nmagnetic torques, Miez\ni\u0002\u0000\nBext\ni+Bint\ni\u0001\n= 0 and the inverse of the transverse magnetic susceptibility can be identi\fed\nas\n\u001f\u00001\ni\u000bj\f(!) =\u000eij\u0012\n\u000e\u000b\fBe\u000b\niz\nMi+i!\n\rMi\u000f\u000b\f\u0016\u0013\n+1\nMiMj(RiJijRT\nj)\u000b\f+ i!(RiGijRT\nj)\u000b\f; (3)\nfrom which it follows that the Gilbert damping is directly related to the linear in frequency imaginary part of the\ninverse susceptibility\nd\nd!=[\u001f\u00001]\u000b\f\nij=\u000eij\u00121\n\rMi\u000f\u000b\f\u0016\u0013\n+ (RiGijRT\nj)\u000b\f: (4)\nNote thatRiandRjare rotation matrices rotating to the local frames of site iandj, respectively, which de\fne the\ncoordinates \u000b;\f=fx;yg(see Appendix B).\nBased on electronic structure theory, the transverse dynamical susceptibility can be extracted from a Dyson-like\nequation:\u001f\u00001(!) =\u001f\u00001\n0(!)\u0000U, where\u001f0is the susceptibility of non-interacting electron while Uis a many-body\ninteraction Kernel, called exchange-correlation Kernel in the context of time-dependent density functional theory30.\nThe Kernel is generally assumed to be adiabatic, which enables the evaluation of the Gilbert damping directly from4\nthe non-interacting susceptibility. Obviously:d\nd!\u001f\u00001(!) =d\nd!\u001f\u00001\n0(!). For small frequencies !,\u001f0has a simple\n!-dependence11:\n\u001f0(!)\u0019<\u001f0(0) +i!=d\nd!\u001f0j!=0 (5)\nand as shown in Ref.33\nd\nd!\u001f\u00001\n0(!)\u0019[<\u001f0(0)]\u00002=d\nd!\u001f0j!=0: (6)\nStarting from the electronic Hamiltonian Hand the corresponding Green functions G(E\u0006i\u0011) = (E\u0000H\u0006i\u0011)\u00001, one\ncan show that the non-interacting magnetic susceptibility can be de\fned via\n\u001f\u000b\f\n0;ij(!+ i\u0011) =\u00001\n\u0019TrZEF\ndE\u0002\n\u001b\u000bGij(E+!+ i\u0011)\u001b\fImGji(E) +\u001b\u000bImGij(E)\u001b\fGji(E\u0000!\u0000i\u0011)\u0003\n;(7)\nwith\u001bbeing the vector of Pauli matrices. Obviously to identify the Gilbert damping and how it reacts to magnetic\nnon-collinearity, we have to inspect the dependence of the susceptibility, and therefore the Green function, on the\nmisalignment of the magnetic moments.\nIII. MULTI-SITE EXPANSION OF THE GILBERT DAMPING\nAssuming the hamiltonian Hconsisting of an on-site contribution H0and an inter-site term encoded in a hopping\ntermt, which can be spin-dependent, one can proceed with a perturbative expansion of the corresponding Green\nfunction utilizing the Dyson equation\nGij=G0\ni\u000eij+G0\nitijG0\nj+G0\nitikG0\nktkjG0\nj+::: : (8)\nWithin the Alexander-Anderson single-orbital impurity model29,H0\ni=Ed\u0000i \u0000\u0000Uimi\u0001\u001b\u0000Bi\u0001\u001b, whereEd\nis the energy of the localized orbitals, \u0000 is the hybridization in the wide band limit, Uiis the local interaction\nresponsible for the formation of a magnetic moment and Biis an constraining or external magnetic \feld. SOC can be\nincorportated as tsoc\nij=i\u0015ijnij\u0001\u001b, where\u0015ijandnij=\u0000njirepresent respectively the strength and direction of the\nanisotropy \feld. It can be parameterized as a spin-dependent hopping using the Rashba-like spin-momentum locking\ntij=t(cos'R\u001b0\u0000i sin'Rnij\u0001\u001b)34.\nDepending on whether the considered Green function is an on-site Green function Giior an inter-site Green function\nGijdi\u000berent orders in the hopping are relevant. On-site Green functions require an even number of hopping processes,\nwhile inter-site Green functions require at least one hopping process.\nThe on-site Green function G0\nican be separated into a spin-less part Niand a spin dependent part Mi,\nG0\ni=Ni\u001b0+Mi\u0001\u001b ; (9)\nwhere the spin dependent part is parallel to the magnetic moment of site i,Mikmi(note that SOC is added later on\nto the hoppings). Using the perturbative expansion, eq. (8), and the separated Green function, eq. (9), to calculate\nthe magnetic susceptibility, eq. (7), one can systematically classify the allowed dependencies of the susceptibility with\nrespect to the directions of the magnetic moments, e.g. by using diagrammatic techniques as shown in Ref.18for a\nrelated model in the context of higher-order magnetic exchange interactions.\nSince our interest is in the form of the Gilbert damping, and therefore also in the form of the magnetic susceptibility, the\nperturbative expansion can be applied to the magnetic susceptibility. The general form of the magnetic susceptibility\nin terms of the Green function, eq. (7), depends on a combination of two Green functions with di\u000berent energy\narguments, which are labeled as !and 0 in the following. The relevant structure is then identi\fed as33,\n\u001f\u000b\f\nij(!)\u0018Tr\u001b\u000b\niGij(!)\u001b\f\njGji(0): (10)\nThe sake of the perturbative expansion is to gather insights in the possible forms and dependencies on the magnetic\nmoments of the Gilbert damping, and not to calculate explicitly the strength of the Gilbert damping from this5\nexpansion. Therefore, we focus on the structure of eq. (10), even though the susceptibility has more ingredients,\nwhich are of a similar form.\nInstead of writing all the perturbations explicitly, we set up a diagrammatic approach, which has the following\ningredients and rules:\n1. Each diagram contains the operators NandM, which are \u001b\u000band\u001b\ffor the magnetic susceptibility. The\noperators are represented by a white circle with the site and spin index: i\u000b\n2. Hoppings are represented by grey circles indicating the hopping from site itoj:ij. The vertex corresponds\ntotij.\n3. SOC is described as a spin-dependent hopping from site itojand represented by: ij;\u000b. The vertex\ncorresponds to tsoc\nij=i\u0015ij^n\u000b\nij\u001b\u000b.\n4. The bare spin-independent (on-site) Green functions are represented by directional lines with an energy at-\ntributed to it: !. The Green function connects operators and hoppings. The line corresponds to Ni(!).\n5. The spin-dependent part of the bare Green function is represented by: !;\u000b .\u000bindicates the spin direction.\nThe direction ensures the right order within the trace (due to the Pauli matrices, the di\u000berent objects in the\ndiagram do not commute). The line corresponds to Mi(!)m\u000b\ni\u001b\u000b.\nNote that the diagrammatic rules might be counter-intuitive, since local quantities (the Green function) are represented\nby lines, while non-local quantities (the hopping from itoj) are represented by vertices. However, these diagrammatic\nrules allow a much simpli\fed description and identi\fcation of all the possible forms of the Gilbert damping, without\nhaving to write lengthy perturbative expansions.\nSpin-orbit coupling independent contributions.\nTo get a feeling for the diagrammatic approach, we start with the simplest example: the on-site susceptibility without\nany hoppings to a di\u000berent site, which describes both the single atom and the lowest order term for interacting atoms.\nThe possible forms are,\n\u001fii\n\u000b\f(!)/\n!0\ni\u000b i\f+\n!;\r0\ni\u000b i\f\n+\n!0;\r\ni\u000b i\f+\n!;\u000e0;\r\ni\u000b i\f; (11)6\nwhich evaluate to,\n!0\ni\u000b i\f= Tr\u001b\u000b\u001b\fNi!)Ni(0) =\u000e\u000b\fNi(!)Ni(0) (12)\n!;\r0\ni\u000b i\f= Tr\u001b\u000b\u001b\r\u001b\fMi(!)Ni(0)m\r\ni= i\u000f\u000b\r\fMi(!)Ni(0)m\r\ni (13)\n!0;\r\ni\u000b i\f= Tr\u001b\u000b\u001b\f\u001b\rNi(!)Mi(0)m\r\ni= i\u000f\u000b\f\rMi(!)Mi(0)m\r\ni (14)\n!;\u000e0;\r\ni\u000b i\f= Tr\u001b\u000b\u001b\u000e\u001b\f\u001b\rMi(!)Mi(0)m\u000e\nim\r\ni\n= (\u000e\u000b\u000e\u000e\f\r+\u000e\u000b\r\u000e\f\u000e\u0000\u000e\u000b\f\u000e\r\u000e)Mi(!)Mi(0)m\u000e\nim\r\ni: (15)\nThe \frst diagram yields an isotropic contribution, the second and third diagrams yield an anti-symmetric contribution,\nwhich is linear in the magnetic moment, and the last diagram yields a symmetric contribution being quadratic in the\nmagnetic moment. Note that the energy dependence of the Green functions is crucial, since otherwise the sum of\neqs. (13) and (14) vanishes. In particular this means that the static susceptibility has no dependence linear in the\nmagnetic moment, while the the slope of the susceptibility with respect to energy can have a dependence linear in\nthe magnetic moment. The static part of the susceptibility maps to the magnetic exchange interactions, which are\nknown to be even in the magnetic moment due to time reversal symmetry.\nCombining all the functional forms of the diagrams, we \fnd the following possible dependencies of the on-site Gilbert\ndamping on the magnetic moments,\nG\u000b\f\nii(fmg)/f\u000e\u000b\f;\u000f\u000b\f\rm\r\ni;m\u000b\nim\f\nig: (16)\nSince we work in the local frames, mi= (0;0;mz\ni), the last dependence is a purely longitudinal term, which is not\nrelevant for the transversal dynamics discussed in this work.\nIf we still focus on the on-site term, but allow for two hoppings to another atom and back, we \fnd the following new7\ndiagrams,\n!00\n0\ni\u000b i\fij ji\n+\n!;\r00\n0\ni\u000b i\fij ji\n+:::+\n!;\r0;\u000e0\n0\ni\u000b i\fij ji\n+:::\n+\n!;\r0;\u000e0;\u0011\n0\ni\u000b i\fij ji\n+:::+\n!;\r0;\u000e0;\u0011\n0;\u0017\ni\u000b i\fij ji\n: (17)\nThe dashed line in the second diagram can be inserted in any of the four sides of the square, with the other possibilities\nomitted. Likewise for the diagrams with two or three dashed lines, the di\u000berent possible assignments have to be\nconsidered. The additional hopping to the site jyields a dependence of the on-site magnetic susceptibility and\ntherefore also the on-site Gilbert damping tensor on the magnetic moment of site j.\nAnother contribution to the Gilbert damping originates from the inter-site part, thus encoding the dependence of the\nmoment site ion the dynamics of the moment of site jviaGij. This contribution is often neglected in the literature,\nsince for many systems it is believed to have no signi\fcant impact. Using the microscopic model, a di\u000berent class\nof diagrams is responsible for the inter-site damping. In the lowest order in t=Um the diagrams contain already two\nhopping events,\n! !0 0\ni\u000b j\f\nijij\n+\n!;\r !0 0\ni\u000b j\f\nijij\n+:::+\n!;\r !;\u000e0 0\ni\u000b j\f\nijij\n+:::\n+\n!;\r !;\u000e0;\u0011 0\ni\u000b j\f\nijij\n+:::+\n!;\r !;\u000e0;\u0011 0;\u0010\ni\u000b j\f\nijij\n: (18)\nIn total, we \fnd that the spin-orbit independent intra-site and inter-site Gilbert damping tensors can be respectively\nwritten as\nGii=\u0010\nSi+Sij;(1)\ni (ei\u0001ej) +Sij;(2)\ni (ei\u0001ej)2\u0011\nI\n+\u0010\nAi+Aij\ni(ei\u0001ej)\u0011\nE(ei);(19)8\nand\nG\u000b\f\nij=\u0000\nSij+Sdot\nij(ei\u0001ej)\u0001\n\u000e\u000b\f\n+\u0000\nAij+Adot\nij(ei\u0001ej)\u0001\n(E(ei) +E(ej))\u000b\f\n+Scross\nij(ei\u0002ej)\u000b(ei\u0002ej)\f+Sba\nije\f\nie\u000b\nj; (20)\nwhere as mentioned earlier SandArepresent symmetric and asymmetric contributions, Iis the 3\u00023 identity while\nE(ei) =0\n@0ez\ni\u0000ey\ni\n\u0000ez\ni0ex\ni\ney\ni\u0000ex\ni01\nA.\nRemarkably, we \fnd that both the symmetric and anti-symmetric parts of the Gilbert damping tensor have a rich\ndependence with the opening angle of the magnetic moments. We identify, for example, the dot and the square\nof the dot products of the magnetic moments to possibly play a crucial role in modifying the damping, similarly to\nbilinear and biquadratic magnetic interactions. It is worth noting that even though the intra-site Gilbert damping can\nexplicitly depend on other magnetic moments, its meaning remains unchanged. The anti-symmetric precession-like\nterm describes a precession of the moment around its own e\u000bective magnetic \feld, while the diagonal damping-like\nterm describes a damping towards its own e\u000bective magnetic \feld. The dependence on other magnetic moments\nrenormalizes the intensity of those two processes. The inter-site Gilbert damping describes similar processes, but with\nrespect to the e\u000bective \feld of the other involved magnetic moment. On the basis of the LLG equation, eq. (2), it can be\nshown that the term related to Sba\nijwith a functional form of e\f\nie\u000b\njdescribes a precession of the i-th moment around\nthej-th moment with a time- and directional-dependent amplitude, @tmi/(mi\u0002mj) (mi\u0001@tmj). The double\ncross product term yields a time dependence of @tmi/(mi\u0002(mi\u0002mj)) ((mi\u0002mj)\u0001@tmj). Both contributions\nare neither pure precession-like nor pure damping-like, but show complex time- and directional-dependent dynamics.\nSpin-orbit coupling contributions. The spin-orbit interaction gives rise to new possible dependencies of the\ndamping on the magnetic structure. In particular, the so-called chiral damping, which in general is the di\u000berence\nof the damping between a right-handed and a left-handed opening, rises from SOC and broken inversion symmetry.\nUsing our perturbative model, we can identify all possible dependencies up to second order in SOC and third order\nin the magnetic moments.\nIn the diagramms SOC is added by replacing one spin-independent hopping vertex by a spin-dependent one,\n!00\n0\ni\u000b i\fij ji\n!\n!00\n0\ni\u000b i\fij\r ij\n: (21)\nUp to \frst-order in SOC, we \fnd the the following dependencies were found for the on-site Gilbert damping\nGii(fmg)/f\u000f\u000b\f\r^n\r\nij;^n\u000b\nij^n\f\nji;^n\f\nijm\u000b\ni;^n\u000b\nijm\f\ni;\u000e\u000b\f(^nij\u0001mi);\u000e\u000b\f(^nij\u0001mj);\n^n\f\nijm\u000b\nj;^n\u000b\nijm\f\nj;m\u000b\ni(^nij\u0002mi)\f;m\f\ni(^nij\u0002mi)\u000b;\n\u000e\u000b\f^nij\u0001(mi\u0002mj);m\u000b\ni(^nij\u0002mj)\f;m\f\ni(^nij\u0002mj)\u000b;(^nij\u0001mj)\u000f\u000b\f\rm\r\ni;\nm\u000b\nim\f\ni(^nij\u0001mj);(m\u000b\nim\f\nj\u0000m\f\nim\u000b\nj)(^nij\u0001mj);^n\f\nijm\u000b\ni(mi\u0001mj);^n\u000b\nijm\f\ni(mi\u0001mj)g: (22)\nWe identi\fed the following contributions for the on-site and intersite damping to be the most relevant one after the\nnumerical evaluation discussed in the next sections:\nGsoc\nii=Ssoc;ij\ni nij\u0001(ei\u0002ej)I\n+Ssoc;ij;(2)\ni (nij\u0001ei)(nij\u0001ej)I\n+Asoc;ij\ni nij\u0001(ei\u0002ej)E(ei)\n+Asoc;ij;(2)\ni (nij\u0001ej)E(nij); (23)9\nand\nGsoc;\u000b\f\nij =Ssoc\nijnij\u0001(ei\u0002ej)\u000e\u000b\f+Ssoc;ba\nijn\f\nij(ei\u0002ej)\u000b\n+Asoc\nijE\u000b\f(nij): (24)\nThe contributions being \frst-order in SOC are obviously chiral since they depend on the cross product, ei\u0002ej. Thus,\nsimilar to the magnetic Dzyaloshinskii-Moriya interaction, SOC gives rise to a dependence of the Gilbert damping\non the vector chirality, ei\u0002ej. The term chiral damping used in literature refers to the dependence of the Gilbert\ndamping on the chirality, but to our knowledge it was not shown so far how this dependence evolves from a microscopic\nmodel, and how it looks like in an atomistic model.\nExtension to three sites. Including three di\u000berent sites i,j, andkin the expansions allows for a ring exchange\ni!j!k!iinvolving three hopping processes, which gives rise to new dependencies of the Gilbert damping on\nthe directions of the moments.\nAn example of a diagram showing up for the on-site Gilbert damping is given below for the on-site Gilbert damping\nthe diagram,\n!00 0;\r\n0\ni\u000b i\fijjk\nki(25)\nApart from the natural extensions of the previously discussed 2-site quantities, the intra-site Gilbert damping of site i\ncan depend on the angle between the sites jandk,ej\u0001ek, or in higher-order on the product of the angles between site\niandjwithiandk, (ei\u0001ej)(ei\u0001ek). In sixth-order in the magnetic moments the term ( ei\u0001ej)(ej\u0001ek)(ek\u0001ei) yields\nto a dependence on the square of the scalar spin chirality of the three sites, [ ei\u0001(ej\u0002ek)]2. Including SOC, there are\ntwo interesting dependencies on the scalar spin chirality. In \frst-order one \fnds similarly to the recently discovered\nchiral biquadratic interaction18and its 3-site generalization19, e.g. ( nij\u0001ei) (ei\u0001(ej\u0002ek)), while in second order a\ndirect dependence on the scalar spin chirality is allowed, e.g. n\u000b\nijn\f\nki(ei\u0001(ej\u0002ek)). The scalar spin chirality directly\nrelates to the topological orbital moment35{37and therefore the physical origin of those dependencies lies in the\ntopological orbital moment. Even though these terms might not be the most important ones in our model, for speci\fc\nnon-collinear con\fgurations or for some realistic elements with a large topological orbital moment, e.g. MnGe20, they\nmight be important and even dominant yielding interesting new physics.\nIV. APPLICATION TO THE ALEXANDER-ANDERSON MODEL\nMagnetic dimers. Based on a 2-site Alexander-Anderson model, we investigated the dependence of the Gilbert\ndamping on the directions of the magnetic moments using the previously discussed possible terms (see more details\non the method in Appendix C). The spin splitting Ude\fnes the energy scale and all other parameters. The energy of\norbitals is set to Ed= 1:0. The magnetization is self-consistently constrained in a range of m= 0:2 tom= 0:8 using\nmagnetic constraining \felds. The corresponding spin-resolved local density of states is illustrated in Fig. 1b, where\nthe inter-site hopping is set to t= 0:2 and the hybridization to \u0000 = 0 :2. We performed two sets of calculations: one\nwithout spin-dependent hopping, 'R= 0 °, and one with a spin-dependent hopping, 'R= 20 °.\nThe di\u000berent damping parameters are shown in Fig. 2 as function of the magnetization. They are obtained from a\nleast-squares \ft to several non-collinear con\fgurations based on a Lebedev mesh for `= 238. The damping, which is\nindependent of the relative orientation of the two sites, is shown in Fig. 2a. The symmetric damping-like intra-site\ncontributionSidominates the damping tensor for most magnetizations and has a maximum at m= 0:3. The anti-\nsymmetric intra-site contribtuion Ai, which renormalizes the gyromagnetic ratio, approximately changes sign when\nthe Fermi level passes the peak of the minority spin channel at m\u00190:5 and has a signi\fcantly larger amplitude\nfor small magnetizations. Both contributions depend mainly on the broadening \u0000, which mimics the coupling to an10\nMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,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\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,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\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,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\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,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\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,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\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,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\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)abc\nab\nFIG. 2. Damping parameters as function of the magnetization for the dimers described within the Alexander-Anderson\nmodel including spin orbit coupling. A longitudinal magnetic \feld is used to self-consistently constrain the magnetization. The\nparameters are extracted from \ftting to the inverse of the transversal susceptibility for several non-collinear con\fgurations\nbased on a Lebedev mesh. Model parameters in units of U:Ed= 1:0; t= 0:2;\u0000 = 0:2; 'R= 20 °.\nelectron bath and is responsible for the absorption of spin currents, which in turn are responsible for the damping of\nthe magnetization dynamics15,16.\nThe directional dependencies of the intra-site damping are shown in Fig. 2b. With our choice of parameters, the\ncorrection to the damping-like symmetric Gilbert damping can reach half of the direction-independent term. This\nmeans that the damping can vary between \u00190:4\u00001:0 for a ferromagnetic and an antiferromagnetic state at m= 0:4.\nAlso for the renormalization of the gyromagnetic ratio a signi\fcant correction is found, which in the ferromagnetic case\nalways lowers and in the antiferromagnetic case enhances the amplitude. The most dominant contribution induced\nby SOC is the chiral one, which depends on the cross product of the moments iandj, which in terms of amplitude is\ncomparable to the isotropic dot product terms. Interestingly, while the inter-site damping term is in general known\nto be less relevant than the intra-site damping, we \fnd that this does not hold for the directional dependence of the\ndamping. The inter-site damping is shown in Fig. 2c. Even though the directional-independent term, Sij, is nearly\none order of magnitude smaller than the equivalent intra-site contribution, this is not necessarily the case for the\ndirectional-dependent terms, which are comparable to the intra-site equivalents.\nV. APPLICATION TO FIRST-PRINCIPLES SIMULATIONS\nTo investigate the importance of non-collinear e\u000bects for the Gilbert damping in realistic systems, we use DFT and\ntime-dependent DFT to explore the prototypical example of monoatomic 3 dtransition metal adatoms and dimers\ndeposited on a heavy metal surface hosting large SOC (see Fig. 3a for an illustration of the con\fguration). We\nconsider a Cr, Mn, Fe and Co atoms deposited on the fcc-Au(111) surface (details of the simulations are described in\nAppendix D). The parameters and the corresponding functional forms are \ftted to our \frst-principles data using 196\nnon-collinear states based on a Lebedev mesh for `= 238.\nAdatoms on Au(111). To illustrate the di\u000berent e\u000bects on the Gilbert damping, we start by exploring magnetic\nadatoms in the uniaxial symmetry of the Au(111) surface. For the adatoms no non-local e\u000bects can contribute to the\nGilbert damping.\nThe Gilbert damping tensor of a single adatom without SOC has the form shown in relation to eq. (16),\nG0\ni=SiI+AiE(ei): (26)\nNote that SOC can induce additional anisotropies, as shown in eq. (22). The most important ones for the case of a\nsingle adatom are f\u000f\u000b\f\r^n\r\nij;^n\u000b\nij^n\f\njig, which in the C3vsymmetry result in\nGi=G0\ni+Ssoc\ni0\n@0 0 0\n0 0 0\n0 0 11\nA+Asoc\ni0\n@0 1 0\n\u00001 0 0\n0 0 01\nA; (27)11\nDampingCr / Au(111) Mn / Au(111) Fe / Au(111) Co / Au(111)parameters\nSi 0:083 0 :014 0 :242 0 :472\nAi 0:204 0 :100 0 :200 0 :024\nSsoc\ni 0:000 0 :000 0 :116 0 :010\nAsoc\ni 0:000 0 :000\u00000:022 0 :012\n\rrenorm\nx=y 1:42 1 :67 1 :43 1 :91\n\rrenorm\nz 1:42 1 :67 1 :48 1 :87\nTABLE I. Gilbert damping parameters of Cr, Mn, Fe and Co adatoms deposited on the Au(111) surface as parametrized in\neqs. (26) and (27). The SOC \feld points in the z-direction due to the C3vsymmetry. The renormalized gyromagnetic ratio\n\rrenormis calculated according to eqs. (28) for an in-plane magnetic moment and an out-of-plane magnetic moment.\nsince the sum of all SOC vectors points in the out-of-plane direction with ^nij!ez. Thus, the Gilbert damping tensor\nof adatoms deposited on the Au(111) surface can be described by the four parameters shown in eqs. (26) and (27),\nwhich are reported in Table I for Cr, Mn, Fe and Co adatoms. Cr and Mn, being nearly half-\flled, are characterized\nby a small damping-like contribution Si, while Fe and Co having states at the Fermi level show a signi\fcant damping\nof up to 0:47 in the case of Co. The antisymmetric part Aiof the Gilbert damping tensor results in an e\u000bective\nrenormalization of the gyromagnetic ratio \r, as shown in relation to eq. (A5), which using the full LLG equation,\neq. (2), and approximating mi\u0001dmi\ndt= 0 is given by,\n\rrenorm=\r1\n1 +\r(ei\u0001Ai); (28)\nwhere Aidescribes the vector Ai=\u0000\nAi;Ai;Ai+Asoc\ni\u0001\n. For Cr and Fe there is a signi\fcant renormalization of the\ngyromagnetic ratio resulting in approximately 1 :4. In contrast, Co shows only a weak renormalization with 1 :9 being\nclose to the gyromagnetic ratio of 2. The SOC e\u000bects are negigible for most adatoms except for Fe, which shows a\nsmall anisotropy in the renormalized gyromagentic ratio ( \u001910 %) and a large anisotropy in the damping-like term of\nnearly 50 %.\nDimers on Au(111). In contrast to single adatoms, dimers can show non-local contributions and dependencies on\nthe relative orientation of the magnetic moments carried by the atoms. All quantities depending on the SOC vector\nare assumed to lie in the y-z-plane due to the mirror symmetry of the system. A sketch of the dimer and its nearest\nneighboring substrate atoms together with adatoms' local density of states are presented in Fig. 3.\nThe density of states originates mainly from the d-states of the dimer atoms. It can be seen that the dimers exhibit\na much more complicated hybridization pattern than the Alexander-Anderson model. In addition the crystal \feld\nsplits the di\u000berent d-states resulting in a rich and high complexity than assumed in the model. However, the main\nfeatures are comparable: For all dimers there is either a fully occupied majority channel (Mn, Fe, and Co) or a fully\nunoccupied minority channel (Cr). The other spin channel determines the magnetic moments of the dimer atoms\nf4:04;4:48;3:42;2:20g\u0016Bfor respectively Cr, Mn, Fe and Co. Using the maximal spin moment, which is according\nto Hund's rule 5 \u0016B, the \frst-principles results can be converted to the single-orbital Alexander-Anderson model\ncorresponding to approximately m=f0:81;0:90;0:68;0:44g\u0016Bfor the aforementioned sequence of atoms. Thus by\nthis comparison, we expect large non-collinear contributions for Fe and Co, while Cr and Mn should show only weak\nnon-local dependencies.\nThe obtained parametrization is given in Table II. The Cr and Mn dimers show a weak or nearly no directional\ndependence. While the overall damping for both nanostructures is rather small, there is a signi\fcant correction to\nthe gyromagnetic ratio.\nIn contrast, the Fe and Co dimers are characterized by a very strong directional dependence. Originating from the\nisotropic dependencies of the damping-like contributions, the damping of the Fe dimer can vary between 0 :21 in\nthe ferromagnetic state and 0 :99 in the antiferromagnetic state. For the Co dimer the inter-site damping is even\ndominated by the bilinear and biquadratic term, while the constant damping is negligible. In total, there is a very\ngood qualitative agreement between the expectations derived from studying the Alexander-Anderson model and the\n\frst-principles results.12\nDampingCr / Au(111) Mn / Au(111) Fe / Au(111) Co / Au(111)parameters\nSi 0:0911 0 :0210 0 :2307 0 :5235\nSij;(1)\ni 0:0376 0 :0006\u00000:3924\u00000:2662\nSij;(2)\ni 0:0133\u00000:0006 0 :3707 0 :3119\nAi 0:2135 0 :1158 0 :1472 0 :0915\nAij\ni 0:0521 0 :0028\u00000:0710\u00000:0305\nSij\u00000:0356 0 :0028 0 :2932 0 :0929\nSdot\nij\u00000:0344\u00000:0018\u00000:3396\u00000:4056\nSdot;(2)\nij 0:0100 0 :0001 0 :1579 0 :2468\nAij\u00000:0281\u00000:0044 0 :0103 0 :0011\nAdot\nij\u00000:0175 0 :0000\u00000:0234\u00000:0402\nScross\nij 0:0288 0 :0002\u00000:2857\u00000:0895\nSba\nij 0:0331 0 :0036 0 :2181 0 :2651\nSsoc;ij;y\ni 0:0034 0 :0000 0 :0143\u00000:0225\nSsoc;ij;z\ni 0:0011 0 :0000\u00000:0104 0 :0156\nAsoc;ij;y\ni 0:0024\u00000:0001\u00000:0036 0 :0022\nAsoc;ij;z\ni 0:0018\u00000:0005 0 :0039\u00000:0144\nSsoc;y\nij 0:0004 0 :0001 0 :0307 0 :0159\nSsoc;z\nij\u00000:0011 0 :0000\u00000:0233 0 :0206\nSba,soc ;y\nij\u00000:0027 0 :0000\u00000:0184\u00000:0270\nSba,soc ;z\nij 0:0005\u00000:0001 0 :0116\u00000:0411\nTABLE II. Damping parameters of Cr, Mn, Fe and Co dimers deposited on the Au(111) surface. The possible forms of the\ndamping are taken from the analytic model. The SOC \feld is assumed to lie in the y-zplane and inverts under permutation\nof the two dimer atoms.\n\u00003\u00002\u000010123E\u0000EF[eV]\u00006\u0000303DOS [#states/eV]Cr dimerMn dimerFe dimerCo dimersurface\nab\nFIG. 3. aIllustration of a non-collinear magnetic dimer (red spheres) deposited on the (111) facets of Au (grey spheres).\nFrom the initial C3vspatial symmetry of the surface the dimers preserve the mirror plane (indicated grey) in the y-zplane. b\nLocal density of states of the Cr, Mn, Fe and Co dimers deposited on the Au(111) surface. The grey background indicates the\nsurface density of states. The dimers are collinear in the z-direction.\nVI. CONCLUSIONS\nIn this article, we presented a comprehensive analysis of magnetization dynamics in non-collinear system with a special\nfocus on the Gilbert damping tensor and its dependencies on the non-collinearity. Using a perturbative expansion\nof the two-site Alexander-Anderson model, we could identify that both, the intra-site and the inter-site part of the\nGilbert damping, depend isotropically on the environment via the e\u000bective angle between the two magnetic moments,\nei\u0001ej. SOC was identi\fed as the source of a chiral contribution to the Gilbert damping, which similarly to the\nDzyaloshinskii-Moriya and chiral biquadratic interactions depends linearly on the vector spin chirality, ei\u0002ej. We\nunveiled dependencies that are proportional to the three-spin scalar chirality ei\u0001(ej\u0002ek), i.e. to the chiral or\ntopological moment, and to its square. Using the Alexander-Anderson model, we investigated the importance of the13\ndi\u000berent contributions in terms of their magnitude as function of the magnetization. Using the prototypical test\nsystem of Cr, Mn, Fe and Co dimers deposited on the Au(111) surface, we extracted the e\u000bects of the non-collinearity\non the Gilbert damping using time-dependent DFT. Overall, the \frst-principles results agree qualitatively well with\nthe Alexander-Anderson model, showing no dependence for the nearly half-\flled systems Cr and Mn and a strong\ndependence on the non-collinearity for Fe and Co having a half-\flled minority spin-channel. The realistic systems\nindicate an even stronger dependence on the magnetic texture than the model with the used parameters. The Fe and\nthe Co dimer show signi\fcant isotropic terms up to the biquadratic term, while the chiral contributions originating\nfrom SOC have only a weak impact on the total Gilbert damping. However, the chiral contributions can play the\ndeciding role for systems which are degenerate in the isotropic terms, like e.g. spin spirals of opposite chirality.\nWe expect the dependencies of the Gilbert damping on the magnetic texture to have a signi\fcant and non-trivial\nimpact on the spin dynamics of complex magnetic structures. Our \fndings are readily implementable in the LLG\nmodel, which can trivially be amended with the angular dependencies provided in the manuscript. Utilizing multiscale\nmapping approaches, it is rather straightforward to generalize the presented forms for an implementation of the\nmicromagnetic LLG and Thiele equations. The impact of the di\u000berent contributions to the Gilbert damping, e.g. the\nvector (and/or scalar) chiral and the isotropic contributions, can be analyzed on the basis of either free parameters\nor sophisticated parametrizations obtained from \frst principles as discussed in this manuscript. It remains to be\nexplored how the newly found dependencies of the Gilbert damping a\u000bect the excitations and motion of a plethora of\nhighly non-collinear magnetic quasi-particles such as magnetic skyrmions, bobbers, hop\fons, domain walls and spin\nspirals. Future studies using atomistic spin dynamics simulations could shed some light on this aspect and help for\nthe design of future devices based on spintronics.\nACKNOWLEDGMENTS\nThis work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research\nand innovation program (ERC-consolidator grant 681405 { DYNASORE) and from Deutsche Forschungsgemeinschaft\n(DFG) through SPP 2137 \\Skyrmionics\" (Project LO 1659/8-1). The authors gratefully acknowledge the computing\ntime granted through JARA-HPC on the supercomputer JURECA at the Forschungszentrum J ulich39.\nVII. METHODS\nAppendix A: Analysis of the Gilbert damping tensor\nThe Gilbert damping tensor Gcan be decomposed into a symmetric part Sand an anti-symmetric part A,\nA=G\u0000GT\n2andS=G+GT\n2: (A1)\nWhile the symmetric contribution can be referred to as the damping-like contribution including potential anisotropies,\nthe anti-symmetric Atypically renormalizes the gyromagnetic ratio as can be seen as follows: The three independent\ncomponents of an anti-symmetric tensor can be encoded in a vector Ayielding\nA\u000b\f=\u000f\u000b\f\rA\r; (A2)\nwhere\u000f\u000b\f\ris the Levi-Cevita symbol. Inserting this into the LLG equation yields\ndmi\ndt=\u0000\rmi\u00020\n@Be\u000b\ni+X\njAijdmj\ndt1\nA (A3)\n\u0019\u0000\rmi\u00020\n@Be\u000b\ni\u0000\rX\njAij\u0000\nmj\u0002Be\u000b\nj\u00011\nA: (A4)\nThe last term can be rewritten as\n(Aij\u0001mj)Be\u000b\nj\u0000\u0000\nAij\u0001Be\u000b\nj\u0001\nmj: (A5)14\nFor the local contribution, Aii, the correction is kmiandkBe\u000b\niyielding a renormalization of \rmi\u0002Be\u000b\ni. However,\nthe non-local parts of the anti-symmetric Gilbert damping tensor can be damping-like.\nAppendix B: Relation between the LLG and the magnetic susceptibility\nThe Fourier transform of the LLG equation is given by\n\u0000i!mi=\u0000\rmi\u00020\n@Bext\ni\u0000X\njJijmj\u0000i!X\njGijmj1\nA: (B1)\nTransforming this equation to the local frames of site iandjusing the rotation matrices RiandRjyields\ni!\n\rMimloc\ni=mloc\ni\nMi\u00020\n@RiBext\ni\u0000X\njRiJijRT\njmloc\nj\u0000i!X\njRiGijRT\njmloc\nj1\nA; (B2)\nwhere mloc\ni=Rimiandmloc\nj=Rjmj. The rotation matrices are written as R(#i;'i) = cos(#i=2)\u001b0+\ni sin(#i=2)\u0000\nsin('i)\u001bx\u0000cos('i)\u001by\u0001\n, with (#i;'i) being the polar and azimuthal angle pertaining to the moment\nmi. In the ground state the magnetic torque vanishes. Thus, denoting mloc\ni= (mx\ni; my\ni; Mi), wheremx=y\niare\nperturbations to the ground states, yields for the ground state\n0\n@(RiBext\ni)x\u0000P\nj(RiJijRT\njMjez)x\n(RiBext\ni)y\u0000P\nj(RiJijRT\njMjez)y\n(RiBext\ni)z\u0000P\nj(RiJijRT\njMjez)z1\nA=0\n@0\n0\n(RiBe\u000b\ni)z1\nA: (B3)\nLinearizing the LLG and using the previous result and limiting our expansion to transveral excitations yield\ni!\n\rMimx\ni=my\ni(RiBe\u000b\ni)z\nMi\u0000(RiBext\ni)y+X\nj(RiJijRT\njmloc\nj)y+ i!X\nj(RiGijRT\njmloc\nj)y(B4)\ni!\n\rMimy\ni=\u0000mx\ni(RiBe\u000b\ni)z\nMi+ (RiBext\ni)x\u0000X\nj(RiJijRT\njmloc\nj)x\u0000i!X\nj(RiGijRT\njmloc\nj)x; (B5)\nwhich in a compact form gives\nX\nj\n\f=x;y0\n@\u000eij\u0012\n\u000e\u000b\f(RiBe\u000b\ni)z\nMi+\u000f\u000b\f\u0016i!\n\rMi\u0013\n+X\nj(RiJijRT\nj)\u000b\f+ i!X\nj(RiGijRT\nj)\u000b\f1\nAm\f\nj= (RiBext\ni)\u000b; (B6)\nand can be related to the inverse of the magnetic susceptibility\nX\nj\n\f=x;y\u001f\u00001\ni\u000b;j\f(!)m\f\nj= (RiBext\ni)\u000b: (B7)\nThus, the magnetic susceptibility in the local frames of site iandjis given by\n\u001f\u00001\ni\u000b;j\f(!) =\u000eij\u0012\n\u000e\u000b\f(RiBe\u000b\ni)z\nMi+\u000f\u000b\f\u0016i!\n\rMi\u0013\n+X\nj(RiJijRT\nj)\u000b\f+ i!X\nj(RiGijRT\nj)\u000b\f(B8)\nAppendix C: Alexander-Anderson model{more details\nWe use a single orbital Alexander-Anderson model,\nH=X\nij[\u000eij(Ed\u0000i \u0000\u0000Uimi\u0001\u001b\u0000Bi\u0001\u001b)\u0000(1\u0000\u000eij)tij]; (C1)15\nwhereiandjsum over all n-sites,Edis the energy of the localized orbitals, \u0000 is the hybridization in the wide band\nlimit,Uiis the local interaction responsible for the formation of a magnetic moment, miis the magnetic moment of site\ni,Biis an constraining or external magnetic \feld, \u001bare the Pauli matrices, and tijis the hopping parameter between\nsiteiandj, which can be in general spin-dependent. SOC is added as spin-dependent hopping using a Rashba-like\nspin-momentum locking tij=t(cos'R\u001b0\u0000i sin'Rnij\u0001\u001b), where the spin-dependent hopping is characterized by its\nstrength de\fned by 'Rand its direction nij=\u0000nji34. The eigenenergies and eigenstates of the model are given by,\nHjni= (En\u0000i \u0000)jni: (C2)\nThe single particle Green function can be de\fned using the eigensystem,\nG(E+ i\u0011) =X\nnjnihnj\nE\u0000En+ i\u0011; (C3)\nwhere\u0011is an in\fnitesimal parameter de\fning the retarded ( \u0011!0+) and advanced ( \u0011!0\u0000) Green function. The\nmagnitude of the magnetic moment is determined self-consistently using\nmi=\u00001\n\u0019Im TrZ\ndE\u001bGii(E); (C4)\nwhereGii(E) is the local Green function of site idepending on the magnetic moment. Using the magnetic torque\nexerted on the moment of site i,\ndH\nd^ei=\u0000miBe\u000b\ni; (C5)\nmagnetic constraining \felds can be de\fned ensuring the stability of an arbitrary non-collinear con\fguration,\nBconstr=\u0000Pm\n?mi\njmijBe\u000b\ni) Hconstr=\u0000Bconstr\u0001\u001b ; (C6)\nwherePm\n?is the projection on the plane perpendicular to the moment m. The constraining \felds are added to the\nhamiltonian, eq. (C1), and determined self-consistently.\nAppendix D: Density functional theory{details\nThe density functional theory calculations were performed with the Korringa-Kohn-Rostoker (KKR) Green function\nmethod. We assume the atomic sphere approximation for the the potential and include full charge density in the\nself-consistent scheme40. Exchange and correlation e\u000bects are treated in the local spin density approximation (LSDA)\nas parametrized by Vosko, Wilk and Nusair41, and SOC is added to the scalar-relativistic approximation in a self-\nconsistent fashion42. We model the pristine surfaces utilizing a slab of 40 layers with the experimental lattice constant\nof Au assuming open boundary conditions in the stacking direction, and surrounded by two vacuum regions. No\nrelaxation of the surface layer is considered, as it was shown to be negligible43. We use 450\u0002450k-points in the\ntwo-dimensional Brillouin zone, and the angular momentum expansions for the scattering problem are carried out up\nto`max= 3. Each adatom is placed in the fcc-stacking position on the surface, using the embedding KKR method.\nPreviously reported relaxations towards the surface of 3 dadatoms deposited on the Au(111) surface44indicate a\nweak dependence of the relaxation on the chemical nature of the element. Therefore, we use a relaxation towards the\nsurface of 20 % of the inter-layer distance for all the considered dimers. The embedding region consists of a spherical\ncluster around each magnetic adatom, including the nearest-neighbor surface atoms. 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B 94(12) 121114 URL https:\n//link.aps.org/doi/10.1103/PhysRevB.94.121114\n37dos Santos Dias M and Lounis S 2017 Spintronics X 10357 136 { 152 URL https://doi.org/10.1117/12.2275305\n38Lebedev V I and Laikov D 1999 Doklady Mathematics 59477{481\n39J ulich Supercomputing Centre 2018 Journal of large-scale research facilities 4URL http://dx.doi.org/10.17815/\njlsrf-4-121-1\n40Papanikolaou N, Zeller R and Dederichs P H 2002 Journal of Physics: Condensed Matter 142799{2823\n41Vosko S H, Wilk L and Nusair M 1980 Canadian Journal of physics 581200{1211\n42Bauer D S G 2014 Development of a relativistic full-potential \frst-principles multiple scattering Green function method\napplied to complex magnetic textures of nano structures at surfaces (Forschungszentrum J ulich J ulich)\n43B lo\u0013 nski P and Hafner J 2009 Journal of Physics: Condensed Matter 21426001 ISSN 0953-8984\n44Brinker S, dos Santos Dias M and Lounis S 2018 Phys. Rev. B 98(9) 094428 URL https://link.aps.org/doi/10.1103/\nPhysRevB.98.094428" }, { "title": "0801.0549v1.Spin_orbit_precession_damping_in_transition_metal_ferromagnets.pdf", "content": "arXiv:0801.0549v1 [cond-mat.mtrl-sci] 3 Jan 2008Spin-orbit precession damping in transition metal ferroma gnets\nK. Gilmore1,2, Y.U. Idzerda2, and M.D. Stiles1\n1National Institute of Standards and Technology, Gaithersb urg, MD 20899-6202\n2Physics Department, Montana State University, Bozeman, MT 59717\n(Dated: November 1, 2018, Journal of Applied Physics )\nWe provide a simple explanation, based on an effective field, f or the precession damping rate due\nto the spin-orbit interaction. Previous effective field trea tments of spin-orbit damping include only\nvariations of the state energies with respect to the magneti zation direction, an effect referred to\nas the breathing Fermi surface. Treating the interaction of the rotating spins with the orbits as\na perturbation, we include also changes in the state populat ions in the effective field. In order to\ninvestigate the quantitative differences between the dampi ng rates of iron, cobalt, and nickel, we\ncompute the dependence of the damping rate on the density of s tates and the spin-orbit parameter.\nThere is a strong correlation between the density of states a nd the damping rate. The intraband\nterms of the damping rate depend on the spin-orbit parameter cubed while the interband terms are\nproportional to the spin-orbit parameter squared. However , the spectrum of band gaps is also an\nimportant quantity and does not appear to depend in a simple w ay on material parameters.\nI. INTRODUCTION\nMagnetic memory devices are useful if they can be re-\nliably switched between two stable states. The fidelity of\nthis switching process depends sensitively on the damp-\ning rate of the system. Despite decades of research and\nthe relentless industrial push toward smaller and faster\ndevices, many questions about the damping process re-\nmain unanswered, particularly for metallic ferromagnets.\nRecent experimental efforts have investigated the extent\nto which the damping rate of NiFe alloys can be tuned\nthrough doping, particularly with the addition of rare\nearth [1] and transition metal elements [2]. While these\ninvestigations found a general trend suggesting damp-\ning increases with increasing spin-orbit coupling of the\ndopant, the details behind this effect remained elusive.\nTo aid this effort, this article provides a simple descrip-\ntion of the damping process and investigates how some\nmaterial properties affect the damping rate.\nPrecession damping in metallic ferromagnets results\npredominantly from a combined effort of spin-orbit cou-\npling and electron-lattice scattering [3, 4]. The role of\nlattice scattering was studied in early experimental work\nthrough the temperature dependence of damping rates\n[5, 6]. Measurement of damping rates versus tempera-\nture revealed two primary contributions to damping, an\nexpected part that increased with temperature, and an\nunexpected part that decreased with temperature. In\ncobalt these two opposing contributions combine to pro-\nduce a minimum damping rate near 100 K, for nickel the\nincreasing term is weaker leading to a temperature inde-\npendent damping rate above 300 K, while for iron the\ndamping rate becomes independent of temperature be-\nlow room temperature. Heinrich et al. later noted that\nthe temperature dependence of the increasing and de-\ncreasingcontributionsmatchedthat ofthe resistivityand\nconductivity, respectively [7, 8], and so dubbed the two\ncontributions conductivity-like for the decreasing piece\nand resistivity-like for the increasing part.\nAmong the many theories on intrinsic precessiondamping[3, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], Kam-\nbersk´ y’storque-correlationmodel [3] is unique in qualita-\ntivelymatchingtheobservednon-monotonictemperature\ndependence that we just described. We recently evalu-\nated this model for iron, cobalt, and nickel, and showed\nthat it accurately predicts the precession damping rates\nofthesesystems[4]. Whilethis modelsucceedsincaptur-\ning the important physical effects involved in precession\ndamping, it does not easily identify the important physi-\ncal processes or give insight into how one might alter the\ndampingratethroughsamplemanipulation. In sectionII\nwe briefly describe the torque-correlation model. In or-\nder to provide a more tangible explanation of precession\ndamping we rederive the damping rate from an effective\nfield approach in section III. This discussion is followed\nin section IV by a quantitative analysis of the effect on\nthe damping rate of tuning the density of states and the\nspin-orbit parameter.\nII. TORQUE-CORRELATION MODEL\nKambersk´ y’s theory describes damping in terms of the\nspin-orbit torque correlation function, finding a damping\nrate of\nλ=π/planckover2pi1γ2\nµ0/summationdisplay\nnm/integraldisplayd3k\n(2π)3|Γ−\nmn(k)|2\n×/integraldisplay\ndǫ1Ank(ǫ1)Amk(ǫ1)η(ǫ1). (1)\nThe gyromagneticratio is γ=gµ0µB//planckover2pi1,gis the Land´ e g\nfactor,µ0is the permeability of space, nandmare band\nindices, and kis the electronwavevector. The matrix ele-\nments|Γ−\nmn(k)|2describe a torque between the spin and\norbital moments that arises as the spins precess. η(ǫ)\nis the derivative of the Fermi function −df/dǫ, which\nis a positive distribution peaked about the Fermi level\nthat restricts scattering events to the neighborhood of\ntheFermi surface. The electronspectralfunctions Ank(ǫ)2\nareLorentziansin energyspace centeredat band energies\nwithwidthsdeterminedbythescatteringrate. Theyphe-\nnomenologically account for electron-lattice scattering.\nEquation (1) includes two processes: the decay of\nmagnons into electron-hole pairs and the scattering of\nthe electrons and holes with the lattice. This expres-\nsion is similar in structure to sp-dmodels that have\nproven successful in describing dissipation in semicon-\nductors [20]. However, the physics of the magnon decay\nprocess is very different. In the present case, there is no\ndistinction between spanddelectrons. The spin-orbit\ntorque annihilates a uniform mode magnon and gener-\nates an electron-hole pair. The electron-hole pair is then\ncollapsed through lattice scattering. The electron and\nhole are dressed through lattice interactions and are best\nthought of as a single quasiparticle with indeterminant\nenergy and a lifetime given by the electron-lattice scat-\ntering time. The dressed electron and hole can occupy\nthe same band ( m=n), which we call an intraband\ntransition, or two different bands ( m/negationslash=n), aninter-\nbandtransition. For intraband transitions, the integra-\ntion over the spectral functions is proportional to the\nscattering time, just like the conductivity. For interband\ntransitions, the intregration over the spectral functions\nis roughly inversely proportional to the scattering time,\nas is the resistivity. Therefore, the intraband terms in\nEq. (1) give the conductivity-like contributions to damp-\ning that decrease with temperature while the interband\nterms yield the resistivity-likecontributionsthat increase\nwith temperature.\nIII. EFFECTIVE FIELD DERIVATION\nAn effective field for the magnetization dynamics is de-\nfinedasthevariationoftheelectronicenergywithrespect\nto the magnetization direction µ0Heff=−∂E/∂M. The\nmagnitude of the magnetization Mis considered con-\nstant within the Landau-Lifshitz formulation, only the\ndirection ˆMof the magnetization changes. The total\nelectronic energy of the system can be approximated by\nE=/summationtext\nnkρnkǫnk, which is a summation over the single\nelectron energies ǫnkweighted by the state occupancies\nρnk. If the state occupancies are held at their equilbrium\nvalues, the resulting effective field is equivalent to that of\nthe magnetocrystalline anisotropy [21], which describes\nreversible processes. If however, the state occupancies\nare allowed to deviate from the equilibrium populations\ninresponsetotheoscillatingperturbation, anirreversible\ncontribution also arrises, which we show produces the\ndamping in Eq. (1).\nAs the magnetization precesses the energies of the\nstates change through variations in the spin-orbit con-\ntribution and transitions between states occur. These\ntwo effects, the changing energies of the states and the\ntransitions between states, produce a contribution to theeffective field\nHeff=−1\nµ0M/summationdisplay\nnk/bracketleftbigg\nρnk∂ǫnk\n∂ˆM+∂ρnk\n∂ˆMǫnk/bracketrightbigg\n.(2)\nThe first term in the brackets describes the variation in\nthe spin-orbit energies of the states as the magnetization\ndirection changes. This effect, which has been discussed\nand evaluated before [10, 13, 22], is generally referred\nto as the breathing Fermi surface model. The spin-orbit\ntorque does not cause transitions between states in this\npicture, but does cause the Fermi surface to swell and\ncontract as the magnetization precesses. We will show\nthat this portion of the effective field gives the intraband\ntermsofEq.(1). Thesecondterminthebracketshaspre-\nviously been neglected in effective field treatments, but\naccounts for changes in the system energy due to tran-\nsitions between states. This term does not change the\nenergies of the states, but does create electron-hole pairs\nby exciting electrons from lower bands to higher bands.\nThis process can be pictured as a bubbling of individ-\nual electrons on the Fermi surface. We will demonstrate\nthat this portion of the effective field gives the interband\nterms of Eq. (1).\nA. Intraband terms\nThe first term in the effective field Eq. (2) accounts for\nthe effects of the breathing Fermi surface (bfs) model.\nSince this model has previously been discussed in detail\n[10, 13, 22] we will give only a very brief review of it here,\nfocusing instead on connecting it to the intraband terms\nof Eq. (1).\nAs the magnetization precesses the spin-orbit energy\nof each state changes. Some occupied states originally\njust below the Fermi level get pushed above the Fermi\nlevel and simultaneously some unoccupied state origi-\nnally above the Fermi level may be pushed below it.\nThis process takes the system, which was originally in\nthe ground state, and drives it out of equilibrium into an\nexcited state creating electron-hole pairs in the absence\nof any scattering events. Scattering, which occurs with a\nrate given by the inverse of the relaxation time τ, brings\nthe system to a new equilibrium. The relaxation time\napproximation determines how far from equilibrium the\nsystem can get.\nρnk=fnk−τdfnk\ndt. (3)\nThe occupancy ρnkof each state ψnkdeviates from its\nequilibrium value fnkby an amount proportional to the\nscattering time. How quickly the system damps depends\non the magnitude of this deviation.\nThe rate of change of the equilibrium distribution\ndfnk/dtdepends on how much the distribution changes\nas the energy of the state changes dfnk/dǫnk, how much\nthe state energy changes as the precession angle changes3\nFIG. 1: Schematic description of precession geometry. With in\nthebreathingFermi surface model (a) thedampingrate is cal -\nculatedas themagnetization passesthroughaspecific point in\na given direction. The torque correlation model (b) gives th e\ndamping rate for precessing about a given direction. Dashed\ncurves indicate the precession trajectory.\ndǫnk/dˆM, and how quickly the spin direction is precess-\ningdˆM/dt. These can be combined with a chain rule\ndfnk\ndt=dfnk\ndǫnkdǫnk\ndˆMdˆM\ndt. (4)\nCombining this result with the relaxation time approxi-\nmation Eq. (3) and substituting these state occupancies\ninto the first term of the effective field in Eq. (2) gives\nHeff\nbfs=Hani\nbfs+Hdamp\nbfs, (5)\nHani\nbfs=−1\nµ0M/summationdisplay\nnkfnk∂ǫnk\n∂ˆM, (6)\nHdamp\nbfs=−1\nµ0M/summationdisplay\nnkτ/parenleftbigg\n−dfnk\ndǫnk/parenrightbigg/parenleftbiggdǫnk\ndˆM/parenrightbigg2dˆM\ndt.(7)\nHani\nbfsis a contribution to the magnetocrystalline\nanisotropy field and Hdamp\nbfsthe damping field from the\nbreathing Fermi surface model. When we compare this\ndamping field to the damping field postulated by the\nLandau-Lifshitz-Gilbert equation\nHdamp\nLLG=−λ\nγ2MdˆM\ndt(8)\nwe find that the damping rate is\nλbfs=τγ2\nµ0/summationdisplay\nnkη(ǫnk)/parenleftbigg∂ǫnk\n∂ˆM/parenrightbigg2\n. (9)\nAs in Eq. (1), η(ǫ) is the negative derivative of the Fermi\nfunction and is a positive distribution peaked about the\nFermi energy.\nAs described in Fig. (1a), the result of the breathing\nFermi surface model Eq. (9) describes the damping rate\nof a material as the magnetization rotates through a par-\nticular point ˆ zabout a given axis ˆϑ. When ˆMis instan-\ntaneously aligned with ˆ zthe direction of the change in\nthe magnetization dˆMwill be perpendicular to ˆ z, in the\nˆx-ˆyplane. On the other hand, the torque correlationmodel Eq. (1) gives the damping rate when the magne-\ntization is undergoing small angle precession about the\nˆzdirection (see Fig.(1b)). When ˆ zis a high symmetry\ndirection the change in the magnetization will stay in the\nˆx-ˆyplane. In each scenario – rotating ˆMthroughˆzin the\nbreathing Fermi surface model and rotating ˆMaboutˆz\nin the torque correlation model – dˆMis confined to the\nˆx-ˆyplane. Therefore, rotating through ˆ zand rotating\nabout ˆzare equivalent in the small angle limit when ˆ z\nis a high symmetry direction. With this observation we\nnow show that the intraband contributions of the torque\ncorrelation model are equivalent to the breathing Fermi\nsurface result under these conditions.\nThe only energy that changes as the magnetization\nrotates is the spin-orbit energy Hso. As the spin of the\nstate|nk/angbracketrightrotates about the ˆϑdirection by angle ϑits\nspin-orbit energy is given by\nǫ(ϑ) =/angbracketleftnk|eiσ·/vectorϑHsoe−iσ·/vectorϑ|nk/angbracketright (10)\nwhere/vectorϑ=ϑˆϑ. Taking the derivative of this energy with\nrespect toϑin the limit that ϑgoes to zero shows that\nthe energy derivatives are\n∂ǫ\n∂ϑ=i/angbracketleftnk|[σ·ˆϑ,Hso]|nk/angbracketright. (11)\nFigure (1) shows that the derivative ∂ǫ/∂ϑis identical to\n∂ǫ/∂ˆMandthatwhen ˆM= ˆztherotationdirection ˆϑlies\nin thex−yplane. The two components of the transverse\ntorque operatorΓxand Γycan be obtained (up to factors\nofi) by setting ˆϑequal to ˆxor ˆy, respectively. From this\nobservation we find\n|/angbracketleftnk|Γ−|nk/angbracketright|2=/parenleftbigg∂ǫ\n∂x/parenrightbigg2\n+/parenleftbigg∂ǫ\n∂y/parenrightbigg2\n.(12)\nWhen the magnetization direction ˆ zis pointed along a\nhigh symmetry direction the transverse directions ˆ xand\nˆyare equivalent and |Γ−|2= 2(∂ǫ/∂ˆM)2.\nSubstituting the torque matrix elements for the energy\nderivatives in Eq.(9) gives a damping rate of\nλbfs=τγ2\n2µ0/summationdisplay\nnk/vextendsingle/vextendsingleΓ−\nn(k)/vextendsingle/vextendsingle2η(ǫnk). (13)\nFor the intraband terms in Eq. (1) the integration over\nthe spectralfunctions reduces to τη(ǫnk)/2π/planckover2pi1so we find\nλbfs=π/planckover2pi1γ2\nµ0/summationdisplay\nn/integraldisplayd3k\n(2π)3/vextendsingle/vextendsingleΓ−\nn(k)/vextendsingle/vextendsingle2\n×/integraldisplay\ndǫ1Ank(ǫ1)Ank(ǫ1)η(ǫ1),(14)\nwhich matches the intraband terms of Eq. (1).4\nB. Interband terms\nAs the magnetization precesses, the spins rotate and\nthe spin-orbit energy changes. This variation acts as a\ntime dependent perturbation\nV(t) =eiσ·ϕ(t)Hsoe−iσ·ϕ(t)−Hso(0)≈i[σ·ϕ(t),Hso].\n(15)\nThis approximation results from linearizing the expo-\nnents, which is appropriate in the small angle limit.\nThe time dependence of the spin direction is ˆ ϕ(t) =\ncosωtˆx+sinωtˆy, up to a phase factor. This perturba-\ntion causes band transitions between the states ψnkand\nψmk. The initial and final states have the same wavevec-\ntor because these transitions are caused by the uniform\nprecession, which has a wavevector of zero. The transi-\ntion rate between states due to this perturbation is\nWmn(k) =2π\n/planckover2pi1/vextendsingle/vextendsingleΓ−\nmn(k)/vextendsingle/vextendsingle2δ(ǫmk−ǫnk−/planckover2pi1ω).(16)\nThe variations of the occupancies of the states with\nrespect to the magnetization direction are given by the\nmaster equation\n∂ρnk\n∂t=/summationdisplay\nm/negationslash=nWmn(k)[ρmk−ρnk].(17)\nThe second term in the effective field Eq. (2) contains\nthe factor∂ρnk/∂ˆMwhich is (∂ρnk/∂t)/(∂ϕ/∂t) where\n∂ϕ/∂t=ω. Inserting these expressions into the second\nterminthe effectivefield andrearrangingthe sumsgives\nHeff=−1\n2µ0M/summationdisplay\nnk/summationdisplay\nm/negationslash=nWmn(k)\nω2[ρnk−ρmk][ǫmk−ǫnk]dˆM\ndt.\n(18)\nComparing this result to the effective field predicted by\nthe Landau-Lifshitz-Gilbertequation(8) wefind adamp-\ning rate of\nλ=γ2\n2µ0/summationdisplay\nnk/summationdisplay\nm/negationslash=nWmn(k)[ρnk−ρmk]\nω[ǫmk−ǫnk]\nω.(19)\nThe finite lifetime of the states is introduced with the\nspectral functions\nλ=/planckover2pi12γ2\n2µ0/summationdisplay\nnk/summationdisplay\nm/negationslash=n/integraldisplay\ndǫ1Ank(ǫ1)/integraldisplay\ndǫ2Amk(ǫ2)\n×Wmn(k)[f(ǫ1)−f(ǫ2)]\n/planckover2pi1ω[ǫ2−ǫ1]\n/planckover2pi1ω. (20)\nInserting the transition rate Eq. (16), integrating over ǫ2,\nand taking the limit that ωgoes to zero leaves\nλ=π/planckover2pi1γ2\nµ0/summationdisplay\nn/summationdisplay\nm/negationslash=n/integraldisplayd3k\n(2π)3/vextendsingle/vextendsingleΓ−\nmn(k)/vextendsingle/vextendsingle2\n×/integraldisplay\ndǫ1Ank(ǫ1)Amk(ǫ1)η(ǫ1), (21)which are the interband terms of Eq. (1).\nIn this derivation of the bubbling Fermi surface contri-\nbution to the damping we have ignored an additional, re-\nversible term that contributes to the magnetocrystalline\nanisotropy. This contribution arises from changes in the\nequilibrium state occupancies as the magnetization di-\nrection changes. This contribution to the magnetocrys-\ntalline anisotropy is localized to the Fermi surface while\nthe contribution dicussed in IIIA is spread over all of the\noccupied levels.\nIV. TUNING THE DAMPING RATE\nWe have previously demonstrated that the mechansim\nof thetorque correlation model Eq. (1) accounts for the\nmajority of the precession damping rates of the transi-\ntion metals iron, cobalt, and nickel [4]. In the present\nwork we have so far shown that this expression for the\ndamping rate can be described simply within an effective\nfield picture. We now investigate the degree to which\nthe damping rate may be modified by adjusting certain\nmaterial parameters. Inspection of Eq. (1) reveals that\nthe damping rate depends on the convolution of two fac-\ntors: thetorquematrixelementsandtheintegraloverthe\nspectral functions. We separate the quantitative analy-\nsis of the damping rates into their dependencies on these\ntwo factors, beginning with the spectral weight.\nThe calculations for the damping rate of Eq. (1) dis-\ncussed below are performed using the linear augmented\nplanewave method in the local spin density approxima-\ntion. The details of the computational technique may be\nfound in [4], [21], and the included references.\nA. Spectral overlap\nFor the intraband terms, the integral over the spec-\ntral functions is essentially proportional to the density of\nstates at the Fermi level. Therefore, it appears reason-\nable to suspect that the intraband contribution to the\ndamping rate of a given material should be roughly pro-\nportional to the density of states of that material at the\nFermi level. To test this claim numerically, we artificially\nvaried the Fermi level of the metals within the d-bands\nand calculated the intraband damping rate as a function\nof the Fermi level. The results of these calculations are\nsuperimposed on the calculated densities of states of the\nmaterials in Fig. 2. The correlation between the damp-\ning rates and the densities of states, while not exact, is\ncertainly strong, indicating that increasing the density\nof states of a system at the Fermi level will generally\nincrease the intraband contribution to damping.\nThe dependence of the interband terms on the spectral\noverlap is more complicated than that of the intraband\nterms. The spectral overlap depends on the energy dif-\nferencesǫm−ǫn, which can vary significantly between\nbands and over k-points. When the scattering rate /planckover2pi1/τ5\nFIG. 2: Intraband damping rate versus Fermi level superim-\nposed upon density of states. A strong correlation between\nthe intraband damping rate versus Fermi level ( •) and the\ndensity of states (solid curves) is observed. Vertical blac k\nlines indicates true Fermi energy calculated by density fun c-\ntional theory.\nis much less than these energy gaps the interband terms\nare proportional to the scattering rate. However, this\nproportionality only holds at low scattering rates when\nthe interband contribution is much less than the intra-\nband contribution. The proportionality breaks down at\nhigher scattering rates when /planckover2pi1/τbecomes comparable to\nthe band gaps. After this point the damping rate gradu-\nally plateaus with respect to the scattering rate. Unfor-\ntunately, this complicated functional dependence of the\nspectral overlap on the scattering rate makes it difficult\nto obtain a simple description of the effect of the spec-\ntral overlap on the interband damping rate in terms of\nmaterial parameters.B. Torque matrix elements\nThe damping rate also depends on the square of the\ntorquematrixelements. Agoalofdopingisoftentomod-\nify the effective spin-orbit coupling of a sample. While\ndoping does more than this, such as intoducing strong lo-\ncal scattering centers, it is nevertheless useful to estimate\nthe dependence of the matrix elements on the spin-orbit\nparameter ξ. We begin with pure spin states ψ0\nnand\ntreat the spin-orbit interaction V=ξV′as a perturba-\ntion. The states can be expanded in powers of ξas\nψn=ψ0\nn+ξψ1\nn+ξ2ψ2\nn+... . (22)\nThe superscripts refer to the unperturbed wavefunction\n(0) and the additions ( i) due to the perturbation to the\nith order while the subscript nis the band index, which\nincludes the spin direction, up or down. Since the torque\noperator also contains a factor of the spin-orbit parame-\nter the matrix elements haveterms in everyorderof ξbe-\nginning with the first order. Therefore, the squared ma-\ntrix elements have contributions of order ξ2and higher.\nTo determine the importance of these terms we arti-\nficially tune the spin-orbit interaction from zero to full\nstrength, calculating the damping rate over this range.\nWe then fit the intraband and interband damping rates\nseparately to polynomials. In each material, this fitting\nshowed that for the intraband terms the ξdependence of\nthe damping rate was primarily third order, with smaller\ncontributions from the second and fourth order terms.\nRestricting the fit to only the third order term produced\na very reasonable result, shown in Fig. (3). For the in-\nterband terms, polynomial fitting was dominated by the\nsecond order term, with all other powers contributing\nonly negligibly. The second order fit is shown in Fig. (3).\nTo understand the difference in the ξdependence of\nthe intraband and interband contributions it is useful to\ndefine the torque operator\nΓ−=ξ(ℓ−σz−ℓzσ−). (23)\nThe torque operator lowers the angular momentum of\nthe state it acts on. This can be accomplished either by\nlowering the spin momentum ℓzσ−, a spin flip, or low-\nering the orbital momentum ℓ−σz, an orbital excitation.\nTherefore, both the intraband and interband contribu-\ntionseachhavetwosub-mechanism: spinflipsandorbital\nexcitations.\nThe second order terms for the intraband case are\nξ2|/angbracketleftψ0\nn|(ℓzσ−−ℓ−σz)|ψ0\nn/angbracketright|2. Sincetheunperturbedstates\nψ0\nnare pure spin states the spin flip part ℓzσ−of the\ntorque returns zero. Therefore, only the orbital excita-\ntions exist to lowest order in ξ, reducing the strength\nof the second order term in the intraband case. How-\never, the interband terms contain matrix elements be-\ntween several states, some with the same spin direction,\nbut others with opposite spin direction. Therefore, both\nspin flips and orbital excitations contribute in second or-\nder to the interband contribution.6\nFIG. 3: ξdependence of intraband and interband damping\nrates. Damping rates were calculated for a range of spin-orb it\ninteraction strengths between off ( ξ= 0) and full strength\n(ξ= 1).ξ2fits were made to the interband damping rates\n(left axes and ◭symbols) and ξ3fits to the intraband rates\n(right axes and ◮symbols).\nV. CONCLUSIONS\nThebreathingFermisurfacemodelhasprovidedasim-\nple and understandable effective field explanation of pre-\ncession damping in metallic ferromagnets. However, it is\nonlyapplicabletoverypuresystemsatlowtemperatures.\nOn the other hand, the torque correlation model accu-\nrately predicts damping rates of systems with imperfec-\ntions from low temperatures to above room temperature.\nThe shortcoming of the torque correlation model is that\nit does not illuminate the phyiscal mechanisms responsi-\nble for damping. We have pointed out that the breath-\ning Fermi surface model accounts for only one of the two\nterms in the effective field. By constructing an effective\nfield with the previously studied breathing Fermi surface\ncontribution and also the new bubbling effect we haveshown that this simpler picture may be mapped onto the\ntorque correlation model such that the breathing terms\nmatchtheintrabandcontributionandthebubblingterms\nmatch the interband contribution.\nSince there is considerable interest in understanding\nhow to manipulate the damping rates of materials we\ninvestigated the dependence of the intraband and inter-\nband damping rates on both the spectral overlap inte-\ngral and the torque matrix elements. For the intraband\nterms, the spectral overlap is proportional to the density\nof states and we found a strong correlation between the\nintraband damping rate and the density of states of the\nmaterial. The interband case is significantly complicated\nby the range of band gaps present in materials. No sim-\nple relation was found between the strength or scattering\nratedependence ofthe interband terms and common ma-\nterial parameters. The importance of the torque matrix\nelements to the damping rates was characterizedthrough\ntheir dependence on the spin-orbitparameter. The intra-\nband damping rates were found to vary as the spin-orbit\nparametercubed while the interbanddamping rateswent\nas the spin-orbit parameter squared. This difference was\nexplained by noting that the torque operatorchanges the\nangular momentum of states either through spin flips, or\nby changing their orbital angular momentum. Spin-flip\nexcitations do not occur to second order in ξfor the in-\ntraband terms, but do contribute at second order for the\ninterband terms.\nIt is desirable to understand the relative differences in\ndamping rates amoung various materials, such as why\nthe damping rate for nickel is higher than that for cobalt\nand iron. We have shown that the relative damping rates\nof these materials depend in part on the differences of\ntheirdensitiesofstatesandspin-orbitcouplingstrengths.\nHowever, they also depend in an intricate way on the en-\nergy gap spectra of each metal. For the interband terms\nthe dependence on the gap spectrum enters through the\nspectral overlap integral. For the intraband terms the\nenergy gaps appear in the denominators of the matrix\nelements. Therefore, states with very small splittings\ncan dominate the k-space convolution. The abundance\nofsuchstates in nickel appearsto contribute to the larger\ndamping rate in this material [23].\nDoping is a common technique for modifying damp-\ning rates. Doping has a number of consequences on a\nsample and these effects vary with the method of dop-\ning. Dopants can increase the electron-lattice scattering\nrate, introduce magnetic inhomogeneities that act as lo-\ncal scattering centers, alter the density of states, and\nchange the effective spin-orbit parameter. We have in-\nvestigated the consequences of modifying the densities of\nstatesandspin-orbitparameteronthe dampingrate, and\npreviously demonstrated the scattering rate dependence\nof the damping rate; however, it is not clear what new\ndamping mechanisms arise when rare-earth elements are\nadded to a transition metal host.\nThis work was supported in part by the Office of Naval\nResearch through grant N00014-03-1-0692 and through7\ngrant N00014-06-1-1016.\n[1] W. Bailey, P. Kabos, F. Mancoff, and S. Russek, IEEE\nTrans. Mag. 37, 1749 (2001).\n[2] J. Rantschler, R. McMichael, A. Castiello, A. Shapiro,\nW.F. Egelhoff, Jr., B. Maranville, D. Pulugurtha,\nA. Chen, and L. Conners, J. Appl. Phys. 101, 033911\n(2007).\n[3] V. Kambersk´ y, Czech. J. Phys. B 26, 1366 (1976).\n[4] K. Gilmore, Y. Idzerda, and M. Stiles, Phys. Rev. Lett.\n99, 027204 (2007).\n[5] S. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974).\n[6] B. Heinrich and Z. Frait, Phys. Stat. Sol. 16, K11 (1966).\n[7] B. Heinrich, D. Meredith, and J. Cochran, J. Appl. Phys.\n50, 7726 (1979).\n[8] J.F.Cochran and B. Heinrich, IEEE Trans. Magn. 16,\n660 (1980).\n[9] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y,\nPhys. Stat. Sol. 23, 501 (1967).\n[10] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).\n[11] V. Korenman and R. Prange, Phys. Rev. B 6, 2769\n(1972).\n[12] V. Kambersk´ y and C. Patton, Phys. Rev. B 11, 2668\n(1975).\n[13] J. Kuneˇ s and V. Kambersk´ y, Phys. Rev. B 65, 212411(2002).\n[14] J. Ho, F. Khanna, and B. Choi, Phys. Rev. Lett. 92,\n097601 (2004).\n[15] Y. Tserkovnyak, G. Fiete, and B. Halperin,\nAppl. Phys. Lett. 84, 5234 (2004).\n[16] E. Rossi, O. G. Heinonen, and A. H. MacDonald,\nPhys. Rev. B 72, 174412 (2005).\n[17] B. Heinrich, Ultrathin magnetic structures III (Springer,\nBerlin, 2005).\n[18] M. F¨ ahnle and D. Steiauf, Phys. Rev. B 73, 184427\n(2006).\n[19] H.J. Skadsem, Y. Tserkovnyak, A. Brataas, and\nG.E.W. Bauer, Phys. Rev. B 75, 094416 (2007).\n[20] J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J. Furdyna,\nW. Atkinson, and A. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n[21] M. Stiles, S. Halilov, R. Hyman, and A. Zangwill,\nPhys. Rev. B 64, 104430 (2001).\n[22] D. Steiauf and M. F¨ ahnle, Phys. Rev. B 72, 064450\n(2005).\n[23] V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007)." }, { "title": "1811.04821v1.Choking_non_local_magnetic_damping_in_exchange_biased_ferromagnets.pdf", "content": "1 \n Choking non -local magnetic damping in exchange biased ferromagnets \n \nTakahiro Moriy amaa), Kent Oda, and Teruo Ono \nInstitute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto, 611-0011, Japan. \na ) corresponding to :mtaka@scl.kyoto -u.ac.jp \n \n \nAbstract \nWe investigated the temperature dependence of the magnetic damping in the \nexchange biased Pt/ Fe 50Mn 50 /Fe 20Ni80 /SiO x multilayer s. In sample s having a stro ng \nexchange bias, we observed a drastic decrease of the magnetic damping of the FeNi with \nincreasing temperature up to the blocking temperature . The results essentially indicate \nthat the non -local enhancement of the magnetic damping can be choked by the adjacent \nantiferromagne t and its temperature dependent exchange bias. We also point ed out that \nsuch a strong temperature dependent damping may be very beneficial for spintronic \napplications. \n 2 \n The Gilbert damping constant , α, appearing in the Landau -Lifshitz -Gilbert equation is \none of the important parameters characteri zing the magnetization dynamics [1 .. It \ninfluences the switching speed of the magnetization [2 , 3 . and also determines the \nthreshold current for various spin-torque -related phenomena [4, 5, 6, 7, 8.. With those \nnumerous examples, i t is probably not an exagger ation to say that the performance of \nevery spintronic device relies on α. It is, howev er, unfortunate that the magnetic damping \nis one of the least con trollable magnetic parameters among those including magnetic \nanisotropy, saturation magnetization, and gyromagnetic ratio which a material \nengineering as well as some exogenous controls (e.g. temperature, electric field bias , \nstructural strain , etc. ) [9, 10, 11, 12, 13, 14, 15. can easily tune over a wide range . \nAlthough some efforts have been made to minimize α by carefully engineering the band \nstructure [ 16., an exogenous control of it, which is very much desirable for spintronic \napplications, has rarely been explored. \n The magnetic damping is rooted in various relaxation processes of the spin \nangular momentum [17 .. The majority of such process es locally occur within the \nmagnetic material via, e.g., spin -phonon and spin -spin relaxations [18.. On the other hand, \nan interaction between the magnetization dynamics and the itinerant electron spin , i.e. the \nspin pumping effect [ 19., results in a non -local magnetic damp ing of the spin angular \nmomentum. In other words, when the spin angular momentum carried by the itinerant \nelectron diffuses into an adjacent non -magnetic material, an additional damping \nenhancement can occur depending on the spin relaxation properties of the non -magnetic \nmaterial . For instance, a strong damping enhancement due to the spin pumping effect has \nbeen observed Pt/ ferrom agnet (FM) multilayers in which the Pt works as a good spin \ndissipative material [ 20 .. This non -local dampin g enhancement has been recently 3 \n revisited for the system of the exchange biased FM/antiferromagnet (AFM) multialyers \n[21.. In Ref. 21, it was shown that the non-local dam ping was modified by the N éel order \nin the AFM . In particular, the linear relation between the strength of the exchange bias \n(EB) and the en hanced damping strongly indicates that an additional spin relaxation takes \neffect due to th e orientation of the N éel order . Furthermore, t he relatively gradual AFM \nthickness dependence of the damping enhancement , suggesting a spin current \ntransmission through the AFM [22 , 23 , 24 , 25 , 26 , 27 , 28 ., was also certainly \nintriguing . \nFrom another point of view, t hese re sults indeed offer a control of the magnetic \ndamping by the magnetic order in the adjacent AFM. It is therefore worthwhile to further \ninvestigate and understand the magnetic damping in exchange biased FM/AFM systems \nand to gain a control of it for the sake of advancement in spintronic application s. In this \nwork, we extend ed our investigations to the temperature dependence of the non -local \ndamping in the exchange biased AFM/FM multilayer . We particularly paid attention to \nthe damping variation s toward the blocking temperature TB of the EB. It is found that the \nnon-local magnetic damping drastically decreases with increasing temperature when EB \nis strong . Moreover, the linear relation between the strength of the EB and the enhanced \ndamping, previously found in terms of the AFM thickness dependence of EB [ 21., is \nfound to also hold in terms of the temperature dependence. \nWe investigated Pt 5 nm/ Fe 50Mn 50 tFeMn nm/Fe 20Ni80 4 nm/SiO x 2nm ( tFeMn = 0, \n3, 20, and 60 nm) multilayers grown on a thermally oxidized Si substrate as illustrated in \nFig. 1 (a) . The samples were photolithographically patterned into a 10 μm wide strip \nattached to a coplanar waveguide made of Ti/Au layer facilitating a high frequency \nmeasurement . EB was set by a field cooling process with a n annealing temperature of TFC 4 \n = 200 Cº and a field of 2 kOe. No irreversible degradations of the film, e.g. due to atom \ndiffusions, were found in this field cooling process [21.. High sensitivity f erromagnetic \nresonance (FMR) measurements were carried out by a homodyne detection technique [29, \n30 , 31 . with an identical circuitry and methodology used in Ref. 21. The homodyne \nvoltage Vdc were measured with a fixed frequency f of the r.f. current and with a swept \nexternal magnetic field applied either along or against t he direction of the exchange bias \nfield Heb which are denoted by “along EB ” and “agains t EB”, respectively . The FMR \nmeas urements were at first performed at 298K and the measurement temperature was \nstepped up to 393 K. \nFigure 1 (b) sh ows representative FMR spectra obtained at 298 K and 393 K for \ntFeMn = 20 nm with f = 8 GHz and the external field along and against EB . The \ndisplacement between the two spectra at 298 K with the fields along EB and against EB \nmanifests a unidirectional magnetic anisotropy, i.e. the exchange bias field. One can also \nsee that the exchange bias is diminished at 393 K. FMR spectra are fitted by the \ncombination of symmetric and anti -symmetric Lorentzians by which the resonant \nparame ters, such as the resonant field Hres and the spectral linewidth ΔH, are extracted \n[29.. \nFigure 2 shows f vs. Hres and ΔH vs. f at 298 K and 393 K for tFeMn = 0, 3, 20, \nand 60 nm. In order to extract the effective demagnetizing field 4𝜋𝑀eff and the exchange \nbias field Heb, the f vs. Hres curves are fitted by the Kittel ’s equation 𝑓=\n(𝛾2𝜋⁄ )√(𝐻res+𝐻𝑢)(𝐻res+4𝜋𝑀eff), where 𝛾 is the gyromagnetic ratio and Hu is the \nanisotropy field from which both uniaxial anisotropy field and the exchange bias field Heb \nare derived . The Gilbert damping 𝛼 is extracted from the ΔH vs. f plot by using ∆𝐻=\n∆𝐻0+2𝜋𝛼𝑓 𝛾⁄ [32., where ∆𝐻0 is a frequency independent -linewidth known as the 5 \n inhomogeneous broadening [ 33.. We note that our linewidth analyses explicitly separate \nthe frequency dependent damping , or the Gilbert damping 𝛼, from the independent on e, \nor inhomogeneous broadening which may be related to the two -magnon scattering at the \nFM/AFM interface [ 34,35.. As shown in Fig. 2, a ll the samples exhibit a good Kittel ’s \nfitting as well as a linear fitting for the α extraction . We note that, for tFeMn = 3 nm at 298 \nK, there was an exceptionally large uniaxial magnetic anisotropy field of 390 Oe, which \nmay lead to significantly large ∆𝐻0 compared to samples with other tFeMn which \nessentially read ∆𝐻0 = 0. All the discussion s about the magnetic damping thereafter will \nbe referring to α which reflects the intrinsic property of the system [33.. \nFigure s 3 (a), (b), and (c) summarize 4𝜋𝑀eff , Heb, and α, respectively, as a \nfunction of temperature. For all the samples, 4𝜋𝑀eff is ranging from 0.88 to 0.96 Tesla \nwhich is considered reasonable for the Fe 20Ni80 with a possible interfacial perpendicular \nanisotropy [ 36.. 4𝜋𝑀eff overall decreases by ~10 % with increasing temperature from \n298 to 393 K, which can be attributed to the thermal decay of the magnetization as well \nas a change in the interfacial anisotropy . The exchange bias field i s found to be largest \nHeb = 79 Oe with tFeMn = 20 nm and it monotonically decreases with increasing \ntemperature. The blocking temperature for the exchange bias is found to be TB = 393 K \nand 373 K for tFeMn = 20 and 60 nm, respectively. No exchange bias field was observed \nfor tFeMn = 0 and 3 nm in the measurement temperature range. It is remarkable that , for \ndifferent tFeMn , α behaves quite differently with respect to temperature . Namely, the \nsamples with tFeMn = 0 and 60 nm show a slight upward trend of α with increasing \ntemperature but one with tFeMn = 3 shows almost constant with respect to temperature . It \nis noticeable that α for tFeMn = 20 nm undergoes a dra stic decrease, almost by a half , with \nincreasing temperature up to TB. We also emphasize that α measured with the field along 6 \n EB was found to be always smaller than that measured with the field against EB. \nFigure 3 (d) plot s Δα as a function of Heb wher e Δα is the difference of α \nmeasured with the field along EB and against EB. It is found that Δα increases lineally \nwith respect to Heb. Although Heb varies due to temperature in this present case, the \nobservation of the linear relation between Heb and Δα is essentially same as what was \nobserved previously with respect to Heb which varies with respect to tFeMn [21.. The inset \nof Fig. 3 (d) shows α as a function of the field angle φ with respect to the exchange bias \ndirection ( φ = 0 corresponds to the direction along EB.) measured for tFeMn = 20 nm at \n298 and 343 K , which depicts the φ dependence of α similar to that observed in the \nprevious report [ 21.. \nNow, o ne find s two peculiar point s in our results , especially when focus ing on \ntFeMn = 20 nm . One is the strong temperature dependence of α with the exchange bias . The \nother is the temperature dependence of Δα which results in the linear relation between Δα \nand Heb. While the latter is consistent with our previous report and originates from the \nNéel order twisting [ 21., the former is quite intriguing in both physics and application \npoints of views. Looking at Figs. 3 (c) and (d), one can presume that the reduction of α \nmay be correlated with the strength of the exchange bias. \nIt has been shown that the intrinsic damping of a single layer of FeNi only \nslightly varies in this temperature range [37.. A non -local damping enhancement by an \nadjacent Pt due to the spin pumping effect should not significantly vary since the relevant \nparameters such as the mixing conductance and the spin diffusion length in Pt do not vary \nmuch in this temperature range. Very little temperature variation of α in our control \nsample (tFeMn = 0 nm) is therefore consistent with those previous observation s as well as \nthe expectations . It is now very clear that the observed temperature dependence of α with 7 \n tFeMn =20 nm is solely associated with the FeMn insertion and the exchange bias between \nthe FeMn and FeNi [38. (Also, see Supplementary Information) . \nThe mechanism of the exchange bias is generally explained by two major factors \n[39.; i.e. the exchange coupling at the FM/AFM interface and the magnetic anisotropy \nenergy of t he AFM . Because a thermal agitation effectively reduces the magnetic \nanisotropy, the exchange bias field diminishes toward TB at which the effective magnetic \nanisotropy essentially becomes zero. Considering the spin transmission in \nantiferromagnets by the form of the N éel order dynamics, there is a negative correlation \nbetween the magnetic anisotropy and the spin diffusion length (or the so-called healing \nlength for the spin transmission) λ𝐹𝑒𝑀𝑛 [24,40]. Ther efore, our observations shown in \nFigs. 3(b) and (c) can be considered representing a negative correlation between α and \nλ𝐹𝑒𝑀𝑛 . \nAccording to Ref. 40, α as a function of λ𝐹𝑒𝑀𝑛 behaves differently depending \non either the strong or weak damping limit of the antiferromagnet . Our experimental \nresults are more consistent with the strong damping limit in which α increases with \ndecreasing λ𝐹𝑒𝑀𝑛 . We should note here that the present analysis neglects a direct thermal \neffect on the N éel order dynamics , such as thermal magnons, which could be more \nimportant than considering the effective magnetic anisotropy but is difficult to be taken \ninto account analytically at this moment. \nFinally , we highlight our results in the engineering point of view. As we pointed \nout above, t he magnetic damping of FeNi should generally exhibit only a slight \ntemperature dependence , at around room temperature, regardless of the local or non -local \nmechanisms. Our results suggest that, by making use of the exchange biased bilayer, one \ncan drastically “choke ” the non-local enhancement of the magnetic damping in a 8 \n relatively narrow temperature range . For those spintronic applications i nvolving a Joule \nheating up on the operation ( e.g. the spin torque magnetic memory [6. requires quite a bit \nof current density to write information bits which heats up the ferromagnetic bits.), this \ntemperature control of α may be beneficial since it is reduced only when they are in \noperation. \nIn summary, we investigated the temperature dependence of the magnetic \ndamping in the exchange biased Pt/ Fe 50Mn 50 /Fe 20Ni80 /SiO x multilayer in the \ntemperature range of 298 ~ 393 K. We reconfirmed the linear relation between Heb and \nΔα, which have been seen in our previous report [21., by the present experimental \napproach which is different f rom that in the previous report . In sample s having a strong \nexchange bias, we observed the drastic decrease of the magnetic dampin g with increasing \ntemperature up to the blocking temperature. The results lead us to the negative correlation \nbetween the magnetic anisotropy of the FeMn and the damping enhancement, implying \nthe mechanism of spin current transmission through the FeMn with a strong damping \nlimit [ 40.. Furthermore, w e pointed out that this strong temperature dependent damping \nmay be very beneficial for spintronic applications. \n \n \nAcknowledgements \nThis work was supported in part by JSPS KAKENHI Grant Numbers 26870300, \n17H04924, 15H05702 , and 17H05181 (“Nano Spin Conversion Science ”). We also \nacknowledge the support from Center for Spintronics Research Network (CSRN) . \n 9 \n Figure captions: \n \nFig. 1 (a) Schematic illustration of the layer structure under investigat ion by the \nhomodyne FMR measurement . (b) Representative FMR spectra (the homodyne voltage \nVdc as a function of the applied field either along (red markers ) or against EB (blue \nmarkers )) for tFeMn = 20 nm at f = 8 GHz measured at 298 K (upp er panel) and 393 K \n(lower panel). The continuous lines are the fitting by the combination of symmetric and \nanti-symmetric Lorentzians. \n \nFig. 2 Resonant field Hres and the spectral linewidth ΔH as a function of frequency f \nmeasured at 298 K and 393 K for (a, e) tFeMn = 0 nm, (b, f) tFeMn = 3 nm, (c, g) tFeMn = 20 \nnm, and (d, h) tFeMn = 60 nm. The data with the applied field along and against EB are \nplotted. The continuous lines are the fitting by the Kittel ’s equation and ∆𝐻=∆𝐻0+\n2𝜋𝛼𝑓 𝛾⁄ described in the main text. \n \nFig. 3 (a) 4𝜋𝑀𝑒𝑓𝑓 as a function of temperature , (b) Heb as a function of temperature, (c) \nα as a function of temperature, and (d) Δα as a function of Heb for tFeMn = 0 nm (dark blue), \n3 nm, (light blue), 20 nm (green), and 60 nm (red) . 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Demidov1 \n1Institute for Applied Physics and Center for Nonlinear Science, University of Muenster, \n48149 Muenster, Germany \n2Department of Physics, Emory University, Atlanta, GA 30322, USA \n \nLarge-amplitude magnetization dy namics is substantially more co mplex compared \nto the low-amplitude linear regim e, due to the inevitable emerg ence of \nnonlinearities1-3. One of the fundamental nonlinear phenomena is the nonlinear \ndamping enhancement, which impos es strict limitations on the op eration and \nefficiency of magnetic nanodevices2,4,5. In particular, nonlinear damping prevents \nexcitation of coherent magnetiz ation auto-oscillations driven b y the injection of \nspin current into spatially extended magnetic regions6,7. Here, we propose and \nexperimentally demonstrate that nonlinear damping can be contro lled by the \nellipticity of magnetization precession. By balancing different contributions to \nanisotropy, we minimize the ellip ticity and achieve coherent ma gnetization \noscillations driven by spatially extended spin current injectio n into a microscopic \nmagnetic disk. Our results pr ovide a novel route for the implem entation of \nefficient active spintronic and magnonic devices driven by spin current. 2A large spin-Hall effect (SHE)8,9 exhibited by certain materials with strong spin-\norbit interaction res ults in the generation of significant pure spin currents, enabling the \nimplementation of a variety of effi cient active magnetic nanode vices7,10,11. Pure spin \ncurrents produced by SHE enable generation of incoherent magnon s12,13 and coherent \nmagnetic auto-oscillations6,7,11,14-16, as well as excitation of propagating spin waves17,18 \nin conducting and insulating magne tic materials. These applicat ions take advantage of \nthe compensation of the natural magnetic damping by the antidam ping effect of spin \ncurrent19-21. \nThe antidamping torque is proporti onal to spin current, at smal l currents resulting \nin a linear decrease of the net effective damping. In addition to the effect on damping, \nspin current enhances the fluct uation amplitudes of all the spi n-wave modes in the \nsystem12. The strongest enhancement is ge nerally observed for the lowes t-frequency \nmode that exhibits the s mallest relaxation rate7. This dominant mode is expected to \ntransition to the auto-oscillati on regime at its damping compen sation point. However, \nthe increase of the amplitudes of the dynamical modes leads to their nonlinear coupling. \nThe resulting onset of strong nonlin ear relaxation of the energ y and the angular \nmomentum of the dominant mode i nto other spin-wave modes can be described as \namplitude-dependent nonlinear damping2,4,5. Starting with the fir st experiments on the \ninteraction of pure spin current s with the magnetization, this mechanism has become \nrecognized as the main culprit ge nerally preventing complete da mping compensation \nand excitation of coherent magneti zation auto-oscillations in s patially extended \nmagnetic systems12,22. \nThe adverse effects of nonlinear damping can be reduced either by suppressing \nthe amplitudes of parasitic sp in-wave modes, or by directly con trolling the mechanisms 3responsible for the nonlinear mode coupling. The former approac h was recently \nimplemented by using local injec tion of spin current into a nan oscale region of an \nextended magnetic system6,7. In this geometry, parasitic incoherent spin waves are \nradiated from the localized act ive area, resulting in reduced n onlinear damping of the \ndominant mode. However, in this a pproach, most of the angular m omentum delivered \nby the spin current is lost to the parasitic spin waves radiate d from the active region, \nwhich requires large currents for device operation. Additionall y, this approach requires \nthat the active region is limited to nanoscale, limiting the ac hievable dynamical \ncoherence and the possibilities for the magnonic device integra tion. Intense efforts have \nbeen dedicated to improving the dynamical coherence by utilizin g arrays of mutually \nsynchronized nanodevices, which i ncreases the effective size of the active region11,23, \nbut achieving consistency and s trong coupling of nanodevices ha s proven challenging. \nHere, we demonstrate a novel a pproach based on the direct contr ol of the mode \ncoupling mechanisms, which all ows minimization of the nonlinear damping, without \nconstraining the geometry or the efficiency of spin current-dri ven nanostructures. We \nshow experimentally and by microma gnetic simulations that the n onlinear spin wave \ncoupling is determined by the el lipticity of mag netization prec ession, which is \ncontrolled by the magnetic anisotr opy. We achieve almost circul ar precession by \ntailoring the perpendicular ma gnetic anisotropy (PMA) of the ma gnetic film to \ncompensate the dipolar anisotropy, resulting in suppression of nonlinear damping, and \nenabling coherent magnetizati on dynamics driven by the injectio n of spin current \ngenerated by the spin Hall eff ect into an extended magnetic reg ion. The demonstrated \neffects are confirmed by the mic romagnetic simulations, which p rovide additional 4information about the mechanisms of energy flow associated with the nonlinear \ndamping. \nOur test devices are based on the 8 nm-thick and 1.3 m-wide Pt strip, and a 5 \nnm-thick CoNi bilayer disk with the diameter of 0.5 m on top, Fig. 1a. The relative \nthicknesses of Co and Ni were adj usted so that the PMA of the d isk, determined mostly \nby the Pt/Co and Co/Ni interface anisotropies, nearly compensat es the dipolar \nanisotropy of the magnetic film. T he saturation magnetization w as 4MCoNi=6.9 kG and \nthe PMA anisotropy field was Ha=6.6 kG, as determined by separate magnetic \ncharacterization. To verify that th e demonstrated effects origi nate from PMA, we have \nalso studied a control sample utilizing a 5 nm-thick Permalloy (Py) disk with negligible \nPMA, instead of the CoNi. \nBecause of SHE in Pt, current I produces an out-of-plane spin current IS, which is \ninjected into the ferromagnet (i nset in Fig. 1a), exerting anti damping spin torque on its \nmagnetization M 19. To maximize the antidamping e ffect of spin current, the \nmagnetization in the studied devi ces was saturated by the in-pl ane static magnetic field \nH0 applied perpendicular to the d irection of the current flow24. \nAt sufficiently large I, spin current-induced torque may be expected to completely \ncompensate the natural damping, resulting in the excitation of coherent magnetization \nauto-oscillations. However, experi ments have shown that this do es not occur if spin \ncurrent is injected into an ex tended magnetic region. Instead, the effects of spin current \nsaturate due to the onset of non linear damping, and compensatio n is never achieved12. \nThe effects of magnetic aniso tropy on the magnetization precess ion characteristics \nare illustrated in Figs. 1b and 1c. The precessing magnetizatio n vector M induces 5dynamic demagnetizing field hd antiparallel to the out-of-plane component Mx of \nmagnetization, resulting in an e lliptical precession trajectory with the short axis normal \nto the film plane (Fig. 1b). The elliptical precession is accom panied by the oscillation of \nthe component of magnetization p arallel to the precession axis mz2, at twice the \nfrequency of precession. This oscillation acts as a parametric pump25,26 that drives \nenergy transfer from the dominant excited spin-wave mode into o ther modes, resulting \nin nonlinear damping of the former. \nIn contrast to hd, the effective dynamical field hPMA associated with PMA is \nparallel to Mx. If the magnitude of hPMA is close to that of hd, the two fields compensate \neach other, and precession becomes circular (Fig. 1c). As follo ws from the arguments \ngiven above, nonlinear damping is expected to become suppressed in films with PMA \ncompensating dipolar anisotropy. \nWe experimentally characterize the effects of PMA on the curren t-induced \nmagnetization dynamics by using micro-focus Brillouin light sca ttering (BLS) \nspectroscopy27. The probing laser light is fo cused on the surface of the magn etic disk \n(Fig. 1a), and the modulation of the scattered light by high-fr equency magnetization \noscillations is detected. The B LS signal intensity is proportio nal to the intensity of \nmagnetization oscillations a t the selected frequency. \nFigures 1d and 1e show the BLS sp ectra of magnetic oscillations detected in Py \n(Fig. 1d) and CoNi (Fig. 1e) disks with current I close to the critical value IC, at which \nthe spin current is expected to c ompletely compensate the natur al linear magnetic \ndamping (see Supplementary Note 1 for the determination of the critical currents). 6At IIC. In the Py disk, th e intensity of \nfluctuations saturates, while t heir spectral width significantl y increases (Fig. 1d). In \ncontrast, a narrow intense pea k emerges in CoNi, marking a tran sition to the auto-\noscillation regime (Fig. 1e). The se results indicate that the p henomena preventing the \nonset of auto-oscillations in t he Py disk are suppressed in CoN i. \nThe differences between the tw o systems are highlighted by the quantitative \nanalysis of their chara cteristics, Fig. 2. At IIC, while the spectral width r apidly increases, indicating \nthe onset of nonlinear damping. In c ontrast, in CoNi the intens ity rapidly increases at \nI>IC, while the spectral linewidth c ontinues to decrease. We note t hat the BLS spectra \nare broadened by the finite freque ncy resolution of the techniq ue, increasing the \nmeasured values particularly for small linewidths. At large cur rents, the BLS intensity \nin CoNi somewhat decreases and t he spectral width increases, in dicating an onset of \nhigher-order nonlinear processes that cannot be completely avoi ded in real systems. The \ntwo systems also exhibit a quali tatively different dependence o f the BLS peak frequency \non current – the nonlinear freque ncy shift (Fig. 2c). For CoNi, the frequency slightly \nincreases with current, while f or Py, it exhibits a redshift th at becomes increasingly \nsignificant above IC. The large frequency nonlineari ty in Py is likely associated w ith the \nnonlinear excitation of a broad spe ctrum of spin-wave modes, wh ich is directly related \nto the nonlinear suppression of aut o-oscillation, as discussed in detail below. 7The effects of the nonlinear damp ing can be also clearly seen i n the time-domain BLS \nmeasurements. In these measuremen ts, the current was applied in pulses with the \nduration of 1 s and period of 5 s, and the temporal evolution of the BLS intensity was \nanalyzed. Figure 3 shows the temporal traces of the intensity r ecorded for the Py and \nCoNi disks at I=1.07 IC, corresponding to the maximum intensity achieved in CoNi \nsample (Fig. 2a). For the CoNi disk, the intensity monotonicall y increases and then \nsaturates. In contrast, for the Py disk, the intensity saturate s at a much lower level \nshortly after the start of the pulse, followed by a gradual dec rease over the rest of the \npulse duration, indicating the onset of energy flow into other spin-wave modes. \nThe mechanisms underlying the obser ved behaviors are elucidated by the \nmicromagnetic simulations, whic h were performed using the MuMax 3 software29. The \nlinear spin wave dispersions, calculated using the small-amplit ude limit Mymax/M=0.01, \nare qualitatively similar for Py and CoNi (symbols in Figs. 4a and 4b). The two \nbranches corresponding to spin w aves propagating perpendicular and parallel to the \nstatic field H0 merge at the wavevector k=0, at the frequency fFMR of the uniform-\nprecession ferromagnetic resonance (FMR). The frequency of the branch with kH0 \nmonotonically increases with k, while the branch with k||H0 exhibits a minimum fmin at \nfinite k due to the competition between the dipolar and the exchange in teractions. The \nfrequencies obtained from the simulations are in a good agreeme nt with the results of \ncalculations using analytical spin- wave theory (solid curves in Figs. 4a and 4b)30. \nThe qualitative differences between the nonlinear characteristi cs of the two \nsystems are revealed by the dependence of frequency on the ampl itude of magnetization \noscillations, Fig. 4c. In the Py film, both fFMR and fmin exhibit a strong negative \nnonlinear frequency shift. In contr ast, in the CoNi film the fr equency fFMR slightly 8decreases, while the frequency fmin increases with increasing amplitude. This difference \nallows us to identify the auto- oscillation mode excited in the CoNi sample. Since the \nexperimentally observed oscillat ion frequency for CoNi sample i ncreases with \nincreasing amplitude (Fig. 2c), we conclude that current-induce d auto-oscillations \ncorrespond not to the quasi-unifor m FMR mode, but rather to the lowest-frequency \nspin-wave mode. This conclusion i s in agreements with the recen t experimental \nobservations31, which showed that the injecti on of the pure spin current resu lts in the \naccumulation of magnons in the sta te with the lowest frequency. \nWe now analyze the relationship b etween the dispersion characte ristics and the \nnonlinear damping effects. The l owest-frequency state at fmin is non-degenerate in both \nPy and CoNi (Figs. 4a, 4b). The absence of degeneracy is common ly viewed as a \nsufficient condition for the suppression of nonlinear damping, since it prohibits resonant \nfour-wave interactions15. However, this view is incons istent with our experimental \nresults, Figs. 1d, and 1e, as also confirmed by the micromagnet ic simulations illustrated \nin Fig. 5. In these simulations, we use artificially small line ar damping to emulate \ndamping compensation by the spin current, and analyze the dynam ics of the lowest-\nfrequency mode excited at time t=0. Figure 5a shows the projections of the \nmagnetization precession trajectories on the My-Mx and Mz-Mx planes, for a relatively \nlarge precession amplitude Mymax/M=0.17. As expected, the precession is nearly circular \nin CoNi, and elliptical in Py. The ellipticity in Py results in the oscillation of the \nprojection mz2 of magnetization on the equilibriu m direction at twice the osc illation \nfrequency, which plays the role of a parametric pump for other spin-wave modes. In \nsimulations performed for zero temperature (Fig. 5b), precessio n initiated at t=0 \ncontinues indefinitely, i.e. ene rgy is not transferred to other modes. This result is 9consistent with the parametric m echanism of mode coupling, whic h requires non-zero \namplitudes of all the involved modes. In contrast to T=0, at finite temperatures all the \nspin-wave modes have non-zero amplitudes due to thermal fluctua tions, enabling their \nparametric excitation. In the simulations performed at T=300 K, the amplitude of \nprecession excited in Py at t=0 abruptly drops at about 100 ns and continues to decrease \nthereafter, indicating the ons et of nonlinear damping (Fig. 5c) . Spectral analysis \nconfirms that the in itially monochromatic o scillation at freque ncy fmin transitions to a \nbroad spectrum of spin-wave mode s excited at longer times due t o their nonlinear \ncoupling to the initially exc ited mode (Fig. 5d). We emphasize that this nonlinear \ncoupling must be non-resonant2,32, since it cannot be describe d in terms of energy- and \nmomentum-conserving magnon-magnon interactions26. \nThe parametric mechanism of t he nonlinear mode coupling is conf irmed by the \nsimulation results for the CoNi f ilm, where the oscillations of the longitudinal \nmagnetization component are neglig ibly small, and the precessio n amplitude remains \nconstant (Fig. 5c). These results clearly show that the compens ation of the precession \nellipticity by the PMA enables suppression of the nonlinear dam ping, supporting our \ninterpretation of the e xperimental findings. \nThe simulations described above wer e performed for a model syst em - an \nextended magnetic film charac terized by a continuous spin wave spectrum. The \ngenerality of the nonlinear damping mechanism revealed by these simulations, and its \nrelevance to our experimental results, was confirmed by additio nal simulations \nperformed for 0.5 m Py and CoNi disks matching thos e studied in our experiments. \nThe temporal evolution of the lo west-frequency modes is very si milar to that obtained \nfor extended films (see Supplementary Fig. 2). However, the spe ctrum excited due to 1 0the nonlinear damping of the low est-frequency mode becomes disc rete, consistent with \nthe quantization of spin wave sp ectrum expected for a confined magnetic system. \nFinally, we discuss the conse quences of residual precession ell ipticity, which can \nresult from incomplete compens ation of the dipol ar anisotropy b y PMA. To characterize \nthese effects, we performed add itional measurements for CoNi sa mples with the \nanisotropy field deviating from the saturation magnetization by about 10% in both \ndirections (Supplementary Fig. 3) . For these samples, the trans ition to the auto-\noscillation regime becomes noticeably suppressed by the nonline ar damping, confirming \nthe importance of precise suppr ession of the pr ecession ellipti city to achieve auto-\noscillations in an extended magne tic region (Supplementary Note 2). \nIn conclusion, our experiments and s imulations show that the ad verse nonlinear \ndamping can be efficiently suppres sed by minimizing the ellipti city of magnetization \nprecession, using magnetic materials where in-plane dipolar ani sotropy is compensated \nby the perpendicular magnetic a nisotropy. This allows one to ac hieve complete \ncompensation of the magnetic damping, and excitation of coheren t magnetization auto-\noscillations by the spin current without confining the spin-cur rent injection area to \nnanoscale. Our findings open a rout e for the implementation of spin-Hall oscillators \ncapable of generating microwave signals with t echnologically re levant power levels and \ncoherence, circumventing the c hallenges of phase locking a larg e number of oscillators \nwith nano-scale dimensions. The y also provide a route for the i mplementation of \nspatially extended amplificati on of coherent propagating spin w aves, which is vital for \nthe emerging field of magnonics u tilizing these waves as the in formation carrier. The \nproposed approach can also facil itate the experimental realizat ion of spin current-driven 1 1Bose-Einstein condensation of ma gnons, which has not be achieve d so far due to the \ndetrimental effects of nonlinear damping31. \n \n \nMethods \n \nSample fabrication and characterization. The samples were fabricated on annealed \nsapphire substrates with pre-pa tterned Au electrodes. First, 1. 3 m-wide \nTa(4)Pt(8)FM(5)Ta(3) strips conne cted to the Au electrodes were fabricated by a \ncombination of e-beam lithography and high-vacuum sputtering. H ere, thicknesses are \nin nanometers, and the ferromagnetic layer layer FM(5) is eithe r Co(1.2)/Ni(3.8) or \nPy(5). The bottom Ta(4) was us ed as a buffer layer, and the top Ta(3) as a capping layer \nto prevent oxidation of the magne tic structures. A circular 500 nm AlO x(10) mask was \nthen defined by Al evaporati on in 0.02 mTorr of oxygen, followe d by Ar ion milling \ntimed to remove the part of the multilayer down to the top of t he Pt(8) layer. Finally, the \nstructures were passivated with an AlO x(10) capping layer. \nThe values of the saturation magnetization 4 MCoNi=6.9 kG and 4 MPy=10.1 kG were \ndetermined from vibrating-sample magnetometry measurements. The PMA anisotropy \nfield Ha=6.6 kG of CoNi was obtained from the FMR frequency and the kno wn \nmagnitude of the magnetization. \n Measurements. All measurements were performe d at room temperature. In micro -\nfocus BLS measurements, the probing las er light with the wavelength o f 532 nm and the \npower of 0.1 mW was focused int o a diffraction-limited spot on the surface of the \nmagnetic disk by using a high-numer ical-aperture 100x microscop e objective lens. The \nposition of the probing spot was actively stabilized by using c ustom-designed software \nproviding long-term spatial sta bility of better than 50 nm. The spectrum of light \ninelastically scattered from ma gnetization oscillations was ana lysed by a six-pass Fabry-\nPerot interferometer TFP-2HC (JRS Scientific Instruments, Switz erland). The obtained \nBLS intensity at a given freque ncy is proportional to the squar e of the amplitude of the \ndynamic magnetization at this fr equency. Therefore, BLS spectra directly represent the \nspectral distribution of the ma gnetization oscillations in the magnetic disk. 1 2 \nMicromagnetic simulations. The micromagnetic simulations were performed by using \nthe software package MuMax3 (Re f. 29). We modelled the magnetiz ation dynamics in a \n10 m 10 m large and 5 nm thick magnetic f ilm. The computational area wa s \ndiscretised into 5 nm 5 nm 5 nm cells. Periodic boundary conditions at all lateral \nboundaries were used. Magnetiza tion dynamics was excited by ini tially deflecting \nmagnetic moments from their equi librium orientation in the plan e of the film. The \ndeviation was either spatiall y uniform or periodic. By analysin g the free dynamics of \nmagnetization, the frequency c orresponding to the given spatial period was determined, \nwhich allowed us to reconstruct t he frequency vs wavevector dis persion curves (see \nSupplementary Fig. 4). By varyi ng the angle of the initial defl ection, we determined the \ndependence of the frequency on the oscillation amplitude and an alysed the instability of \nprecession caused by the nonlinear damping at large amplitudes (see Supplementary \nFig. 5). Simulations were perfor med using artificially small Gi lbert damping parameter \n10-12, which emulates the conditions of the compensation of the line ar damping by the \npure spin current. The values of the saturation magnetization 4 MCoNi=6.9 kG and \n4MPy=10.1 kG were determined from the vibrating-sample magnetometry \nmeasurements. The PMA anisotr opy constant of CoNi film 0.181 J/ m3 was calculated \nbased on the experimentally determ ined values of the anisotropy field and the \nmagnetization. Standard values of t he exchange stiffness consta nt of 13 pJ/m and 9 pJ/m \nwere used for Py and CoN i, respectively. \n \nAcknowledgements: We acknowledge support from Deutsche Forschungsgemeinschaft \n(Project No. 423113162) and the Nati onal Science Foundation of USA. \n Author Contributions: B.D. and V.E.D. performed meas urements and data analysis. \nB.D. additionally performed micr omagnetic simulations. S.U. des igned and fabricated \nthe samples. S.U., S.O.D., and V.E .D. managed the project. All authors co-wrote the \nmanuscript. 1 3 \nAdditional Information: The authors have no compe ting financial interests. \nCorrespondence and requests for materials should be addressed t o V.E.D. \n(demidov@uni-muenster.de). Data availability. The data that support the findi ngs of this study are available from the \ncorresponding author upon reasonable request. \n \nReferences \n1. Mohseni, S. M. et al. Spin t orque–generated magnetic droplet solitons. Science 339, \n1295 (2013). \n2. 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JETP Lett. 13, 467 (1971). \n9. Hirsch, J. E. Spin Hall Effect. Phys. Rev. Lett. 83, 1834 (1999). \n10. Kajiwara, Y. et al. Transmi ssion of electrical signals by s pin-wave interconversion \nin a magnetic insulator. Nature 464, 262–266 (2010). \n11. Awad, A. A. et al. Long-range m utual synchronization of spi n Hall nano-oscillators. \nNat. Phys. 13, 292–299 (2017). \n12. Demidov, V. E. et al. Control of magnetic fluctuations by s pin current. Phys. Rev. \nLett. 107, 107204 (2011). \n13. Cornelissen, L. J., Liu, J., Dui ne, R. A., Ben Youssef, J. & van Wees, B. J. Long-\ndistance transport of magnon spi n information in a magnetic ins ulator at room \ntemperature. Nature Phys. 11, 1022-1026 (2015). \n14. Liu, L., Pai, C.-F., Ralph, D. C . & Buhrman, R. A. Magnetic oscillations driven by \nthe spin hall effect in 3-term inal magnetic tunnel junction dev ices. Phys. Rev. Lett. \n109, 186602 (2012). \n15. Duan, Z. et al. Nanowire spin torque oscillator driven by s pin orbit torques. Nat. \nCommun. 5, 5616 (2014). \n16. Collet, M. et al. Generation of coherent spin-wave modes in yttrium iron garnet \nmicrodiscs by spin-orbit torque. Nat. Commun. 7, 10377 (2016). \n17. Divinskiy, B. et al. Excitati on and amplification of spin w aves by spin-orbit torque. \nAdv. Mater. 30, 1802837 (2018). \n18. Evelt, M. et al. Emission of coherent propagating magnons b y insulator-based spin-\norbit-torque oscillators. Phys. Rev. Appl. 10, 041002 (2018). 1 519. Ando, K. et al., Electric mani pulation of spin relaxation u sing the spin Hall effect. \nPhys. Rev. Lett. 101, 036601 (2008). \n20. Liu, L. et al., Spin-torque fe rromagnetic resonance induced by the spin Hall effect. \nPhys. Rev. Lett. 106, 036601 (2011). \n21. Hamadeh, A. et al. Full control of the spin-wave damping in a magnetic insulator \nusing spin-orbit torque. Phys. Rev. Lett. 113, 197203 (2014). \n22. Evelt, M. et al. High-efficien cy control of spin-wave propa gation in ultra-thin \nyttrium iron garnet by th e spin-orbit torque. Appl. Phys. Lett. 108, 172406 (2016). \n23. Zahedinejad, M. et al. Two-dimensional mutual synchronizati on in spin Hall nano-\noscillator array. Preprint at https://arxi v.org/abs/1812.09630 (2018). \n24. The measurement were performed for H0 ranging from 1000 to 2000 Oe. The found \nbehaviours were found to be qualitati vely the same over the ent ire range of fields. \n25. Suhl, H. The theory of ferroma gnetic resonance at high sign al powers. J Phys. \nChem. Sol. 1, 209 (1957). \n26. Gurevich, A. G. & Melkov, G. A. Magnetization Oscillations and Waves (CRC, \nNew York, 1996). \n27. Demidov, V. E. & Demokritov, S. O. Magnonic waveguides stud ied by microfocus \nBrillouin light scattering. IEEE Trans. Mag. 51, 0800215 (2015). \n28. Slavin, A. & Tiberkevich, V. N onlinear Auto-O scillator Theo ry of Microwave \nGeneration by Spin-Polarized Current. IEEE Trans. Magn. 45, 1875-1918 (2009). \n29. Vansteenkiste, A. et al. T he design and verification of MuM ax3. AIP Advances 4, \n107133 (2014). \n30. Kalinikos, B. A. Excitation of pr opagating spin waves in fe rromagnetic films. IEE \nProc. H 127, 4-10 (1980). 1 631. Demidov, V. E. et al. Chemical p otential of quasi-equilibri um magnon gas driven by \npure spin current, Nat. Commun. 8, 1579 (2017). \n32. Melkov, G. A. et al. Nonlinear f erromagnetic resonance in n anostructures having \ndiscrete spectrum of spin-wave modes, IEEE Magn. Lett. 4, 4000504 (2013). \n \n Figure legends Figure 1. Schematics of the experiment a nd spectra of current-induced mag netic \noscillations. a , Layout of the test devices. Mag netic disks are fabricated on top of a Pt \nstrip either from Py, or from the Co/Ni bilayer with PMA tailor ed to compensate the \ndipolar anisotropy of the film. Inset illustrates the device op eration principle, based on \nthe injection into the ferroma gnetic disk of pure spin current generated due to the SHE \nin Pt. b, The ellipticity of the magne tization precession in Py is caus ed by the dipolar \nanisotropy. c, In CoNi, the ellipticity is mi nimized due to PMA compensating the \ndipolar anisotropy. d and e, BLS spectra of magnetic osci llations vs current for Py and \nCoNi disks, respectively. I\nC marks the critical current, a t which the spin current is \nexpected to completely compensat e the natural linear magnetic d amping. The data were \nrecorded at H0=2.0 kOe. \nFigure 2. Characterization of the current-dependent magnetizati on dynamics in Py \nand CoNi disks. a, Maximum intensities of the BLS spectra vs current. b, Current \ndependences of the spectral widt h of the BLS peaks at half the maximum intensity. c, \nCenter frequency of the detecte d spectral peaks vs current. Sym bols are the \nexperimental data, lines are guide s for the eye. The data were recorded at H0=2.0 kOe. 1 7Figure 3. Evolution of the nonlinear damping in the time domain . Time dependence \nof the peak BLS intensity in response to the 1 s long pulse of current obtained for the \nPy and CoNi disks, as labelle d. The data were recorded at H0=2.0 kOe and I=1.07 IC. \nFigure 4. Calculated spin wave dispersion spectra and their dep endence on spin \nwave amplitude. a and b, Dispersion spectra for the Py a nd CoNi films, respectively, \ncalculated in the small-am plitude linear regime. fFMR and fmin label the frequencies of the \nquasi-uniform FMR and of the low est-frequency spin-wave mode, r espectively. \nSymbols are the results of mic romagnetic simulations, curves – calculations based on \nthe analytical theory. c, Dependences of the cha racteristic frequencies fFMR and fmin on \nthe normalized precession amplitude. Symbols are the results of micromagnetic \nsimulations, curves – guides for th e eye. All calculations were performed at H0=2.0 \nkOe. \nFigure 5. Analysis of nonlinear da mping based on the micromagne tic simulations. \na, Calculated magnetiza tion trajectories for the lowest-frequenc y spin-wave states in Py \nand CoNi, as labeled. mz2 labels the double-freque ncy dynamic component of \nmagnetization, which serves as a parametric pumping source for the nonlinear spin-\nwave excitation. b and c, Temporal evolution of the free precession amplitude starting \nwith a large initial amplitude at t=0, at T=0 (b) and T=300 K (c). The simulations were \nperformed with negligible linear damping, emulating the damping compensation by the \nspin current. d, Fourier spectra of magnetizati on oscillations in Py before ( t=0-50 ns) \nand after ( t=150-200 ns) the onset of nonlinear damping. Fig. 1Probing laser light\nIH0Co(1.2 nm)/Ni(3.8nm)\nor Py(5 nm)Py\nCoNi\nPt(8 nm)M\n1.3 mμ0.5 mμx\nzyIsab\nc\nPMA axisM\nMhd\nhd\nhPMAmz2/c119\nBLS intensity, a.u.\nBLS intensity, a.u.\nFrequency, GHz Frequency, GHzd e\n0 0200 200400 400600 600800 800\n18\n171616\n1515\n141314\nI, mA I, mAIcIc 108\n99\n8411512613714Py CoNia\nbPeak intensity, a.u.\nSpectral width,GHz0\n0.85\n0.85\n0.850.9\n0.9\n0.90.95\n0.95\n0.951\n1\n11.05\n1.05\n1.051.1\n1.1\n1.11.15\n1.15\n1.15250\n6810121411.52\n00.55007501000\nII/ccCoNi\nCoNi\nCoNiPy\nPy\nPy\nFig. 2Center frequencyGHz,CoNiPy\nTime, ns0 500 1000 1500Peak intensity, a.u.\n02505007501000\nFig. 3b\nc\nFig. 4k,mμ-1k,mμ-166.26.6\n6.4\n13.7\n-0.3\n0 0.1 0.2 0.3 0.4 0.5-0.2-0.100 10 20 5 15 01020304013.813.914 6.8\nFrequency,G H z\nMMymax/Frequency shift,G H z\nΔffFMR\nΔfFMRfFMR\nΔfFMRfmin\nΔfminfmin\nΔfmina\nkk kk H0 H0 H0 H0CoNi\nCoNi:Py\nPy:b\nc\nd\nFig. 5a\nPy, = 0T\nPy, = 0 50 nst −\nPy, = 150 200 nst −Py, = 300 KTCoNi, = 0 T\nCoNi, = 300 K T\nt, ns0\n0\n0 25 7500\n0 0.98\n0\n1e-51e-41e-31e-21e-110.05\n0.050.10.1\n0.1 0.99-0.1\n-0.1\n0.10.15\n0.150.20.2\n0.2 1-0.2\n-0.2\n0.250\n50\n50100\n100\n100150\n150200\n200\nFFT power, a.u.\nFrequency, GHzMMx/\nMMy/ MMz/PyPyCoNi CoNi\nMMxmax/\nMMxmax/\nfminmz2/c119 1Supplementary information \n \nSupplementary Note 1. Determinatio n of the critical currents. \nAccording to the spin-transfer t orque theory, injection of spin current into ferromagnets \nresults not only in the compensation of the natural damping, bu t also in enhancement of \nmagnetic fluctuations1. A t a c e r t a i n c r i t i c a l c u r r e n t v a l u e IC, natural damping becomes \ncompletely compensated, while the intensity of fluctuations div erges. At currents I below IC the \ninverse of the fluctuation intensity exhibits a linear dependen ce on current, extrapolating to zero \nat I=IC. This dependence provides a precise method for the determinati on of IC by the BLS \nspectroscopy, whose high sensitivity allows one to detect magne tic fluctuations even at I=0. \nSupplementary Figure 1 shows the inverse of the measured integr al BLS intensiti es of current-\ndependent fluctuations for the Py and CoNi disks. As expected, the data for both types of \nsamples exhibit a linear dependence on current, yielding IC=16 mA for Py, and 14.5 mA for \nCoNi. \n \n \nSupplementary Figure 1. Enhancement of magnetic fluctuations by spin current. Normaliz ed \ncurrent dependences of the inverse of the integral BLS intensit y for the Py and CoNi disks, as \nlabelled. Symbols: experimental data, line: linear fit. IC marks the extrapolated value of current, \nat which the intensity of fluctuations is expected to diverge. The data were obtained at H0=2 \nkOe. \n 2 \n \nSupplementary Figure 2. Results of micromagnetic simulations performed for magnetic di sks \nwith the diameter of 0.5 m. a, Spatial distributions of the static magnetization. b, Spatial \ndistributions of the dynamic ma gnetization for the dynamic mode with the lowest frequency. c, \nTemporal evolution of the free precession amplitude starting wi th a large initial amplitude at t=0. \nThe simulations were performed with negligible linear damping, emulating the damping \ncompensation by the spin current. T=300 K. d, Fourier spectra of magnetization oscillations in \nthe Py disk before ( t=0-50 ns) and after ( t=350-400 ns) the onset of nonlinear damping. \n 3Supplementary Note 2. Effects of res idual precession ellipticit y \nAs seen from the data of Supplementary Fig. 3a, auto-oscillatio ns become noticeably \nsuppressed by the nonlinear damping in two samples, where the P MA anisotropy differs from \nthe saturation magnetization by a bout 10% in one or the other d irection. In both samples, the \noscillation intensities significan tly decrease, and the maximum intensity is achieved at larger \ncurrents. These behaviors are consistent with our interpretatio n. In particular, the current value, \nat which the maximum intensity is achieved, is determined by tw o factors: (i) the value of the \nintensity of magnetization oscill ations necessary for the onset of strong nonlinear relaxation \nand (ii) the rate, at which the intensity increases with the in crease of current above the threshold \nvalue. The intensity (i) depends on the efficiency of the nonli near coupling determined by the \ndegree of precession ellipticity. In agreement with this pictur e , t h e s a m p l e s w i t h l a r g e r \nellipticity exhibit an onset of strong nonlinear relaxation at smaller intensities. The rate (ii) is \nalso expected to depend on the e fficiency of nonlinear mechanis ms. Since nonlinear scattering \ncounteracts the energy flow due t o the injection of the pure sp in current, in samples with larger \nellipticity, the rate (ii) should be small er, in agreement with the data of Supplementary Fig. 3a. \nFinally, in samples with very large ellipticity, such as the Py sample (Fig. 2 in the main text), \nthe strong nonlinear relaxation develops at very small intensit ies at currents close to the \nthreshold current, resulting i n complete suppression of auto-os cillations. \n 4 \n \nSupplementary Figure 3. Transition to the auto-oscillations in CoNi disks with differe nt \nPMA anisotropy fields, as labelled. a, Maximum intensities of the BLS spectra vs current. \nb, Current dependences of the spectral width of the BLS peaks at half the maximum intensity. \nSymbols: experimental data, line: guide for the eye. The data were obtained at H0=2 kOe \n \n \n 5 \n \n \nSupplementary Figure 4. Procedure used for the calculations of the linear-regime spin- wave \nspectra. a, Magnetic moments ar e initially deflected from their equilibr ium orientation in the y-\ndirection by a small angle. The d eflection is sinusoidal with t he period corresponding to the \nselected wavevector k. b, The free dynamics of magnetization is calculated for artific ially small \nGilbert damping parameter. c, By performing the Fourier transform of the obtained temporal \ntrace, the frequency corres ponding to the wavevector k is determined. By varying the spatial \nperiod of the initial deflecti on pattern and the deflection ang le, the dependence of the frequency \nof spin waves on the wavevector and the amplitude is determined . \n 6 \n \nSupplementary References \n1. Slavin, A. & Tiberkevich, V. Non linear Auto-Oscillator Theor y of Microwave Generation \nby Spin-Polarized Current. IEEE Trans. Magn . 45, 1875-1918 (2009). \nSupplementary Figure 5. Analysis of the instability at large precession angles. a, Initial state \nwith the selected spatial period and a relatively large deflect ion angle is defined. b, The free \ndynamics exhibits a transition from periodic oscillations to th e multimodal regime. c, Fourier \ntransform calculated using the entire calculated time interval exhibits a broad spectrum (compare \nto the Supplementary Figure 4c). By performing Fourier transfor m of 50 ns-long intervals at \ndifferent delays (see Fig. 5d in the main text), the time-depen dent flow of energy from the \ninitially excited mode to other modes due to the nonlinear coup ling is analyzed. The temporal \nevolution of the amplitudes of specific modes (see Fig. 5c in t he main text) is determined by \nanalyzing the correspon ding Fourier harmonics. \n" }, { "title": "1502.00027v1.Intrinsic_Damping_of_Collective_Spin_Modes_in_a_Two_Dimensional_Fermi_Liquid_with_Spin_Orbit_Coupling.pdf", "content": "arXiv:1502.00027v1 [cond-mat.str-el] 30 Jan 2015Intrinsic Damping of Collective Spin Modes\nin a Two-Dimensional Fermi Liquid with Spin-Orbit Coupling\nSaurabh Maiti1,2and Dmitrii L. Maslov1\n1Department of Physics, University of Florida, Gainesville , FL 32611 and\n2National High Magnetic Field Laboratory, Tallahassee, FL 3 2310\n(Dated: December 1, 2018)\nA Fermi liquid with spin-orbit coupling (SOC) is expected to support a new kind of collective\nmodes: oscillations of magnetization in the absence of the m agnetic field. We show that these modes\nare damped by the electron-electron interaction even in the limit of an infinitely long wavelength\n(q= 0). The linewidth of the collective mode is on the order of ¯∆2/EF, where¯∆ is a characteristic\nspin-orbit energy splitting and EFis the Fermi energy. Such damping is in a stark contrast to\nknown damping mechanisms of both charge and spin collective modes in the absence of SOC, all of\nwhich disappear at q= 0, and arises because none of the components of total spin is conserved in\nthe presence of SOC.\nElectron systems with spin-orbit coupling (SOC) ex-\nhibit rich physics, some of which may have technological\napplications;1,2equally rich is the physics of cold-atom\nsystems with synthetic SOC.3,4Combining many-body\ninteractions with broken SU(2) symmetry, one obtains\na special–“chiral”–kind of Fermi liquid (FL)5–7that sup-\nportsanewtypeofcollectivemodes, “chiral-spinwaves”–\noscillations of the spin density in zero magnetic field.8–11\nIn the absence of SOC, electron-electron interaction\n(eei)12does not affect certain properties of an electron\nsystem given that some symmetries are preserved. For\nexample, the conductivity and cyclotron-resonance fre-\nquency of a Galilean-invariant system are not affected\nbyeei; same is true for the de Haas-van Alphen (dHvA)\nfrequency in an isotropic system13and for the Larmor\nfrequency in the presence of an SU(2)-symmetric in-\nteraction. Being a relativistic effect, SOC breaks both\nGalilean (but not necessarily rotational) invariance and\nSU(2) symmetry and thus lifts the protection ensured by\nthese symmetries. As a result, several physical quanti-\nties become dependent on eei. The list of such quantities\nincludes the optical conductivity,16Drude weight,17, and\nfrequencies of collective spin modes, which play the role\nof Larmor frequencies in zero magnetic field.8–11\nIn this Letter, we discuss another fundamentally new\neffect induced solely by SOC: intrinsic damping of collec-\ntive spin modes in the uniform ( q= 0) limit. Interaction-\ninduced damping of collective modes is not, by itself,\na new effect. For example, plasmons in 3D,202D,21\nand 1D (Ref. 22) electron systems, the Silin-Leggett\n(SL) mode18,19in a partially spin-polarized FL,23,25and\nmagnons in a ferromagnetic FL24,25are all damped by\ninteraction processes involving excitations of multiple\nparticle-hole pairs. (This mechanism is different from\nLandau damping which involves only a single particle-\nhole pair: damping by multiple pairs occurs even outside\nthe single-particle continuum.) However, Galilean invari-\nance, in the case of charge modes,26and conservation of\nthe total spin component along the field ( S3), in the case\nof spin modes,27ensure that this kind of damping van-\nishes atq= 0. We show here that this is not the case forelectron systems with SOC.\nFIG. 1: Left: Schematics of the Silin-Leggett mode in a par-\ntially spin-polarized FL. Right: The chiral-spin modes in a\nFL with Rashba spin-orbit coupling. The shaded regions de-\nnote the particle-hole continua, ∆ Bis the Larmor frequency,\n˜∆Bis the quasiparticle Zeeman energy, and ∆ min/maxis the\nlower/upper boundary of the continuum at q= 0.\n(a) (b) (c)K+QK+QK\nKK\nK+QK\nK+QK+Q+PK+P\nK+P\nK+Q+P\n(a) (b) (c)K+QK+QK\nKK\nK+QK\nK+QK+Q+PK+P\nK+P\nK+Q+P\n(e) (d) (e) (d)(d)\nFIG. 2: Top: The ladder (RPA) series for the spin suscepti-\nbility. The boxed wavy line is the static effective interacti on\nUx. Bottom: Diagrams contributing to damping of the collec-\ntive modes. The wavy line in diagrams a-edenotes a dynamic\ninteraction, Veff(P).\nTwo-dimensional (2D) electron systems with\nmomentum-dependent SOC, e.g., of Rashba or Dres-\nselhaus types, bear certain similarity to a partially\nspin-polarized Fermi gas. The latter has a transverse SL\nmode in the spin sector (see Fig. 1, left),18,19while the\nformer has three (two transverse and one longitudinal)\nchiral-spin modes (Ω 1...Ω3in Fig. 1, right),8–11which2\ncorrespond to oscillations of the three components of\nmagnetization. Indices 1 −3 label the Cartesian system\nwith the 3 axis along the normal to the plane of a 2D\nelectron gas. (Although SL-mode has been studied\npreviously in 3D, the same mode should occur in 2D\nas well.) Conservation of S3ensures that the frequency\nof the SL mode at q= 0 coincides with the Larmor\nfrequency in the absence of eei. In the presence of SOC,\nnone of the three spin components is conserved. As a\nresult, one obtains three distinct modes with frequencies\nrenormalized by eei. In addition, as we show here,\nthese modes have finite linewidth which, in order of\nmagnitude, is given by the inverse transport lifetime\nof a quasiparticle with energy equal to the spin-orbit\nsplitting. That the SL mode at q= 0 is not affected by\neeifollows already from the exact equations of motion\nfor magnetization.23Diagrammatically, this occurs due\nto a cancellation between the self-energy and vertex\ngraphs for the spin susceptibility.23Such a cancellation,\nhowever, does not occur in the presence of SOC.\nThe single-particle Hamiltonian of a 2D system with\nSOC can be written as (we set /planckover2pi1= 1)\nHk=/parenleftbiggk2\n2m−µ/parenrightbigg\nσ0+λ/vector σ·/vectorf(/vectork), (1)\nwhereµis the chemical potential, σ0is the 2×2unit ma-\ntrix,/vector σis the three-dimensional vector of Pauli matrices,\nλistheSOCconstant, and /vectorf(/vectork) =−/vectorf(−/vectork) isa2Dvector\nthat depends on the details of SOC; e.g., /vectorf= (k2,−k1,0)\nfor linear Rashba SOC. The single-particle Green’s func-\ntion is given by\nG(K) =/summationdisplay\nsΩs(/vectork)gs(K),Ωs(/vectork) =1\n2/bracketleftbig\nσ0+sˆη/vectork/bracketrightbig\n,(2)\nwheregs(K) = (ik0−k2\n2m−s∆/vectork/2+µ)−1,K≡(ik0,/vectork),\ns=±1 labels either spin projection or chirality, ˆ η/vectork≡\n/vector σ·/vectorf/|/vectorf|, and ∆ /vectork= 2λ|/vectorf|is the spin-orbit splitting which,\nin general, depends not only on the magnitude but also\non the direction of /vectork. We will be primarily interested in\nthe case of weak SOC, when ∆ /vectork≪EFfor any/vectork. In\nwhat follows, we will be comparing the Ω 3chiral-spin\nmode to the (2D) SL mode, as both modes are trans-\nverse to the SOC-induced/Zeeman magnetic field. The\nlatter can be described by the same Hamiltonian with\n/vectorf= (0,0,∆B/2λ), where ∆ B≡gµBB,gis the effective\ng−factor,µBis the Bohr magneton, and /vectorBis the mag-\nnetic field chosen to be along the 3-direction. The orbital\neffect of the field is not considered here.\nWithin the Random Phase Approximation (RPA), the\nspin susceptibility tensor is given by the ladder series in\nFig. 2, where the boxed wavy line is a short-range inter-\naction,Ux, which mimics the exchange interaction in the\nspin channel. As shown in Ref. 11, the frequencies of the\ncollective modes correspond to the roots of the equation\nDet/parenleftbig\nσ0⊗σ0+Ux\n2Π0/parenrightbig\n, where the elements of the 4 ×4spin-charge polarization matrix Π0are given by\nΠ0\nij(Q) =/integraldisplay\nKTr[σiG(K)σjG(K+Q)],(3)\nwhere/integraltext\nK≡T/summationtext\nk0/integraltextd2k\n(2π)2,i,j∈0,1,2,3, and 0 corre-\nsponds to the charge component. At q= 0, all mixed\nspin-charge susceptibilities, Π0\n0jwithj∝ne}ationslash= 0, vanish by\ncharge conservation, while the matrix of spin suscepti-\nbilities can always be transformed to a diagonal form.\nIn general, there are three spin modes, whose frequen-\ncies are found from the equations 1+ δjUxΠ0\nii= 0, with\nδ1=δ2= 1 and δ3= 1/2. The Green’s functions in\nΠ0\nijcontain the self-energy parts. However, for the spe-\ncial case of Ux=constant, they drop out (see below) and,\nafter analytic continuation ( iq0→Ω+iδ), one obtains\nΠ0\n33(Ω) = 2ν/angbracketleftBigg∆2\n/vectorkF\n(Ω+iδ)2−∆2\n/vectorkF/angbracketrightBigg\nFS,(4)\nwhere we have already assumed that SOC is weak in the\nsense specified above, νis the density of states per spin\nprojection, ∝an}bracketle{t...∝an}bracketri}htdenotesaveragingovertheFermisurface\n(FS),/vectorkF=kF/vectork/k, andkFis the Fermi momentum in\nthe absence of SOC. In general, ∆ /vectorkvaries from ∆ min\nto ∆maxalong the FS. The continuum of inter-subband\nparticle-hole excitations, where ImΠ0\n33∝ne}ationslash= 0, is confined\nto the interval ∆ min≤Ω≤∆max. At the boundaries\nof the continuum, ReΠ0\n33has square root singularities,28\nwhich guarantee a solution of the eigenmode equation\n1 +UxΠ0\n33/2 = 0 for Ω <∆mineven at weak coupling.\nIf SOC is isotropic, ∆ min= ∆max≡∆, the two square-\nroot singularities merge into a single pole at Ω = ∆, the\ninter-subband continuum shrinks to a single point, and\nthe mode frequency is given by Ω 3= ∆√1−u, where\nu≡Uxν.11\nRenormalization of the mode frequency by eeiin the\nSOC case and the lack thereof in the SL case is an impor-\ntant difference, which we discuss now as it will help us\nto understand the differences in damping later on. This\ndifference occurs because the Greens’ functions in the\nRPA series in Fig. 1 include chirality- or spin-dependent\nshifts in the chemical potential, which are given by the\nmomentum- and frequency-independent parts of the self-\nenergy. Each rung of the ladder diagram (Π0\nij) contains\na difference of the self-energies\nδΣs= Σs−Σ−s=−Ux/summationdisplay\ns′/integraldisplay\nP(Bs,s′−B−s,s′)gs′(P),\n(5)\nwhereBs,s′is the matrix element for the transition\ns→s′. Within a given rung, δΣsrenormalizes the spin-\nsplitting perturbation, be it SOC or the magnetic field.\nSinces=s′in the SL case, the self-energy of an electron\nwith given spin is proportional to the number density of\nelectrons with the same spin. Hence, δΣsis proportional\nto magnetization [ δΣs=su∆B/(1−u)], and eachrung of\nthe diagram contains the renormalized Zeeman energy of3\na quasiparticle, ˜∆B= ∆B/(1−u). The boundary of the\ncontinuum at q= 0 is shifted from ∆ Bto˜∆B(cf. Fig. 1,\nleft) as can be seen, e.g., from the 11 component of the\nrung: Π0\n11(Ω) =−ν2˜∆2\nB\n(Ω+iδ)2−˜∆2\nB. The Larmor theorem\nis effected via a cancellation between the self-energy and\nvertex contributions:23when the rung is substituted into\nthe eigenmode equation, the factor of 1 −ucancels out\nand the frequency of the mode coincides with bare ∆ B.\nThe SOC case is different in that chirality, in contrast\nto spin, is not conserved by eei, and the sum over s′in\nEq. (5) contains both the s=s′ands′=−sterms.\nForUx= const, this implies that the self-energy of an\nelectron with given chirality is proportional to the total\nnumber density, and thus ∆Σ s= 0.6The vertex part\nis, however, non-zero. Therefore, there is no cancellation\nbetweenthe self-energyandvertexcontributions, andthe\nfrequencies of the modes are renormalizedby eei. IfUx∝ne}ationslash=\nconst, one can show29,30thatδΣsdoes not contain the\nzeroth angular harmonic of the interaction (which is why\nδΣs= 0 forUx= const), whereas the vertex contribution\ndoes. Thus, there is no cancellation between the two\ncontributions in the general case as well.\nDamping of collective modes in the region of fre-\nquencies and momenta outside the single-particle contin-\nuum occurs via generationof multiple particle-holepairs,\nwhich requires a dynamic interaction, e.g., a dynamically\nscreened Coulomb potential. The self-energy of collec-\ntive modes is depicted diagrammatically in Fig. 2 a-e.\nThe same set of diagrams has been encountered in the\nanalysis of various two-particle correlation functions in\nthe case of an RPA-type interaction.31–34Although the\nAslamazov-Larkin (AL) diagrams dandecontain two\nwavy lines, they are of the same order in the bare cou-\npling constant of the theory (the electron charge in our\ncase), as diagrams a-c. However, the contribution of the\nAL diagramsto damping vanishes within the approxima-\ntions made in this work.36,37\nIn the case of the SL mode, renormalization of the\ntransverse spin susceptibility ( χ⊥∼χ11+χ22) by dia-\ngramsa-cis given by δχ⊥(Q) =−µ2\nB(Πa+Πb+Πc),\nwhere\nΠa=/integraldisplay\nK/bracketleftBig\ng2\n−(K)g+(K+Q)Σ−(K)+/parenleftBig\n∆/vectorkF→ −∆/vectorkF/parenrightBig/bracketrightBig\n,Πb=/integraldisplay\nK/bracketleftbig\ng2\n−(K+Q)g+(K)Σ−(K+Q)\n+/parenleftBig\n∆/vectorkF→ −∆/vectorkF/parenrightBig/bracketrightBig\n,\nΠc=/integraldisplay\nK/parenleftbiggg−(K+Q)g+(K)\niq0+∆B[Σ+(K)−Σ−(K+Q)]\n+/bracketleftBig\n∆/vectorkF→ −∆/vectorkF/bracketrightBig/parenrightBig\n, (6)\n±denote up/down spins, Q= (iq0,0), Σ ±(K) =\n−/integraltext\nPg±(K+P)Veff(P), andVeff(P) is some dynamic in-\nteraction. In the last line of Eq. (6), we used the identity\ng±(K+P+Q)g∓(K+P) =g±(K+P)−g∓(K+P+Q)\niq0±∆B\n(7)\nand integrated over P. Because the denominator in\nEq.(7) does not depend on P, this last step produced the\nsame self-energies, Σ ±, as in diagrams aandb. Adding\nup the three lines of Eq. (6), we arrive at\nΠa+Πb+Πc=2∆B\nq2\n0+∆2\nB(A+−A−),(8)\nwhereA±≡/integraltext\nKg2\n±(K)Σ±(K). Recalling that Veff(P) is\nreal on the Matsubara axis and changing the variables as\nk0→ −k0andp0→ −p0, we find that A±=A∗\n±. Thus\nthe frequency-independent prefactorin Eq.(8), A+−A−,\nis real. Continuing iq0to the real axis, we see that the\nimaginarypartof δχ⊥comesonlyfromaresonanceatthe\nbare Larmor frequency, Ω = ∆ B, which coincides with\nthepoleof χ⊥intheRPAapproximation. Theonlyeffect\nof the interaction processes represented by diagrams a-\ncis thus to renormalize the amplitude of the SL mode\nwithout either shifting its frequency or smearing it.\nFor the case of SOC, it is convenient to consider\nrenormalization of the out-of-plane spin susceptibility,\nδχ33(Q) =−µ2\nB(Πa+Πb+Πc), where now\nΠa=/integraldisplay\nK1\n2/bracketleftbig\ng2\n−(K)g+(K+Q){Σ+−(K)+Σ−+(K)}+g2\n+(K)g−(K+Q){Σ++(K)+Σ−−(K)}/bracketrightbig\n,\nΠb=/integraldisplay\nK1\n2/bracketleftbig\ng2\n−(K+Q)g+(K){Σ+−(K+Q)+Σ−+(K+Q)}+g2\n+(K+Q)g−(K){Σ++(K+Q)+Σ−−(K+Q)}/bracketrightbig\n,\nΠc=−/integraldisplay\nK/integraldisplay\nP1\n2Veff(P)/parenleftBigg\nN−+M+−−\niq0+∆/vectork+/vector p+N+−M+−+\niq0+∆/vectork+/vector p+N−+M−++\niq0−∆/vectork+/vector p+N+−M−+−\niq0−∆/vectork+/vector p/parenrightBigg\n,\nMrts= [gr(K+P)−gt(K+P+Q)]/bracketleftBig\n1+scos/parenleftBig\nφ/vectork−φ/vectork+/vector p/parenrightBig/bracketrightBig\n,Nrt=gr(K)gt(K+Q). (9)4\nHere,r,t,s=±label the spin-split bands, φ/vectorkde-\npends on the azimuthal angle θ/vectorkof/vectork(and is equal\ntoθ/vectorkfor linear Rashba SOC),38and the partial\nself-energies are defined as Σ rt(K) =−/integraltext\nPgr(K+\nP)Veff(P)/bracketleftBig\n1+tcos(φ/vectork−φ/vectork+/vector p)/bracketrightBig\n, such that the total self-\nenergies of the Rashba subbands are Σ += Σ+++Σ−−\nand Σ −= Σ+−+ Σ−+. In the last line of Eq. (9), we\nonly used the identity (7). In contrast to the SL case,\nhowever, the spin-orbit splittings in the denominators of\nΠcdepend on /vectork+/vector p, and thus integrationover Pdoesnot,\nin general, produce the self-energies. This already tells\nus that, in general, the imaginary part of δχ33cannotcancel out between the self-energy and vertex diagrams.\nHowever, there are two realistic approximations,\nnamely, of a long-rangeinteraction ( p≪kF) and ofweak\nSOC (∆ /vectork≪EF), within which the momentum depen-\ndence of ∆ /vectork+/vector pcan be neglected. Assuming that these\ntwo conditions are satisfied, the vertex part can again be\nrewritten in terms of the partial self-energies. Even in\nthis limit, however, there is no complete cancellation be-\ntween the self-energy and vertex diagrams. Namely, we\nfindthatΠ a+Πb+ΠccanberewrittenasΠ R+ΠD, where\nΠR=/integraltext\nK∆/vectorkF\nq2\n0+∆2\n/vectorkF/braceleftbig\nΣ+(K)g2\n+(K)−Σ−(K)g2\n−(K)/bracerightbig\nand\nΠD=−/integraldisplay\nK∆/vectorkF\nq2\n0+∆2\n/vectorkF/braceleftBig\n[Σ−−(K+Q)−Σ+−(K)]g−(K)g+(K+Q)−/parenleftBig\n∆/vectorkF→ −∆/vectorkF/parenrightBig/bracerightBig\n. (10)\nThe first term, Π R, has the same structure as in Eq. (8);\nusing the same arguments as before, we conclude that\nΠRdoes not contribute to damping. In contrast, the\nsecond term, Π D, does have, in general, an imaginary\npart at all frequencies, which means damping. To cal-\nculate Π Dexplicitly, one needs to specify the interac-\ntion, which we choose to be in the form of a dynami-\ncally screened Coulomb potential. Deferring the compu-\ntationaldetailstoSec.IVoftheSupplementaryMaterial,\nwe quote here only the final result for χ33; near the res-\nonance at Ω = Ω 3,\nχ−1\n33(Ω) = (2νµ2\nB)−1A/bracketleftbig\nΩ2\n3−(Ω+iΓ/2)2/bracketrightbig\nA=/angbracketleftBigg/parenleftbigg\n∆/vectorkFξ/vectorkF\n∆2\n/vectorkF−Ω2\n3/parenrightbigg2/angbracketrightBigg\nFS/angbracketleftbigg\n∆2\n/vectorkFξ2\n/vectorkF\n∆2\n/vectorkF−Ω2\n3/angbracketrightbigg−2\nFS;\nΓ =ω2\nC\n2EF/parenleftBig∆/vectorkF\nEF/parenrightBig2\n, (11)\nwhereω2\nC=r2\nsE2\nFlnr−1\ns/12π,rs=√\n2e2/vFis the cou-\npling constant of the Coulomb interaction, and ξ2\n/vectorkFis a\ndimensionless form-factor which depends on the details\nof SOC; for isotropic SOC, ξ2\n/vectorkF= 1.\nThe damping rate Γ has an expected FL form. Notice\nthough that the quasiparticle damping rate in 2D scales\nas Γqp∝Ω2lnΩ, as opposed to just Ω2, with a prefactor\nwhich does not depend on rs.35Being a gauge-invariant\nquantity, Γ contains the differences of the single-particle\nself-energies [see Eq. (26)], while Γ qpis related to the\nself-energy itself. The infrared singularity in the self-\nenergy, which gives rise to the lnΩ factor in Γ qp, cancels\nout in Γ. As a result, Γ is on the order of the transport\ndecay rate, which is much smaller than Γ qp. The FL\nnature of the result for Γ indicates that it would not\nchange substantially if, instead of a FL with Coulomb\ninteraction, we would consider a FL of neutral particles\nwith short-range interaction. The only change would beinω2\nCwhich, for the case of a contact interaction with\ncoupling U, should be replaced by ∼(UνEF)2.\nSince the frequencies of chiral-spin modes are propor-\ntionaltothespin-orbitsplitting, onemightbetemptedto\nconcludethat it isbetter tolook forthese modesin mate-\nrials with strong SOC. Our result in Eq. (11) shows that\nthe advantage of strong SOC has its limits. Indeed, the\nratio of the linewidth to the mode frequency, γ≡¯Γ/¯∆,\nscales as C¯∆/EF, where ¯Γ and¯∆ are the appropriate\nangular averages of Γ and ∆ /vectorkF, correspondingly, and C\na dimensionless prefactor. In a material with sufficiently\nstrongeei, one should expect that C∼1. If, in addi-\ntion, SOC is also strong ( ¯∆∼EF), thenγ∼1 and the\nmode is overdamped. We emphasize that this effect is a\nunique feature of SOC; in contrast, the SL mode remains\nundamped (at q= 0) even if the Zeeman energy becomes\ncomparable to the Fermi energy.\nIn a particular model of the screened Coulomb poten-\ntial, the effect of damping appears to be rather weak.\nFor isotropic SOC, Eq. (11) yields C=r2\nslnr−1\ns/12π. Us-\ning parameters for an InGaAs/InAlAs quantum well, we\nthen find γ≈2×10−3¯∆/EFfor an electron number den-\nsity of 1.6×1012cm−2. On the other hand, chiral-spin\nwaves are also damped by disorder via the Dyakonov-\nPerel’ mechanism.8For a mobility of 2 ×105cm2/V·s,\ndamping due to disorder is stronger than that by eeiby\na factor of ten. One should not forget, however, that\nEq. (11) is valid only for rs≪1 and the actual numbers\nmay differ from quoted above as rsincreases. Although\ndamping from disorder appears to be the dominant effect\nin solid-state systems, damping due to interaction should\nbe dominant in (fermionic) cold-atom systems with syn-\nthetic SOC,3,4which have virtually no disorder. In this\ncase, the interaction is short-ranged but, as we have al-\nready mentioned, this should only affect the prefactor in\nEq. (11).\nIn conclusion, we showed that eeiin the presence of5\nSOC not only gives rise to a new type of collective modes\nbut also leads to their damping. This damping occurs\neven atq= 0 and its rate scales as the squareof the spin-\norbit splitting. This effect occurs because neither of the\nthree components of magnetization is a good quantum\nnumber in the presence of SOC.\nWe would like to thank M. Imran and V. Zyuzin for\nuseful discussions. SM is a Dirac Post-DoctoralFellow atthe National High Magnetic Field Laboratory, which is\nsupported by the National Science Foundation via Co-\noperative agreement No. DMR-1157490, the State of\nFlorida, and the U.S. Department of Energy. DLM ac-\nknowledges support from the National Science Founda-\ntion via grant NSF DMR-1308972.\nSupplementary Material\nI. THRESHOLD SINGULARITIES IN EQUATION (4) OF THE MAIN TEXT ( MT) FOR GENERIC\nSPIN-ORBIT COUPLING (SOC)\nFor a generic (but still weak compared to the Fermi energy SOC) th e spin-orbit band splitting, ∆ /vectorkkF, is anisotropic.\nSuppose that ∆ /vectorkkFvaries from ∆ minto ∆maxalong the Fermi surface (FS). Quite generally, the angular depend ence\nof ∆/vectorkkFnear the extremal points is ∆ min≈∆−+β2\n+θ2and ∆ max≈∆+−β2\n−θ2. Near each of these extremal points,\nthe diverging part of the angular integral in Eq. (4) of MT can be writ ten as\n/angbracketleftBigg∆2\n/vectorkF\nΩ2−∆2\n/vectorkF/angbracketrightBigg\nFS≈\n\n1\n2/integraltextdθ\n2π∆+\n(Ω−∆++β2\n+θ2)∝1√\nΩ−∆+,near maximum ,\n1\n2/integraltextdθ\n2π∆+\n(Ω−∆−−β2\n−θ2)∝ −1√\n∆−−Ω,near minimum .\nNote that the angular averageis imaginary when ∆ −<Ω<∆+, which correspond to the continuum of inter-subband\nparticle-holeexcitations. Thesingularityin therealpartofthesus ceptibility atthe lowerend ofthe continuumensures\nthat the collective modes exist even for infinitesimally weak electron- electron interaction ( eei). For an isotropic SOC,\ne.g., either for Rashba or Dresselhaus SOC, ∆ += ∆−≡∆ and the two square-root branch cuts merge into a simple\npole 1/(Ω−∆).\nII. STATIC PART OF THE SELF-ENERGY FOR MOMENTUM-DEPENDENT I NTERACTION\nIn the MT, we argued that the difference of the self-energies of ch iral subbands does not contain the zeroth angular\nharmonic of the interaction potential. In this section, we prove this statement. The self-energy of a subband with\nchirality s=±1 due to eeivia a static but momentum-dependent potential, U|/vector q|, is equal to\nΣs(K) =−1\n2/integraldisplay\nK′/braceleftbig\ng+(K′)+g−(K′)+s[g+(K′)−g−(K′)]cosφ/vectork/vectork′/bracerightbig\nU|/vectork−/vectork′|. (12)\nFor the case of linear Rashba SOC, φ/vectork/vectork′is the angle between /vectorkand/vectork′(≡θ/vectork/vectork′); for a general case, φ/vectork/vectork′is some\nfunction of θ/vectork/vectork′such that/integraltext\ncosφ/vectork/vectork′dθ/vectork/vectork′= 0. We first consider the case of Rashba SOC in some detail and then\npoint out the qualitative features that remain the same for arbitra ry SOC. In addition, we also assume that Rashba\nSOC is weak ( λ≪vF), in which case the spin-orbit splitting can be approximated by its valu e projected on the FS\nin the absence of SOC: ∆ ≈2λkF. The difference of the self-energies of the chiral subbands, δΣs≡Σs−Σ−s, is now\ngiven by\nδΣs(K) =−s/integraldisplay\n/vectork′/bracketleftbigg\nnF/parenleftbigg\nε/vectork′+∆\n2/parenrightbigg\n−nF/parenleftbigg\nε/vectork′−∆\n2/parenrightbigg/bracketrightbigg\ncosθ/vectork/vectork′U|/vectork−/vectork′|, (13)\nwherenF(ǫ) is the Fermi function. Setting T= 0 and expanding the Fermi functions in ∆, we find\nδΣs(K) =sν∆/integraldisplaydθ/vectork/vectork′\n2πcosθ/vectork/vectork′U|/vectork−/vectork′\nF|, (14)\nwhere/vectork′\nF=kF/vectork′//vectork. To find the frequency of the collective mode, one needs to projec t/vectorkon the FS upon which δΣs\nbecomes proportional to the first harmonic of the interaction pot ential.\nA similar argument can be made for an arbitrary (but still weak) SOC. The Green’s function in Eq. (2) of MT can\nbe separated into symmetric (S) and asymmetric (A) parts: G=GS+GA. Any crystalline plane has at least a C26\nsymmetry upon which /vectork→ −/vectork. This guarantees that ∆−/vectork= ∆/vectorkand thus the subband Green’s functions, g±, are\neven on/vectork→ −/vectork. ThenGSis even while GAis odd on this operation. Consider the self-energy in the spin basis (a\n2×2 matrix)\nΣ(K) =−/integraldisplay\nK′U|/vectork−/vectork′|G(K′), (15)\nwhich can be also decomposed into symmetric and asymmetric parts: Σ = ΣS+ΣA. The zeroth angular harmonic of\nthe interaction potential, U{0}, survives only in the part of the integral associated with GSand enters the diagonal\nelements of Σ as −U{0}n, wherenis the total number density. When Σ is transformed to the (diagona l) chiral basis,\nthe−U{0}nterms remain on the diagonal and cancel out in the difference Σ +−Σ−.\nIII. ASLAMAZOV-LARKIN (AL) DIAGRAMS\nIn the presence of a dynamic and long range interaction, the AL diag rams are of the same order as the self energy\nand vertex diagrams (See Refs. 29-32 of MT). In the absence of S OC and for a spin-invariant interaction, however,\nAL diagrams for the spin susceptibility vanish for a trivial reason: Tr [σiGGG]= 0, (i= 1,2,3) because G∝σ0.\nIn the presence of the magnetic field (along the 3-axis), G=aσ0+bσ3. The relevant quantity in this case in the\ntransverse spin susceptibility, which contains the trace Tr[ σiGGG] withi= 1,2. Again, this trace is equal to zero.\nThe vanishing of the AL diagrams in both these cases is a consequenc e of conservation of either total spin (in the\nformer case) or its component along the magnetic field (in the latter case). In the presence of SOC, this reasoning\ndoes not apply because Gnow has off-diagonal elements. However, some general statemen ts about the AL diagrams\ncan still be made for q= 0 case.\n(d)K+Q\n(e)K\nK+P\nK+QK\nK+Q-P\nK1+QK1\nK1+P\u0001i\n\u0000i\n\u0002i\n\u0003iK1+QK1\nK1+PP P-Q\nP-Q\nP\n(d)K+Q\n(e)K\nK+P\nK+QK\nK+Q-P\nK1+QK1\nK1+P\u0004i\n\u0005i\n\u0006i\n\u0007iK1+QK1\nK1+PP P-Q\nP-Q\nP\nFIG. 3: Detailed AL diagrams from Fig. 2 of MT\nFor convenience, we present the AL diagrams from Fig. 2, dandeof MT here in Fig. 3. It suffices to consider only\nthe diagonal components of the spin susceptibility, χii, withi= 1...3, in which case the diagrams read\nΠd\nii=/integraldisplay\nP/integraldisplay\nKTr[σiG(K)G(K+P)G(K+Q)]VPVP−Q/integraldisplay\nK1Tr[σiG(K1+Q)G(K1+P)G(K1)],\nΠe\nii=/integraldisplay\nP/integraldisplay\nKTr[σiG(K+Q)G(K+Q−P)G(K)]VPVP−Q/integraldisplay\nK1Tr[σiG(K1+Q)G(K1+P)G(K1)].\n(16)\nHere,Qhas only the temporal component: Q= (iq0,/vector0). We can define\nhi(P,Q)≡/integraldisplay\nKTr[σiGK+QGK+PGK],\n˜hi(P,Q)≡/integraldisplay\nKTr[σiGKGK+PGK+Q],\nfi(P,Q)≡/integraldisplay\nKTr[σiGK+QGK+Q−PGK]. (17)\nUsing the matrix structure of G(K), we observe that\nh1,2(P,Q) = +˜h1,2(P,Q);h3(P,Q) =−˜h3(P,Q) andfi(P,Q) =−˜h∗\ni(P,Q) fori= 1,2,3. (18)7\nWith the help of these properties, the AL diagrams can now be writte n as\nΠd\nii=/integraldisplay\nPhi(P,Q)˜hi(P,Q)VPVP−Q=/integraldisplay\nPh2\ni(P,Q)VPVP−Qfori= 1,2 and\n=−/integraldisplay\nPh2\ni(P,Q)VPVP−Qfori= 3, (19)\nΠe\nii=/integraldisplay\nPf(P,Q)˜hi(P,Q)VPVP−Q=−/integraldisplay\nP|hi(P,Q)|2VPVQ−P, (20)\nand thus their sum is reduced to\nΠd\nii+Πe\nii=/integraldisplay\nP/parenleftbig\nh2\ni(P,Q)−|hi(P,Q)|2/parenrightbig\nVPVP−Q,fori= 1,2 and\n=−/integraldisplay\nP/parenleftbig\nh2\ni(P,Q)+|hi(P,Q)|2/parenrightbig\nVPVP−Q,fori= 3. (21)\nWe now study two particular cases–that of Rashba SOC and of comb ined Rashba-Dresselhaus SOC– and show that\nthe AL diagrams do not contribute to damping in either of these two c ases.\nRashba SOC : Fori= 3,\nh3(P,Q) =−i\n4/summationdisplay\na,b,c=±b(a−c)/integraldisplay\nKsin(θ/vectork−θ/vectork+/vector p)ga(K+Q)gb(K+P)gc(K). (22)\nSinceQhas no spatial component, we can choose the direction of /vector pas a reference. On reflecting vector /vectorkabout this\ndirection, both θ/vectorkandθ/vectork+/vector pchange signs and so does the factor of sin( ...). On the other hand, since the energy\nspectrum is isotropic, each of the three subband Green’s function in the formula above is a function of the magnitude\nof the corresponding momentum. It is then easy to see that all the three Green’s function are even on reflection\nk1→k1,k2→ −k2. Therefore, the integral over θ/vectorkvanishes, and the AL diagrams give no contribution to χ33.\nThe same reasoning applies for i= 1. In this case,\nh1(P,Q) =1\n4/summationdisplay\na,b,c=±/integraldisplay\nK/bracketleftBig\n(a+c)sinθ/vectork+abcsin(2θ/vectork+θ/vectork+/vector p)+bsinθ/vectork+/vector p/bracketrightBig\nga(K+Q)gb(K+P)gc(K) (23)\ncontains again an angular factor, which is odd on reflection, and a co mbination of the Green’s functions, which is\neven on reflection. Therefore, h1(P,Q) = 0 and there is no contribution to χ11from the AL diagrams either.\nThe same argument cannot be used for i= 2, because the angular factor in\nh2(P,Q) =−1\n4/summationdisplay\na,b,c=±/integraldisplay\nK/bracketleftBig\n(a+c)cosθ/vectork+abccos/parenleftBig\n2θ/vectork+θ/vectork+/vector p/parenrightBig\n+bcosθ/vectork+/vector p)/bracketrightBig\nga(K+Q)gb(K+P)gc(K).(24)\nis even on reflection. Nevertheless, in-plane rotational symmetry of the Rashba Hamiltonian ensures that χ11=χ22.\nTherefore, one does not need to consider χ22separately: if there is no damping of resonance associated with χ11, the\nsame is true for χ22.\nCombined Rashba and Dresselhaus SOC : In the presence of both Rashba and Dresselhaus SOC, the only ch ange in\nEqs. ((22-24)) is that θ/vectorkneeds to replaced by φ/vectork, as defined after Eq. (12). It is convenient to choose the Cartes ian\nsystem rotated by π/4 compared to the conventional one. In such a system, the combin ed Rashba plus Dresselhaus\nHamiltonian is described by in Eq. (1) of MT with λ/vectorf(k) = ([β+α]k2,[β−α]k1,0) and\ncosφ/vectork=−(α−β)cosθ/vectork\nΛ/vectork,\nsinφ/vectork=−(β+α)sinθ/vectork\nΛ/vectork, (25)\nwhere Λ /vectork=/radicalbigα2+β2−2αβcos2θ/vectork. Note that φ/vectork= tan−1/parenleftBig\nα+β\nα−βtanθ/vectork/parenrightBig\n. The spin-orbit splitting ∆ /vectorkF= 2kFΛ/vectork\nis even on reflection about the k1-axis. As discussed in MT one can approximate ∆ /vectork+/vector pby ∆/vectorkFfor a long-range\ninteraction; therefore, the products of the Green’s functions a re still even on reflection. Because of the relation\nbetween φ/vectorkandθ/vectork, the same symmetries that led to the vanishing of h1andh3in the case with only Rashba SOC\nensure that h1andh3vanish is the case when both Rashba and Dresselhaus SOC are prese nt as well.8\nIV. DERIVATION OF EQUATION (11) OF THE MAIN TEXT\nIn this section, we present the evaluation of Π Dfrom Eq. (10) of the MT. For convenience, we present this equatio n\nbelow in an expanded form:\nΠD=−/integraldisplay\nK∆/vectorkF\nq2\n0+∆2\n/vectorkF{[Σ−−(K+Q)−Σ+−(K)]g−(K)g+(K+Q)−[Σ+−(K+Q)−Σ−−(K)]g+(K)g−(K+Q)},\n(26)\nwhere Σ ab(K) =−/integraltext\nPga(K+P)Veff(P)/bracketleftBig\n1+bcos(φ/vectork−φ/vectork+/vector p)/bracketrightBig\nand the angle φ/vectorkis defined after Eq. (12). Since Π D\nis already proportional to ∆ /vectorkF, we can neglect the effect of SOC on the interaction potential and t ake it to be a\ndynamically screened Coulomb potential in 2D:\nVeff(P) =2πe2\np+κ/parenleftbigg\n1−|p0|√\np2\n0+v2\nFp2/parenrightbigg, (27)\nwhereκ=√\n2rskFis the inverse Thomas-Fermi screening length and rs=√\n2e2/vF. The momentum transfer /vector p\nis decomposed into two components: along the normal to the FS at p oint/vectork(p||) and along the tangent ( p⊥). We\nassume and then verify that the quasiparticle interaction is determ ined by processes with typical p||on the order\nofp0/vF, whereas typical p⊥are on the order of κ. In turn, typical p0are on the order of the external frequency\nwhich, in our case, is fixed by spin-orbit splitting, ¯∆, averaged over the FS. We choose to work in a realistic regime\nof¯∆≪vFκ≪EF. [Although the form of the potential in Eq. (27) is, strictly speaking , valid for rs≪1 which\nimplies that κ≪kF, these strong inequalities are never satisfied in real materials. The refore, the interval of energies\nin between vFκandEFis never wide enough to consider a possibility of ¯∆ being within this interval.] Since\n|p||| ∼ |p0|/vF≪ |p⊥| (28)\ninthisregime,wecansimplifytheLandau-dampingterminEq.(27)to |p0|/vF|p⊥|andalsoexpandthedenominatorof\nthis equation in this parameter. Subtracting off the static screene d potential which gives no contribution to damping,\nwe obtain for the dynamical ( p0-dependent) part of the interaction\nUdyn\neff(P) =/parenleftbigg2πe2\n|p⊥|+κ/parenrightbigg2ν\nvF|p0|\n|p⊥|. (29)\nThanks to the inequality (28), the integrals over the fermionic ( K) and bosonic ( P) momenta become separable. Let\nus pick one of the terms in Eq. (26):\n/integraldisplay\nKΣ−−(K+Q)g−(K)g+(K+Q) =−/integraldisplay\nK/integraldisplay\nPUdyn\neffg−(K+Q+P)g−(K)g+(K+Q)/bracketleftBig\n1−cos/parenleftBig\nφ/vectork−φ/vectork+/vector p/parenrightBig/bracketrightBig\n.(30)\nSince it is expected that typical |p⊥| ∼κ≪kFwhile/vectorkis expected to be near the FS, the angular factor can be\nexpanded as 1 −cos(φ/vectork−φ/vectork+/vector p)≈ξ2\n/vectorkp2\n⊥\n2k2\nF, whereξ/vectorkis a form-factor that depends on the details of the SOC.\nFor example, in the case of linear Rashba SOC, φ/vectorkcoincides with the angle θ/vectorkthat/vectorkmakes with, e.g., the 1-axis.\nAlso, due to rotational symmetry, the result of the integral over Pdoes not depend on the direction of /vectorkand thus we\ncan choose θ/vectork= 0 in 1−cos(θ/vectork−θ/vectork+/vector p). An expansion of 1 −cos(θ/vectork+/vector p) then yieldsp2\n⊥\n2k2\nF, which means that ξ/vectork= 1 for\na system with linear Rashba SOC. Notice that the condition |p||| ≪ |p⊥|implies that /vector pis almost perpendicular to /vectork,\ni.e., that |θ/vector p−θ/vectork| ≈π/2.\nIn the presence of both Rashba and Dresselhaus SOC, φ/vectorkis given by Eq. (25). Expanding in pand using that\n|θ/vector p−θ/vectork| ≈π/2, we get\n1−cos(φ/vectork−φ/vectork+/vector p)≈p2\n⊥\n2k2\nF/bracketleftBigg\n1+/parenleftbiggγ\n1−γcos2θ/vectork/parenrightbigg2/bracketrightBigg\n,whereγ=2αβ\nα2+β2<1 (31)\nThusξ2\n/vectork= 1+/parenleftBig\nγ\n1−γcos2θ/vectork/parenrightBig2\nfor this system. We avoid the special case of |α|=|β|, whenγ= 1 and ξ/vectorkhas a pole: it\ncan be shown that collective modes are completely covered by the co ntinuum in this case.?9\nNext, we write g−(K)g+(K+Q) as [g−(K)−g+(K+Q)]/(iq0−∆/vectork) and integrate the products of two Green’s\nfunctions over ε/vectork,k0andp||; for one of the products, we obtain\n/integraldisplaydp/bardbl\n2π/integraldisplaydk0\n2π/integraldisplay\ndε/vectorkga(K+P)gb(K) =/integraldisplaydp/bardbl\n2πip0\nip0−vFp/bardbl−a−b\n2∆/vectork=|p0|\n2vF. (32)\nThe rest of the integrals are evaluated in the similar manner. Collectin g all terms, we arrive at\nΠD=−/angbracketleftBigg/parenleftBigg\n∆/vectorkFξ/vectorkF\nq2\n0+∆2\n/vectorkF/parenrightBigg2/angbracketrightBigg\n/vectork2ν\nvFk2\nF/integraldisplaykF\n0dp⊥\n2πp2\n⊥/integraldisplay∞\n0dp0\n2πUdyn\neff(P){|q0+p0|+|q0−p0|−2|p0|}. (33)\nThe factor in {...}constraints the range of integration over p0to (0,q0). This leads to\nΠD=−ν/angbracketleftBigg/parenleftBigg\n∆/vectorkξ/vectorkF\nq2\n0+∆2\n/vectork/parenrightBigg2/angbracketrightBigg\n/vectorkr2\ns\n2πEF/integraldisplayq0\n0dp0(q0−p0)p0/integraldisplaykF\n0dp⊥p⊥\n(p⊥+κ)2(34)\nTo logarithmic accuracy, the integral over p⊥gives lnkF/κ≈ −lnrswhile the integral over p0givesq3\n0/6. Note that\ntypicalp0∼q0while typical p⊥∼κ(in the logarithmic sense), and thus our initial assumption is verified.\nAfter analytic continuation, we get\nΠD=−iν/angbracketleftBigg/parenleftBigg\n∆/vectorkFξkF\nΩ2−∆2\n/vectorkF/parenrightBigg2/angbracketrightBigg\n/vectorkFω2\nC/parenleftbiggΩ\nEF/parenrightbigg3\n, (35)\nwhereω2\nC=r2\nsE2\nFlnr−1\ns/12πandrs=√\n2e2/vF. This is the correction to Π0\n33that is responsible for damping.\nRecalling that the physical susceptibility is given by χ33=−µ2\nBΠ33/(1 +Ux\n2Π33) Π33differs from Π0\n33due to the\ndamping correction. To account for this correction, we may write\nΠ33= 2ν/angbracketleftBigg∆2\n/vectorkFξ2\n/vectorkF\n∆2\n/vectorkF−(Ω+iΓ/2)2/angbracketrightBigg\nFS, (36)\nwhere Γ =ω2\nC\n2EF/parenleftBig∆/vectorkF\nEF/parenrightBig2\n. Using the fact that Ω 3is the pole in χ33when Γ = 0, we can eliminate the Uxdependence.\nWriting ∆2\n/vectorkF−(Ω+iΓ/2)2as ∆2\n/vectorkF−Ω2\n3+Ω2\n3−(Ω+iΓ/2)2and expanding in Ω2\n3−(Ω+iΓ/2)2near the resonance,\nwe obtain Eq. (11) of MT.\n1D. Awschalom and N. Samarth, Physics 2, 50 (2009).\n2W. P. McCray, Nature Nanotech. 4, 2 (2009).\n3P. Wang, Z-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai,\nH. Zhai, and J. Zhang, Phys. Rev. Lett., 109, 095301\n(2012); Lawrence W. Cheuk, Ariel T. Sommer, Z. Hadz-\nibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys.\nRev. Lett., 109, 095302 (2012).\n4J. Quintanilla, S. T. Carr, and J. J. Betouras, Phys. Rev.\nA79, 031601 (2009); B. M. Fregoso and E. Fradkin, Phys.\nRev. Lett. 103, 205301 (2009); Y. Li and C. Wu, Phys.\nRev. 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Pines, The Theory of Quantum Liquids ,(New York: Benjamin), 1966.\n27P. S. Kondratenko, JETP 19, 972 (1964); JETP 20, 1032\n(1965).\n28See supplementary material (SM), section I.\n29G.H.ChenandM.E.Raikh, Phys.Rev.B 60, 4826(1999).\n30See SM, Sec. II.\n31M. Yu. Reizer and V. M. Vinokur, Phys. Rev. B 62,\nR16306 (2000).\n32I. V. Gornyi and A. D. Mirlin, Phys. Rev. B 69, 045313\n(2004).\n33J. Rech, C. P´ epin, and A. V. Chubukov, Phys. Rev. B 74,\n195126 (2006).\n34A. V. Chubukov and D. L. Maslov, Phys. Rev. Lett. 103,\n216401 (2009).\n35See B. N. Narozhny, G. Zala, and I. L. Aleiner, Phys. Rev.\nB65, 180202(R) (2002) and references therein.\n36R. A.˙Zak, D. L. Maslov, and D. Loss, Phys. Rev. 85,\n115424 (2012).\n37See SM, Sec. III.\n38See SM, Sec. IV." }, { "title": "0910.5184v1.Rabi_type_oscillations_in_damped_single_2D_quantum_dot.pdf", "content": "arXiv:0910.5184v1 [cond-mat.mes-hall] 27 Oct 2009Rabi type oscillations in damped single 2D-quantum dot\nMadhuri Mukhopadhyaya, Ram Kuntal Hazraa, Manas\nGhoshb, S. Mukherjeea, S.P. Bhattacharyyaa∗† ‡\naDepartment of Physical Chemistry,\nIndian Association for the Cultivation of Science Jadavpur , Kolkata, India\nbDepartment of Chemistry, Physical Chemistry section,\nVisva Bharati University Santiniketan, Birbhum, India\nAbstract\nWe present a quantized model of harmonically confined dot ato m with inherent damping in the\npresence of a transverse magnetic field. The model leads to a n on hermitian Hamiltonian in real\ncoordinate. We have analytically studied the effects that dam pinghas on the Rabitype oscillations\nof the system. The model explains the decoherence of Rabi osc illation in a Josephson Junction.\nPACS numbers: 78.67.-n, 78.67.Hc, 03.65.Yz\nkeywords : damped quantum dot, quantization of damping, Rabi oscilla tion.\n∗e-mail address: pcspb@iacs.res.in\n†Phone : (91)(33)2473 4971, 3372, 3073, 5374\n‡Fax : (91)(33)2473 2805\n1I. INTRODUCTION\nRabi oscillation [1] is one of the fundamental observations in light mat ter interaction that\noccurs coherently and nonlinearly [2] and which has no classical analo gue. The generation of\ncoherent superposition of quantum states using ultra short laser pulses and the subsequent\ndecoherence due to some inherent damping or interaction with the e nvironment is of great\ninterest especially in semiconductor quantum dots due to the prosp ect of future applications\n[3, 4, 5] in quantum information processing and making novel laser de vices [6]. Rabi oscil-\nlations using excitons in single quantum dots [7, 8, 9, 10] have been st udied successfully by\ndifferent groups in the past few years [11, 12, 13, 14]. Control of t he decoherence of Rabi\noscillation in quantum dot, the mechanism of which is still a matter of inv estigation, has\nattracted wide attention [15, 16]. There has been rapid progress in experimental control\nof dephasing of coherent states in quantum dots [17]. On the cont rary, theoretical studies\non quantum dots leading to the dynamics in the presence of damping a re scarce. Till date\ntheoretical studies in dots have been done mainly on the basis of dam ping that has been in-\ntroduced phenomenologically. To the best of our knowledge, no qua ntum theoretical model\nof the dot has been developed with the inherent damping incorporat ed in the model.\nWe develop a model of a damped quantum dot going beyond the pheno menological de-\nscription used so far. The model, we believe provides some insight into the Rabi dynamics.\nOur analytical results lead to an understanding of the experimenta l observation of decoher-\nence in Josepshon Junction [18].\nII. MODEL\nWe show, in what follows, that the damped one electron dot can be de scribed by the\neigenstates of a quantum Hamiltonian H that is non hermitian. The art ificial atom that we\nhave modeled is composed of a single electron confined in 2-D by harmo nic potential with\nsomeinherent dampingandahomogeneousmagneticfieldappliednorm altotheconfinement\nplane. Let us start with the classical equation of motion of the damp ed harmonic oscillator\nwhich reads\nme¨− →r+γ˙− →r+k− →r= 0 (1)\n2where k is the harmonic force constant and γis the damping constant and meis oscillator\nmass. The system described by equation (1) is known to have a time d ependent Lagrangian\nandHamiltonian[19, 20, 21]. Therehave beenmany attemptsto quan tize thedampedlinear\noscillator [22, 23, 24] but a completely satisfactory solution been elu sive. The stumbling\nblock has been the lack of a time independent Hamiltonian formalism. Re cently, however\nsuch a formalism has been proposed making a definite progress [25, 2 6]. We proposed a\ndifferent strategy that brings a non hermitian Hamiltonian formalism. From equation (1)\nwe start by noting that it isimmediately possible towrite down the Euler -Lagrangeequation\nfor the dissipative system by defining a velocity dependent force Fgenand setting\nd\ndt/parenleftBigg∂L\n∂˙r/parenrightBigg\n−∂L\n∂r=Fgen (2)\nwhereFgenis defined as the negative derivative of Rayleigh dissipative function′f′with\nrespect to ˙r[19].\nFgen=−∂\n∂˙r(f) (3)\nfis determined by the damping constant γand the velocity (˙ r) as follows:\nf=1\n2γ˙r2(4)\nEquations (3) and (4) suggest that the time dependent damping fo rce (Fd) is linearly related\nto the velocity:\nFd=−γ˙− →r (5)\nWith equation (3) the Euler Lagrange equation (2) now reads\nd\ndt/parenleftBigg∂L\n∂˙r/parenrightBigg\n−∂L\n∂r+∂f\n∂˙r= 0 (6)\nEquation (6) requires that the Lagrangian L is chosen as\nL=1\n2me˙r2+γr˙r−1\n2kr2(7)\nClearly the Lagrangian of equation (7) is consistent with the equatio n of motion of the\ndamped harmonic oscillator equation (1). Since the momentum p=∂L\n∂˙rthe modified mo-\nmentum for the damped harmonic oscillator becomes\np=me˙− →r+γ− →r (8)\n3Let the damped oscillator have a charge ’q’ and let it experience an ele ctric field (E) and a\ntransverse magnetic field (B). The Lorentz force acting on it is\nF=q/bracketleftbigg\nE+1\nc(v×B)/bracketrightbigg\n=q/bracketleftBigg\n−∇φ−1\nc/parenleftBigg∂A\n∂t/parenrightBigg\n+1\nc(v×B)/bracketrightBigg\n(9)\nwhereE=−∇φ(φ=scalar potential) and B=∇×A(A= Vector potential).\nThe electric and magnetic fields bring in additional terms in the Lagran gian (L=¯L, say)\nwhere\n¯L=1\n2me˙r2+γr˙r−qφ+q\nc− →A˙− →r (10)\n, whereqφ=1\n2kr2, scalar potential. The modified momentum (¯ p) for the system (described\nby¯L)\n¯p=me˙− →r+γ− →r+q\nc− →A. (11)\nThe modified momentum ¯ pleads to the Hamiltonian ( ¯H) of the system represented by a\nsingle carrier electron in a damped quantum dot as follows:\n¯H=1\n2me/bracketleftbigg\n(me˙− →r+γ− →r+q\nc− →A)·(me˙− →r+γ− →r+q\nc− →A)/bracketrightbigg\n+qφ (12)\nTaking the cyclotron frequency ωc=qB\nmec,the confinement potential qφ=1\n2meω2\n0(x2+y2)\nand replacing the classical operators by their respective quantum analogues, the quantum\nmechanical Hamiltonian of the system in Cartesian coordinates beco mes\n¯H=−¯h2\n2me/parenleftBigg∂2\n∂x2+∂2\n∂y2/parenrightBigg\n−i¯hγ\nme/parenleftBigg\n1+x∂\n∂x+y∂\n∂y/parenrightBigg\n−i¯hωc\n2/parenleftBigg\n−y∂\n∂x+x∂\n∂y/parenrightBigg\n+γ2\n2me(x2+y2)+me\n8ω2\nc(x2+y2)+1\n2meω2\n0(x2+y2). (13)\nTransforming from Cartesian to polar coordinates the Hamiltonian c hanges to\nH=−¯h2\n2me/parenleftBigg∂2\n∂r2+1\nr∂\n∂r+1\nr2∂2\n∂φ2/parenrightBigg\n−i¯hγ\nme/parenleftBigg\n1+r∂\n∂r/parenrightBigg\n−i¯hωc\n2/parenleftBigg∂\n∂φ/parenrightBigg\n+Ω2\ndr2(14)\nwhere Ω2\nd=1\n2me/bracketleftBigω2\nc\n4+γ2\nm2e+ω2\n0/bracketrightBig\n.\nH is manifestly non-hermitian. H may be thought of as defining a set of eigenstates ψn,l(r,φ)\nwith complex energy En,lif we assume that H obeys the energy eigenvalue equation\nHψn,l(r,φ) =En,lψn,l(r,φ) (15)\n4A straight forward series solution of equation (15) (Appendix-A) le ads to the quantized\nenergy eigenvalues of the damped dot:\nEn,l=ωcl\n2+(2n+l+1)Ω−iγ(2n+l+1)Ω (16)\nwhere ‘n’ and ‘l�� are principal, and angular momentum quantum numbers, respectiv ely and\nΩ2=/bracketleftBigω2\nc\n4+γ2\nm2e+ω2\n0/bracketrightBig\n.\nThe energy is clearly complex and the imaginary part of it is related to t he dissipating\nenergy which is given by\nΓn,l=−γ(2n+l+1)Ω (17)\nThus, starting from the classical equation of motion of the damped harmonic oscillator\nquantization has been carried out through a Lagrange-Hamiltonian formalism, where the\nHamiltonianisnon-Hermitian[27, 28] asexpected fora nonconserva tive system [29, 30]. We\nhave described the system in terms of real positional coordinates in contrast with attempts\nto handle the problem in terms of complex coordinate [27].\nThus, proceeding with the assumption that the system described b y the non-Hermitian\nHamiltonian of equation (13) satisfies time-independent Schr¨ odinger equation Hψ=Eψ\n[27], we have obtained all the quasi energy eigenstates.\nψn,l(r,φ) =C\n2√πe−Ω2r2\n2rlL|l|\nn (18)\nwhereL|l|\nnis the Laguerre series and C is the normalization constant. In the ab sence of\ndamping these states merge into Fock-Darwin energy spectrum [31 , 32], while the presence\nof damping makes the energy levels quasi stationary. The importan t outcome is that for a\nknownω0andωccomparison of the energy separation between two states as obse rved from\nexperiment and obtained from the expressions with and without dam ping can lead to the\nrealization of the intrinsic damping coefficient of a dot system. For da mped dot system the\nenergy states are shifted from the energy levels without damping a nd the shifts are more\npronounced for stronger damping whereas for greater effective mass of the carrier electron\nthe effect of damping is somewhat quenched. Since the non-hermitia n Hamiltonian obtained\nfor the damped dot has complex eigenvalues that correlate with the energy eigenvalues of\nthe dot in the limit of zero damping, it could be interesting to investigat e the dynamics of\nthe damped dot in response to perturbation by laser light.\n5III. DYNAMICS OF DAMPED QUANTUM DOT:\nLet us consider the time-dependent Schr¨ odinger equation for the complex energy eigen\nstates of H;\ni¯h∂Ψn,l(r,t)\n∂t= (ER−iΓ)n,lΨn,l(r,t), (19)\nThe corresponding wave function is decaying and the probability P(r ,t) is proportional\nto|ψn,l(r,0)|2e−2Γt\n¯h. The exponential function accounts for the exponential fall-off o f the\namplitude with time, the first factor being the the amplitude of the init ial state which is\nnow damped. The intrinsic life time τn,l=1\nΓn,lof these quasi-stationary states are therefore\ndetermined by the damping coefficient and the quantum number char acterizing the states.\nWe now consider the two energy levels ( gande) of the damped quantum dot system, the\ntwo states aredesignated as ψgandψeareassumed to bewell separated fromall otherstates.\nThe system interacts with a laser of frequency ωLandωa≡ωegis the resonance frequency\n(Fig.1). The effect of perturbationproduced by the laser can betr eated semiclassically using\nthe eigenfunctions of the damped dot Ψ n,las zeroth order wave function. The perturbed\nHamiltonian is partitioned into H0and V, where the unperturbed dot Hamiltonian H0\nis given by the equation (13) and the perturbation in the dipole appro ximation:− →V=\ne.rE0cos(ωLt).\nWemaynowconsider thesemiclassical perturbationtreatmentbas edonthedampedwave\nfunctions of the damped dot already obtained. The time-dependen t Schrodinger equation\nfor the perturbed system is\n|˙Ψ(r,t)/an}b∇acket∇i}ht=−i\n¯hH(r,t)|Ψ(r,t)/an}b∇acket∇i}ht (20)\nwhile the solution is (k=e,g)\nΨ(r,t) =/summationdisplay\nkCk(t)ψk(r)e−iωR\nkte−γωI\nkt(21)\nProjecting on to the states |e/an}b∇acket∇i}htand|g/an}b∇acket∇i}htand integrating over spatial coordinates in each case\nwe arrive at the equations governing the time development of the am plitudes (Cg, andCe);\ni˙Cg=Cg(t)(ωR\ng−iγωI\ng)+Ce(t)ΩR− →V(t)e−γωI\nate−iωR\nat(22)\ni˙Ce=Ce(t)(ωR\ne−iγωI\ne)+Cg(t)ΩR− →V(t)eγωI\nate−iωR\nat(23)\nwhere the Rabi frequency is defined as Ω R=eE0\n¯h/an}b∇acketle{te|r|g/an}b∇acket∇i}htand the dipole approximation has\nbeen used.\n6Introducing the transformations ˜Cg= [Cge(iωR\ng+γωI\ng)t] and˜Ce= [Cee(iωR\ne+γωI\ne)t]\ni˜Cg=˜CeΩR− →E(t)e(−iωR\na−γωI\na)t(24)\ni˜Ce=˜CgΩR− →E(t)e(iωR\na+γωI\na)t(25)\n− →E(t) =eiωLt+e−iωLt\n2;ωL+ωa=ω+;ωL−ωa=δdetuning frequency\nInvoking the rotating wave approximation we get\ni˙˜Cg=ΩR\n2Cee−γωI\nateiδt(26)\ni˙˜Ce=ΩR\n2CgeγωI\nate−iδt(27)\nEquation (26) and (27) can be uncoupled by the standard route, le ading to\n¨˜Cg−(γωI\na−iδ)˙˜Cg+Ω2\nR\n4˜Cg= 0 (28)\n¨˜Ce+(γωI\na−iδ)˙˜Ce+Ω2\nR\n4˜Ce= 0 (29)\nTaking the initial condition that Cg(0) = 1 ;Ce(0) = 0 we get the solutions\n˜Cg=e−∆\n2t/bracketleftbigg\ncosΩRdt\n2+i∆\nΩRdsinΩRdt\n2/bracketrightbigg\n(30)\n˜Ce=e∆\n2t/bracketleftbigg\niΩR\nΩRdsinΩRdt\n2/bracketrightbigg\n(31)\nwhere ∆ = γωI\na−iδand Ω Rd=/radicalBig\nΩ2\nR−(γωIa−iδ)2.\nHence,\nCe=e−γ(ωIg+ωIe\n2)te−iδ\n2te−iωR\net/bracketleftbigg\niΩR\nΩRdsinΩRdt\n2/bracketrightbigg\n(32)\nHence, the excited state population Pe=|Ce|2is given by\n|Ce|2=e−γ(ωI\ng+ωI\ne)t/bracketleftBiggΩR2\nΩRd2sin2ΩRdt\n2/bracketrightBigg\n(33)\nThe result shows that the contribution of a given state to the evolv ing wave function (Ψ) of\nthe system at a particular time is given in terms of the damping coefficie nt and the sum of\ntheenergies ofthetwo levels coupled by laser light. The coherent te mporal oscillations ofthe\npopulation in the excited state obtained above matches with the exp erimental observations\nmade by Yu et al [18] in Jopsepshon phase qubit. The observed oscilla tory behaviour of the\ndecaying amplitude reported by them is successfully explained by our model based on the\n7description of a damped quantum dot by a non-hermitian Hamiltonian in real Coordinates.\nTheprobabilityofbeinginthestate’k’(eorg)atanygiventimeisther eforegivenby Pk(t) =\n|Ck(t)|2. For the excited state ’e’ Figure2 shows thenature of the time dep endence ofPe. As\nexpected it is coherently oscillatory and exponentially damped.We not e that the equation\n33 was earlier developed by Yu et al [18] as the asymtotic limit of solutio n of the appropriate\nLioville equation for the density operator under the rotating wave a pproximation, and used\nto interpret their experimental observation. We have arrived at t he same results based on\nthe non-hermitian Hamiltonian.\nIV. CONCLUSION\nIn summary the proposed model describes correctly the effects o f inherent damping in\na quantum dot. The amount of dissipating energy in a particular stat e in a quantum dot\nis naturally related to the damping coefficient. The decoherence of R abi oscillations shows\nthat the rate of decoherence is exponentially related not only to th e damping coefficient but\nalso to the energy separation between the two levels. Again one inte resting point is that the\ninherent life times of all the different states is predictable assuming t hat the life time of any\none particular state are known from experiment. We also note that the temporal coherent\noscillation of population in Josephson junction is correctly explained b y the present model.\nV. APPENDIX\nIn atomic units the Hamiltonian of equation (14) reads\nH=−1\n2me/parenleftBigg∂2\n∂r2+1\nr∂\n∂r+1\nr2∂2\n∂φ2/parenrightBigg\n−iγ\nme/parenleftBigg\n1+r∂\n∂r/parenrightBigg\n−iωc\n2/parenleftBigg∂\n∂φ/parenrightBigg\n+Ω2\ndr2(34)\nwhere Ω2\nd=1\n2me/bracketleftBigω2\nc\n4+γ2\nm2e+ω2\n0/bracketrightBig\n.\nSubstituting, f(r,φ) =e(ilφ)Ψ(r)√\n2πand multiplying both sides by√\n2πe−ilφr1\n2leads to radial\nSchrodinger equation,\n−1\n2me/bracketleftBiggd2\ndr2+1\n4r2−l2\nr2+Ω2\ndr2/bracketrightBigg\nf(r)+lwc\n2−iγ\nme/braceleftbigg\nf(r)+rf′(r)−1\n2f(r)/bracerightbigg\n=Ef(r)(35)\nor\n/bracketleftBiggd2\ndr2+(1\n4−l2)1\nr2−Ω′2\ndr2−meωcl+2iγ(1\n2+r∂\n∂r)+2Eme/bracketrightBigg\nf(r) = 0 (36)\n8Where Ω′2\nd=m2\ne(ω2\nc\n4+γ2\nm2e+ω2\n0)\nSubstituting r=x√\nΩ′\ndthe radial function f(r) changes to g(x).\n/bracketleftBiggd2\ndx2+/parenleftbigg1\n4−l2/parenrightbigg1\nx2−me(ωcl−2E)\nΩ′\nd+2iγ\nΩ′\nd/parenleftBigg1\n2+xd\ndx/parenrightBigg\n+2Eme/bracketrightBigg\ng(x) = 0 (37)\nAsymptotic analysis leads to\ng(x) =g0(x)V(x)g∞(x) ;g0(x) =e−x2/2,g∞(x) =x1\n2+|l|\ng(x) =e−x2/2V(x)x1\n2+|l|.\nWhereV(x) =/summationtext\njbjxjsatisfies laguure series. Again taking z=x2the function V(x)\nchanges to function q(z), such that q(z) =/summationtext\nkakzksatisfy the Laguure series.\n[zd2\ndz2+(l+1−z)d\ndz−{l+1\n2+me(ωcl−2E)\n4Ω′\nd}\n+2iγ\n4(1++2zd\ndz+l−z)]q(z) = 0 (38)\nwhere E is complex (= ER−iΓ). The Lageurre series satisfies equation (36). Accordingly\nthe total wave function reads\nψn,l(r,φ) =C\n2√πeiφe−Ω2\ndr2\n2rln/summationdisplay\n0bnr2n(39)\nVI. ACKNOWLEDGEMENT\nM. Mukhopadhyay would like to thank the CSIR of Goverment of India , New Delhi for\nthe award senior research fellowship. S.P. Bhattacharyya thanks the DST for a generous\nresearch grant.\n[1] I. I. Rabi, Phys. Rev. 51 (1937) 652.\n[2] M. O. Scully, M. S. Zubairy, Quantum Optics. Cambridge Un iversity Press, Cambridge,1997.\n[3] Yu. A. Pashkin T. Yamamoto O. Astafiev, Y. Nakamura D. V. Av erin, J. S. Tsai, Nature. 421\n(2003) 823.\n[4] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huan g, J. Majer, S. Kumar, S. M.\nGirvin, R. J. Schoelkofpt, Nature. 431 (2004) 162.\n9[5] D. Vion A. Assime, A.Cottet, P.joyez H. Pothier, C. Urbin a, D. Esteve, M. H. Dovert, Science.\n296 (2002) 886.\n[6] D. O’Brien, S. P. Hegarty, G. Huyet, A. V. Uskov, Optics Le tters. 29 (2007) 1072.\n[7] M. Ghosh, R. K. Hazra, S. P. Bhattacharyya, J. Theo. Comp. Chem. 5 (2006) 25.\n[8] V. S. C. M. Rao, S. Hughes, Phys. Rev. Lett. 99 (2007) 19390 1.\n[9] R. K. Hazra, M. Ghosh, S. P. Bhattacharyya, Chem. Phys. 33 3 (2007) 18.\n[10] D. P. Fussell, M. M. Dignam, Phys. Rev. A 76 (2007) 053801 .\n[11] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer , M.H. Devoret, S. M. Girvin, R. J.\nSchoelkopf, Phys. Rev. Lett. 95 (2005) 060501.\n[12] J. M. Martinis, S. Nam, J. Aumentado, Phys. Rev. Lett. 89 (2002) 117901.\n[13] A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. W undke, H.M. Gibbs, G. Khitrova,\nN. Peyghambarian. Phys. Rev. Lett. 82 (1999) 2346.\n[14] T. H. Stievater, X. Li, D. Gammon, D. Park, C. Piermarocc hi, L. J. Sham, Phys. Rev. Lett.\n87 (2001) 13363.\n[15] A. Zrenner, E.Beham, S.Stufler, F.Findeis, M.Biehler, G.Abstreiter, Nature. 418 (2002) 612.\n[16] N. Kosugi, S. Matsuo, K. Konno, N. Hatakenaka, Phys. Rev . B. 72 (2005) 172509.\n[17] J. M. Villas-Boas, S. E. Ulloa, A. O. Govorov, Phys. Rev. Lett. 94 (2005) 057404.\n[18] Y. Yu, S. Han, Xi. Chu, S. I. chu, Z. Wang, Science. 296 (20 02) 889.\n[19] H. Goldstein, Classical Mechanics. 2nd ed. Narosa Publ ishing House, New Delhi, 2001.\n[20] J.R. Ray, Am. J. Phys. 47 (1979) 626.\n[21] F. Riewe, Phys. Rev. E. 53 (1996) 1890.\n[22] H. Dekker, Phys. Rev. A. 16 (1977) 2126.\n[23] H. Dekker, Phys.Rep. 80 (1981) 1.\n[24] E. G. Harris, Phys. Rev. A. 42 (1990) 3685.\n[25] D. C. Latimer, J. Phys. A: Math. Gen. 38 (2005) 2021.\n[26] V. K. Chandrasekar, M. Stenthilvelan, M. Lakshmanan, J . Math. Phys. 48 (2007) 032701.\n[27] H. Dekker, Z. Physik B. 21 (1975) 295.\n[28] M.S. Wartak, J. Phys. A. Math. Gen. 22 (1989) 361.\n[29] M. Razavy, Z. Physik B. 26 (1977) 201.\n[30] S.Pal, Phys. Rev. A. 39 (1989) 3825.\n[31] V. Fock, Phys Z. 47 (1928) 446.\n10[32] C. G. Darwin, Proc. Camb. Philos. Soc. 27 (1930) 86.\n11FIG. 1: Energy level diagram for two-level system showing de cay rates for ground and excited\nstates Γ g, Γerespectively. The excited and ground state frequency differe nce is denoted by ωeg.\nThe Rabi oscillation Ω Ris introduced by an external laser frequency ωL.\nFIG. 2: A plot of total population of the excited states versu s real time (in a.u) with me=1 a.u.,\nωc= 10−3a.u.,ω0=0.0141 a.u., Ω R= 0.01 a.u., ( ωg+ωe) = 0.04 a.u. and γ= 0.0001 a.u.\n12" }, { "title": "1906.10326v2.Conductivity_Like_Gilbert_Damping_due_to_Intraband_Scattering_in_Epitaxial_Iron.pdf", "content": "1 \n Conductivity -Like Gilbert Damping due to Intraband Scattering in Epitaxial Iron \n Behrouz Khodadadi1, Anish Rai2,3, Arjun Sapkota2,3, Abhishek Srivastava2,3, Bhuwan Nepal2,3, \nYoungmin Lim1, David A. Smith1, Claudia Mewes2,3, Sujan Budhathoki2,3, Adam J. Hauser2,3, \nMin Gao4, Jie-Fang Li4, Dwight D. Viehland4, Zijian Jiang1, Jean J. Heremans1, Prasanna V. \nBalachandran5,6, Tim Mewes2,3, Satoru Emori1* \n1 Department of Physics, Virginia Tech , VA 24061, U.S.A \n2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA \n 3 Center for Materials for Information Technology (MINT), University of Alabama, Tuscaloosa, \nAL 35487, U .S.A. \n4 Department of Material Science and Engineering, Virginia Tech , \n Blacksburg, VA 24061, U.S.A . \n5 Department of Material Science and Engineering, University of Virginia, \n Charlottesville, VA 22904, U.S.A . \n6 Department of Mechanical and Aerospace Engineering , Univer sity of Virginia, \n Charlottesville, VA 22904, U.S.A. \n*email: semori@vt.edu \n \nConfirming the or igin of Gilbert damping by experiment has remained a challenge for \nmany decades , even for simple ferromagnetic metals . In this Letter, we experimentally \nidentify Gilbert damping that increases with decreasing electronic scattering in epitaxial \nthin films of pure Fe . This observation of conductivity -like damping, which cannot be \naccounted for by classical eddy current loss , is in excellent quantitative agreement with \ntheoretical predictions of Gilbert damping due to intraband scatte ring. Our results resolve 2 \n the longstanding question about a fundamental damping mechanism and offer hints for \nengineering low -loss magnetic metals for cryogenic spintronic s and quantum devices. \n \nDamping determines how fast the magnetization relaxes towards the effective magnetic \nfield and plays a central role in many aspects of magnetization dynamics [1,2] . The magnitude of \nviscous Gilbert damping governs the threshold current for spin -torque magnetic switching and \nauto-oscillations [3,4] , mobility of magnetic domain walls [5,6] , and decay leng ths of diffusive \nspin waves and superfluid -like spin current s [7,8] . To enable spintronic technologies with low \npower dissipation , there is currently much interest in minimizing Gilbert damping in thin films of \nmagnetic m aterials [9–13], especially ferromagnetic metals [14–23] that are compatible with \nconventional device fabrication schemes . Despite the fundamental and technological importance \nof Gilbert damping, its physical mechanisms in various magnetic materials have yet to be \nconfirmed by experiment . \nGilbert damping is generally attributed to spin-orbit coupling that ultimately dissipates \nthe energy of the magnetic system to the lattice [1,2] . Kambersky’s torque correlation model [24] \nqualitatively captures the temperature dependence of damping in some experiments [25–28] by \npartitioning Gilbert damping into two mechanisms due to spin -orbit coupling, namely interband \nand intraband scattering mechanisms, each with a distinct dependence on the elect ronic \nmomentum scattering tim e e. For the interband scattering mechanism where magnetization \ndynamics can excite electron -hole pairs across dif ferent bands, the resulting Gilbert damping is \n“resistivity -like” as its magnitude scales with e-1, i.e., increased electronic scattering results in \nhigher damping [29,30] . By contrast, the intraband scattering mechanism is typically understood \nthrough the breathing Fermi surface mode l [31], where electron -hole pairs are excited in the 3 \n same band , yielding “conductivity -like” Gilbert damping that scales with e, i.e., reduced \nelectronic scattering results in higher damping. \nConductivity -like Gilbert damping was reported experimentally more than 40 years ago \nin bulk crystals of pure Ni and Co at low temperatures , but surprisingly not in pure Fe [25]. The \napparent absence of co nductivity -like damping in Fe has been at odds with many theoretical \npredictions that intraband scattering should dominate at low temperatures [32–38], although \nsome theoretical studies have suggested that intraband scattering may be absent alt ogether in \npure metals [39,40] . To date, no experimental work has conclusively addressed the role of \nintraband scattering in pure Fe1. There thus remains a significant gap in the fundamental \nunderstanding of damping in one of the simplest ferromagnetic metals. Intrinsic conductivity -\nlike Gilbert damping in Fe is also technologically relevant, since minimizing damping in \nferromagnetic metals at low temperatures is crucial for cryogenic superconducting spintronic \nmemories [41,42] and quantum information transduction schemes [43,44] . \nIn this Letter, we experimentally demonstrate the presence of conductivity -like Gilbert \ndamping due to intr aband scattering in epitaxial thin films of body -centered -cubic (BCC) Fe. By \ncombining broadband ferromagnetic resonance (FMR) measurements with characterization of \nstructural and transport properties of these model -system thin films, we show that conductivity -\nlike Gilbert damping dominates at lo w temperatures in epitaxial Fe . These experimental results \n \n1 Ref. [36] includes experimental data that suggest the presence of conductivity -like Gilbert damping in an ultrathin \nFe film, although no detailed information is given about the sample and t he experimental results deviate \nconsiderably from the calculations. An earlier study by Rudd et al. also suggests an increase in Gilbert damping with \ndecreasing temperature [27], but quantification of the Gilbert damping parameter in this experiment is difficult. \n 4 \n agree remarkably well with the magnitude of Gilbert damping derived from first -principles \ncalculations [32,33,36] , thereby providing evidence for intraband scatterin g as a key mechanism \nfor Gilbert damping in pure BCC Fe. Our experiment thus resolves the longstanding question \nregarding the origin of damping in the prototypical ferromagnetic metal . Our results also confirm \nthat – somewhat counterintuitively – disorder can partially suppress intrinsic damping at low \ntemperatures in ferromagnetic metals, such that optimally disordered films may be well suited \nfor cryogenic spintronic and quantum applications [41–44]. \nEpitaxial BCC Fe thin films were sputter deposited on (001) -oriented MgAl 2O4 (MAO) \nand MgO single crystal substrates. The choices of substrates were inspired by the recent \nexperiment by Lee et al. [20], where epitaxial growth is enabled with t he [100] axis of a B CC \nFe-rich alloy oriented 45o with respect to the [100] axis of MAO or MgO. MAO with a lattice \nparameter of a MAO /(2√2) = 0.2858 nm exhibits a lattice mismatch of less than 0.4% with Fe (a Fe \n≈ 0.287 nm) , whereas the lattice mismatch between MgO ( aMgO/√2 = 0.2978 nm) and Fe is of the \norder 4%. Here , we focus on 25 -nm-thick Fe films that were grown simultaneously on MAO and \nMgO by confocal DC magnetron sputtering [45]. In the Supplemental Material [45], we report \non additional films depos ited by off -axis magnetron sputtering. \nWe verified the crystalline quality of the epitaxial Fe films by X -ray diffraction, as s hown \nin Fig. 1( a-c). Only (00X )-type peaks of the substrate and film are found in each 2θ-ω scan, \nconsistent with the single -phase epitaxial growth of the Fe films. The 2θ-ω scans reveal a larger \namplitude of film peak for MAO/Fe, suggesting higher crystalline quality than that of MgO/Fe. \nPronounced Laue oscillations, indicative of atomically smooth film interfaces, are o bserved \naround the film peak of MAO/Fe, whereas they are absent for MgO/Fe. The high crystalline \nquality of MAO/Fe is also evidenced by its narrow film -peak rocking curve with a FWHM of 5 \n only 0.02o, comparable to the rocking curve F WHM of the substrate2. By contrast, the film -peak \nrocking curve of MgO/Fe has a FWHM of 1o, which indicates substantial mosaic spread in the \nfilm due to the large lattice mismatch with the MgO substrate. \nResults of 2θ -ω scans for different film thicknesses [45] suggest that the 25 -nm-thick Fe \nfilm may be coherently strained to the MAO substrate , consistent with the smooth interfaces and \nminimal mosaic spread of MAO/Fe . By contrast, i t is likely that 25 -nm-thick Fe on MgO is \nrelaxed to accommodate the large film-substrate lattice mismatch. Static magnetometry provides \nfurther evidence that Fe is strained on MAO a nd relaxed on MgO [45]. Since strained MAO/Fe \nand relaxed MgO/Fe exhibit distinct crystalline quality, as evidenced by an approximately 50 \ntimes narrower rocking FWHM for MAO/Fe , we have two model systems that enable \nexperimental investigation of the impact of structural disorder on Gilbert damping. \nThe residual electrical resistivity also reflects the structural quality of metal s. As shown \nin Fig. 1(d ), the residual resistivity is 20 % lower for MAO/Fe compared to MgO/Fe, which \ncorroborates the lower defect density in MAO/Fe. The resistivity increases by nearly an order of \nmagnitude with increasing temperature, reaching 1.1×10-7 m for both samples at room \ntemperature , consistent with behavior expected for pure metal thin film s. \nWe now examine how the difference in crystalline quality correlates with magnetic \ndamping in MAO/Fe and MgO /Fe. Broadband FMR measurements were performed at room \ntemperature up to 65 GHz with a custom spectrometer that empl oys a coplanar waveguide \n(center conductor width 0.4 mm ) and an electromagnet (maximum field < 2 T) . For each \nmeasurement at a fixed excitat ion frequency, an external bias magnetic field was swept parallel \nto the film plane along the [110] axis of Fe , unless otherwise noted. I n the Supplemental \n \n2 The angular resolu tion of the diffractometer is 0.0068o. 6 \n Material [45], we show similar results with the field applied along the [110] and [100] axes of \nFe; Gilbert damping is essentially isotropic within the film plane for our epitaxial Fe films , in \ncontrast to a recent report of anisotropic damping in ultrathin epitaxial Fe [22]. \nFigure 2 shows that the peak -to-peak FMR linewidth Hpp scales linearly with frequency \nf, enabling a precise determination of the measured Gilbert damping parameter 𝛼𝑚𝑒𝑎𝑠 from the \nstandard equation, \n𝜇0∆𝐻𝑝𝑝=𝜇0∆𝐻0+2\n√3𝛼𝑚𝑒𝑎𝑠\n𝛾′𝑓, (1) \nwhere Hpp,0 is the zero -frequency linewidth and 𝛾′=𝛾/2𝜋≈29.5 GHz/T is the reduced \ngyromagnetic ratio . Despite the difference in crystalline quality , we find essentially the same \nmeasured Gilbert damp ing parameter of 𝛼𝑚𝑒𝑎𝑠 ≈ 2.3×10-3 for MAO/Fe and MgO/Fe. We note \nthat t his value of 𝛼𝑚𝑒𝑎𝑠 is comparable to the lowest damping parameters reported for epitaxial Fe \nat room temperature [15–17]. Our results indicate that Gilbert damping at room temperature is \ninsensitive to the strain state or structural disorder in epitaxial Fe.3 \n The measured damping parameter 𝛼𝑚𝑒𝑎𝑠 from in-plane FMR can generally include a \ncontribution from non-Gilbert relaxation , namely two -magnon scattering driven by defects [46–\n49]. However, two-magnon scattering is suppressed when the film is magnetized out-of-\nplane [19,48] . To isolate any two -magnon scattering contribution to d amping, we performed out-\nof-plane FMR measurements under a sufficiently large magnetic field (>4 T) for complete \nsaturation of the Fe film, using a custom W-band shorted waveguide combined with a \n \n3 However, the crystallographic texture of Fe has significant impact on damping; for example, non -epitaxial Fe films \ndeposited directly on amorphous SiO 2 substrates exhibit an order of magnitude wider linewidths, due to much more \npronounced non -Gilbert damping (e.g., two -magnon scattering), compared to (001) -oriented epitaxial Fe films. \n 7 \n superconducting magnet. As shown in Fig. 2, the out -of-plane and in -plane FMR data yield the \nsame slope and hence 𝛼𝑚𝑒𝑎𝑠 (Eq. 1) to within < 8%. This finding indicates that two -magnon \nscattering is negligible and that frequency -dependent magnetic relaxation is dominated by \nGilbert damping in epitaxial Fe examined here. \nThe insensitivity of Gilbert damping to disorder found in Fig. 2 can be explained by the \ndominance of the interband (resistivity -like) mechanis m at room temperature, with phonon \nscattering dominating over defect scattering. Indeed, since MAO/Fe and MgO/Fe have the same \nroom -temperature resistivity (Fig. 1(d )), any contributions to Gilbert damping from electronic \nscattering should be identical for both samples at room temperature. Moreover, according to our \ndensity functional theory calculations [45], the density of states of BCC Fe at the Fermi energy, \nD(EF), does not depend significantly on the strain state of the crystal. Therefore, i n light of the \nrecent reports that Gilbert damping is proportional to D(EF) [18,50,51] , the different strain states \nof MAO/Fe and MgO/Fe are not expe cted to cause a significant difference in Gilbert damping. \n However, since MAO/Fe and MgO/Fe exhibit distinct resistivities (electronic scattering \ntimes e) at low temperatures, one might expect to observe distinct temperature dependence in \nGilbert damping for these two samples. To this end, we performed variable -temperature FMR \nmeasurements using a coplanar -waveguide -based spectrometer (maximum frequency 40 GHz, \nfield < 2 T) equipped with a clos ed-cycle cryostat4. Figure 3(a,b) shows that meas is enhanced \nfor both samples at lower temperatures. Notably, this damping enhancement with decreasing \ntemperature is significantl y greater for MAO/Fe . Thus, at low temperatures, we find a \n \n4 The W -band spectrometer for out -of-plane FMR (Fig. 2) could not be cooled below room temperature due to its \nlarge thermal mass , limiting us to in -plane FMR measurements at low temperatures. 8 \n conductivity -like damping increase that is evidently more pronounced in epitaxial Fe with less \nstructural disorder. \nWhile this increased damping at low temperatures is reminiscent of intrinsic Gilbert \ndamping from intraband scattering [31–38], we first consider other possible contributions. One \npossibility is two -magnon scattering [46–49], which we have ruled out at room temperature (Fig. \n2) but could be present in our low -temperature in-plane FMR measurements . From Fig. 3(a,b), \nthe zero -frequency linewidth H0 (Eq. 1 ) – typically attributed to magnetic inhomogeneity – is \nshown to increase along with meas at low temperatures [45], which might point to the emergence \nof two -magnon scattering [48,49] . However, our mean -field model calculations (see \nSupplemental Material [45]) shows that H0 correlates with meas due to interactions among \ndifferent regions of the inhomogeneous film [52]. The increase of H0 at low temperatures is \ntherefore readily accounted for by increased Gilbert damping , rather than two -magnon scattering . \nWe are also not aware of any mechanism that enhance s two-magnon scattering with \ndecreasing temperature, particularly given that the saturation magnetization (i.e., dipolar \ninteractions) is constant across the measured temperature range [45]. Moreover, the isotropic in -\nplane damping found in our study is inconsistent with typically anisotropic two-magnon \nscattering tied to the crystal symmetry of epitaxial films [46,47] , and the film thickness in our \nstudy (e.g., 25 nm) rules out t wo-magnon scattering of interfacial origin [49]. As such, we \nconclude that two -magnon scattering does not play any essential role in our experimental \nobservations. \n Another possible contribution is dissipation due to classical eddy current s, which \nincrease s proportionally with the increasing conductivity 𝜎 at lower temperatures . We estimate \nthe eddy current contribution to the measured Gilbert damping with [15,53] 9 \n 𝛼𝑒𝑑𝑑𝑦 =𝜎\n12𝛾𝜇02𝑀𝑠𝑡𝐹2, (2) \nwhere 𝜇0𝑀𝑠≈2.0 T is the saturation magnetization and tF is the film thickness . We find that \neddy curr ent damping accounts for only ≈20% (≈ 30%) of the total measured damping of \nMAO/Fe (MgO/Fe) even at the lowest measured temperature (Fig. 3(c)) . Furthermore, a s shown \nin the Supplemental Material [45], thinner MAO/Fe film s, e.g., tF = 11 nm , with negligible eddy \nstill exhibit a significant increase in damping with decreasing temperature. Our results thus \nindicate a substantial contribution to conductivity -like Gilbert damping that is not accounted for \nby classica l eddy current damping. \n For further discussion , we subtract the eddy -current damping from the measured damping \nto denote the Gilbert damping parameter attributed to intrinsic spin-orbit coupling as \n𝛼𝑠𝑜= 𝛼𝑚𝑒𝑎𝑠 − 𝛼𝑒𝑑𝑑𝑦. To correlate electronic transport and magnetic damping across the entire \nmeas ured temperature range, we perform a phenomenological fit of the temperature dependence \nof Gilbert damping with [26] \n𝛼𝑠𝑜=𝑐𝜎(𝑇)\n𝜎(300 𝐾)+𝑑𝜌(𝑇)\n𝜌(300 𝐾), (3) \nwhere the conductivity -like (intraband) and resistivity -like (interband) terms are scaled by \nadjustable parameters c and d, respectively. As shown in Fig. 4(a),(b), t his simple \nphenomenological model using the experimental transport results (Fig. 1(d)) agrees remarkably \nwell with the temperature dependence of Gilbert damping for both MAO/Fe and MgO/Fe. \nOur fi nding s that Gilbert damping can be phenomenologically partit ioned into two \ndistinct contributions (Eq. 3 ) are in line with Kambersky’s torque correlation model . We \ncompare our experimental resul ts to first-principles calculations by Gilmore et al. [32,33] that \nrelate electronic momentum scattering rate e-1 and Gilbert damping through Kambersky’s torque \ncorrelation model. We use the experimentally measured resistivity ρ (Fig. 1(d)) to convert the 10 \n temperature to e-1 by assuming the constant conversion factor ρ e = 1.30×10-21 m s [33]. To \naccount for the difference in electronic scattering time for the minority spin and majority spin \n, we take the calculated curve from Gilmore et al. with / = 4 [33], which is close to the \nratio of D(EF) of the spin-split bands for BCC Fe , e.g., derived from our density functional \ntheory calculations [45]. For explicit comparison with Refs. [32,33] , the Gilbert damping \nparameter in Fig. 4(c) is converted to the magnetic relaxation rate 𝜆= 𝛾𝛼𝑠𝑜𝜇0𝑀𝑠. The \ncalculated prediction is in excellent quantitative agreement with our experimental results for both \nstrained MAO/Fe and relaxed MgO/Fe (Fig. 4(c)) , providing additional experimental evidence \nthat intraband scattering predominately contribute s to Gilber t damping at low temperatures. \n We also compare our experimental results to a more recent first -principles calculation \nstudy by Mankovsky et al., which utilizes the linear response formalism [36]. This approach \ndoes not rely on a phenomenological electronic scattering rate and instead allows for explicitly \nincorporating thermal effects and structural disorder . Figure 4(d ) shows the calculated \ntemperature dependence of the Gilbert damping parameter for BCC Fe with a small density of \ndefects, i.e., 0.1% vacancies , adapted from Ref. [36]. We again find good quantitative agreement \nbetween the ca lculations and our experimental results for MAO/Fe. On the other hand, the \nGilbert damping parameter s at low temperatures for relaxed MgO/Fe are significantly below the \ncalculated values . This is consistent with the reduction of intraband scattering due to enhanced \nelectronic scattering (enhanced e-1) from defects in relaxed MgO/Fe . \n Indeed, significant defect -mediated electronic scattering may explain the absence of \nconductivity -like Gilbert damping for crystalline Fe in prior experiments. For example, Ref. [25] \nreports an upper limit of only a two -fold increase of the estimated Gilbert damping parameter \nfrom T = 300 K to 4 K . This relatively small damping enhancement is similar to that for MgO/Fe 11 \n in our study (Fig. 4(b)) , suggesting that intraband scattering may have been suppressed in Fe in \nRef. [25] due to a similar degree of structural disorder to MgO/Fe. We therefore conclude that \nconductivity -like Gilbert damping from intraband scattering is highly sensitive to disorder in \nferromagnetic metals. \nMore generally , the presence of defects in all real metals – evidenced by finite residual \nresistivity – ensures that the Gilbert damping parameter is finite even in the zero -temperature \nlimit . This circumvents the theoretical deficiency of Kambersky’s torque correlation model \nwhere Gilbert damping would diverge in a perfectly clean ferromagnetic metal at T 0 [39,40] . \nWe also remark that a fully quantum mechanical many -body theory of magnetization dynami cs \nyields finite Gilbert damping even in the clean, T = 0 limit [54]. \n In summary, we have demonstrated the dominance of conductiv ity-like Gilbert damping \ndue to intraband scattering at low temperatures in high-quality epitaxial Fe . Our experimental \nresults also validate the longstanding theoretical prediction of intraband scattering as an essential \nmechanism for Gilbert damping in pure ferromagnetic metals [32–38], thereby advancing the \nfundamental understanding of magnetic relaxation in real materials . Moreover, we have \nconfirmed that, at low temperatures, a ma gnetic metal with imperfect crystallinity can exhibit \nlower Gilbert damping (sp in decoherence) than its cleaner counterpart. This somewhat \ncounterintuitive finding suggests that magnetic thin film s with optimal structural or chemical \ndisorder may be useful for cryogenic spintronic memories [41,42] and spin-wave -driven \nquan tum information systems [43,44] . \n \n \n 12 \n Acknowledgements \nThis research was funded in part by 4 -VA, a collaborative partnership for advancing the \nCommonwealth of Vir ginia , as well as by the ICTAS Junior Faculty Award . A. Sapkota and C. \nMewes would like to acknowledge support by NSF-CAREER Award No. 1452670 , and A. \nSrivastava would like to acknowledge support by NASA Award No. CAN80NSSC18M0023 . \nWe thank M. D. Stiles , B. K. Nikolic , and F. Mahfouzi for helpful discussions on theoretical \nmodels for computing Gilbert damping , as well as R. D. McMichael for his input on the mean -\nfield modeling of interactions in inhomogeneous ferromagnetic films. \n \n \n1. B. 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B \n96, 214421 (2017). \n \n 19 \n \nFigure 1. (a,b) 2θ -ω X-ray diffraction scans of MAO/Fe and MgO/Fe (a) over a wide angle range \nand (b) near the BCC Fe (002) film peak. (c) Rocking curve scans about the film peak. (d) \nTemperature dependence of resistivity plotted on a log -log scale. \n \n \nFigure 2. Frequency dependence of FMR linewidth Hpp for MAO/Fe and MgO/Fe at room \ntemperature. Linewidths measured under in -plane field are shown as open symbols, whereas \nthose measured under out -of-plane (OP) field are shown as filled symbols . \n62 64 66 68log(intensity) [a.u.]\n2q [deg.]\n30 40 50 60 70log(intensity) [a.u.]\n2q [deg.]\n-1.0 -0.5 0.0 0.5 1.0intensity [a.u.]\nw002 [deg.]\n10 10010-810-7r [ m]\nT [K] MgO/Fe\n MAO/Fe\nMAO (004)MgO (002)\nFe (002)MAO/Fe\nMgO /Fe\n( 4)(a) (b) (c) (d)MAO/Fe\nMgO /Fe\n(a)\n0 20 40 60 80 100 120024681012 MAO/Fe (OP)\nMgO/Fe\nMAO/Fem0Hpp [mT]\nf [GHz]20 \n \nFigure 3. (a,b) Frequency dependence of FMR linewidth for MA O/Fe and MgO/Fe at (a) T = 100 \nK and (b) T = 10 K. (c) Temperature dependence of measured Gilbert damping parameter meas \nand estimated eddy -current damping parameter eddy. \n \n \n0 50 100 150 200 250 3000246810\n meas MAO/Fe \n eddy estimate\n meas MgO/Fe\n eddy estimatemeas, eddy [10-3]\nT [K]\n0 10 20 30 40051015\nMAO/Fe\nMgO/Fem0Hpp (mT)\nf (GHz)T = 100 K\n0 10 20 30 40051015m0Hpp (mT)\nf (GHz)T = 10 K(c)(a) (b)21 \n \nFigure 4. (a,b) Temperature dependence of the spin-orbit -induced Gilbert damping parameter \nso, fit phenomenologically with the experimentally measured resistivity for (a) MAO/Fe and (b) \nMgO/Fe. The dashed and dotted curves indicate the conductivity -like and resistivity -like \ncontributions, respectively; the solid curve represents the fit curve for the total spin -orbit -induced \nGilbert damping parameter. (c,d) Comparison of our experimental results with calculated Gilbert \ndamping parameters by (c) Gilmore et al. [32,33] and (d) Mankovsky et al. [36]. \n \n \n0 100 200 30002468\nr-liker-likeso [10-3]\nT [K]s-likeMAO/Fe\n0 100 200 30002468\nMgO/Fe\nr-likes-likeso [10-3]\nT [K](a) (b)\n0 100 200 30002468\n MAO/Fe\n MgO/Fe\n calculated [Mankovsky]so [10-3]\nT [K]\n0 50 1000123\nr-like MAO/Fe\n MgO/Fe\n calculated [Gilmore]l [109 s-1]\ne-1 [1012 s-1]s-like0.0 0.5 1.0\n02468\nso [10-3]r [10-7 m]\n(c)\n(d)" }, { "title": "1104.1625v1.Magnetization_Dissipation_in_Ferromagnets_from_Scattering_Theory.pdf", "content": "arXiv:1104.1625v1 [cond-mat.mes-hall] 8 Apr 2011Magnetization Dissipation in Ferromagnets from Scatterin g Theory\nArne Brataas∗\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nYaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\nGerrit E. W. Bauer\nInstitute for Materials Research, Tohoku University, Send ai 980-8577, Japan and\nKavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\nThe magnetization dynamicsofferromagnets are often formu lated intermsof theLandau-Lifshitz-\nGilbert (LLG) equation. The reactive part of this equation d escribes the response of the magnetiza-\ntion in terms of effective fields, whereas the dissipative par t is parameterized by the Gilbert damping\ntensor. We formulate a scattering theory for the magnetizat ion dynamics and map this description\non the linearized LLG equation by attaching electric contac ts to the ferromagnet. The reactive part\ncan then be expressed in terms of the static scattering matri x. The dissipative contribution to the\nlow-frequency magnetization dynamics can be described as a n adiabatic energy pumping process\nto the electronic subsystem by the time-dependent magnetiz ation. The Gilbert damping tensor\ndepends on the time derivative of the scattering matrix as a f unction of the magnetization direction.\nBy the fluctuation-dissipation theorem, the fluctuations of the effective fields can also be formulated\nin terms of the quasistatic scattering matrix. The theory is formulated for general magnetization\ntextures and worked out for monodomain precessions and doma in wall motions. We prove that the\nGilbert damping from scattering theory is identical to the r esult obtained by the Kubo formalism.\nPACS numbers: 75.40.Gb,76.60.Es,72.25.Mk\nI. INTRODUCTION\nFerromagnets develop a spontaneous magnetization\nbelow the Curie temperature. The long-wavelengthmod-\nulations of the magnetization direction consist of spin\nwaves, the low-lying elementary excitations (Goldstone\nmodes) of the ordered state. When the thermal energy is\nmuch smaller than the microscopic exchange energy, the\nmagnetization dynamics can be phenomenologically ex-\npressed in a generalized Landau-Lifshitz-Gilbert (LLG)\nform:\n˙ m(r,t) =−γm(r,t)×[Heff(r,t)+h(r,t)]+\nm(r,t)×/integraldisplay\ndr′[˜α[m](r,r′)˙ m(r′,t)],(1)\nwhere the magnetization texture is described by m(r,t),\nthe unit vector along the magnetization direction at po-\nsitionrand timet,˙ m(r,t) =∂m(r,t)/∂t,γ=gµB//planckover2pi1is\nthe gyromagnetic ratio in terms of the g-factor (≈2 for\nfree electrons) and the Bohr magneton µB. The Gilbert\ndamping ˜αis a nonlocal symmetric 3 ×3 tensor that is\na functional of m. The Gilbert damping tensor is com-\nmonly approximated to be diagonal and isotropic (i), lo-\ncal (l), and independent of the magnetization m, with\ndiagonal elements\nαil(r,r′) =αδ(r−r′). (2)\nThe linearized version of the LLG equation for small-\namplitude excitations has been derived microscopically.1It has been used very successfully to describe the mea-\nsured response of ferromagnetic bulk materials and thin\nfilms in terms of a small number of adjustable, material-\nspecific parameters. The experiment of choice is fer-\nromagnetic resonance (FMR), which probes the small-\namplitude coherent precession of the magnet.2The\nGilbertdampingmodelinthelocalandtime-independent\napproximationhasimportantramifications, suchasalin-\near dependence of the FMR line width on resonance fre-\nquency, that have been frequently found to be correct.\nThe damping constant is technologically important since\nit governs the switching rate of ferromagnets driven by\nexternal magnetic fields or electric currents.3In spatially\ndependent magnetization textures, the nonlocal charac-\nter of the damping can be significant as well.4–6Moti-\nvated by the belief that the Gilbert damping constant is\nanimportantmaterialproperty, weset outheretounder-\nstand its physical origins from first principles. We focus\non the well studied and technologically important itiner-\nant ferromagnets, although the formalism can be used in\nprinciple for any magnetic system.\nThe reactive dynamics within the LLG Eq. (1) is de-\nscribed by the thermodynamic potential Ω[ M] as a func-\ntional of the magnetization. The effective magnetic field\nHeff[M](r)≡ −δΩ/δM(r) is the functional derivative\nwith respect to the local magnetization M(r) =Msm(r),\nincluding the external magnetic field Hext, the magnetic\ndipolar field Hd, the texture-dependent exchange energy,\nand crystal field anisotropies. Msis the saturation mag-\nnetization density. Thermal fluctuations can be included\nby a stochastic magnetic field h(r,t) with zero time av-2\nleft\nreservoirF N Nright\nreservoir\nFIG. 1: Schematic picture of a ferromagnet (F) in contact\nwith a thermal bath (reservoirs) via metallic normal metal\nleads (N).\nerage,/an}b∇acketle{th/an}b∇acket∇i}ht= 0, and white-noise correlation:7\n/an}b∇acketle{thi(r,t)hj(r′,t′)/an}b∇acket∇i}ht=2kBT\nγMs˜αij[m](r,r′)δ(t−t′),(3)\nwhereMsis the magnetization, iandjare the Cartesian\nindices, and Tis the temperature. This relation is a con-\nsequence ofthe fluctuation-dissipation theorem (FDT) in\nthe classical (Maxwell-Boltzmann) limit.\nThe scattering ( S-) matrix is defined in the space of\nthe transport channels that connect a scattering region\n(the sample) to real or fictitious thermodynamic (left\nand right) reservoirs by electric contacts with leads that\nare modeled as ideal wave guides. Scattering matri-\nces are known to describe transport properties, such as\nthe giant magnetoresistance, spin pumping, and current-\ninducedmagnetizationdynamicsinlayerednormal-metal\n(N)|ferromagnet (F).8–10When the ferromagnet is part\nof an open system as in Fig. 1, also Ω can be expressed\nin terms of the scattering matrix, which has been used\nto express the non-local exchange coupling between fer-\nromagnetic layers through conducting spacers.11We will\nshow here that the scattering matrix description of the\neffective magnetic fields is valid even when the system is\nclosed, provided the dominant contribution comes from\nthe electronic band structure, scattering potential disor-\nder, and spin-orbit interaction.\nScattering theory can also be used to compute the\nGilbert damping tensor ˜ αfor magnetization dynamics.15\nThe energy loss rate of the scattering region can be ex-\npressedin termsofthe time-dependent S-matrix. To this\nend, the theory of adiabatic quantum pumping has to be\ngeneralizedtodescribedissipationinametallicferromag-\nnet. The Gilbert damping tensor is found by evaluating\nthe energy pumping out of the ferromagnet and relat-\ning it to the energy loss that is dictated by the LLG\nequation. In this way, it is proven that the Gilbert phe-\nnomenology is valid beyond the linear response regime\nof small magnetization amplitudes. The key approxima-\ntion that is necessary to derive Eq. (1) including ˜ αis the\n(adiabatic) assumption that the ferromagnetic resonance\nfrequencyωFMRthat characterizesthe magnetizationdy-\nnamics is small compared to internal energy scale set by\nthe exchange splitting ∆ and spin-flip relaxation rates\nτs. The LLG phenomenology works well for ferromag-\nnets for which ωFMR≪∆//planckover2pi1, which is certainly the case\nfor transition metal ferromagnets such as Fe and Co.\nGilbert damping in transition-metal ferromagnets is\ngenerally believed to stem from the transfer of energy\nfromthemagneticorderparametertotheitinerantquasi-particle continuum. This requires either magnetic disor-\nder or spin-orbit interactions in combination with impu-\nrity/phonon scattering.2Since the heat capacitance of\nthe ferromagnet is dominated by the lattice, the energy\ntransferred to the quasiparticles will be dissipated to the\nlattice as heat. Here we focus on the limit in which elas-\ntic scattering dominates, such that the details of the heat\ntransfer to the lattice does not affect our results. Our ap-\nproachformallybreaks down in sufficiently clean samples\nat high temperatures in which inelastic electron-phonon\nscattering dominates. Nevertheless, quantitative insight\ncan be gained by our method even in that limit by mod-\nelling phonons by frozen deformations.12\nIn the present formulation, the heat generated by the\nmagnetization dynamics can escape only via the contacts\nto the electronic reservoirs. By computing this heat cur-\nrent through the contacts we access the total dissipa-\ntion rate. Part of the heat and spin current that es-\ncapes the sample is due to spin pumping that causes\nenergy and momentum loss even for otherwise dissipa-\ntion less magnetization dynamics. This process is now\nwellunderstood.10For sufficiently largesamples, the spin\npumping contribution is overwhelmed by the dissipation\nin the bulk of the ferromagnet. Both contributions can\nbe separated by studying the heat generation as a func-\ntion of the length of a wire. In principle, a voltage can be\nadded to study dissipation in the presence of electric cur-\nrents as in 13,14, but we concentrate here on a common\nand constant chemical potential in both reservoirs.\nAlthough it is not a necessity, results can be simpli-\nfied by expanding the S-matrix to lowest order in the\namplitude of the magnetization dynamics. In this limit\nscattering theory and the Kubo linear response formal-\nism for the dissipation can be directly compared. We\nwill demonstrate explicitly that both approaches lead to\nidentical results, which increases our confidence in our\nmethod. The coupling to the reservoirs of large samples\nis identified to play the same role as the infinitesimals in\nthe Kubo approach that guarantee causality.\nOur formalism was introduced first in Ref. 15 lim-\nited to the macrospin model and zero temperature. An\nextension to the friction associatedwith domain wall mo-\ntion was given in Ref. 13. Here we show how to handle\ngeneral magnetization textures and finite temperatures.\nFurthermore, we offer an alternative route to derive the\nGilbert damping in terms of the scattering matrix from\nthe thermal fluctuations of the effective field. We also\nexplain in more detail the relation of the present theory\nto spin and charge pumping by magnetization textures.\nOur paper is organized in the following way. In Sec-\ntion II, we introduce our microscopic model for the fer-\nromagnet. In Section III, dissipation in the Landau-\nLifshitz-Gilbert equation is exposed. The scattering the-\nory of magnetization dynamics is developed in Sec. IV.\nWe discuss the Kubo formalism for the time-dependent\nmagnetizationsin Sec. V, before concluding our article in\nSec. VI. The Appendices provide technical derivations of\nspin, charge, and energy pumping in terms of the scat-3\ntering matrix of the system.\nII. MODEL\nOur approach rests on density-functional theory\n(DFT), which is widely and successfully used to describe\nthe electronic structure and magnetism in many fer-\nromagnets, including transition-metal ferromagnets and\nferromagnetic semiconductors.16In the Kohn-Sham im-\nplementation of DFT, noninteracting hypothetical par-\nticles experience an effective exchange-correlationpoten-\ntial that leads to the same ground-statedensity as the in-\nteractingmany-electronsystem.17Asimpleyetsuccessful\nscheme is the local-densityapproximationto the effective\npotential. DFT theory can also handle time-dependent\nphenomena. We adopt here the adiabatic local-density\napproximation (ALDA), i.e. an exchange-correlationpo-\ntential that is time-dependent, but local in time and\nspace.18,19As the name expresses, the ALDA is valid\nwhen the parametric time-dependence of the problem is\nadiabatic with respect to the electron time constants.\nHere we consider a magnetization direction that varies\nslowly in both space and time. The ALDA should be\nsuited to treat magnetization dynamics, since the typical\ntime scale ( tFMR∼1/(10 GHz) ∼10−10s) is long com-\nparedtothethat associatedwith theFermi andexchange\nenergies, 1 −10 eV leading to /planckover2pi1/∆∼10−13s in transition\nmetal ferromagnets.\nIn the ALDA, the system is described by the time-\ndependent effective Schr¨ odinger equation\nˆHALDAΨ(r,t) =i/planckover2pi1∂\n∂tΨ(r,t), (4)\nwhere Ψ( r,t) is the quasiparticle wave function at posi-\ntionrand timet. We consider a generic mean-field elec-\ntronic Hamiltonian that depends on the magnetization\ndirection ˆHALDA[m] and includes the periodic Hartree,\nexchange and correlation potentials and relativistic cor-\nrectionssuchasthe spin-orbitinteraction. Impurityscat-\ntering including magnetic disorder is also represented by\nˆHALDA.The magnetization mis allowed to vary in time\nand space. The total Hamiltonian depends additionally\non the Zeeman energy of the magnetization in external\nHextand dipolar Hdmagnetic fields:\nˆH=ˆHALDA[m]−Ms/integraldisplay\ndrm·(Hext+Hd).(5)\nFor this general Hamiltonian (5), our task is to de-\nduce an expression for the Gilbert damping tensor ˜ α. To\nthis end, from the form of the Landau-Lifshitz-Gilbert\nequation (3), it is clear that we should seek an expansionin terms of the slow variations of the magnetizations in\ntime. Such an expansion is valid provided the adiabatic\nmagnetization precession frequency is much less than the\nexchange splitting ∆ or the spin-orbit energy which gov-\nerns spin relaxation of electrons. We discuss first dissi-\npation in the LLG equation and subsequently compare\nit with the expressions from scattering theory of electron\ntransport. This leads to a recipe to describe dissipation\nby first principles. Finally, we discuss the connection to\nthe Kubo linear response formalism and prove that the\ntwo formulations are identical in linear response.\nIII. DISSIPATION AND\nLANDAU-LIFSHITZ-GILBERT EQUATION\nThe energy dissipation can be obtained from the solu-\ntion of the LLG Eq. (1) as\n˙E=−Ms/integraldisplay\ndr[˙ m(r,t)·Heff(r,t)] (6)\n=−Ms\nγ/integraldisplay\ndr/integraldisplay\ndr′˙ m(r)·˜α[m](r,r′)·˙ m(r′).(7)\nThescatteringtheoryofmagnetizationdissipationcanbe\nformulated for arbitrary spatiotemporal magnetization\ntextures. Much insight can be gained for certain special\ncases. In small particles or high magnetic fields the col-\nlective magnetization motion is approximately constant\nin space and the “macrospin” model is valid in which\nall spatial dependences are disregarded. We will also\nconsider special magnetization textures with a dynamics\ncharacterized by a number of dynamic (soft) collective\ncoordinates ξa(t) counted by a:20,21\nm(r,t) =mst(r;{ξa(t)}), (8)\nwheremstis the profile at t→ −∞.This representation\nhas proven to be very effective in handling magnetiza-\ntion dynamics of domain walls in ferromagnetic wires.\nThe description is approximate, but (for few variables)\nit becomes exact in special limits, such as a transverse\ndomain wall in wires below the Walker breakdown (see\nbelow); it becomes arbitrarily accurate by increasing the\nnumber of collective variables. The energy dissipation to\nlowest (quadratic) order in the rate of change ˙ξaof the\ncollective coordinates is\n˙E=−/summationdisplay\nab˜Γab˙ξa˙ξb, (9)\nThe (symmetric) dissipation tensor ˜Γabreads4\n˜Γab=Ms\nγ/integraldisplay\ndr/integraldisplay\ndr′∂mst(r)\n∂ξaα[m](r,r′)·∂mst(r′)\n∂ξb. (10)\nThe equation of motion of the collective coordinates un-\nder a force\nF=−∂Ω\n∂ξ(11)\nare20,21\n˜η˙ξ+[F+f(t)]−˜Γ˙ξ= 0, (12)\nintroducing the antisymmetric and time-independent gy-\nrotropic tensor:\n˜ηab=Ms\nγ/integraldisplay\ndrmst(r)·/bracketleftbigg∂mst(r)\n∂ξa×∂mst(r)\n∂ξb/bracketrightbigg\n.(13)\nWe show below that Fand˜Γ can be expressed in terms\nof the scattering matrix. For our subsequent discussions\nit is necessary to include a fluctuating force f(t) (with\n/an}b∇acketle{tf(t)/an}b∇acket∇i}ht= 0),which has not been considered in Refs. 20,21.\nFrom Eq. (3) if follows the time correlation of fis white\nand obeys the fluctuation-dissipation theorem:\n/an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBT˜Γabδ(t−t′). (14)\nIn the following we illustrate the collective coordinate\ndescription of magnetization textures for the macrospin\nmodel and the Walker model for a transverse domain\nwall. The treatment is easily extended to other rigid\ntextures such as magnetic vortices.\nA. Macrospin excitations\nWhen high magnetic fields are applied or when the\nsystem dimensions are small the exchange stiffness dom-\ninates. In both limits the magnetization direction and\nits low energy excitations lie on the unit sphere and its\nmagnetization dynamics is described by the polar angles\nθ(t) andϕ(t):\nm= (sinθcosϕ,sinθsinϕ,cosθ).(15)\nThe diagonal components of the gyrotropic tensor vanish\nby (anti)symmetry ˜ ηθθ= 0, ˜ηϕϕ= 0.Its off-diagonal\ncomponents are\nηθϕ=MsV\nγsinθ=−ηϕθ. (16)\nVis the particle volume and MsVthe total magnetic\nmoment. We now have two coupled equations of motion\nMsV\nγ˙ϕsinθ−∂Ω\n∂θ−/parenleftBig\n˜Γθθ˙θ+˜Γ��ϕ˙ϕ/parenrightBig\n= 0,(17)\n−MsV\nγ˙θsinθ−∂Ω\n∂ϕ−/parenleftBig\n˜Γϕθ˙θ+˜Γϕϕ˙ϕ/parenrightBig\n= 0.The thermodynamic potential Ω determines the ballistic\ntrajectories of the magnetization. The Gilbert damping\ntensor˜Γabwill be computed below, but when isotropic\nand local,\n˜Γ =˜1δ(r−r′)Msα/γ, (18)\nwhere˜1 is a unit matrix in the Cartesian basis and α\nis the dimensionless Gilbert constant, Γ θθ=MsVα/γ,\nΓθϕ= 0 = Γ ϕθ, and Γ ϕϕ= sin2θMsVα/γ.\nB. Domain Wall Motion\nWe focus on a one-dimensional model, in which the\nmagnetization gradient, magnetic easy axis, and external\nmagnetic field point along the wire ( z) axis. The mag-\nnetic energy of such a wire with transverse cross section\nScan be written as22\nΩ =MsS/integraldisplay\ndzφ(z), (19)\nin terms of the one-dimensional energy density\nφ=A\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m\n∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n−Hamz+K1\n2/parenleftbig\n1−m2\nz/parenrightbig\n+K2\n2m2\nx,(20)\nwhereHais the applied field and Ais the exchange stiff-\nness. Here the easy-axis anisotropy is parametrized by\nan anisotropy constant K1. In the case of a thin film\nwire, there is also a smaller anisotropy energy associated\nwith the magnetization transverse to the wire governed\nbyK2. In a cylindrical wire from a material without\ncrystal anisotropy (such as permalloy) K2= 0.\nWhen the shape of such a domain wall is pre-\nserved in the dynamics, three collective coordinates\ncharacterize the magnetization texture: the domain\nwall position ξ1(t) =rw(t), the polar angle ξ2(t) =\nϕw(t), and the domain wall width λw(t). We con-\nsider a head-to-head transverse domain wall (a tail-\nto-tail wall can be treated analogously). m(z) =\n(sinθwcosϕw,sinθwsinϕw,cosθw), where\ncosθw= tanhrw−z\nλw(21)\nand\ncscθw= coshrw−z\nλw(22)\nminimizes the energy (20) under the constraint that the\nmagnetization to the far left and right points towardsthe5\ndomain wall. The off-diagonal elements are then ˜ ηrl=\n0 = ˜ηlrand ˜ηrϕ=−2Ms/γ=−˜ηϕr.The energy (20)\nreduces to\nΩ =MsS/bracketleftbig\nA/λw−2Har+K1λw+K2λwcos2ϕw/bracketrightbig\n.\n(23)\nDisregarding fluctuations, the equation of motion Eq.\n(12) can be expanded as:\n2˙rw+αϕϕ˙ϕ+αϕr˙rw+αϕλ˙λw=γK2λwsin2ϕw,\n(24)\n−2 ˙ϕ+αrr˙rw+αrϕ˙ϕ+αrλ˙λw= 2γHa, (25)\nA/λ2\nw+αλr˙rw+αλϕ˙ϕ+αλλ˙λw=K1+K2cos2ϕw,\n(26)\nwhereαab=γΓab/MsS.\nWhen the Gilbert dampingtensorisisotropicandlocal\nin the basis of the Cartesian coordinates, ˜Γ =˜1δ(r−\nr′)Msα/γ\nαrr=2α\nλw;αϕϕ= 2αλw;αλλ=π2α\n6λw.(27)\nwhereas all off-diagonal elements vanish.\nMost experiments are carried out on thin film ferro-\nmagnetic wires for which K2is finite. Dissipation is es-\npecially simple below the Walker threshold, the regime\nin which the wall moves with a constant drift velocity,\n˙ϕw= 0 and23\n˙rw=−2γHa/αrr. (28)\nThe Gilbert damping coefficient αrrcan be obtained di-\nrectly from the scattering matrix by the parametric de-\npendence of the scattering matrix on the center coordi-\nnate position rw. When the Gilbert damping tensor is\nisotropic and local, we find ˙ rw=λwγHa/α. The domain\nwall width λw=/radicalbig\nA/(K1+K2cos2ϕw) and the out-\nof-plane angle ϕw=1\n2arcsin2γHa/αK2. At the Walker-\nbreakdownfield ( Ha)WB=αK2/(2γ) the sliding domain\nwall becomes unstable.\nIn a cylindrical wire without anisotropy, K2= 0,ϕwis\ntime-dependent and satisfies\n˙ϕw=−(2+αϕr)\nαϕϕ˙rw (29)\nwhile\n˙rw=2γHa\n2/parenleftBig\n2+αϕr\nαϕϕ/parenrightBig\n+αrr. (30)\nFor isotropic and local Gilbert damping coefficients,22\n˙rw\nλw=αγHa\n1+α2. (31)\nInthe nextsection, weformulatehowthe Gilbert scatter-\ning tensor can be computed from time-dependent scat-\ntering theory.IV. SCATTERING THEORY OF MESOSCOPIC\nMAGNETIZATION DYNAMICS\nScattering theory of transport phenomena24has\nproven its worth in the context of magnetoelectronics.\nIt has been used advantageously to evaluate the non-\nlocal exchange interactions multilayers or spin valves,11\nthe giantmagnetoresistance,25spin-transfertorque,9and\nspin pumping.10We first review the scattering theory\nof equilibrium magnetic properties and anisotropy fields\nand then will turn to non-equilibrium transport.\nA. Conservative forces\nConsidering only the electronic degrees of freedom in\nour model, the thermodynamic (grand) potential is de-\nfined as\nΩ =−kBTlnTre−(ˆHALDA−µˆN), (32)\nwhileµis the chemical potential, and ˆNis the number\noperator. The conservative force\nF=−∂Ω\n∂ξ. (33)\ncan be computed for an open systems by defining a scat-\nteringregionthat isconnectedby idealleadstoreservoirs\nat common equilibrium. For a two-terminal device, the\nflow of charge, spin, and energy between the reservoirs\ncan then be described in terms of the S-matrix:\nS=/parenleftbigg\nr t′\nt r′/parenrightbigg\n, (34)\nwhereris the matrix of probability amplitudes of states\nimpinging from and reflected into the left reservoir, while\ntdenotes the probability amplitudes of states incoming\nfrom the left and transmitted to the right. Similarly,\nr′andt′describes the probability amplitudes for states\nthat originate from the right reservoir. r,r′,t, andt′are\nmatricesin the space spanned by eigenstates in the leads.\nWe areinterested in the free magnetic energymodulation\nby the magnetic configuration that allows evaluation of\nthe forces Eq. (33). The free energy change reads\n∆Ω =−kBT/integraldisplay\ndǫ∆n(ǫ)ln/bracketleftBig\n1+e(ǫ−µ)/kBT/bracketrightBig\n,(35)\nwhere ∆n(ǫ)dǫis the change in the number of states at\nenergyǫand interval dǫ, which can be expressed in terms\nof the scattering matrix45\n∆n(ǫ) =−1\n2πi∂\n∂ǫTrlnS(ǫ). (36)\nCarrying out the derivative, we arrive at the force\nF=−1\n2πi/integraldisplay\ndǫf(ǫ)Tr/parenleftbigg\nS†∂S\n∂ξ/parenrightbigg\n,(37)6\nwheref(ǫ) is the Fermi-Dirac distribution function with\nchemical potential µ. This established result will be re-\nproducedandgeneralizedtothedescriptionofdissipation\nand fluctuations below.\nB. Gilbert damping as energy pumping\nHere we interpretGilbert damping asan energypump-\ning process by equating the results for energy dissipa-\ntion from the microscopic adiabatic pumping formalism\nwith the LLG phenomenology in terms of collective co-\nordinates, Eq. (9). The adiabatic energy loss rate of a\nscattering region in terms of scattering matrix at zero\ntemperature has been derived in Refs. 26,27. In the ap-\npendices, we generalize this result to finite temperatures:\n˙E=/planckover2pi1\n4π/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ,t)\n∂t∂S†(ǫ,t)\n∂t/bracketrightbigg\n.(38)\nSince we employ the adiabatic approximation, S(ǫ,t) is\nthe energy-dependent scattering matrix for an instanta-\nneous (“frozen”)scattering potential at time t. In a mag-\nnetic system, the time dependence arises from its magne-\ntization dynamics, S(ǫ,t) =S[m(t)](ǫ). In terms of the\ncollective coordinates ξ(t),S(ǫ,t) =S(ǫ,{ξ(t)})\n∂S[m(t)]\n∂t≈/summationdisplay\na∂S\n∂ξa˙ξa, (39)\nwhere the approximate sign has been discussed in the\nprevious section. We can now identify the dissipation\ntensor (10) in terms of the scattering matrix\nΓab=/planckover2pi1\n4π/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ)\n∂ξa∂S†(ǫ)\n∂ξb/bracketrightbigg\n.(40)In the macrospin model the Gilbert damping tensor can\nthen be expressed as\n˜αij=γ/planckover2pi1\n4πMs/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ)\n∂mi∂S†(ǫ)\n∂mj/bracketrightbigg\n,(41)\nwheremiis a Cartesian component of the magnetization\ndirection..\nC. Gilbert damping and fluctuation-dissipation\ntheorem\nAt finite temperatures the forces acting on the mag-\nnetization contain thermal fluctuations that are related\nto the Gilbert dissipation by the fluctuation-dissipation\ntheorem, Eq. (14). The dissipation tensor is therefore ac-\ncessible via the stochastic forces in thermal equilibrium.\nThe time dependence of the force operators\nˆF(t) =−∂ˆHALDA(m)\n∂ξ(42)\nis caused by the thermal fluctuations of the magneti-\nzation. It is convenient to rearrange the Hamiltonian\nˆHALDAinto an unperturbed part that does not de-\npend on the magnetization and a scattering potential\nˆHALDA(m) =ˆH0+ˆV(m). In the basis of scattering\nwave functions of the leads, the force operator reads\nˆF=−/integraldisplay\ndǫ/integraldisplay\ndǫ′/an}b∇acketle{tǫα|∂ˆV\n∂ξ|ǫ′β/an}b∇acket∇i}htˆa†\nα(ǫ)ˆaβ(ǫ′)ei(ǫ−ǫ′)t//planckover2pi1, (43)\nwhere ˆaβannihilates an electron incident on the scatter-\ning region, βlabels the lead (left or right) and quantum\nnumbers of the wave guide mode, and |ǫ′β/an}b∇acket∇i}htis an associ-\nated scatteringeigenstateat energy ǫ′. We takeagainthe\nleft and rightreservoirsto be in thermal equilibrium with\nthe same chemical potentials, such that the expectation\nvalues\n/angbracketleftbig\nˆa†\nα(ǫ)ˆaβ(ǫ′)/angbracketrightbig\n=δαβδ(ǫ−ǫ′)f(ǫ).(44)\nTherelationbetweenthematrixelementofthescattering\npotential and the S-matrix\n/bracketleftbigg\nS†(ǫ)∂S(ǫ)\n∂ξ/bracketrightbigg\nαβ=−2πi/an}b∇acketle{tǫα|∂ˆV\n∂ξ|ǫβ/an}b∇acket∇i}ht(45)follows from the relation derived in Eq. (61) below as\nwell as unitarity of the S-matrix,S†S= 1. Taking these\nrelationsintoaccount,the expectationvalueof ˆFisfound\nto be Eq. (37). We now consider the fluctuations in the\nforceˆf(t) =ˆF(t)− /an}b∇acketle{tˆF(t)/an}b∇acket∇i}ht, which involves expectation\nvalues\n/angbracketleftbig\nˆa†\nα1(ǫ1)ˆaβ1(ǫ′\n1)ˆa†\nα2(ǫ2)ˆaβ2(ǫ′\n2)/angbracketrightbig\n−/angbracketleftbig\nˆa†\nα1(ǫ1)ˆaβ1(ǫ′\n1)/angbracketrightbig/angbracketleftbig\nˆa†\nα2(ǫ2)ˆaβ2(ǫ′\n2)/angbracketrightbig\n=δα1β2δ(ǫ1−��′\n2)δβ1α2δ(ǫ′\n1−ǫ2)f(ǫ1)[1−f(ǫ2)],\n(46)\nwhere we invoked Wick’s theorem. Putting everything7\ntogether, we finally find\n/an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBTδ(t−t′)Γab, (47)\nwhere Γ abhas been defined in Eq. (40). Comparing with\nEq. (14), we conclude that the dissipation tensor Γ ab\ngoverningthe fluctuationsisidentical tothe oneobtained\nfrom the energy pumping, Eq. (40), thereby confirming\nthe fluctuation-dissipation theorem.\nV. KUBO FORMULA\nThe quality factor of the magnetization dynamics of\nmost ferromagnets is high ( α/lessorsimilar0.01). Damping can\ntherefore often be treated as a small perturbation. In\nthe presentSectionwedemonstratethat the dampingob-\ntained from linear response (Kubo) theory agrees28with\nthat ofthe scattering theory ofmagnetization dissipation\nin this limit. At sufficiently low temperatures or strong\nelastic disorder scattering the coupling to phonons may\nbe disregarded and is not discussed here.\nThe energy dissipation can be written as\n˙E=/angbracketleftBigg\ndˆH\ndt/angbracketrightBigg\n, (48)\nwhere/an}b∇acketle{t/an}b∇acket∇i}htdenotes the expectation value for the non-\nequilibrium state. We are interested in the adiabatic\nresponse of the system to a time-dependent perturba-\ntion. In the adiabatic (slow) regime, we can at any time\nexpand the Hamiltonian around a static configuration at\nthe reference time t= 0,\nˆH=ˆHst+/summationdisplay\naδξa(t)/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r).(49)\nThe static part, ˆHst, is the Hamiltonian for a magneti-\nzation for a fixed and arbitrary initial texture mst, as,\nwithout loss of generality, described by the collective\ncoordinates ξa. Since we assume that the variation of\nthe magnetization in time is small, a linear expansion in\nterms of the small deviations of the collective coordinate\nδξi(t) is valid for sufficiently short time intervals. We can\nthen employ the Kubo formalism and express the energy\ndissipation as\n˙E=/summationdisplay\naδ˙ξa(t)/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r),(50)\nwhere the expectation value of the out-of-equilibrium\nconservative force\n/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r)≡∂aˆH (51)consists of an equilibrium contribution and a term linear\nin the perturbed magnetization direction:\n/angbracketleftBig\n∂aˆH/angbracketrightBig\n(t) =/angbracketleftBig\n∂aˆH/angbracketrightBig\nst+/summationdisplay\nb/integraldisplay∞\n−∞dt′χab(t−t′)δξb(t′).\n(52)\nHere, we introduced the retarded susceptibility\nχab(t−t′) =−i\n/planckover2pi1θ(t−t′)/angbracketleftBig/bracketleftBig\n∂aˆH(t),∂bˆH(t′)/bracketrightBig/angbracketrightBig\nst,(53)\nwhere/an}b∇acketle{t/an}b∇acket∇i}htstis the expectation value for the wave functions\nof the static configuration. Focussing on slow modula-\ntions we can further simplify the expression by expand-\ning\nδξa(t′)≈δξa(t)+(t′−t)δ˙ξa(t), (54)\nso that\n/angbracketleftBig\n∂aˆH/angbracketrightBig\n=/angbracketleftBig\n∂aˆH/angbracketrightBig\nst+/integraldisplay∞\n−∞dt′χab(t−t′)δξb(t)+\n/integraldisplay∞\n−∞dt′χab(t−t′)(t′−t)δ˙ξb(t). (55)\nThe first two terms in this expression, /an}b∇acketle{t∂aˆH/an}b∇acket∇i}htst+/integraltext∞\n−∞dt′χab(t−t′)δξb(t),correspond to the energy vari-\nation with respect to a change in the static magnetiza-\ntion. These terms do not contribute to the dissipation\nsince the magnetic excitations are transverse, ˙ m·m= 0.\nOnly the last term in Eq. (55) gives rise to dissipation.\nHence, the energy loss reduces to29\n˙E=i/summationdisplay\nijδ˙ξaδ˙ξb∂χS\nab\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0, (56)\nwhereχS\nab(ω) =/integraltext∞\n−∞dt[χab(t)+χba(t)]eiωt/2. The\nsymmetrized susceptibility can be expanded as\nχS\nab=/summationdisplay\nnm(fn−fm)\n2/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}ht+(a↔b)\n/planckover2pi1ω+iη−(ǫn−ǫm),\n(57)\nwhere|n/an}b∇acket∇i}htis an eigenstate of the Hamiltonian ˆHstwith\neigenvalueǫn,fn≡f(ǫn),f(ǫ) is the Fermi-Dirac distri-\nbution function at energy ǫ, andηis a positive infinites-\nimal constant. Therefore,8\ni/parenleftbigg∂χS\nab\n∂ω/parenrightbigg\nω=0=π/summationdisplay\nnm/parenleftbigg\n−∂fn\n∂ǫ/parenrightbigg\n/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm), (58)\nand the dissipation tensor\nΓab=π/summationdisplay\nnm/parenleftbigg\n−∂fn\n∂ǫ/parenrightbigg\n/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm). (59)\nWe nowdemonstratethatthe dissipationtensorobtained\nfrom the Kubo linear response formula, Eq. (59), is\nidentical to the expression from scattering theory, Eq.\n(40), following the Fisher and Lee proof of the equiv-\nalence of linear response and scattering theory for the\nconductance.36\nThe static Hamiltonian ˆHst(ξ) =ˆH0+ˆV(ξ) can be\ndecomposed into a free-electron part ˆH0=−/planckover2pi12∇2/2m\nand a scattering potential ˆV(ξ). The eigenstates of ˆH0\nare denoted |ϕs,q(ǫ)/an}b∇acket∇i}ht,with eigenenergies ǫ, wheres=±\ndenotes the longitudinal propagation direction along the\nsystem (say, to the left or to the right), and qa trans-\nverse quantum number determined by the lateral con-\nfinement. The potential ˆV(ξ) scatters the particles be-tween the propagating states forward or backward. The\noutgoing (+) and incoming ( −) scattering eigenstates\nof the static Hamiltonian ˆHstare written as/vextendsingle/vextendsingle/vextendsingleψ(±)\ns,q(ǫ)/angbracketrightBig\n,\nwhichform anothercomplete basiswith orthogonalityre-\nlations/angbracketleftBig\nψ(±)\ns,q(ǫ)/vextendsingle/vextendsingle/vextendsingleψ(±)\ns′,q′(ǫ′)/angbracketrightBig\n=δs,s′δq,q′δ(ǫ−ǫ′).33These\nwave functions can be expressed as/vextendsingle/vextendsingle/vextendsingleψ(±)\ns,q(ǫ)/angbracketrightBig\n= [1 +\nˆG(±)\nstˆV]|ϕs,q/an}b∇acket∇i}ht, where the retarded (+) and advanced ( −)\nGreen’s functions read ˆG(±)\nst(ǫ) = (ǫ±iη−ˆHst)−1. By\nexpanding Γ abin the basis of outgoing wave functions,\n|ψ(+)\ns,q/an}b∇acket∇i}ht, the energy dissipation (59) becomes\nΓab=π/summationdisplay\nsq,s′q′/integraldisplay\ndǫ/parenleftbigg\n−∂fs,q\n∂ǫ/parenrightbigg/angbracketleftBig\nψ(+)\ns,q/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns′,q′/angbracketrightBig/angbracketleftBig\nψ(+)\ns′,q′/vextendsingle/vextendsingle/vextendsingle∂bˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns,q/angbracketrightBig\n, (60)\nwhere wave functions should be evaluated at the energy ǫ.\nLet us now compare this result, Eq. (60), to the direct scattering matrix expression for the energy dissipation,\nEq. (40). The S-matrix operator can be written in terms of the T-matrix as ˆS(ǫ;ξ) = 1−2πiˆT(ǫ;ξ), where the\nT-matrix is defined recursively by ˆT=ˆV[1+ˆG(+)\nstˆT]. We then find\n∂ˆT\n∂ξa=/bracketleftBig\n1+ˆVˆG(+)\nst/bracketrightBig\n∂aˆH/bracketleftBig\n1+ˆG(+)\nstˆV/bracketrightBig\n.\nThe change in the scattering matrix appearing in Eq. (40) is then\n∂Ss′q′,sq\n∂ξa=−2πi/an}b∇acketle{tϕs,q|/bracketleftBig\n1+ˆVˆG(+)\nst/bracketrightBig\n∂aˆH/bracketleftBig\n1+ˆG(+)\nstˆV/bracketrightBig\n|ϕs′,q′/an}b∇acket∇i}ht=−2πi/angbracketleftBig\nψ(−)\ns′,q′/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns′,q′/angbracketrightBig\n. (61)\nSince\n/angbracketleftBig\nψ(−)\ns,q(ǫ)/vextendsingle/vextendsingle/vextendsingle=/summationdisplay\ns′q′Ssq,s′q′/angbracketleftBig\nψ(+)\ns′q′(ǫ)/vextendsingle/vextendsingle/vextendsingle(62)\nandSS†= 1, we can write the linear response result,\nEq. (60), as energy pumping (40). This completes our\nproof of the equivalence between adiabatic energy pump-\ningintermsofthe S-matrixandtheKubolinearresponse\ntheory.VI. CONCLUSIONS\nWe have shown that most aspects of magnetization\ndynamics in ferromagnets can be understood in terms of\nthe boundary conditions to normal metal contacts, i.e.\na scattering matrix. By using the established numerical\nmethods to compute electron transport based on scatter-\ning theory, this opens the way to compute dissipation in\nferromagnets from first-principles. In particular, our for-9\nmalism should work well for systems with strong elastic\nscattering due to a high density of large impurity poten-\ntials or in disordered alloys, including Ni 1−xFex(x= 0.2\nrepresents the technologically important “permalloy”).\nThe dimensionless Gilbert damping tensors (41) for\nmacrospin excitations, which can be measured directly\nin terms of the broadening of the ferromagnetic reso-\nnance, havebeen evaluated for Ni 1−xFexalloysby ab ini-\ntiomethods.42Permalloy is substitutionally disordered\nand damping is dominated by the spin-orbit interaction\nin combination with disorder scattering. Without ad-\njustable parameters good agreement has been obtained\nwith the available low temperature experimental data,\nwhich is a strong indication of the practical value of our\napproach.\nIn clean samples and at high temperatures, the\nelectron-phonon scattering importantly affects damping.\nPhonons are not explicitly included here, but the scat-\ntering theory of Gilbert damping can still be used for\na frozen configuration of thermally displaced atoms, ne-\nglecting the inelastic aspect of scattering.12\nWhile the energy pumping by scattering theory has\nbeen applied to described magnetization damping,15it\ncan be used to compute other dissipation phenomena.\nThis has recently been demonstrated for the case of\ncurrent-induced mechanical forces and damping,43with\na formalism analogous to that for current-induced mag-\nnetization torques.13,14\nAcknowledgments\nWe would like to thank Kjetil Hals, Paul J. Kelly, Yi\nLiu, Hans Joakim Skadsem, Anton Starikov, and Zhe\nYuan for stimulating discussions. This work was sup-\nported by the EC Contract ICT-257159 “MACALO,”\ntheNSFunderGrantNo.DMR-0840965,DARPA,FOM,\nDFG, and by the Project of Knowledge Innovation Pro-\ngram(PKIP) of Chinese Academy of Sciences, Grant No.\nKJCX2.YW.W10\nAppendix A: Adiabatic Pumping\nAdiabatic pumping is the current response to a time-\ndependent scattering potential to first order in the time-\nvariation or “pumping” frequency when all reservoirsare\nat the same electro-chemical potential.38A compact for-\nmulation of the pumping charge current in terms of the\ninstantaneous scattering matrix was derived in Ref. 39.\nIn the same spirit, the energy current pumped out of the\nscattering region has been formulated (at zero tempera-\nture) in Ref. 27. Some time ago, we extended the charge\npumping concept to include the spin degree of free-\ndomandascertainedits importancein magnetoelectronic\ncircuits.10More recently, we demonstrated that the en-\nergyemitted byaferromagnetwith time-dependentmag-\nnetizations into adjacent conductors is not only causedby interface spin pumping, but also reflects the energy\nloss by spin-flip processes inside the ferromagnet15and\ntherefore Gilbert damping. Here we derive the energy\npumping expressions at finite temperatures, thereby gen-\neralizing the zero temperature results derived in Ref. 27\nand used in Ref. 15. Our results differ from an earlier ex-\ntension to finite temperature derived in Ref. 40 and we\npoint out the origin of the discrepancies. The magneti-\nzation dynamics must satisfy the fluctuation-dissipation\ntheorem, which is indeed the case in our formulation.\nWe proceed by deriving the charge, spin, and energy\ncurrentsintermsofthetimedependenceofthescattering\nmatrix of a two-terminal device. The transport direction\nisxand the transverse coordinates are ̺= (y,z). An\narbitrary single-particle Hamiltonian can be decomposed\nas\nH(r) =−/planckover2pi12\n2m∂2\n∂x2+H⊥(x,̺), (A1)\nwhere the transverse part is\nH⊥(x,̺) =−/planckover2pi12\n2m∂2\n∂̺2+V(x,̺).(A2)\nV(̺) is an elastic scattering potential in 2 ×2 Pauli\nspin space that includes the lattice, impurity, and\nself-consistent exchange-correlation potentials, including\nspin-orbit interaction and magnetic disorder. The scat-\nteringregionisattachedtoperfect non-magneticelectron\nwave guides (left α=Land rightα=R) with constant\npotential and without spin-orbit interaction. In lead α,\nthe transverse part of the 2 ×2 spinor wave function\nϕ(n)\nα(x,̺) and its corresponding transverse energy ǫ(n)\nα\nobey the Schr¨ odinger equation\nH⊥(̺)ϕ(n)\nα(̺) =ǫ(n)\nαϕ(n)\nα(̺), (A3)\nwherenis the spin and orbit quantum number. These\ntransverse wave guide modes form the basis for the ex-\npansion of the time-dependent scattering states in lead\nα=L,R:\nˆΨα=/integraldisplay∞\n0dk√\n2π/summationdisplay\nnσϕ(n)\nα(̺)eiσkxe−iǫ(nk)\nαt//planckover2pi1ˆc(nkσ)\nα,(A4)\nwhere ˆc(nkσ)\nαannihilates an electron in mode nincident\n(σ= +) or outgoing ( σ=−) in leadα. The field opera-\ntors satisfy the anticommutation relation\n/braceleftBig\nˆc(nkσ)\nα,ˆc†(n′k′σ′)\nβ/bracerightBig\n=δαβδnn′δσσ′δ(k−k′).\nThe total energy is ǫ(nk)\nα=/planckover2pi12k2/2m+ǫ(n)\nα. In the leads\nthe particle, spins, and energy currents in the transport10\ndirection are\nˆI(p)=/planckover2pi1\n2mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†∂ˆΨ\n∂x−∂ˆΨ†\n∂xˆΨ/parenrightBigg\n,(A5a)\nˆI(s)=/planckover2pi1\n2mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†σ∂ˆΨ\n∂x−∂ˆΨ†\n∂xσˆΨ/parenrightBigg\n,(A5b)\nˆI(e)=/planckover2pi1\n4mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†H∂ˆΨ\n∂x−∂ˆΨ†\n∂xHˆΨ/parenrightBigg\n+H.c.,\n(A5c)\nwhere we suppressed the time tand lead index α,σ=\n(σx,σy,σz) is a vector of Pauli matrices, and Tr sdenotes\nthe trace in spin space. Note that the spin current Is\nflows in the x-direction with polarization vector Is/Is.\nTo avoid dependence on an arbitrary global potential\nshift, it is convenient to work with heat ˆI(q)rather than\nenergy currents ˆI(ǫ):\nˆI(q)(t) =ˆI(ǫ)(t)−µˆI(p)(t), (A6)\nwhereµis the chemical potential. Inserting the waveg-uide representation (A4) into (A5), the particle current\nreads41\nˆI(p)\nα=/planckover2pi1\n4πm/integraldisplay∞\n0dkdk′/summationdisplay\nnσσ′(σk+σ′k′)×\nei(σk−σ′k′)xe−i/bracketleftBig\nǫ(nk)\nα−ǫ(nk′)\nα/bracketrightBig\nt//planckover2pi1ˆc†(nk′σ′)\nαˆc(nkσ)\nα.(A7)\nWeareinterestedinthelow-frequencylimitoftheFourier\ntransforms I(x)\nα(ω) =/integraltext∞\n−∞dteiωtI(x)\nα(t). Following Ref.\n41 we assume long wavelengths such that only the inter-\nvals withk≈k′andσ=σ′contribute. In the adiabatic\nlimitω→0 this approach is correct to leading order in\n/planckover2pi1ω/ǫF,whereǫFis the Fermi energy. By introducing the\n(current-normalized) operator\nˆc(nσ)\nα(ǫ(nk)\nα) =1/radicalBig\ndǫ(nkσ)\nα\ndkˆc(nkσ)\nα, (A8)\nwhich obey the anticommutation relations\n/braceleftBig\nˆc(nσ)\nα(ǫα),ˆc†(n′σ′)\nβ(ǫβ)/bracerightBig\n=δαβδnn′δσσ′δ(ǫα−ǫβ). (A9)\nThe charge current can be written as\nˆI(c)\nα(t) =1\n2π/planckover2pi1/integraldisplay∞\nǫ(n)\nαdǫdǫ′/summationdisplay\nnσσe−i(ǫ−ǫ′)t//planckover2pi1ˆc†(nσ)\nα(ǫ′)ˆc(nσ)\nα(ǫ). (A10)\nWeoperateinthe linearresponseregimeinwhichapplied\nvoltages and temperature differences as well as the exter-\nnally induced dynamics disturb the system only weakly.\nTransport is then governed by states close to the Fermi\nenergy. We may therefore extend the limits of the en-\nergy integration in Eq. (A10) from ( ǫ(n)\nα,∞) to (−∞\nto∞). We relabel the annihilation operators so that\nˆa(nk)\nα= ˆc(nk)\nα+denotes particles incident on the scattering\nregion from lead αandˆb(nk)\nα= ˆc(nk)\nα−denotes particles\nleavingthe scatteringregionbylead α. Using the Fourier\ntransforms\nˆc(nσ)\nα(ǫ) =/integraldisplay∞\n−∞dtˆc(nσ)\nα(t)eiǫt//planckover2pi1, (A11)\nˆc(nσ)\nα(t) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫˆc(nσ)\nα(ǫ)e−iǫt//planckover2pi1,(A12)\nwe obtain in the low-frequency limit41\nˆI(p)\nα(t) = 2π/planckover2pi1/bracketleftBig\nˆa†\nα(t)ˆaα(t)−ˆb†\nα(t)ˆbα(t)/bracketrightBig\n,(A13)\nwhereˆbαis a column vector of the creation operators forall wave-guidemodes {ˆb(n)\nα}. Analogouscalculations lead\nto the spin current\nˆI(s)\nα= 2π/planckover2pi1/parenleftBig\nˆa†\nασˆaα−ˆb†\nασˆbα/parenrightBig\n(A14)\nand the energy current\nˆI(e)\nα=iπ/planckover2pi12/parenleftBigg\nˆa†\nα∂ˆaα\n∂t−ˆb†\nα∂ˆbα\n∂t/parenrightBigg\n+H.c..(A15)\nNext, we express the outgoing operators ˆb(t) in terms\nof the incoming operators ˆ a(t) via the time-dependent\nscattering matrix (in the space spanned by all waveguide\nmodes, including spin and orbit quantum number):\nˆbα(t) =/summationdisplay\nβ/integraldisplay\ndt′Sαβ(t,t′)ˆaβ(t′).(A16)\nWhen the scattering region is stationary, Sαβ(t,t′) only\ndepends on the relative time difference t−t′, and its\nFourier transform with respect to the relative time is\nenergy independent, i.e.transport is elastic and can11\nbe computed for each energy separately. For time-\ndependent problems, Sαβ(t,t′) also depends on the total\ntimet+t′and there is an inelastic contribution to trans-\nport as well. An electron can originate from a lead with\nenergyǫ, pick up energy in the scattering region and end\nup in the same or the other lead with different energy ǫ′.\nThe reservoirs are in equilibrium with controlled lo-\ncal chemical potentials and temperatures. We insert the\nS-matrix (A16) into the expressions for the currents,Eqs. (A13), (A14), (A15), and use the expectation value\nat thermal equilibrium\n/angbracketleftBig\nˆa†(n)\nα(t2)ˆa(m)\nβ(t1)/angbracketrightBig\neq=δnmδαβfα(t1−t2)/2πℏ,(A17)\nwherefβ(t1−t2) = (2π/planckover2pi1)−1/integraltext\ndǫ−iǫ(t1−t2)//planckover2pi1fα(ǫ) and\nfα(ǫ) is the Fermi-Dirac distribution of electrons with\nenergyǫin theα-th reservoir. We then find\n2π/planckover2pi1/angbracketleftBig\nˆb†\nα(t)ˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2S∗\nαβ(t,t2)Sαβ(t,t1)fβ(t1−t2), (A18)\n2π/planckover2pi1/angbracketleftBig\nˆb†\nα(t)σˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2S∗\nαβ(t,t2)σSαβ(t,t1)fβ(t1−t2), (A19)\n2π/planckover2pi1/angbracketleftBig\n/planckover2pi1∂tˆb†\nα(t)ˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2/bracketleftbig\n/planckover2pi1∂tS∗\nαβ(t,t2)/bracketrightbig\nSαβ(t,t1)fβ(t1−t2). (A20)\nNext, we use the Wigner representation (B1):\nS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫS/parenleftbiggt+t′\n2,ǫ/parenrightbigg\ne−iǫ(t−t′)//planckover2pi1, (A21)\nand by Taylor expanding the Wigner represented S-matrix S((t+t′)/2,ǫ) aroundS(t,ǫ), S((t+t′)/2,ǫ) =/summationtext∞\nn=0∂n\ntS(t,ǫ)(t′−t)n/(2nn!), we find\nS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2S(t,ǫ) (A22)\nand\n/planckover2pi1∂tS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2/parenleftbigg1\n2/planckover2pi1∂t−iǫ/parenrightbigg\nS(t,ǫ). (A23)\nThe factor 1 /2 scaling the term /planckover2pi1∂tS(t,ǫ) arises from commuting ǫwithei/planckover2pi1∂ǫ∂t/2. The currents can now be evaluated\nas\nI(c)\nα(t) =−1\n2π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/planckover2pi1/2S†\nβα(ǫ,t)/parenrightBig/parenleftBig\nei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−fα(ǫ)/bracketrightBig\n(A24a)\nI(s)\nα(t) =−1\n2π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/planckover2pi1/2S†\nβα(ǫ,t)/parenrightBig\nσ/parenleftBig\nei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)/bracketrightBig\n(A24b)\nI(ǫ)\nα(t) =−1\n4π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/2/planckover2pi1(−i/planckover2pi1∂t/2+ǫ)S†\nβα(ǫ,t)/parenrightBig/parenleftBig\ne+i∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−ǫfα(ǫ)/bracketrightBig\n−1\n4π/planckover2pi1/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/2/planckover2pi1S†\nβα(ǫ,t)/parenrightBig/parenleftBig\nei∂ǫ∂t/2/planckover2pi1(i/planckover2pi1∂t/2+ǫ)Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−ǫfα(ǫ)/bracketrightBig\n,(A24c)\nwhere the adjoint of the S-matrix has elements S†(n′,n)\nβα=S∗(n,n′)\nαβ.\nWe are interested in the average (DC) currents, where simplified ex pressions can be found by partial integration\nover energy and time intervals. We will consider the total DC curren tsout ofthe scattering region, I(out)=−/summationtext\nαIα,\nwhen the electrochemical potentials in the reservoirs are equal, fα(ǫ) =f(ǫ) for allα. The averaged pumped spin and12\nenergy currents out of the system in a time interval τcan be written compactly as\nI(c)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−f(ǫ)/bracerightbigg\n, (A25a)\nI(s)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg\nσ/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†/bracerightbigg\n, (A25b)\nI(ǫ)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−ǫf(ǫ)/bracerightbigg\n+1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg/parenleftbigg\n−i/planckover2pi1∂S†\n∂t/parenrightbigg/bracerightbigg\n, (A25c)\nwhere Tr is the trace over all waveguide modes (spin\nand orbital quantum numbers). As shown in Ap-\npendix C the charge pumped into the reservoirs vanishes\nfor a scattering matrix with a periodic time dependence\nwhen,integrated over one cycle:\nI(p)\nout= 0. (A26)\nThis reflects particle conservation; the number of elec-\ntrons cannot build up in the scattering region for peri-\nodic variations ofthe system. We can showthat a similar\ncontribution to the energy current, i.e.the first line in\nEq. (A25c), vanishes, leading to to the simple expression\nI(e)\nout=−i\n2π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg∂S†\n∂t/bracerightbigg\n.\n(A27)\nExpanded to lowest order in the pumping frequency the\npumped spin current (A25b) becomes\nI(s)\nout=1\n2π/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg\nSS†f−i/planckover2pi1\n2∂S\n∂tS†∂ǫf/parenrightbigg\nσ/bracerightbigg\n(A28)\nThis formula is not the most convenient form to com-\npute the current to specified order. SS†also contains\ncontributions that are linear and quadratic in the pre-\ncession frequency since S(t,ǫ) is theS-matrix for a time-\ndependent problem. Instead, wewouldliketoexpressthe\ncurrent in terms of the frozenscattering matrix Sfr(t,ǫ).\nThe latter is computed for an instantaneous, static elec-\ntronic potential. In our case this is determined by a mag-\nnetization configuration that depends parametrically on\ntime:Sfr(t,ǫ) =S[m(t),ǫ]. Using unitarity of the time-dependentS-matrix (as elaborated in Appendix C), ex-\npand it to lowest order in the pumping frequency, and\ninsert it into (A28) leads to39\nI(s)\nout=i\n2π/summationdisplay\nβ/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/braceleftbigg∂Sfr\n∂tS†\nfrσ/bracerightbigg\n.\n(A29)\nWe evaluate the energy pumping by expanding (A27)\nto second order in the pumping frequency:\nI(e)\nout=/planckover2pi1\n4π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg\n−ifS∂S†\n∂t−(∂ǫf)1\n2∂S\n∂t∂S†\n∂t/bracerightbigg\n.\n(A30)\nAs a consequence of unitarity of the S-matrix (see Ap-\npendix C), the first term vanishes to second order in the\nprecession frequency:\nI(e)\nout=/planckover2pi1\n4π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/braceleftBigg\n∂Sfr\n∂t∂S†\nfr\n∂t/bracerightBigg\n,(A31)\nwhere,at this point , we may insert the frozen scattering\nmatrix since the current expression is already propor-\ntional to the square of the pumping frequency. Further-\nmore, since there is no net pumped charge current in\none cycle (and we are assuming reservoirs in a common\nequilibrium), the pumped heat current is identical to the\npumped energy current, I(q)\nout=I(e)\nout.\nOur expression for the pumped energy current (A31)\nagrees with that derived in Ref. 27 at zero temperature.\nOur result (A31) differs from Ref. 40 at finite tempera-\ntures. The discrepancy can be explained as follows. In-\ntegration by parts over time tin Eq. (A27), using\n/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\ni/planckover2pi1∂S\n∂t/bracketrightbigg\nS†= 2/bracketleftbigg\nǫf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−2/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†,(A32)\nand the unitarity condition from Appendix C,\n/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†=/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫǫf(ǫ), (A33)13\nthe DC pumped energy current can be rewritten as\nI(ǫ)\nout=1\nπ/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nǫf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−ǫf(ǫ)/bracerightbigg\n. (A34)\nNext, we expand this to the second order in the pumping frequency and find\nI(ǫ)\nout=1\nπ/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg\nǫf(ǫ)/parenleftbig\nSS†−1/parenrightbig\n−ǫ(∂ǫf)i/planckover2pi1\n2∂S\n∂tS†−ǫ(∂2\nǫf)/planckover2pi12\n8∂2S\n∂t2S†/bracerightbigg\n. (A35)\nThis form of the pumped energy current, Eq. (A35),\nagrees with Eq. (10) in Ref. 40 if one ( incorrectly ) as-\nsumesSS†= 1. Although for the frozen scattering ma-\ntrix,SfrS†\nfr= 1, unitarity does not hold for the Wigner\nrepresentation of the scattering matrix to the second or-\nder in the pumping frequency. ( SS†−1) therefore does\nnot vanish but contributes to leading order in the fre-\nquency to the pumped current, which may not be ne-\nglected at finite temperatures. Only when this term is\nincluded our new result Eq. (A31) is recovered.\nAppendix B: Fourier transform and Wigner\nrepresentation\nThere is a long tradition in quantum theory to trans-\nform the two-time dependence of two-operator correla-\ntion functions such as scattering matrices by a mixed\n(Wigner)representationconsistingofaFouriertransform\nover the time difference and an average time, which has\ndistinct advantages when the scattering potential varies\nslowlyintime.44Inordertoestablishconventionsandno-\ntations, we present here a short exposure how this works\nin our case.\nThe Fourier transform of the time dependent annihi-\nlation operators are defined in Eqs. (A11) and (A12).Consider a function Athat depends on two times t1\nandt2,A=A(t1,t2). The Wigner representation with\nt= (t1+t2)/2 andt′=t1−t2is defined as:\nA(t1,t2) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫA(t,ǫ)e−iǫ(t1−t2)//planckover2pi1,(B1)\nA(t,ǫ) =/integraldisplay∞\n−∞dt′A/parenleftbigg\nt+t′\n2,t−t′\n2/parenrightbigg\neiǫt′//planckover2pi1.(B2)\nWe also need the Wigner representation of convolutions,\n(A⊗B)(t1,t2) =/integraldisplay∞\n−∞dt′A(t1,t′)B(t′,t2).(B3)\nBy a series expansion, this can be expressed as44\n(A⊗B)(t,ǫ) =e−i(∂A\nt∂B\nǫ−∂B\nt∂A\nǫ)/2A(t,ǫ)B(t,ǫ) (B4)\nwhich we use in the following section.\nAppendix C: Properties of S-matrix\nHere we discuss some general properties of the two-\npoint time-dependent scattering matrix. Current conser-\nvation is reflected by the unitarity of the S-matrix which\ncan be expressed as\n/summationdisplay\nβn′s′/integraldisplay\ndt′S(α1β)\nn1s1,n′s′(t1,t′)S(α2β)∗\nn2s2,n′s′(t′,t2) =δn1n2δs1s2δα1α2δ(t1−t2). (C1)\nPhysically, this means that a particle entering the scattering region from a lead αat some time tis bound to exit the\nscattering region in some lead βat another (later) time t′. Using Wigner representation (B1) and integrating over\nthe local time variable, this implies (using Eq. (B4))\n1 =/parenleftbig\nS⊗S†/parenrightbig\n(t,ǫ) =e−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S(t,ǫ)S†(t,ǫ), (C2)\nwhere 1 is a unit matrix in the space spanned by the wave guide modes ( labelled by spin sand orbital quantum\nnumbern). Similary, we find\n1 =/parenleftbig\nS†⊗S/parenrightbig\n(t,ǫ) =e+i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S†(t,ǫ)S(t,ǫ). (C3)\nTo second order in the precession frequency, by respectively sub tracting and summing Eqs. (C2) and (C3) give\nTr/braceleftbigg∂S\n∂t∂S†\n∂ǫ−∂S\n∂ǫ∂S†\n∂t/bracerightbigg\n= 0 (C4)14\nand\nTr/braceleftbig\nSS†−1/bracerightbig\n= Tr/braceleftbigg∂2S\n∂t2∂2S†\n∂ǫ2−2∂2S\n∂t∂ǫ∂2S†\n∂t∂ǫ+∂2S\n∂ǫ2∂2S†\n∂t2/bracerightbigg\n. (C5)\nFurthermore, foranyenergydependent function Z(ǫ)andarbitrarymatrixin thespacespannedbyspinandtransverse\nwaveguide modes Y, Eq. (C2) implies\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫZ(ǫ)Tr/braceleftbigg/bracketleftbigg\ne−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S(t,ǫ)S†(t,ǫ)−1/bracketrightbigg\nY/bracerightbigg\n= 0. (C6)\nIntegration by parts with respect to tandǫgives\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\ne−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S\nt∂ZS†\nǫ/parenrightBig\n/2S(t,ǫ)Z(ǫ)S†(t,ǫ)−Z(ǫ)/bracketrightbigg\nY/bracerightbigg\n= 0, (C7)\nwhich can be simplified to\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg/bracketleftbigg\nZ/parenleftbigg\nǫ+i\n2∂\n∂t/parenrightbigg\nS(t,ǫ)/bracketrightbigg\nS†(t,ǫ)−Z(ǫ)/parenrightbigg\nY/bracerightbigg\n= 0. (C8)\nSimilarly from (C3), we find\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg\nS†(t,ǫ)/bracketleftbigg\nZ/parenleftbigg\nǫ−i\n2∂\n∂t/parenrightbigg\nS(t,ǫ)/bracketrightbigg\n−1/parenrightbigg\nY/bracerightbigg\n= 0. (C9)\nUsing this result for Y= 1 andZ(ǫ) =f(ǫ) in the\nexpression for the DC particle current (A25a), we see\nthat unitarity indeed implies particle current conserva-\ntion,/summationtext\nαI(c)\nα= 0 for a time-periodic potential. 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Pandya,3 Sujeet Chaudhary,4 \nRimantas Brucas,1 Peter Svedlindh,1* and Ankit Kumar1,,* \n1Department of Materials Science and Engineering , Uppsala University, Box 35, SE-751 03 \nUppsala, Sweden \n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, \nSweden \n3Departments of Physics and Materials Engineering, Indian Institute of Technology Jammu, \nJammu 181221, India \n4Thin Film Laboratory, Department of Physics, Indian Institute of Technology Delhi, \nNew Delhi 110016, India \n \nA large anti -damping spin -obit torque (SOT) efficiency in magnetic heterostructures is a \nprerequisite to realize energy efficient spin torque based magnetic memories and logic \ndevices . The efficiency can be characterized in terms of the spin-orbit fields generated by \nanti-damping torque s when an electric current is passed through the non-magnet ic layer . \nWe report a giant spin-orbit field of 48.96 (27.50) mT at an applied current density of \n 𝟏 𝐌𝐀𝐜𝐦−𝟐 in β-W interfaced Co 60Fe40 (Ni81Fe19)/TiN epitaxial structures due to an anti-\ndamping like torque , which result s in a magnetization auto -oscillation current density as \nlow as 1.68 (3.27) 𝐌𝐀𝐜𝐦−𝟐. The spin -orbit field value increases with decrease of β-W layer \nthickness , which affirms th at epitaxial surface states are responsible for the extraordinary \nlarge efficiency . SOT induced energy efficient in-plane magnetization switching in large \n𝟐𝟎×𝟏𝟎𝟎 𝛍𝐦𝟐 structure s has been demonstrated by Kerr microscopy and the finding s are \nsupported by results from micromagnetic simulations . The observed giant SOT efficiencies \nin the studied all -epitaxial heterostructure s are comparable to values reported for \ntopological insulators . These results confirm that by utilizing epitaxial material \ncombinations an extraordinary large SOT efficiency can be achieved using semiconducting \nindustry compatible 5d heavy metals , which provides immediate solutions for the \nrealization of energy efficient spin-logic devices . \n 2 Pure spin current based spintronic devices have advantages over conventional \nmicroelectronic devices owing to low energy dissipation, fast switching, and high -speed data \nprocessing., and can be integrated with microelectronic semiconductor devices for better \nfunctionality1-4. These new spintronic devices mainly work on the principle of spin manipulation \nemploying the spin Hall effect (SHE) and the Rashba -Edelstein effect (REE). The basic building \nblock of the spin devices is comprised of ferromagnetic (FM)/ non-magnetic (NM) bilayers; and \nthe NM layers and their interface s should possess strong relativistic spin-orbit interaction (SOI). \nThe SOI can be of bulk as well as of interfacial origin , generating damping -like (DL) and field-\nlike (FL) spin-orbit torques (SOTs)4-8. A charge current applied to SHE and REE based devices \ngenerates a transverse spin current and therefore a SOT at the FM/NM interface , which can be \nused to manipulate the FM state4-7. In contrast, devices based on the inverse effects, the inverse \nSHE (ISHE) and inverse REE (IREE) convert a spin current generated by spin pumping into a \ncharge current in the NM layer by the ISHE and at the FM/NM interface by the IREE6-12. Bulk \nSOI of the NM layer is responsible for the SHE and ISHE mechanism s2, 4, 5, 8, while interfacial \nSOI is responsible for the REE and IREE mechanism s6, 7, 9, 10. A strong DL SOT in FM \n(Ga,Mn) As thin films7 and Py/CuO x heterostructures10 has been reported , the origin of which lies \nin crystal inversion asymmetry and interfacial SOI induced Berry curvature7, 10, respectively . \nHigh value s of the spin Hall angle, which is a measure of the SHE , have been reported , of the \norder of 100% in Ni 0.6Cu0.4 13, 1880 % in the topological insulator BixSe1–x 14, and 5200% in the \nconducting topological insulator Bi0.9Sb0.1 15 thin films . The topological insulators exhibit the \nhighest value s of the spin Hall angle , however their thin film fabrication and integration in \nembedded memories are challenging. Therefore, present technological focus is to find CMOS \n(complementary metal -oxide semiconductor) technology compatible material combinations with 3 even large r SOTs , or more specifically large r spin angular momentum transfer across the \ninterfaces in magnetic heterostructures. Spin manipulation in the bulk or at the interface plays a \ndecisive role in building the next generation of spintronic devices, viz. spin torque magnetic \nrandom -access memories (ST-MRAMs), spin logic devices, ST -transistors and ST nano -\noscillators 4-10. \nTo optimize the bilayer stacks for devices there is a need to control interfacial properties like \nspin backflow, spin memory loss and magnetic proximity induced effect s that may arise at/near \nthe interface in FM/NM bilayer structures, properties that demean the overall spin transport in \nsuch structures3, 16-22. It is known that the spin current at the interface also exhibits spin memory \nloss in the presence of interfacial SOI, by creating a parallel relaxation path for the spin current3, \n19-22. The spin memory loss can be quantified as the relative amount of spin current which is \ndissipated while passing through the FM/NM interface. Recently it was reported that interface \nalloying and abrupt interfaces may enhance the spin memory loss22. Therefore , it is apparent that \nto reduce the spin memory loss epi taxial interfaces constitute a potential solution. \nIn a quest to enhance the SOT efficiency of the 5d metal heterostructures , in this work we \nhave fabricated 𝛽-W/Co 60Fe40, Ni81Fe19/TiN /Si all- epitaxial heterostructures in which all the \nlayers are epitaxial . The SOT efficiency of these structures has been determined by performing \nspin torque ferromagnetic resonance (STFMR) and planar Hall effect (PHE) measurements. In-\nplane magnetization switching has been evidenced by recording Kerr microscopy images. The dc \nbias dependent changes of the effective damping and the PHE results confirm the presence of \nsignificantly large interfacial DL torques in these heterostructures whose origin lie s both at the \n𝛽-W and TiN interface s with the epitaxial ferromagnet ic layer. The SOT induced magnetization 4 auto-oscillations and magnetization switching current density at which the effective damping \nreverses sign is comparable to the values reported for conductive topological insulators15. \nStructural Characterization To confirm the epitaxial growth of CoFe and NiFe (henceforth \nreferred to a s Py) on the epi-TiN buffered Si substrate, texture analyses were performed by \nrecording X-ray diffraction ( XRD ) pole figures and scanning transmission electron microscopy \n(STEM) images . \nFigure 1 (a) show s the pole figure XRD patterns of the CoFe(022) plane at 2=45.2o for the \nW/CoFe/TiN/Si thin films , while Fig. 1(b) show s the pole figure XRD patterns of the Py( 111) \nplane at 2 = 44.2o for the W/Py/TiN/Si thin films confirming the epitaxial quality of the thin \nfilms ; here W is read as β-W. Figures 1(c ) and ( d) show the cross -section STEM image s of the \nW/CoFe/TiN/Si and W/Py/TiN/Si heterostructures, respectively. The STEM images give \nevidence of sharp epitaxial interfaces in both the structures . The interface roughness of each \nindividual layer determined by X-ray reflectivity is in the range of 1 nm, and these values are \nalso closely match ing with previous ly reported values23-25. \nSpin -orbit torque ferromagnetic resonance To determine the SOT s, STFMR measurements \nwere performed on patterned (20 100 m2) all-epitaxial W/CoFe , Py /TiN/Si structures . \nSchematic figures of the STFMR setup and the torques acting on the magnetization are shown in \nFigs. 2(a) and (b) (see Ref. 26 for measurement details ). In the STFMR measurements, the \nmicrowave current 𝐼𝑟𝑓 of a constant frequency ƒ generates an Oersted field and a transverse spin \ncurrent in the NM layer and at its interface with the FM layer . The torques due to the Oersted \nfield (𝜏𝑂𝑒) and due to the transverse spin current (𝜏𝐴𝐷) contribute with anti -symmetric and \nsymmetric profiles, respectively, to the FMR line-shape . The anti -damping 𝜏𝐴𝐷 torque , 5 dominantly generated by the transverse spin current, acts against the intrinsic damping torque 𝜏𝐷 \nin the FM layer . In case of interfacial SOI, the REE contributes dominantly with a field-like \ntorque (𝜏𝐹𝐿) with direction opposite to that of 𝜏𝑂𝑒. At resonance , the temporal variation of the \nmagnetization vector with respect to the direction of 𝐼𝑟𝑓, in combination with the anisotropic \nmagnetoresistance of the FM layer and spin magnetoresistance of the interface , generates a time \nvarying resistance that mixes with 𝐼𝑟𝑓 yielding a dc voltage output . In our case, u sing a low -\nfrequency (1 kHz) modulation of 𝐼𝑟𝑓 the STFMR signal is detected using a lock -in amplifier. The \nobserved STFMR spectra exhibit a combination of symmetric and anti -symmetric Lorentz ian \nweight factors27-33, as shown in Fig s. 2(c) and (d). The STFMR spect rum at constant ƒ is \nexpressed as 𝑉𝑚𝑖𝑥=𝑉0 [𝑆𝐹𝑆(𝐻)+𝐴𝐹𝐴(𝐻)], where 𝑉0 is the amplitude of the mixing voltage. 𝑆 \nand 𝐴 are symmetric and anti -symmetric Lorentzian weight factors, accounting for anti -damping \nand f ield-like torques, respectively . 𝐹𝑆(𝐻) is the symmetric and 𝐹𝐴(𝐻) is the anti-symmetric \nLorentzian function (for details see Ref. 28). \nIn order to make a reliable determination of the SOT efficiency , one ca n use the dc induced \nchange of the damping , often referred to as the modulation of damping (MOD) method , for \nwhich the spin pumping (ISHE ) and field-like contribution s are absent . In the MOD method , one \nmeasures th e change of the STFMR linewidth (Δ𝐻) applying a constant dc to the patterned \nstructure. From the linewidth versus frequency behavior we can determine how the effective \ndamping value varies with the applied dc 27-29. \nThe 𝐼𝑑𝑐 induced change of the effective Gilbert damping 𝛼𝑒𝑓𝑓(𝐼𝑑𝑐) is given by28,29 \n𝛼𝑒𝑓𝑓(𝐼𝑑𝑐)−𝛼𝑒𝑓𝑓(𝐼𝑑𝑐=0)=(sin𝜑\n(𝐻𝑟+0.5𝑀𝑒𝑓𝑓)𝜇0𝑀𝑆𝑡𝐹𝑀ħ\n2𝑒)𝐽𝑆, (1) 6 where 𝜑, 𝐻𝑟, 𝑀𝑒𝑓𝑓, 𝑀𝑆, ħ and 𝑒 are the angle between the current and the magnetization, \nresonance field, effective magnetization, saturation magnetization, reduced Planck’s constant and \nelectr ic charge of an electron , respectively. T he spin current density , 𝐽𝑆=𝑆𝐻𝑀𝑂𝐷𝐽𝐶,𝑑𝑐(𝑊+𝑇𝑖𝑁); \n𝐽𝐶,𝑑𝑐(𝑊+𝑇𝑖𝑁)=𝐼𝑑𝑐[(1\n𝐴𝑊𝑅𝐹𝑀×𝑅𝑇𝑖𝑁\n(𝑅𝑊×𝑅𝑇𝑖𝑁)+(𝑅𝐹𝑀×𝑅𝑇𝑖𝑁)+(𝑅𝑊× 𝑅𝐹𝑀))\n+(1\n𝐴𝑇𝑖𝑁𝑅𝐹𝑀×𝑅𝑊\n(𝑅𝑊×𝑅𝑇𝑖𝑁)+(𝑅𝐹𝑀×𝑅𝑇𝑖𝑁)+(𝑅𝑊×𝑅𝐹𝑀))]. \nHere 𝐽𝐶,𝑑𝑐 is the charge current density in the non-magnetic layer s. 𝐴𝑊 and 𝐴𝑇𝑖𝑁 are the cross -\nsectional area s of the W and TiN layer s, and 𝑅𝑊, 𝑅𝑇𝑖𝑁 and 𝑅𝐹𝑀 are the resistance s of the W, TiN \nand FM layer s, respectively . The measured resistivity values of W, TiN, CoFe and Py are 3 -\ncm, 38 -cm, 32-cm, and 31-cm, respectively. The low resistivity of W is due to the \npresence of a conductive native WO x in β-W. In the W(6nm)/CoFe(10 nm)/TiN/Si structure 88% \nof 𝐼𝑑𝑐 passes through the W and TiN/Si layer s, while in the W(8nm)/Py(15nm)/TiN/Si structure \n84% of 𝐼𝑑𝑐 passes through the W and TiN/Si layer s. The spin Hall angle 𝑆𝐻𝑀𝑂𝐷 can be estimated \nby measuring the 𝐼𝑑𝑐 dependent rate of change of the effective damping , 𝜕𝛼𝑒𝑓𝑓(𝐼𝑑𝑐)𝜕𝐼𝑑𝑐⁄ . \nHence , 𝑆𝐻𝑀𝑂𝐷 can be expressed as29 \n𝑆𝐻𝑀𝑂𝐷=[𝜕𝛼𝑒𝑓𝑓\n𝜕𝐽𝐶,𝑑𝑐(𝑊+𝑇𝑖𝑁)⁄\n(sin𝜑\n(𝐻𝑟+𝑀𝑒𝑓𝑓)𝜇0𝑀𝑆𝑡𝐹𝑀ħ\n2𝑒)]. (2) \nSTFMR spectra were recorded for different 𝐼𝑑𝑐 in the range +5 mA to ‒5mA on all-\nepitaxial W(6nm)/CoFe(10nm)/TiN/Si and W(8nm)/Py(15nm)/TiN/Si . The recorded spectra \nwere fitted by using Lorentzian functions to determine the line -shape parameters . Figure 2(e) \nshows ƒ vs. 𝐻𝑟 data and figure 2(f) shows Δ𝐻 vs. ƒ data at respective applied dc for the two \nmulti -layer structures. The ƒ vs. 𝐻𝑟 data was fitted by using the in-plane Kittel equation to 7 determine the gyromagnetic ratio and 𝑀𝑒𝑓𝑓. The Δ𝐻 vs. ƒ data was fitted by using the standard \nexpression to determine 𝛼𝑒𝑓𝑓(𝐼𝑑𝑐), which are presented in Fig. 3(a). 𝛼𝑒𝑓𝑓(𝐼𝑑𝑐) increases from \n0.002 50.000 6 to 0.01 170.000 4 for W(6 nm)/ CoFe(10 nm)/TiN/Si as 𝐼𝑑𝑐 is varied from +5 mA \nto ‒5mA . To make this data more readable , the change in 𝐻 and 𝛼𝑒𝑓𝑓 with 𝐼𝑑𝑐, i.e. 𝐻(𝐼𝑑𝑐)−\n𝐻(𝐼𝑑𝑐=0) and 𝛼𝑒𝑓𝑓(𝐼𝑑𝑐)−𝛼𝑒𝑓𝑓(𝐼𝑑𝑐=0) vs. 𝐼𝑑𝑐, are plotted in Figs. 3(b) and ( c). A linear \ndecrease in 𝐻(𝐼𝑑𝑐)−𝐻(𝐼𝑑𝑐=0) with increasing 𝐼𝑑𝑐 for both samples clearly depicts the \nabsence of heating in our measurements. The observed variation of 𝛼𝑒𝑓𝑓(𝐼𝑑𝑐) is larger in \nW(6nm)/CoFe(10nm) /TiN/Si in comparison to W(8nm)/Py(15nm)/TiN/Si ,. The percentage \ncurrent -induced change of 𝛼𝑒𝑓𝑓(𝐼𝑑𝑐), defined as , 𝛼𝑒𝑓𝑓(𝐼𝑑𝑐)−𝛼𝑒𝑓𝑓(𝐼𝑑𝑐=0)\n𝛼𝑒𝑓𝑓(𝐼𝑑𝑐=0)×100% , at 𝐼𝑑𝑐=±5 mA \nis 66% and 26% for the W based CoFe and Py hetero structure s, respectively. Here, 𝐼𝑑𝑐=5𝑚𝐴 \ncorresponds to a dc current density 𝐽𝑑𝑐=1.07×1010 𝐴\n𝑚2 and 6.61×109 𝐴\n𝑚2 in the W layer for \nthe CoFe and Py based heterostructures , respectively . The observed current -induced change of \n𝛼𝑒𝑓𝑓(𝐼𝑑𝑐), 6.14% (3.89%), in all-epitaxial W/CoFe ,Py/TiN/Si is significantly larger than that \nreported by Pai et al.31, where a current -induced change of 0.17% of the effective damping \nconstant at 𝐽𝑑𝑐=±1.0×109 𝐴\n𝑚2 was reported for the W(6nm)/Co 40Fe40B20(5nm)/SiO x/Si \nsystem. Our current -induced change of 𝛼𝑒𝑓𝑓(𝐼𝑑𝑐) is also significantly larger than that reported by \nLiu et al.27, who achieved a current -induced change of 𝛼𝑒𝑓𝑓(𝐼𝑑𝑐) of 0.003 % at 𝐽𝑑𝑐=±8.95×\n109 𝐴\n𝑚2 in Py (4nm)/Pt(6nm) structures. The calculated value of |𝑆𝐻𝑀𝑂𝐷| averaged over all \nmeasured frequencies for the W/CoFe (Py)/TiN/Si heterostructure using Eq. (4) is 39.010.84 \n(8.450.27). The SOT efficiency from the TiN/ CoFe( Py) interface is 1.22±0.47 (0.250.19). The \ncritical switching current density of the all-epi W/CoFe (Py)/TiN /Si heterostructure at which \neffective damping reverse s sign is 1.68 (3.27) MA/cm2, while it i s 5.99 (9.76) MA/cm2 for a 8 heterostructure not including the TiN buffer layer . The SOT efficiency determined using Eq. ( 2) \ndepends on the thickness and magnetization of the magnetic layer . The spin -orbit field, which is \ngenerated when feeding a charge current through the NM layer due to the SHE or REE , is \nindependent of magnetic layer thickness and magnetiz ation. By measuring spin -orbit field per \nunit applied current density , one can predict the real feature of the heterostructures . \nAngle dependent planar Hall effect measurements to determine spin-orbit torques To \nevaluate the SOT efficiency we have determined the out-of-plane spin -orbit field in W/CoFe , \nPy/TiN/Si heterostructures by performing angle dependent planar Hall effect (PHE) \nmeasurements. An optical image of the Hall bar structure is shown in Fig. 4 (a), where an in-\nplane dc is applied along the length of the bar and the voltage is recorded across the bar while \nvarying the in -plane magnetic field direction. The vector representation of the spin-orbit field in \nthe PHE measurements is presented in Fig. 4 (b), where 𝜑 is the angle between the current \ndirection and the in-plane applied field 𝐻 and 𝜃 is the angle between the current direction and the \nmagnetization vector 𝑀. The rotation angle 𝜙 is considered here with the initial alignment 𝜙=\n𝜋/2, yielding 𝜑=𝜙−𝜋/2. The angle dependent planar Hall resistance recorded at opposite \ncurrent polarities can be expressed as14 \n𝑅(±𝐼𝑑𝑐,𝜑)=𝑅0+𝑅𝐴𝐻𝐸cos𝜑+𝑅𝑃𝐻𝐸sin2𝜑, (3) \nwhere 𝐼𝑑𝑐 is the applied dc, 𝑅0 is the resistance offset accounting for the Hall bar imbalance, \n𝑅𝑃𝐻𝐸 and 𝑅𝐴𝐻𝐸 are the resistances due to the PHE and anomalous Hall effect (AHE ), \nrespectively. The spin-orbit field 𝐻𝑆𝑂 was determined from the differential Hall resistance 𝑅𝐷𝐻 \ndefined as14, 9 𝑅𝐷𝐻(𝐼𝑑𝑐,𝜑)=𝑅(𝐼𝑑𝑐,𝜑)−𝑅(−𝐼𝑑𝑐,𝜑)= 2𝑅𝑃𝐻𝐸𝐻𝑂𝑒+𝐻𝐹𝐿\n𝐻(cos𝜑+cos3𝜑)+\n+2𝑑𝑅𝐴𝐻𝐸\n𝑑𝐻𝑜𝑝𝐻𝑆𝑂cos𝜑+𝐶, (4) \nwhere 𝐻𝑂𝑒 is the Oersted field due to 𝐼𝑑𝑐, 𝐻𝐹𝐿 is the field due to the REE contributing with a \nfield-like torque (𝜏𝐹𝐿), 𝑑𝑅𝐴𝐻𝐸\n𝑑𝐻𝑜𝑝 was obtained from a separate measurement of the AHE resistance \n𝑅𝐴𝐻𝐸 in an out -plane applied magnetic field 𝐻𝑜𝑝 and 𝐶 is the resistance offset. The extracted \nvalues of 𝑑𝑅𝐴𝐻𝐸\n𝑑𝐻𝑜𝑝 are 0.026 /T and 0.096 /T for the CoFe and Py films , respectively (see \nSupplementary section S5 and ref# 34). The 𝑅𝐷𝐻 data was recorded at 𝐻=0.5 T, which is large \nenough to saturate the magnetization of the FM (CoFe, Py) layer s and to suppress the \ncontribution from the field-like torque . Figures 4 (c) and (d) show the PHE resistance 𝑅(𝐼𝑑𝑐,𝜑) \nversus 𝜑 recorded with opposite current polarit ies for W/CoFe /TiN/Si and W/Py/TiN /Si, \nrespectively. The experimental data were fitted using Eq. ( 3) to determine the values of 𝑅𝑃𝐻𝐸 and \n𝑅𝐴𝐻𝐸. The variation of t he spin -orbit field 𝐻𝑆𝑂 with 𝐼𝑑𝑐 was derived by fitting the angle \ndependent 𝑅𝐷𝐻(𝐼,𝜑) data to Eq. ( 4) as shown in Figs. 4 (e) and ( f) for the W(8nm) \n/CoFe(10 nm)/TiN/Si and W(8nm)/Py(15 nm)/TiN/Si structures, respectively. The results from \nthis fitting are presented in Fig. 4 (g), showing 𝐻𝑆𝑂 versus 𝐽𝑑𝑐. \nThe PHE determine d SOT values depend on thickness and magnetization of the ferromag netic \nlayer, therefore we should rely more on the 𝐻𝑆𝑂 derived values to understand the heterostructure \nSOT efficiency . To dig deeper into the origin of these extraordinary large 𝐻𝑆𝑂 values , we have \ninvestigated W-thickness (𝑡𝑊) dependence of 𝐻𝑆𝑂 in W(4-10nm)/Py(15 nm)/TiN(15 nm); the \nresults are shown in Figs. 4 (h-k). We have not varied the TiN thickness here since the TiN/Py \ninterface spin torque efficiency is very small compared to that of the Py(15 nm)/W interface. The 10 angle dependent planar Hall resistance at 𝐼𝑑𝑐=±4.5 mA dc is presented in Fig. 4 (h) , while the \ndifferential Hall resistance is plo tted in Fig. 4(i) for the W(4nm)/Py(15nm)/TiN(15nm) /Si \nstructure for different dc amplitudes . Figure 4(j) shows 𝜇0𝐻𝑆𝑂 versus 𝐽𝑑𝑐 for different 𝑡𝑊, while \nFig. 4(k) shows 𝜇0𝐻𝑆𝑂 versus 𝑡𝑊 at constant 𝐽𝑑𝑐=1×1010 A/m2. It is evident from the results \nthat 𝜇0𝐻𝑆𝑂 at constant 𝐽𝑑𝑐 increases with decreas ing W-thickness. \nIn-plane magneti zation switching To show SOT driven magnetization switching we have \nrecorded MOKE microscope images on rectangular bar structures of size 20×100 𝜇𝑚2 for W/ \nCoFe(Py)/ TiN/Si by varying the amplitude of the applied dc pulse . Since the device structure is \nlarge , possessing a multidomain magnetic microstructure in the remanent magnetization s tate, we \nhave applied a magnetic field of 4.5 mT (0.35 mT) along the ±𝑦 direction s for the W/CoFe \n(Py)/ TiN/Si structures while applying the dc pulse along the ±𝑥 direction s. Figure 5 (a) show s \nMOKE images recorded for the W(7nm)/Py(15 nm)/TiN hetero structure. A black contrast in the \nMOKE image represents a domain magnetization along the +𝑦-direction , while a white contrast \nrepresents a domain magnetization along the −𝑦-direction. By applying the magnetic field along \nthe +𝑦-direction a mixed dark -light contrast appears in the MOKE images , which represents a \nmultidomain magnetic state. The contrast gradually changes to dark by increasing the amplitude \nof the dc pulse along the +𝑥-direction. To further confirm the SOT switchi ng, the magnetic field \nwas applied along the 𝑦 direction and the pulse d dc was applied along the 𝑥 direction. The \nchange in magnetization contrast by changing the direction of the applied dc pulse confirm s the \nin-plane magnetization switching. The applied current density value required for room \ntemperature SOT induced in-plane magnetization switching was 𝐽𝑑𝑐=±5.78×1010 𝐴\n𝑚2 for the \nW(8nm)/Py(15nm)/TiN structure. The MOKE switching for CoFe is described in supplementary \ninformation S6 . The absence of magnetization switching by changing the applied dc pulse 11 direction while keeping the applied magnetic field the in −𝑦-direction affirms the absence of \nJoule heating or Oersted field induced magnetization switching. \nSOT induced magnetizatio n switching was also studied in the remnant magnetization state \nwith the result that only a small part of the magnetic microstructure could be switched (see \nsupplementary S6). The remanent magnetization state exhibits a multidomain magnetic \nmicrostructure and the SOT acting on the domain magnetizations will only switch domains with \na domain magnetization along the hard -axis of the bar shaped structure (±𝑦-directions). The \nSOT magnetization switching has also been modeled by using micromagnetic simulations on a \nrectangular cuboid structure model with dimensions 200×1000×15nm3 (the same aspect \nratio in -plane as used in the STFMR and Hall experiments) . The experimentally determined layer \nparameters, i.e., effect ive damping constant, anti -damping and field -like torques, effective \nanisotropy, saturation magnetization and thicknesses, were used in the simulations . The dc pulse \ndependent magnetization switching is presented in Figs. 5 (b) and ( c) for the W/Py/TiN structure . \nThe simulated in -plane magnetization switching current density value was 𝐽𝑑𝑐=±7.0×\n1010 𝐴\n𝑚2 and is comparable to the experimentally determined value. \nDiscussion The determined giant value of the spin-orbit field is comparable to value s \nreported for conducting topological insulators ; Bi0.9Sb0.1/MnGa bilayers15 and BixSe(1–\nx)(tBSnm)/CoFeB(5nm)14. Therefore, in a quest to find the origin of the observed giant spin-orbit \ntorque efficiency in all-epi hetero structures, we have delved into the literature on interfacial SOT \nand/or presence of topological surface states in W. \nThonig et. al .35 have reported Dirac -type surface state s in the (110) plane of strained W, \nwhich are topologically protected by mirror symmetry and, thus, exhibiting nonzero topological 12 invariants. The presence of these topological states near the Fermi level can yield large spin -orbit \ntorque efficiency. Very recently , theoretical calculations have also revealed that β -W is a \ntopological massive Dirac metal36. In the absence of SOC, in calculations, Dirac nodal lines exist \nin the Brillouin zone and their existence is associated with the band inversion at high -symmetry \npoint s. However, once the SOC is switched -on in the calculations massive Dirac states appea r. \nFurthermore, the nontrivial topologically protected surface states and Fermi arcs have also been \nobserved36 in single crystal s. This work pro posed that epitaxial β-tungsten is a pure topological \nmassive Dirac metal. Both these theoretical reports suggest that both the epitaxial and strained W \n(110) plane can exhibit massive Dirac surface states. In our studied all- epitaxial system s β-W is \nalso epitaxial and exhibit s a strain with respect to the FM epitaxial interface. Therefore, a \npresence of the Dirac surface states at the β -W interface in our studied all -epitaxial \nheterostructures should be considered . \nA giant spin torque efficiency was in the topological insulator Bi xSe1-x, which increased with \ndecreasing film thickness owing to topological surface states induced momentum locking14. We \nhave also observed an enhancement of the spin torque efficiency and the spin-orbit field by \nreducing the thickness of the β-W layer, which clearly evidences the presence of topological \nsurface states. Therefore , the observed giant SOT efficiency and the spin-orbit field are due to \nthe presence of surface topological states providing a large spin Berry curvature and hence a \nlarge intrinsic SHE in the all-epitaxial W based heterostructure s36,37. Moreover, light element , \ne.g. Ti, interfaces with in -plane magnetized FM generates large interfacial spin -orbit to rques as \ndemonstrated in CoFeB /Ti, NiFe(4nm)/Ti(3nm)38. TiN also generates a non-negligible spin \ncurrent at the FM /TiN interface and we find that the FM/ TiN interface also plays a role in the 𝐼𝑑𝑐 \ndependent change of the effective damping, originating from anti-damping -like torques due to 13 interfacial symmetry breaking and SOI. The interfacial spin torques are very sensitive to the \ncrystallographic structure and therefore to the orbital ordering at the interface . Hence, it appears \nthat it is the epitaxial nature of the heterostructures and the topological surface states that are \nresponsible for the observed giant spin-orbit torque efficiency. \nConclusions \nIn summary, spin-orbit torque studies have been performed on all-epitaxial β-W/CoFe /TiN/Si \nand β -W/Py/TiN/Si heterostructures. The observed giant spin torque efficiency is comparable to \nvalue s reported for topological insulators, and the origin of this giant spin-orbit torque efficiency \nlies in the creation of topological surface states at the epitaxial interfaces. T he extraordinary \nspin-orbit torque efficiency in the studied semiconducting technology compatible W and TiN \ninterfaced FM heterostructures open s up a new avenue for the realization of ultralow power ed \nspintronic devices by utilizing epitaxial magnetic heterostr uctures. \nMaterials and Methods \nThe epitaxial Co 60Fe40 and Ni 81Fe19(Py) films were deposited on TiN(200)[100](10nm)/Si \nsubstrates by pulsed dc magnetron sputtering technique. Layers of W with different thickness 𝑡𝑊 \nin the range 1 -15 nm were deposited on CoFe/TiN/Si and Py/TiN/Si . The FM layer thickness 𝑡𝐹𝑀 \nwas varied in the range 3 -17nm. It was noted that the grown W layers mostly exhibit the desired \n-phase (i.e. , A-15 cubic for W) as reported in our previous work23,39. X-ray reflectivity, X-ray \ndiffraction pole figure s and transmission electron microscopy measurements were performed to \nascertain the quality of the films. 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B 96, \n241105(R) (2017). \n38. S. C. Baek, V. P. Amin, Y. W. Oh, G. Go, S. J. Lee, G. H. Lee, K. J. Kim, M. D. Stiles, B. \nG. Park and K. J. Lee, Nat. Mater. 17, 509 (2018). \n39. N. Behera, A. Kumar, S. Chaudhary, and D. K. Pandya, RSC Adv. 7, 8106 (2017). \n \nAcknowledgements \nThis work was supported by the Swedish Research Council (VR), grant no 2017 -03799. Prof. \nMikael Ottosson is acknowledged for help during XRD measurements. \nAuthors Contributions: \nNB fabricated all films with a support from AK, SC, and DP . NB performed STFMR with \nsupport from AK. NB performed XRR measurements. NB performed pole figure measurements. \nS.H and R.G performed Hall measurements. SH performed micr omagnetic simulations. R B \nfabricated all the devices. JS and RP performed Kerr microscopic measurements with support \nfrom GA. NB, SH, PS and AK analysed the data. NB and AK wrote the manuscript. AK \nconceived the idea. AK and PS designed and supervised this project. All authors discussed the \nresults, reviewed and co mmented on the manuscript. \n*Correspondence and request for materials should be addressed to Email: \nchainutyagi@gmail.com \npeter.svedlindh@angstrom.uu.se 17 Competing financial interest \nThe authors declare no competing financial interests. \n 18 \n \n \nFigure 1 Pole figure XRD patterns of (a) Co 60Fe40 (022) in W/Co60Fe40/TiN/Si and (b) Ni 81Fe19 \n(111) in W/Py/TiN/Si thin films, confirming the epitaxial quality of the Co60Fe40 and \nNi81Fe19 thin films grown on TiN buffered Si (100) substrates. STEM images for (c) \nW/Co60Fe40/TiN/Si , and (d ) W/Py/TiN/Si thin film s. \n19 \n \nFigure 2 (a) Schematic of the patterned structure and the STFMR setup. (b) Schematic of \ntorque induced precession of the magnetization 𝑀 around its equilibrium direction, \nwhere 𝜃 is the angle of 𝑀 with respect to the 𝑥𝑦-plane. 𝜏𝐷, 𝜏𝐴𝐷, 𝜏𝑂𝑒 and 𝜏𝐹𝐿 are intrinsic \ndamping torque, spin current anti -damping torque, Oersted torque, and REE field-like \ntorque, respectively. STFMR spectr a of (c) W(6nm)/ CoFe (10nm)/TiN/Si at 16 GHz \nand (d) W( 8nm)/ Py(15nm)/TiN/Si at 15 GHz. The red lines are fits to the \nexperimental data ; green and blue lines are symmetric and anti -symmetric Lorentzian \ncomponents of the STFMR signal. (e) 𝑓 vs. 𝜇0𝐻𝑟 and (f) 𝜇0(∆𝐻−∆𝐻0) vs. 𝑓 of \nW(6nm)/ Co60Fe40(10nm)/TiN/Si and W( 8nm)/Py(15nm)/TiN/Si at 𝐼𝑑𝑐=0, +4.0 and \n-4.0 mA, respectively. The color ed solid lines are fits to the experimental data (see \nmain text for explanation). \n 20 \n \nFigure 3 (a) Effective damping parameter 𝛼𝑒𝑓𝑓 vs. 𝐼𝑑𝑐 and. (b) 𝛼𝑒𝑓𝑓(𝐼𝑑𝑐)−𝛼𝑒𝑓𝑓(𝐼𝑑𝑐=0) \nvs. 𝐼𝑑𝑐 for positive applied fields for W(6 nm)/CoFe(10 nm)/TiN /Si and \nW(8nm)/Py(15 nm)/TiN /Si. The solid lines are linear fits to the experimental data. \n \n 21 \nFigure 4: Hall ba r device measurements \n \nFigure 4 (a) Optical image of Hall -bar device with circuit measurement geometry. (b) In -\nplane magnetic field 𝐻⃗⃗ , magnetization 𝑀⃗⃗ and fields generated by SOTs; anti -damping \n𝐻⃗⃗ 𝑆𝑂 and field -like 𝐻⃗⃗ 𝐹𝐿. Angle dependent PHE resistance for (c) \nW(8nm)/CoFe(10 nm)/TiN at 𝐼𝑑𝑐=±4.0 mA and ( d) W(8nm)/Py(15 nm)/TiN at 𝐼𝑑𝑐=\n±4.5 mA and corresponding 𝑅𝐷𝐻 plots in (e ) and (f) where solid lines are fits to Eq. (4) . \n(g) μ0𝐻𝑆𝑂 vs. 𝐽𝑑𝑐 for both structures , where solid lines are linear fits to the data . (h) Angle \ndependent PHE resistance at 𝐼𝑑𝑐=±4.5 mA and corresponding 𝑅𝐷𝐻 plots in ( i) at \ndifferent 𝐼𝑑𝑐 for W(4 nm)/Py(15 nm)/TiN. Solid l ines are fits to Eq. (4). (j) μ0𝐻𝑆𝑂 vs. 𝐽𝑑𝑐 \nfor W(4 -8nm)/Py(15 nm)/TiN structures. Solid lines represent linear fits to the data. (k) \nμ0𝐻𝑆𝑂 vs. 𝑡𝑊 for W(𝑡𝑊)/Py(15nm)/TiN structures at 𝐽𝑑𝑐=1×1010 A/m2. \n \n 22 Figure 5: Kerr microscopy image s and results from micromagnetic simulations \n \nFigure 5 MOKE images of the SOT -driven magnetization switching by applying (a) a \nmagnetic field of 0.35 mT along +(−)𝑦-axis and a pulsed dc of increasing amplitude \nalong +(−)𝑥-axis for the W(8nm)/Py(15 nm)/TiN structure. (b, c) Micromagnetic \nsimulated magnetization (𝑦-component ) switching behavior of a 0.2×1.0 μm2 structure \nmodel ed by applying a constant magnetic field along the +𝑦-axis and a pulsed dc of \nincreasing amplitude along the +𝑥-axis for the W(8nm)/Py(15 nm)/TiN structure . \nChanges in contrast visualizes the magnetization switching. \n \n \n \n" }, { "title": "1406.3630v2.Magnetic_Field_Amplification_in_the_Thin_X_ray_Rims_of_SN1006.pdf", "content": "arXiv:1406.3630v2 [astro-ph.HE] 24 Jun 2014Magnetic-Field Amplification in the Thin X-ray Rims of SN100 6\nSean M. Ressler1, Satoru Katsuda2, Stephen P. Reynolds3, Knox S. Long4, Robert Petre5,\nBrian J. Williams5, & P. Frank Winkler6\n1Department of Physics, University of California Berkeley, Berkeley , CA 94720, USA\n2Institute of Space and Astronautical Science (ISAS), Japan Aer ospace Exploration\nAgency (JAXA), 3-1-1 Yoshinodai, Chuo, Sagamihara, Kanagawa 2 52-5210, Japan\n3Physics Department, North Carolina State University, Raleigh, NC 2 7695, USA\n4Space Telescope Science Institute, 3700 San Martin Drive, Baltimor e, MD 21218, USA\n5NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA\n6Department of Physics, Middlebury College, Middlebury, VT 05753, U SA\nReceived ; accepted– 2 –\nABSTRACT\nSeveral young supernova remnants (SNRs), including SN1006, em it syn-\nchrotron X-rays in narrow filaments, hereafter thin rims, along th eir periphery.\nThe widths of these rims imply 50 to 100 µG fields in the region immediately be-\nhind the shock, far larger than expected for the interstellar mediu m compressed\nby unmodified shocks, assuming electron radiative losses limit rim width s. How-\never, magnetic-field damping could also produce thin rims. Here we re view the\nliterature on rim width calculations, summarizing the case for magnet ic-field\namplification. We extend these calculations to include an arbitrary po wer-law\ndependence of the diffusion coefficient on energy, D∝Eµ. Loss-limited rim\nwidths should shrink with increasing photon energy, while magnetic-d amping\nmodels predict widths almost independent of photon energy. We use these re-\nsults to analyze Chandra observations of SN 1006, in particular the southwest\nlimb. We parameterize the full widths at half maximum (FWHM) in terms o f\nenergy as FWHM ∝EmEγ. Filament widths in SN1006 decrease with energy;\nmE∼ −0.3 to−0.8, implying magnetic field amplification by factors of 10 to 50,\nabove the factor of 4 expected in strong unmodified shocks. For S N 1006, the\nrapid shrinkage rules out magnetic damping models. It also favors sh ort mean\nfree paths (small diffusion coefficients) and strong dependence of Don energy\n(µ≥1).\nSubject headings: acceleration of particles – ISM: individual objects (SN 100 6) – ISM:\nmagnetic fields – ISM: supernova remnants – X-rays: ISM– 3 –\n1. Introduction\nCosmic synchrotron sources, such as jets in active galactic nuclei, radio halos and\nrelics in clusters of galaxies, pulsar-wind nebulae, and shell superno va remnants (SNRs),\ndemonstrate the ubiquity of power-law distributions of relativistic e lectrons. Understanding\nthe origins of these fast particles is necessary to learn about thes e objects’ energy budgets\nand evolution. The synchrotron flux density emitted by a source de pends roughly on the\nproduct of the energy density of relativistic electrons ueand the magnetic field uB, but an\nindependent determination of magnetic-field strengths in synchro tron sources has proven\nelusive. The minimum energy of a synchrotron source occurs when t he two energy densities\nare roughly equal (“equipartition;” actually, ue= (4/3)uB; e.g., Pacholczyk 1970). However,\nit is not clear whether the unseen population of relativistic protons s hould also be included,\nand if so, what the proton-to-electron energy ratio should be. Fu rthermore, there is no\nobvious physical reason to expect equipartition. The argument fo r equipartition derives\nfrom attempts to explain extragalactic radio sources in which the to tal energy budget is\nso large that it was of interest to find a lower bound (Burbidge 1956) . However, many\nother synchrotron sources, including SNRs, release a relatively sm all fraction of their total\nenergy content as synchrotron emission, so could easily afford to b e far from equipartition\n(in either direction).\nAlthough magnetic fields are not dynamically important in SNRs (e.g., Ju n & Jones\n1999), their strength is critical in determining the maximum energy t o which particles can\nbe accelerated. For the diffusive shock acceleration process (DSA ; e.g., Blandford & Eichler\n1987), the time τ(E) to accelerate particles to energy Edepends on the diffusion coefficient\nDand the shock velocity vshockbyτ(E)∼D/v2\nshock. For relativistic particles, D=λc/3.\nThen in “Bohm-like” diffusion, where the mean free path λis assumed proportional to the\nparticle gyroradius ( λ=ηrg=ηE/eB),τ(E)∝1/B, and higher magnetic fields result– 4 –\nin more rapid acceleration and higher maximum energies. This is indepen dent of which\nof several competing mechanisms ultimately limits acceleration (finite time since onset of\nacceleration, radiative losses, or escape). Note that for the abo ve description of the diffusion\ncoefficient, taking η= 1 is called Bohm diffusion or the “Bohm limit.” It corresponds to\nλ=rg, often assumed to be the shortest physically plausible mean free pa th. However,\nin a turbulent wave field, it is not clear whether this is a true limit, or eve n what kind of\naverage value for the magnetic-field strength should be used to ca lculaterg(e.g., Reville &\nBell 2013).\nLargely by exclusion of competing hypotheses, Galactic cosmic ray a cceleration is now\nwidely attributed to SNR shocks. The consensus is that SNRs can ac celerate particles\nup to the “knee,” the slight inflection and steepening around 3 PeV (3 ×1015eV). (No\nplausible version of SNR-based DSA produces the maximum energies o bserved in cosmic\nrays of above 1019eV [e.g., Abraham et al. 2008], for which an extragalactic origin is\npresumed.) However, since the work of Lagage & Cesarsky (1983) , it has been clear that\ntypical estimates of magnetic-field strengths behind SNR shocks o f a fewµGauss (the mean\ninterstellar magnetic field multiplied by the shock compression ratio, r, taken to be 4 for\nstrong nonrelativistic shocks), result in maximum energies that fall short of the “knee”\nby an order of magnitude or more. These estimates of Bare based on measurements of\nthe interstellar magnetic field strength within a few kpc of the Sun, w hich is about 2 – 3\nµGauss (Lyne & Smith 1989). Simple compression in a strong shock with adiabatic index\nγ= 5/3 (i.e., unmodified by cosmic rays) would produce downstream values la rger by a\nfactor of up to 4 (no amplification if the shock velocity is parallel to th e field, a factor of\n4 increase if perpendicular). Thus magnetic fields larger than about 12µGauss require\nan additional process of amplification. Independent methods of es timating interstellar\nmagnetic-field strengths, such as Zeeman splitting in molecular lines, are not relevant to\nSNR environments.– 5 –\nThus the plausibility of models in which SNR shocks produce Galactic cos mic rays\nup to the “knee” may depend on observational determinations of t he post-shock magnetic\nfield strength. Fortunately, the realization that young SNRs can p roduce synchrotron\nemission into the X-ray band has made available a new, potentially powe rful method\nfor such determinations. Several young SNRs show thin (a very sm all fraction of their\ndiameter), synchrotron-emitting filaments along their edges. The widths of these rims in\nthe radial direction have been used to infer estimates for the post -shock magnetic field\nstrength and presented as evidence for significant magnetic field a mplification by strong\nshocks, as we shall review in detail below. The complexity of the calcu lations used to\ninfer the magnetic field magnitude varies significantly from simple analy tic approximations\nto detailed numerical calculations, but the consensus from these s tudies is that field\namplification well beyond a factor of four is required to explain the X- ray observations.\nThin synchrotron rims present a well-defined problem. While synchro tron emissivity\nmay suddenly turn on at the shock front due to particle acceleratio n and magnetic-field\ncompression (or amplification) there, turning it off again only about a tenth of a parsec\ndownstream is not so simple: one must either eliminate the radiating pa rticles or the\nmagnetic field. Both possibilities have been suggested. Radiative ene rgy losses as particles\nadvect and diffuse downstream will eventually lower electron energie s below the level at\nwhich synchrotron X-rays can be produced. More rapid synchrot ron losses for higher\nelectron energies then predict that rims will become thinner at highe r photon energies.\nHowever, it is also possible that magnetic fields somehow decay behind the shock, with a\nlength scale (almost) independent of electron energy, predicting r ims whose thickness is\nrelatively constant with photon energy.\nIn the energy-loss scenario, the rate at which rims become thinner as observing energy\nrises depends on electron transport. If electrons are simply adve cted downstream, rim– 6 –\nwidthsldepend only on the magnetic-field strength, and drop quite rapidly w ith increasing\nphoton energy, l∝(hν)−1/2in the simplest approximation, as we describe below. However,\ndiffusion allows electrons of the same energy to spread out spatially, diluting this effect\nsomewhat and predicting slower drops in rim widths with energy. Meas uring the rim widths\nat several different photon energies is thus key to discriminating am ong models.\nThe remnant of the Type Ia Supernova of AD 1006 is well suited for t his analysis,\nas its large angular size coupled with Chandra’s high spatial resolution allows accurate\nmeasurements of radial profiles of the filaments: the remnant rad ius of about 15′\ncorresponds to over 1800 Chandra ACIS pixels. Furthermore, the shock speeds in the\nsynchrotron-dominated northeast (NE) and southwest (SW) ed ges are about 5000 km s−1,\nas measured from their proper motion (Katsuda et al. 2009, Winkler et al. 2014), and\nthe SNR has been detected as a TeV source (Acero et al. 2010), su ggesting that SN1006\nproduces very high-energy cosmic rays.\nOur purpose here is to consider theories of particle diffusion and mag netic-field\namplification in the light of new deep observations of SN1006 made with Chandra. In\nSection 2 (summarized in Table 1), we review earlier work on filament ca lculations and the\nevidence for field amplification to establish a firm background for the new work presented\nhere. In Section 3, we generalize previous work by allowing different e nergy dependence\nof the diffusion coefficient from Bohm-like, including Kolmogorov and Kr aichnan-type\ndiffusion, among others. We first neglect any cut-off in the electron spectrum and\ncalculate model profiles and their energy dependence for the loss- limited (Section 3.1) and\nmagnetically damped (Section 3.2) scenarios. We then add the effect s of an electron cut-off\nenergy (Section 3.3), and examine the effects of making the δ-function approximation for\nthe emissivity (Section 3.4). In Section 4, we describe measurement s of the widths of\nthe nonthermal filaments in SN1006, including for the first time the S W region, making– 7 –\nuse of new high-resolution Chandra measurements. We find that rim widths decrease\nwith increasing photon energy, quite rapidly in some cases. We use th ese measurements\nto constrain both the post-shock magnetic field and diffusion coeffic ient. The general\nnarrowing of rims eliminates the magnetic-damping model for rim width s; quantitatively,\nwe find values of post-shock magnetic field of 70 – 200 µG, comparable to those obtained in\nearlier work. However, the rapidity of rim shrinkage suggests that diffusion mean free paths\nin some areas are quite small, perhaps less than the gyroradius. We d iscuss these results in\nSection 5, including reviewing theoretical and observational work o n sub-Bohm diffusion\nand implications for particle acceleration to high energies. We summar ize our conclusions in\nSection 6. Finally, in the appendix, we offer a list of the results of our e xperience in applying\nthe various rim models to observations, as a guide for potential fut ure investigations.\n2. Previous Work\nPrior work bifurcates into two eras, an earlier one in which it was assu med that the\nonly influence on filament shapes was synchrotron losses, followed b y one beginning in\n2005 when additional effects such as magnetic field damping began to be introduced. We\nconsider first the former case; decay of magnetic field is considere d in Section 2.2.\n2.1. Loss-limited models\nBefore 2005, it was universally assumed that the shapes of nonthe rmal X-ray filaments\nobserved in SNRs were due to synchrotron losses by high energy ele ctrons (see the many\nreferences described below and in Table 1). The idea was that an elec tron could only travel\na certain distance before losing enough energy that its radiation dr opped below the X-ray\nband. This distance is determined by two competing transport mech anisms: advection (bulk– 8 –\nmotion of plasma) and diffusion (random motion of electrons on the sc ale of gyroradii).\nConsidered separately, one can obtain simple expressions for the a ppropriate length scale\nfor each in terms of the diffusion coefficient, D, the downstream plasma speed in the shock\nframe,vd, and the synchrotron cooling time, τsynch. We can estimate τsynch= 1/(bB2E),\nwhereb= 1.57×10−3in cgs, from the relation ˙E∝E2. Thus, for vd=vshock/4, given by\nthe Rankine-Hugoniot conditions for a strong adiabatic shock, we h ave an advective length\noflad≈vdτsynch= (vd)/(bB2E). For a diffusion coefficient that is taken as a constant\nmultiple, η, of the Bohm value D=ηCdE/B, whereCd≡c/(3e), we arrive at a diffusive\nlength of ldiff≈/radicalbig\nDτsynch=/radicalbig\nD/(bB2E). For the values of the constants b,Cd,cm, andc1\nused here and throughout, see Table 8. Now, an electron of energ yEin a magnetic field\nradiates primarily at the frequency νm=cmE2Bso that the advection and diffusion lengths\nas a function of frequency are\nlad=vd√cm\nbB−3/2ν−1/2\nm (1)\nand\nldiff=/radicalbigg\nηCd\nbB−3/2. (2)\nThe approximation that the electron radiates all its energy at νmis called the delta-function\napproximation. The important result here is that ladvaries as ν−1/2whileldiffis\nindependent of frequency. Thus above some critical energy, Ec, and an associated\nphoton frequency, νc, electrons will be able to diffuse further in a loss time than they could\nadvect, and electron diffusion will become the dominant method of tr ansport. Ecis found\nsimply by equating the expressions for ladandldiff:\nEc=vd√ηCdbB≈69.12 ergs/parenleftbiggvd\n1250 km s−1/parenrightbigg/parenleftbiggB\n100µG/parenrightbigg−1/2\nη−1/2(3)\nhνc=cmv2\nd\nηCdb≈3.61 keV/parenleftbiggvd\n1250 km s−1/parenrightbigg21\nη. (4)\n(We have taken vd=vshock/4 = 1250 km s−1, assuming no shock modification by cosmic\nrays.) Near this photon energy, both advection and diffusion are imp ortant. This simple– 9 –\napproach was taken by Ballet (2006), Bamba et al. (2003), and Yam azaki et al. (2004)\nto infer magnetic field strengths of 14–87 µG in SN 1006. Vink & Laming (2003) used a\nsimilar technique for Cas A, and estimated Bto be∼100µG.\nParizot et al. (2006) adopted a somewhat more sophisticated appr oach, combining\nboth processes in the steady state form of the one-dimensional t ransport equation to solve\nfor the post-shock electron distribution f(p,x) (V¨ olk et al. 1981):\nv∂f\n∂x−D∂2f\n∂x2+f\nτsynch= 0, (5)\nwhere the loss term f/(τsynch) assumes that an electron maintains constant energy as it\ntravels away from the injection site until a catastrophic dump at tim eτsynch. Here the\nshock is at x= 0 and x >0 is the distance downstream. The solution to this equation is\nf(p,x)∝e−|x|/a, with the scale length agiven by:\na=2D/vd/radicalBig\n1+4D\nv2\ndτsynch−1(6)\n(Berezkho & V¨ olk 2004, Parizot et al. 2006). More explicitly, in term s of electron energy E\nthe scale length is\na=2ηCdE/Bv d/radicalBig\n1+4bηCdE2B\nv2\nd−1. (7)\nAt a given observation frequency ν,Ewill depend on the magnetic field B, by\nE=/radicalbig\nν/(cmB), so\na=2ηCdc−1/2\nmν1/2B−3/2/vd/radicalBig\n1+4bηCdν\ncmv2\nd−1. (8)\nIn order to estimate the strength of the magnetic field, we invert t his expression so that it\nis a function of observables:\nB=\nac1/2\nmvd/parenleftBig/radicalBig\n1+4bηCdν\ncmv2\nd−1/parenrightBig\n2ηCdν1/2\n−2/3\n. (9)– 10 –\nIn this equation ais the scale length of the electron distribution, whereas we observe\nthe scale length of the emitted synchrotron intensity, including line- of-sight projection\neffects. In the δ-function approximation of the synchrotron emissivity, jν∝√\nνBf(E,x)\n(E=pcfor relativistic electrons), and the radial intensity for a spherical shock is\nIν(r) = 2√\nr2s−r2/integraldisplay\n0jν/parenleftBig\nrs−√\ns2+r2/parenrightBig\nds. (10)\nHereris the sky-plane radius ( rsthe shock radius), and sis the line-of-sight coordinate.\nThe resulting profile will have a FWHM = βa,βa projection factor. Ballet (2006) showed\nthat in the case of a purely exponential (in space) electron distribu tion and for a spherical\nsource, the result of this integral will give β= 4.6, that is, a filament with a Full Width at\nHalf Maximum (FWHM) of 4 .6a. Thus in terms of the observed filament width, wobs, we\nhave\nB=\nwobsc1/2\nmvd/parenleftBig/radicalBig\n1+4bηCdν\ncmv2\nd−1/parenrightBig\n2βηCdν1/2\n−2/3\n. (11)\nUsing this result, the post-shock magnetic field strength in the NE r im of SN1006 was\nestimated to be around 91-110 µG forwobs= 20′′, an amplification of roughly 30-37 for an\nambient 3 µG field (Parizot et al. 2006). While those authors did not make use of t he fact,\nwe note that the inferred value of Bdepends on observing frequency.\nIt should be noted that the projection factor β= 4.6 is entirely dependent on the\nexponential form of the synchrotron emissivity given from the solu tion of (5), which may\nnot be valid, as well as on the assumption of exact sphericity. This is a n important caveat,\nas the width of the rims scales as B−3/2(from equation (8)), so, since the width is inversely\nproportional to the projection factor, β, the above estimates for the post-shock magnetic\nfield strength are proportional to β2/3. In our later calculations, however, we do not assume\na simple constant projection factor and perform the full numerica l line-of-sight integration.– 11 –\nFinally, the most sophisticated synchrotron-loss based models of B erezhko et al. (2003),\nBerezhko & V¨ olk (2004), Cassam-Chena¨ ı et al. (2007), Morlino e t al. (2010) and Rettig\n& Pohl (2012) use an electron distribution obtained by solving the co ntinuous energy loss\nconvection-diffusion equation (properly, the advection-diffusion e quation):\nv∂f\n∂x−∂\n∂x/parenleftbigg\nD∂f\n∂x/parenrightbigg\n−∂\n∂E/parenleftbig\nbB2E2f/parenrightbig\n=K0E−se−E/Ecutδ(x), (12)\nwhere it is assumed that electrons are injected at the shock and fo llow a power-law energy\ndistribution with an exponential cut-off: ( N(E)∝E−seE/Ecut, wheres= 2.2 – the value\nappropriate for SN 1006). The electron distribution obtained from solving this equation is\nconvolved with the single-particle emissivity and then integrated alon g lines of sight (see\nequation 10) to compute radial intensity profiles. The magnetic-fie ld strength for SN 1006\npredicted using this method is in the range of 90-130 µG, an amplification of roughly 30–43\nfor an ambient field of 3 µG (Berezhko et al. 2003, Morlino et al. 2010, Rettig & Pohl 2012).\nTable 1: Previous Magnetic Field Strength Estimates for SN1006\nNote:B0≡Bdirectly behind the shock\nPaper Technique SN1006 B0estimate Amplification Factor\n(For ISM Field of 3 µG)\nAraya et al. (2010) Catastrophic Dump C-D equation - -\nBallet (2006) Equated ldiffto rim sizes 87 µG 29\nBamba et al. (2003) Equated max( ldiff,lad) to rim sizes - -\nBerezhko et al. (2003) Time dependent continuous loss C-D equatio n ∼100µG 33\nBerezhko & V¨ olk (2004) Time dependent continuous loss C-D equat ion - -\nCassam Chena¨ ı et al. (2007) CR modified numerical solution to C-D e quation - -\nMorlino et al. (2010) Nonlinear DSA Model Fit 90 µG 30\nParizot et al. (2006) Catastrophic Dump C-D equation + δ-function 91-110 µG 30-37\nRettig & Pohl (2012) Continuous loss C-D equation 130 µG (loss limited) 43\n∼65µG (B-limited) 22\nVink & Laming (2003) Equated max( ldiff,lad) to rim sizes - -\nYamazaki et al. (2004) Equated max( ldiff,lad) to rim sizes 14-85 µG 5-28– 12 –\n2.2. Energy Dependence of the Filament Widths\nIn almost all previous calculations, the energy dependence of the fi lament width was\nignored and profiles were fit at a single photon energy. One exceptio n to this is the work\nof Araya et al. (2010) in their analysis of the shapes of the rims of Ca s A. Interestingly,\nwhile Araya et al. found no significant energy dependence in the rim wid ths between the\nenergy ranges 3–6 keV and 6–10 keV, they did report a small but no n-negligible difference\nbetween widths at 0.3–2 and 3–6 keV. However, they did not use this result as a parameter\nconstraint. Here we show that this dependence has important phy sical consequences for\nparameter estimation.\nIn all the models we shall consider, the diffusion coefficient rises with e nergy. This\nmeans that electrons with lower energies will stay closer to their orig inal fluid element\nwhile those with higher energies move about more freely. Specifically, as can be seen from\nEquation 6, for ν≫νc,a≈ldiff, while for ν≪νc,a≈lad. This behavior shows up in\nhow the width, a, varies with energy, which we can parameterize as a∝EmEγfor a photon\nenergy of Eγ≡hv. Written in this way, we have\nmE=−1\n2/parenleftBigg\n1−4D/(v2\ndτ)\n1+4D/(v2\ndτ)−/radicalbig\n1+4D/(v2\ndτ)/parenrightBigg\n, (13)\nwhere 4D/(v2\ndτ)∝E2B∝ν, with the last proportion coming from the δ-function\napproximation, meaning that mEis independent of magnetic field strength. It is also clear\nfrom equation (13) that mEwill go from −1/2→0, or in other words that the scale length,\na, will go from a∝E−1/2\nγ→a∝E0\nγ, asνgoes from 0 → ∞(forD∝E, or in our later\nnotation, µ= 1).– 13 –\n2.3. Magnetic-Field Damping\nIn 2005 Pohl, Yan, & Lazarian introduced a more sophisticated appr oach, which\nsuggested that claims of strong field amplification might be prematur e. They proposed\nseveral processes that could lead to an exponentially decaying mag netic field, as well as\naccount for the narrow filamentation. In this case, rim profiles wou ld reflect the spatial\ndistribution of the magnetic field. Unfortunately, there are no simp le predictions for the\nmagnetic-field damping length (or detailed spatial dependence). Th ere are a variety of\nphysically possible damping mechanisms, so the damping length is a free parameter in\nmodels of this type (although its dependence on the immediate post- shock value of Bcan\nbe preserved).\nCassam-Chena¨ ı et al.(2007) used this idea to fit intensity profiles o f the filaments in\nTycho’s SNR, employing a hydrodynamics code that included cosmic-r ay shock modification\n(increased compression ratios due to energetically important part icles becoming relativistic\nand/or escaping). They generated model profiles assuming a magn etic-field profile with\nexponential damping, similar to our expression in Section 3.4 below. By incorporating radio\nobservations, they concluded that the synchrotron loss-limited m odel provides a slightly\nbetter fit than the magnetically damped model, though neither comp letely reproduces\nthe radio profiles. More recently, Rettig and Pohl (2012) followed u p by probing the\nobservational consequences of both a magnetically damped model and a constant field\nmodel by using differences in the spectral index between the emissio n at the rim peak (i.e.\nthe emission from the shock front to a FWHM distance away)and in th e “plateau” (i.e. the\nemission from regions beyond the FWHM). Their magnetic-field estima tes for both models\nstill favor /greaterorsimilar60µG for SN1006.\nMarcowith & Casse (2010) performed detailed calculations to invest igate the magnetic-\ndamping model, studying the amplification process due to linear and no nlinear cosmic-ray– 14 –\nstreaming instabilities, and identifying processes to damp the turbu lent magnetic field.\nThey report that a damping model could explain rims in the younger re mnants Cassiopeia\nA, Tycho, and Kepler, but not in SN 1006 or G347.3–0.5 (RX J1713.7-3 946). For the objects\nwhich satisfy their conditions for magnetic damping, they deduce qu ite high magnetic-field\nstrengths of 200 – 300 µG.\n3. Generalized Diffusion Model\nHere, we will consider the case of diffusion coefficients of the form\nD=ηDB(Eh)(E/Eh)µ,whereDB(Eh) is the Bohm diffusion coefficient at an arbi-\ntrary fiducial energy Ehand a magnetic field B0,ηis a constant scaling factor taken in\nconjunction with DB(Eh) as a free parameter, and µparameterizes the energy dependence\nofD. (So for Bohm diffusion, η= 1 and µ= 1.) Forµ <1,Ehmust be above the relevant\nenergy range for X-ray emitting electrons, so that Dremains greater than the minimum\nBohm value at all energies. On the other hand, for µ >1, this energy is the lower threshold\nenergy for this exotic type of diffusion to occur. We expect µto be related to the power-law\nindexnof hydromagnetic turbulence, I(k)∝k−nwhereI(k) is the wave power per unit\nwavenumber. Then n= 5/3 corresponds to a Kolmogorov spectrum, and n= 3/2 to a\nKraichnan spectrum. In quasi-linear theory, particles have a mean free path inversely\nproportional to the energy density of MHD waves with wavelength c omparable to the\nparticle gyroradius, resulting in µ= 2−n(e.g., Reynolds 2004). So Kolmogorov turbulence\npredictsµ= 1/3 and Kraichnan, µ= 1/2.\nIn all of the ensuing discussion we will be concerning ourselves with th e consequences\nof these models observable in the X-ray filaments of SN1006, which w e will characterize by\ntheir FWHM, x1/2, and its energy dependence, again parametrized by mE. This is explicitly\nwritten as x1/2∝EmE.– 15 –\nFor the case of µ= 1 theEhindependent case, we adopt Rettig & Pohl’s solution to\nequation (12) for the electron spatial distribution, assuming the in jected spectrum to be an\nexponentially cut off power law with index s, integrated over n≡E′/E:\nf(x,E) =Q0E−(s+1)∞/integraldisplay\n1n−s\n/radicalbig\nln(n)\n×exp/bracketleftBigg\n−nE\nEcut−/bracketleftbig\nlad/parenleftbig\n1−1\nn/parenrightbig\n−z(x)/bracketrightbig2\n4l2\ndiffln(n)/bracketrightBigg\ndn,(14)\nwhile for µ∝negationslash= 1, we adopt Lerche & Schlickeiser’s (1980) solution to equation (12 ),\nf(x,E) =Q0/radicalbig\n(1−µ)E−(s+1/2+µ/2)1/integraldisplay\n0n(s+µ−2)/(1−µ)\n√1−n\n×exp/bracketleftBigg\n−n1/(1−µ)E\nEcut−(1−µ)/bracketleftbig\nlad/parenleftbig\n1−n1/(1−µ)/parenrightbig\n−z(x)/bracketrightbig2\n4l2\ndiff(1−n)/bracketrightBigg\ndn,(15)\nforµ <1, and\nf(x,E) =Q0/radicalbig\n(1−µ)E−(s+1/2+µ/2)∞/integraldisplay\n1n(s+µ−2)/(1−µ)\n√n−1\n×exp/bracketleftBigg\n−n1/(1−µ)E\nEcut−(1−µ)/bracketleftbig\nlad/parenleftbig\n1−n1/(1−µ)/parenrightbig\n−z(x)/bracketrightbig2\n4l2\ndiff(1−n)/bracketrightBigg\ndn,(16)\nforµ >1. HereQ0is a normalization constant that does not factor into our calculation s.\nIn this formalism, all the information about the spatial dependence of the magnetic field is\ncontained in the function z(x), defined as\nz(x) =1\nB2\n0x/integraldisplay\n0B(u)2du. (17)\nwhereB0is the magnetic field immediately behind the shock, not the far upstre am value.\nIt is presumably amplified from its initial value to the extent demanded by the data.\nFurthermore, the cut-off energy, Ecut, is found by equating loss times and acceleration\ntimes, which gives, in the Bohm Limit (Rettig & Pohl 2012):\nEcut= 8.3 TeV/parenleftbiggB0\n100µG/parenrightbigg−1/2/parenleftbiggvs\n1000 km s−1/parenrightbigg\n. (18)– 16 –\nFor arbitrary diffusion coefficients, it is a straightforward generaliz ation to show:\nEcut∝/parenleftbiggB0\n100µG/parenrightbigg−1\n1+µ/parenleftbiggvs\n1000 km s−1/parenrightbigg2\n1+µ/parenleftbiggEh\nη/parenrightbigg1\n1+µ\n. (19)\nThese analytic solutions were derived under the assumption that B(x)2D(x) is a constant\nwith respect to x, the distance from the shock. This condition is somewhat peculiar in\nthat it is only naturally satisfied if Bis constant, since the spatial dependence of D(x) is\ncontained in its dependence on B. For a field that evolves due to flux conservation, we\nexpect this to be a reasonable approximation within a thin rim. On the o ther hand, for a\nrapidly varying magnetic field (e.g. one that exponentially decays in sp ace), this imposes\na rapid variation in D(x) withx, which may or may not be realistic. However, we do\nnot expect this to affect our results at the qualitative level, and som e justification for this\nassumption can be found in Rettig & Pohl (2012).\nUsing this spectrum of electrons, intensity profiles are then obtain ed by first evaluating\nthe synchrotron emissivity\njν=c3B∞/integraldisplay\n0G(y)f(x,E)dE (20)\nwithy≡ν/c1E2B, andG(y)≡y/integraltext∞\nyK5/3(z)dz, in a slightly different notation from\nPacholczyk (1970); here K5/3(z) is a Bessel function of the second kind with imaginary\nargument. Then, integrating along lines of sight:\nIν(r) = 2√\nr2s−r2/integraldisplay\n0jν/parenleftBig\nrs−√\ns2+r2/parenrightBig\nds. (21)\nTo characterize the size of the filaments, we use the FWHM of this ra dial intensity,\ndenotedx1/2. Then, we write x1/2∝EmEγto characterize the energy dependence of the\nFWHM at each photon energy by\nmE=log(x1/2/x′\n1/2)\nlog(Eγ/E′γ). (22)– 17 –\nIn a brief aside, we note that one can get qualitative results for the behavior of mE\nwhenµ∝negationslash= 1 by noting that ldiff=/radicalbig\nDτsynch∝E(µ−1)/2∝ν(µ−1)/4in the delta function\napproximation. Furthermore, we can generalize equation (13) by u sing the new diffusion\ncoefficient in equation (6). This gives\nmE=−1\n2/parenleftBigg\nµ−4(µ+1)D/(v2\ndτ)\n2+8D/(v2\ndτ)−2/radicalbig\n1+4D/(v2\ndτ)/parenrightBigg\n. (23)\nNow at a photon energy of 2 keV, we can write D(2 keV) = η2DB(2keV), and coupled with\na modified equation (9), we can solve uniquely for η2and the maximum field strength B0\nby measuring wobsandmEat 2 keV. The results of using this approximate result are shown\nin Table 2.\nWhen performing the full numerical calculation of FWHMs, we distingu ish between\ntwo parametrizations of the magnetic field, called the “loss-limited mo del” and the\n“magnetically damped model.” That is, we can use the appropriate elec tron distribution\n(14), (15), or (16) for both cases, varying the spatial depende nce ofBto select either\nloss-limited or magnetically damped situations. We will initially neglect the cut-off in the\ninjected electron spectrum in order to more clearly highlight the ene rgy dependence of each\nmodel; we include the effects of cut-offs in Section 3.4.\n3.1. Loss-Limited Model\nIn the loss-limited model, we assume the magnetic field is spatially unifor m, a good\napproximation if we expect it to evolve by flux conservation in the nar row region behind\nthe shock. Then the function z(x) as defined in Equation (17) reduces to just x. With only\ntwo free parameters, the scaling factor for the diffusion coefficien t at 2 keV, η2and the\nmaximum field strength B0, we can fit the observed filaments and their energy dependence\nuniquely for any value of µ. For the case of Bohm diffusion ( µ= 1), the simple estimate of– 18 –\nTable 2. Best fit parameters for the Filaments in varying values of µ(Analytic Results)\nFilament 1 Filament 2 Filament 3\nµ η 2 B0 η2 B0 η2 B0\n0 7 ±4 165 ±21 0 ±0.04 143 ±52 0 ±0.007 81 ±3\n1/3 2.4 ±0.9 130 ±8 0 ±1.2 143 ±39 0 ±0.008 80.7 ±0.9\n1/2 1.8 ±0.6 123 ±7 0 ±1.2 144 ±35 0 ±0.007 80.7 ±1.2\n1 1.1 ±0.4 113 ±4 0 ±1.1 145 ±26 0 ±0.018 80.7 ±2\n1.5 .8 ±0.3 108 ±3 0 ±1.0 145 ±21 0 ±0.03 80.7 ±1\n2 .7 ±0.3 105 ±3 .2 ±0.9 150 ±21 0±3×10−780.7±1.1\nFilament 4 Filament 5\nµ η 2 B0 η2 B0\n0 0 ±0.001 118 ±1.1 0 ±1.3×1051000±500\n1/3 0 ±2×10−5117.9±0.8 0 ±2×1040±60000\n1/2 0 ±0.0003 117.9 ±1.4 0 ±120 300 ±300\n1 0 ±0.016 117.9 ±0.8 5 ±5 160 ±40\n1.5 0 ±0.0012 117.9 ±0.8 2.5 ±1.7 140 ±20\n2 0 ±1×10−8117.9±1.5 1.7 ±0.9 125 ±9\n∗Results of fitting equation (6) to the data using a Levenberg- Marquardt least-squares al-\ngorithm. The stated uncertainties are estimated as 1 σ.B0is in units of µG while η2is a\ndimensionless quantity representing the ratio of the fitted diffusion coefficient to the Bohm-\nlimit diffusion coefficient at a photon energy of 2 keV.– 19 –\nsection 1.1.1 resulted in an equation (13) for mEwhich is independent of B0. This behavior\nis preserved in the full calculation, and even for values of µdiffering from 1 the energy\ndependence only weakly depends on the magnetic field strength.\n3.1.1. Energy Dependence of the Loss-Limited Model\nIn the loss limited model, we still see the same general behavior of the FWHM with\nenergy as the calculation of Section 1.2. There is a clear transition be tween energies where\nadvection is dominant to those where diffusion is dominant, with mEdropping from –1/2\nto (µ−1)/4. A plot of this behavior for several values of µis shown in Figure 1 and an\nexample of calculated profiles is shown in Figure 2. A crucial point to ma ke is that the\nmagnitude of the diffusion coefficient is by far the most important fac tor in determining mE.\nFor Bohm-type diffusion (i.e. µ= 1) this mapping is 1-1, while for µ∝negationslash= 1 there exists only\na weak dependence of mEon the magnitude of the post-shock magnetic field. Thus, the\nobservation of mEis a direct probe of the properties of the diffusion coefficient, includin g\nboth its magnitude and behavior with energy, as can be clearly seen in Figure 3.– 20 –\n10−1100101102\nPhotonEnergy(keV)101102103FWHM(arcsec)\nFig. 1.— Energy dependence of filament widths for different diffusion c oefficients, for pure\npower-law electron spectra without a cut-off. Solid line is Kolmogorov -like (D∝E1/3),\ndashed line is Bohm-like ( D∝E), and dot-dashed line is for µ= 2 (D∝E2).\n0.92 0.94 0.96 0.98 1.00\nr/rs0.20.40.60.81.0NormalizedIntensity\nFig. 2.— Calculated profiles in the loss-limited model for B0= 100µG,µ= 1/3. The solid\nline represents a photon energy of 1 keV, the dashed line represen ts a photon energy of 2\nkeV, and the dot-dashed line represents a photon energy of 8 keV .– 21 –\n5 10 15 20 25 30 35 40\nD/DBohm(2keV)−0.5−0.4−0.3−0.2−0.10.00.1mE\nFig. 3.— Dependence of the parameter mEon the magnitude of the diffusion coefficient,\nD, measured in units of the Bohm value at a photon energy of 2 keV. Re call that mEis\ndefined such that the FWHM of the rim is ∝EmE. The lines are, from bottom to top:\nµ= 0,1/3,1/2,1 (Bohm), 1 .5,and 2. The calculations were done with B0= 100µG. For\nsmall values of Dwe find that all models converge to just below mE=−0.5, while for larger\nvalues of Dclear limits can be placed on the range of mE, an observable quantity, for each\nvalue ofµ(whereD∝Eµ).\n3.2. Magnetically Damped Model\nIn this model, we assume that the magnetic field amplification decays e xponentially\nbehind the shock, in the form of B(x) =Bmin+(B0+Bmin)e−x/ab, withBmintaken to be\n5µG(conservatively taken to be slightly higher with the above quoted va lue of 3µG in the\nISM). For this damped form of the magnetic field,\nz(x) =/parenleftbiggBmin\nB0/parenrightbigg2\nx+2abBmin(B0−Bmin)\nB2\n0(1−e−x/ab)+ab\n2/parenleftbiggB0−Bmin\nB0/parenrightbigg2\n(1−e−2x/ab).(24)\nIn order to both produce the observed filament profiles and to be d istinguishable from\nthe loss-limited model, B0must be roughly at least four times Bmin. There are three free– 22 –\nparameters, η,B0, andab, so with only two observational constraints the fits are not unique .\n3.2.1. Energy Dependence in the Magnetically Damped Model\nAgain, neglecting the energy cut-off for a moment, we see key feat ures develop in the\nenergy dependence of the FWHM. At low photon energies, where los ses are negligible over\nthe small region ab, rim sizes are energy-independent. Radial profiles at these energ ies\nreflect the spatial dependence of the magnetic field, and so, if this model is correct, we\nwould expect thin filaments to be observed in high resolution radio imag es. At higher\nenergies, the maximum value that mEreaches is determined by the competition between\nab,ldiff, andlad. This can be roughly expressed by the equation\nleff=min[max(lad,ldiff),ab]. (25)\nThus there are three possibilities. If ldiffis small enough, then as energy increases,\nsynchrotron losses will “catch up” to aband there will be a clear transition between\nloss-limited rims and magnetically limited rims. If ldiffis large enough, the rims will be\ndamping limited at all photon energies. Finally, if abis large enough, the rims will be\nloss-limited at all photon energies. It is worth noting that low-energ y (i.e., radio) thin\nsynchrotron filaments would be a clear signature of field damping. Ex amples of how the\ncalculated profiles can vary with energy in ’strong’ and ’weak’ dampin g are plotted in\nFigure 4.– 23 –\n0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00\nr/rs0.20.40.60.81.0NormalizedIntensity\n0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00\nr/rs0.20.40.60.81.0NormalizedIntensity\nFig. 4.— Calculated profiles in the magnetically damped model for B0= 70µG,ab=.005rs\n(left) and ab=.05rs(right). The solid line represents a photon energy of 1 keV, the das hed\nline represents a photon energy of 2 keV, and the dot-dashed line r epresents a photon energy\nof 8 keV.\n3.3. Electron Cut-Off Energy and the Energy Dependence of Rim Widths\nThe fact that the injected spectrum of electrons has a cut-off ab ove some energy Ecut\nhas an impact on the energy dependence of the FWHMs of the obser ved intensity profiles.\nThis is most easily seen in the δ-function approximation to the synchrotron emissivity\nfor the case of a constant magnetic field B0in the absence of a cut-off in the electron\ndistribution. Here z(x) is justx,E=/radicalbig\nν/(cmB0)≡Eν,0and, using\nf(E) =K(E0(E))−sdE0\ndE(26)\nand\nE0=E\n1−EbB2\n0t=Ev\nv−EbB2\n0x≡Eν,0\n1−x/lad(27)\nfrom Reynolds (2009), we get for the spatial dependence of the e missivity (recalling that\njν∝√\nνBf(ν,x) in this case)\njν=Cj/parenleftbigg\n1−x\nlad/parenrightbiggs−2\ne−Eν,0\nEcut/parenleftbigg\n1−x\nlad/parenrightbigg\n. (28)– 24 –\nWhenE>∼Ecut, the exponential term dominates the spatial behavior, resulting in an\nemissivity that will decay to half its peak at a distance x1/2given by\nx1/2=lad\n1+Eν,0\nEcutln(2). (29)\nwith an energy index of\nmE=∂log(x1/2)\n∂log(ν)=−1\n2/parenleftbigg\n1+ν1/2\nEcut√cmB0+ν1/2/parenrightbigg\n=−1\n2/parenleftBigg\n1+E1/2\nγ\nE1/2\nrolloff+E1/2\nγ/parenrightBigg\n(30)\nSo near the rolloff photon energy Erolloff≡hνm(Ecut),|mE|is higher that the value of 1 /2\nexpected from pure advection. In the full numerical calculation, t his arises as a shift in\nthe expected mEby some negative constant above some energy, even when diffusion is the\ndominant method of transport (see Figure 5).\n100101\nPhotonEnergy(keV)101102FWHM(arcsec)\n100101\nPhotonEnergy(keV)101102FWHM(arcsec)\nFig. 5.— Demonstration of the effect of including a cut-off in the inject ed electron spectrum\nontheenergydependenceofthefilamentwidthsfor η2=2.6,B0= 100µG,andforµ= 1(left)\nandµ= 2(right). Solid lines represent calculations with a cut-off in the elect ron spectrum\nand dashed lines represent calculations without a cut-off in the elect ron spectrum.– 25 –\n3.4. Shifts in Intensity Peak Location\nIn Figures 2 and 4 we see that in the presence of filament widths that shrink with\nenergy there is an associated outward motion of the location of pea k emission. Regardless\nof the underlying mechanism responsible for reducing the widths, th at same mechanism\nexplains this peak shift as it focuses the emission to increasingly narr ow regions at higher\nenergies. Thus this effect is a model-independent prediction.\nHowever, determining the location of the peak emission from observ ations is a task\nmuch more uncertain than determining the FWHM of radial profiles. F urthermore, the\npredicted shift in location can be quite small ( <∼1′) for the best-fit parameters found in\nsection 4. The combination of these two limitations leads us to ignore t his effect in the rest\nof our analysis.\n3.5. Comparison to the δ-Function Approximation\nWe first considered using the delta function approximation of the em issivity, namely\njν∝√\nBνf/parenleftbigg\nx,E=/radicalbiggν\ncmB/parenrightbigg\n. (31)\nHowever, when compared with the full convolution of the particle dis tribution with the\nsingle electron emissivity, the results for the FWHMs disagree consid erably (see Figure 6).\nWhat is worse, the difference is dependent on the electron energy s o that a simple constant\ncorrection factor could not be employed. The combination of the δ-function approximation\nwith the catastrophic dump form of the convection-diffusion equat ion seems to provide\na much better approximation, but it does not account for the cut- off in the injected\nspectrum of electrons. Thus, we were compelled to use the full syn chrotron emissivity in\nour numerical calculations coupled with the integral solution to the c ontinuous energy loss\nconvection-diffusion equation.– 26 –\n100101\nPhotonEnergy(keV)2×101102FWHM(arcsec)\nFig. 6.— Comparison of the energy dependence of the FHWM for Bohm diffusion at\nB0= 100µGandη=2.6 in different possible approximations without an electron cut-off.\nSimilar differences were seen for other input parameters. Solid line re presents a full con-\nvolution without a cut-off in the electron spectrum, dashed line repr esents the δ-function\napproximation used with the catastrophic dump convection-diffusio n equation, and dot-\ndashed line represents the δ-function approximation used with the continuous energy loss\nconvection-diffusion equation.\n4. Results\nIn this section we first summarize our observational methodology in measuring the\nfilament widths and spectra. Then we detail our fitting procedure f or applying our model\nto the data and describe our findings.\nTo extract radial profiles of the NE and SW limbs of SN 1006, we use six Chandra\nobservations, the parameters of which are summarized in Table 3. T he observations of the\nSW and some of the NE were performed as part of a Chandra Large Program (Winkler et\nal. 2014). These new observations provide the first high quality imag e of the SW quadrant,\ncomparable in quality with previous images of the NE. We reprocessed the level-1 event– 27 –\nfiles with CIAO ver. 4.4 and CALDB ver.4.5.1. After correcting for vign etting effects and\nexposure times for all of the data sets, we extract radial profiles in three energy bands:\n0.7–1keV, 1–2keV, and 2–7keV from 22 regions shown in Figure 7. Ea ch profile is binned\nby 1′′. When combining the NE profiles from different epochs, we take into a ccount the\nexpansion of the remnant by 4” according to the literature (Katsu da et al. 2009; Winkler et\nal. 2014). Region 8 was excluded from this analysis because in the lowe st energy bin there\nwas spatial overlap between two filaments.\n4.1. Profile Modeling\nTo estimate rim widths, we fit each profile with an empirical model defin ed as,\nh(x) =\n\nAuexp/parenleftBig\nx0−x\nwu/parenrightBig\n+Cu(upstream)\nAdexp/parenleftBig\nx−x0\nwd/parenrightBig\n+Bexp/parenleftBig\n−(x−x1)2\n2πσ2/parenrightBig\n+Cd(downstream)(32)\nwhereAu,x0,wu,Cu,Ad,wd,B,x1,σ, andCdare all free parameters. We note that\neitherx0orx1can correspond to the peak of the X-ray profile, and that Curepresents\nthe background level. The best-fit models are plotted as solid lines in F igures 8 and 9.\nBased on the best-fit model, we calculate a full width at half maximum ( FWHM) for\neach profile. The model accounts for plateaus of emission upstrea m and downstream\nof the peak; the Gaussian component describes possible downstre am features due to\nTable 3. Chandra observations of SN 1006\nObsID Array R.A. (J2000) Decl. (J2000) Roll Obs. Date Exposu re (ks) PI\n732 ACIS-S 15:03:51.7 -41:51:16 280.2◦2000 Jul 10 55.3 K.S. Long\n9107 ACIS-S 15:03:51.5 -41:51:19 280.4◦2008 Jun 24 68.9 R. Petre\n13738 ACIS-I 15:01:43.7 -41:57:55 25.3◦2012 Apr 23 73.5 P.F. Winkler\n13739 ACIS-I 15:02:14.9 -42:06:49 9.1◦2012 May 4 100.1 P.F. Winkler\n13743 ACIS-I 15:03:01.8 -41:43:05 19.9◦2012 Apr 25 92.6 P.F. Winkler\n14424 ACIS-I 15:01:43.7 -41:57:55 253.1◦2012 Apr 27 25.4 P.F. Winkler– 28 –\nprojection effects. Since our primary interest is in the energy-dep endence of widths, the\nmost important consideration is the consistency of a filament model among the three energy\nbins. To estimate the uncertainties of FWHMs, the best-fit profiles are artificially re-scaled\n(stretched or shrunk) along the x-axis, so that a new x-position of the model profile ( x′)\nbecomes x/parenleftBig\n1+ξ×x−x0\n200′′−x0/parenrightBig\n, wherexis the original x-position of the model profile and ξis\na variable stretch factor. For various ξ-values,χ2values between the re-scaled model profile\nand the data are calculated, resulting in statistical uncertainties o na(and FWHM).\nThe best-fit FWHMs and their statistical uncertainties (ranges co rresponding to\n∆χ2= 2.7) are listed in Tables 4, 5, and 6. The results are categorized into fo ur groups\nthat appear to be along the same filaments. Also listed in Tables 4, 5, a nd 6 is the average\nvalue ofmE≡log(FWHM/FWHM′)/log(ν/ν′) = either log(FWHM/FWHM′)/log(2/1) or\nlog(FWHM/FWHM′)/log(1/0.7) for our purposes, taking the FWHM for each energy range\nas that of the lower limit and using the lower adjacent energy bin for t he primed variables.\nThis quantity characterizes the energy dependence of the FWHMs by writing them as ∝\nEmE. To get the uncertainties on the calculation of mEand the averages, we took the\nuncertainty on each data point as approximately symmetric with σ= (σ++σ−)/2.\nTo check the energy-dependence of rim widths, we extract two X- ray spectra from\neach region: one is taken from a filament region (covering from a sho ck front to a FWHM\nposition downstream) and the other is taken from a plateau region n ext to the filament\nregion up to a 2 ×FWHM position downstream. These spectra together with the best -fit\nmodels (srcutin XSPEC: Reynolds 1998) are presented in Figures 8 and 9, where bla ck\nand red are responsible for the filament and the plateau regions, re spectively. In some\nregions (e.g., region #3), spectral softening downstream is clearly seen. This is consistent\nwith the fact that the higher the energy band, the narrower the r im widths become, as\nshown in Figure 10.– 29 –\nTable 4. Measured Filament FWHM (arcseconds) vs. Energy Band - N ortheast Limb\nFilament 1 Filament 2\nRegion 0.7-1 keV 1-2 keV 2-7 keV Region 0.7-1 keV 1-2 keV 2-7 ke V\n1 36+2.3\n−2.133.4+1.5\n−1.033.3+2.3\n−2.85 25 .8+.9\n−119.5+0.3\n−0.617.5+0.7\n−0.7\n2 9 .4+0.8\n−0.76.0+0.2\n−0.34.9+0.5\n−0.37 13 .8+0.4\n−0.39.7+0.1\n−0.210.2+0.1\n−0\n3 10 .3+0.6\n−0.610.1+0.4\n−0.46.5+0.3\n−0.29 26 .9+0.3\n−0.616.2+0.1\n−011.1+0.1\n−0.1\n4 78 .1+7.3\n−6.876.7+4.6\n−4.648.4+2.4\n−2.210 33 .4+1.5\n−1.330.7+0.5\n−0.526.8+0.7\n−0.6\n6 43 .7+5.5\n−3.333.5+0.8\n−0.933.6+1.6\n−1.611 15 .2+0.3\n−0.211.2+0.1\n010.9+0.9\n−0.7\nAverage 36 ±1.7 32 ±1.0 25 ±1.7 Average 23.0 ±0.4 17.5 ±0.14 15.3 ±0.6\nAverage mE -0.30±0.16 -0.3 ±0.11 Average mE -0.78±0.05 -0.19 ±0.05\n∗When calculating the uncertainties on the average FWHM and t he average mE, the uncertainties on each individual\nFWHM were treated as symmetric with uncertainty ( σ++σ−)/2. The average mEis defined as the mEcalculated from\nthe average FWHMs.\nTable 5. Measured Filament FWHM (arcseconds) vs. Energy Band - S outhwest Limb\nFilament 3 Filament 4\nRegion 0.7-1 keV 1-2 keV 2-7 keV Region 0.7-1 keV 1-2 keV 2-7 ke V\n12 12 .4+1.7\n−1.614.2+1.0\n−1.011.0+0.9\n−0.917 35 .2+2.8\n−327.1+1.2\n−1.120.5+1.6\n−1.5\n13 50 .8+2.6\n−1.954.9+2.3\n−1.738.0+2.9\n−0.818 24 .2+1.5\n−2.029.9+0.9\n−0.919.0+1.2\n−1.1\n14 38 .6+2.2\n−1.933.6+1.3\n−1.127.7+3.2\n−0.619 13 .7+1.1\n−1.115.1+0.6\n−0.66.9+0.6\n−0.5\n15 69 .9+3.8\n−4.547.5+1.1\n−1.923.7+1.5\n−1.020 34 .2+3.0\n−2.939.8+1.5\n−1.627.0+0.1.6\n−1.3\n16 74 .0+5.2\n−5.163.6+2.1\n−2.046.3+2.3\n−2.321 35 .0+1.7\n−2.114.0+0.9\n−0.112.3+0.1\n−0.5\n22 31 .7+2.2\n−1.917.5+0.5\n−0.813.9+0.9\n−1.2\nAverage 49 ±1.5 42.8 ±0.7 29.3 ±0.8 Average 29.0 ±0.9 23.9 ±0.4 16.6 ±0.5\nAverage mE -0.4±0.10 -0.54 ±0.04 Average mE -0.5±0.10 -0.53 ±0.05\n∗See note on Table 4\nTable 6. Measured Filament FWHM (arcseconds) vs. Energy Band - S outhwest Limb\nFilament 5\nRegion 0.7-1 keV 1-2 keV 2-7 keV\n8 23 .8+2.0\n−1.520.9+1.0\n−0.815.9+0.8\n−0.9\n10 33 .4+2.5\n−1.330.7+5\n−0.526.8+0.7\n−0.6\nAverage 24 ±2 27.2 ±0.6 24.8 ±0.6\nAverage mE -0.6±0.2 -0.14 ±0.05\n∗See note on Table 4– 30 –\n22\n2021\n19\n18\n17\n1615\n14\n13\n121110987654321\nFig. 7.— Chandra image at 2-7keV showing the regions where radial profiles were extr acted.\nFilament 1:Regions1-4and6; Filament 2: Regions5, 7, and9-11; Filame nt 3: Regions12-16;\nFilament 4: Regions 17-22; Filament 5: Regions 6 and 8– 31 –\n40 60 80 1000 5×10−910−81.5×10−82×10−8X−ray intensity (A.U.)\nRadial position (arcsec)69.5 − 72.5 degrees\nP1= 42.63 , P2= 42.20 , P3= 1.8066E−08, P4= 45.80 , P5= 2.000P6= 1.9804E−09, P7=−9.1953E−10, P8= 1.818 , P9= 1.4599E−08, P0= 2.9449E−0930 40 50 60 700 5×10−910−81.5×10−8X−ray intensity (A.U.)\nRadial position (arcsec)59.5 − 62.5 degrees\nP1= 45.77 , P2= 8.851 , P3= 1.1313E−08, P4= 55.22 , P5= 6.450P6= 3.0141E−09, P7= 3.7829E−09, P8= 1.145 , P9= 1.8379E−08, P0= 5.1951E−10\n40 60 80 1000 10−8 2×10−8X−ray intensity (A.U.)\nRadial position (arcsec)25 − 28 degrees\nP1= 42.59 , P2= 74.05 , P3= 6.4972E−09, P4= 37.97 , P5= 6.521P6= 2.2843E−08, P7=−1.0678E−09, P8= 0.5633 , P9= 2.8406E−08, P0= 1.9415E−09\n50 100 150 2000 5×10−910−81.5×10−8X−ray intensity (A.U.)\nRadial position (arcsec)79 − 82 degrees\nP1= 44.18 , P2= 30.33 , P3= 6.2005E−09, P4= 45.76 , P5= 2.738P6= 1.0734E−08, P7= 6.5624E−10, P8= 6.535 , P9= 8.7642E−09, P0= 4.9742E−0940 50 60 70 800 5×10−910−81.5×10−8X−ray intensity (A.U.)\nRadial position (arcsec)55 − 58.5 degrees\nP1= 46.19 , P2= 0.1681 , P3=−2.2484E−07, P4= 57.57 , P5= 13.67P6= 8.7007E−09, P7= 4.0235E−09, P8= 0.4510 , P9= 7.2497E−09, P0= 6.7412E−10\n50 100 150 2000 5×10−910−81.5×10−8X−ray intensity (A.U.)\nRadial position (arcsec)59 − 62.5 degrees\nP1= 47.59 , P2= 616.0 , P3= 1.6353E−08, P4= 19.79 , P5= 14.53P6= 6.2732E−08, P7=−1.1532E−08, P8= 4.057 , P9= 1.6257E−08, P0= 4.7171E−09\n50 100 150 2000 5×10−910−81.5×10−82×10−8X−ray intensity (A.U.)\nRadial position (arcsec)62.5 − 66 degrees\nP1= 38.54 , P2= 71.38 , P3= 7.3464E−09, P4= 31.83 , P5= 8.120P6= 1.3115E−08, P7= 2.4805E−10, P8= 0.6673 , P9= 1.4546E−08, P0= 5.3461E−0940 60 80 1000 10−92×10−93×10−9X−ray intensity (A.U.)\nRadial position (arcsec)6.5 − 9.5 degrees\nP1= 36.00 , P2= 1.3927E+05, P3= 1.3290E−08, P4= 53.12 , P5= 11.62P6= 1.8292E−09, P7=−1.2700E−08, P8= 2.0730E−02, P9= 1.3377E−09, P0= 2.3849E−10\n40 60 80 1000 5×10−910−81.5×10−8X−ray intensity (A.U.)\nRadial position (arcsec)22 − 25 degrees\nP1= 44.91 , P2= 5.145 , P3= 1.5085E−08, P4= 76.58 , P5= −62.48P6= 2.6022E−09, P7=−4.4529E−10, P8= 1.092 , P9= 1.4901E−08, P0= 1.5238E−09\n40 60 80 1000 5×10−910−81.5×10−8X−ray intensity (A.U.)\nRadial position (arcsec)48 − 51.5 degrees\nP1= 47.10 , P2= 14.25 , P3= 1.2548E−08, P4= 10.00 , P5= 0.1000P6= 0.000 , P7= 1.9478E−09, P8= 2.429 , P9= 8.0522E−09, P0= 5.4645E−0950 100 1500 5×10−9 10−8X−ray intensity (A.U.)\nRadial position (arcsec)33.5 − 36.5 degrees\nP1= 49.93 , P2= 62.98 , P3= 7.3624E−09, P4= 68.83 , P5= −6.940P6= 3.6669E−09, P7= 2.1605E−09, P8= 2.542 , P9= 1.0107E−08, P0= 3.4518E−104\n576\n8\n9\n1110−610−510−410−3Counts s−1 keV−1 cm−26.5 − 9.5 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n1010−610−510−410−3Counts s−1 keV−1 cm−222 − 25 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−225 − 28 degrees\n1 2 5−4−2024χ\nEnergy (keV)10−610−510−410−3Counts s−1 keV−1 cm−255 − 58.5 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−259.5 − 62.5 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−259 − 62.5 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−233.5 − 36.5 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−248 − 51.5 degrees\n1 2 5−4−2024χ\nEnergy (keV)10−610−510−410−3Counts s−1 keV−1 cm−262.5 − 66 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−279 − 82 degrees\n1 2 5−4−2024χ\nEnergy (keV)10−610−510−410−3Counts s−1 keV−1 cm−269.5 − 72.5 degrees\n1 2 5−4−2024χ\nEnergy (keV)1\n2\n3\nFig. 8.— Left: Filament radial intensity profiles at 2-7 keV. The dashe d, black vertical lines\noccur at a length of one FWHM on each side of the emission’s peak, while the dashed, red\nvertical lines enclose the region extending towards the center of t he remnant that starts at\nthe edge of the black region to a distance of 2 ×FWHM away from the peak. Right: Energy\nspectra of the filaments separated into the same two regions– 32 –\n80 100 120 140 160 1800 2×10−94×10−96×10−9X−ray intensity (A.U.)\nRadial position (arcsec)268.5 − 271.5 degrees\nP1= 164.9 , P2= −2811. , P3=−3.4422E−08, P4= 145.6 , P5= 7.645P6= 3.0516E−09, P7= 3.6170E−08, P8= 13.19 , P9= 1.0089E−09, P0= 1.7328E−10100 150 2000 5×10−9 10−8X−ray intensity (A.U.)\nRadial position (arcsec)253 − 256 degrees\nP1= 165.9 , P2= 20.17 , P3= 4.5265E−09, P4= 157.0 , P5= 5.480P6= 4.3016E−09, P7= 1.1999E−09, P8= 3.942 , P9= 3.2893E−09, P0= 1.8935E−09\n100 120 140 160 1800 5×10−9 10−8X−ray intensity (A.U.)\nRadial position (arcsec)226 − 229 degrees\nP1= 159.0 , P2= 26.02 , P3= 1.3772E−08, P4= 164.4 , P5= −4.077P6=−1.7630E−08, P7= 7.9324E−10, P8= 1.410 , P9= 1.1623E−08, P0= 4.5096E−10\n100 120 140 160 1800 2×10−94×10−96×10−9X−ray intensity (A.U.)\nRadial position (arcsec)272 − 277 degrees\nP1= 159.8 , P2= 12.83 , P3= 4.1633E−09, P4= 1114. , P5= −256.2P6=−2.1512E−07, P7= 7.8072E−10, P8= 2.008 , P9= 5.5568E−09, P0= 5.6165E−10100 150 2000 2×10−94×10−96×10−9X−ray intensity (A.U.)\nRadial position (arcsec)249 − 252 degrees\nP1= 163.1 , P2= 15.08 , P3= 4.8883E−09, P4= 0.000 , P5= 1.0000E−10P6= 0.000 , P7= 1.3546E−09, P8= 3.883 , P9= 4.6929E−09, P0= 1.8155E−09\n80 100 120 140 160 1800 5×10−9 10−8X−ray intensity (A.U.)\nRadial position (arcsec)258 − 261 degrees\nP1= 164.8 , P2= 16.16 , P3= 4.2663E−09, P4= 156.6 , P5= 4.499P6= 7.5628E−09, P7= 1.5008E−09, P8= 0.4168 , P9= 5.0000E−09, P0= 1.0382E−09\n80 100 120 140 160 1800 2×10−94×10−96×10−98×10−9X−ray intensity (A.U.)\nRadial position (arcsec)261 − 264 degrees\nP1= 161.7 , P2= 31.11 , P3= 6.3991E−09, P4=−4.8313E+10, P5= 3.8616E+10P6= 1.0411E−10, P7= 5.0016E−10, P8= 3.468 , P9= 6.1374E−09, P0= 8.5192E−1080 100 120 140 1600 5×10−910−81.5×10−8X−ray intensity (A.U.)\nRadial position (arcsec)211 − 214 degrees\nP1= 161.6 , P2= 9.166 , P3= 1.2685E−08, P4= 134.4 , P5= −13.94P6= 0.000 , P7= 3.6833E−09, P8= 0.6187 , P9= 2.4606E−08, P0= 4.5498E−09\n80 100 120 140 160 1800 2×10−94×10−96×10−98×10−9X−ray intensity (A.U.)\nRadial position (arcsec)221.5 − 224.5 degrees\nP1= 160.9 , P2= 43.90 , P3= 7.7490E−09, P4= 134.4 , P5= −13.94P6= 0.000 , P7= 1.2144E−09, P8= 0.9169 , P9= 5.0006E−09, P0= 5.1325E−10\n50 100 1500 2×10−94×10−96×10−98×10−9X−ray intensity (A.U.)\nRadial position (arcsec)235 − 238 degrees\nP1= 161.2 , P2= 72.67 , P3= 8.7290E−09, P4= 134.4 , P5= −13.94P6= 0.000 , P7=−8.5612E−10, P8= 0.9046 , P9= 9.8641E−09, P0= 1.6287E−0950 100 1500 5×10−9 10−8X−ray intensity (A.U.)\nRadial position (arcsec)231.5 − 234.5 degrees\nP1= 161.8 , P2= 330.5 , P3= 1.9103E−08, P4= 154.7 , P5= 3.516P6= 4.5434E−09, P7=−1.2593E−08, P8= 6.7419E−02, P9= 3.0824E−08, P0= 1.5328E−0915\n1617\n19\n20\n21\n2210−610−510−410−3Counts s−1 keV−1 cm−2211 − 214 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n18\n10−610−510−410−3Counts s−1 keV−1 cm−2221.5 − 224.5 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−2226 − 229 degrees\n1 2 5−4−2024χ\nEnergy (keV)10−610−510−410−3Counts s−1 keV−1 cm−2249 − 252 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−2253 − 256 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−2258 − 261 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−2231.5 − 234.5 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−2235 − 238 degrees\n1 2 5−4−2024χ\nEnergy (keV)10−610−510−410−3Counts s−1 keV−1 cm−2261 − 264 degrees\n1 2 5−4−2024χ\nEnergy (keV)\n10−610−510−410−3Counts s−1 keV−1 cm−2272 − 277 degrees\n1 2 5−4−2024χ\nEnergy (keV)10−610−510−410−3Counts s−1 keV−1 cm−2268.5 − 271.5 degrees\n1 2 5−4−2024χ\nEnergy (keV)12\n13\n14\nFig. 9.— See Figure 8 caption– 33 –\n0 5 10 15 200123Normalized FWHM\nRegion Number\nFig. 10.— Observed energy dependence of the FHWMs of SN1006, plo tted vs. region\nnumber and normalized to the middle (1–2 keV) energy band. Circles r epresent 0.7–1 keV\nwhile squares represent 2–7 keV\n4.2. Fitting procedure\nFor each choice of spectral index for the power law dependence of the diffusion\ncoefficient, µ, we constructed a two dimensional grid in the parameter space of ( B0,ηE1−µ\nh)\nfor which we calculated radial profiles at 0.7, 1, and 2 keV for each po int in the grid\n(recall that ηE1−µ\nhis the constant scaling factor of the diffusion coefficient, D). From\nthis, we obtained both the FWHM at 2 keV and the specific value of mEat 2 keV from\nlog(FWHM(2keV)/FWHM(1keV))/log(2). Thus we had numerical re sults for a large\nnumber of discrete points over this parameter space and manually f ound the point that\nsimultaneously reproduced the values of both B0andηobtained from our observations.\nWe obtained the stated uncertainties by varying the parameters ηandB0around this\nbest-fit value to find the domain in which the observations were still s atisfied within their– 34 –\nrespective uncertainties. While not a formal error-analysis, this p rocedure is adequate to\nput approximate lower and upper bounds on our estimates.\n4.3. Loss-Limited Model\nThe best-fit parameters are shown in Table 7, where η2is the strength of the\ndiffusion coefficient divided by the Bohm diffusion coefficient at an energ y of/radicalbig\nν/cmB0,\nwhich depends also on the fitted value for B0. Note that this calculation included the\nconvolution of the single particle emissivity with the solution of the con tinuous energy loss\nconvection-diffusion equation for a spectrum of electrons expone ntially cut off at the shock.\n.\n4.4. Magnetically Damped Model\nThe magnetically damped model predicts that the data should show F HWMs that are\nonly weakly dependent on energy, caused by the electron distribut ion’s cut-off, with values\nofmEon the order of −0.1. This is decidedly not what we observe (see Figure 9 and Tables\n3-5), as the averaged filaments all display values of |mE|>∼0.14 at 2 keV and >∼0.3 at 1\nkeV. This does not demonstrate that post-shock magnetic field da mping cannot occur, or\nthat rims might not be magnetically damped at much lower observation energies, but it\nprovides sufficient evidence that the damping length must at least be large enough to be\nunimportant, i.e., larger than the synchrotron-loss length, for ele ctrons radiating at keV\nenergies. Therefore, we confidently conclude that the X-ray rims of SN1006 are notwell\ndescribed by the magnetically damped model.– 35 –\nTable 7. Best fit parameters for the Filaments in varying values of µ(Numerical Results)\nFilament 1 Filament 2 Filament 3\nµ η 2 B0 η2 B0 η2 B0\n0 7.5 ±2 142 ±5 - - <∼0.1 77 ±.8\n1/3 4 ±1.3 120 ±5 - - <∼0.1 76 ±1.4\n1/2 3 ±1.1 112 ±4 - - <∼0.1 75 ±1.0\n1 2 ±1.0 100 ±3 22±3 214 ±4<∼0.1 74 ±1.1\n1.5 1.9 ±1.2 95 ±3 9±1.2 167 ±4<∼0.1 74 ±1.2\n2 2 ±1.0 92 ±4 7±1.1 152 ±4<∼0.1 73 ±1.2\nFilament 4 Filament 5\nµ η 2 B0 η2 B0\n0 <∼0.2 113 ±2 - -\n1/3 <∼0.2 112 ±2 - -\n1/2 <∼0.2 111 ±2 - -\n1 <∼0.2 109 ±2 80+∞\n−4206±3\n1.5 <∼0.2 108 ±2 19 ±2 140 ±2\n2 <∼0.2 107 ±2 12 ±1.0 120 ±2\n∗Results of fitting the data using our generalized diffusion mo del for the\nloss-limited case outlined in Section 3. Dashes denote plac es where fits were\nunobtainable. See note on Table 2 for B0andη2– 36 –\n5. Discussion\nIn this section we will analyze the results of applying our model of Sec tion 3 to the\nSN1006 data presented in section 4.\nIn fitting the data for various values of µ, we were able to acquire the best fit diffusion\ncoefficient energy relation in the region in which electron energies are relevant for keV\nemission.\nFitting the data allowed us to constrain both the magnitude and the e nergy-dependence\nof the diffusion coefficient D, where the latter is reflected in the values of µfor which\nwe could obtain fits. However, fixing the magnitude of Dat some energy only fixes the\ncombination ηE1−µ\nh, so there is a degeneracy in the choice of ηandEh. There are two\nrestricting conditions. The first is that D(E)> DB(E) at all energies, as the Bohm\ncoefficient is the minimum allowable. In other words,\nη/parenleftbiggE\nEh/parenrightbiggµ−1\n>1. (33)\nForµ >1, this restricts Eµdiffusion to above some threshold energy Eh, while for µ <1,\nthis restricts Eµdiffusion to below some maximum energy Eh. It also means that the\nconstant ηmust always be greater than 1 as D(Eh)/Db(Eh) =η. The second restriction is\nthatEhshould be outside the relevant energy range for electrons emitting keV X-rays, as\nall of our calculations had a fixed value for µ. What we can find, however, is the minimum\n(forµ <1) or maximum (for µ >1) value of this energy bound by fixing ηat 1. For our\nsuccessful fits that have non-negligible diffusion coefficients, the r esults of doing this give\nvalues of Ehthat are so far outside the 0.7–7 keV photon range that we can eas ily find an\nappropriate Ehto satisfy the above conditions.\nFrom Tables 2 and 7, we see that several of our averaged filaments require very small\ndiffusion coefficients, well below the Bohm value, even using the amplifie d magnetic field.– 37 –\nThis obviously implies that varying the parameter µwill have no effect on the fits, as is\nevident in the fitted values for B0. Qualitatively, this means that the transport of electrons\nis being carried out dominantly by the convection of plasma away from the shock, and that\neach electron will stay attached to its particular fluid element. This r esult is required by the\nstrong energy dependence ( mE∼ −0.5) of the filament widths, as the presence of diffusion\nwill always drive mEtowards 0 (or beyond to positive numbers in the case of µ >1. On\nthe other hand, filaments with non-negligible diffusion coefficients are all consistent with\nthe condition that D > D Bohm, within their respective uncertainties. This may suggest that\nsome mechanism is severely limiting electron diffusion in various regions o f the remnant,\nprimarily the SW.\nOne shortcoming of the magnetically damped model is the requiremen t thatB(x)2D\nis constant. This implies that the diffusion coefficient varies as 1 /B(x)2, when we have\nexplicitly written the diffusion coefficient as proportional to 1 /B(x) in our formalism.\nHowever, this does not affect our conclusion that the magnetically lim ited model is a poor\nfit to the data, as the qualitative behavior of the FWHMs as a functio n of energy would be\nthe same. This requirement presents no issues in the case of the los s-limited model, as both\nDandBare spatially uniform in the narrow region behind the shock.\nOur finding that rim widths drop too rapidly with energy to allow significa nt diffusion\nsuggests the possibility of “sub-Bohm diffusion” ( λ < r g, orη <1) in astrophysical\nsources. This possibility has important implications for acceleration t imes, since much\nsmaller diffusion coefficients Dwould result in much shorter acceleration times to a given\nenergy. There has been considerable discussion of the possibility of sub-Bohm diffusion\nin the literature. For instance, Zank et al. (2006) find that in perpe ndicular shocks in\nthe solar wind, effective mean free paths can be an order of magnitu de or more less than\nthe gyroradius. They find some supporting evidence in heliospheric o bservations. Using– 38 –\na 3D hybrid MHD-kinetic code, Reville & Bell (2013) studied the develop ment of shock\nprecursors generated by accelerated particles, finding sub-Boh m behavior for both parallel\nand oblique shocks, but more pronounced for parallel shocks. How ever, some of their\nsimulations find that the Bohm limit is still respected using the amplified m agnetic field.\nReville & Bell discuss other possible ambiguities in the definition of the Bo hm limit,\nincluding the possibility of highly inhomogeneous magnetic fields on small scales. At any\nrate, it seems clear that the complexities of the propagation of par ticles in the presence of\ndynamic self-generated magnetic turbulence are such that the su ppression of diffusion to\nlevels considerably below those implied by the Bohm limit is not ruled out by theoretical\nconsiderations. We emphasize that in spite of the elaborate theore tical structure we have\npresented, our limits on the diffusion coefficient are closely related to the rapid drop in\nfilament widths with photon energy that we see in Figure 10. The rate of shrinkage is too\nlarge to tolerate much particle diffusion, independently of detailed mo deling. However,\ndetailed quantitative statements are dependent on the details of o ur mechanism for fitting\nfilament widths, and on inevitable projection and curvature effects . Our models do predict\nconsiderably thinner filaments at 4 keV than at 2; while our current o bservations do not\nhave adequate photon statistics to test this prediction, future s tudies should be performed\nto clarify this important issue.\nFinally, we call attention to the µdependence in the electron cut-off energy used in our\nmodel as described in Equation (19). For a synchrotron emitting so urce, this cut-off energy\ncorresponds to a rolloff frequency of νroll∝E2\ncutB0∝B(µ−1)/(µ+1)\n0 v4/(µ+1)\ns. Ifµ= 1 as in the\nstandard Bohm assumption, we find a rolloff frequency that is indepe ndent of the magnetic\nfield and solely a function of the shock speed. In that case, we would expect constant rolloff\nfrequencies along the same filaments and only a relatively weak azimut hal dependence,\npredictable from observed proper motions. On the other hand, if µ∝negationslash= 1, then we recover\naB0dependence, which could account for the some of the systematic o rder-of-magnitude– 39 –\nazimuthal variation of the measured rolloff frequencies seen in both SN1006 (Katsuda et al.\n2010, Miceli et al. 2009, Reynolds et al. 2012) and G1.9+0.3 (Reynolds et al. 2009). For\nµ= 2, one would require a very large variation in B0, largest at the brightness maxima, to\nexplain the observed factor of 10 range in rolloff frequency, howev er.\n5.1. Comparisons to Cas A\nA detailed application of our results to other SNRs such as Cas A will re quire much\nmore extensive analysis, but we can use the published filament widths of Araya et al.\n(2010) for Cas A to get preliminary estimates of the magnetic field st rength and diffusion\ncoefficient by applying our model. In their data, it appears that the fi laments in Cas A\nshrink by a factor of ∼0.8 between 0.3 and 3 keV, while the filament widths appear to\nbe energy-independent between 3 and 6 keV. Qualitatively, this is co nsistent with the\nloss-limited model, as our parameter mEis predicted to decrease with energy. For the\nlower energy range of 0.3–3 keV, reproducing mE∼ −0.1 (equivalent to the factor of 0.8\ndrop in size) requires magnetic fields on the order of 200-500 µG and diffusion coefficients\nabout 5×DBohm(3 keV), about an order of magnitude higher than the values one ob tains\nby neglecting the energy dependence. One can also see directly fro m Figure 3 that µ <1\nmodels of the diffusion coefficient are excluded for mE∼ −0.1.\n6. Summary and Conclusions\nWe have outlined a generalized diffusion model that solves the continu ous energy-loss\nconvection-diffusion equation for electrons subject to both conv ection and diffusion as they\ntravel away from the shock. This model is able to incorporate arbit rary power-law energy\ndependence of the diffusion coefficient D, in the form of D∝Eµ, as well as arbitrary– 40 –\nspatial dependence of the magnetic field strength. Assuming sphe rical symmetry, we then\nconvolved this electron distribution with the single electron power sp ectrum to obtain\na non-thermal emissivity, which we integrated along lines of sight to o btain the specific\nintensity of the source as a function of radial distance from its cen ter. We specialized\nthis model into two general categories: a “Magnetically Damped Mod el,” which assumes\nB∝exp−x/abforxdefined as the distance behind the shock, and a “Loss-limited Model”\nwhich assumes a constant post-shock magnetic field strength. Fu rthermore, we selected\nspecific values of the parameter µ(namely 0,1/3, 1/2, 1, 1.5, and 2) to analyze. We found\nthat independent of model details, magnetically damped models pred ict rim widths almost\nindependent of photon energy, while loss-limited models predict rim wid ths to shrink with\nincreasing photon energy at a rate dependent on the diffusion coeffi cient. Our quantitative\nresults are summarized in Figure 3.\nWith this model as our guide, we used Chandra observations of SN 1006 to measure\nthe energy dependence of the thin, non-thermal rims in the NE and SW quadrants. For\nthe SW, these are the first such measurements, utilizing data from a recent Large Chandra\nProject. The SW filament profiles are of similar width to those in the NE . This is consistent\nwith the results of Winkler et al. (2014), who find the conditions in bot h regions to be\nsimilar. Furthermore, the filament widths of SN1006 show a decreas e with photon energy,\nwhich we have shown has important physical consequences for bot h the diffusion coefficient\nand the post-shock magnetic field, and is incompatible with a magnetic damping model.\nIn the other SNRs for which magnetic fields have been inferred from rim thicknesses, the\nenergy dependence of the widths should be examined similarly, as evid enced by our quick\ncomparison with the results of Araya et al. (2010) for Cas A.\nUsing our generalized diffusion model and its subsets outlined above, we find the\nmeasured widths of the SW filaments of SN 1006, like those previously reported for the– 41 –\nNE, favor magnetic fields on the order of 100 µG, significantly amplified above the typical\ninterstellar medium value of about 3 µG. The strength of our model is that it encompasses\nall effects included by previous authors in this type of investigation.\nWe also conclude that for the filaments of SN 1006 , and even the filam ents of Cas A,\nvalues of µ <1 are, for the most part, less able to reproduce the data. This is du e to the\nresult that the lowest possible mEfor a given diffusion coefficient is ( µ−1)/4, which requires\nµ>∼1 in Filaments 2 and 5 for SN 1006 and the majority of filaments in Cas A ( Araya\net al. 2010). While not conclusive, this result is in agreement with the c alculations of\nReynolds (2004), which predict SNR images that do not resemble obs erved SNRs for µ <1.\nThus both Kolmogorov turbulence ( µ= 1/3) and Kraichnan ( µ= 1/2) are disfavored.\nIn an application to other SNRs, values of µ >1 allow completely energy-independent\nfilament widths even with a cut-off in the injected spectrum of electr ons. Thus, the results\nof Araya et al. (2010), who found energy independent FWHMs in Cas A between 3-6\nkeV, could result from strong diffusion and a very high electron cut- off energy, or from\nnon-Bohm-like diffusion with µ >1. In the latter case, our model also predicts that the\nrolloff frequency should depend on magnetic field strength and thus could have significant\nazimuthal dependence.\nWe also find that that the magnitudes of the diffusion coefficients in th e filaments\nof SN1006 are split into two distinct categories. One group, due to t he strong energy\ndependence of their filament widths, requires negligible amounts of d iffusion (i.e. less than\nthe Bohm limit) in order to reproduce the observations. The occurr ence of sub-Bohm\ndiffusion is thus far undocumented and would be a groundbreaking re sult. However, there\nare inherent uncertainties in our measurements of the FWHMs due t o projection, overlap,\nand averaging effects which may have influenced our numerical calcu lations. Thus, we do\nnot claim strong evidence of this theoretically hard to explain phenom enon and note that– 42 –\nfurther study is needed. On the other hand, we are much more con fident in the rest of our\nconclusions, which are fairly robust and do not depend on sub-Bohm diffusion. The second\ngroup, with less energy-dependent filament widths, is consistent w ith diffusion coefficients\nclose to but above the Bohm limit. This result is of interest because th e strength of diffusion\nis directly tied to the maximum energy attainable by electrons being ac celerated at the\nshock front. Diffusion coefficients much larger than the Bohm value w ould have suggested\nweaker scattering, which in turn would reduce the maximum energy ( Reynolds 1998).\nHowever, in our model of the filaments, the radial profiles are prod uced by electrons in the\npost-shock region, so pre-shock electrons could have much differ ent diffusive properties.\nOur fits predict continuing shrinkage of filament widths at higher ene rgies than 2 keV,\nthough photon statistics in our current observations are not ade quate to test this. (Our 2–7\nkeV band is dominated by photons near 2 keV.) A longer observation o f the SW region of\nSN 1006 with Chandra could allow division of that band into 2 – 4 and 4 – 7 keV bands,\npermitting this important test. Our surprising result of rapid shrink age of some filaments\nrequiring sub-Bohm diffusion coefficients can be searched for in othe r thin-rim remnants\nsuch as Tycho.\nFinally, we find that the results of applying our generalized diffusion mo del are\nremarkably consistent with the results obtained by simply fitting Equ ation 6 to the data\n(recall that Equation 6 was the result of applying the δ-function approximation for the\nelectron spectra to the catastrophic dump version of the convec tion-diffusion equation). This\nis in spite of the fact that our model solves the continuous energy lo ss convection-diffusion\nequation for the electron distribution, uses the full synchrotron emissivity, and includes a\ncutoff in the injected spectrum of electrons, all of which we have sh own to have important\neffects on the FWHMs and their energy dependence. This may simply b e a unique result\nfor the observational data from SN1006, or it could suggest that the the effects of adding\neach of these more detailed considerations cancel each other out when combined. On the– 43 –\nother hand, Equation 6 is incapable of describing magnetically damped filaments, which\nmay occur in other remnants (e.g., Marcowith & Casse 2010), thoug h we have ruled this\nout for SN1006.\nThe general formalism presented here is applicable to the thin, non- thermal filaments\nobserved in nearly all historical SNRs, and has the potential to pro vide a consistent estimate\nof magnetic-field amplification across the variety of ambient environ ments into which these\nremnants are expanding.\nSupport for this work was provided by the National Aeronautics an d Space\nAdministration through Chandra Grant Number GO2-13066, issued by the Chandra X-ray\nObservatory Center, which is operated by the Smithsonian Astrop hysical Observatory for\nand on behalf of NASA under contract NAS8-03060.\nWe thank the anonymous referee for an extremely careful readin g of this paper, and\nfor suggestions that have led to substantial improvements.\nA. Appendix\nWe summarize here some details of the relation of our models to obser vations. Our\nbasic conclusion is that the variation of filament widths with energy co ntains essential\ninformation required to compare models, and to obtain quantitative estimates of the\nmagnetic field and diffusion coefficient. Without this information, ther e are several\ncompeting models that can allow for a wide range of magnetic-field str engths and diffusion\ncoefficients for the same filament width.\nWe assume throughout a spherical shock surface, in which the pea k of synchrotron\nemission occurs at a radius slightly behind the shock due to the geome try of the line of\nsight integration. If instead a plane shock with velocity exactly in the plane of the sky is– 44 –\nassumed, derived quantities will vary somewhat. In addition, we find that the δ-function\napproximation gives a poor representation of the spatial distribut ion of high-energy\nelectrons, resulting in an underestimate of filament widths. It is also essential to consider\nthe cutoff in the electron spectrum above some maximum energy as it causes filament\nwidths to shrink with photon energy in a model-independent way. At p hoton energies close\nto the rolloff frequency, the only way to have truly energy-indepen dent rim sizes is with\nµ >1.In energy-loss models with µ≤1, or damping models, |mE|will always be at least ∼\n0.1 if the cutoff is not exceptionally high (well above the keV band).\nThen the strength of the energy-dependence of filament widths s erves as the essential\ndiscriminant among models. If a weak energy dependence (0 .1<∼|mE|<∼0.2) is observed\nfor photon energies near the synchrotron rolloff frequency, the behavior at lower photon\nenergies should be inspected as it will have greater discriminatory po wer. Here “lower\nenergies” means energies lower than those where diffusion and the e lectron cutoff start to\nbecome important, which depend on the source parameters. Near the rolloff frequency,\nmany effects can combine to cause weak energy-dependence of fila ment widths.\nIf moderately strong energy-dependence of filament widths is obs erved (0.2<∼|mE|<∼\n0.5) at a specific energy, then a magnetic-field damping model can be r uled out at that\nenergy and above. This is the region in which it can be assumed that diff usion is important\nin competing with advection. The details of this then depend on the as sumed model of\ndiffusion. And if very strong energy-dependence of filament widths is observed ( |mE|>∼.5)\nthen the only explanation is weak diffusion and the predominance of ad vection as the\nelectron transport mechanism.\nFinally, if filaments widths are ever observed to be growing with energ y then the only\nknown explanation would be µ >1 diffusion. Higher values of µallow for more rapid\nchanges in mEas a function of energy.– 45 –\nREFERENCES\nAbraham, J., Abreu, P., Aglietta, M., et al. 2008, PhRvL, 101, 06110 1\nAcero, F., Aharonian, F., Akhperjanian, A. G., et al. 2010, A&A, 516 , A62\nAraya, M., Lomiashvili, D., Chang, C., Lyutikov, M. & Cui, W. 2010, ApJ, 714, 396\nBallet, J., 2006, Adv. Sp. Res., 37, 1902\nBamba, A., Yamazaki, R., Ueno, M., & Koyama, K. 2003, ApJ, 589, 827 ApJ, 632, 294\nBerezhko, E. G., & V¨ olk, H. J., 2004, A&A, 419, L27\nBerezhko, E. G., Ksenofontov, L. T., & V¨ olk, H. J. 2003, A&A, 412 , L11\nBlandford, R., & Eichler, D. 1987, PhR, 154, 1\nBurbidge, G.R. 1956, ApJ, 124, 416\nCassam-Chena¨ ı, G., Hughes, J. P., Ballet, J., & Decourchelle, A. 200 7, ApJ, 665, 315\nGreen, D. A. 2009, Bulletin of the Astronomical Society of India, 37 , 45\nJun, B.-I., & Jones, T.W. 1999, ApJ, 511, 774\nKatsuda, S., Petre, R., Long, K. 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Symbol Glossary (Numerical values in CGS)\nSymbol Expression/Value Explanation\nDiffusion Coefficient, D = ηDB(Eh)/parenleftBig\nE\nEh/parenrightBigµ\nDBCdE\nB“Bohm-limit” of D\nCd2.083×1019\nη Scaling factor of D\nµ Power index for D with E\nEh Arbitrary energy (keeps ηdimensionless)\nη2D\nDB(2 keV) Relative D at hν=2 keV\nSynchrotron Parameters\nτsynch1\nbB2ESynchrotron lifetime\nladvs\n4τsynch Advective length\nldiff/radicalbig\nDτsynch Diffusive length\nνm cmE2B δ -function synchrotron frequency\ncm 1.82×1018\nc1 6.27×1018\nb 1.57 ×10−3\nDiffusion Model Parameters\nB0 Immediate Post-shock B-field\nx1/2 FWHM of radial intensity profile\nmE∂log(x1/2)\n∂log(Eγ)Power index of x1/2withEγ\nf(x,E) e−spatial and energy distribution\nab Length scale in B-damping model– 49 –\nTable 8—Continued\nSymbol Expression/Value Explanation\nSN1006 Parameters\nva\ns 5×108Shock Velocity\nrb\ns 2.96×1019Shock Radius\ns 2.2 e−spectral index\naKatsuda et al. 2009\nbCalculated using the angular size in Green’s\ncatalog (2009) and the distance to the remnant\ngiven by Winkler et al. (2003)" }, { "title": "2004.12243v1.Pulse_assisted_magnetization_switching_in_magnetic_nanowires_at_picosecond_and_nanosecond_timescales_with_low_energy.pdf", "content": "Pulse -assisted magnetization switching in magnetic nanowires at picosecond and \nnanosecond timescales with low energy \n \nFurkan Şahbaz, Mehmet C. Onbaşlı* \nKoç University, Department of Electrical and Electronics Engineering, Sarıyer, 34450 Istanbul \n*corresponding author: monbasli@ku.edu.tr \n \nDetailed understanding of spin dynamics in magnetic nanomaterial s is necessary for develop ing \nultrafast , low -energy and high -density spintronic logic and memory. Here, we develop \nmicro magnetic models and analytical solutions to elucidate the effect of increasing damping and \nuniaxial anisotropy on magnetic field pulse -assisted switching time, energy and field requirements \nof nanowires with perpendicular magnetic anisotropy and yttrium iron garnet -like spin transport \nproperties. A nanowire is initially magnetized using an external magnetic field pulse (write) and \nself-relaxation. Next, magnetic moments exhibit deterministic switching upon receiving 2.5 ns -\nlong external magnet ic pulses in both vertical polarities . Favorable damping (α~ 0.1-0.5) and \nanisotropy energies (104-105J·m-3) allow for as low as picosecond magnetization switching times. \nMagnetization reversal with fields below coercivity was observed using spin precession \ninstabilities. A competition or a nanomagnetic trilemma arises among the switching rate, energy \ncost and external field required. Developing magnetic nanowires with optimized damping and \neffective anisotropy could reduce the switching energy barrier down to 3163 ×kBT at room \ntemperature . Thus, pulse -assisted picosecond and low energy switching in nanomagnets could \nenable ultrafast nanomagnetic logic and cellular automata . \n I. Introduction \nAn in-depth understanding of spin relaxation in magnetic nanostructures is necessary to \ndevelop highly efficient and ultrafast switching methods . The interplay between external magnetic \nfield and magnetic material properties remains to be understood at sub -100 nanosecond and \nnanometer length scales for fundamental studies of spin -spin, spin -electric field and spin -magnetic \nfield interactions for developing future spintronic devices . The effects of external field amplitude \n[1], frequency [ 2] and polarization [3] on the spin relaxation of nano magnetic media have been \ninvestigated. Previous studies indicate that switching field decreases when the polarization \ndirection and frequency of the circularly polarized microwave field matches that of the \nferromagnetic re sonance of the nanomagnet [3 ]. Applying a microwave magnetic field with \noptimal frequency at or near ferromagnetic resonance reduces the coercive field by helping \novercome the effective energy barrier of the domain nucl eation [1 ]. The coercivity reduction is \nlarger than the microwave field magnitude H rf at a cert ain frequency and input power [2 ]. \nWhile microwave, laser or heat -assisted switching effects that determine switching energy \nhave been investigated [4,5], the intrinsic magnetic material property dependence of spin \nrelaxation has not been studied extensively. Previous spin relaxation studies include nanodot \nmodels [ 6], permalloy rectangle models [ 7], nan owire models [ 8], and ferromagnetic nano particles \n[9]. The key magnetic properties that affect spin relaxation include Gilbert damping constant, \nsaturation magnetic moment, exchange stiffness, anisotropy, dimensions and aspect ratios . In this \nstudy, we use analytical and numerical micromagnetic models to qu antify the regimes under which \nincreasing damping, uniaxial anisotropy and external pulse field can switch magnetism in sub-100 \nnm nanowires . The results of these analyses prompted us to propose a n external magnetic field \npulse -assisted magnetization rever sal mechanism that could enable sub -coercivity and sub -nanosecond nonvolatile switching with low energy (a few thousand k BT per bit at room \ntemperature) . \nPrevious studies show that damping plays a key role in magnetization dynamics of \nnanostructures [ 10,11]. Ref. [ 10] inspected the effect of damping on reversal time without \nanisotropy and showed that magnetization reversal time can increase (decrease) with increasing \n(decreasing) damping constant. Since realistic materials have nonzero intrinsic magnetic \nanisotropy, magnetization reversal models must include anisotropy . A generalized analysis of \nmagnetization reversal [12] highlights the significa nce of demagnetization factors and anisotropy \nparameters . In nanostructures, magnetoelastic [1 3], magnetocrystalline [ 14], off -stoichiometry \n[15], growth -induced anisotropy [ 16, 17] and surface -induced anisotropy (especially for large \nsurface area -to-volume ratio nanostructures ) can be modeled with an overall uniaxial anisotropy \nterm, which alters switching times significantly [18,19 ]. One could engineer these terms to achieve \nperpendicular magnetic anisotropy (PMA) preferred in high-density memory [20]. Large \nperpendicular anisotropy increases the effective field and causes precession -driven magnetization \ndynamics with high precession frequencies [ 21]. \nWe present the results of our analytical spin relaxation model and numerical methods in \nSection II. In section III , we present Gilbert damping constant and uniaxial anisotropy dependence \nof spin relaxation and magnetization reversal time. In section IV, magnetization switching time \nand energy are modeled function s of external magnetic field pulse intensi ty and width. \nII. Numerical Modeling and Analytical Solutions of Spin Relaxation \n1. Numerical model details \nNumerical models were developed to understand the magnetic relaxation and reversal in \nnanowires . We used Object -oriented Micromagnetic Framework (OOMMF) to obtain magnetic nanowire hysteresis loops and spin relaxation dynamics as function of damping (α) and uniaxial \nanisotropy constant (K u). A rectangular 20×100×10 nm3 Y3Fe5O12 (YIG) nanowire (width, length, \nthickness) was used for all models in this study . The time evolution of m x, my and m z vectors were \ncalculated with minimum temporal step sizes of 2.14 fs. These nanowire dimensions were chosen \nto elucidate the effect of damping (α= 10-4–10) and uniaxial anisotropy (K u=103–106J·m-3) in the \nnear single dom ain regime . These dimensions are experimentally feasible with state -of-the-art \nfabrication techniques [22-26]. We chose YIG (exchange stiffness A ex=3.65± 0.38pJ·m-1, \nsaturation magnetization M s=140 kA·m-1) due to its very low and tunable damping [27-29] and \ndue to its lower exchange stiffness compared with permalloy (13pJ·m-1) [7], cobalt -platinum \nmultilayers as well as Heusler alloys (15pJ·m-1) [30, 31]. We focus on m agnetic insulators (MI) \nlike YIG over metals due to their reduced Joule dissipation , lower damping and lower exchange \nstiffness. L ower exchange stiffness and exchange energy in MI allow write energy per bit could \nbe lower for MI than for metals. The magnetic field pulses applied on nanowire were chosen to be \n2 ns wide, as Si CMOS can operate at similar periods for read/write memory pulses. \nThe switching models were prepared in three steps: \n1) Self-relaxation (0-15ns) . First, nanowires with different uniaxial anisotropy constants but \nidentical geometries were allowed to equilibrat e into minimum energy states in absence of external \nmagnetic field or initial magnetization. The magnetization profiles after self-relaxation for \nnanowires with increasing uniaxial anisotropy constants were calculated and are shown in Fig. 1 . \nWhen uniaxial anisotropy constant is low (103 J·m-3), shape anisotropy renders the nanowire an \nin-plane easy axis material. When uniaxial anisotropy is large enough to overcome shape \nanisotropy, the nanowire becomes PMA. For lower field and lower energy switching, \nperpendicular ma gnetic anisotropy (PMA) in the nanowires is desired. 2) Initialization (15-45ns) . In the second step, we applied an external magnetic field pulse of 2.2 \nTesla for initialization of magnetic moment s along + z axis . \n3) Deterministic switching (45-100 ns) . In this third and final step, 2 ns -wide and apart field pulses \nwere applied to investigate the effect of anisotropy and damping on switching time and energy of \nthe nanowire . \n \n2. Analytical model results \nThe Landau -Lifshitz -Gilbert equation (Supplementary Materials Part 1) captures the time \nevolution in nanomagnets. Its analytical solutions yield three general cases based on the main \nparameters, which determine switching likelihood and time constants: prec ession -driven, \ndamping -driven and effective field -driven regimes. In the precession -driven regime (α ≪ 1), \nmagnetization reversal cannot settle since the nanomagnet undergoes precession indefinitely: \n𝐌(𝐫,t)=Ms𝑒−𝜅𝑡(−𝐲̂sin(γ̅Hefft)+𝐱̂cos(γ̅Hefft)) (10) \n \nIn the damping -driven regime ( α ≫ 1), the spins dissipate the injected pulse energy before \ntriggering any magnetization reversal : \n∂𝐦\n∂t≈(−|γ̅|α𝐦×(𝐦×𝐇𝐞𝐟𝐟)) (14) \n𝐌(𝐫,t)=𝑒−|γ̅|αΔ∗𝑡(𝐱̂Mx0+𝐲̂My0)+𝐳̂Mz0 (15) \nThe effective field -driven case contains multiple in and out -of-plane anisotropy field terms that \nassist magnetization reversal. Here, the reversal time constant is determined by the external field, \ndemagnetizing field and damping constant . Overall, an o ptimal window of damping and uniaxial \nanisotropy constant s were found to enable deterministic magnetization reversal in picoseconds . \n \n III. Uniaxial anisotropy and Gilbert Damping dependence of magnetization reversal \nIn this section, we investigate the effect of Ku on the self-relaxation and pulse -assisted \nswitching. Fig. 1(a) -(d) show the time evolution for magnetic moments of the nanowires with K u \n= 103, 104, 105 and 106 J·m-3, respectively, during self -relaxation (no external field applied: 0 -15 \nns) and during applied external magnetic field pulse (15 -45 ns) and after the pulse is applied (45 -\n100 ns). The nanowires were first set to an infinitesimally small magnetic moment and they were \nallowed to relax their magnetic moments in absence of external magnetic field until 15 ns. This \ninitialization numerical ly demonstrates the easy axis for each case before applying the magnetic \nfields. The magnetic moment of the nanowire in Fig. 1(a) with K u = 103 J·m-3 self-relaxes towards \n+y direction, which indicates that its magnetic easy axis is along the long axis (y) of the structure \nand that K u < K shape. For Fig. 1(b), the structure relaxes to –z direction, indicating that uniaxial \nanisotr opy now overcomes shape and renders the nanowire PMA. In Fig. 1(c), although Ku = 105 \nJ·m-3 > K shape, the nanowire cannot relax to a vertical direction since the spins form a transient \ndomain wall (Spin profiles in Supplementary Figure S1). When an additional external pulse was \napplied, the multi -domain structure overcomes the domain wall energy barrier , aligns and \nstabilizes along +z direction. In Fig. 1(d), the structure is clearly PMA and it relaxes to –z direction \nin 40 ps. Increasing uniaxial anisotropy energy from 103 to 106 J·m-3 changes self-relaxation times \nfrom 2 ns (in -plane) down to 4 ns (PMA, single domain), 2 ns (PMA but two transient domains) \nto 40 ps (PMA, single domain), respectively. \nWhen an external magnetic field pulse along +z axis has been applied for initialization, in \neach case except Fig. 1(d), the magnetic moment aligns with the external field first. Since the \nstructure in Fig. 1(a) has in -plane easy axis, it cannot retain its moment along +z and it relaxes to \nsurface plane. Since the structure in Fig. 1(b) is PMA, it switches to +z and retains its remanent state (Hexternal = 2.2 T > Hsat ~ 2K u/Ms = 0.143 T ). The applied field on the nanowire in Fig. 1(c) \nhelps overcome the domain wall energy and helps align the domains along +z axis as the saturation \nfield for this structure (H sat ~ 2K u/Ms = 1.43 T) is less than the applied pu lse intensity. Since the \nstructure is intrinsically PMA, the structure retains its magnetic moment along +z. In Fig. 1(d), \nsince the calculated Hsat is about 14.3 T, the structure is not magnetically saturated and does not \nswitch although it is PMA. The nanowire size determines the shape anisotropy and the minimum \nuniaxial anisotropy energy needed for PMA. When PMA is achieved with sufficiently large K u, \nincreasing Keff reduces the self -relaxation time down to sub -100 ps ranges although i ncreasing K u \nto as high as 106 J·m-3 increases the saturation field beyond feasible magnetic field intensities. \n \nFIG. 1. Magnetic initialization steps of the nanowires with α = 0.1 for (a -d) K u = 103, 104, 105, \nand 106 J·m-3, respectively. \nDeterministic switching has been shown by applying six consecutive positive and negative \nswitching pulses with 2200 mT intensity and 2 ns width each. Fig. 2 shows the corresponding \nswitching time as a function of K u and α. The colored regions indicate deterministic switching with \ntimes corresponding to their color codes. The gray regions indicate no deterministic switching. \nSwitching time is defined as the time it takes for transitioning from m z = -1 to +1 (or vice versa) \nupon receiving an external magnetic field pulse. For α > 0.1, a s uniaxial anisotropy increases, the \ntotal effective field H eff of nanowire increases and the switching time increases due to longer \nprecession. As α increases, switching time decreases as the damping term starts balancing the \nprecession term in the LLG eq uation. In the ideal case of no damping, the spins would have \nprecessed indefinitely at ⍵ = γH eff without aligning with the applied external magnetic field. With \nfinite or increasing damping term, the precession energy is absorbed and the spins equilibrate \nfaster. Deterministic switching was not observed for materials with low damping (α < 10-1) as \nprecession prevented switching. \nFor α > 10-1, relaxation timescales are reduced to below mostly 400 ps. When uniaxia l \nanisotropy energy exceeds 5× 104 Jᐧm-3 until 1.5 ×105 Jᐧm-3, the nanowire starts forming domain \nwalls, which prevents from or delays reaching steady state reversal. As the uniaxial anisotropy \nenergy increases towa rds 5 ×104 Jᐧm-3, the domain wall width δDW = 2√(A/K u), decreases to 17 nm \nwhich is below the nanowire width (20 nm). With higher uniaxial anisotropy energies, domain \nwall width decreases and domains form within the nanowire. For Ku > 1.5× 105 Jᐧm-3, saturation \nfield exceeds the applied external field pulse (2200 mT) and the nanowires are not fully saturated. \nTherefore, for multi -domain grains or nanostructures, anisotropy energy must be large enough to \nachieve PMA and K u should be sufficiently small such that realistic ex ternal field pulse intensities \ncould reverse magnetic orientatio n. \nFIG. 2. Gilbert damping and uniaxial anisotropy constant dependence of switching time \n(Hext=2200 mT) . \n \nIV. Dependence of switching energy and rate on pulse width and intensity \nFig. 3(a) shows the switching energy of the nanowire in units of k BT (T = 300 K) for pulse \nwidths between 50 and 3000 ps and external magnetic field intensities between 1000 and 10000 \nmT. The energy barriers were calculated in micromagnetic models based on the energy magnitudes \nthe nanowire overcomes after applying the ex ternal magnetic field pulse. In these calculations, the \nenergy difference accounts for the Zeeman, demagnetizing, exchange and uniaxial anisotropy \nenergies. In the Hamiltonian ( Suppl. eqn. 3 -6), the time evolution of the energy is driven mainly \nby the chan ges in the Zeeman energy due to reorientation of the nanowire spins upon applying \nfield pulse and the demagnetizing field of the geometry. In this figure, the gray regions show no \nswitching (N/S) and the other regions have switching energies corresponding to their color codes. \nThe figure indicates that the switching energy of the nanowire could be lowered from over 12000 \nkBT to 3163 kBT by tuning the applied field pulse . This effect indicates low Zeeman energy (due \nto its reduced volume ), low uniaxial aniso tropy and low exchange stiffness (of YIG ) make \nmagnetization reversal energetically favorable. Thus, d ecreasing the field intensity decreases the \nswitching energy. \nThe hysteresis loop calculated for the mz component indicate that nanowires have a \ncoercivity of 1950 mT with PMA (Supplementary Figure S4) . The nanowire switches its magnetic \norientation even below this coercivity with the external field pulse . Decr easing the pulse width \nhelps reduce switching energy unti l applied field pulses of 4394 mT, since it reduces the average \nZeeman energy injected into the nanowire. Therefore, Fig. 3(a) indicates that pulse -assisted and \nsub-coercivity switching with lower energy costs can be achieved for magnetic nanowires. Fig. \n3(b) shows nanowire switching time for the same pulse widths and intensities used in Fig. 3(a). \nThe switching time was calculated based on the time the nanowire takes for a complete steady -\nstate reversal of its vertical magnetization. The fastest magnetizat ion reversal occurs at 0.150 ns \nfor 10000 mT and 368.4 ps pulse width. The large field intensity and short pulses enable fast \nmagnetization reversal and minimal time spent in transient precession motion. Decreasing the field \nintensity increases the switchi ng time. While the nanowire switches faster for fields above its \ncoercive field (1950 mT), one could achieve complete magnetization reversal with sub -coercivity \npulses with longer switching times. Switching with sub -coercivity pulses relies on the dynamic \ninsta bilities of the magnetic moment and most sub -coercivity switching cases in Fig. 3(b) have \nextended switching times. Low field intensities cause precession for extended periods (damping -\ndominated regime) , thus preventing or significantly delaying reversal . Decreasing the pulse width \nhelps reduce switching time as it reduces the interaction time between the magnetic moment and \nthe field. Pulse-assisted and sub -coercivity switching could be achieved for magnetic nanowires if longer transient reversal times are allowed. Thus , a trade -off between optimal switching energy/ \ntime and pulse width/ field intensity could be established. \nFIG. 3. Pulse width and intensity dependence of (a) s witching energy (units o f kBT at T = \n300K ) and (b) switching time for the nanowire with Ku = 104 Jᐧm-3 and α = 0.01. \n \n \nFIG. 4. Deterministic switching for cases with short and long er switching times. External field \ndependence of relaxation rate for (a) and (b) with external field intensity: 2276 mT, pulse width: \n2564.3 ps (faster) ( 𝛂 = 0.01, Ku = 104 Jᐧm-3), and for (c and d) external field intensity: 1638 mT, \npulse width: 2564.3 ps (sub -coercivity, slower) ( 𝛂 = 0.01, Ku = 104 Jᐧm-3), \nBased on the deterministic switching results shown on Fig. 3, we investigate further two \ncases from Fig. 3(a) and (b): (pulse intensity, pulse width) = (2276 mT, 2564.3 ps) and (1638 mT, \n2564.3 ps). For this condition, the calculated hysteresis loops indicate that the saturation field is \n1950 mT ( Suppl. Fig. S4). These two cases were chosen to investigate the deterministic switching \ndynamics for above and below -coercivity switching, respectively. The switching dynamics of the \nfirst and second cases are shown on Fig. 4(a,b) and Fig. 4(c,d), respectively. Fig. 4(a,b) \ndemon strate shorter switching times due to the higher external field intensity on the nanowire. Fig. \n4(c,d) show deterministic magnetization switching at sub -coercivity external field s (below 1950 \nmT). As shown on Fig. 3, sub -coercivity deterministic switching requires the pulse duration to be \ngreater than a minimum threshold (i.e. 2192 ps for 1931 mT). This threshold depends on both the \nextrinsic factors (external field intensity , nanowire dimensions ) and intrinsic factors (Ku, α, A ex \nand M s). Magnetic r eversal is delayed due to precession -driven switching dynamics . These results \nshow that deterministic sub -coercivity switching in magnetic nanowires is feasible and allows for \nreduced switching fields with lower energy barrier materials and geometries. \n \nV. Conclusions \nThe temporal and spatial evolution of magnetization switching in nanowires were \ninvestigated as functions of pulse width, pulse intensity, uniaxial anisotropy constant and damping. \nDamping, precession and effective field -driven regimes have been identified in the analytical \nmodels of magnetization reversal in nanowires. These simulations and models indicate that the \nmagnetization states of these magnetic nanowires could be reversed under external pulses with \nsufficient pulse intensity and width for optimal damping ( 𝛂 > 0.1) and uniaxial anisotropy (K u < \n105 Jᐧm-3). In high aspect ratio nanowires (in plane x:y = 100:20), sufficiently high uniaxial \nanisotropy constants K u (at least 104 Jᐧm-3) are needed to obtain perpendicular magnetic anisotropy \nby overcoming shape anisotropy. When Ku becomes too high (≥ 105 Jᐧm-3), the effective anisotropy \nof nanowire increases beyond feasible magnetic field pulse intensities (2.2 T or less). For optimal \ndamping, anisotropy and pulse properties ( 𝛂 ∈ [0.1, 0.5], Ku ∈ [104, 105 Jᐧm-3), 0.5 to 3ns-wide \npulses), the nanowires could switch with picosecond timescales and low energy consumption per \nbit as low as 3.163 ×103 kBT at T = 300 K. In all switching cases, PMA is necessary for \ndeterministic pulse -assisted magnetization reversal and dense memory bits. The effective field \nprovides the energy barrier needed for both stable memory and low -power logic functionalities. Two key outcome s emerge from this study : First is the observation of a nanomagnetic \nswitching trilemma or the competition between nanowire (i) switching rate, (ii) energy cost of \nswitching per bit and (iii) external field required for switching. In this trilemma, high switching \nrate requires either high e xternal field or high switching energy. Lower switching energy requires \nan optimal external magnetic field intensity and pulse width per bit. Minimizing the external \nmagnetic field and reducing switching time requires optimal pulse width at the cost of inc reasing \nenergy per bit. This trilemma originates from a more general competition between the energy -\ndelay product between the external magnetic field and the damping -driven magnetization reversal \n(Suppl . Fig. S3). The second key outcome is s ub-coercivity s witching observed under appropriate \nuniaxial anisotropy (i.e. K u=104Jᐧm-3), damping ( 𝛂 = 0.1) and pulse properties (163 8K shape). We attribute the origin of this effect in the main manuscript to a transient domain wall \n(DW) formati on. In Suppl ementary Figure S1, we present the calculated spin profile for K u = 105 \nJ·m-3. Fig. S1 shows a DW , which traverses the nanowire along its short axis. \n \nSupplementary Figure S1. For K u = 105 J·m-3, a domain wall prevents relaxation to the \nvertical axis (z) (the units are in nm for all three axes). \nNondeterministic reversal of nanomagnets over a wider window of uniaxial anisotropy and Gilbert \ndamping values: Fig. 2 of the main manuscript includes results on Gilbert damping (α = 0.01 -0.5) \nand uniaxial anisotropy (K u = 104-105 J·m-3) dependence of relaxation rate. We calculated the \nrelaxation rates for the range of α = 10-4-10-1 and K u = 103-106 J·m-3. We observed that switching \nis not deterministic or cannot happen at all under these conditions. In Supplementary Fig ure 2, we \nindicate this result as a lack of deterministic switching case (N/S: no switching). For K u > 106 J·m-\n3, the applied external magnetic field is not sufficient to overcome the effective magnetic field. For \nKu < 106 J·m-3, damping constant is not sufficient to stabilize the spin precession during \nmagnetization reversal. \n \nSupplementary Figure S2. The calculated relaxation rates for the range of α = 10-4-10-1 and \nKu = 103-106 J·m-3. For K u > 1.5 × 105 J·m-3, the saturation field exceeds the applied external \nfield pulse and the nanostructures are not saturated. For lower anisotropy values, although \nmagnetization reversal occurs, the damping is not sufficient to end the precession motion for \ndeterministic switching. \n \n \n \nEnergy -delay product and the nanomagnetic trilemma: In the main text, we mentioned the \ncompetition between (i) switching rate, (ii) energy cost of switching per bit and (iii) external field \nrequired for switching. We named this competition the nanomagnetic trilemma since this effect \noriginates from a more general competition between the energy -delay product between the external \nmagnetic field intensity and the internal precession/damping -driven reversal mechanisms of \nmagnetic nanostructures. In Supplementary Fig S3, we provide the calculated energy -delay \nproduct (units of fJ·ps) for different pulse widths and external field intensities. \n \nSupplementary Figure S3. Energy -delay product (EDP, fJ·ps) for the external field -driven \nmagnetization rever sal and the internal precession/damping -driven reversal mechanisms of \nmagnetic nanostructures. \n \n \n \n \n \n \n \nEnergy -delay product and the nanomagnetic trilemma: In the main text, we mentioned that the \ncalculated hysteresis loops indicate that the saturation field is 1950 mT. In Supplementary Fig. S4, \nwe present the calculated hysteresis loop for the normalized vertical magnetic moment component \nmz as a function of applied magnetic field H z. \n \nSupplementary Figure S4. Simulated magnetic hysteresis loop of the nanow ires when an \nexternal magnetic field is applied on the nanowire along vertical axis (z). The hysteresis loop \nshows the normalized vertical magnetization component (m z), a perpendicular magnetic easy \naxis with a vertical remanent state and coercive and saturation field of H c = H sat = 1950 mT. \n \n \n \n \n \n \n \n1. Theoretical Analysis \nDynamic evolution of spin vector components in a magnetic material is described using \nthe Landau -Lifshitz -Gilbert ( LLG ) shown in equation s (1) and (2 ). Here, m is the normalized \nmagnetization vector with 𝐦= 𝐌(x,y,z,t)\nMs, where M(x,y,z ,t) is the magnetization profile throughout \nthe magnetic nanostructure and M s is the saturation magnetic moment of the material. \n∂𝐦\n∂t=−γ𝐦×𝐇𝐞𝐟𝐟+𝛼𝐦×∂𝐦\n∂t (1) \nor \n∂𝐌\n∂t=−|γ̅|𝐌×𝐇𝐞𝐟𝐟−|γ̅|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟) (2) \nwhere γ̅:Landau −Lifshitz gyromagnetic ratio ,α:damping coefficient \n \nIn this equation, time evolution of magnetic moment vectors sampled within a rectangular \ngrid of 5 nm size is calculated over the rectangular magnetic nanostructure presented above. Heff \nis the effective magnetic field vector. As the total energy of the nanostructure is minimized along \nperpendicular axis, Equation (3) describes the magnetic anisotropy and perpendicular easy axis of \nthe magnetic nanostructure: \n𝐇𝐞𝐟𝐟=𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 +𝐇𝐝𝐞𝐦𝐚𝐠 +𝐇𝐞𝐱𝐜𝐡𝐚𝐧𝐠𝐞 +𝐇𝐮𝐧𝐢𝐚𝐱𝐢𝐚𝐥 (3) \n𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 =𝐳̂H0sin(ωt) (4) \n𝐇𝐝𝐞𝐦𝐚𝐠 =𝐲̂H1 (5) \n𝐇𝐞𝐱𝐜𝐡𝐚𝐧𝐠𝐞 =𝐳̂2A0\nμ0Ms∇2𝑚 (6) \n𝐇𝐮𝐧𝐢𝐚𝐱𝐢𝐚𝐥 =𝐳̂(K1sin2θ+K2sin2θsin2α)≈𝐳̂K1sin2θ (6) \n \nThe first term of th e right hand side of Equation (2 ) is known as the precession term that drives \noscillations within nanowires under magnetic fields. The second term , also called the damping \nterm, drives the alignment rate of the magnetic moment with the external magnetic field. This \ndissipative term is one of the energy loss channels in the relaxation process. The “Hamiltonian” \nfor the LLG equation or the effective field is defined in Equation 3. This equation includes the \ncontributions from external magnetic field, demagnetization field (shape anisotropy term), \nexchange field term and uniaxial anisotropy field. The external field, H external , is applied perpendicular to the nanostructure surface al ong z axis. The demagnetization term, H demag, is the \nfield, which originates due to the absence of magnetic monopoles (divergence -free magnetic flux \ndensity) and the resulting field distribution within the nanowire geometry along its long in -plane \ny axis . Demagnetizing field is one of the terms , which capture s the effect of geometry on \nanisotropy and magnetization dynamics . The exchange term, H exchange , is a field generated due to \nthe Heisenberg exchange interaction between adjacent spins. This field can be come particularly \nimportant when metallic magnetic nanowires are used (permalloy: A ex = 13 pJ·m-1) [1] and Cobalt -\nPlatinum multilayers (A ex = 15 pJ·m-1) [2]) and less significant for YIG nanostructures with A ex = \n3.65 ± 0.38 pJ·m-1 [3-5] although the presence of exchange interaction is not essential for spin \nwave propagation or magnetization reversal [ 6]. The last term is the uniaxial anisotropy energy \nterm, which indicates the vertical directional preference of magnetic moment during relaxation \nand switching. This term could originate from a variety of sources includin g magnetocrystalline \nanisotropy , magnetoelastic anisotropy fo r thin epitaxial nanostructures , strain doping as well as \nother gro wth-induced uniaxial anisotropy . \nOne can a nalyze magnetic relaxation and switching in nanostructures in three regimes: \n(1) precession -driven (when damping term is negligible ), \n(2) damping -driven (α is large such that the damping term prevents magnetization reversal) \n(3) effective field -driven \nConsidering the combined effects of material constants and the anisotropy terms, we derive and \ninvestigate these regime s in further detail below . \nI. Precession -driven magnetization dynamics \nWhen Gilbert damping α is small such that damping term is much smaller than the precession \nterm, magnetic relaxation and r eversal is driven by precession: \n|−|γ̅|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟)|≪|−|γ̅|𝐌×𝐇𝐞𝐟𝐟| (7) \n|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟)|≪|𝐌×𝐇𝐞𝐟𝐟| \n|α𝐌\nMs||(𝐌×𝐇𝐞𝐟𝐟)||sin(90°)|≪|𝐌×𝐇𝐞𝐟𝐟| \nα|m|≪1 \nα≪1 (8) Here, the angle between the magnetization vector M and the M×Heff vector is 90° . Since the \nmagnitude of the normalized magnetization vector m is always one, precession term dominates \nwhen the damping coefficient is much smaller than one. When magnetization dynamics is driven \nby precession, relaxation or reversal processes continue indefinitely or much longer than otherwise \nin absence of damping . As a result, we do not observe any magnetization reversal : \n∂𝐌\n∂t≈−|γ̅|𝐌×𝐇𝐞𝐟𝐟 (9) \n𝐌(𝐫,t)=Ms𝑒−𝜅𝑡(−𝐲̂sin (γ̅Hefft)+𝐱̂cos (γ̅Hefft)) (10) \n \nFor κ = 0 (zero damping limit), the oscillation continues indefinitely. For nonzero and small κ, the \noscillations continue for extended p eriods with an evanescent decaying envelope. Gilbert damping \nparameter α is low for YIG [4,5] (3-7×10-4), magnetostrictive spinel ferrites [7] (< 3×10-3), Heusler \n[8] (10-3) or other low -damping metallic alloys such as CoFe [9] (10-4-10-3). A very small damping \ncoefficient, regardless of the H eff magnitude, triggers the precession -driven regime . The terms \npresent in the effective field determine the Larmor precession frequency (around few GHz for \nferromagnets and potentially towards THz for antiferromagnets). \n \nII. Damping -driven magnetization dynamics \nWhen Gilbert damping α is large such that the damping term is much larger than the precession \nterm, magnetic relaxation an d reversal is driven by damping: \n \n|−|γ̅|𝐌×𝐇𝐞𝐟𝐟|≪|−|γ̅|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟)| (11) \n|𝐌×𝐇𝐞𝐟𝐟|≪|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟)| \n|𝐌×𝐇𝐞𝐟𝐟|≪|α\nMs𝐌||𝐌×𝐇𝐞𝐟𝐟||sin(θ)| \n1≪α|𝑚||sin(θ)| \n1≪α (12) \n \nWhen the magnetization dynamics is driven by damping due to large damping coefficient , \nrelaxation or reversal processes occur with very short evanescent lifetimes : ∂𝐌\n∂t≈−|γ̅|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟) (13) \n∂𝐌\n∂t≈−|γ̅|α 𝐦×(𝐌×𝐇𝐞𝐟𝐟) \n1\nMs∂𝐌\n∂t≈1\nMs(−|γ̅|α 𝐦×(𝐌×𝐇𝐞𝐟𝐟)) \n∂𝐦\n∂t≈(−|γ̅|α 𝐦×(𝐦×𝐇𝐞𝐟𝐟)) (14) \n \nEquation 14 shows that the decay rate in the damping -driven regime is driven by the gyromagnetic \nratio, damping constant , effective field and the orientation of the effective field with respect to the \ninitial magnetization orientation. For a magnetic moment initially oriented along +z, large damping \ndecays the in -plane excitations due to the pulse and the initial magnetization along +z is retained: \n \n𝐌(𝐫,t)=𝑒−|γ̅|α Δ∗𝑡(𝐱̂Mx0+𝐲̂My0)+𝐳̂Mz0 (15) \n \nIn order to trigger damping -driven regime described by equations 11, 14 and 15, Gilbert damping \nconstant α should be larger than 1. The effective field should also not be parallel to the initial \nmagnetization, since this configuration would not trigger any reversal. \n \nWhen a material has ultralow damping and has high effective fields as in Yttrium iron garnet \nnanowires , precession regime prevails and relaxation timescales could be extended indefinitely as \nlong as damping is negligible or compensated. For ultrafast magnetization reversal, damping must \nnot be negligible or effective field (anisotropy and external field) must not be too high. \n \nIII. Effective field -driven magnetization dynamics \nWhen neither damping nor precession term dominates time -dependent magnetic relaxation, \ndynamic control of individual terms in the effective field determines the time evolution of \nmagnetization reversal process. In th is case, the LLG equation follows the standard form in \nequation 2 . The vectors in the effective field term determine the switching time scales together \nwith damping. The external field or the uniaxial anisotropy determine the timescales in the LLG \nequation when they are much large r with respect to the other terms in the effective field . The other in-plane terms in the effective field, such as the demagnetizing fields, are necessary to trigger \nmagnetization reversal : \n \n∂𝐌\n∂t=−|γ̅|𝐌×(𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 +𝐇𝐝𝐞𝐦𝐚𝐠 +𝐇𝐞𝐱𝐜𝐡𝐚𝐧𝐠𝐞 +𝐇𝐮𝐧𝐢𝐚𝐱𝐢𝐚𝐥 )−|γ̅|α\nMs𝐌×(𝐌\n×(𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 +𝐇𝐝𝐞𝐦𝐚𝐠 +𝐇𝐞𝐱𝐜𝐡𝐚𝐧𝐠𝐞 +𝐇𝐮𝐧𝐢𝐚𝐱𝐢𝐚𝐥 )) (16) \n∂𝐌\n∂t≈−|γ̅|𝐌×(𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 +𝐇𝐢𝐧 𝐩𝐥𝐚𝐧𝐞 )−|γ̅|α\nMs𝐌×(𝐌×(𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 +𝐇𝐢𝐧 𝐩𝐥𝐚𝐧𝐞 ) (17) \n \nHere, achieving ultrashort reversal time constants depend on the damping co nstant and the external \nfield intensity along the direction of the new magnetic state. Large external field and some nonzero \nbuilt-in in -plane field drives a faster preces sion and magnetization reversal, while a sizable \ndamping is necessary to stabilize the moments along the final orientation. \n \nAppendix 1. OOMMF Source Code \n(rect_structure_field .mif and rect_structure_hysteresis .mif) \nAppendix 2. Calculated domain wall movie for Supplementary Fig. S1 & Figure 1(c) \n(1E5 Sample .mov) \n \nReferences \n1. R. Hertel, Thickness dependence of magnetization structures in thin Permalloy rectangles, Z. \nMetallkd. 93, 957 (2002). \n2. H. T. Fook, W. L. Gan, and W. S. Lew, Gateable Skyrmion Transport via Field -induced \nPotential Barrier Modulation , Sci. Rep. 6, 21099 (2016). \n3. S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. \nHillebrands and A. Conca, Measurements of the exchange stiffness of YIG films using \nbroadband ferromagnetic resonance techniques, J. Phys. D: Appl. Phys. 48, 015001 (2015). \n4. A. Kehlberger, K. Richter, M. C. Onbasli, G. Jakob, D. H. Kim, T. Goto, C. A. Ross, G. Götz, \nG. Reiss, T. Kuschel, and M. Kläui, Enhanced Magneto -optic Kerr Effect and Magnetic \nProperties of CeY 2Fe5O12 Epitaxial Thin Films, Phys. Rev. Appl. 4, 014008 (2015). 5. M. C. Onbasli, A. Kehlberger, D. H. Kim, G. Jakob, M. Kläui, A. V. Chumak, B. Hillebrands, \nC. A. Ross, Pulsed laser deposition of epitaxial yttrium iron garnet films with low Gilbert \ndamping and bulk -like magnetization, APL Mater. 2, 1061 02 (2014). \n6. K. Oyanagi, S. Takahashi, L. J. Cornelissen, J. Shan, S. Daimon, T. Kikkawa, G. E. W. Bauer, \nB. J. van Wees & E. Saitoh, Spin transport in insulators without exchange stiffness, Nat. \nCommun. 10, 4740 (2019). \n7. S. Emori, B. A. Gray, Jeon, H.‐M., J. Peoples, M. Schmitt, K. Mahalingam, M. Hill, M. E. \nMcConney, T. M. Gray, U. S. Alaan, A. C. Bornstein, P. Shafer, A. T. N'Diaye, E. Arenholz, \nG. Haugstad, K.‐Y. Meng, F. Yang, F., D. Li, S. Mahat, D. G. Cahill, P. D hagat, A. Jander, \nN. X. Sun, Y. Suzuki, B. M. Howe, Coexistence of Low Damping and Strong Magnetoelastic \nCoupling in Epitaxial Spinel Ferrite Thin Films , Adv. Mater. 29, 1701130 (2017). \n8. A. Conca, A. Niesen, G. Reiss, and B. Hillebrands, Low damping magnet ic properties and \nperpendicular magnetic anisotropy in the Heusler alloy Fe1.5CoGe, AIP Advances 9, 085205 \n(2019) . \n9. M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J. Silva, H. T. Nembach, O. Eriksson, O. \nKaris & J. M. Shaw, Ultra -low magnetic damping of a metallic ferromagnet, Nat. Phys. 12, \n839 (2016). " }, { "title": "2302.10758v1.Quasinormal_modes__Hawking_radiation_and_absorption_of_the_massless_scalar_field_for_Bardeen_black_hole_surrounded_by_perfect_fluid_dark_matter.pdf", "content": "arXiv:2302.10758v1 [physics.gen-ph] 11 Feb 2023Quasinormal modes, Hawking radiation and absorption\nof the massless scalar field for Bardeen black hole\nsurrounded by perfect fluid dark matter\nQi Sun1, Qian Li1, Yu Zhang1∗, Qi-Quan Li1\n1Faculty of Science, Kunming University of Science and Techn ology,\nKunming, Yunnan 650500, People’s Republic of China\nAbstract We study the quasinormal modes, Hawking radiation and absorption cross section of the\nBardeen black hole surrounded by perfect fluid dark matter for a m assless scalar field. Our result shows that\nthe oscillation frequency of quasinormal modes is enhanced as magn etic charge gor the dark matter param-\neterαincreases. For damping rate of quasinormal modes, the influence o f them is different. Specifically, the\nincrease of dark matter parameter αmakes the damping rate increasing at first and then decreasing. Wh ile\nthe damping rate is continuously decreasing with the increase of the magnetic charge g. Moreover, we find\nthat the increase of the dark matter parameter αenhances the power emission spectrum whereas magnetic\nchargegsuppresses it. This means that the lifespan of black holes increases for smaller value of αand larger\nvalue of gwhen other parameters are fixed. Finally, the absorption cross se ction of the considered black\nhole is calculated with the help of the partial wave approach. Our res ult suggests that the absorption cross\nsection decreases with the dark matter αor the magnetic charge gincreasing.\nKey words: Black hole; quasinormal modes; Hawking radiation; absorption cros s section; massless\nscalar field\nPACS:04.70.-s, 44.40.+a, 11.80.-m\n1 Introduction\nIn the past few years, LIGO-Virgo detected gravitational waves from a binary black merger [1–3] and Event\nHorizon Telescope captured the shadow image of supermassive blac k hole in the center of M87, which\nis a giant elliptical galaxy [4, 5]. These observations have excellent con cordance with the prediction of\ngeneral relativity concerning the existence of black hole. But black hole spacetimes having singularities is an\nattractive problem in general relativity. At the singularity of space -time, the curvature and density tend to\ninfinity and the laws of physics are invalid. The existence of singularitie s denotes that physical phenomena\n∗Corresponding Author(Y. Zhang): Email: zhangyu 128@126.com\n1cannot predict, which marks the incompleteness of general relativ ity. Fortunately, regular black holes (non-\nsingular)is exhibited toavoidthesingularitiesbysomeattempts, for example, the theoryofquantumgravity,\nmatter backgrounds take the place of central singularity and so o n. Inspired by some ideas, Bardeen firstly\nconstructed a regular black hole [6], called Bardeen black hole. Ayon- Beato et al. pointed out that the\nsource of Bardeen black hole is a nonlinear electrodynamic field [7]. Sin ce the first regular black hole model\nwas proposed, the study of regular black hole has never stopped. The more regular black holes is derived\nsuch as Dymnikova black hole [8], Bronnilov black hole [9] and Hayward bla ck hole [10]. The proposal of\nthese regular black holes has stimulated further research on the p roperties of these black holes. For example,\nthe shadow of black hole [11–13], black hole thermodynamics [14–16], etc.\nWhen Oort used a telescope to observe stars in the Milky Way, he fou nd that they were moving so fast\nthat they could escape the gravitational effects of all the visible ma tter in the galaxy [17]. It denotes that\nthere needs more mass in our universe for the sake of explaining this astronomical observations. Besides,\nthis object having mass does not emit light and cannot be detected d irectly, which called dark matter by\nresearchers. Then Rubin et al. found that stars far from the cen ter of the galaxy were spinning significantly\nfaster than expected [18] and the image of Bullet Cluster observed by Observatories in 2006 [19], verifing\nthe existence of dark matter. Up to now, it is generally accepted th at our current universe consists mainly\nof 4.9% baryon matter, 26.8% dark matter, and 68.3% dark energy [2 0]. And scholars further define dark\nmatter being non-baryonic as well as non-luminous. After that the re are a lot of papers about dark matter\n[21–23]. As mentioned above, it is thus clear that dark matter has alw ays been a hot topic. Therefore, it\nis necessary to study black holes under dark matter. Perfect fluid dark matter (PFDM) as a candidate of\ndark matter was proposed by Kiselev [24], which has the character o f fluid. And in some extent, PFDM\ncan regard as a model that is a scalar field with an exponential poten tial. Then Heydarzade et al. derived\ncharged/uncharged Kiselev-like black holes in Rastall theory [25]. Xu et al. obtained Kerr-Newman-anti-de\nSitter black holes surrounded by PFDM in Rastall gravity [26]. In this p aper, we take one of regular black\nholes immersed in dark matter into consideration, i.e., Bardeen black h ole surrounded by PFDM proposed\nby Zhang et al [27].\nNow, the model of Bardeen black hole surrounded by PFDM has been built. But how should we know\nthe properties of the black hole? A black hole is not an isolated object and it interacts with its environment.\nWhen binary black hole mergers, it can produces gravitational wave s. And the signal of it can be divided\ninto three phases: inspire, merger and ringdown. At ringdown phas e, the black hole emits gravitational\nwaves having the form of discrete and complex frequencies, which c alled quasinormal modes (QNMs) [28].\n2As the ”characteristic sound” of black holes, the real part of QNM s indicates the oscillation frequency while\nthe imaginary part of it represents damping rate. It is noted that Q NMs only rely on the background and\nthe type of perturbation field, which is classical ”fingerprints” of b lack holes. Therefore, we can obtain the\ninformation of black holes by solving the wave equation under special boundary, i.e., QNMs. Vishveshwara\nfirst pointed out that the signal decays with the curve of exponen tial for most of the time, which comes from\nperturbed black hole [29]. And Chandrasekhar et al. discussed in de tail the QNMs of Schwarzschild black\nhole [30]. After that, a number of studies have sprung up on QNMs [31 –41], which are aimed to analyze the\nparameter of black hole, detect the connection of QNMs with area s pectrum and the relation of QNMs with\ngravitational lens.\nOn the other hand, Hawking proposed black holes can emit thermal r adiation when quantum effects\nare considered and this radiation is called Hawking radiation. As quant um ”fringerprints” of black holes,\nHawking radiation has the information about the evaporation dynam ics of black holes. And the process\nof Hawking radiation is that negative particles enter black holes and p ositive particles escape to infinity,\nwhich produced by a vacuum fluctuation near the event horizon of b lack holes [42]. It should be pointed\nthat the emergence of Hawking radiation is an combination of genera l relativity, quantum mechanics and\nthermodynamics. Since the concept of Hawking radiation was propo sed, the research on the topic has not\nstopped in the last few decades. In 1975, Damour and Mann develop ed a method for computing Hawking\nradiation [43]. In 2000, Parikh and Wilczek proposed a quantum tunne ling method to research Hawking\nradiation [44] and Kerner et al. further improved the method [45, 46 ]. In 2011, Kokkotas et al. used 6th\nWKB method formula proposed by Konoplya to compute Hawking radia tion by grey-body factors [47, 48].\nMoreover, due to the feature of versatility and flexibility, the 6th W KB method is adopt to obtain Hawking\nradiation in many papers [40, 49, 50] and we also employ the method in this paper.\nThe forms of general relativity and quantum mechanics are incompa tible, which are theories of gravita-\ntional interaction and atomic scale, respectively. However, since H awking discovered that black holes can\nabsorbparticles, the topic of absorptioncrosssection of black ho le for different test fields has spurred interest\nof scholars [39, 51–53]. After obtaining the analytical solution of th e wave equation, Sanchez discovered that\nthe absorption cross section for a massless scalar field oscillates in t he vicinity of geometric-optics limit [54].\nUnruh investigated the absorption cross section of nonrotating b lack hole. And he found the absorption\ncross section for Dirac particles is 1/8 of massive scalar particles in t he low-energy situation [55]. In the\nSchwarzschild spacetime, Crispino et al. numerically calculated the ab sorption of electromagnetic waves for\narbitrary frequencies [56]. Furtherly, Macedo and Crispino studied the absorption of Bardeen black hole and\n3compared it with Reissner-Nordstr¨ om black hole for a massless sca lar field. Then they found the absorption\ncross section of the two black holes might be the same at high-frequ encies [57].\nThe remainder of the paper is organized as follows. In section 2, we r eview the Bardeen black hole\nsurrounded by PFDM and discuss the effective potential with respe ct to different parameters. In section 3,\n6th WKB method is briefly introduced and used to calculate QNMs of a m assless scalar field for the black\nhole under consideration. In section 4, we consider the Hawking tem perature and calculate the energy\nemission rate in the black hole. In section 5, using the partial wave me thod, we investigate the absorption\ncross section of the black hole. Conclusions are presented in the las t section. Moreover, we set M=G=c=1\nin this paper.\n2 Massless scalar field equation in static and spherically sy mmet-\nric Bardeen Black Hole Surrounded by PFDM\nFor reasonably explaining astronomical observation at different sc ales, such as the Cosmic Microwave Back-\nground and galaxy rotation curve, the concept of dark matter ha s been proposed. Kiselev [24] proposed\nPFDM as one of alternative models of dark matter, which is regarded as quintessence scalar field. Assuming\nthat the black hole is immersed in PFDM, Zhang et al. [27] considered th e coupling of the gravitational and\na non-linear electromagnetic field, and obtained the solution of Bard een balck hole surrounded by PFDM.\nNow let us briefly review their work. After the mentioned above two fi elds are coupled, the Einstein-Maxwell\nequations can be described as follows:\nGν\nµ= 2/parenleftbigg∂L(F)\n∂FFµλFνλ−δν\nµL/parenrightbigg\n+8πTν\nµ, (1)\n∇µ/parenleftbigg∂L(F)\n∂FFνµ/parenrightbigg\n= 0, (2)\n∇µ(∗Fνµ) = 0, (3)\nwhereFµν= 2∇[µAν]andLis a function of F≡1\n4FµνFµνdefined by [7]\nL(F) =3M\n|g|3/parenleftBigg/radicalbig\n2g2F\n1+/radicalbig\n2g2F/parenrightBigg5\n2\n, (4)\nin which gdenotes magnetic charge, and Mindicates the black hole mass. The energy momentum tensor\ncan be expressed as Tµ\nν= diag(−ǫ,pr,pθ,pφ) for a black hole surrounded by PFDM, with the density, radial\nand tangential pressures of the dark matter being [24, 58]\n−ǫ=pr=α\n8πr3andpθ=pφ=−α\n16πr3. (5)\n4Assume that the metric is static and spherically symmetric, which can be represented in the following\nform:\nds2=−f(r)dt2+f(r)−1dr2+r2dΩ2, (6)\nwith\nf(r) = 1−2Mr2\n(r2+g2)3\n2+α\nrlnr\n|α|. (7)\nαrepresents the parameter for the density and pressure of PFDM , called the dark matter parameter.\nAnd He et al. have shown that the weak energy condition is satisfied w henα <0 [27]. From above equation,\nit can be seen that the black hole recoveres to Schwarzschild black h ole when dark matter parameter α= 0\nand magnetic charge g= 0. When dark matter parameter αvanishs, it reduces to Bardeen black hole. And\nit restores to Schwarzschild black hole immered by PFDM when g= 0.\nThe general covariant equation of a massless scalar field in curved s pacetime can be given by\n1√−g∂µ/parenleftbig√−ggµν∂νΦ/parenrightbig\n= 0. (8)\nFor separating the radial and angular directions, we introduce\nΦ(t,r,θ,φ) =1\nre−iωtYl(r,θ)Ψs(r), (9)\nand\ndr∗=dr\nf(r). (10)\nHereYl(r,θ) andr∗are the spherical harmonics and tortoise coordinate, respective ly. And then we\nsubstitute Eqs. (6), (9) and (10) into the Eq. (8), so a Schr¨ odin ger-like equation representing a massless\nscalar field perturbation in the black hole space-time under consider ation can be expressed as\nd2Ψs\ndr2∗+/parenleftbig\nω2−Vs(r)/parenrightbig\nΨs= 0, (11)\nwhere\nVs(r) =\n1−2Mr2\n(g2+r2)3/2+αlog/parenleftBig\nr\n|α|/parenrightBig\nr\n\nl(l+1)\nr2+6Mr2\n(g2+r2)5/2−4M\n(g2+r2)3/2+α\nr3−αlog/parenleftBig\nr\n|α|/parenrightBig\nr3\n.\n(12)\nWe can see that the effective potential depends on multiple number l, dark matter parameter αand\nmagnetic charge g. In Figs. 1, 2 and 3, we show the variation of the e ffective potential with runder different\n5l=1\nl=2\nl=3\nl=4\n2 4 6 8 100.00.10.20.30.40.50.6\u0001V(\n\u0000)\nFig. 1The effective potential versus rforl= 1,2,3,4 with fixed g= 0.1,α=−0.1\nα=0\nα= \u00020.1\nα\u0003 \u00040.3\nα\u0005 \u00060.5\n2 4 6 8 100.00.20.40.60.81.01.2\nV()\nFig. 2The effective potential versus rforα=\n−0.5,−0.3,−0.1,0 (Bardeen black hole) with\nfixedg= 0.1,l= 5g=0\ng=0.1\ng=0.3\ng=0.5\n3.003.053.103.153.200.750.760.770.780.790.80\nV()\n0 2 4 6 8 100.00.20.40.60.81.01.2\nV()\nFig. 3 The effective potential versus rfor\ng= 0 (Schwarzschild black hole surrounded by\nPFDM), 0.1, 0.3, 0.5 with fixed α=−0.1,l= 5\nparameters. At the same value of the parameters αandg, the peak value of potential barrier increases and\nthe peak location moves to right with the increase of l. For fixed gandl, the barrier height of the effective\npotential decreases obviously with the increase of the absolute va lue ofα, and the peak of the effective\npotential shifts to the right. The peak value of the potential incre ases asgincreases, and the location of\npeak moves to the left for fixed α=−0.1 andl= 5. Compared Fig. 2 and Fig. 3, we can find that dark\nmatter parameter has greater influence than magnetic charge fo r the effective potential.\n3 QNMs of a massless scalar field perturbation\nIn this section, we plan to study the relationship of QNMs with dark ma tterα, magnetic charge gand\nspherical harmonic index l, respectively. In order to calculate the QNMs of Bardeen black hole surrounded\nby PFDM for the massless scalar field, we adopt the WKB method. Sch utz and Will who considered that the\nperturbation equation for a particle has the form of Schr¨ odinger -like equation, proposed a semi-analytical\nmethod to obtain the QNMs, i.e., WKB method and found its accuracy is about 6% [59]. Iyer and Will\nexpanded it to the third order letting its accuracy up to 1% [60]. The n Konoplya promoted it to the sixth\n6order [47] and the thirteenth order is developed by Matyjasek and Opala [61]. But it should be noted that\nWKB method has a characteristic of asymptotic convergence, its a ccuracy cannot be improved simply by\nincreasing the order of formula. To pursue the efficiency and accur acy of calculation, we exploit the 6th\nWKB method to compute the QNMs of the black hole, which can be writt en as [47]\niω2\nn−V0/radicalbig\n−2V′′\n0−6/summationdisplay\ni=2Λi=n+1\n2. (13)\nWhere Λ idenotes a higher-order correction, Λ 2and Λ 3are given by [60]\nΛ2=1/radicalbig\n−2V′′\n0/bracketleftBigg\n1\n8/parenleftBigg\nV(4)\n0\nV′′\n0/parenrightBigg/parenleftbigg1\n4+α2/parenrightbigg\n−1\n288/parenleftBigg\nV(3)\n0\nV′′\n0/parenrightBigg2/parenleftbig\n7+60α2/parenrightbig/bracketrightBigg\n, (14)\nΛ3=n+1\n2\n−2V′′\n0/bracketleftBigg\n5\n6912/parenleftBigg\nV(3)\n0\nV′′\n0/parenrightBigg4/parenleftbig\n77+188α2/parenrightbig\n−1\n384/parenleftBigg\nV′′′2\n0V(4)\n0\nV′′3\n0/parenrightBigg\n/parenleftbig\n51+100α2/parenrightbig\n+1\n2304/parenleftBigg\nV(4)\n0\nV′′\n0/parenrightBigg2/parenleftbig\n67+68α2/parenrightbig\n+1\n288/parenleftBigg\nV′′′\n0V(5)\n0\nV′′2\n0/parenrightBigg\n/parenleftbig\n19+28α2/parenrightbig\n−1\n288/parenleftBigg\nV(6)\n0\nV′′\n0/parenrightBigg\n/parenleftbig\n5+4α2/parenrightbig/bracketrightBigg\n, (15)\nand the other corrections can be found in Ref. [47]. In the above eq uations, α=n+1\n2, andnis overtone\nnumber. V0represents the maximum of the effective potential and V(n)\n0denotes n-th derivative of V0\nto tortoise coordinate r∗. Plugging Eq. (12) into Eq. (13), we obtain the QNMs of Bardeen blac k hole\nsurrounded by PFDM for the massless scalar field. Through this pro cess, we can see the changes of the real\nand imaginary parts of the quasinormal frequencies with the param eters of the black hole from Figs. 4, 5\nand 6.\nFig. 4 plots the real part and imaginary part of quasinormal freque ncies varing with spherical harmonic\nindexlfor the massless scalar field when M= 1,n= 0, and g= 1. Based on the figure, we obtain that the\nreal part of the quasinormal frequencies monotonously increase s while the imaginary part of the quasinormal\nfrequencies decreases with the increase of spherical harmonic ind exl. It means that the increase of spherical\nharmonic index lmakes the massless scalar field oscillation more quickly and decay more slowly.\nFixedM= 1,n= 0, and g= 1, we depicts the dependence of QNMs on dark matter parameter α\nfor the massless scalar field in Fig. 5. According to the figure, we find that the increase of dark matter\nparameter αcauses the real part of the quasinormal frequencies to increase , while the imaginary part of\nthe quasinormal frequencies firstly increases and then decrease s. It implies that the real oscillation of the\nquasinormal frequencies increases and the decay rate increases firstly and then decreases as dark matter\nparameter αincreases for the massless scalar field.\nFig. 6 drawsthe behavior of Re ωand Imωwith respect to magnetic charge gfor the masslessscalarfield\n7α=-0.1\nα=-0.2\nα=-0.3\nα=-0.4\nα=-0.5\n2 4 6 8 100.51.01.52.0\nlRe(\n\u0007)α=-0.1\nα=-0.2\nα=-0.3\nα=-0.4\nα=-0.5\n2 4 6 8 100.0450.0500.0550.0600.0650.0700.0750.080\nl-\nI\b (ω)\nFig. 4The real part (left panel) and the imaginary part (right panel) of qu asinormal frequencies versus l\nfor the massless scalar field with M= 1,n= 0 and g= 1\n++++++++++++++l=1\n+l=5\nl=10\n-0.5 -0.4 -0.3 -0.2 -0.10.51.01.52.02.5\nαRe(ω)+++++++++++\n+\n+\n+l=1\n+l=5\nl=10\n-0.370-0.365-0.360-0.355-0.3500.05520.05530.05540.05550.05560.05570.0558-\n\t\n (\n\u000b)\n-0.5 -0.4 -0.3 -0.2 -0.10.0500.0520.0540.056\n0 \f \r \u000e \u000f0.0600.062\nα-\n\u0010\u0011 (ω)\nFig. 5The real part (left panel) and the imaginary part (right panel) of qu asinormal frequencies versus α\nfor the massless scalar field with M= 1,n= 0 and g= 1\nwhenM= 1,n= 0, and α=−0.3. By analyzing the picture, we observe that the magnitude of the r eal part\nof the quasinormal frequencies increases and the imaginary part o f the quasinormal frequencies decreases as\nmagnetic charge gincreases. It indicates that the larger magnetic charge gis, the faster oscillates and the\nslower decays for the massless scalar field.\n4 Hawking radiation\nIn this section, we adopt the WKB method to study the Hawking radia tionof Bardeen blackhole surrounded\nby PFDM for the massless scalar field, and discuss the influence of da rk matter parameter, magnetic charge\nand spherical harmonic index on the Hawking radiation.\nTaking account of quantum effects, Hawking thought that black ho les are not ”black” and can produce\nthermal radiation, known as Hawking radiation. We should note that the difference between radiation at the\nevent horizon which is pure blackbody spectrum with the radiation re ceiving at infinity which is corrected by\n8l=1\nl=5\nl=10\n0.10.20.30.40.50.60.70.00.51.01.52.02.5\ngRe(ω)l=1\nl=5\nl=10\n0.10.20.30.40.50.60.70.06250.06300.06350.06400.06450.06500.06550.0660\ng-\n\u0012\u0013 (ω)\nFig. 6The real part (left panel) and the imaginary part (right panel) of qu asinormal frequencies versus g\nfor the massless scalar field with M= 1,n= 0 and α=−0.3\nthe spacetime curvature. The difference can be expressed by a co efficient, i.e., gray-body factor. Using gray-\nbody factor, we can calculate the Hawking radiation of the black hole . Therefore, we impose the following\nboundary conditions on the wave equation,\nΨs=e−iωr∗+Reiωr∗, r∗→+∞,\nΨs=Te−iωr∗, r ∗→ −∞.(16)\nTandRrepresent transmission and reflection coefficients respectively an d their relationship is as follows\n|T|2+|R|2= 1. (17)\nEmploying the 6th WKB method, which describes in Section 3, we can ge t the reflection coefficient\nR=/parenleftbig\n1+e−2iπK/parenrightbig−1\n2, (18)\nwhere\nK−i/parenleftbig\nω2−V0/parenrightbig\n/radicalbig\n−2V′′\n0−i=6/summationdisplay\ni=2Λi(K) = 0. (19)\nThen using Eq. (17) and Eq. (18), we can obtain gray-body factor |Al|2easily,\n|Al|2= 1−|Rl|2=|Tl|2. (20)\nNow, we can calculate the power emission rate of Bardeen black hole s urrounded by PFDM for the\nmassless scalar field. We suppose that the black hole and its environm ent are a canonical ensemble, that is,\nthe Hawking temperature of the black hole does not change betwee n the subsequent emission of two particles\n[62]. Then the power emission rate of the massless scalar field can be w ritten as [42]\ndE\ndt=/summationdisplay\nlNl|Al|2ω\nexp(ω/TH)−1dω\n2π, (21)\n9l=1\nl=2\nl=30.00.20.40.60.81.001.×10-92.×10-93.×10-94.×10-9\nωd2Edω\n\u0014\u0015\n0.0 0.2 0.4 0.6\u0016 \u0017 \u0018 1.00.0000005.×10-60.0000100.0000150.000020\nωd2E\u0019ω\n\u001a\u001b\nFig. 7TheHawkingradiationtaken lasthevariable\nwithg= 0.1,α=−0.3α=0\nα\u001c\u001d0.1\nα\u001e\u001f0.3\nα !0.5\n0.0 0.2 0.4 0.6 0.8 1.002.×10-74.×10-76.×10-78.×10-71.×10-61.2×10-6\nωd2E\"ω\n#$\nFig. 8The Hawking radiation taken αas the vari-\nable with l= 2,g= 0.1\nwhere[63]\nTH=−3αg2log/parenleftBig\nr+\n|α|/parenrightBig\n+α/parenleftbig\ng2+r2\n+/parenrightbig\n+r+/parenleftbig\nr2\n+−2g2/parenrightbig\n4πr2\n+/parenleftbig\ng2+r2\n+/parenrightbig . (22)\nHereNl= 2l+ 1 for the massless scalar field. THandr+represent Hawking temperature and event\nhorizon, respectively. Figs. 7, 8 and 9 show the energy emission rat e with different parameters of Bardeen\nblack hole surrounded by PFDM for the massless scalar field. Fixed gandα, it is obvious from Fig. 7, the\nenergy emission rate is suppressed with the increase of spherical h armonic index land is almost negligible\nwhenl≥3. It means that l= 1 and l= 2 can well denote the full picture of massless scalar radiation.\nBesides, the peak of the energy emission rate shifts to high freque ncy. Then in Fig. 8 we consider that the\nemission of scalar radiation changes with dark matter parameter αwhen we fix landg. We obtain that\nthe increase of parameter αpromotes the emission of scalar radiation and the position of peak mo ves to\nright. As can be seen from Fig. 9, which depicts the effect of magnet ic charge gon the power emission\nspectrum for given landα, we find that the power emission spectrum is depressed as paramet ergincreases\nand the location of peak slowly shifts to high frequency. We draw a co nclusion that the increase of spherical\nharmonic index land magnetic charge gboth decrease the energy emission rate but dark matter paramet er\nαenhances the energy emission rate. Hence, it is easily to understan d that the values of αis smaller and g\nis larger, the black hole lifespan will be longer.\n5 Absorption cross section\nIn 2014, Benoneet al. [64] developedthe partialwavemethod to c ompute the absorptioncrosssection, which\nuses the gray-body factor |Al|2in process of calculation. In this section, we adopt the partial wave method\nto calculate the absorption cross section of Bardeen black hole sur rounded by PFDM for the massless scalar\n10field. The total absorption cross section can be written as\nσabs=∞/summationdisplay\nl=0σl\nabs, (23)\nand the expression of the partial absorption cross section is\nσl\nabs=π(2l+1)\nω2/parenleftBig\n1−/vextendsingle/vextendsinglee2iδl/vextendsingle/vextendsingle2/parenrightBig\n=π(2l+1)\nω2|Al|2. (24)\nIn Fig. 10, we exhibit the partial absorption cross section of Barde en black hole surrounded by PFDM\nfor the massless scalar field with fixed α=−0.1 andg= 0.1. Analyzing the graph, we discover that with\nthe increase of frequency, the partial absorption cross section first starts from zero and climbs the peak, and\nthen diminishes. Besides, the maximum value of the partial absorptio n cross section decreases as lincreases.\nFor exploring the influence of dark matter parameter αon the partial absorption cross section, we show the\nbehavior of the partial absorption cross section with different valu es ofαand take g= 0.1 andl= 5 in Fig.\n11. As shown in the picture, the bigger values of α, the lower partial absorption cross section. It means\nthat dark matter parameter suppresses the partial absorption cross section. When α=−0.1 andl= 5,\nthe curves of the partial absorption cross section for the differe nt values of magnetic charge gare presented\nin Fig. 12. It can be seen that the peak value of the partial absorpt ion cross section decreases with the\nincrease of g. In other words, magnetic charge also depresses the partial abs orption cross section. On the\nbasis of the above discussion, we obtain that l,αandgall suppress the partial absorption cross section. The\nresult is compatible to the variation of effective potential, which is high er withl,αandgincreasing. And\nthe essence of the phenomenon is that the higher potential barrie r causes the less massless scalar wave from\ninfinity transmitting to black hole.\nTo better investigate the properties of the considered black hole, we also depict the total absorption cross\nsection for the massless scalar field in Fig. 13 and Fig. 14, which contr ibutes from l= 0 up to l=10. It can\nbe observed that the total absorption cross section decreases asαandgincrease. Its variation is consistent\nwith the partial absorption cross section.\nFig. 15 is drawn to compare the total absorption cross section of t he Bardeen black hole surrounded by\nPFDM ( α/negationslash= 0) with Bardeen black hole ( α= 0). We find that the total absorption cross section increases\nbecause of the presence of α. In Fig. 16, we compare the total absorption cross section of Bar deen black\nhole surrounded by PFDM ( g/negationslash= 0) with Schwarzschild black hole surrounded by PFDM ( g= 0), and the\npicture shows that the total absorption cross section diminishes d ue to the contribution of g. By comparing\nFig. 15 and Fig. 16, it is found that the influence of dark matter para meter on the total absorption cross\nsection is greater than that of the magnetic charge.\n11g=0\ng=0.1\ng=0.3\ng=0.50.240.250.260.270.289.×10-81.×10-71.1×10-71.2×10-71.3×10-71.4×10-71.5×10-71.6×10-71.7×10-7\nωd2E%ω\n&'\n0.0 0.2 0.4 0.6( ) * 1.005.×10-81.×10-71.5×10-+\nωd2E,ω\n-.\nFig. 9The Hawking radiation taken gas the vari-\nable with l= 2,α=−0.5l=1\nl=2\nl=3\nl=4\n0.00.20.40.60.81.01.21.4020406080\nωσl\nFig. 10 The partial absorption cross section taken\nlas the variable with α=−0.1,g= 0.1\nα=0\nα/ 10.1\nα2 30.3\nα4 50.5\n0.0 0.5 1.0 1.50102030405060\nω\n6l\nFig. 11 The partial absorption cross section\ntakenαas the variable with g= 0.1,l= 5g=0\ng=0.1\ng=0.3\ng=0.5\n1.0001.0021.0041.0061.00833.5033.5533.6033.6533.7033.7533.80\nωσl\n0.0 0.5 1.0 1.501020304050\nωσl\nFig. 12 The partial absorption cross section\ntakengas the variable with α=−0.1,l= 5\nα=- 9 : ; < >=0.1\nα=- ? @ A B C=0.1\nα=-D E F G H=0.1\nSchwarzschild\n0.00.20.40.60.81.01.21.4050100150200250300350\nω\nJtotal\nFig. 13 The total absorption cross section taken\nαas the variable with g= 0.1α=-0.1,g=0.1\nα=-0.1,g=0.3\nα=-0.1,g=0.5\n0.5 1.0 1.56080100120140\nω\nKtotal\nFig. 14The total absorption cross section taken\ngas the variable with α=−0.1\n12α=0,g=0.1\nαL M0.3,gN0.1\nαO P0.5,gQ0.1\n0.20.40.60.81.01.21.4050100150200250300\nω\nRtotal\nFig. 15 The total absorption cross section of\nBardeen black hole surrounded by PFDM ( α/negationslash=\n0) and Bardeen black hole ( α= 0) with g= 0.1α=-0.1,g=0\nα=-0.1,g=0.3\nα=-0.1,g=0.5\n1.021.041.061.081.10100105110115120\nω\nStotal\n0.20.40.60.81.01.21.4050100150200\nω\nTtotal\nFig. 16 The total absorption cross section of\nBardeen black hole surrounded by PFDM ( g/negationslash=\n0) and Schwarzschild black hole surrounded by\nPFDM (g= 0) with α=−0.1\n6 Conclusion\nIn summary, this paper has argued that QNMs, Hawking radiation an d absorption cross section for the\nmasslessscalar field ofBardeen black hole surroundedby PFDM , whic h described by dark matter parameter\nand magnetic charge. Firstly, we have discussed the effective pote ntial taking l,α, andgas variables. Our\nanalysis implies that the barrier height of effective potential increas es with the increase of l,αandg. Then,\nwe have carefully performed a study of QNMs of the considered blac k hole for the massless scalar field\nwith the help of 6th WKB method. It has been discovered that the re al part of quasinormal frequencies\nbecomes more positive whereas the imaginary part decreases when the value of spherical harmonic index l\nincreases. This indicates that the actual frequency of the oscillat ion increases and the damping decreases\nwithlincreasing. And it has been found that the real part of quasinorma l frequencies grows with dark\nmatter parameter αincreasing but the imaginary part first increases and then decreas es. In other words,\nthe oscillation frequency becomes faster and the decay rate show s a trend of first increasing then decreasing\nwhen the value of αincreases. As magnetic charge gincreases, the real part of quasinormal frequencies\nincreases while the imaginary part reduces. Namely, galso enhances the actual frequency of the oscillation\nand depresses decay rate.\nMoreover, we have calculated the power emission rate of the black h ole for the massless scalar field using\n6th WKB method. The result shows that the energy emission rate is im peded by the increase of multiple\nnumberland magnetic charge gwhile dark matter parameter αenhances it. And the location of peak shift\nto high frequency with the three parameters increasing. Besides, it is worth mentioning that l= 1 and l= 2\ndominant the energy emission rate.\nFinally, we have devoted to computing absorption cross section for the massless scalar field by employing\n13the partial wave method. It is obvious that l,α, andgdepress the partial cross section which is the results\nthat correspondto the effective potential. From previous analysis , we have known that the effective potential\nis enhanced by the increase of l,α, andg. Then the higher height of effective potential, the less waves is\ntransmitted. In further analysis, we have compared the total ab sorption cross section of three black holes,\ni.e., Bardeen black hole, Schwarzschild black hole surrounded by PFDM and Bardeen black hole surrounded\nby PFDM from l= 0 tol= 10. Fixed g= 0.1, we have explored the existence of dark matter parameter\naffecting the total absorption cross section. It has been investig ated that the total absorption cross section\nof Bardeen black hole surrounded by PFDM ( α/negationslash= 0) is bigger than that of Bardeen black hole ( α= 0). And\nwhenα=−0.1, we have studied the existence of dark matter parameter influen cing the total absorption\ncross section. 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Shcherbakov∗\nHarvard-Smithsonian Center for Astrophysics, 60 Garden St reet, Cambridge, MA 02138\n(Dated: November 12, 2018)\nThe dispersion laws of Langmuir and transverse waves are cal culated in the relativistic non-\nmagnetized formalism for several isotropic particle distr ibutions: thermal, power-law, relativistic\nLorentzian κ,and hybrid β. For Langmuir waves the parameters of superluminal undampe d, sub-\nluminal damped principal and higher modes are determined fo r a range of distribution parameters.\nThe undamped and principal damped modes are found to match sm oothly. Principal damped and\nsecond damped modes are found not to match smoothly. The pres ence of maximum wavenumber is\ndiscovered above that no longitudinal modes formally exist . The higher damped modes are discov-\nered to be qualitatively different for thermal and certain no n-thermal distributions. Consistently\nwith the known results, the Landau damping is calculated to b e stronger for non-thermal power-\nlaw-like distributions. The dispersion law is obtained for the single undamped transverse mode.\nThe analytic results for the simplest distributions are pro vided.\nI. INTRODUCTION\nThe non-thermal distributions of electrons are as im-\nportant as thermal for astrophysical plasmas. Shocked\nand turbulent medium is likely to accelerate the elec-\ntrons into the power-laws. Gamma-Ray bursts1(GRBs),\njets from compact sources2, low luminosity active galac-\ntic nuclei3(LLAGNs) show evidence of the non-thermal\nrelativistic distributions. The electrons are non-thermal\nand mildly relativistic near Earth4and in solar corona5.\nThe properties of EM waves propagating through such\nmedium are worth knowing even in a non-magnetized\ncase. EM waves can be generated in turbulence, propa-\ngate some distance and dump via the Landau damping.\nThus energy is redistributed. Whereas the realistic tur-\nbulenceisnon-linear,onlythelinearwavesareconsidered\nin this work. No back-reaction of waves on the electron\ndistributions is assumed.\nThe reasonable relativistic electron distributions are\nthermal (10), power-law (11), Lorentzian κ(12) and hy-\nbridβ(13). They are taken to be normalized to unity\nwhen integrated over momenta. I take all the quantities\nin the paper to be dimensionless for the sake of clarity\nand brevity. Temperature Tof thermal distribution is\nevaluated in the units of mc2/kBfor the particles with\nmassm.Herecis the speed of light and kBis Boltzmann\nconstant. The inverse temperature\nρ=1\nT(1)\nis sometimes denoted as µin the literature. Momenta of\nparticles pis measured in mc.The speed of light cis set\nto unity. Thus\nγ=/radicalbig\n1+p2 (2)\nisthe particlesdimensionlessenergy. The non-relativistic\nplasma frequency in CGS units (for particles with charge\nq)\nωp=/radicalbigg\n4πnq2\nm(3)is employed to normalize the absolute values of a\nwavenumber kand frequency ωas6\nΩ =ω\nωp,K=k\nωp. (4)\nThe scalar Kis the absolute value of the vector K.\nThe general characteristics of relativistic plasma are\nthe following. Transverse waves have the phase veloc-\nity greater than the speed of light Ω /K >1 and so are\nundamped. They only have the single mode. The longi-\ntudinal waves exhibit the wider range of phenomena6,7.\nThe phase velocity goes from >1 to<1 asKincreases,\nthe undamped mode at small Kmatches the principal\ndamped mode that exists at higher Kup toKmax.8Be-\nsides the principal mode, the finite set of higher damped\nmodes exists. The present paper elaborates on all these\nmodes for various isotropic particle distribution. Either\na mixture of electrons and positrons with same distri-\nbutions or immobile ions are considered, so that the ion\nsound does not appear.\nThe paper is organized as follows. The formalism of\nlinear plasma dispersion is reviewed in §II. The results\nof numeric evaluations are presented in §III. Some ana-\nlytic formulas can be found in §IV. I conclude with the\ndiscussion and the summary in §V.\nII. LINEAR DISPERSION LAWS\nWavespropagatingin plasma feel the plasmaresponse,\nwhich can be characterized by the permittivity tensor\nε(K,Ω) in the linear regime. Permittivity tensor6,9\nε(K,Ω) = 1+/summationdisplay\nspecies1\nΩ2/integraldisplay1\np∂f\n∂ppp\nγ−K·p/Ωd3p(5)2\nleads6,10after the integration over the polar angles to\nlongitudinal permittivity\nεL=2π2iσΩ\nK3/integraldisplay∞\nγ0∂f\n∂γγ2dγ+1 (6)\n−2π\nΩK/integraldisplay∞\n0p2∂f\n∂p/braceleftBigg\n2γΩ\nKp+γ2Ω2\nK2p2ln/bracketleftbiggγΩ−Kp\nγΩ+Kp/bracketrightbigg/bracerightBigg\ndp\nand transverse permittivity\nεT= 1 +π2iσ\nΩK/integraldisplay∞\nγ0∂f\n∂γ/parenleftbigg\n1−γ2\nγ2\n0/parenrightbigg\ndγ+ (7)\nπ\nΩK/integraldisplay∞\n0p2∂f\n∂p/braceleftBigg\n2γΩ\nKp+/parenleftbiggγ2Ω2\nK2p2−1/parenrightbigg\nln/bracketleftbiggγΩ−Kp\nγΩ+Kp/bracketrightbigg/bracerightBigg\ndp,\nwhereγ0= (1−Ω2/K2)−1/2. TheLandauruleisapplied,\nso thatσ= 0 for Re[Ω2]≥K2and for Re[Ω2]< K2one\nhas6\nσ=/braceleftBigg\n0 for Im[Ω] >0,\n2 for Im[Ω] <0.(8)\nThecasewithIm[Ω] = 0forRe[Ω2]< K2isunphysical11.\nOnly one sort of species is considered in the above for-\nmulas without the loss of generality. The plasma density\nnis that of the mobile species in the the case of immobile\nions or the total density for electron-positron plasma.\nIII. NUMERIC RESULTS\nThe dispersion laws Ω( K) of waves propagating in\nplasma are determined as solutions of the equations\nεL= 0, ε T=K2\nΩ2. (9)\nA. Thermal distribution\nThe thermal relativistic distribution of particles\nfT(p) =ρexp(−ργ)\n4πK2(ρ)(10)\ndescribes highly collisional relaxed plasma. Here and be-\nlowKnrepresents n-th modified Bessel function of the\nsecond kind.\nThe longitudinal waves in thermal plasma have variety\nof features. The non-relativistic theory12predicts the in-\nfinite number of damped modes, whereas the relativistic\nanalysis7limits the damped modes to a finite set. In ad-\ndition, the superluminal undamped part of the spectrum\nappears. The boundary Kfor these modes are shown on\nFig. 1. These are the solutions of the dispersion relation\n(9) for real Ω and K.The transition between the damped\n(highK) and undamped (low K) modes is indicated by1.00.5 2.0 0.2 5.010.020.01.0\n0.52.0\n0.25.010.0\nΡK\nFIG. 1: (Color online) Upper solid (red) - maximum Kfor\nprincipal damped longitudinal harmonic, lower solid (blue )\n- for undamped harmonic; dashed (green) - Kboundaries\nfor second damped harmonic, dot-dashed (cyan) - for third\ndamped harmonic.\nthe lower solid blue line. The principal damped mode\nexists for Klimited by the upper solid red line, whose K\ngrows exponentially fast as T→0.No modes formally\nexist above that line, as the ion sound is absent. How-\never, the calculations in this large- Karea are unphysical,\nbecause the set of particles constituting plasma does not\nbehave coherently at very low wavelengths, as the wave-\nlength approaches the particle effective mean free path\nabout Debye radius λD.Thus the branch ceases to exist\nat wavenumbers k/greaterorsimilar1/λD13. Every realization of the\ndisribution function f(p) gives rise to different evolution\nof the imposed initial condition. The dashed green lines\nindicate the upper and lower liming Kfor the second\ndamped mode. This mode exists for any temperature.\nIts region of allowed Koverlaps with that for the princi-\npal mode. The lower limiting Kgoes to 0 at ρ∼2 and is\nzero for higher ρ(lower temperature). The same is true\nfor the third damping mode (boundaries in dot-dashed\ncyan): the lower limiting Kgoes to 0 at ρ∼8.The third\ndamped mode exists only for ρ >6.7.\nThe mode completion effect was claimed to exist by\nSchlikeiser7. It consists of the smooth transition of\nRe(Ω(K)) between the principal and second damped\nmodes. This is quite surprising given the fact that\nIm(Ω(K)) is discontinuous, thus Ω( K) is discontinuous.\nThe careful analysis shows that the real part is also dis-\ncontinuous. The real and imaginary parts of Ω( K) for\nρ= 2 are shown on Fig. 2 (see Fig. 3 in Ref. (7) for com-\nparison). In agreement with K-overlappingof the princi-\npal and second damped modes (Fig. 1), Kmax≈1.060for\ntheseconddampedmodesislargerthan Kmin≈1.000for\nthe principal damped mode. Thus, the mode completion\neffect does not occur.\nThe isocontours of constant Re(Ω) and −Im(Ω) for\nthe principal mode are shown on, respectively, Fig. 3\nand Fig. 4. These are the extensions of Figs. 6 and3\n0.0 0.5 1.0 1.5 2.0 2.50.51.01.52.02.5\nKRe/LParen1/CapOmeΓa/RParen1\n0.5 1.0 1.5 2.0 2.50.0010.010.1\nK/MinusIm/LParen1/CapOmeΓa/RParen1\nFIG. 2: (Color online) Dispersion relation for longitudina l\nwaves for ρ= 2.Undampedmode is dot-dashed(blue), princi-\npal damped mode is dark solid (black), second damped mode\nis light solid (cyan).\n7 in Ref. (6) to higher temperatures (lower ρ) with\nthe maximum Kboundary (see Fig. 1) employed. No\nmodes formally exist below the dark solid line denoted\nby ”max(K)”. The modes are undamped to the left from\nthe lightsolid cyanline ”Ω = K ,Im(Ω) = 0”and damped\nto the right. The plasma frequency at the infinite wave-\nlength is shown in dashed green. The correspondent plot\nfor the transverse waves is given in Ref. (6) along with\nthe approximations.\nB. Power-law distribution\nThe distribution\nfP(p) =Γ(κ)\nπ3/2Γ(κ−3\n2)γ−2κ(11)\nrepresents a simplest f(p). Also, it produces many ana-\nlytic results (see §IV). It is often used to interpret the\nresults ofastronomicalobservationsofjets2or any object\nwhere shockaccelerationtakesplace. In applicationsthis\ndistribution is usually applied with the limited range of\nγ,but for my calculation the entire range of γis taken.\nThe migration of critical points of longitudinal disper-\nsion relation is shown on Fig. 5. The superluminal mode0 1 2 3 4 5/Minus1.0/Minus0.50.00.51.01.52.0\nKlog10ΡIsocontoursof constantRe /LParen1/CapOmeΓa/RParen1\n0.70.91.05\n1.1\n1.2 1.52\n2.53\n4/CapOmeΓa/EqualK,Im/LParen1/CapOmeΓa/RParen1/Equal0\n/CapOmeΓaforK/Equal0max/LParen1K/RParen1\nFIG. 3: (Color online) Dispersion contours for longitudina l\nwaves, showing Re(Ω) in ( K,log10ρ) plane (solid blue lines),\nzero damping boundary (light solid cyan line), maximum K\ncurve (dark solid black line), Ω as a function of ρforK= 0\n(dashed green line).\n0 1 2 3 4 5/Minus1.0/Minus0.50.00.51.01.52.0\nKlog10ΡIsocontoursof constant /MinusIm/LParen1/CapOmeΓa/RParen1\n1.6 1.2 0.80.40.2\n0.10.05\n0.02\n0.007/CapOmeΓa/EqualK,Im/LParen1/CapOmeΓa/RParen1/Equal0\n/CapOmeΓaforK/Equal0max/LParen1K/RParen1\nFIG. 4: (Color online) Dispersion contours for longitudina l\nwaves, showing −Im(Ω) in ( K,log10ρ)plane(solid bluelines),\nzero damping boundary (light solid cyan line), maximum K\ncurve (dark solid black line), Ω as a function of ρforK= 0\n(dashed green line).\nsmoothly converts to a damped principal mode at crit-\nicalKabout 1,shown in solid blue on the figure. The\nmaximum Kof the principal mode (dashed black curve)\ngrowsexponentiallyfastasafunctionof κsimilarlytothe\nbehaviorofmaximum Kboundaryfor the thermal distri-\nbution Fig. 1. The behavior of higher damped modes is,\nin contrast, different. The second damped mode appears\natκ= 4.58.At higher κsecond damped mode can as-\nsumeKfrom 0toacritical K,shownindot-dashedgreen\non the figure. The presence of the second and higher or-\nderdamped modes at K= 0 is afeature ofthe power-law\ndistribution and is not observed for the thermal distribu-4\ntion.\n2 3 4 5 6 7 81101001000\nΚK\nFIG. 5: (Color online) Maximum Kfor principal damped lon-\ngitudinal mode (dashed black line), for second damped mode\n(dot-dashed green line), for undamped mode (solid blue line ).\nThe isocontours of constant Re(Ω) and −Im(Ω) for\nthe principal mode are shown on, respectively, Fig. 6\nand Fig. 7. The choosen small range of κ∈[2,3] re-\nflects the astronomical need to consider distributions\nN(γ)∼γ−xdγforγ≫1 withx∈[2,4].As in the\ncase of thermal plasma, no modes formally exist below\nthe solid black line denoted by ”max(K)”. The modes\nare undamped to the left from the light solid cyan line\n”Ω = K ,Im(Ω) = 0” and damped to the right. The\nplasma frequency at the infinite wavelength is shown\nin dashed green. The isocontour Kappears to depend\nonly weakly on κ,however the isocontour −Im(Ω) val-\nues (Fig. 7) are at least 1 .5 times larger for the same K,\nthan in case of thermal distribution (Fig. 4). The com-\nparison of damping coefficients for thermal and hybrid\ndistributions is given in the next subsection.\nC. Hybrid distribution\nI choose the relativistic isotropic Lorentzian κ\ndistribution4,14\nfH(p)∝/parenleftBigg\n1+/radicalbig\n1+p2−1\nκθ2/parenrightBigg−(κ+1)\n,(12)\nwhich tends to relativistic thermal distribution with\nT=θ2asκapproaches infinity. The normalization\ncoefficient is calculated numerically from the condition/integraltext∞\n04πp2fH(p)dp= 1.This distribution is the realistic\nchoice when the acceleration of particles is balanced by\nthe relaxation processes: almost thermal distribution at\nhighκand almost power-law at low κ.Thus, it is possi-\nble tocomparethe wavepropagationforthermal, slightly\nnon-thermal and highly non-thermal distributions.\nThe comparison of distributions is made on Fig 8. The\ninverse temperature ρforfT(p) and the temperature-\nlike parameter θ2forfH(p) are choosen in such a way0 1 2 3 4 52.02.22.42.62.83.0\nKΚIsocontoursof constantRe /LParen1/CapOmeΓa/RParen1\n0.75\n1.251.5\n1.752\n2.252.5\n2.753\n3.253.5\n3.754.\n4.254.5\n4.75/CapOmeΓa/EqualK,Im/LParen1/CapOmeΓa/RParen1/Equal0\n/CapOmeΓaforK/Equal0\nmax/LParen1K/RParen1\nFIG. 6: (Color online) Dispersion contours for longitudina l\nwaves, showing Re(Ω) in the ( K,κ) plane (solid blue lines),\nzero damping boundary (light solid cyan line), maximum K\ncurve (dark solid black line), Ω as a function κforK= 0\n(dashed green line).\n0 1 2 3 4 52.02.22.42.62.83.0\nKΚIsocontoursof constant /MinusIm/LParen1/CapOmeΓa/RParen1\n0.03\n0.10.2\n0.40.8\n1.21.6\n2.2.4\n/CapOmeΓa/EqualK,Im/LParen1/CapOmeΓa/RParen1/Equal0/CapOmeΓaforK/Equal0\nmax/LParen1K/RParen1\nFIG. 7: (Color online) Dispersion contours for longitudina l\nwaves, showing −Im(Ω) in the ( K,κ) plane (solid blue lines),\nzero damping boundary (light solid cyan line), maximum K\ncurve (dark solid black line), Ω as a function κforK= 0\n(dashed green line).\nthat the average kinetic energy of particle EK=/integraltext\n(γ−\n1)4πp2f(p)dpis equal to 1 .The choice of constant EK=\n1 is made to show the typical behavior of dispersion re-\nlations for plasmas with approximately the same ener-\ngetics. The thermal distribution is shown in dot-dashed\nblack. The transition from light green to dark blue\ncorresponds to the increase of κthrough the discrete\nsetκ= 4.5,5,5.5,6,7,10,20.The relativistic Lorentzian\nκdistribution appears to have more particles at lower\nγ≈1.2 and higher γ/greaterorsimilar5,but fewer particles in the in-\ntermediate region γ∼3.For the contrast I added the5\n10.0 5.0 2.0 3.0 1.5 7.010/Minus40.0010.010.11\nΓN/LParen1Γ/RParen1RelativisticLorentzian Κ,thermaland hybrid Βdistributions\nFIG. 8: (Color online) Distributions of particles with kine tic\nenergyEK= 1 : thermal ( κ→ ∞) (dot-dashed black line),\nLorentzian κwithκ= 4.5,5,5.5,6,7,10,20 (light green to\ndark blue lines), hybrid βdistribution (infinite EK) (dashed\nred line).\nhybridβdistribution\nfH2(p) =β3\nπ2(1+β2p2)2(13)\nwithβ= 2 that is shown by the dashed red line. It was\nnot normalized to EK= 1,since it has the divergent\ntotal energy. However, the dispersion laws exist for it.\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.51.01.52.02.5\nKRe/LParen1/CapOmeΓa/RParen1Re/LParen1/CapOmeΓa/RParen1for longitudinal waves\nFIG. 9: (Color online) Re(Ω( K)) dependence for longitudinal\nwaves for Lorentzian κdistribution: limiting thermal case\n(κ→ ∞) (dot-dashed black line), various κfrom 4.5 to 20\n(light green to dark blue lines), hybrid βdistribution (dashed\nred line).\nThe dependencies of Re(Ω( K)) for longitudinal waves,\n−Im(Ω(K)) for longitudinal waves and Ω( K) for trans-\nverse waves are shown for principal modes on, respec-\ntively, Fig. 9, Fig. 10, Fig. 11. The thin black line on\nFig. 9 separates the undamped mode (on the left) from1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0\nK/MinusIm/LParen1/CapOmeΓa/RParen1/MinusIm/LParen1/CapOmeΓa/RParen1for longitudinal waves\nFIG. 10: (Color online) −Im(Ω(K)) dependencefor longitudi-\nnal waves for Lorentzian κdistribution: limiting thermal case\n(κ→ ∞) (dot-dashed black line), various κfrom 4.5 to 20\n(light green to dark blue lines), hybrid βdistribution (dashed\nred line).\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.51.01.52.02.53.0\nK/CapOmeΓa/CapOmeΓafor transversewaves\nFIG. 11: (Color online) Re(Ω( K)) dependence for transverse\nwaves for Lorentzian κdistribution: limiting thermal case\n(κ→ ∞) (dot-dashed black line), various κfrom 4.5 to 20\n(light green to dark blue lines), hybrid βdistribution (dashed\nred line).\nthe damped one (on the right). The long-wavelength be-\nhavior of longitudinal modes is predictable: more mobile\nlowerγspecies in thermal distribution have the lowest\nplasma frequency. By the same reason Ω at K= 0 for\ntransverse waves (Fig. 11) is larger for smaller κ.The\nvariation of Re(Ω) is within 20% for both longitudinal\nand transverse waves. The Landau damping (Fig. 10)\nshows larger variation of up to 3 times. The electrons\nresponsible for Landau damping mainly have high γ/greaterorsimilar5,\nwhat makes distributions with smaller κdissipate waves\nmore effectively. The similar result is observed for non-\nrelativistic Lorentzian κdistribution15. The behavior of\nplasma waves for hybrid βdistribution is very different6\nfrom that for thermal or Lorentzian κdistribution.\nIV. ANALYTIC FORMULAS\nA. Thermal distribution\nThe Trubnikov’s form16of the dielectric tensor reads\nεµν=δµν+i\nΩ2ρ2\nK2(ρ)/integraldisplay∞\n0dα/parenleftbiggK2(r)\nr2δµν−α2KµKν\nΩ2K3(r)\nr3/parenrightbigg\n,\n(14)\nwhere\nr=/bracketleftbigg\nρ2−2iαρ+α2/parenleftbiggK2\nΩ2−1/parenrightbigg/bracketrightbigg1/2\n.(15)\nI denote\n∆ = 1−K2\nΩ2(16)\nand expandthe Besselfunctions in equation (14) in series\nin ∆.The resultant integrals can be evaluated analyti-\ncally knowing that\n/integraldisplay∞\n0dα/bracketleftBigg\nαnKm(/radicalbig\nρ2−2iρα)\n(ρ2−2iρα)m/2/bracketrightBigg\n=n!in+1Kn−m+1(ρ)\nρm.\n(17)\nThen the implicit formula for ∆\n∆ = Ω−2∞/summationdisplay\nj=0/parenleftbigg\n−∆\n2ρ/parenrightbiggj(2j)!\nj!Kj−1(ρ)\nK2(ρ)(18)\ncan be derived9. It can be rewritten as the explicit for-\nmula for ∆ .The expression for the transverse permittiv-\nity\nεT= 1−∆ = 1−∞/summationdisplay\nn=0Ω−2(n+1)n/productdisplay\nl=0Kl−1(ρ)\nK2(ρ)(2l)!\n(−2ρ)ll!(19)\ncan easily give the dependence K(Ω).Takingn= 0 one\nobtains\nΩ2=K2+K1(T−1)\nK2(T−1)(20)\nin a high-frequency limit Ω ≫1, coincident with the\nexpression in Ref. (10). As the frequency goes down,\nterms with higher nbecome important. The consecu-\ntive approximations are expected to work well for any\ntemperature. The author believes this elegant derivation\nof the high-frequency approximation was not previously\noutlined.\nB. Power-law distribution\nThe permittivity tensor for the power-law distribution\n(11) of particles can be calculated analytically.1. General case\nThe permittivity components are\nεL= 1+2Γ(κ−1)\nΓ(κ+1\n2)Γ(κ−3\n2)K2/braceleftbigg\nΓ(κ+1) (21a)\n−κcsc(πκ)/bracketleftbigg√π(1−∆)1\n2−κ∆κ−1Γ/parenleftbigg\nκ+1\n2/parenrightbigg\n+π/parenleftbigg\nκ−1\n2/parenrightbigg/parenleftbigg\n22˜F1/parenleftbigg\n2,−1\n2,2−κ,∆/parenrightbigg\n−32˜F1/parenleftbigg\n1,−1\n2,2−κ,∆/parenrightbigg/parenrightbigg/bracketrightbigg\n+√π∆κ−1(∆−1)1\n2−κκσΓ/parenleftbigg\nκ+1\n2/parenrightbigg/bracerightbigg\n,\nεT= 1−Γ(κ−1)\n2Γ(κ+1\n2)Γ(κ−3\n2)K2/braceleftbigg\n2Γ(κ+1) (21b)\n−csc(πκ)/bracketleftbigg\n2√π∆κ(1−∆)1\n2−κΓ/parenleftbigg\nκ+1\n2/parenrightbigg\n+πκ(2−3∆−2κ)2˜F1/parenleftbigg\n1,−1\n2,2−κ,∆/parenrightbigg\n+ 2π(∆κ+κ−1)2˜F1/parenleftbigg\n2,−1\n2,2−κ,∆/parenrightbigg/bracketrightbigg\n−2√π∆κ(∆−1)1\n2−κσΓ/parenleftbigg\nκ+1\n2/parenrightbigg/bracerightbigg\n,\nwhereσis determined by Landau rule (see equation (8)\nand discussion therein). The notation 2˜F1represents the\nregularized hypergeometric function. For the integer κ\nthe limit of κgoing to that singular integer value should\nbe considered. The non-singular expressions were also\nderived, but are longer and are not provided here. The\nMathematica 6 convention of branch cuts should be used\nto evaluate the values of roots and non-integer powers.\nIt sets the branch cuts to be on arg( z) =πline in the\ncomplex z plane.\nThe high-frequency limit for the transverse waves is\nΩ2=K2−πcsc(πκ)Γ(κ)\nΓ(2−κ)Γ(κ−3\n2)Γ(κ+1\n2).(22)\nAgain the limit must be considered for integer κ.\n2. Special cases\nMuchshorterformulasforpermittivitiescanbederived\nfor certain κ.The shortest ones are for κin the observa-\ntionally motivated range κ∈[2,3].Let me choose κ= 2\nandκ= 5/2 as the examples.\nThe case κ= 2 (equivalent to dN(γ)∼γ−2dγat high7\nγ) leads to\nεT= 1+(4−10∆)√\n∆−1+3σπ∆2+6∆2arcsec(√\n∆)\n3πK2(∆−1)3/2,\n(23a)\nεL= 1+(8+16∆)√\n∆−1+12σπ∆2−24∆arcsec(√\n∆)\n3πK2(∆−1)3/2.\n(23b)\nThe case κ=5\n2(equivalent to dN(γ)∼γ−3dγat high\nγ) leads to\nεT= 1+π∆(10−15∆+8∆3/2(1−2σ))−3\n16K2(∆−1)2,(24a)\nεL= 1+5π∆(6+3∆ −8∆1/2(1−2σ))−1\n16K2(∆−1)2.(24b)\nThe longitudinal dispersion relation (24b) for κ= 5/2\nremarkably gives the compact analytic form for Ω( K).\nThe undamped mode at low Kexists in this case along\nwith the single damped mode at higher Kas\nΩ =1\n8/radicalBigg\n48K2+5π+/radicalbigg\n(5π−16K2)3\n5π(25a)\nforK= 0..√\n5π\n4,\nΩ =1\n8/radicalBigg\n48K2+5π−i/radicalbigg\n(16K2−5π)3\n5π(25b)\nforK=√\n5π\n4..1\n4/radicalBig\n5π(9+4√\n5).\nThe relation Ω( K) can be expanded near the point Kx=√\n5π/4 to give the powers of ( Kx−K)3/2so that Ω K\nstays real for K < K xand gains the imaginary part for\nK > K x.The second derivative of Ω( K) is discontinuous\natKx.The explicit dispersion law Ω( K) (25) is plotted\nforκ= 5/2 on Fig. 12.\nC. Hybrid βdistribution\nThe analytic expressions for transverse and longitudi-\nnal permittivities need some terms to be abbreviated for\nbrevity in the case of hybrid βdistribution (13). They\nread\nεT= 1+β\nπA3BCK2/bracketleftbigg\nC(2AB+β4((B−2C2)(26a)\n×arctanA−Barcsecβ))−2A3∆2arctanhC/bracketrightbigg\n−∆2βσi\nBCK2,0 1 2 3 401234\nKRe/LParen1/CapOmeΓa/RParen1\n0 1 2 3 4012\nK/MinusIm/LParen1/CapOmeΓa/RParen1\nFIG. 12: (Color online) Longitudinal dispersion law for κ=\n5/2: undamped mode (dot-dashed blue line), damped mode\n(solid red line), separated by Re(Ω) = K(thin black line).\nεL= 1+2β\nπA3B2CK2/bracketleftbigg\nB2Cβ4arcsecβ−AC(26b)\n×(B(B+∆A2)+Aβ4(B+2C2)arctanA)\n+ 2A3∆(B+C2β2)arctanhC/bracketrightbigg\n−2β∆σi(B+C2β2)\nB2CK2,\nwhere\nA=/radicalbig\nβ2−1, B=β2−∆, C=√\n1−∆.(26c)\nThe expressions for β= 1 coincide with those for the\npower-law distribution with κ= 2.\nV. DISCUSSION & CONCLUSION\nThe paper derives and reconsiders the broad range of\nlinearwaveeffects in non-magnetizedone species plasma.\nThe various distributions are employed for comparison\nand to consider the realistic non-equilibrium plasmas.\nThe thermal (10), power-law (11), relativistic Lorentzian\nκ(12) and hybrid β(13) distributions are employed with\nthe possible inclusion of the simulated distributions17in\nthe future.8\nFor longitudinal waves the maximum Kwas found to\nexist above that neither the undamped mode nor the\ndamped mode survive. The mode completion effect7\nwas closed. The regions of the principal and the second\ndamped modes overlap in the temperature-wavenumber\nplane for thermal plasma. The second damped mode al-\nwaysexist. The lower possible Kis zero for low tempera-\nturesforsecondandthirddampedmodes, but isnon-zero\nfor higher temperatures. In contrast, for the power-law\ndistribution (11) the second damped mode does not ex-\nist for low κ. When it exists at higher κ, its lowest Kis\nalways zero.\nSome analytic results are derived that can accelerate\nthe calculations and provide some insight. The full an-\nalytic result for longitudinal Ω( K) was derived for the\npower-law distribution (11) with κ= 5/2.The transitionatKxbetween the undamped mode and the principle\ndamped mode was found to be smooth with discontinu-\nous second derivative of Ω( K) because of ( Kx−K)3/2\nterm.\nAcknowledgments\nThe author is grateful to Ramesh Narayan for fruit-\nful discussions and to anonymous referee, whose com-\nments helped to improve the manuscript. The paper was\nsupported by NASA Earth and Space Science Fellowship\nNNX08AX04H and partially supported by NASA grant\nNNX08AH32G.\n∗Electronic address: rshcherbakov@cfa.harvard.edu;\nURL:http://www.cfa.harvard.edu/ ~rshcherb/\n1A. Panaitescu and P. Kumar, Astrophys. J. 571, 779\n(2002).\n2J. F. C. Wardle, D. C. Homan, R.Ojha, andD. H.Roberts,\nNature395, 457 (1998).\n3F. Yuan, E. 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Trubnikov, Thesis, Moscow Institute of Engineering\nand Physics (1958) [English translation in AEC-tr-4073,\nUS Atomic Energy Commission, Oak Ridge, Tennessee,\n1960].\n17W. J. Liu , P. F. Chen, M. D. Ding, and C. Fang, ”En-\nergy spectrum of the electrons accelerated by reconnection\nelectric field: exponential or power-law?”, submitted to\nAstrophys. J. ." }, { "title": "2106.14858v3.Stability_of_a_Magnetically_Levitated_Nanomagnet_in_Vacuum__Effects_of_Gas_and_Magnetization_Damping.pdf", "content": "Stability of a Magnetically Levitated Nanomagnet in Vacuum: E\u000bects of Gas and\nMagnetization Damping\nKatja Kustura,1, 2Vanessa Wachter,3, 4Adri\u0013 an E. Rubio L\u0013 opez,1, 2and Cosimo C. Rusconi5, 6\n1Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria.\n2Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria.\n3Max Planck Institute for the Science of Light, Staudtstra\u0019e 2, 91058 Erlangen, Germany\n4Department of Physics, University of Erlangen-N urnberg, Staudtstra\u0019e 7, 91058 Erlangen, Germany\n5Max-Planck-Institut f ur Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany.\n6Munich Center for Quantum Science and Technology,\nSchellingstrasse 4, D-80799 M unchen, Germany.\n(Dated: June 1, 2022)\nIn the absence of dissipation a non-rotating magnetic nanoparticle can be stably levitated in a\nstatic magnetic \feld as a consequence of the spin origin of its magnetization. Here we study the\ne\u000bects of dissipation on the stability of the system, considering the interaction with the background\ngas and the intrinsic Gilbert damping of magnetization dynamics. At large applied magnetic \felds\nwe identify magnetization switching induced by Gilbert damping as the key limiting factor for\nstable levitation. At low applied magnetic \felds and for small particle dimensions magnetization\nswitching is prevented due to the strong coupling of rotation and magnetization dynamics, and\nthe stability is mainly limited by the gas-induced dissipation. In the latter case, high vacuum\nshould be su\u000ecient to extend stable levitation over experimentally relevant timescales. Our results\ndemonstrate the possibility to experimentally observe the phenomenon of quantum spin stabilized\nmagnetic levitation.\nI. INTRODUCTION\nThe Einstein{de Haas [1, 2] and Barnett e\u000bects [3] are\nmacroscopic manifestations of the internal angular mo-\nmentum origin of magnetization: a change in the mag-\nnetization causes a change in the mechanical rotation\nand conversely. Because of the reduced moment of in-\nertia of levitated nano- to microscale particles, these ef-\nfects play a dominant role in the dynamics of such sys-\ntems [4{10]. This o\u000bers the possibility to harness these\ne\u000bects for a variety of applications such as precise magne-\ntometry [11{16], inertial sensing [17, 18], coherent spin-\nmechanical control [19, 20], and spin-mechanical cool-\ning [21, 22] among others. Notable in this context is\nthe possibility to stably levitate a ferromagnetic parti-\ncle in a static magnetic \feld in vacuum [23, 24]. Stable\nlevitation is enabled by the internal angular momentum\norigin of the magnetization which, even in the absence of\nmechanical rotation, provides the required angular mo-\nmentum to gyroscopically stabilize the system. Such a\nphenomenon, which we refer to as quantum spin stabi-\nlized levitation to distinguish it from the rotational stabi-\nlization of magnetic tops [25{27], relies on the conserva-\ntive interchange between internal and mechanical angular\nmomentum. Omnipresent dissipation, however, exerts\nadditional non-conservative torques on the system which\nmight alter the delicate gyroscopic stability [26, 28]. It\nthus remains to be determined if stable levitation can\nbe observed under realistic conditions, where dissipative\ne\u000bects cannot be neglected.\nIn this article, we address this question. Speci\f-\ncally, we consider the dynamics of a levitated magnetic\nnanoparticle (nanomagnet hereafter) in a static magnetic\n\feld in the presence of dissipation originating both fromthe collisions with the background gas and from the\nintrinsic damping of magnetization dynamics (Gilbert\ndamping) [29, 30], which are generally considered to be\nthe dominant sources of dissipation for levitated nano-\nmagnets [8, 13, 31{33]. Con\fned dynamics can be ob-\nserved only when the time over which the nanomagnet is\nlevitated is longer than the period of center-of-mass os-\ncillations in the magnetic trap. When this is the case, we\nde\fne the system to be metastable . We demonstrate that\nthe system can be metastable in experimentally feasible\nconditions, with the levitation time and the mechanism\nbehind the instability depending on the parameter regime\nof the system. In particular, we show that at weak ap-\nplied magnetic \felds and for small particle dimensions\n(to be precisely de\fned below) levitation time can be\nsigni\fcantly extended in high vacuum (i.e. pressures be-\nlow 10\u00003mbar). Our results evidence the potential of\nunambiguous experimental observation of quantum spin\nstabilized magnetic levitation.\nWe emphasize that our analysis is particularly timely.\nPresently there is a growing interest in levitating and con-\ntrolling magnetic systems in vacuum [9, 34, 35]. Current\nexperimental e\u000borts focus on levitation of charged para-\nmagnetic ensembles in a Paul trap [19, 36, 37], diamag-\nnetic particles in magneto-gravitational traps [38{40], or\nferromagnets above a superconductor [14, 20, 41]. Lev-\nitating ferromagnetic particles in a static magnetic trap\no\u000bers a viable alternative, with the possibility of reaching\nlarger mechanical trapping frequencies.\nThe article is organized as follows. In Sec. II we in-\ntroduce the model of the nanomagnet, and we de\fne\ntwo relevant regimes for metastability, namely the atom\nphase and the Einstein{de Haas phase. In Sec. III and\nIV we analyze the dynamics in the atom phase and thearXiv:2106.14858v3 [cond-mat.mes-hall] 31 May 20222\nFigure 1. (a) Illustration of a spheroidal nanomagnet levi-\ntated in an external \feld B(r) and surrounded by a gas at the\ntemperature Tand the pressure P. (b) Linear stability dia-\ngram of a non-rotating nanomagnet in the absence of dissipa-\ntion, assuming a= 2b. Blue and red regions denote the stable\natom and Einstein{de Haas phase, respectively; hatched area\nis the unstable region. Dashed lines show the critical values\nof the bias \feld which de\fne the two phases. In particular,\nBEdH,1\u00115\u0016=[4\r2\n0(a2+b2)M],BEdH,2\u00113 [\u0016B02=(4\r0M)]1=3,\nandBatom = 2kaV=\u0016. Numerical values of physical parame-\nters used to generate panel (b) are given in Table I.\nEinstein{de Haas phase, respectively. We discuss our re-\nsults in Sec. V. Conclusions and outlook are provided in\nSec. VI. Our work is complemented by three appendices\nwhere we de\fne the transformation between the body-\n\fxed and laboratory reference frames (App. A), analyze\nthe e\u000bect of thermal \ructuations (App. B), and provide\nadditional \fgures (App. C).\nII. DESCRIPTION OF THE SYSTEM\nWe consider a single domain nanomagnet levitated in\na static1magnetic \feld B(r) as shown schematically in\nFig. 1(a). We model the nanomagnet as a spheroidal\nrigid body of mass density \u001aMand semi-axes lengths a;b\n(a > b ), having uniaxial magnetocrystalline anisotropy,\nwith the anisotropy axis assumed to be along the major\nsemi-axisa[42]. Additionally, we assume that the mag-\nnetic response of the nanomagnet is approximated by a\npoint dipole with magnetic moment \u0016of constant mag-\nnitude\u0016\u0011j\u0016j, as it is often justi\fed for single domain\nparticles [42, 43]. Let us remark that such a simpli\fed\nmodel has been considered before to study the classical\ndynamics of nanomagnets in a viscous medium [31, 44{\n49], as well as to study the quantum dynamics of mag-\nnetic nanoparticles in vacuum [5, 13, 50, 51]. Since the\nmodel has been successful in describing the dynamics of\nsingle-domain nanomagnets, we adopt it here to inves-\ntigate the stability in a magnetic trap. In particular,\nour study has three main di\u000berences as compared with\n1We denote a \feld static if it does not have explicit time depen-\ndence, namely if @B(r)=@t= 0.Table I. Physical parameters of the model and the values used\nthroughout the article. We calculate the magnitude of the\nmagnetic moment as \u0016=\u001a\u0016V, where\u001a\u0016=\u001aM\u0016B=(50amu),\nwith\u0016Bthe Bohr magneton and amu the atomic mass unit.\nParameter Description Value [units]\n\u001aM mass density 104[kg m\u00003]\na;b semi-axes see main text [m]\n\u001a\u0016 magnetization 2 :2\u0002106[J T\u00001m\u00003]\nka anisotropy constant 105[J m\u00003]\n\r0 gyromagnetic ratio 1 :76\u00021011[rad s\u00001T\u00001]\nB0 \feld bias see main text [T]\nB0\feld gradient 104[T m\u00001]\nB00\feld curvature 106[T m\u00002]\n\u0011 Gilbert damping 10\u00002[n. u.]\nT temperature 10\u00001[K]\nP pressure 10\u00002[mbar]\nM molar mass 29 [g mol\u00001]\n\u000bc re\rection coe\u000ecient 1 [n. u.]\nprevious work. (i) We consider a particle levitated in\nhigh vacuum, where the mean free path of the gas parti-\ncles is larger than the nanomagnet dimensions (Knudsen\nregime [52]). This leads to gas damping which is gen-\nerally di\u000berent from the case of dense viscous medium\nmostly considered in the literature. (ii) We consider\ncenter-of-mass motion and its coupling to the rotational\nand magnetic degrees of freedom, while previous work\nmostly focuses on coupling between rotation and mag-\nnetization only (with the notable exception of [48]). (iii)\nWe are primarily interested in the center-of-mass con\fne-\nment of the particle, and not in its magnetic response.\nWithin this model the relevant degrees of freedom of\nthe system are the center-of-mass position r, the linear\nmomentum p, the mechanical angular momentum L, the\norientation of the nanomagnet in space \n, and the mag-\nnetic moment \u0016. The orientation of the nanomagnet\nis speci\fed by the body-\fxed reference frame Oe1e2e3,\nwhich is obtained from the laboratory frame Oexeyez\naccording to ( e1;e2;e3)T=R(\n)(ex;ey;ez)T, where\n\n= (\u000b;\f;\r )Tare the Euler angles and R(\n) is the\nrotational matrix. We provide the expression for R(\n)\nin App. A. The body-\fxed reference frame is chosen such\nthate3coincides with the anisotropy axis. The magnetic\nmoment\u0016is related to the internal angular momentum F\naccording to the gyromagnetic relation \u0016=\r0F, where\n\r0is the gyromagnetic ratio of the material2.\n2The total internal angular momentum Fis a sum of the individ-\nual atomic angular momenta (spin and orbital), which contribute\nto the atomic magnetic moment. For a single domain magnetic\nparticle, it is customary to assume that Fcan be described as\na vector of constant magnitude, jFj=\u0016=\r0(macrospin approxi-\nmation) [43].3\nA. Equations of Motion\nWe describe the dynamics of the nanomagnet in the\nmagnetic trap with a set of stochastic di\u000berential equa-\ntions which model both the deterministic dissipative evo-\nlution of the system and the random \ructuations due to\nthe environment. In the following it is convenient to de-\n\fne dimensionless variables: the center-of-mass variables\n~r\u0011r=a,~p\u0011\r0ap=\u0016, the mechanical angular momen-\ntum`\u0011\r0L=\u0016, the magnetic moment m\u0011\u0016=\u0016, and\nthe magnetic \feld b(~r)\u0011B(a~r)=B0, whereB0denotes\nthe minimum of the \feld intensity in a magnetic trap,\nwhich we hereafter refer to as the bias \feld. Note that\nwe choose to normalize the position r, the magnetic mo-\nment\u0016and the magnetic \feld B(r) with respect to the\nparticle size a, the magnetic moment magnitude \u0016, and\nthe bias \feld B0, respectively. The scaling factor for an-\ngular momentum, \u0016=\r0, and linear momentum, \u0016=(a\r0),\nfollow as a consequence of the gyromagnetic relation.\nThe dynamics of the nanomagnet in the laboratory\nframe are given by the equations of motion\n_~r=!I~p; (1)\n_e3=!\u0002e3; (2)\n_~p=!Lr~r[m\u0001b(~r)]\u0000\u0000cm~p+\u001ep(t); (3)\n_`=!Lm\u0002b(~r)\u0000_m\u0000\u0000rot`+\u0018l(t); (4)\n_m=m\n1 +\u00112\u0002[!e\u000b\u0000\u0011m\u0002(!+!e\u000b+\u0011!\u0002m)\n+\u0010b(t)]: (5)\nHere we de\fne the relevant system frequencies: !I\u0011\n\u0016=(\r0Ma2) is the Einstein{de Haas frequency, with M\nthe mass of the nanomagnet, !L\u0011\r0B0is the Larmor\nfrequency,!A\u0011kaV\r0=\u0016is the anisotropy frequency,\nwithVthe volume of the nanomagnet and kathe ma-\nterial dependent anisotropy constant [43], !\u0011I\u00001L\nis the angular velocity, with Ithe tensor of inertia,\nand!e\u000b\u00112!A(m\u0001e3)e3+!Lb(~r). Dissipation is\nparametrized by the dimensionless Gilbert damping pa-\nrameter\u0011[29, 53], and the center-of-mass and rotational\nfriction tensors \u0000 cmand \u0000 rot, respectively [32]. The e\u000bect\nof stochastic thermal \ructuations is represented by the\nrandom variables \u001ep(t) and\u0018l(t) which describe, respec-\ntively, the \ructuating force and torque exerted by the\nsurrounding gas, and by \u0010b(t) which describes the ran-\ndom magnetic \feld accounting for thermal \ructuations\nin magnetization dynamics [54]. We assume Gaussian\nwhite noise, namely, for X(t)\u0011(\u001ep(t);\u0018l(t);\u0010b(t))Twe\nhavehXi(t)i= 0 andhXi(t)Xj(t0)i\u0018\u000eij\u000e(t\u0000t0).\nEquations (1-4) describe the center-of-mass and rota-\ntional dynamics of a rigid body in the presence of dis-\nsipation and noise induced by the background gas [32].\nThe expressions for \u0000 cmand \u0000 rotdepend on the parti-\ncle shape { here we take the expressions derived in [32]for a cylindrical particle3{, and on the ratio of the sur-\nface and the bulk temperature of the particle, which\nwe assume to be equal to the gas temperature T. Fur-\nthermore, they account for two di\u000berent scattering pro-\ncesses, namely the specular and the di\u000busive re\rection\nof the gas from the particle, which is described by a\nphenomenological interpolation coe\u000ecient \u000bc. The or-\nder of magnitude of the di\u000berent components of \u0000 cmand\n\u0000rotis generally well approximated by the dissipation\nrate \u0000\u0011(2Pab=M )[2\u0019M=(NAkBT)]1=2, wherePand\nMare, respectively, the gas pressure and molar mass,\nkBis the Boltzmann constant and NAis the Avogadro\nnumber. The magnetization dynamics given by Eq. (5)\nis the Landau-Lifshitz-Gilbert equation in the laboratory\nframe [8, 57], with the e\u000bective magnetic \feld !e\u000b=\r0.\nWe remark that Eqs. (1-5) describe the classical dynam-\nics of a levitated nanomagnet where the e\u000bect of the\nquantum spin origin of magnetization, namely the gy-\nromagnetic relation, is taken into account phenomeno-\nlogically by Eq. (5). This is equivalent to the equations\nof motion obtained from a quantum Hamiltonian in the\nmean-\feld approximation [24].\nLet us discuss the e\u000bect of thermal \ructuations on\nthe dynamics of the nanomagnet at subkelvin temper-\natures and in high vacuum. These conditions are com-\nmon in recent experiments with levitated particles [58{\n60]. The thermal \ructuations of magnetization dy-\nnamics, captured by the last term in Eq. (5), lead\nto thermally activated transition of the magnetic mo-\nment between the two stable orientations along the\nanisotropy axis [54, 61]. Such process can be quan-\nti\fed by the N\u0013 eel relaxation time, which is given by\n\u001cN\u0019(\u0019=! A)p\nkBT=(kaV)ekaV=(kBT). Thermal acti-\nvation can be neglected when \u001cNis larger than other\ntimescales of magnetization dynamics, namely the pre-\ncession timescale given by \u001cL\u00111=j!e\u000bj, and the Gilbert\ndamping timescale given by \u001cG\u00111=(\u0011j!e\u000bj). Con-\nsidering for simplicity j!e\u000bj\u00182!A, for a particle size\na= 2b= 1 nm and temperature T= 1 K, and the\nvalues of the remaining parameters as in Table I, the ra-\ntio of the timescales is of the order \u001cN=\u001cL\u0018103, and\nit is signi\fcantly increased for larger particle sizes and\nat smaller temperatures. We remark that, for the val-\nues considered in this article, \u001cNis much larger than the\nlongest dynamical timescale in Eqs. (1-5) which is associ-\nated with the motion along ex. Thermal activation of the\nmagnetic moment can therefore be safely neglected. The\nstochastic e\u000bects ascribed to the background gas, cap-\ntured by the last terms in Eqs. (3-4), are expected to be\nimportant at high temperatures (namely, a regime where\nMkBT\r2\n0a2=\u00162&1 [32]). At subkelvin temperatures and\nin high vacuum these \ructuations are weak and, con-\nsequently, they do not destroy the deterministic e\u000bects\n3The expressions for \u0000 cmand \u0000 rotfor a cylindrical particle\ncapture the order of magnitude of the dissipation rates for a\nspheroidal particle [55, 56].4\ncaptured by the remaining terms in Eqs. (1-5) [33]. In-\ndeed, for the values of parameters given in Table I and\nfora= 2b,MkBT\r2\n0a2=\u00162\u00190:8T=(a[nm]). For sub-\nkelvin temperatures and particle sizes a>1 nm, thermal\n\ructuations due to the background gas can therefore be\nsafely neglected.\nIn the following we thus neglect stochastic e\u000bects by\nsetting\u001ep=\u0018l=\u0010b= 0, and we consider only the de-\nterministic part of Eqs. (1-5) as an appropriate model\nfor the dynamics [8, 33, 54]. In App. B we carry out\nthe analysis of the dynamics including the e\u000bects of gas\n\ructuations in equations (1-5), and we show that the\nresults presented in the main text remain qualitatively\nvalid even in the presence of thermal noise. For the mag-\nnetic \feld B(r) we hereafter consider a Io\u000be-Pritchard\nmagnetic trap, given by\nB(r) =ex\u0014\nB0+B00\n2\u0012\nx2\u0000y2+z2\n2\u0013\u0015\n\u0000ey\u0012\nB0y+B00\n2xy\u0013\n+ez\u0012\nB0z\u0000B00\n2xz\u0013\n;(6)\nwhereB0;B0andB00are, respectively, the \feld bias, gra-\ndient and curvature [62]. We remark that this is not a\nfundamental choice, and di\u000berent magnetic traps, pro-\nvided they have a non-zero bias \feld, should result in\nsimilar qualitative behavior.\nB. Initial conditions\nThe initial conditions for the dynamics in Eqs. (1-5),\nnamely at time t= 0, depend on the initial state of the\nsystem, which is determined by the preparation of the\nnanomagnet in the magnetic trap. In our analysis, we\nconsider the nanomagnet to be prepared in the thermal\nstate of an auxiliary loading potential at the temperature\nT. Subsequently, we assume to switch o\u000b the loading\npotential at t= 0, while at the same time switching\non the Io\u000be-Pritchard magnetic trap. The choice of the\nauxiliary potential is determined by two features: (i) it\nallows us to simply parametrize the initial conditions by a\nsingle parameter, namely the temperature T, and (ii) it is\nan adequate approximation of general trapping schemes\nused to trap magnetic particles.\nRegarding point (i), we assume that the particle is lev-\nitated in a harmonic trap, in the presence of an external\nmagnetic \feld applied along ex. This loading scheme\nprovides, on the one hand, trapping of the center-of-mass\ndegrees of freedom, with trapping frequencies denoted by\n!i(i=x;y;z ). On the other hand, the magnetic moment\nin this case is polarized along ex. The Hamiltonian of the\nsystem in such a con\fguration reads Haux=p2=(2M) +P\ni=x;y;zM!2\nir2\ni=2+LI\u00001L=2\u0000kaVe2\n3;x\u0000\u0016xBaux, where\nBauxdenotes the magnitude of the external magnetic\n\feld, which we for simplicity set to Baux=B0in all our\nsimulations. At t= 0 the particle is released in the mag-\nnetic trap given by Eq. (6). For the degrees of freedomx\u0011(~r;~p;`;mx)T, we take as the initial displacement\nfrom the equilibrium the corresponding standard devia-\ntion in a thermal state of Haux. More precisely, xi(0) =\nxi;e+ (hx2\nii\u0000hxii2)1=2, wherexi;edenotes the equilib-\nrium value, andhxk\nii\u0011Z\u00001R\ndxxk\niexp[\u0000Haux=(kBT)],\nwithk= 1;2 and the partition function Z. For the Eu-\nler angles \nwe use \n 1(0)\u0011cos\u00001[\u0000p\nhcos2\n1i] and\n\ni(0)\u0011cos\u00001[p\nhcos2\nii] (i= 2;3). The initial condi-\ntions for e3follow from \nusing the transformation given\nin App. A.\nRegarding point (ii), the initial conditions obtained in\nthis way describe a trapped particle prepared in a ther-\nmal equilibrium in the presence of an external loading\npotential where the center of mass is decoupled from the\nmagnetization and the rotational dynamics. It is outside\nthe scope of this article to study in detail a particular\nloading scheme. However, we point out that an auxil-\niary potential given by Hauxcan be obtained, for exam-\nple, by trapping the nanomagnet using a Paul trap as\ndemonstrated in recent experiments [19, 21, 37, 63{70].\nIn particular, trapping of a ferromagnetic particle has\nbeen demonstrated in a Paul trap at P= 10\u00002mbar,\nwith center-of-mass trapping frequency of up to 1 MHz,\nand alignment of the particle along the direction of an\napplied \feld [19]. We note that particles are shown to\nremain trapped even when the magnetic \feld is varied\nover many orders of magnitudes or switched o\u000b. We re-\nmark further that alignment of elongated particles can\nbe achieved using a quadrupole Paul trap even in the\nabsence of magnetic \feld [55, 71].\nC. Linear stability\nIn the absence of thermal \ructuations, an equilibrium\nsolution of Eqs. (1-5) is given by ~re=~pe=`e= 0 and\ne3;e=me=\u0000ex. This corresponds to the con\fguration\nin which the nanomagnet is \fxed at the trap center, with\nthe magnetic moment along the anisotropy axis and anti-\naligned to the bias \feld B0. Linear stability analysis of\nEqs. (1-5) shows that the system is unstable, as expected\nfor a gyroscopic system in the presence of dissipation [28].\nHowever, when the nanomagnet is metastable, it is still\npossible for it to levitate for an extended time before\nbeing eventually lost from the trap, as in the case of a\nclassical magnetic top [25{27]. As we show in the fol-\nlowing sections, the dynamics of the system, and thus its\nmetastability, strongly depend on the applied bias \feld\nB0. We identify two relevant regimes: (i) strong-\feld\nregime, de\fned by bias \feld values B0> B atom, and\n(ii) weak-\feld regime, de\fned by B0< B atom, where\nBatom\u00112kaV=\u0016. This di\u000berence is reminiscent of the\ntwo di\u000berent stable regions which arise as a function of\nB0in the linear stability diagram in the absence of dis-\nsipation [see Fig. 1(b)] [23, 24]. In Sec. III and Sec. IV\nwe investigate the possibility of metastable levitation by\nsolving numerically Eqs. (1-5) in the strong-\feld and\nweak-\feld regime, respectively.5\nIII. DYNAMICS IN THE STRONG-FIELD\nREGIME: ATOM PHASE\nThe strong-\feld regime, according to the de\fnition\ngiven in Sec. II C, corresponds to the blue region in the\nlinear stability diagram in the absence of dissipation,\nshown in Fig. 1(b). This region is named atom phase\nin [23, 24], and we hereafter refer to the strong-\feld\nregime as the atom phase. This parameter regime corre-\nsponds to the condition !L\u001d!A;!I. In this regime, the\ncoupling of the magnetic moment \u0016and the anisotropy\naxise3is negligible, and, to \frst approximation, the\nnanomagnet undergoes a free Larmor precession about\nthe local magnetic \feld. In the absence of dissipation,\nthis stabilizes the system in full analogy to magnetic trap-\nping of neutral atoms [72, 73].\nIn Fig. 2(a-c) we show the numerical solution of\nEqs. (1-5) for nanomagnet dimensions a= 2b= 20\nnm and the bias \feld B0= 200 mT. As evidenced by\nFig. 2(a), the magnetization mxof the particle changes\ndirection. During this change, the mechanical angu-\nlar momentum lxchanges accordingly in the manifesta-\ntion of the Einstein{de Haas e\u000bect, such that the to-\ntal angular momentum m+`is conserved4. The dy-\nnamics observed in Fig. 2(a) is indicative of Gilbert-\ndamping-induced magnetization switching, a well-known\nphenomenon in which the projection of the magnetic mo-\nment along the e\u000bective magnetic \feld !e\u000b=\r0changes\nsign [30]. This is expected to happen when the applied\nbias \feldB0is larger than the e\u000bective magnetic \feld\nassociated with the anisotropy, given by \u0018!A=\r0. Mag-\nnetization switching displaces the system from its equi-\nlibrium position on a timescale which is much shorter\nthan the period of center-of-mass oscillations, estimated\nfrom [24] to be \u001ccm\u00181\u0016s. The nanomagnet thus shows\nno signature of con\fnement [see Fig. 2(b)].\nThe timescale of levitation in the atom phase is given\nby the timescale of magnetization switching, which we\nestimate as follows. As evidenced by Fig. 2(a-b), the\ndynamics of the center of mass and the anisotropy axis\nare approximately constant during switching, such that\n!e\u000b\u0019!e\u000b(t= 0). Under this approximation and as-\nsuming\u0011\u001c1, the magnetic moment projection mk\u0011\n!e\u000b\u0001m=j!e\u000bjevolves as\n_mk\u0019\u0011[!L+ 2!Amk](1\u0000m2\nk): (7)\nAccording to Eq. (7) the component mkexhibits switch-\ning ifmk(t= 0)&\u00001 and!L=2!A>1 [30], both of\nwhich are ful\flled in the atom phase. Integrating Eq. (7)\nwe obtain the switching time \u001c[de\fned as mk(\u001c)\u00110],\nwhich can be well approximated by\n\u001c\u0019ln\u0000\n1 +jmk(t= 0)j\u0001\n2\u0011(!L+ 2!A)\u0000ln\u0000\n1\u0000jmk(t= 0)j\u0001\n2\u0011(!L\u00002!A):(8)\n4We always \fnd the transfer of angular momentum to the center\nof mass angular momentum r\u0002pto be negligible.\nFigure 2. Dynamics in the atom phase. (a) Dynamics of\nthe magnetic moment component mx, the mechanical angular\nmomentum component lx, and the anisotropy axis component\ne3;xfor nanomagnet dimensions a= 2b= 20 nm and the bias\n\feldB0= 200 mT. For the initial conditions we consider\ntrapping frequencies !x= 2\u0019\u00022 kHz and!y=!z= 2\u0019\u000250\nkHz. Unless otherwise stated, for the remaining parameters\nthe numerical values are given in Table I. (b) Center-of-mass\ndynamics for the same case considered in (a). (c) Dynamics of\nthe magnetic moment component mk. Line denoted by circle\ncorresponds to the case considered in (a). Each remaining\nline di\u000bers by a single parameter, as denoted by the legend.\nDotted vertical lines show Eq. (8). (d) Switching time given\nby Eq. (8) as a function of the bias \feld B0and the major\nsemi-axisa. In the region left of the thick dashed line the\ndeviation from the exact value is more than 5%. Hatched\narea is the unstable region in the linear stability diagram in\nFig. 1.(b).\nThe estimation Eq. (8) is in excellent agreement with\nthe numerical results for di\u000berent parameter values [see\nFig. 2(c)].\nMagnetization switching characterizes the dynamics of\nthe system in the entire atom phase. In particular, in\nFig. 2(d) we analyze the validity of Eq. (8) for di\u000berent\nvalues of the bias \feld B0and the major semi-axis a, as-\nsumingb=a=2. The thick dashed line shows the region\nwhere Eq. (8) di\u000bers from the exact switching time, as\nestimated from the full dynamics of the system, by 5%;\nleft of this line the deviation becomes increasingly more\nsigni\fcant, with Eq. (8) predicting up to 20% larger val-\nues close to the stability border (namely, for bias \feld\nclose toBatom = 90 mT). We believe that the signi\f-\ncant deviation close to the border of the atom phase is\ndue to the non-negligible coupling to the anisotropy axis,6\nFigure 3. Dynamics in the Einstein{de Haas phase. (a) Motion of the system in the ey-ezplane until time t= 5\u0016s for\nnanomagnet dimensions a= 2b= 2 nm and the bias \feld B0= 0:5 mT. For the initial conditions we consider trapping\nfrequencies !x= 2\u0019\u00022 kHz and !y=!z= 2\u0019\u00021 MHz. For the remaining parameters the numerical values are given\nin Table I. (b) Dynamics of the projection mkand (c) dynamics of the anisotropy axis component e3;x, for the same case\nconsidered in (a). (d) Dynamics of the center-of-mass component ryand (e) dynamics of the magnetic moment component\nmxon a longer timescale, for the same values of parameters as in (a). (f) Escape time t?as a function of gas pressure P, for\ndi\u000berent con\fgurations in the Einstein{de Haas phase. Circles correspond to the case considered in (a). Each remaining case\ndi\u000bers by parameters indicated by the legend. (g) Escape time t?as a function of the major semi-axis a, with the values of\nthe remaining parameters as in (a). Dashed vertical line denotes the upper limit of the Einstein{de Haas phase, given by the\ncritical \feld BEdH,1 [see Fig. 1(b)].\nwhich results in additional mechanisms not captured by\nthe simple model Eq. (7). In fact, it is known that cou-\npling between magnetization and mechanical degrees of\nfreedom might have an impact on the switching dynam-\nics [74]. As demonstrated by Fig. 2(d), the switching\ntime is always shorter than the center-of-mass oscillation\nperiod\u001ccm, and thus no metastability can be observed in\nthe atom phase.\nLet us note that the conclusions we draw in Fig. 2\nremain valid if one varies the anisotropy constant ka,\nGilbert damping parameter \u0011, and the temperature T,\nas we show in App. C. Finally, we note that the dis-\nsipation due to the background gas has negligible ef-\nfects. In particular, for the values assumed in Fig. 2(a-b)\nthe timescale of the gas-induced dissipation is given by\n1=\u0000 = 440\u0016s.\nIV. DYNAMICS IN THE WEAK-FIELD\nREGIME: EINSTEIN{DE HAAS PHASE\nWe now focus on the regime of weak bias \feld, cor-\nresponding to the condition !L\u001c!A. In this regime\nmagnetization switching does not occur, and the dynam-\nics critically depend on the particle size. In the follow-\ning we focus on the regime of small particle dimensions,i.e.!L\u001c!I, which, as we will show, is bene\fcial for\nmetastability. In the absence of dissipation, this regime\ncorresponds to the Einstein{de Haas phase [red region\nin Fig. 1(b)] [23, 24]. The hierarchy of energy scales in\nthe Einstein{de Haas phase (namely, !L\u001c!A;!I) man-\nifests in two ways: (i) the anisotropy is strong enough to\ne\u000bectively \\lock\" the direction of the magnetic moment \u0016\nalong the anisotropy axis e3(!A\u001d!L), and (ii) accord-\ning to the Einstein{de Haas e\u000bect, the frequency at which\nthe nanomagnet would rotate if \u0016switched direction is\nsigni\fcantly increased at small dimensions ( !I\u001d!L),\nsuch that switching can be prevented due to energy con-\nservation [4]. In the absence of dissipation, the combina-\ntion of these two e\u000bects stabilizes the system.\nIn Fig. 3(a-c) we show the numerical solution of\nEqs. (1-5) for nanomagnet dimensions a= 2b= 2 nm\nand the bias \feld B0= 0:5 mT. The nanomagnet is\nmetastable, as evidenced by the con\fned center-of-mass\nmotion shown in Fig. 3(a). In Fig. 3(b-c) we show the\ndynamics of the magnetic moment component mkand\nthe anisotropy axis component e3;x, respectively, which\nindicates that no magnetization switching occurs in this\nregime. We remark that the absence of switching can-\nnot be simply explained on the basis of Eqs. (7-8). In\nfact, the simple model of magnetization switching, given7\nby Eq. (7), assumes that the dynamics of the rotation\nand the center-of-mass motion happen on a much longer\ntimescale than the timescale of magnetization dynam-\nics. However, in this case rotation and magnetization\ndynamics occur on a comparable timescale, as evidenced\nby Fig. 3(b-c). The weak-\feld condition alone ( !L\u001c!A)\nis thus not su\u000ecient to correctly explain the absence of\nswitching, and the role of particle size ( !L\u001c!I) needs\nto be considered.\nLet us analyze the role of Gilbert damping in this case.\nSince in the Einstein{de Haas phase mk\u00181, we de\fne\nm\u0011e3+\u000em, where\u000emrepresents the deviation of m\nfrom the anisotropy axis e3, and we assumej\u000emj\u001cje3j\n[see Fig. 3(b)]. This allows us to simplify Eq. (5) as\n\u000e_m\u0019!e\u000b\u0002\u000em\u0000\u0011[2!A+!3e3\u0001(m+`)]\u000em;(9)\nwhere!3\u0011\u0016=(\r0I3), withI3the principal moment of\ninertia along e3. As evidenced by Eq. (9), the only e\u000bect\nof Gilbert damping is to align mande3on a timescale\ngiven by\u001c0\u00111=[\u0011(2!A+!3)], irrespective of the dy-\nnamics of e3. For the values of parameters considered in\nFig. 3(a-c), \u001c0= 5 ns, and it is much shorter than the\ntimescale of center-of-mass dynamics, given by \u001ccm\u00181\n\u0016s. For all practical purposes, the magnetization in the\nEinstein{de Haas phase can be considered frozen along\nthe anisotropy axis. The nanomagnet in the presence of\nGilbert damping is therefore equivalent to a hard magnet\n(i. e.ka!1 ) [24].\nThe main mechanism behind the instability in the\nEinstein{de Haas phase is thus gas-induced dissipation.\nIn Fig. 3(d-e) we plot the dynamics of the center-of-\nmass component ryand the magnetic moment compo-\nnentmxon a longer timescale, for two di\u000berent values of\nthe pressure P. The e\u000bect of gas-induced dissipation is\nto dampen the center-of-mass motion to the equilibrium\nposition, while the magnetic moment moves away from\nthe equilibrium. Both processes happen on a timescale\ngiven by the dissipation rate \u0000. When ex=mx\u00190, the\nsystem becomes unstable and ultimately leaves the trap\n[see arrow in Fig. 3(d)]. We de\fne the escape time t?as\nthe time at which the particle position is y(t?)\u00115y(0),\nand we show it in Fig. 3(f) as a function of pressure Pfor\ndi\u000berent con\fgurations in the Einstein{de Haas phase,\nand forb=a=2. Fig. 3(f) con\frms that the dissipation\na\u000bects the system on a timescale which scales as \u00181=P.\nThe metastability of the nanomagnet in the Einstein{de\nHaas phase is therefore limited solely by the gas-induced\ndissipation, which can be signi\fcantly reduced in high\nvacuum. Finally, in Fig. 3(g) we analyze the e\u000bect of\nparticle size on metastability. Speci\fcally, we show the\nescape time t?as a function of the major semi-axis aat\nthe bias \feld B0= 0:5 mT, forb=a=2. The escape time\nis signi\fcantly reduced at increased particle sizes. This\ncon\frms the advantage of the Einstein{de Haas phase to\nobserve metastability, even in the presence of dissipation.V. DISCUSSION\nIn deriving the results discussed in the preceding sec-\ntions, we assumed (i) a single-magnetic-domain nanopar-\nticle with uniaxial anisotropy and constant magnetiza-\ntion, with the values of the physical parameters summa-\nrized in Table I, (ii) deterministic dynamics, i. e. the\nabsence of thermal \ructuations, (iii) that gravity can be\nneglected, and (iv) a non-rotating nanomagnet. Let us\njustify the validity of these assumptions.\nWe \frst discuss the values of the parameters given in\nTable I, which are used in our analysis. The material pa-\nrameters, such as \u001aM,\u001a\u0016,kaand\u0011, are consistent with,\nfor example, cobalt [75{78]. We remark that the uniax-\nial anisotropy considered in our model represents a good\ndescription even for materials which do not have an in-\ntrinsic magnetocrystalline uniaxial anisotropy, provided\nthat they have a dominant contribution from the uniaxial\nshape anisotropy. This is the case, for example, for fer-\nromagnetic particles with a prolate shape [75]. We point\nout that the values used here do not correspond to a spe-\nci\fc material, but instead they describe a general order\nof magnitude corresponding to common magnetic materi-\nals. Indeed, our results are general and can be particular-\nized to speci\fc materials by replacing the above generic\nvalues with exact numbers. As we show in App. C, the re-\nsults and conclusions presented here remain unchanged\neven when di\u000berent values of the parameters are con-\nsidered. The values used for the \feld gradient B0and\nthe curvature B00have been obtained in magnetic mi-\ncrotraps [62, 79{82]. The values of the gas pressure P\nand the temperature Tare experimentally feasible, with\nnumerous recent experiments reaching pressure values as\nlow asP= 10\u00006mbar [58, 68, 70, 83{85]. All the values\nassumed in our analysis are therefore consistent with cur-\nrently available technologies in levitated optomechanics.\nThermal \ructuations can be neglected at cryogenic\nconditions (as we argue in Sec. II A), as their e\u000bect is\nweak enough not to destroy the deterministic e\u000bects cap-\ntured by Eqs. (1-5). In particular, thermal activation of\nthe magnetization, as quanti\fed by the N\u0013 eel relaxation\ntime, can be safely neglected due to the large value of\nthe uniaxial anisotropy even for the smallest particles\nconsidered. As for the mechanical thermal \ructuations,\nwe con\frm that they do not modify the deterministic\ndynamics in App. B, where we simulate the associated\nstochastic dynamics.\nGravity, assumed to be along ex, can be safely ne-\nglected, since the gravity-induced displacement of the\ntrap center from the origin is much smaller than the\nlength scale over which the Io\u000be-Pritchard \feld signi\f-\ncantly changes [24]. Speci\fcally, the gravitational poten-\ntialMgx shifts the trap center from the origin r= 0\nalong exby an amount rg\u0011Mg= (\u0016B00), wheregis\nthe gravitational acceleration. On the other hand, the\ncharacteristic length scales of the Io\u000be-Pritchard \feld\nare given by \u0001 r0\u0011p\nB0=B00for the variation along\nex, and \u0001r0\u0011B0=B00for the variation o\u000b-axis. When-8\neverrg\u001c\u0001r0;\u0001r0, gravity has a negligible role in the\nmetastable dynamics of the system. In the parameter\nregime considered in this article, this is always the case.\nWe note that the condition to neglect gravity is the same\nas for a magnetically trapped atom, since both Mand\u0016\nscale with the volume.\nFinally, we remark that the analysis presented here\nis carried out for the case of a non-rotating nanomag-\nnet5. The same qualitative behavior is obtained even in\nthe presence of mechanical rotation (namely, considering\na more general equilibrium con\fguration with `e6= 0).\nThe analysis of dynamics in the presence of rotation is\nprovided in App. C. In particular, the dynamics in the\nEinstein{de Haas phase remains largely una\u000bected, pro-\nvided that the total angular momentum of the system is\nnot zero. In the atom phase, mechanical rotation leads to\ndi\u000berences in the switching time \u001c, as generally expected\nin the presence of magneto-mechanical coupling [74, 88].\nVI. CONCLUSION\nIn conclusion, we analyzed how the stability of a nano-\nmagnet levitated in a static magnetic \feld is a\u000bected by\nthe most relevant sources of dissipation. We \fnd that in\nthe strong-\feld regime (atom phase) the system is un-\nstable due to the Gilbert-damping-induced magnetiza-\ntion switching, which occurs on a much faster timescale\nthan the center-of-mass oscillations, thereby preventing\nthe observation of levitation. On the other hand, the sys-\ntem is metastable in the weak-\feld regime and for small\nparticle dimensions (Einstein{de Haas phase). In this\nregime, the con\fnement of the nanomagnet in a mag-\nnetic trap is limited only by the gas-induced dissipation.\nOur results suggest that the timescale of stable levitation\ncan reach and even exceed several hundreds of periods of\ncenter-of-mass oscillations in high vacuum. These \fnd-\nings indicate the possibility of observing the phenomenon\nof quantum spin stabilized magnetic levitation, which we\nhope will encourage further experimental research.\nThe analysis presented in this article is relevant for\nthe community of levitated magnetic systems. Speci\f-\ncally, we give precise conditions for the observation of\nthe phenomenon of quantum spin stabilized levitation\nunder experimentally feasible conditions. Levitating a\nmagnet in a time-independent gradient trap represents a\nnew direction in the currently growing \feld of magnetic\nlevitation of micro- and nanoparticles, which is interest-\ning for two reasons. First, the experimental observation\nof stable magnetic levitation of a non-rotating nanomag-\nnet would represent a direct observation of the quantum\nnature of magnetization. Second, the observation of such\n5Rotational cooling might be needed to unambiguously identify\nthe internal spin as the source of stabilization. Subkelvin cooling\nof a nanorotor has been recently achieved [86, 87], and cooling\nto\u0016K temperatures should be possible [56].phenomenon would be a step towards controlling and us-\ning the rich physics of magnetically levitated nanomag-\nnets, with applications in magnetometry and in tests of\nfundamental forces [9, 11, 34, 35].\nACKNOWLEDGMENTS\nWe thank G. E. W. Bauer, J. J. Garc\u0013 \u0010a-Ripoll, O.\nRomero-Isart, and B. A. Stickler for helpful discussions.\nWe are grateful to O. Romero-Isart, B. A. Stickler and\nS. Viola Kusminskiy for comments on an early ver-\nsion of the manuscript. C.C.R. acknowledges funding\nfrom ERC Advanced Grant QENOCOBA under the EU\nHorizon 2020 program (Grant Agreement No. 742102).\nV.W. acknowledges funding from the Max Planck So-\nciety and from the Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation) through Project-\nID 429529648-TRR 306 QuCoLiMa (\"Quantum Cooper-\nativity of Light and Matter\"). A.E.R.L. thanks the AMS\nfor the \fnancial support.\nAppendix A: Rotation to the body frame\nIn this appendix we de\fne the transformation ma-\ntrix between the body-\fxed and the laboratory reference\nframes according to the ZYZ Euler angle convention,\nwith the Euler angles denoted as \n= (\u000b;\f;\r )T. We\nde\fne the transformation between the laboratory frame\nOexeyezand the body frame Oe1e2e3as follows,\n0\n@e1\ne2\ne31\nA=R(\n)0\n@ex\ney\nez1\nA; (A1)\nwhere\nR(\n)\u0011Rz(\u000b)Ry(\f)Rz(\r) =0\n@cos\rsin\r0\n\u0000sin\rcos\r0\n0 0 11\nA\n0\n@cos\f0\u0000sin\f\n0 1 0\n\u0000sin\f0 cos\f1\nA0\n@cos\u000bsin\u000b0\n\u0000sin\u000bcos\u000b0\n0 0 11\nA:(A2)\nAccordingly, the components vj(j= 1;2;3) of a vector\nvin the body frame Oe1e2e3and the components v\u0017\n(\u0017=x;y;z ) of the same vector in the laboratory frame\nOexeyezare related as\n0\n@v1\nv2\nv31\nA=RT(\n)0\n@vx\nvy\nvz1\nA: (A3)\nThe angular velocity of a rotating particle !can be writ-\nten in terms of the Euler angles as != _\u000bez+_\fe0\ny+ _\re3,\nwhere ( e0\nx;e0\ny;e0\nz)T=Rz(\u000b)(ex;ey;ez)Tdenotes the\nframeOe0\nxe0\nye0\nzobtained after the \frst rotation of the9\nlaboratory frame Oexeyezin the ZYZ convention. By\nusing (A1) and (A2), we can rewrite angular velocity in\nterms of the body frame coordinates,\n!= _\u000b2\n4R(\n)\u000010\n@e1\ne2\ne31\nA3\n5\n3+_\f2\n4R(\r)\u000010\n@e1\ne2\ne31\nA3\n5\n2+ _\re3;\n(A4)\nwhich is compactly written as ( !1;!2;!3)T=A(\n)_\n,\nwith\nA(\n) =0\n@\u0000cos\rsin\fsin\r0\nsin\fsin\rcos\r0\ncos\f 0 11\nA: (A5)\nAppendix B: Dynamics in the presence of thermal\n\ructuations\nIn this appendix we consider the dynamics of a lev-\nitated nanomagnet in the presence of stochastic forces\nand torques induced by the surrounding gas. The dy-\nnamics of the system are described by the following set\nof stochastic di\u000berential equations (SDE),\nd~r=!I~pdt; (B1)\nde3=!\u0002e3dt; (B2)\nd~p= [!Lr~r[m\u0001b(~r)]\u0000\u0000cm~p] dt+p\nDcmdWp;(B3)\nd`= [!Lm\u0002b(~r)\u0000_m\u0000\u0000rot`] dt+p\nDrotdWl;\n(B4)\ndm=m\n1 +\u00112\u0002[!e\u000b\u0000\u0011m\u0002(!+!e\u000b+\u0011!\u0002m)]dt;\n(B5)\nwhere we model the thermal \ructuations as uncorrelated\nGaussian noise represented by a six-dimensional vector\nof independent Wiener increments (d Wp;dWl)T. The\ncorresponding di\u000busion rate is described by the tensors\nDcmandDrotwhich, in agreement with the \ructuation-\ndissipation theorem, are related to the corresponding dis-\nsipation tensors \u0000 cmand \u0000 rotasDcm\u00112\u0000cm\u001f;D rot\u0011\n2\u0000rot\u001f, where\u001f\u0011MkBT\r2\n0a2=\u00162.\nIn the following we numerically integrate Eqs. (B1-B5)\nusing the stochastic Euler method implemented in the\nstochastic di\u000berential equations package in MATLAB. As\nthe e\u000bect of thermal noise is more prominent for small\nparticles at weak \felds, we focus on the Einstein-de Haas\nregime considered in Sec. IV. We show that even in this\ncase the e\u000bect of thermal \ructuations leads to dynamics\nwhich are qualitatively very close to the results obtained\nin Sec. IV. In Fig. 4 we present the results of the stochas-\ntic integrator by averaging the solution of 100 di\u000berent\ntrajectories calculated using the same parameters consid-\nered in Fig. 3(a-c). The resulting average dynamics agree\nqualitatively with the results obtained by integrating the\ncorresponding set of deterministic equations Eqs. (1-5)\nFigure 4. Stochastic dynamics of a nanomagnet for the same\nparameter regime as considered in Fig. 3. (a) Average motion\nof the system in the y-zplane until time t= 5\u0016s. (b) Dy-\nnamics of center of mass along the ey(top) and ez(bottom)\ndirections. (c) Dynamics of the anisotropy axis component\ne3;x. (d) Numerical error as function of time. The simulations\nshow the results of the average of 100 di\u000berent realizations of\nthe system dynamics. In panels (b-d) the solid dark lines are\nthe average trajectories, while the shaded area represents the\nstandard deviation.\n[cfr. Fig. 3(a-c)]. The main e\u000bect of thermal excitations\nis to shift the center of oscillations of the particle's de-\ngrees of freedom around the value given by the thermal\n\ructuations. This is more evident for the dynamics of\ne3[cfr. Fig. 4(c) and Fig. 3(c)]. We thus conclude that\nthe deterministic equations Eqs. (1-5) considered in the\nmain text correctly capture the metastable behavior of\nthe system. We emphasize that the results presented in\nthis section include only the noise due to the surround-\ning gas. Should one be interested in simulating the ef-\nfect of the \ructuations of the magnetic moment, the Eu-\nler method used here is not appropriate, and the Heun\nmethod should be used instead [89].\nLet us conclude with a technical note on the numerical\nsimulations. In the presence of dissipation and thermal\n\ructuations the only conserved quantity of the system is\nthe magnitude of the magnetic moment ( jmj= 1). We\nthus use the deviation 1 \u0000jmj2as a measure of the numer-\nical error in both the stochastic and deterministic sim-\nulations presented in this article. For the deterministic\nsimulations the error stays much smaller than any other\nphysical degree of freedom of the system during the whole\nsimulation time. The simulation of the stochastic dynam-\nics shows a larger numerical error [see Fig. 4(d)], which\ncan be partially reduced by taking a smaller time-step\nsize. We note that, for the value of magnetic anisotropy\ngiven in Table I, the system of SDE is sti\u000b. This, together\nwith the requirement imposed on the time-step size by10\nthe numerical error, ultimately limits the maximum time\nwe can simulate to a few microseconds. However, this is\nsu\u000ecient to validate the agreement between the SDE and\nthe deterministic simulations presented in the article.\nAppendix C: Additional \fgures\nIn this appendix we provide additional \fgures.\n1. Dynamics in the atom phase\nIn Fig. 5 we analyze magnetization dynamics in the\natom phase as a function of di\u000berent system parame-\nters. In Fig. 5(a) we show how magnetization switching\nchanges as the anisotropy constant kais varied. We con-\nsider the bias \feld B0= 1100 mT, which is larger than\nthe value considered in the main text. This is done to en-\nsure thatB0>B atom for all anisotropy values. Fig. 5(a)\ndemonstrates that the switching time \u001c, given by Eq. (8),\nis an excellent approximation for the dynamics across a\nwide range of values for the anisotropy constant ka. The\nlarger discrepancy between Eq. (8) and the line showing\nthe case with ka= 106J/m3is explained by the prox-\nimity of this point to the unstable region (in this case\ngiven by the critical \feld Batom = 900 mT), and better\nagreement is recovered at larger bias \feld values.\nIn Fig. 5(b) we analyze the validity of Eq. (8) for dif-\nferent values of the Gilbert damping parameter \u0011and the\ntemperature T. The thick dashed line shows the region\nwhere Eq. (8) di\u000bers from the exact switching time by\n5%; below this line the deviation becomes increasingly\nmore signi\fcant. As evidenced by Fig. 5(b), \u001cshows lit-\ntle dependence on T; its order of magnitude remains con-\nstant over a wide range of cryogenic temperatures. On\nthe other hand, the dependence on \u0011is more pronounced.\nIn fact, reducing the Gilbert parameter signi\fcantly de-\nlays the switching time, leading to levitation times as\nlong as\u00181\u0016s.\nAdditionally, we point out that \u001cdepends on the \feldgradientB0and curvature B00through the initial con-\nditionmk(t= 0). In particular, magnetization switch-\ning can be delayed by decreasing B0, as this reduces\nthe initial misalignment of the magnetization and the\nanisotropy axis (i. e. jmk(t= 0)j!1).\n2. Dynamics in the presence of rotation\nIn Fig. 6 we consider a more general equilibrium con-\n\fguration, namely a nanomagnet initially rotating such\nthat in the equilibrium point Le=\u0000I3!Sex, with!S>0\ndenoting the rotation in the clockwise direction. This\nequilibrium point is linearly stable in the absence of dis-\nsipation [23, 24], with additional stability of the system\nprovided by the mechanical rotation, analogously to the\nclassical magnetic top [25{27].\nIn Fig. 6(a) we analyze how magnetization switching\nin the atom phase changes in the presence of rotation for\ndi\u000berent values of parameters. The rotation has a slight\ne\u000bect on the switching time \u001c, shifting it forwards (back-\nwards) in case of a clockwise (counterclockwise) rotation.\nThis is generally expected in the presence of magneto-\nmechanical coupling [74, 88].\nIn Fig. 6(b) we show the motion in the y-zplane in\nthe Einstein{de Haas phase for both directions of rota-\ntion. This can be compared with Fig. 3(a). The rotation\ndoes not qualitatively a\u000bect the dynamics of the system.\nThe di\u000berence in the two trajectories can be explained\nby a di\u000berent total angular momentum in the two cases,\nas in the case of a clockwise (counterclockwise) rotation\nthe mechanical and the internal angular momentum are\nparallel (anti-parallel), such that the total angular mo-\nmentum is increased (decreased) compared to the non-\nrotating case. This asymmetry arises from the linear\nstability of a rotating nanomagnet, and it is not a conse-\nquence of dissipation. In fact, we con\frm by numerical\nsimulations that the escape time t?as a function of the\npressurePshows no dependence on the mechanical ro-\ntation!S. Namely, even in the presence of mechanical\nrotation one recovers the same plot as shown in Fig. 3(f).\n[1] A. Einstein and W. J. de Haas, Experimental proof of\nthe existence of Amp\u0012 ere's molecular currents, Proc. K.\nNed. Akad. Wet. 18, 696 (1915).\n[2] O. W. 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In the region\nbelow the thick dashed line the deviation from the exact\nvalue is more than 5%.\nFigure 6. Dynamics of a nanomagnet initially rotat-\ning around the axis exwith frequencyj!Sj=(2\u0019) = 100\nMHz. (a) Magnetization switching in the atom phase.\nLine denoted by circle corresponds to the same set of pa-\nrameters as in Fig. 2(a). Each remaining line di\u000bers by a\nsingle parameter, as denoted by the legend. Dotted verti-\ncal lines show Eq. (8). (b) Motion in the y-zplane in the\nEinstein{de Haas phase, using the same numerical values\nof the parameters as in Fig. 3. Left panel: Clockwise\nrotation. Right panel: counterclockwise rotation.\n[11] D. F. Jackson Kimball, A. O. Sushkov, and D. Budker,\nPrecessing ferromagnetic needle magnetometer, Phys.\nRev. Lett. 116, 190801 (2016).\n[12] P. Kumar and M. Bhattacharya, Magnetometry via spin-\nmechanical coupling in levitated optomechanics, Opt.\nExpress 25, 19568 (2017).\n[13] Y. B. Band, Y. Avishai, and A. 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Mech. , P09008\n(2014)." }, { "title": "0902.3779v1.Noise_and_dissipation_in_magnetoelectronic_nanostructures.pdf", "content": "arXiv:0902.3779v1 [cond-mat.mes-hall] 22 Feb 2009Noise and dissipation in magnetoelectronic nanostructure s\nJørn Foros and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, 7491 Trondheim, Norway\nGerrit E. W. Bauer\nKavli Institute of NanoScience, Delft University of Techno logy, 2628 CJ Delft, The Netherlands\nYaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n(Dated: November 10, 2018)\nWe study the coupled current and magnetization noise in magn etic nanostructures by magneto-\nelectronic circuit theory. Spin current fluctuations, whic h depend on the magnetic configuration,\nare found to be an important source of magnetization noise an d damping in thinly layered systems.\nThe enhanced magnetization fluctuations in spin valves can b e directly measured by their effect on\nthe resistance noise.\nPACS numbers: 72.70.+m, 72.25.Mk, 75.75.+a\nI. INTRODUCTION\nNew functionalities can be realized by integrating fer-\nromagnetic elements into electronic circuits and devices.\nThe interplay between magnetism and electric currents\nin these structures is utilized by the giant magnetoresis-\ntance(GMR),theoperatingprincipleofthereadheadsin\nmodernmagneticharddiskdrives. Considerableprogress\nhas been made in improving magnetic random access\nmemories.1Efforts to further miniaturize and improve\nthe performance of magnetoelectronic devices are ongo-\ning in academic and corporate laboratories. Low power\nconsumption and noise levels areessential. In spite ofthe\ntechnological relevance, a comprehensive understanding\nof coupled current and magnetization noise and the re-\nlated energy dissipation in nanoscale magnetoelectronic\ncircuits is lacking.\nFrom the early studies of Johnson2and Nyquist,3we\nknow that the equilibrium voltage noise power in con-\nductors is proportional to the electric resistance. This\nrelation between the equilibrium noise and the out-of-\nequilibrium energy dissipation is a standard example of\nthe fluctuation-dissipation theorem (FDT).4,5In recent\nyears, important advances have been made in the un-\nderstanding of electronic equilibrium (thermal) and non-\nequilibrium (shot) noise in mesoscopic conductors.6\nThe electron spin plays an important role in elec-\ntrical noise phenomena in magnetic multilayers. In\nearly theoretical studies7,8,9,10,11,12of charge and spin-\npolarized current noise in such systems, magnetizations\nwere assumed to be static. However, the magnetiza-\ntion itself fluctuates as well. Thermal fluctuations of\nthe magnetization vector in isolated single-domain fer-\nromagnets have been analyzed by Brown,13who intro-\nduced a stochastic Langevin field acting on the magne-\ntization to account for thermal agitation. His proof that\nthis field’s (white-noise) correlator is proportional to the\nmagnetization damping (see below) is another manifes-\ntation of the FDT.14,15The stochastic field can be in-troduced into the spatiotemporal equation of motion for\nthe magnetization (Landau-Lifshitz-Gilbert equation),\naffecting, e.g., current-driven magnetization dynamics\nand reversal.16,17,18,19\nA moving magnetization vector in ferromagnets un-\ndergoes viscous damping that relaxes the magnetization\ntoward the lowest (free-)energy configuration. This pro-\ncess is in practice well described by a phenomenological\ndamping constant, introduced by Gilbert.20,21Despite\nsome progress,22,23,24,25,26,27a rigorous quantitative un-\nderstanding of the magnetic damping in transition-metal\nferromagnets has not yet been achieved. The theory of\nthe enhanced Gilbert damping in ferromagnets in good\nelectrical contact with a conducting environment is in a\nbetter shape. The loss of angular momentum due to spin\ncurrent pumping into the environment agrees with the\nGilbert phenomenology,28,29and experiment and theory\naddressing the additional damping agree well with each\nother.29\nThe electronic and magnetic fluctuations in magne-\ntoelectronic structures are intimately coupled to each\nother.30,31For example, the magnetization noise in fer-\nromagnetic films in good electric contact with normal\nmetals has been predicted to increase due to spin cur-\nrent fluctuations: The spin current components polar-\nized perpendicularly to the magnetization are absorbed\nat the interface, leading to a fluctuating spin-transfer\ntorque32,33,34,35that induces additional magnetization\nnoise. This noise is related to the excess Gilbert damp-\ning caused by the angular momentum loss due to spin\npumping, in accordance with the FDT.\nHere we investigate the interplay between (zero fre-\nquency) current and magnetization noise in multilayers\nof alternating magnetic and non-magnetic films. We\ntake advantage of the FDT to relate the equilibrium\nelectric (current and voltage) and magnetic (magneti-\nzation and field) noise to the corresponding dissipation\nof energy. We start by reviewing the noise in a single\nmonodomain ferromagnet sandwiched by normal met-2\nals, including technical details that were omitted in Ref.\n30. Both thermal equilibrium (Johnson-Nyquist) current\nnoise and nonequilibrium shot noise are taken into ac-\ncount. Next, we consider spin valves, i.e., two ferromag-\nnetic films separated by a normal metal spacer.36We\nconsider both a symmetric structure in which both lay-\ners fluctuate, as well as an asymmetric one, in which\none layer is assumed fixed. Magnetoelectronic circuit\ntheory37,38,39is used to calculate the charge and spin\ncurrent fluctuations. The resulting enhanced magnetiza-\ntion noise and Gilbert damping in principle are tensors\nthat depend on the magnetic configuration.\nSpin valves provide an opportunity to indirectly mea-\nsure magnetization noise via resistance fluctuations,\nwhicharemanifestedbyvoltagenoiseforacurrent-biased\nsystem or current noise for a voltage-biased system.40,41\nThis offers an experimental test of our theory. We ob-\ntain analytical expressions for the magnetic contribution\nto the induced electric noise for different magnetic con-\nfigurations. The noise is of potential importance for the\nperformance of spin valve read heads.41For symmetric\nstructures in which both layers fluctuate, dynamic cross\ntalk between the layers becomes important, causing a\npossibly large difference in noise level between the paral-\nlel and antiparallel magnetic configurations. Our results\nforthesespinvalvesincludepreviouslypresentedfindings\nas a limiting case.36After the completion of this work,42\nit was shown that spin-valves in equilibrium also exhibit\ncolored voltage fluctuations caused by spin pumping of\nthe moving magnetizations.43\nThe paperis organizedas follows. We begin by review-\ning the fluctuation-dissipation theorem, applied to mag-\nnetic systems. In sectionIII, the noisepropertiesofa sin-\ngle ferromagnetic thin film sandwiched by normal metals\nis worked out in detail, emphasizing the relation of the\nnoise to the damping. In section IV, we consider current\nnoise, magnetization noise, and magnetization damping\nin spin valves, and use the results to calculate the resis-\ntance noise induced by GMR. In section V we summarize\nour conclusions.\nII. FLUCTUATION-DISSIPATION THEOREM\nThe fluctuation-dissipation theorem (FDT) relates the\nspontaneous time-dependent changes of an observable of\na given system in thermal equilibrium its linear response\nto an external perturbation that couples to that observ-\nable. For example, in an electric conductor the sponta-\nneous fluctuations in the electric currentareproportional\nto the dissipative (real) part of the conductivity, i.e., the\nresponsefunction to anapplied electricfield.2,3Similarly,\nthe equilibrium fluctuations of the magnetization vector\nin a ferromagnet are proportional to the dissipative part\nof the magnetic susceptibility, i.e., imaginary part of the\nresponse function to an applied magnetic field. In the\nfollowing, we briefly recapitulate this FDT for magnetic\nsystems.Sufficiently below the Curie temperature, changes in\nthe modulus ofthe magnetizationare energeticallycostly\nand may be disregarded. For sufficiently small magnetic\nstructures spin waves freeze out of the problem. Hence,\na small ferromagnetic particle or thin film is well de-\nscribed in terms of a single magnetization vector Msm,\nwhereMsis the magnitude of the magnetization and m\na unit vector (“macrospin” model). The time-dependent\nequilibrium fluctuations of the magnetization are charac-\nterized by the autocorrelation function /an}bracketle{tδmi(t)δmj(t′)/an}bracketri}ht,\nwhereδmi(t) =mi(t)−/an}bracketle{tmi(t)/an}bracketri}htare transverse fluctua-\ntions. Here the brackets denote statistical averaging at\nequilibrium, and iandjdenote Cartesian components\nperpendicular to the equilibrium/average magnetization\ndirection. The classical FDT states that these fluctua-\ntions are related to the magnetic susceptibility:\n/an}bracketle{tδmi(t)δmj(t′)/an}bracketri}ht=kBT\n2πMsV/integraldisplay\ndωe−iω(t−t′)\n×χij(ω)−χ∗\nji(ω)\niω, (1)\nwhereTis the temperature, Vthe volume of the ferro-\nmagnet, and χij(ω) theij-component of the transverse\nmagnetic susceptibility at frequency ω. The latter is\nthe linear (causal) response function that describes the\nchanges of the magnetization, ∆ mi(t), caused by an ex-\nternal driving field H(dr)(t):\n∆mi(t) =/summationdisplay\nj/integraldisplay\ndt′χij(t−t′)H(dr)\nj(t′).(2)\nAn alternative form of the FDT that turns out useful in\nthe course of this paper can be derived by introducing\na stochastic magnetic field h(0)(t) with zero mean. This\nfield effectively represents the coupling of the magnetiza-\ntion to the dissipative degrees of freedom, and is viewed\nas the cause of the thermal fluctuations δm(t). The mi-\ncroscopicoriginof h(0)(t) doesnot concernus here, but it\nmight, e.g., represent thermally excited phonons that de-\nform the crystal anisotropyfields. From Eq. (2) it follows\nthatδmi(ω) =/summationtext\njχij(ω)h(0)\nj(ω) in frequency domain.\nInverting this relation, the correlator of the stochastic\nfield has to obey the relation\n/an}bracketle{th(0)\ni(t)h(0)\nj(t′)/an}bracketri}ht=kBT\n2πMsV/integraldisplay\ndωe−iω(t−t′)\n×[χ−1\nji(ω)]∗−χ−1\nij(ω)\niω,(3)\nwhereχ−1\nij(ω) is theij-component of the Fourier trans-\nformed inverse susceptibility.\nIII. SINGLE FERROMAGNET\nThe magnetization dynamics of an isolated single-\ndomain ferromagnet is well described by the Landau-3\nLifshitz-Gilbert (LLG) equation20,44\ndm\ndt=−γ0m×Heff+α0m×dm\ndt,(4)\nwhereγ0is the gyromagnetic ratio, Heffthe effective\nmagnetic field, and α0the Gilbert damping constant.\nThe effective field has contributions due to crystal and\nform anisotropies, as well as externally applied magnetic\nfields. By linearizing this LLG equation we can evaluate\nthe magnetic susceptibility and the equilibrium magne-\ntization noise. The average equilibrium direction of the\nmagnetization is aligned with Heffto minimize the en-\nergy:m0=Heff/|Heff|. A weak external driving field\nis included by substituting Heff→Heff+H(dr)(t).In\nthe present model only the component of H(dr)trans-\nverse to the magnetization will solicit a response m(t)≈\nm0+ ∆m(t) of the magnetization. Here ∆ m(t) is nor-\nmal tom0. To lowest order in ∆ m(t), the LLG equation\ngives the inverse susceptibility tensor matrix\nχ−1=1\nγ0/bracketleftbigg\nγ0|Heff|−iωα0 iω\n−iω γ 0|Heff|−iωα0/bracketrightbigg\n(5)\nin the plane normal to m0. A dependence of the effective\nfieldHeffonmdoes not affect the noise properties.\nThe magnetization noise follows from substituting\nEq. (5) into Eq. (1). The correlatorof the stochastic field\nis obtained from Eqs. (3) and (5) and does not depend\non the effective field:13\n/an}bracketle{th(0)\ni(t)h(0)\nj(t′)/an}bracketri}ht= 2kBTα0\nγ0MsVδijδ(t−t′).(6)\nThe relation between the equilibrium magnetization fluc-\ntuations and the dissipation in the form of the Gilbert\ndamping is evident.\nUp to now we considered a ferromagnet isolated from\nthe outside world. Its dynamics is altered by embed-\nding into a conducting environment.28A ferromagnet\nwithtime-dependentmagnetization“pumps”anangular-\nmomentum (spin) current\nIpump\ns=/planckover2pi1\n4π/parenleftbigg\nReg↑↓m×dm\ndt+Img↑↓dm\ndt/parenrightbigg\n,(7)\ninto an adjacent conductor. Here g↑↓is the dimension-\nless transverse spin (“spin mixing” ) conductance that\ndepends on the interface transparencybetween ferromag-\nnet and proximate metal.37,38,39When the spin current\nis efficiently dissipated in the conductor, thus does not\nbuild up a spin accumulation close to the interface, the\nlossofangularmomentumcorrespondstoanextratorque\nγIpump\ns/(MsV) on the right hand side of the Eq. (4).\nThis is equivalent to an increased Gilbert damping and\na modified gyromagnetic ratio:28\n1\nγ0→1\nγ=1\nγ0/parenleftbigg\n1−γ0/planckover2pi1Img↑↓\n4πMsV/parenrightbigg\n, (8)\nα0→α=γ\nγ0/parenleftbigg\nα0+γ0/planckover2pi1Reg↑↓\n4πMsV/parenrightbigg\n. (9)In the strong coupling limit (intermetallic interfaces),\nImg↑↓≪Reg↑↓and we are allowed to disregard the dif-\nference between γandγ0.\nAnother term which modifies the magnetization dy-\nnamics is the so-called spin-transfer torque.32,33,34,35\nIt is also proportional to the spin-mixing conduc-\ntance introduced above37,38,39and represented by adding\n−γ0Is,abs/(MsV) to the right hand side of Eq. (4). Here\nIs,absis the spin-polarized current transversely polarized\nto the magnetization, which is absorbed by the ferromag-\nnet on an atomic length scale, thereby transferring its\nangular momentum to the magnetization. Spin pump-\ning and spin-transfer torque are related by an Onsager\nreciprocity relation.45\nRecentlywehaveshown30thatthemagnetizationnoise\nin magnetoelectronic nanostructures can be considerably\nincreased as compared to an isolated ferromagnet. At\nelevated temperatures, thermal fluctuations in the spin\ncurrent exert a fluctuating torque on the magnetization,\nincreasing the noise. For a ferromagnet sandwiched by\nnormal metals, the enhancement of the noise is described\nby a stochastic field h(th)(t) similar to the intrinsic field\nh(0)(t). Its correlation function reads30\n/an}bracketle{th(th)\ni(t)h(th)\nj(t′)/an}bracketri}ht= 2kBTα′\nγMsVδijδ(t−t′),(10)\nwhere\nα′=γ/planckover2pi1Reg↑↓\n4πMsV(11)\nis the enhancement of the Gilbert damping due to spin\npumping (see Eq. (9)). Assuming that h(0)(t) and\nh(th)(t) are statistically independent, the total magne-\ntization noise is thus given by h(t) =h(0)(t) +h(th)(t).\nWe know that the total damping is determined by α=\nα0+α′, and from Eqs. (6) and (10) we see that the total\nnoise is related to the total damping, in agreement with\nthe FDT. Hence, the thermal spin current noise is the\nstochastic process related to the enhanced dissipation of\nenergy by spin pumping. By calculating the noise power\nwe also know the damping and vice versa. In thin ferro-\nmagnetic films, α′can be of the same orderor even larger\nthanα0.29In the following subsections we will give a de-\ntailed derivation of Eq. (10). We also evaluate the shot\nnoise contribution to the magnetization noise, which is\nimportant at low temperatures.30We note here that Eq.\n(10) may be found also by direct application of Eq. (3)\nto the LLG equation with spin-pumping included.\nA. Scattering theory\nWe study a thin ferromagnetic film connected to two\nnormal reservoirs, as shown in Fig. 1. The reservoirs\nare perfect spin sinks and the ferromagnet is taken to\nbe thicker than the magnetic coherence length λc=4\naL\nFbLbR\naRN Nt\nt’r r’\nFIG. 1: A thin ferromagnetic (F) film is sandwiched by nor-\nmal metals (N). The current fluctuations in the system are\nevaluated in terms of transmission probabilities for the el ec-\ntron states, with the aid of second quantized annihilation\nand creation operators. The operators shown in the figure\nare annihilation operators, with the a-operators annihilat-\ning electrons moving towards the ferromagnet, and the b-\noperators annihilating electrons moving away from the fer-\nromagnet. Also shown are the reflection and transmission\nmatrices r,r′,t,t′(see Eq. (14)), for simplicity without spin\nindices.\nπ/(k↑−k↓), where k↑(↓)are spin-dependent Fermi mo-\nmenta. For transition metals, λcis of the order of mono-\nlayers. The normal metals are characterized by Fermi-\nDirac distribution functions fLandfRwith chemical\npotentials µLandµR, whereLandRrefer to the left\nand right sides at a common temperature T. We use the\nLandauer-B¨ uttiker (LB) scattering theory6to evaluate\nthe spin current fluctuations, and the LLG equation to\ncalculate the resulting magnetization noise.\nIn the LB approach electron transport is expressed in\nterms of transmission probabilities between the electron\nstates on different sides of a scattering region. Here we\ninterpret the ferromagnetic film as a scatterer that limits\nthe propagation of electrons between the normal reser-\nvoirs. The scattering properties of the ferromagnet and\nthe bias between the reservoirs determine the transport\nproperties of the system. The transport channels in the\nleads are modelled as ideal electron wave guides in which\nthe transverse and longitudinal motions are separable.\nThe transport channels at a given energy Eare then la-\nbeled by the discrete mode index for the quantized trans-\nversemotion,bywhichthecontinuouswavevectorforthe\nlongitudinal motion is fixed. The LB formalism6,46gen-\neralized to describe spin transport leads to the current\noperator\nˆIαβ\nA(t) =e\nh/integraldisplay\ndEdE′ei(E−E′)t//planckover2pi1\n×[a†\nAβ(E)aAα(E′)−b†\nAβ(E)bAα(E′)].(12)\nat timeton sideA[=L(left) or R(right)] of the fer-\nromagnetic film. Here, αandβdenote components in\n2×2 spin space. aAα(E) andbAα(E) are operators for\nall transport channels at energy Ethat annihilate elec-\ntrons with spin αin lead A that move towards and away\nfrom the ferromagnet, respectively (see Fig. 1). The a-operators are related to the b-operators by the scattering\nproperties of the ferromagnet:\nbAα(E) =/summationdisplay\nBβsABαβ(E)aBβ(E), (13)\nwheresABαβis the scattering matrix for incoming elec-\ntrons with spin βin leadB(=LorR) scattered to\noutgoing states in lead Awith spin α. The summation is\noverB=L,Rand over spin β=↑,↓. A similar relation\nholds for the creation operators. Current conservation\nimplies that the scattering matrix is unitary. Suppress-\ning spin indices for simplicity (see Fig. 1)\n/parenleftbigg\nbL\nbR/parenrightbigg\n=/parenleftbigg\nr t′\nt r′/parenrightbigg/parenleftbigg\naL\naR/parenrightbigg\n, (14)\nwherer=sLL,r′=sRR,t=sRLandt′=sLR. In\nthe following we disregard spin-flip processes in the fer-\nromagnet. Choosing the spin quantization z-axis in the\ndirection of the average magnetization, this implies that\nsABαβ=sABαδαβ.\nThe outgoing charge and spin currents are given\nrespectively by Ic,A(t) =/summationtext\nαˆIαα\nA(t) andIs,A(t) =\n−(/planckover2pi1/2e)/summationtext\nαβˆσαβˆIβα\nA(t), where ˆσ= (ˆσx,ˆσy,ˆσz) is\nthe vector of Pauli matrices. The expectation val-\nues for charge and spin currents are evaluated using\nthe quantum statistical average /an}bracketle{ta†\nAmα(E)aBnβ(E′)/an}bracketri}ht=\nδABδmnδαβδ(E−E′)fA(E) ofthe productofonecreation\nand one annihilation operator, where mandnlabel the\ntransport channels. The creation and annihilation oper-\nators obey the anticommutation relation\n{a†\nAmα(E),aBnβ(E′)}=δABδmnδαβδ(E−E′),(15)\nwhereas the anticommutators of two creation or two an-\nnihilation operatorsvanish. Similar relationshold for the\nboperators. The average\n/an}bracketle{ta†\nAkα(E1)aBlβ(E2)a†\nCmγ(E3)aDnδ(E4)/an}bracketri}ht\n−/an}bracketle{ta†\nAkα(E1)aBlβ(E2)/an}bracketri}ht/an}bracketle{ta†\nCmγ(E3)aDnδ(E4)/an}bracketri}ht\n=δADδBCδknδlmδαδδβγδ(E1−E4)δ(E2−E3)\n×fA(E1)[1−fB(E2)], (16)\nwhere the subscripts A,B,C,D denote leads, k,l,m,n\ntransport channels, and α,β,γ,δ spin, is needed for the\ncalculation of the current fluctuations. We also need the\nidentity\n/summationdisplay\nCDTr(s†\nACαsADβs†\nBDβsBCα) =δABMA,(17)\nwhich follows from the unitarity of the scattering matrix.\nHerethe traceis overthe space ofthe transportchannels,\nandMAis the number of transverse channels in lead A,\nall at a given energy.\nThe charge and spin current correlation functions read\nSc,AB(t−t′) =/an}bracketle{tδIc,A(t)δIc,B(t′)/an}bracketri}ht(18)5\nand\nSij,AB(t−t′) =/an}bracketle{tδIsi,A(t)δIsj,B(t′)/an}bracketri}ht,(19)\nwhereδIc,A(t) =Ic,A(t)−/an}bracketle{tIc,A(t)/an}bracketri}htdenotes the deviation\nof the charge current from its average value in lead A\nat time t, andδIsi,A(t) is the deviation of the vector\ncomponent i(i=x,yorz) of the spin current. We\nare interested mainly in the low-frequency noise, i.e., the\ntime integrated value of the correlation functions:\nSc,AB(ω= 0) =/integraldisplay\nd(t−t′)Sc,AB(t−t′).(20)\nTwo fundamentally different types of current noise have\nto be distinguished: Thermal (equilibrium) noise and\n(non-equilibrium) shot noise. In general, the total noise\nis not simply a linear combination of both types. Nev-\nertheless, it is convenient to treat the two noise sources\nindependently, by separatelyinvestigatingthe noise of an\nunbiased system at finite temperatures in Sec. IIIB and\nthe shot noise under an applied bias at zero temperature\nin Sec. IIIC.\nB. Thermal current noise\nAt equilibrium fL=fR=f, and the average current\nvanishes. However, at finite temperatures, the occupa-\ntion numbers of the electron channels incident on the\nsample fluctuate in time and so does the current. Using\nEqs. (12), (13), (16), (17) and f(1−f) =kBT(−∂f/∂E),\nwe recover the well-known Johnson-Nyquist noise\nS(th)\nc,AA(ω= 0) =2e2\nhkBT(g↑+g↓) (21)\nin the zero-frequency limit. Here gα= Tr(1−r†\nαrα),\nwhere the trace indicates again a summation over trans-\nport channels, is the spin-dependent dimensionless con-\nductanceoftheferromagnet,tobeevaluatedattheFermi\nenergy. The superscript (th) emphasizes that the fluctu-\nations are caused by thermal agitation. The result for\nS(th)\nc,AB(ω= 0), where B/ne}ationslash=A, differs from the above ex-\npression only by a minus sign, since current direction is\ndefined positive towards the ferromagnet on both sides,\nand charge current is conserved. The Johnson-Nyquist\nnoise, Eq. (21), is a manifestation of the FDT, since it\nrelates the equilibrium current noise to the dissipation of\nenergy prarameterized by the conductance.\nThe thermal spin current noise can be obtained in a\nsimilar way. At zero frequency\nS(th)\nij,AB(0) =/planckover2pi1kBT\n8π/summationdisplay\nαβσαβ\niσβα\nj\n×Tr[2δAB−s†\nBAαsBAβ−s†\nABβsABα],(22)\nwhere the scattering matrices should again be evaluated\nat the Fermi energy. The noise power of the zcompo-\nnent (polarized parallel to the magnetization) of the spincurrent\nS(th)\nzz,AA=/planckover2pi1\n4πkBT(g↑+g↓) (23)\ndiffers from the charge current noise only by the squared\nconversion factor, ( /planckover2pi1/2e)2, from charge to spin currents.\nThe transverse (polarized perpendicular to the magneti-\nzation) spin-current components fluctuate as\nS(th)\nxx,AA=S(th)\nyy,AA=/planckover2pi1\n4πkBT(g↑↓\nA+g↓↑\nA).(24)\nThe “spin mixing” conductances g↑↓\nL= Tr[1−r↑(r↓)†] =\n(g↓↑\nL)∗andg↑↓\nR= Tr[1−r′\n↑(r′\n↓)†] = (g↓↑\nR)∗parametrize\nthe absorbtivity of the ferromagnetic interfaces for\ntransverse-polarized spin currents. We see that also the\nspin-current noise obeys the FDT, since the spin-current\ncorrelators are proportional to the conductances for the\nrespective spin current components.\nThe cross correlation S(th)\nzz,LR=−S(th)\nzz,LLreflects con-\nservation of the longitudinal spin current in the ferro-\nmagnet, since spin-flip scattering is disregarded. On the\nother hand, S(th)\nxx,LR=S(th)\nyy,LR= 0, because the transverse\nspin current is absorbed at the interfaces to a ferromag-\nnet thicker than the magnetic coherence length.\nC. Shot noise\nShot noise of the electronic charge current is an out-\nof-equilibrium phenomenon proportional to the current\nbias. Shot noise is due to the discreteness of the electron\ncharge,and the probabilisticincidence ofelectronson the\nscatterer/resistor. Let µL−µR=eUwithUthe applied\nvoltage,andtakethetemperaturetobezero. Wearehere\nonly concerned with the current fluctuations, although in\nthis case also the average charge current is nonzero. The\naverage spin current accompanying the average charge\ncurrent does not exert a torque on a single ferromagnet,\nsince the spin current is polarized along the direction of\nmagnetization. From Eqs. (12), (13), (16), and making\nuse of the zero temperature relations fA(1−fA) = 0 and/integraltext\ndE(fL−fR)2=e|U|, we reproduce the well-known\ncharge shot noise expression6\nS(sh)\nc,AA(0) =e3\nh|U|[Tr(r†\n↑r↑t†\n↑t↑)+Tr(r†\n↓r↓t†\n↓t↓)] (25)\nAgain, the scattering matrices should be evaluated at the\nFermi energy, and the superscript (sh) emphasizes that\nthis is shot noise. S(sh)\nc,AB(0) =−S(sh)\nc,AA(0), where B/ne}ationslash=A.\nThe spin current shot noise power is\nS(sh)\nij,AB(0) =/planckover2pi1\n8π/summationdisplay\nαβˆσαβ\niˆσβα\nj/integraldisplay\ndE/summationdisplay\nCDfC(1−fD)\n×Tr[s†\nACαsADβs†\nBDβsBCα]. (26)\nFrom this we find, S(sh)\nzz,LR=−S(sh)\nzz,LLandS(th)\nxx,LR=\nS(th)\nyy,LR= 0, which hold for the same reasons as for the\nthermal noise.6\nD. Magnetization noise and damping\nThe absorption of fluctuating transverse spin currents\nat the ferromagnet’sinterfaces implies a fluctuating spin-\ntransfer torque on the magnetization. The resulting in-\ncrement of the magnetization noise can be calculated us-\ning Eq. (4), which by conservationofangularmomentum\nismodified bythespintorque −γ0Is,abs(t)/(MsV). Here\nIs,abs=Is,L+Is,Ris the (instantaneously) absorbed spin\ncurrent. (Recall that on both sides of the ferromagnet,\npositive currentdirection is defined towardsthe magnet.)\nSinceIs,absis perpendicular to m, we may in general\nwriteIs,abs=−m×[m×Is,abs], such that the modified\nstochastic LLG equation reads\ndm\ndt=−γ0m×[Heff+h(0)(t)]+α0m×dm\ndt\n+γ0\nMsVm×[m×Isabs]. (27)\nFor the single ferromagnetic scatterer /an}bracketle{tIs,abs/an}bracketri}ht= 0, but\nδIs,abs(t)/ne}ationslash= 0. We can thus define h(t) =−1/(MsV)m×\nδIs(t) to be a stochastic ”magnetic” field that takes into\naccount the (thermal or shot) spin current noise that\ncomes in addition to the intrinsic noise field h(0)(t). The\ncorrelators of the field\n/an}bracketle{thi(t)hi(t′)/an}bracketri}ht=1\nM2sV2/summationdisplay\nABSjj,AB(t−t′) (28)\nand\n/an}bracketle{thi(t)hj(t′)/an}bracketri}ht=−1\nM2sV2/summationdisplay\nABSji,AB(t−t′) (29)\nfori,j=x,y;i/ne}ationslash=jare directly obtained from the cur-\nrent noise. h(t) per definition has no component parallel\nto the magnetization. In the limit that the current noise\nis ‘white’ on the relevant energy scales (temperature, ap-\nplied voltage, and exchange splitting), we can approxi-\nmateSij,AB(t−t′)≈Sij,AB(ω= 0)δ(t−t′). Using Eq.\n(22) we then find the already advertised result\n/an}bracketle{th(th)\ni(t)h(th)\nj(t′)/an}bracketri}ht= 2kBTα′\nγ0MsVδijδ(t−t′),(30)\nfor the thermally (th) induced stochastic field. Here\nα′=γ0/planckover2pi1Re(g↑↓\nL+g↑↓\nR)/(4πMsV) is the spin-pumping\nenhancement of the Gilbert damping constant. This re-\nsult is in agreement with the FDT [Eq. (3)] with a total\nGilbert damping α=α0+α′.\nUsing Eq. (26) and the unitarity of the scattering ma-\ntrix we find for the stochastic field generated by the shot\nnoiseh(sh):\n/an}bracketle{th(sh)\ni(t)h(sh)\nj(t′)/an}bracketri}ht=/planckover2pi1\n4πe|U|\nM2sV2δijδ(t−t′)[Tr(r↑r†\n↑t′\n↓t′†\n↓)\n+Tr(r′\n↓r′†\n↓t↑t†\n↑)]. (31)F1\nIs,1R Is,1LL F2 R N\nIs,2L Is,2R\nm1 m2z\nxy\nFIG. 2: A spin valve with two ferromagnets F1andF2with\nunit magnetization vectors m1andm2, here shown in the\nparallel (P) configuration m1=m2=z. The magnetization\nofF2is fixed. The currents in the system are evaluated by\nmagnetoelectronic circuit theory on the normal side of the\ninterfaces, with positive directions defined by the arrows.\nFor a simple Stoner model it can be shown that for typi-\ncal experimental voltage drops in nanoscale metallic spin\nvalves,h(sh)can dominate h(th)at temperatures of the\norder of 10 K.30In the following section we concentrate\non room temperature, at which shot noise may be disre-\ngarded.\nIV. SPIN VALVES\nWe now proceed to consider the noise properties of\nspin valve nanopillars, i.e., layered structures consisting\nof two ferromagnets F1andF2with respective unit mag-\nnetization vectors m1andm2that are separated by a\nthin normal metal spacer N, as sketched in Fig. 2. We\nfirst assume that F2is highly coercive, such that the fluc-\ntuations of it’s magnetization vector are small. Such a\n‘pinning’ is routinely achieved in spin valves, e.g., by ‘ex-\nchange biasing.’ We relax this condition in Sec IVE.\nThe magnetization noise of the free layer F1is caused\nby intrinsic processes as well as by fluctuating spin cur-\nrents in the neighbouring normal metals. The latter\nsource is affected by the presence of the second ferro-\nmagnet. Magnetoelectronic circuit theory37,38,39enables\nus to compute the currentfluctuations and thus the mag-\nnetizations noise of composite structures such as spin\nvalves.\nFluctuations of m1cause an easily measurable elec-\ntrical noise, since the resistance of a spin valves depends\non the relative orientation of the magnetizations (GMR).\nResistance noise is also interesting from a technological\npoint of view, since it affects the sensitivity of spin valve\nread heads in magnetic storage devices.\nIn the following, we briefly explain the spin current\nnoise calculation by magnetoelectronic circuit theory.\nThe stochastic field that acts on the free layer F1and\nthe related Gilbert damping are found for different mag-\nnetic configurations. Using the LLG equation, we then\ncalculatethefluctuationsofthemagnetizationvectorand\nthe resulting resistance noise. We finish this section by\nconsidering spin valves in which both ferromagnets are\nidentically susceptible to fluctuations.7\nA. Circuit theory\nMagnetoelectronic circuit theory37,38,39is a tool to de-\ntermine transport properties of magnetoelectronic het-\nerostructures such as the spin valve shown in Fig. 2. It is\nbasedon the divisionofa givenstructureinto resistiveel-\nements (scatterers), nodes (low resistance interconnects),\nand reservoirs (voltage sources). The current through lo-\ncal resistors is calculated by LB scattering theory, which\nrequires that nodes and reservoirs are characterized by\n(semiclassical) distribution functions. Here we take the\nferromagnetic inserts as scatterers, the central normal\nmetal layer as a node, and the outer normal metals L\n(left) and R(right) as large reservoirs. The reservoirs\nare in thermal equilibrium, and hence characterized by\nFermi-Dirac distribution functions fL=f(E−µL) and\nfR=f(E−µR), where µLandµRare the respective\nchemical potentials. Depending on the relative orienta-\ntion of the magnetization vectors m1andm2, there can\nbe a non-equilibrium accumulation of spins on the nor-\nmal metal node, thus characterized by a scalar (charge)\ndistribution function fcN, and a vector spin distribution\nfunction fsN.fcNandfsNform the distribution matrix\nˆfN=ˆ1fcN+ˆσ·fsNin 2×2 spin space. As before, the\nferromagnets are thicker than λcbut thin enough such\nthat spin-flip processes can be disregarded. We also as-\nsume that spin-flip in the central normal metal node is\nnegligible. We are in the diffuse scattering regime, so ˆfN\nis isotropic and constant in space.\nReferring back to Eq. (12), we need now quantum sta-\ntistical averages /an}bracketle{ta†\nAmα(E)aBnβ(E′)/an}bracketri}ht=δABδmnδ(E−\nE′)fβα\nA(E), where aBnβis the annihilation operator for\nelectrons moving in normal metal A(A=L,RorN)\ntowards one of the ferromagnets, and fβα\nAis theβα-\ncomponent of the 2 ×2 semiclassical distribution matrix\nˆfAinspinspace. Forthereservoirs( A=LorR), wesim-\nply have fβα\nA=δβαf(E−µA). In contrast, in the central\nnode the spin accumulation is not necessarily parallel to\nthe spin quantization axis in either of the ferromagnets,\nmeaning that non-diagonal ( β/ne}ationslash=α) terms in the distri-\nbution matrix do not vanish. The averagecharge current\nflowing from the right into ferromagnet F1can then be\nexpressed by the generalized LB expressions37,39\n/an}bracketle{tIc,1R/an}bracketri}ht=e\nh/integraldisplay\ndE/bracketleftBig\ng↑\n1(fcN+fsN·m1−fL)\n+g↓\n1(fcN−fsN·m1−fL)/bracketrightBig\n,(32)\nwhereas the average spin current reads\n/an}bracketle{tIs,1R/an}bracketri}ht=1\n4π/integraldisplay\ndE/braceleftBig\nm1/bracketleftBig\ng↑\n1(fcN+fsN·m1−fL)\n−g↓\n1(fcN−fsN·m1−fL)/bracketrightBig\n+2Reg↑↓\n1Rm1×(fsN×m1)\n+2Img↑↓\n1RfsN×m1/bracerightBig\n. (33)\nHeregα\n1is the spin-dependent dimensionlessconductanceofF1andg↑↓\n1Ris the mixing conductance of the inter-\nface between F1and the middle normal metal. The\naverage charge current and the component of the spin\ncurrent polarized along the magnetization are conserved\nthrough the ferromagnet. Hence /an}bracketle{tIc,1L/an}bracketri}ht=−/an}bracketle{tIc,1R/an}bracketri}htand\n/an}bracketle{tIs,1L/an}bracketri}ht·m1=−/an}bracketle{tIs,1R/an}bracketri}ht·m1. The transverse spin current\nis absorbed in the ferromagnet, leading to\n/an}bracketle{tIs,1L/an}bracketri}ht=1\n4π/integraldisplay\ndEm1/bracketleftBig\ng↑\n1(fL−fcN−fsN·m1)\n−g↓\n1(fL−fcN+fsN·m1)/bracketrightBig\n.(34)\nSimilarexpressionshold forthe currentsevaluatedon the\nleft and rightsides of F2. In ordertokeep the expressions\nsimple we adopt from now on the parameters gα\n1=gα\n2=\ngα, andg↑↓\n1L=g↑↓\n1R=g↑↓\n2L=g↑↓\n2R=g↑↓.\nSince spin-flip processes are disregarded, both charge\nand spin are conservedon the middle normal metal node:\n/an}bracketle{tIc,1R/an}bracketri}ht+/an}bracketle{tIc,2L/an}bracketri}ht= 0 (35)\n/an}bracketle{tIs,1R/an}bracketri}ht+/an}bracketle{tIs,2L/an}bracketri}ht= 0 (36)\nEqs. (32)-(36) come down to four equations for the four\nunknown components of the distribution matrix ˆfNas a\nfunction of the angle θ= cos−1(m1·m2) and the applied\nvoltageU= (µL−µR)/e.Eq. (32) then yields /an}bracketle{tIc,1L/an}bracketri}ht=\n−/an}bracketle{tIc,1R/an}bracketri}ht=/an}bracketle{tIc,2L/an}bracketri}ht=−/an}bracketle{tIc,2R/an}bracketri}ht≡Ic=GvU,where37\nGv=e2g\n2h/parenleftbigg\n1−P21−cosθ\n1−cosθ+η+ηcosθ/parenrightbigg\n(37)\nis the spin valve conductance with material parameters\ng=g↑+g↓,P= (g↑−g↓)/g, andη= 2g↑↓/g.\nB. Current noise\nWe combine spin and charge current fluctuations, e.g.,\n∆Ic,1R(t) and ∆Is,1R(t), respectively, on the right side of\nF1,into a 2×2 matrix in spin space:\n∆ˆI1R(t) =ˆ1∆Ic,1R(t)−(2e//planckover2pi1)ˆσ·∆Is,1R(t).(38)\nSince we focus on the zerofrequency noise, instantaneous\ncharge and spin conservation in the central node may be\nassumed, i.e.\n∆ˆI1R(t)+∆ˆI2L(t) = 0, (39)\nwhich requires that the distribution matrix in the node\nfluctuates. The current fluctuations can then be written\n∆ˆI1R(2L)(t) =δˆI1R(2L)(t)+∂/an}bracketle{tˆI1R(2L)/an}bracketri}ht\n∂ˆfNδˆfN(t),(40)\nwhereδˆfN(t) are the fluctuations of the distribution ma-\ntrix, and δˆI1R(2L)(t) are the intrinsic fluctuations [when\nδˆfN(t) = 0], coinciding with the fluctuations calculated8\nfor single ferromagnets in the previous section. Expres-\nsion (40) applies also to the current fluctuations evalu-\nated on the left side of ferromagnet F1and the right side\nof ferromagnet F2. In the following, we focus on thermal\ncurrent noise, recalling from Sec. IIID that for typical\nvoltage drops in spin valves, shot noise is only important\nat low temperatures.\nFrom Eqs. (32), (33), (39) and (40) and results from\nSec. III, we can evaluate the charge and spin current\nfluctuations in the spin valve. The correlator Sc(0) =/integraltext\nd(t−t′)/an}bracketle{t∆Ic(t)∆Ic(t′)/an}bracketri}htof the charge current fluctua-\ntions is simply related to the conductance (37) by the\nfollowing configuration-dependent FDT:\nSc(ω= 0;θ) = 2kBTGv(θ). (41)\nIn the low-frequency regime considered here, charge cur-\nrent noise is the same anywhere in the spin valve. Gv\ncan vary easily by a factor of two as a function of θ,\nwhich corresponds to the same variation in noise power.\nResistance noise via magnetization fluctuations is an ad-\nditional source of electric noise that is treated below\nThe spin current correlator /an}bracketle{t∆Isi,A(t)∆Isj,B(t′)/an}bracketri}ht,\nwhereiandjdenote Cartesian components and A(B) =\n1L,1R,2Lor 2R, can be found analogously. Since spin\ncurrent is not conserved at the ferromagnetic interfaces,\nthe spin current correlator depends on the location in\nthe spin valve and is not directly observable. We there-\nfore proceed to evaluate the magnetization fluctuations\ncaused by the spin current noise in the next subsection.\nC. Magnetization noise and damping\nThe current-induced stochastic field acting on F1fol-\nlows from the spin current fluctuations as explained in\nSec. IIID. Here we discuss this field and, by using the\nFDT, the corresponding Gilbert damping enhancement\nin spin valves. In order to keep the algebra manageable,\nwe focus on the most relevant parallel, antiparallel and\nperpendicular configurations (cos θ= 0,±1). The mix-\ning conductances are taken to be identical for all four\nF|N-interfaces. In the semiclassical approach, intrinsic\ncurrent fluctuations are not correlated across the node,\nimplying that/an}bracketle{tδIsi,1R(t)δIsj,2L(t′)/an}bracketri}ht= 0.\n1. Parallel configuration\nFor the parallel (P) magnetic configuration, m1·m2=\n1, the thermal spin current-induced stochastic magnetic\nfield in ferromagnet F1reads\n/an}bracketle{th(th)\ni(t)h(th)\nj(t′)/an}bracketri}htP= 2kBTαsv\nγ0MsVδijδ(t−t′) (42)\nwherei,jlabel vector components perpendicular to the\nmagnetization, and\nαsv=3γ0/planckover2pi1Reg↑↓\n8πMsV. (43)By the FDT, αsvis identical to the spin-pumping en-\nhancement of the Gilbert damping of the F1magnetiza-\ntion. This can be checked by following the steps outlined\nfor a single ferromagnet, Eqs. (3)-(5). A possible ex-\nchange coupling between the ferromagnets modifies the\ndynamics via Heffin the LLG equation, but does not\naffect the stochastic field and Gilbert damping.\nThe field correlator and damping for the parallel con-\nfiguration is reduced by a factor 3 /4 compared with (30)\nfor the single ferromagnet sandwiched by normal met-\nals. This result may be found also in a more direct way:\nUsing Eqs. (7) and (33) we can compute the net spin an-\ngular momentum leaving each of the ferromagnets when\nthemagnetizationsareslightlyoutofequilibrium, andby\nconservation of angular momentum infer the correspond-\ning enhancement of the Gilbert damping constant. The\nfactor 3/4 follows from the diffuse/chaotic nature of the\nnode: Half of the spin current that is pumped into the\nnode is reflected back and reabsorbed by F1.\nOne subtle point needs to be noted in this discussion:\nWhen the F-N interfaces are nearly transparent, the in-\nterfacial conductance parameters from scattering theory\nshould be corrected for spurious so-called Sharvin con-\nductances (See Sec. II.B. of Ref. 29). In practice, this\nwill correct (43) only by a numerical prefactor close to\none.\n2. Antiparallel configuration\nFortheantiparallel(AP)configuration( m1·m2=−1),\n/an}bracketle{th(th)\ni(t)h(th)\nj(t′)/an}bracketri}htAP=/an}bracketle{th(th)\ni(t)h(th)\nj(t′)/an}bracketri}htP,(44)\ni.e., thecurrent-inducednoiseanddampingisthesameas\nin the P configuration. This result holds only when the\nimaginary part of the mixing conductance is negligibly\nsmall.\n3. Perpendicular configuration\nWhen the F2magnetization is pinned along the x-\ndirection and m1points along the z-axis\n/an}bracketle{th(th)\nx(t)h(th)\nx(t′)/an}bracketri}ht⊥= 2kBTα′\nxx\nγ0MsVδ(t−t′),(45)\n/an}bracketle{th(th)\ny(t)h(th)\ny(t′)/an}bracketri}ht⊥= 2kBTα′\nyy\nγ0MsVδ(t−t′),(46)\nwhere the subscript ⊥emphasizes that this is valid for\nthe perpendicular configuration, and, according to the\nFDT,\nα′\nxx=3γ0/planckover2pi1Reg↑↓\n8πMsV,\nα′\nyy=γ0/planckover2pi1Reg↑↓\n4πMsV/bracketleftbigg\n2−η(2−P2+2η)\n2(1+η)(1−P2+η)/bracketrightbigg\n(47)9\nis the spin pumping-induced enhancement of the Gilbert\ndamping. The cross correlators /an}bracketle{th(th)\nx(t)h(th)\ny(t′)/an}bracketri}ht⊥=\n/an}bracketle{th(th)\ny(t)h(th)\nx(t′)/an}bracketri}ht⊥= 0. In non-collinear spin valves, the\nnoise correlators and the Gilbert damping are therefore\ntensors. This can be accommodated by the LLG equa-\ntion form1by a damping torque m1×← →α dm1/dt, where\nthe Gilbert damping tensor (in the plane perpendicular\nto the magnetization) reads:\n← →α=/parenleftbigg\nα0+α′\nxx0\n0α0+α′\nyy/parenrightbigg\n. (48)\nNote that the damping tensor must be written inside the\ncross product in the damping torque to ensure that the\nLLG equation preserves the length of the unit magneti-\nzation vector.\nIn our evaluation of the Gilbert damping (47), we have\nassumed that the outer left and right reservoirs have a\nfixed chemical potential which allows chargecurrent fluc-\ntuations into the reservoirs. This is valid when the reser-\nvoirsareconnectedtoexternalcircuitelementswithsuffi-\nciently long RC-times compared to the FMR precession\nperiod. In the opposite limit, when the reservoirs are\nfully decoupled from other circuit elements, charge cur-\nrent into the reservoirs must vanish at any time, and the\nchemical potentials fluctuate. This regime was consid-\nered in Ref. 47 with the result\nα′\nxx=3γ0/planckover2pi1Reg↑↓\n8πMsV,\nα′\nyy=γ0/planckover2pi1Reg↑↓\n4πMsV/bracketleftbigg\n2−η\n1−P2+η/bracketrightbigg\n(49)\nD. Resistance noise\nThe fluctuations of the magnetization vector can be\ncalculated by the LLG equation that incorporates the\nstochastic fields. Fluctuations in the magnetic config-\nuration affect the electrical resistance that depends on\nthe dot product m1·m2. Resistance noise is an im-\nportant issue for application of spin-valve read heads.41\nCovington et al.40measured resistance noise in current-\nperpendicular-to-the-plane (CPP) spin valves, which are\nconsidered an alternative for the conventional current-\nin-the-plane spin valve read heads. We focus here on the\nzero-frequency resistance noise\nSR(ω= 0) =/integraldisplay\nd(t−t′)/an}bracketle{t∆R(t)∆R(t′)/an}bracketri}ht,(50)\nwhere ∆ R(t) is the time dependent deviation of the re-\nsistance from the time-averaged value.\nResistance noise can be measured e.g. as voltage noise\nforconstantcurrentbiasorascurrentnoiseforaconstant\nvoltage bias. The resistance noise comes on top of the\nJohnson-Nyquist noise discussed in Sec IVB and in Ref.\n43. We find that at relatively high current densities, themagnetization-induced noise can be the dominant contri-\nbution to the electric noise. The current densities con-\nsidered are not so high that shot noise dominates over\nJohnson-Nyquist noise, consistent with our assumption\nthat shot noise may be neglected.\nIn the following, we derive the resistance noise in the\nparallel, antiparallel and perpendicular configurations.\nRecall that the magnetization in ferromagnet F2is as-\nsumed pinned. The analysis of resistance noise in the\ncase of two fluctuating magnetizations is left for the next\nsection.\n1. Parallel configuration\nThe total stochastic field in F1causes fluctuations\nδm1(t) =m1(t)−/an}bracketle{tm1/an}bracketri}htrelative to its time-averaged\nequilibrium value /an}bracketle{tm1/an}bracketri}ht. For the parallel configuration\n/an}bracketle{tm1/an}bracketri}ht=m2, such that the dot product of the magne-\ntizations is cos θ=m1·m2= 1−δm2\n1/2, withθthe\nangle between the magnetization directions. For small\nfluctuations we can expand the resistance to first order\ninδm2\n1\nR(m1·m2)≈R(1)−1\n2δm2\n1∂R(1)\n∂cosθ,(51)\nsuch that the resistance noise correlator becomes\n/an}bracketle{t∆R(t)∆R(t′)/an}bracketri}htP=/an}bracketle{tR(t)R(t′)/an}bracketri}htP−/an}bracketle{tR(t)/an}bracketri}htP/an}bracketle{tR(t′)/an}bracketri}htP\n=1\n4/parenleftbigg∂R(1)\n∂cosθ/parenrightbigg2/bracketleftbig\n/an}bracketle{tδm2\n1(t)δm2\n1(t′)/an}bracketri}htP\n−/an}bracketle{tδm2\n1(t)/an}bracketri}htP/an}bracketle{tδm2\n1(t′)/an}bracketri}htP/bracketrightbig\n,(52)\nwhere the brackets denote statistical averaging around\nthe parallel configuration. Assuming that the stochastic\nfields are Gaussian distributed, so are the fluctuations of\nthe magnetization vectors, since the magnetization is a\nlinear function of the stochastic fields. We may then em-\nploy Wick’s theorem,48according to which fourth order\nmoments of the fluctuations can be expressed in terms of\nthe sum of products of second order moments. We then\narrive at\n/an}bracketle{t∆R(t)∆R(t′)/an}bracketri}htP=1\n2/parenleftbigg∂R(1)\n∂cosθ/parenrightbigg2\n×/summationdisplay\nij/an}bracketle{tδm1,i(t)δm1,j(t′)/an}bracketri}ht2\nP,(53)\nwhereiandjdenote Cartesian components. From\nEq. (37) we find\n∂R(1)\n∂cosθ=−hP2\ne2gη. (54)\nSince the magnetization fluctuations are small, we may\ndisregard their longitudinal component, whereas the cor-\nrelatorof the transversefluctuations can be computed by\nthe LLG equation.10\nWe use the coordinate system in Fig. 2 with interfaces\nin thexz-plane. The LLG equation reads\ndm1\ndt=−γ0m1×[Heff+h(t)]\n+(α0+αsv)m1×dm1\ndt, (55)\nwherethetotalstochasticfield h(t) =h(0)(t)+h(th)(t)in-\ncludes both the intrinsic field h(0)(t) (see section III) and\nthe current induced field h(th)(t) from the previous sec-\ntion.α0andαsvare the corresponding Gilbert damping\nparameters. Theeffective field Heff=H0+Ha+Hd+He\ncontains the external field H0, the in-plane anisotropy\nfieldHa, the out-of-plane demagnetizing field Hd, and\nthe sum of dipolar and exchange fields He. The external\nand anisotropyfields areboth taken alongthe z-axis. We\nparametrize these fields by ω0andωaasγH0=ω0zand\nγHa=ωa(m1·z)z. The demagnetizing field is directed\nnormal to the plane, i.e. along the y-axis, such that\nγHd=−ωd(m1·y)ythereby introducing the parameter\nωd. The dipolar and exchange couplings are described\nin terms of a Heisenberg coupling −Jm1·m2, which fa-\nvors a parallel magnetic configuration for J >0 and an\nantiparallel one for J <0. This translates into the field\nγHe=ωem2, whereωe=γJ/Msd.\nIn the P configuration /an}bracketle{tm1/an}bracketri}htis aligned with the pinned\nm2in the +zdirection, which can always be enforced by\na sufficiently strong external field. Linearizing the LLG\nequation in the amplitude of the transverse fluctuations\nδm(t)≈��mx(t)x+δmy(t)y, we find the magnetization\nnoise correlator\n/an}bracketle{tδmi(t)δmj(t′)/an}bracketri}htP=γ0kBTα\nπMsV/integraldisplay\ndωe−iω(t−t′)Uij,(56)\nby using the correlators of the stochastic fields. Here\nUxx=[ω2+(ωt+ωd)2]\n[ω2−ωt(ωt+ωd)]2+ω2α2(2ωt+ωd)2,(57)\nUxy=−iω(2ωt+ωd)\n[ω2−ωt(ωt+ωd)]2+ω2α2(2ωt+ωd)2,(58)\nUyy=(ω2+ω2\nt)\n[ω2−ωt(ωt+ωd)]2+ω2α2(2ωt+ωd)2,(59)\nUyx=−Uxy, (60)\nwithα=α0+αsvandωt=ω0+ωa+ωe. The above\nexpressions hold for small damping, i.e.,α2\n0,α2\nsv≪1.\nThe zero-frequency resistance noise SP(0) =/integraltext\nd(t−\nt′)/an}bracketle{t∆R(t)∆R(t′)/an}bracketri}htPis obtained by inserting Eq. (56) into\nEq. (53):\nSP(0) =1\nπ/parenleftbigghP2\ne2gη/parenrightbigg2/parenleftbiggγ0kBTα\nMsV/parenrightbigg2\n×/integraldisplay\ndω(U2\nxx+U2\nyy−2U2\nxy).(61)\nTo gaininsight into this rathercomplicated expression, it\nis convenientto make some simplifications. Although thedemagnetizing field, which serves to stabilize the magne-\ntization in the plane of the film, is important to get the\nright magnitude of the noise, we can gain physical un-\nderstanding by disregarding it. Setting ωd= 0, we find\nSP(0) =/parenleftbiggγ0kBT\nMsV/parenrightbigg2/parenleftbigghP2\ne2gη/parenrightbigg21\nω3\ntα.(62)\nObviously, the resistance noise strongly depends on the\nparameter ωt. The external and anisotropy fields sta-\nbilize the magnetization, hence lowering the noise. The\ndipolar and exchangefield either stabilizes or destabilizes\nthe magnetization, depending on the sign of the coupling\nconstant J. We observe that the Gilbert damping also\nstrongly affects the resistance noise. The resistance noise\ndecreases with increasing damping, because the suppres-\nsion of the magnetic susceptibility by a large alpha turns\nout to be more important than the FDT-motivated in-\ncrease of the stochastic field noise. Since αsvcan be of\nthe same order as α029, the importance of spin current\nnoise and spin pumping is evident.\nWhen a constant voltage bias is applied, the resis-\ntance noise causes current noise. At sufficiently small\nbias, the Johnson-Nyquist current noise (Sec. IVB) al-\nways wins. However, at relatively high current densi-\nties, the effects of the resistance noise are very signifi-\ncant. That noise may be important for the next gen-\neration magnetoresistive spin valve read heads.41For a\nquantitative comparison, which depends on many mate-\nrial parameters, it is important to use Eq. (61) and not\nEq. (62), since the demagnetizing field has a large effect\non the magnitude of the magnetization-induced noise.\nThe magnetization-induced noise is most prominent for\nsmall structures, since the ratio ofJohnson-Nyquistnoise\nto magnetization-induced noise scales with the volume of\nthe ferromagnet.\n2. Antiparallel configuration\nWhenJ <0, the dipolar and exchange coupling fa-\nvors an AP configuration ( /an}bracketle{tm1/an}bracketri}ht=−m2) at zero exter-\nnal magnetic field. Following the recipe of the previous\nsubsection, we find a resistance noise\n/an}bracketle{t∆R(t)∆R(t′)/an}bracketri}htAP=1\n2/parenleftbigg∂R(−1)\n∂cosθ/parenrightbigg2\n×/summationdisplay\nij/an}bracketle{tδm1,i(t)δm1,j(t′)/an}bracketri}ht2\nAP,(63)\nwhere the sensitivity of the resistance to the fluctuations\nis\n∂R(−1)\n∂cosθ=−hP2η\ne2g(1−P2)2. (64)\nUsing the magnetization noise correlators from the lin-\nearized Eq. (55), the zero frequency resistance noise be-11\ncomes\nSAP(0) =/integraldisplay\nd(t−t′)/an}bracketle{t∆R(t)∆R(t′)/an}bracketri}htAP\n=1\nπ/parenleftbigghP2η\ne2g(1−P2)2/parenrightbigg2/parenleftbiggγ0kBTα\nMsV/parenrightbigg2\n×/integraldisplay\ndω(V2\nxx+V2\nyy−2V2\nxy), (65)\nwhereVij=Uij(ωt→ωs) withωs=ωa−ωe(recall\nthatωe<0). Again disregardingthe demagnetizing field\nstrongly simplifies the expression:\nSAP(0) =/parenleftbiggγ0kBT\nMsV/parenrightbigg2/parenleftbigghP2η\ne2g(1−P2)2/parenrightbigg21\nω3sα.(66)\nAs expected, the resistance noise decreases with increas-\ningωs. The anisotropy, dipolar and exchange fields sta-\nbilizes the magnetization, playing a role similar to that\nof the external field in the P configuration. The Gilbert\ndamping enters in the same way as for the P configura-\ntion.\nExcept for the prefactor that reflects the sensitivity of\nthe resistance to the magnetization fluctuations, SP(0)\nandSAP(0)areverysimilar. Forthespecialcase ωt=ωs,\nSP\nSAP=(1−P2)4\nη4. (67)\nFor, e.g., P= 0.7andη= 1, this becomes SP/SAP≈6%\nshowing that the difference in noise level between the P\nand AP configurations can be substantial.\nThis asymmetry in the noise level between the P and\nAP configurations is consistent with the the experimen-\ntal results of Covington et al. on nearly cylindrical mul-\ntilayer pillars.49In these experiments the magnetizationswere aligned parallel when the external magnetic field\nreached about 1500 Oe. Although we treat spin valves\nwith two ferromagnetic films and Covington et al. dealt\nwith multilayers of 4-15 magnetic films, it is likely that\nthedifferencebetweenthenoisepropertiesofbilayersand\nmultilayersis small, astheonlylocalstructuraldifference\nis the number of neighboring ferromagnets. This asser-\ntion is supported by the experiments by Covington et\nal. that did not reveal strong differences for nanopillars\nranging from 4-15 layers.\n3. Perpendicular configuration\nWenowinvestigatetheperpendicularstate /an}bracketle{tm1/an}bracketri}ht·m2=\n0, assuming that m2now has been pinned in the x-\ndirection, whereas m1is on averageparallel to the zaxis,\nas before. In the following we assume that the interlayer\nexchange and dipolar coupling are negligibly small, since\notherwise the algebra and expressions become awkward.\nExpanding the resistance to first order in the fluctua-\ntionsδm1, we find in this case\n/an}bracketle{t∆R(t)∆R(t′)/an}bracketri}ht⊥=/parenleftbigg∂R(0)\n∂cosθ/parenrightbigg2\n/an}bracketle{tδm1x(t)δm1x(t′)/an}bracketri}ht.(68)\nThemagnetizationfluctuationsaffecttheresistancenoise\nintheperpendicularconfigurationtosecondorder,unlike\nfor the P and AP configurations, in which the leading\ntermwasoffourthorder. Thesensitivityoftheresistance\nfor this configuration is according to Eq. (37)\n∂R(0)\n∂cosθ=−4hP2η\ne2g(1+η−P2)2, (69)\nLinearizing Eq. (55) and using the correlators Eqs. (45)\nand (46) for the stochastic field we find\n/an}bracketle{tδm1x(t)δm1x(t′)/an}bracketri}ht=γ0kBT\nπMsV/integraldisplay\ndωe−iω(t−t′)ω2(α0+α′\nyy)+(ωp+ωd)2(α0+α′\nxx)\n[ω2−ωp(ωp+ωd)]2+ω2[ωp(2α0+α′xx+α′yy)+ωd(α0+α′xx)]2,(70)\nwhereωp=ω0+ωc. We then arriveatthe zero-frequency\nresistance noise\nS⊥(0) =2γ0kBT\nMsV/parenleftbigg4hP2η\ne2g(1+η−P2)2/parenrightbigg2α0+α′\nxx\nω2p,(71)\nquite different from that in the collinear configurations.\nIn particular,the dampingappearsherein the numerator\nand there is no dependence on the demagnetizing field.\nNoticethatsince S⊥isquadraticinmagneticfluctuations\n[see Eq.(68)], it becomeslinear in temperature, unlike SP\nandSAP.E. Two identical ferromagnets\nWe now investigate spin valves in which the ferro-\nmagnets are identical and hence equally susceptible to\nfluctuations,36focusing now only on the P and AP con-\nfigurations. The fluctuations of F1areδm1(t) =m1(t)−\n/an}bracketle{tm1/an}bracketri}htand those of F2areδm2(t) =m2(t)−/an}bracketle{tm2/an}bracketri}ht. As\nbefore, we choose the z-axis so that the time-averaged\nequilibrium values are /an}bracketle{tm1/an}bracketri}ht=/an}bracketle{tm2/an}bracketri}ht=zfor the par-\nallel configuration, and /an}bracketle{tm1/an}bracketri}ht=−/an}bracketle{tm2/an}bracketri}ht=zfor the\nantiparallel. The dot product of the magnetizations is\nm1·m2=±1∓(δm∓)2/2, where the upper (lower) sign\nholds for the P (AP) orientation and δm∓=δm1∓δm2.12\nFor small fluctuations, we can expand the resistance to\nfirst order in ( δm∓)2, finding\nR(m1·m2)≈R(±)∓1\n2(δm∓)2∂R(±1)\n∂cosθ.(72)The resistance noise is then\n/an}bracketle{t∆R(t)∆R(t′)/an}bracketri}htP/AP=/an}bracketle{tR(t)R(t′)/an}bracketri}htP/AP−/an}bracketle{tR(t)/an}bracketri}htP/AP/an}bracketle{tR(t′)/an}bracketri}htP/AP\n=1\n4/parenleftbigg∂R(±1)\n∂cosθ/parenrightbigg2/bracketleftbig\n/an}bracketle{t(δm∓)2(δm∓)2/an}bracketri}htP/AP−/an}bracketle{t(δm∓)2/an}bracketri}htP/AP/an}bracketle{t(δm∓)2/an}bracketri}htP/AP/bracketrightbig\n,(73)\nwhich by employing Wick’s theorem becomes\n/an}bracketle{t∆R(t)∆R(t′)/an}bracketri}htP/AP=1\n2/parenleftbigg∂R(±1)\n∂cosθ/parenrightbigg2\n×/summationdisplay\nij/an}bracketle{tδm∓\ni(t)δm∓\nj(t′)/an}bracketri}ht2\nP/AP.(74)\nLetting the subscripts kandlrefer to ferromagnet 1\nor 2, the LLG equation in this case reads\ndmk\ndt=−γ0mk×[Heff+hk(t)]\n+(α0+αsv)mk×dmk\ndt+αsv\n3ml×dml\ndt,(75)\nwhere the effective field Heffis now taken to be equal\nfor both ferromagnets. Due to current conservation, the\nferromagnets respective current-induced stochastic fields\nare not independent of each other. With the spin current\nnoise calculated in Sec. IVB, and following the recipe in\nSec. IIID, we find\n/an}bracketle{th(th)\n1,i(t)h(th)\n2,j(t′)/an}bracketri}htP=−2kBTαsv/3\nγ0MsVδijδ(t−t′) (76)\nfor the P configuration, and\n/an}bracketle{th(th)\n1,i(t)h(th)\n2,j(t′)/an}bracketri}htAP= 2kBTαsv/3\nγ0MsVδijδ(t−t′).(77)\nfor the AP configuration (as before i,jlabel components\nperpendicular to the magnetization direction). αsvis de-\nfined in Eq. (43). Naturally, the bulk fields h(0)\n1andh(0)\n2\nare uncorrelated. The last term in the LLG Eq. (75)\nrepresent the dynamic spin-exchange coupling:29,50It is\nthe spincurrentpumped fromferromagnet l(seeSec. III)that is transmitted to and subsequently absorbed by fer-\nromagnet k. Since the normal metal node is chaotic,\nthis amounts to one third of the net total spin current\npumpedoutofferromagnet l. Thisdynamiccouplingwas\nnot present in spin valves in which one magnetization is\nnot moving at all.\nBy linearizing Eq. (75) in δmk(t) we can evaluate the\ndesired magnetization noise correlators that are to be\ninserted in Eq. (74). The zero-frequency resistance noise\nfor the P and AP configurations then respectively reads\nSP(0) =1\nπ/parenleftbigghP2\ne2gη/parenrightbigg2/parenleftbigg2γ0kBT\nMsV/parenrightbigg2\n×/integraldisplay\ndω(Z2\nxx+Z2\nyy−2Z2\nxy).(78)\nand\nSAP(0) =1\nπ/parenleftbigghP2η\ne2g(1−P2)2/parenrightbigg2/parenleftbigg2γ0kBT\nMsV/parenrightbigg2\n×/integraldisplay\ndω(X2\nx+X2\ny). (79)\nHere\nZxx=αt[ω2+(ωi+ωd)2]\n[ω2−ωi(ωi+ωd)]2+ω2α2\nt(2ωi+ωd)2,(80)\nZxy=−iωαt(2ωi+ωd)\n[ω2−ωi(ωi+ωd)]2+ω2α2\nt(2ωi+ωd)2,(81)\nZyy=αt(ω2+ω2\ni)\n[ω2−ωi(ωi+ωd)]2+ω2α2\nt(2ωi+ωd)2,(82)\nand\nXx=ω2αs+(ωc+ωd)2αt\n[ω2+(ωc+ωd)(2ωe−ωc)]2+ω2(2ωxαs−2ωcα−ωdαt)2, (83)\nXy=ω2αs+ω2\ncαt\n[ω2+ωc(2ωe−ωc−ωd)]2+ω2(2ωxαs−2ωcα−ωdαs)2. (84)13\nFIG. 3: The resistance noise in the P configuration as a\nfunction of the externally applied magnetic field, given in\nunits of (10−7/π)(hP2/e2gη)2(2γ0kBT/MsV)2. The parame-\nters used are α0=αsv= 0.01,ωc/γ0=ωd/γ0= 100 Oe and\nJ=−0.10 erg/cm2.\nFor convenience, we defined αs=α0+ 2αsv/3,αt=\nα0+4αsv/3,α=α0+αsv(note the difference between\nα,αsandαt), andωi=ω0+ωa+ 2ωx. The above\nexpressions hold for small damping, i.e.,α2\n0,α2\nsv≪1.\nCompared to the results in the previous section, we see\nthat Eq. (78) is similar to Eq. (61), whereas Eq. (79)\ndiffers considerably from Eq. (65). This is due to the\nstatic dipolar and exchange couplings, and the dynamic\nspin-exchange coupling, whose effects on the noise are\nmodified by the presence of the second fluctuating fer-\nromagnet. In particular, the latter coupling causes the\nGilbert dampingconstanttoenterEqs. (78)and(79)dif-\nferently. Eq. (78) decreases with the external field and\nEq. (79) decreases with the dipolar and exchange cou-\npling, as expected, and as shown in Figs. 3 and 4. The\nnoise level is in general higher when both ferromagnets\nfluctuate, than when only one does.\nThe resistance noise is governed by a number of ma-\nterial parameters. Depending on these parameters, the\nnoise level in the P configuration can differ substantially\nfrom that in the AP configuration. Note that Eq. (79)\nreduces to that of Ref. 36 when the demagnetizing field\nis disregarded, i.e., when ωd→0, whereas Eq. (78)\ndoes when ωa→0 andωd→ −ωa, since the exter-\nnal field in our earlier work was perpendicular to the\nanisotropy field. The considerable difference between\nSP(0) canSAP(0) in typical experimental spin-valve se-\ntups can partly be explained by the dynamic exchangecoupling.36However,alsothesensitivityoftheresistance\nto magnetic configuration changes can be important, as\nshown in the previous section. The demagnetizing field\nalsosignificantly affects the numericalresult for the noise\nlevel since it stabilizes the magnetization, in both the P\nand AP configurations.\nFIG. 4: The resistance noise in the AP configuration as\na function of the dipolar and exchange coupling between\nthe ferromagnets, given in units of (10−7/π)[hP2η/e2g(1−\nP2)2]2(2γ0kBT/MsV)2. The parameters used are α0=αsv=\n0.01,ωc/γ0=ωd/γ0= 100 Oe, and H0= 0.\nV. CONCLUSIONS\nUsing scattering theory and magnetoelectronic circuit\ntheory, we demonstrate the effect of spin current fluctua-\ntions on the magnetization in ferromagnetic multilayers.\nVia a fluctuating spin-transfer torque, the current noise\ncauses significantly enhanced magnetization noise, which\nin spin valves is a function of the magnetic configuration.\nThe noise is related to the magnetization damping by the\nFDT, and can be experimentally detected as resistance\nnoise. 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Garcia de Andrade\nDepartamento de F´ ısica Te´ orica – IF – Universidade do Esta do do Rio de Janeiro-UERJ\nRua S˜ ao Francisco Xavier, 524\nCep 20550-003, Maracan˜ a, Rio de Janeiro, RJ, Brasil\nElectronic mail address: garcia@dft.if.uerj.br\nAbstract\nRiemann and sectional curvatures of magnetictwisted flux tu bes in Riemannianmanifold are\ncomputedtoinvestigatethestabilityoftheplasmaastroph ysicaltubes. Thegeodesicequations\nare used to show that in the case of thick magnetic tubes, the c urvature of planar (Frenet\ntorsion-free) tubes have the effect ct of damping the flow spee d along the tube. Stability of\ngeodesic flows in the Riemannian twisted thin tubes (almostfi laments), againstconstant radial\nperturbations is investigated by using the method of negati ve sectional curvature for unstable\nflows. No special form of the flow like Beltrami flows is admitte d, and the proof is general\nfor the case of thin magnetic flux tubes. In the magnetic equil ibrium state, the twist of the\ntube is shown to display also a damping effect on the toroidal v elocity of the plasma flow. It\nis found that for positive perturbations and angular speed o f the flow, instability is achieved ,\nsince the sectional Ricci curvature of the magnetic twisted tube metric is negative. Solar flare\nproduction may appear from these geometrical instabilitie s of the twisted solar loops. PACS\nnumbers: 02.40.Hw:Riemannian geometries\n1I Introduction\nThe stability of geodesic flows have been recently investiga ted by Kambe [1] by making use of\nthe technique of Ricci sectional curvature [2], where the ne gative sectional curvature indicates\ninstability of the flow, while positivity or nul indicates st ability. In the case of instability the\ngeodesics deviate from the perturbation of the fluid. Follow ing the work of D. Anosov [3] on\nthe perturbation in geodesic flows in three-dimensional Rie mannian geometry, in this paper\nthe sectional Riemann curvature of the geodesic flow for a Rie mannian flux tube [4, 5], where\nthe axis of the tube flow possesses Frenet curvature and torsi on. In the approximation of a\nthin tube where the radius of the tube is almost null, we show t hat the flows are unstable,\nagainst orthogonal perturbations, which is equivalently d ue to the negativity of the sectional\nRicci sectional curvature. Throughout the paper the ellega nt coordinate-free language of\ndifferential geometry [6] is used. The paper is organized as f ollows: Section II presents a\nbrief review of Riemannian geometry in the coordinate free l anguage. Section III presents the\ngeodesic flow computation of the Christoffel symbols for the t hick flux tube, where we show\nthat the curvature of the tube axihe speed of the flow. Section IV presents the computation\nof the instability of Riemannian tube flow. Section V present s the conclusions.\nII Ricci and sectional Riemann curvatures\nIn this section we make a brief review of the differential geom etry of surfaces in coordinate-free\nlanguage. The Riemann curvature is defined by\nR(X,Y)Z:=∇X∇YZ−∇Y∇XZ−∇[X,Y]Z (II.1)\nwhereXǫTMis the vector representation which is defined on the tangent s paceTMto the\nmanifold M. Here∇XYrepresents the covariant derivative given by\n∇XY= (X.∇)Y (II.2)\nwhich for the physicists is intuitive, since we are saying th at we are performing derivative along\nthe X direction. The expression [X,Y]represents the commutator, which on a vector basis\n2frame/vector elin this tangent sub-manifold defined by\nX=Xk/vector ek (II.3)\nor in the dual basis ∂k\nX=Xk∂k (II.4)\ncan be expressed as\n[X,Y] = (X,Y)k∂k (II.5)\nIn this same coordinate basis now we are able to write the curv ature expression (II.1) as\nR(X,Y)Z:= [Rl\njkpZjXkYp]∂l (II.6)\nwheretheEinstein summationconventionoftensor calculus isused. Theexpression R(X,Y)Y\nwhich we shall compute bellow is called Ricci curvature. The sectional curvature which is very\nuseful in future computations is defined by\nK(X,Y) :=< R(X,Y)Y,X >\nS(X,Y)(II.7)\nwhereS(X,Y)is defined by\nS(X,Y) :=||X||2||Y||2−< X,Y >2(II.8)\nwhere the symbol <,>implies internal product.\n3III Geodesic equations in Riemannian tube metric\nIn this section we shall consider the twisted flux tube Rieman n metric. The metric g(X,Y)\nline element can be defined as [4, 5]\nds2=dr2+r2dθR2+K2(s)ds2(III.9)\nThis line element was used previously by Ricca [4] and the aut hor [5] as a magnetic flux tubes\nwith applications in solar and plasma astrophysics. This is a Riemannian line element\nds2=gijdxidxj(III.10)\nif the tube coordinates are (r,θR,s)[4] where θ(s) =θR−/integraltextτdswhereτis the Frenet torsion\nof the tube axis and K(s)is given by\nK2(s) = [1−rκ(s)cosθ(s)]2(III.11)\nLet us now compute the geodesic equations\ndvi\ndt+Γi\njkvjvk= 0 (III.12)\nwherevs=ds\ndtandvθ=dθ\ndtthe Riemann-Christoffel symbols are given by\nΓi\njk=1\n2gil[glj,k+glk,j−gjk,l] (III.13)\nThe only nonvanishing components of the Christoffel symbols or Levi-Civita [6] connections\nof the flux tube are\nΓ2\n21=1\nr(III.14)\nΓ2\n33=−K(s)κsinθ\nr(III.15)\nΓ3\n11=rκ\n2K[τsinθ+κ′\nκcosθ] (III.16)\nΓ3\n33=K−1K′=−Γ3\n11 (III.17)\nA simple example can be given by writing the geodesic equatio n for the untwisted tube, where\nvθ= 0, as\n¨s+Γ3\n11[˙r2−˙s2] = 0 (III.18)\n4where substitution of the component Γ311above yields\ndlnvs=−d(lnκ) (III.19)\nwhere we use the chain rule of differential calculus,ds\nvs2dvs\ndt=dvs\nvs. Solution of equation (III.19)\nis\nvs=v0κ−1(III.20)\nwherev0is an integration constant. This solution tells us that when the curvature tends to\n∞the velocity along the tube axis vanishes. Physically this m eans that the curvature acts as\na damping to the flow, along the tube. In the next section we inv estigate in some detail the\nstability of an incompressible or volume preserving flow, us ing the method of the sign of the\nRicci sectional curvature.\nIV Sectional curvature and plasma flow stability\nOneofthemostimportantfeaturesoftheinvestigationofth estabilityofflowsintheEuclidean\nmanifold E3, is the comprehension of the fact that the covariant derivat ive in the flow curved\nmanifold is given by the gradient operator in curvilinear co ordinates. Thus to compute the\nRiemann sectional curvature above, we need to make use of the grad operator in the twisted\nflux tube Riemannian metric given by\n∇= [∂r,r−1∂θR,K−1∂s] (IV.21)\nSince the axis of the tube undergoes torsion and curvature, w e need some dynamical relations\nfrom vector analysis and differential geometry of curves [7] such as the Frenet frame (/vectort,/vector n,/vectorb)\nequations\n/vectort′=κ/vector n (IV.22)\n/vector n′=−κ/vectort+τ/vectorb (IV.23)\n/vectorb′=−τ/vector n (IV.24)\nand the other frame vectors are\n/vector er=/vector ncosθ+/vectorbsinθ (IV.25)\n5/vector eθ=−/vector nsinθ+/vectorbcosθ (IV.26)\n∂θ/vector eθ=−/vector n[(1+τ−1κ)sinθ+cosθ]−/vectorb[cosθ+sinθ] (IV.27)\nLet the constant perturbation be given by\nX=ur1/vector er (IV.28)\nThe upper index one in this expression refers to the fact that the background original value of\nurwas considered as ur0= 0to form the tube. The other variable Y is given by\nY=uθ/vector eθ+us/vectort (IV.29)\nTherefore to compute the Ricci tensor step by step we start by the term\n∇XY=ur(1)∂r[uθ/vector eθ+us/vectort] (IV.30)\nwhich vanishes since we addopt here the approximation ur(1)∂r[uθ]≈0along with the same\nrelation to the radial partial derivative of ur. So\n∇Y∇XY≈0 (IV.31)\nNow the second term in the Ricci tensor is\n∇X∇YY=−ur(1)r−2uθ[(1+τ)\nτ(/vector ncosθ+/vectorbsinθ)] (IV.32)\nwhere we have used the approximation of the thin tube where K(s)≈1andr≈0\n[X.Y] =ur(1)[r−1uθ−τus][/vector er−τ−1/vectort] (IV.33)\nwhich implies that\n∇[X,Y]Y=ur(1)[r−1uθ−τus][−τ−1uθ∂s/vector eθ+κ/vector n] (IV.34)\nThe Ricci tensor is\nR(X,Y)Y=−ur(1)r−1uθτ−1∂s/vector eθ (IV.35)\n6In the previous computations we have made use of the imcompre ssibility of the flow\n∇./vector u= 0 (IV.36)\nwhich is\n∂suθ=τrκuθ≈0 (IV.37)\nsincer≈0on the RHS of equation (IV.37). The sectional curvature is thu s\nK(X,Y) =< R(X,Y)Y,X >\nS(X,Y)=−uθ[1+τ(s)κcosθ]\nru(1)r[uθ2+us2](IV.38)\nwhen the tube is strongly twisted, uθ2>> us2thus the sectional Ricci curvature is\nK(X,Y) =< R(X,Y)Y,X >\nS(X,Y)=−[1+τ(s)κcosθ]\nru(1)ruθ(IV.39)\nwhen the tube, besides is planar or torsion vanishes the last expression reduces to\nK(X,Y) =< R(X,Y)Y,X >\nS(X,Y)=−1\nru(1)ruθ(IV.40)\nNote that when both angular velocity and perturbation both k eep the same sign, the sectional\ncurvature K(X,Y)is negative and the flow along the Riemannian flux tube is unsta ble. There\nis singularity in this sectional Riemannian curvature in r≈0. Note that if we consider the\nimcompressibility equation (IV.36) as\n∂\n∂suθ=uθκrτ\nKsinθ (IV.41)\nSince the magnetic field /vectorBis also divergence-free, where Brvanishes in the equilibrium state,\nthe same equation for the poloidal magnetic field component Bθis obtained\n∂\n∂sBθ=Bθκrτ\nKsinθ (IV.42)\nPuttingK≈0for the thin tube approximation, and the equilibrium state m agnetohydrody-\nnamics (MHD) equation\n∇×[/vector u×/vectorB] = 0 (IV.43)\nand taken the most simple solution\n/vector u×/vectorB= 0 (IV.44)\n7one obtains\nuθBs=usBθ (IV.45)\nsinceurandBrboth vanishes before perturbation in the radial direction o f the magnetic flux\ntube. Taken into account the relation derived by Ricca [4] on the ratio between the poloidal\nand toridal components of the magnetic field\nBθ\nBs=2πrTw\nL(IV.46)\nwhere we have taken K= 1. Substitution of this expression into (IV.45) and using the\nequation for the divergence-free of the velocity plasma flow field yields\nus=L[1−cosθ]\n2πrTw(IV.47)\nwhere\nuθ=2πrTwu s\nL(IV.48)\noruθ= 1−cosθ]which upon substitution in the sectional curvature (IV.40) y ields\nK(X,Y) =< R(X,Y)Y,X >\nS(X,Y)=−1\nru(1)r[1−cosθ](IV.49)\nwhich shows that since cosθpe1, the sectional curvature is singular or negative which impl ies\ninstability of the solar loops, as long as the radial perturb ation on the twisted solar loop is\npositive. In other words, if the tube is radially expanding a s happens on the suface of the sun,\nthe solar loop is highly unstable, which is reasonable for th e production of solar flares.\nV Conclusions\nAn important issue in plasma astrophysics as well as in fluid m echanics is to know when a fluid,\ncharged or not, is unstable or not. In this paper we discuss an d present incompressible flows\nand investigate their stability. Instability is obtained e ven before the singularity is achieved.\nInflexional desiquilibrium of solar loops has been also rece ntly investigated by Ricca [4], based\non the above Riemann twisted magnetic flux tube. He proved a th eorem where inflexion\ndesiquilibrium is proved from a state of MHD equilibrium dur ing passage throghout inflexional\npoints in the solar loop. In this sense, though obtained from completely different methods it\n8seems that coincide with those of Ricca’s [4]. Solar flare prod uction may appear from these\ngeometrical instabilities of the twisted solar loops.\n9References\n[1] T. Kambe, Geometrical Theory of Dynamical Systems and Fl uid Flows (2004) World\nScientific.\n[2] R. Hermann (translator), Ricci and Levi-Civita tensor a nalysis paper (1975) MIT press.\n[3] D.V. Anosov, Geodesic Flows on Compact Riemannian Manifo lds of Negative Curvature\n(1967) (Steklov Mathematical Institute, USSR) vol.90.\n[4] R. Ricca, Solar Physics 172 (1997),241.\n[5] L. C. Garcia de Andrade, Physics of Plasmas 13, 022309 (20 06). R. Ricca, Fluid Dynamics\nResearch 36 (2005),319.\n[6] E. Cartan, Riemann Spaces (2000) MIT Press, Boston.\n[7] M. P. do Carmo, Differential Geometry of curves and Surfac es (1992) Springer.\n10" }, { "title": "1902.09896v1.Enhanced_Gilbert_Damping_in_Re_doped_FeCo_Films__A_Combined_Experimental_and_Theoretical_Study.pdf", "content": "Enhanc ed Gilbert Damping in Re doped FeCo Films – a combined experimental and \ntheoretical study \nS. Akansel1, A. Kumar1, V.A.Venugopal2, R.Banerjee3, C. Autieri3, R.Brucas1, N. Behera1, M. \nA. Sortica3, D. Primetzhofer3, S. Basu2, M.A. Gubbins2, B. Sanyal3, and P. Svedlindh1 \n1Department of Engineering Sciences , Uppsala University, Box 534, SE -751 21 Uppsala, Sweden \n2Seagate Technology, BT48 0BF, Londonderry, United Kingdom \n3Department of Physics and Astronomy, Uppsala University, Box 516, SE -751 20 Uppsala, \nSweden \n \nThe effect s of rhenium doping in the range 0 – 10 at% on the static and dynamic magnetic \nproperties of Fe65Co35 thin films have been studied experimentally as well as with first principles \nelectronic structure calculations focussing on the change of the saturation magnetization (𝑀𝑠) and \nthe Gilbert damping parameter ( 𝛼) Both experiment al and theoretical results show that 𝑀𝑠 \ndecreases with increasing Re doping level, while at the same time 𝛼 increases. The experimental \nlow temperature saturation magnetic induction exhibits a 2 9% decrease, from 2.3 1T to 1. 64T, in \nthe investigated doping concentration range , which is more than predicted by the theoretical \ncalculations. The room temperature value of the damping parameter obtained from ferromagnetic \nresonance measurements , correcting for extrinsic contributions to the damping, is for the undoped \nsample 2.7×10−3, which is close to the theoretically calculated Gilbert damping parameter . With \n10 at% Re doping , the damping parameter increases to 9.0×10−3, which is in good agreement \nwith the theoretical value of 7.3×10−3. The increase in damping parameter with Re doping is \nexplained by the increase in density of states at Fermi level, mostly contributed by the s pin-up \nchannel of Re. Moreover, both experimental and theoretical values for the da mping parameter are \nobserved to be weakly decreas ing with decreasing temperature . \n 1. INTRODUCTION \nDuring the last decades , thin films of soft magnetic alloys such as NiFe and FeCo have been in \nfocus due to possible use in applications such as spin valves ,1,2 magnetic tunneling junctions ,3,4,5 \nspin injectors ,6 magnetic storage technologies and in particular in magnetic recording write heads .7 \nBeside s spintronic and magnetic memory devices , such materials are useful for shielding \napplications that are necessary in order to reduce the effect of electromagnetic fields created by \nelectronic devices. The magnetic damping parameter of the material play s a critical role for the \nperformance of such spintronic and memory devices as well as for shielding applications. On the \none hand, a low damping parameter is desired in order to get low critical switching current in \nspintronic devices .8,9,10 On the other hand , a high damping parameter is necessary in order to \nreduce the magetization switching time in magnetic memory devices and to be able to operate \ndevices at high speeds .11 FeCo alloys are promising materials for high frequency spintronic \napplications and magnetic recording devices due to their high saturation magnetization (𝑀𝑠), high \npermeability, thermal stability and comparably high resistivity .12,13,14 One possible drawback is \nthat FeCo alloy s exhibit high coercivity (𝐻𝑐), which is not favorable for such applications , however \nthis problem can be solved by thin film growth on suitable buffer layer s.15,16,12 Except coercivity \nproblems, the damping parameter of these materials should be increased to make them com patible \nfor high speed devices . \nDynamic properties of magnetic materials are highly dependent on the damping parameter. This \nparameter is composed of both intrinsic and extrinsic contributions. The intrinsic contribution is \ncalled the Gilbert damping and depends primarily on the spin-orbit coupling .17 Intrinsic damping \nis explained as scattering of electrons by phonons and magnons .18,19 Beside s electron scattering , \ndue to the close relation between magnetocrystalline anisotropy and spin-orbit coupling , it can be \nassumed that the intrinsic damping is also related to the magnetocrystalline anisotropy constant .20 \nRegarding extrinsic damping , there can be a number of different contributions. The most common \ncontribution originates from two magnon scattering (TMS) .21 However , this contribution vanishes \nwhen ferromagnetic resonance (FMR) measurements are performed by applying the static \nmagnetic field along the film normal in inplane anisotropic thin films .22 Beside s TMS , there are \nsome other extrinsic contributions to the damping that are not possible to get rid of by changing \nthe measurement configuration . One of these contributions is radiative damping , which arises from \ninductive coupling between the precessing magnetization and the waveguide used for FMR \nmeasurem ents.23 Another contribution for metallic ferromagnetic films is the eddy current \ndamping related to microwave magnetic field induced eddy currents in the thin film s during \nmeasurement s.23,24 \nIn order to make a soft magnetic thin film suitible for a specific applica tion, taking into account \nrequirements set by the device application , its damping paramete r should be tailored. As mentioned \nabove , an increased damping parameter is necesssary for devices requiring high switching speed . \nSeveral efforts have been made for enhanching the damping parameter of soft magnetic materials. \nNiFe alloys constitu te one of the most studied systems in this respect . The most common way to enhance the intrinsic damping of an all oy is to dope it with differ ent elements . Rare earth elements \nwith large spin-orbit coupling have revealed promising results as dopant s in terms of increas ed \ndampin g parameter .25,26,27 3d, 4d and 5d transition metals dopants have also been studied \nexperimentally , revealing an increase of the damping parameter .28,29 Beside s experimental results , \ntheoretical calculations support the idea that transition metals and especially 5d elements can \nenhance the damping parameter of NiF e alloys due to scattering in presence of chemical disorde r \n, as well as due to the effect of spin -orbit coupling .30 \nAlthough NiFe alloys have been the focus in several extensive studies, FeCo alloys have so far not \nbeen studied to the same extent . Attempts have been made to dope FeCo with Yb,20 Dy,31 Gd,32 \nand Si ,33 where in all cases an increase of the damping parameter was observed . Apart from doping \nof alloys , the addition of adjacent layers to NiFe and CoFe has also been studied . In particular , \nadding layers consisting of rare earth elem ents with large orbital moment s gave positive results in \nterms of increased damping parameter .34 \nFe65Co35 alloy s are attractive material s because of high 𝑀𝑠 and reduced 𝐻𝑐 values. However , not \nmuch is known about the magnetic damping mechanism s for this composition . Since it is of \ninterest for high data rate magnetic memory devices, the damping parameter should be increased \nin order to make the magnetic switching faster. To the best of our knowledge , systematic doping \nof Fe 65Co35 with 5d elements has not been studied so far experimentally . Some of us have found \nfrom ab initio calculations that 5d transition metal dopants can increase the damping parameter \nand Re is one of the potential candidates.35 Re is particularly interesting as it has a nice compromise \nof having not so much reduced saturation magnetization and a quite enh anced damping parameter. \nIn this work, we have perfomed a systematic ab initio study of Fe65Co35 doped with increasing Re \nconcentration to find an increasing damping parameter . The theoretical prediction s are confirmed \nby results obtained from temperature dependent FMR measurements performed on Re doped \nFe65Co35 films. \n \n2. EXPERIMENTAL AND THEORETIC AL METHOD S \nRhenium doped Fe 65Co35 samples were prepared by varying the Re concentration from 0 to 10.23 \nat%. All samples were deposited using DC magnetron sputtering on Si/SiO 2 substrate s. First a 3 \nnm thick Ru seed layer was deposited on the Si/SiO 2 substrate followed by room temperature \ndeposition of 20 nm and 40 nm thick Re -doped Fe65Co35 films by co -sputtering between Fe 65Co35 \nand Re target s. Finally, a 3 nm thick Ru layer was deposited as a capping layer over the Re -doped \nFe65Co35 film. The nominal Re concentration was derived from the calibrated deposition rate used \nin the deposition system. The nominal Re doping concentration s of the Fe65Co35 samples are as \nfollows ; 0, 2.62, 5.45 and 10.23 at%. \nThe crystalline structure of the fims were investigated by utilizing grazing incident X -Ray \ndiffraction (GIXRD). The i ncidence angle was fixed at 1o during GIXRD measurements and a CuKα source was used. Accurate values for film thickness and interface roughness were \ndetermined by X -ray reflectivity (XRR) measurements. \nBeside XRD , composition and areal density of the films were deduced by Rutherford \nbackscattering spectrometry36 (RBS) with ion beams of 2 MeV 4He+ and 10 MeV 12C+. The beams \nwere provided by a 5 MV 15SDH -2 tandem accelerator at the Tandem Laboratory at Uppsala \nUniversity. The experiments were performed with the incident beam at 5° with respect to the \nsurface normal and scattering angles of 170° and 120° . The experimental data was evaluated with \nthe SIMNRA program .37 \nIn-plane magnetic hysteresis measurments were performed using a Magnetic Property \nMeasurement System (MPMS, Quantum Design) . \nFerromagnetic resonance measurements were performed using two different techniques. First in-\nplane X -band (9.8 GHz) cavity FMR measurements were performed . The setup is equipped with a \ngoniometer making it possible to rotate the sample with respect to the applied magnetic field; in \nthis way the in -plane anisotropy fields of the different samples have been determine d. Beside s \ncavity FMR studies , a setup for broadband out-of-plane FMR measurements have been utilized . \nFor out -of-plane measurements a vector network analyzer (VNA) was used. Two ports of the VNA \nwere connected to a coplanar waveguide (CPW) mounted on a Ph ysical Property Measurement \nSystem (PPMS, Quantum Design) multi -function probe . The PPMS is equipped with a 9 T \nsuperconducting magnet, which is needed to saturate Fe65Co35 films out -plane and to detect the \nFMR signal. The broadband FMR measurements were carried out a t a fixed microwave frequency \nusing the field -swept mode, repeating the measurement for different f requencies in the range 15 – \n30GHz. \nThe theoretical calculations are based on spin -polarized relativistic m ultiple scattering theory using \nthe Korringa -Kohn -Rostoker (KKR) formalism implemented in the spin polarized relativistic \nKKR code (SPR-KKR) . The Perdew -Burke -Ernzerhof (PBE) exchange -correlation functional \nwithin generalized gradient approximation was used. The equilibrium lattice parameter s were \nobtained by energy minimization for each composition. Substitutional disorder was treated within \nthe Coherent Potential Approximation (CPA). The damping parameters were calcu lated by the \nmethod proposed by Mankovsky et al.,38 based on the ab initio Green's function technique and \nlinear res ponse formalism where one takes into consi deration scattering processes as well as spin -\norbit coupling built in Dirac's relativistic formulation. The calculations of Gilbert damping \nparameters at finite temperatures were done using an alloy -analo gy model of atomic displacements \ncorresponding to the thermal average of the root mean square displacement at a given temperature. \n3. RESULTS AND DISCUSSION \nRe concentrations and layer thickness (areal densities) of the 20 nm doped films were obtained by \nRBS experiments. RBS employing a beam of 2 MeV He primary ions was used to deduce the areal \nconcentration of each layer. Additional measurements with 10 MeV C probing particles permit to resolve the atomic fractions of Fe, Co and Re. The spectra for the samples with different Re \nconcentration are shown in Fig. A1 . The measured Re concentrations are 3.0±0.1 at%, 6.6±0.3 at% \nand 12.6±0.5 at%. Moreover, the results for Fe and Co atomic fractions show that there is no \npreferential replacement by Re , implying that the two elements are replaced according to their \nrespective concentration . \nFigure 1 (a) shows GIXRD spectra in the 2𝜃-range from 20o to 120o for the Fe65Co35 films with \ndifferent Re concentration. Diffraction peaks corresponding to the body centered cubic Fe 65Co35 \nstructure have been indexed in the figure; no other diffraction peaks appear in the different spectra. \nDepending on the Re -dopant concentration shi fts in the peak positions are observed, the diffraction \npeaks are suppressed to lower 2𝜃-values with increasing dopant concentration . The shift for the \n(110) peak for the different dopant concentrations is given as an inset in Fig. 1 (a). Similar shifts \nare observed for the other diffraction peaks. This trend in peak shift is an experimental evidence \nof an increasing amount of Re dopant within the deposited thin films. Since the peaks are shifted \ntowards lower 2𝜃-values with increasing amount of Re dopant , the lattice parameter increases with \nincreasing Re concentration.39 Figure 1 (b) shows the experimental as well as theoretically \ncalculated lattice parameter versus Re concentration. The qualitative agreement between theory \nand experiment is obtained. However, t he rate of lattice parameter increase with increasing Re \nconcentration is larger for the theoretically calculated lattice parameter. This is not unexpected as \nthe generalized gradie nt approximation for the exchange -correlation potential has a tendency to \noverestimate the lattice parameter. Another possible explanation for the difference in lattice \nparameter is that the increase of the lattice parameter for the Re -doped Fe 65Co35 films is held back \nby the compressive strain due to lattice mismatch with Si/SiO 2/Ru. XRR measurements revealed \nthat the surface roughness of the Fe 65Co35 films is less than 1 nm , which cannot affect static and \nmagnetic properties drastically. Results from XRR measurements are given in table 1. \nRoom temperature normalized magnetization curves for the Re-doped Fe 65Co35 films are shown \nin Fig. 2 (a) . The coercivity for all films is in the range of 2 mT and all films, except for the 1 2.6 \nat% Re doped film that show a slightly rounded hysteresis loop, exhibit rectangular hysteresis \nloops. The low value for the coercivity is expected for seed layer grown films .15 The \nexperimentally determined low temperature saturation magnetization together with the \ntheoretically calculated magnetization versus Re concentra tion are shown in Fig. 2 (b). As \nexpected, both experimental and theoretical r esults show that the saturation magnetization \ndecreases with increasing Re concentration . A linear decrease in magnetization is observed in the \ntheoretical calculations whereas a non -linear behavior is seen in the experimental data. \nAngle resolved cavity FMR measurements were used to study the in -plane magnetic anisotropy . \nThe angular -dependent resonant field ( 𝐻𝑟) data was analyzed using the following equation ,40 \n 𝑓=µ0𝛾\n2𝜋[{𝐻𝑟cos(𝜙𝐻−𝜙𝑀)+𝐻𝑐\n2cos4(𝜙𝑀−𝜙𝐶)+𝐻𝑢cos2(𝜙𝑀−𝜙𝑢)}{𝐻𝑟cos(𝜙𝐻−\n𝜙𝑀)+𝑀𝑒𝑓𝑓+𝐻𝑐\n8(3+cos4(𝜙𝑀−𝜙𝐶))+𝐻𝑢cos2(𝜙𝑀−𝜙𝑢)}]12⁄\n, (1) where 𝑓 is the cavity resonance frequency and 𝛾 is the gyromagnetic ratio . 𝜙𝐻, 𝜙𝑀, 𝜙𝑢 and 𝜙𝐶 \nare the in -plane directions for the magnetic field, magnetization, uniaxial anisotropy and cubic \nanisotropy, respectively, with respect to the [100 ] direction of the Si substrate. 𝐻𝑢=2𝐾𝑢\nµ0𝑀𝑠 and \n𝐻𝑐=4𝐾𝑐\nµ0𝑀𝑠 are the uniaxial and cubic anisotropy fields, where 𝐾𝑢 and 𝐾𝑐 are the uniaxial and cubic \nmagnetic anisotropy constants , and 𝑀𝑒𝑓𝑓 is the effective magnetization. Fitting parameters were \nlimited to 𝑀𝑒𝑓𝑓, 𝛾 and 𝐻𝑢, since the Hr versus ϕH curves did not give any indication of a cubic \nanisotropy. \nFigure 3 shows 𝐻𝑟 versus 𝜙𝐻 extracted from the angular -dependent FMR measurements together \nwith fits according to Eq. (1), clearly revealing dominant twofold uniaxial in -plane magnetic \nanisotropy. Extracted anisotropy field and effective magnetization values are given in Table 2 . The \nresults show that 𝐻𝑢 is within the accuracy of the experiment independent of the Re concentration . \nTemperature dependent o ut-of-plane FMR measurements were performed in the temperature range \n50 K to 300 K recording the complex transmission coefficient 𝑆21. Typical field -swept results for \nthe r eal and imaginary components of 𝑆21 for the undoped and 1 2.6 at% Re-doped samples are \nshown in Fig. 4. The field -dependent 𝑆21 data was fitted to the following set of equations,41 \n𝑆21(𝐻,𝑡)=𝑆210+𝐷𝑡+𝜒(𝐻)\n𝜒̃0 \n𝜒(𝐻)=𝑀𝑒𝑓𝑓(𝐻−𝑀𝑒𝑓𝑓)\n(𝐻−𝑀𝑒𝑓𝑓)2−𝐻𝑒𝑓𝑓2−𝑖𝛥𝐻 (𝐻−𝑀𝑒𝑓𝑓) . (2) \nIn these equations 𝑆210 corresponds to the non-magnetic contribution to the complex transmission \nsignal , 𝜒̃0 is an imaginary function of the microwave frequency and film thickness and 𝜒(𝐻) is the \ncomplex susceptibility of the magnetic film. The term 𝐷𝑡 accounts for a linear drift of the recorded \n𝑆21 signal. 𝑀𝑒𝑓𝑓=𝑀𝑠−𝐻𝑘⫠, where 𝐻𝑘⫠ is the perpendicular anisotropy field and 𝐻𝑒𝑓𝑓=2𝑓\n𝛾µ0. The \n𝑆21 spectra were fitted to Eq. (2 ) in order to extract the linewidth 𝛥𝐻 and 𝐻𝑟 values. Fits t o Eq. (2) \nare shown as solid lines in Fig. 4. \nThe experimentally measured total d amping parameter ( 𝛼𝑡𝑜𝑡𝑎𝑙 ), including both the intrinsic \ncontribution (Gilbert damping) and extrinsic contributions , was extracted by fitting 𝛥𝐻 versus \nfrequency to the following expression, 41 \nµ0𝛥𝐻=4𝛼𝑡𝑜𝑡𝑎𝑙 𝑓\n𝛾+µ0𝛥𝐻0 , (3) \nwhere 𝛥𝐻0 is the frequency independent linewidth broadening due to sample inhomogeneity . \nBeside s 𝛼𝑡𝑜𝑡𝑎𝑙 , 𝑀𝑒𝑓𝑓 can also be extracted by fitting the 𝐻𝑟 versus frequency results to the \nexpression µ0𝐻𝑟=2𝜋𝑓\n𝛾+µ0𝑀𝑒𝑓𝑓 . (4) \nTypical temperature dependent results for 𝑓 versus 𝐻𝑟 and 𝛥𝐻 versus 𝑓 are shown in Fig. 5 for \nthe 1 2.6 at% Re -doped Fe65Co35 film. Extracted values of 𝑀𝑒𝑓𝑓 at different temperatures are given \nin Table 3 for all samples . As expected, the results show that 𝑀𝑒𝑓𝑓 decreas es with increasing \ndopant concentration. Since 𝑀𝑒𝑓𝑓=𝑀𝑠−𝐻𝑘⫠ and the film thickness is large enough to make a \npossible contribution from out -of-plane anisotropy negligible one can make the justified \nassumption that 𝑀𝑒𝑓𝑓≈𝑀𝑠. The analysis using Eqs. (2) – (4) also give values for the Land é 𝑔-\nfactor ( 𝛾=𝑔µ𝐵\nħ), yielding 2.064 and 2.075 for the undoped and 12.6 at% doped samples, \nrespectively (similar values are obtained at all temperatures). \nAs indicated above, the d amping parameters extracte d from FMR measurements ( 𝛼𝑡𝑜𝑡𝑎𝑙) include \nboth intrinsic and extrinsic contributions. One of the most common extrinsic contribution s is TMS , \nwhich is avoided in this study by measuring FMR with the magnetic field applied out of the film \nplane. Except TMS , extrinsic contributions such as eddy curr ent damping and radiative damping \nare expected to contribute the measured damping . In a metallic ferromagnet, which is placed on \ntop of a CPW , precession of spin waves induces AC currents in the ferromagnet ic film, thereby \ndissipating energy . The radiative damping has similar origin as the eddy current damping, but here \nthe precession of the magnetization induces microwave -frequency currents in the CPW where \nenergy is dissipated. Thus, there are two extrinsic contributions to the measured damping ; one that \nis caused by eddy currents in the ferromagnet ic film (𝛼𝑒𝑑𝑑𝑦) and another one caused by eddy \ncurrents in the CPW ( 𝛼𝑟𝑎𝑑).23 In order to obtain the reduced damping of the films (𝛼𝑟𝑒𝑑), which \nwe expect to be close to the intrinsic damping of the films, the extrinsic contributions should be \nsubtracted from 𝛼𝑡𝑜𝑡. We have neglected any contribution to the measured damping originating \nfrom spin -pumping into seed and capping layers. However, since spin -pumping in low spin -orbit \ncoupling materials like Ru with thickness quite less than the spin -diffusion length is quit e small, \nthe assumption of negligible contribution from spin -pumping is justified. The t otal damping can \nthus be given as 𝛼𝑡𝑜𝑡=𝛼𝑟𝑒𝑑+𝛼𝑟𝑎𝑑+𝛼𝑒𝑑𝑑𝑦 . \nWhen the precession of the magnetization is assumed to be uniform in the sample , the expression \nfor radiative damping can be given as23 \n𝛼𝑟𝑎𝑑=𝜂𝛾µ02𝑀𝑠𝛿𝑙\n2𝑍0𝑤 , (5) \nwhere 𝑍0 =50 Ω is the waveguide impedance, 𝑤=240 µm is the width of the CPW center \nconductor , 𝜂 is a dimensionless parameter that accounts for FMR mode profile, δ is the thickness \nand 𝑙 is the length of the sample. The l ength of all samples were 4mm and the thickness 20nm for \nthe undoped and 1 2.6 at% Re-doped films and 40nm for the 3.0 at% and 6.6 at% Re-doped films. \nTemperature dependent radiative damping contributions for all Fe 65Co35 films are given in Table \n4. Beside s 𝛼𝑟𝑎𝑑, the 𝛼𝑒𝑑𝑑𝑦 contribution should also be calculated and extracted from 𝛼𝑡𝑜𝑡𝑎𝑙 to extract \nthe reduced damping parameter. 𝛼𝑒𝑑𝑑𝑦 can be estimated by the expression23 \n𝛼𝑒𝑑𝑑𝑦 =𝐶𝛾µ02𝑀𝑠𝛿2\n16𝜌 , (6) \nwhere 𝐶 is a parameter describing the distribution of eddy current s within the films and its value \nis 0.5 in our studied samples and 𝜌 is the resistivity of the films. Resistivity is measured for all \nfilms with different dopant concentrations at different temperatures. It is in the range of 8.2×10-8 \nto 5.6 ×10-8 𝛺𝑚 for undoped, 5.7 ×10-7 to 5.3 ×10-7 𝛺𝑚 for 3.0 at% doped , 6.9 ×10-7 to 6.1 ×10-\n7 𝛺𝑚 for 6.6 at% doped and 3.9×10-7 to 3.6 ×10-7 𝛺𝑚 for 12.6 at% doped films. Temperature \ndependent eddy current damping contributions , which are negligible, for all Fe 65Co35 films are \ngiven in Table 5. \n𝛼𝑡𝑜𝑡 (filled symbols) and 𝛼𝑟𝑒𝑑 (open symbols) versus temperature for the differently Re -doped \nFe65Co35 films are shown in Fig. 6 . Both damping parameter s slowly decrease with decreasing \ntempera ture. Moreover, the damping parameter increases with increasing Re concentration; the \ndamping parameter is 4 times as large for the 12.6 at% Re -doped sample compared to the undoped \nsample . Since the damping parameter depends both on disorder induced scattering and spin-orbit \ncoupling, the observed enhanc ement of the damping parameter can emerge from the electronic \nstructure of the alloy and large spin -orbit coupling of Re. \nA c omparison between temperature dependent experimental 𝛼𝑡𝑜𝑡 and 𝛼𝑟𝑒𝑑 values and \ntheoretically calculated intrinsic damping parameters is shown in Fig. 7 for the undoped and 12.6 \nat% Re -doped Fe 65Co35 films. In agreement with the experimental results, the theoretically \ncalculated damping parameters decrease in magnitude with decreasing temperature . It has been \nargued by Schoen et al., 42 that the contribution to the intrinsic Gilbert damping parameter comes \nprimarily from the strong electron -phonon coupling at high temperatures due to interband \ntransition whereas at a low temperature, density of states at Fermi level (𝑛(𝐸𝐹)) and spin -orbit \ncoupling give rise to intraband transition. In Fig. 8, we show the correspondence between the \ncalculated damping parameter at 10 K with the density of states (spin up +spin down) at Fermi \nlevel for varying Re concentration. The increasing trend in both properties is obviously seen. The \nincrease in DOS mainly comes from increasing DOS at Re sites in the spin -up channel. In the \ninset, the calculated spin -polarization as a function of Re concentration is shown. Spin polarization \nis defined as 𝜁=𝑛(𝐸𝐹)↑−𝑛(𝐸𝐹)↓\n𝑛(𝐸𝐹)↑+𝑛(𝐸��)↓ where the contribution from both spin channels are seen. It is \nclearly observed that Re doping decreases the spin polarization. \n \nOne should note that a quantitative comparison between theory and experiment requires more \nrigoro us theoretical considerations. The difference between experimental and theoretical results \nfor the damping parameter may be explained by the incompleteness of the model used to calculate \nthe Gilbert damping parameter by neglecting several complex scatterin g processes. Firstly, the effect of spin fluctuations was neglected, which in principle could be considered in the present \nmethodology if the temperature dependent magnetization and hence information about the \nfluctuations of atomic moments were available from Monte -Carlo simulations. Other effects such \nas non-local damping and more sophisticated treatment of atomic displac ements in terms of \nphonon self -energies40 that may contribute to the relaxation of the magnetization in magnetic thin \nfilm materials have been neglected . Nevertheless, a qualitative agreement has been achieved where \nboth experimental and theoretical results show that there is a significant increase of the damping \nparameter with increasing concentration of Re. \n \n4. CONCLUSION \nStatic and dynamic magnetic properties of rhenium doped Fe 65Co35 thin films have been \ninvestigated and clarified in a combined experimental and theoretical study. Results from first \nprinciples theoretical calculations show that the saturation magnetization gradually decreases with \nincreasing Re concentration, from 2.3T for the undoped sample to 1.95T for the 10% Re -doped \nsample. The experimental results for the dependence of the saturation magnetization on the Re -\ndoping are in agreement with the theoretical results, although indicating a more pronounced \ndecrease of the saturation magnetization for the largest doping concentrations. The theoretical \ncalculations show that the intrinsic Gilbert damping increases with increasing Re concentration; at \nroom temperature the damping parameter is 2.8×10−3, which increases to 7.3×10−3 for the 10 \nat% Re -doped sample. Moreover, temperature dependent calculations of the Gilbert damping \nparameter reveal a weak decrease of the value with decreasing temperature . At a low temperature, \nour theoretical analysis showed the prominence of intra band contribution arising from an increase \nin the density of states at Fermi level. The experimental results for the damping parameter were \ncorrected for radiative and eddy current contributions to the measured damping parameter and \nreveal similar trends as observed in the theoretical results; the damping parameter increases with \nincreasing Re concentration and the damping parameter value decreases with decreasing \ntemperature. The room temperature value for the reduced damping paramet er was 2.7×10−3 for \nthe undoped sample, which increased to 9.0×10−3 for the 1 2.6 at% Re -doped film. The \npossibility to e nhanc e the damping parameter for Fe65Co35 thin films is a promising result since \nthese materials are used in magnetic memory applications and higher data rates are achievable if \nthe damping parameter of the material is increased. \n \nACKNOWLEDGEMENT \nThis work is supported by the Knut and Alice Wallenberg (KAW) Fou ndation, Grant No. KAW \n2012.0031 and by the Marie Curie Action “Industry -Academia Partnership and Pathways” (ref. \n612170, FP7 -PEOPLE -2013 -IAPP). The authors acknowledge financial support from Swedish \nResearch Council (grant no. 2016 -05366) and Carl Tryggers Stiftelse (grant no. CTS 12:419 and \n13:413). The simulations were performed on resources provided by the Swedish National Infrastructure (SNIC) at National Supercomputer Centre at Link öping University (NSC). M. 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Shaw, Nature Physics, 12, 839 –842 (2016). \n \n \nFigure 1 (a) GIXRD plot for Fe 65Co35 films with dif ferent Re concentrations. S hift of (110) peak \ndiffraction peak with Re concentration is given as insert . (b) Lattice parameter versus Re \nconcentration. Circles are lattice parameters extracted from XRD measurements and squares are \ncalculated th eoretical values. Line s are guide to the eye. \n \nFigure 2 (a) Normalized room temperature magnetization versus magnetic field for Fe 65Co35 \nfilms with different Re concentration . (b) Low temperature saturation magnetization versus Re \nconcentration. Circles are experimental data and squares corresponding calculated results. \nExperimental 𝝁𝟎𝑴𝒔 values were extracted from temperature dependent FMR results. Lines are \nguide s to the eye . \n \n \n \nFigure 3 𝝁𝟎𝑯𝒓 versus in -plane angle of magnetic field 𝝓𝑯 for different dopant concentrations of \nRe. Black squares are experimental data and red line s are fits to Eq. (1). \n \n \n \nFigure 4 Room temperature real (a and c) and imaginary (b and d) 𝑺𝟐𝟏 components versus out -\nof-plane magnetic field for Fe65Co35 thin films with 0% and 12.6 at% Re recorded at 20GHz . \nBlack squares are data points and red lines are fit s to Eq. (2). \n \n \n \n \nFigure 5 (a) Frequency versus 𝝁𝟎𝑯𝒓 values at different temperatures for the Fe65Co35 thin film \nwith 12.6 at% Re. Coloured lines correspond to fits to Eq. ( 4). (b) Linewidth 𝝁𝟎∆𝑯 versus \nfrequency at different temperatures for the same Re doping concentration. Coloured lines \ncorrespond to fits to Eq. ( 3). Symbols represent experimental data. \n \n \n \nFigur e 6 𝜶𝒕𝒐𝒕 versus temperature for Fe 65Co35 thin films with different concentration of Re. \nBesides showing 𝜶𝒕𝒐𝒕 , reduced 𝜶𝒓𝒆𝒅 values are also plotted obtained by subtraction of radiative \ndamping and eddy current damping contributions from 𝜶𝒕𝒐𝒕. Error bars are given for measured \n𝜶𝒕𝒐𝒕 (same size as symbol size ). \n \n \nFigur e 7 𝜶𝒕𝒐𝒕 versus temperature for Fe 65Co35 thin films with 0 at% and 1 2.6 at% concentration \nof Re. Beside s showing 𝜶𝒕𝒐𝒕 , reduced 𝜶𝒓𝒆𝒅 values are also plotted obtained by subtraction of \nradiative damping and eddy current damping contribution s from 𝜶𝒕𝒐𝒕. In addition to \nexperimental results theoretically calculated intrinsic damping parameters are given for the \nsimilar concentrations of Re . Error bars are given for measured 𝜶𝒕𝒐𝒕 (same size as symbol size) . \n \n \n \nFigure 8 Calculated density of states at Fermi level (left axis) and damping parameter (right \naxis) are shown as a function of Re concentration. In the inset, spin -polarization is shown as a \nfunction of Re concentration. \n \n0 0.03 0.06 0.09 0.12\nRe concentration0.90.951DOS at EF (States/eV)\n0 0.03 0.06 0.09 0.12\nRe concentration0.350.40.450.50.55Spin polarization\n0123456\nDamping parameter (x 10-3)Re \n(at%) 𝑡𝑅𝑢,𝑐𝑎𝑝 \n(nm) 𝜎 \n(nm) 𝑡𝐹𝑒𝐶𝑜 \n(nm) 𝜎 \n(nm) 𝑡𝑅𝑢,𝑠𝑒𝑒𝑑 \n(nm) \n \n(nm) \n0 2.46 1.89 39.71 0.67 2.74 0.66 \n3.0 2.47 1.80 37.47 0.59 2.45 1.03 \n6.6 1.85 0.50 37.47 0.51 2.13 0.90 \n12.6 2.15 1.49 37.38 0.64 1.89 1.03 \nTable 1 Thickness and roughness (𝝈) values for different layers in films extracted from XRR \ndata. Error margin is 0.02nm for all thickness and roughness values. \n \nRe (at%) 𝜇0𝐻𝑢 (mT) 𝜇0𝑀𝑒𝑓𝑓 (T) \n0 2.20 2.31 \n3.0 2.10 2.12 \n6.6 2.30 1.95 \n12.6 2.20 1.64 \nTable 2 Room temperature 𝝁𝟎𝑴𝒆𝒇𝒇 and 𝝁𝟎𝑯𝒖 values for Fe 65Co35 films with different \nconcentration of Re extracted by fitting the angle dependent cavity FMR data to Eq. (1). \n \n \nTemperature (K) 0% Re 3.0 at% Re 6.6 at% Re 12.6 at% Re \n𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) \n300 2.29 2.16 1.99 1.61 \n200 2.31 2.16 2.04 1.67 \n150 2.33 2.24 2.06 1.70 \n100 2.36 2.25 2.07 1.72 \n50 2.36 2.27 2.08 1.74 \nTable 3 Temperature dependent 𝝁𝟎𝑴𝒆𝒇𝒇 values for Fe65Co35 films with different concentrati on \nof Re extracted by fitting broadband out -of-plane FMR data to Eq. (4). Error margin is about 10 \nmT. \n \n \nTemperature(K) 𝛼𝑟𝑎𝑑 (×10-3) \n0% Re 3.0 at% Re 6.6 at% Re 12.6 at% Re \n300 0.218 0.482 0.438 0.154 \n200 0.222 0.494 0.450 0.160 \n150 0.216 0.499 0.454 0.162 \n100 0.225 0.502 0.456 0.219 \n50 0.221 0.505 0.458 0.166 \nTable 4 Temperature dependent r adiative damping contribution to total damping parameter for \nFe65Co35 films with different concentration of Re calculated using Eq. (5). \n \nTemperature(K) 𝛼𝑒𝑑𝑑𝑦 (×10-3) \n0% Re 3.3 at% Re 6.6 at% Re 12.6 at% Re \n300 0.038 0.077 0.064 0.006 \n200 0.047 0.081 0.067 0.006 \n150 0.050 0.084 0.070 0.006 \n100 0.055 0.084 0.073 0.007 \n50 0.058 0.086 0.075 0.007 \nTable 5 Temperature dependent eddy current damping contribution to total damping parameter \nfor Fe 65Co35 films with different concentration of Re calculated using Eq. ( 6). \n \nFigure A1 RBS spectra for the Re -doped Fe 65Co35 films. \n" }, { "title": "2009.12073v2.Temperature_dependence_of_the_damping_parameter_in_the_ferrimagnet_Gd__3_Fe__5_O___12__.pdf", "content": " Temperature dependence of the damping parameter in the ferrimagnet \nGd 3Fe5O12 \nIsaac Ng,1,2 a) Ruizi Liu1,3 a), Zheyu Ren1,3, Se Kwon Kim,4 and Qiming Shao 1,2,3 b) \n1Department of Electronic and Computer Engineering Department, Hong Kong University of \nScience and Technology, Clear Water Bay, Kowloon, Hong Kong SAR, China \n2Department of Physics, Hong Kong University of Science and Technology , Clear Water Bay, \nKowloon, Hong Kong SAR, China \n3Guangdong -Hong Kong -Macao Joint Laboratory for Intelligent Micro -Nano Optoelectronic \nTechnology, The Hong Kong University of Science and Technology, Hong Kong, China \n4Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, \nRepublic of Korea \na) Contributed equally b) Email: eeqshao@ust.hk \n \nAbstract \nThe damping parameter 𝛼FM in ferrimagnets defined according to the conventional practice for \nferromagnets is known to be strongly temperature dependent and diverge at the angular \nmomentum compensation temperature, where the net angular momentum vanishes. However, \nrecent theoretical and experimental developments on ferrimagnetic metals suggest that the \ndamping parameter can be defined in such a way, which we denote by 𝛼FiM, that it is free of \nthe diverging anomaly at the angular momentum compensation point and is little dependent o n \ntemperature. To further understand the temperature dependence of the damping parameter in \nferrimagnets, we analyze several data sets from literature for a ferrimagnetic insulator, \ngadolinium iron garnet , by using the two different definitions of the damping parameter. Using \ntwo methods to e stimate the individual sublattice magnetization s, which yield results consistent \nwith each other, we found that in all the used data sets, the damping parameter 𝛼FiM does not \nincrease at the angular compensation temperature and shows no anomaly whereas th e \nconventionally defined 𝛼FM is strongly dependent on the temperature. \n \n \n \n Antiferromagnets have been one important focus in spintronics due to their properties distinct from \nmore conventional ferromagnets including the zero stray field, ultrafast dynamics, and immunity \nto external field1,2. Recently, antiferromagnetically coupled ferrimagnets have emerged as a new \nmaterial platform to study antiferromagnetic dynamics as suggested by the recent discoveries of \ncurrent -driven magnetization switching near magnetization compensation point3,4, where the net \nmagnetization vanishes, and fas t domain -wall dynamics at the angular momentum compensation \ntemperature4,5,6,7, where the net angular momentum vanishes. However, magnetic resonance and \ndynamics of ferrimagnets have not been fully understood par tly due to the involvement of multiple \nmagnetic sublattices and the resultant internal complexity. The dissipation rate of angular \nmomentum in magnetic material is manifested as the linewidth in resonance spectrum . One \nquantity of particular importance in the dissipative dynamics of ferrimagnets is the damping \nparameter, which is a characteristic of the magnetic material that determines the Gilber t-like \ndamping of angular momentum and is usually denoted by the dimensionless number 𝛼. Early \nliterature sugge sted that the effective damping parameter αFM for ferrimagnets defined by a value \nthat is proportional to the line width of the resonance response is strongly temperature -dependent \nand increases anomalously near the angular momentum compensation temperature ( TA).8 Recent \nstudies have provided a new interpretation: the damping parameter can be defined in such a way \nthat it is independent of temperature near the TA while the temperature depend ence of the \nferromagnetic resonance (FMR) is attributed to the temperature dependence of the net angular \nmomentum .9,10,11 A. Kamra et al. have theoretically demonstrated this new perspective by \naccounting the Rayleigh dissipation function in a two -sublattice magnetic system, and the resu ltant \nGilbert damping parameter is independent of temperature near the TA.11 This damping parameter \ndenoted by αFiM is defined as follows: \n αFiM=|snet\nstotal|αFM, (1) \nwhere the snet and stotal are the net and total angular momentum, respectively . The snet is calculated \nby the differenc e of the angular momentum between two sublattice s (snet=|𝑠1−𝑠2 |) and stotal is \ncalculated by the total magnitude of the angular momentum (stotal=|𝑠1|+|𝑠2|). D.-H. Kim et al. \nhave experimentally studied the current -driven domain wall motion in ferrimagnetic metal alloy \nGdFeCo and revealed that the damping parameter αFiM is indeed independent of temperature near \nthe TA.10 Furthermore, T. Okuno et al. has reported that αFiM of the GdFeCo is temperature independent when the FMR measurement temperature is approaching the TA.6 The FMR of \nferrimagnetic thin films below the TA is difficult to achieve because of much enhanced \nperpendicular magnetic anisotropy at lower temperatures . It would be desirable that a full \ntemperature range o f FMR can be investigated for ferrimagnets. \nThe divergence of the conventionally defined damping parameter αFM at TA can be understood \neasily by considering the energy dissipation rate given by 𝑃=αFM𝑠net 𝒎̇2 (which is twice the \nRayleigh dissipation function) , where 𝒎 is the unit magnetization vector. For the given power 𝑃 \nthat is pumped into the ferrimagnet by e.g., applying microwave for FMR , as the temperature \napproaches TA, the net spin density 𝑠net decreases and thus αFM increases. Exactly at TA, the net \nspin density vanishes, making αFM diverge and thus ill -defined. Note that the divergence of αFM at \nTA is due to the appearance of the net spin density 𝑠net in the dissipation rate and should not be \ninterpreted to indicate the divergence of the dissipation rate, which is always finite. In terms of the \nalternative damping parameter αFiM, the energy dissipation rate is given by 𝑃=αFiM𝑠tot 𝒎̇2. The \ntotal spin density 𝑠tot is always finite and has weak temperature dependence, and thus αFiM is well -\ndefined at all temperatures with possibly we ak temperature dependence. This suggests that αFiM, \nwhich is well -defined at all temperatures, might be more useful to describe the damping of \nferrimagnetic dynamics, particular ly in the vicinity of TA, than the more conventional αFM which \ndiverges and thus ill -defined at TA. One way to appreciate the physical meaning of αFiM is to \nconsider a special model, where the energy dissipation of a ferrimagnet occurs independently \nthrough the dynamics of each sublattice and all the sublattice s have the same damping parameter . \nIn this case, αFiM is nothing but the damping parameter of the sublattices. So far, the discussion of \nferrimagnetic damping is limited to ferrimagnetic metals, while ferrimagnetic insulators have \nshown the potential for u ltralow -power spintronics .12,13,14,15,16 \nIn this paper, we investigate the temperature dependence of damp ing parameters in ferrimagnetic \ninsulator, gadolinium iron garnet ( Gd3Fe5O12, GdIG) , by surveying the literature of studies on the \ntemperature dependence of FMR. Since the stotal is usually not given in the literature, we adopt \ntwo different methods to cal culate the individual sublattice magnetization ( MFe and MGd) and then \nevaluate stotal. The first method is to use the magnetization of yttrium iron garnet ( Y3Fe5O12, YIG) \nas the MFe as done in Ref.17, where nuclear magnetic resonance experiments show that the \nmagnetization contribution from iron is similar in YIG and GdIG since yttrium does not cont ribute \nthe magnetization in YIG, and then obtai n MGd from the net magnetization and MFe. The second method uses Brillouin -like function to simulate the temperature dependence of GdIG \nmagnetization , the angular momentum of each individual sublattice can be calculated with the \nBrillouin function . We found consistent results between these two different methods that the \ndamping parameter αFiM is almost temperature -independent near the TA, unlike the conventionally \ndefined αFM which is strongly temperature -dependent and diverge at TA. \nThe FMR linewidth ( ΔH) of GdIG is utilized to find the conventional damping parameter αFM: \n ΔH=αFM\ngeffμB/ℏfres+ΔH0 , (2) \nwhere geff is the effective Landé g -factor, μB is the Bohr magneton, ℏ is the reduced Planck \nconstant , ΔH0 is the frequency -independent inhomogeneous broadening linewidth, and fres is the \nresonance frequency. Then, to convert the αFM to the αFiM, we need to find the ratio snet\nstotal. Note that \nαFM diverges as the temperature approaches TA, meaning that Eq. (2) can be used only when it is \nsufficiently far away from the TA. Therefore, we will only employ data sufficiently far away from \nTA in this perspective. The net spin density snet is calculated from the difference between the \nangular momentum of Fe and Gd: \n sFe=MFe\ngFeμB/ℏ , \nsGd=MGd\ngGdμB/ℏ , \nsnet=|sFe-sGd|=Mnet\ngeffμB/ℏ , (3) \nwhere the Mnet is the net magnetization, gFe and gGd is the Landé g-factor of the iron and \ngadolinium sublattice, respectively. The net magnetization is given by \n Mnet=|MFe-MGd| , (4) \nwhich is normally measured by a superconducting quantum interference device or a vibrating -\nsample magnetometer and provided in the literature.18,19,20 \n \nMETHOD 1 \nWe can use the magnetization of YIG as an approx imation for the MFe to calculat e the MFe and \nMGd from GdIG net magnetization, as yttrium does not contribute to the magnetization of YIG , \nwhich we refer to as Method 1. Experimentally, Boyd et al.17 used the nuclear ferromagnetic resonance technique to determine temperature -dependent MFe in YIG and GdIG and found that \nthey are very similar. Thi s approximation has been used in previous literature and has produced \nreasonable results.21 The magnetization of YIG is obtained from Ref.18. With MFe and MGd \nknown, we can determine the angular momentum of each sublattice with it s respective g -factor . \nThe g factors of Fe and Gd are very similar , the g -factor of iron in measured from YIG and is \ndetermined as 𝑔𝐹𝑒=2.0047 .22 The g -factor of Gd sublattice is 𝑔𝐺𝑑=1.994 and is determined by \nmeasurement of GdIG 23. The TM and TA will be very close to each other, with TA slightly higher \nthan TM. We can calculate the total spin density stotal using \nstotal=sFe+sGd . (5) \nThe net spin density snet can be calculated using Eq. (3) and we can obtain effective g -factor \nmeanwhile. Finally, we can calculate the αFM using Eq. (2) and the αFiM using Eq. (1). \n \nMETHOD 2 \nThe second method is to use the Brillouin -like function to simulate the temperature dependence of \nmagnetization.24 Due to the weak coupling of the Gd -Gd interaction, the gadolinium magnetic \nmoments follow a paramagnetic behavior and increase drastically at low temperatures. The net \nmagnetization in GdIG can be describe d by the sum of the three sublattices with a and d sublattice s \ncorrespond ing to Fe an d c sublattice correspond ing to Gd: \nMnet=|Ma+Mc−Md|. (6) \nThe individual magnetization component can be simulate d by the Brillouin function 𝐵𝑆𝑖(𝑥𝑖) \nMi(T)=Mi(0)BSi(xi) . (7) \nThe 𝑀𝑖(0) is the individual m agnetization at 0 K. \nMd(0)=3nmFe=3ngdSdμB , \nMa(0)=2nmFe=2ngaSaμB , \nMc(0)=3nmGd=3ngcScμB , (8) \n𝑛 is the number of GdIG formula unit per unit volume, it can be calculated using 𝑁𝐴/(𝜌𝑀𝑟), where \n𝑁𝐴 is the Avogadro’s number, 𝜌 and 𝑀𝑟 are the density ( 6.45 𝑔𝑐𝑚−3 25) and molar mass (942.97) of GdIG respectively . 𝑆𝑖 is the electron spin of the respective sublattice . For GdIG, 𝑆𝑑 and 𝑆𝑎 are \n5/2 and 𝑆𝑐 is 7/2. 𝑔𝑖 is the individual g factor and 𝑥𝑖 is defined as: \nxd=(μ0SdgdμB\nkBT)(nddMd+ndaMa+ndcMc) , \nxa=(μ0SagaμB\nkBT)(nadMd+naaMa+nacMc) , \nxc=(μ0ScgcμB\nkBT)(ncdMd+ncaMa+nccMc) , (9) \n𝑛𝑖𝑗 are the Weiss coefficients between two sublattice s, which account for the intersublattice \nmolecular field coupling ( 𝑖≠𝑗) or intrasublattice molecular field interactions ( 𝑖=𝑗).24 𝜇0 is \npermeability of vacuum . \n \nTo determine the snet and stotal from the magnetization fitting will require the sublattice g -factor \ngGd and gFe. gFe in a and d sublattice can be experimentally measured from YIG and is \ndetermined as 𝑔𝐹𝑒,𝑑=2.0047 ,𝑔𝐹𝑒,𝑎=2.003.22 The g -factor of Gd c sublattice has the same value \nas the one in Method 1, 𝑔𝐺𝑑=1.994.23 With the value of the individual sublattice g -factor, the \nangular momentum of each sublattice can be calculate d from Eq. (3). Then we can calculate the \neffective gyromagnetic ratio and effective g -factor with the sublattice magnetization and angular \nmomentum . \nγeff=MFe,d−MFe,a−MGd,c \nsFe,d−sFe,a−sGd,c , \n(11) \ngeff=γeffℏ\nμB , \nThe ratio snet/stotal can be calculated where snet=|sFe,d−sFe,a−sGd,c| and stotal =sFe,d+\nsFe,a+sGd,c, with both the 𝑔eff and angular momentum known . Eventually, the value of αFiM is \nobtained from Eq. (1). \n \nFigure 1. The a nalysis of GdIG data from Rodrigue et al.23 and Dionne et al.18 (a) Calculated \nindividual magnetization as a function of temperature using Method 1. (b) The Magnetization \ncurve of GdIG using Brillouin fitting method (Method 2) compare d to the magnetization from \nDionne et al.18 (c) The geff of GdIG calculated from Method 1 as the green cross and from Method \n2 as the red line compared to the grey dot geff from Rodrigue et al. ([100] direction) .23 (d) (e) (f) \nComparing the damping parameter αFM (red dot) to αFiM based on Method 1 (green cross ) and αFiM \nbased on Method 2 (grey dot) for three directions ([100], [110] and [111]) . \n \nFigure 2. The a nalysis of GdIG data from Flaig et al.26 (a) Calculated individual magnetization as \na function of temperature using Method 1. (b) The Magnetization curve of GdIG using Brillouin \nfitting method (Method 2) compare d to the magnet ization from Flaig et al.26 (c) The geff of GdIG \ncalculated from Method 1 as the red cross and from Method 2 as the green line compared to the \nblue dot geff from Flaig et al.26 (d) Comparing the damping parameter αFM (red dot) to αFiM based \non Method 1 (green cross ) and αFiM based on Method 2 (blue dot). \n \nFigure 3. The a nalysis of GdIG data from Calhoun et al19,20. (a) Calculated individual \nmagnetization as a function of temperature using Method 1. (b) The Magnetization curv e of GdIG \nusing Brillouin fitting method (Method 2) compare d to the magnetization from Calhoun et al.20 (c) \nThe 𝑔eff of GdIG calculated from Method 1 as the green cross and from Method 2 as the grey line \ncompared to the yellow dot 𝑔eff from Calhoun et al.19 (d) Comparing the damping parameter αFM \n(red dot) to αFiM based on Method 1 ( green cross and αFiM based on Method 2 ( blue dot). \n \nRESULTS AND DISCUSSION S \nWe analyze three datasets and evaluate the validity of Method 1 and M ethod 2 using the formula \nprovided above. The first dataset is from Rodrigue et al.23, where the ΔH and geff in three \ndirections [100], [110], [111] are provided. Note that the value of geff is calculated using the Kittel \nequation in Rodrigue ’s paper . The Mnet is obtained from Dionne et al.18 where the GdIG has a \nsimilar compensation temperature to Rodrigue et al.23. fres=9.165GHz and we assume that ΔH0 is \nzero since the GdIG is a polished sphere . We analyze t he data using Method 1 and Method 2 and \nplot the results in Fig. 1. For Method 1, w e can observe that the calculat ed temperature dependence \nof the MGd (see Fig. 1a) and the obtained g -factor (see Fig. 1c) are reasonable . Using Method 2, \nwe get the fitting curves for magnetization from each sublattice and g -factor, which fit accurately \nto the experimental data. \n \nThe second dataset of the temperature dependence of FMR below the TA is from Maier -Flaig et \nal.,26 where the g -factor, ΔH, and Mnet are also provided. Again, we can see that the magnetization \nas a function of temperature from two methods are in accordanc e with Flaig’s data (see Fig. 2). \ngeff calculated dots from Method 1 and fitting curves from Method 2 are highly consistent with \nthe data, which illustrates that both two methods are well established. \n \nThe third set of data is from B. A. Calhoun et al. ,19,20 where fres= 9.479GHz. Similar results to the \nabove two datasets are obtained as shown in Fig. 3. \n \nTo directly compare the above two methods, the ferrimagnetic damping parameter αFiM calculated \nfrom the se two methods are plotted against each other in Fig. 1, 2 and 3, using the data from \nRodrigue et al.23, Flaig et al.26 and Calhoun et al.19. For all datasets , two different methods all give \nconsistent results and have similar values: the newly defined damping parameter αFiM of a \nferrimagnetic material is not divergent near the TA and has much lower value than αFM. The αFiM \nin all three datasets is at low value, revealing the achievability of fast domain -wall dynamics in \nferrimagnetic insulator at the angular momentum compensation temperature . \n \nCONCLUSION \nIn this work, we survey the literature dataset of FMR studies on the fe rrimagnet ic insulator GdIG \nand find that the ferrimagnetic damping parameter αFiM does not increase when the temperature \napproaches the TA, differing from the conventionally defined αFM that shows divergence near the \nTA. This validates the recently developed theory about damping in the ferrimagnetic systems and \nreveals that the damping parameter, when it is appropriately defined with no divergence at all \ntemperatures, is not as high as previously thought. Our work suggests that analyzing the dynamics \nof ferrimagnets needs extra caution, that is not required for ferromagnets, in particular in the vicinity of the TA to avoid unphysical divergences . Besides, potentially lower damping in \ninsulators suggests that ferrimagnetic insulators are promising for future ultrafast and ultralow -\npower spintronic applications. \n \nACKNOWLEDGEMENT \nThe authors at HKUST were supported by the Hong Kong Research Grants Council -Early Career \nScheme (Grant No. 26200520 ) and the Research Fund of Guangdong -Hong Kong -Macao Joint \nLaboratory for Intelligent Micro -Nano Optoelectronic Technology (Grant No. \n2020B1212030010) . S.K.K. was supported by Brain Pool Plus Program through the National \nResearch Foundation of Korea funded by the Ministry of Science and ICT (Grant No. NRF -\n2020H1D3A2A03099291) and by the National Research Foundation of Korea funded by the \nKorea Government via the SRC Center for Quantum Coherence in Condensed Matter (Grant No. \nNRF -2016R1A5A1008184). \n \nREFERENCES \n1. Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics. \nNature Nanotechnology vol. 11 231 –241 (2016). \n2. Baltz, V. et al. Antiferromagnetic spintronics. Rev. Mod. Phys. 90, (2018). \n3. Finley, J. & Liu, L. Spin -Orbit -Torque Effici ency in Compensated Ferrimagnetic Cobalt -\nTerbium Alloys. Phys. Rev. Appl. 6, (2016). \n4. Caretta, L. et al. Fast current -driven domain walls and small skyrmions in a compensated \nferrimagnet. Nat Nanotechnol (2018) doi:10.1038/s41565 -018-0255 -3. \n5. Kim, K. J. et al. Fast domain wall motion in the vicinity of the angular momentum \ncompensation temperature of ferri magnets. Nat Mater 16, 1187 –1192 (2017). \n6. Siddiqui, S. A., Han, J., Finley, J. T., Ross, C. A. & Liu, L. Current -Induced Domain Wall \nMotion in a Compensated Ferrimagnet. Phys. Rev. Lett. 121, (2018). \n7. Cai, K. et al. Ultrafast and energy -efficient spin –orbit torque switching in compensated \nferrimagnets. Nat. Electron. 3, 37–42 (2020). 8. Stanciu, C. D. et al. Ultrafast spin dynamics across compensation points in ferrimagnetic \nGdFeCo: The role of angular momentum compensation. Phys. Rev. B - Condens. Matt er \nMater. Phys. 73, (2006). \n9. Okuno, T. et al. Temperature dependence of magnetic resonance in ferrimagnetic GdFeCo \nalloys. Appl. Phys. Express 12, (2019). \n10. Kim, D. -H. et al. Low Magnetic Damping of Ferrimagnetic GdFeCo Alloys. Phys. Rev. \nLett. 122, 127203 (2019). \n11. Kamra, A., Troncoso, R. E., Belzig, W. & Brataas, A. Gilbert damping phenomenology \nfor two -sublattice magnets. (2018) doi:10.1103/PhysRevB.98.184402. \n12. Shao, Q. et al. Role of dimensional crossover on spin -orbit torque efficiency in magn etic \ninsulator thin films. Nat. Commun. 9, 3612 (2018). \n13. Avci, C. O. et al. Current -induced switching in a magnetic insulator. Nat. Mater. 16, 309 –\n314 (2017). \n14. Avci, C. O. et al. Interface -driven chiral magnetism and current -driven domain walls in \ninsulating magnetic garnets. Nat. 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C., Williams, H. J. & Espinosa, G. P. Magnetic \nStudy of the Heavie r Rare -Earth Iron Garnets. Phys. Rev. 137, A1034 –A1038 (1965). \n22. Geschwind, S. Sign of the ground -state cubic crystal field splitting parameter in Fe3+. \nPhys. Rev. Lett. 3, 207 –209 (1959). 23. Rodrigue, G. P., Meyer, H. & Jones, R. V. Resonance measureme nts in magnetic garnets. \nJ. Appl. Phys. (1960) doi:10.1063/1.1984756. \n24. Coey, J. M. D. Magnetism and magnetic materials . Magnetism and Magnetic Materials \nvol. 9780521816144 (Cambridge University Press, 2010). \n25. Espinosa, G. P. Crystal chemical study of the rare -earth iron garnets. J. Chem. Phys. 37, \n2344 –2347 (1962). \n26. Maier -Flaig, H. et al. Perpendicular magnetic anisotropy in insulating ferrimagnetic \ngadolinium iron garnet thin films. (2017). \n27. Caretta, L. et al. Relativistic kinematics of a magne tic soliton. Science (80 -. ). 370, 1438 –\n1442 (2020). \n28. Zhou, H. -A. et al. Compensated magnetic insulators for extremely fast spin -orbitronics. 1 –\n17 (2019). \n " }, { "title": "1105.4148v2.Magnetization_Dissipation_in_the_Ferromagnetic_Semiconductor__Ga_Mn_As.pdf", "content": "Magnetization Dissipation in the Ferromagnetic Semiconductor (Ga,Mn)As\nKjetil M. D. Hals and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway\nWe compute the Gilbert damping in (Ga,Mn)As based on the scattering theory of magnetization\nrelaxation. The disorder scattering is included non-perturbatively. In the clean limit, spin-pumping\nfrom the localized d-electrons to the itinerant holes dominates the relaxation processes. In the\ndi\u000busive regime, the breathing Fermi-surface e\u000bect is balanced by the e\u000bects of interband scattering,\nwhich cause the Gilbert damping constant to saturate at around 0.005. In small samples, the system\nshape induces a large anisotropy in the Gilbert damping.\nI. INTRODUCTION\nThe magnetization dynamics of a ferromagnet can be\ndescribed phenomenologically by the Landau-Lifshitz-\nGilbert (LLG) equation:1,2\n1\n\rdM\ndt=\u0000M\u0002He\u000b+M\u0002\"~G(M)\n\r2M2sdM\ndt#\n:(1)\nHere,\ris the gyromagnetic ratio, He\u000bis the e\u000bective\nmagnetic \feld (which is the functional derivative of the\nfree energy He\u000b=\u0000\u000eF[M]=\u000eM),Mis the magnetiza-\ntion andMsis its magnitude. The Gilbert damping con-\nstant ~G(M) parameterizes the dissipative friction process\nthat drives the magnetization towards an equilibrium\nstate.3In the most general case, ~G(M) is a symmetric\npositive de\fnite matrix that depends on the magnetiza-\ntion direction; however, it is often assumed to be inde-\npendent of Mand proportional to the unit matrix, as-\nsumptions which are valid for isotropic systems. Gilbert\ndamping is important in magnetization dynamics. It de-\ntermines the magnitudes of the external magnetic \felds4\nand the current densities1that are required to reorient\nthe magnetization direction of a ferromagnet. Therefore,\na thorough understanding of its properties is essential for\nmodeling ferromagnetic systems.\nThe main contribution to the Gilbert damping process\nin metallic ferromagnets is the generation of electron-\nhole pairs.1,2,5,6A model that captures this process was\ndeveloped by Kambersky.5In this model, the electrons\nare excited by a time-varying magnetization via electron-\nmagnon coupling. If the ferromagnet is in metallic con-\ntact with other materials, the spin-pumping into the ad-\njacent leads provides an additional contribution to the\nmagnetization relaxation.7A general theory that cap-\ntures both of these e\u000bects was recently developed.8The\nmodel expresses the ~G(M) tensor in terms of the scatter-\ning matrix Sof the ferromagnetic system ( m\u0011M=Ms):\n~Gij(m) =\r2\u0016h\n4\u0019Re\u001a\nTr\u0014@S\n@mi@Sy\n@mj\u0015\u001b\n: (2)\nThe expression is evaluated at the Fermi energy. Instead\nof~G(M), one often parameterizes the damping by the\ndimensionless Gilbert damping parameter ~ \u000b\u0011~G=\rM s.\nEq. (2) allows studying both the e\u000bects of the systemshape and the disorder dependency of the magnetization\ndamping beyond the relaxation time approximation.9\nIn anisotropic systems, the Gilbert damping is ex-\npected to be a symmetric tensor with non-vanishing o\u000b-\ndiagonal terms. We are interested in how this tensor\nstructure in\ruences the dynamics of the precessing mag-\nnetization in (Ga,Mn)As. Therefore, to brie\ry discuss\nthis issue, let us consider a homogenous ferromagnet in\nwhich the magnetization direction m=m0+\u000empre-\ncesses with a small angle around the equilibrium direc-\ntionm0that points along the external magnetic \feld\nHext.10For clarity, we neglect the anisotropy in the\nfree energy and choose the coordinate system such that\nm0= (0 0 1) and \u000em= (mxmy0). For the lowest order\nof Gilbert damping, the LLG equation can be rewritten\nas:_m=\u0000\rm\u0002Hext+\rm\u0002(~\u000b[Hext\u0002m]), where ~\u000b[:::]\nis the dimensionless Gilbert damping tensor that acts on\nthe vector Hext\u0002m. Linearizing the LLG equation re-\nsults in the following set of equations for mxandmy:\n\u0012\n_mx\n_my\u0013\n=\u0000\rHext \n\u000b(0)\nyy (1\u0000\u000b(0)\nxy)\n\u0000(1 +\u000b(0)\nxy)\u000b(0)\nxx!\u0012\nmx\nmy\u0013\n:\n(3)\nHere,\u000b(0)\nijare the matrix elements of ~ \u000bwhen the tensor\nis evaluated along the equilibrium magnetization direc-\ntionm0. For the lowest order of Gilbert damping, the\neigenvalues of (3) are \u0015\u0006=\u0006i\rH ext\u0000\rHext\u000b, and the\neigenvectors describe a precessing magnetization with a\ncharacteristic life time \u001c= (\u000b\rH ext)\u00001. The e\u000bective\ndamping coe\u000ecient \u000bis:10\n\u000b\u00111\n2\u0010\n\u000b(0)\nxx+\u000b(0)\nyy\u0011\n: (4)\nThe value of \u000bis generally anisotropic and depends on\nthe static magnetization direction m0. The magnetiza-\ntion damping is accessible via ferromagnetic resonance\n(FMR) experiments by measuring the linear relation-\nship between the FMR line width and the precession fre-\nquency. This linear relationship is proportional to \u001c\u00001\nand thus depends linearly on \u000b. Therefore, an FMR ex-\nperiment can be used to determine the e\u000bective damping\ncoe\u000ecient\u000b. In contrast, the o\u000b-diagonal terms, \u000b(0)\nxy\nand\u000b(0)\nyx, do not contribute to the lowest order in the\ndamping and are di\u000ecult to probe experimentally.\nIn this paper, we use Eq. (2) to study the anisotropy\nand disorder dependency of the Gilbert damping in thearXiv:1105.4148v2 [cond-mat.mtrl-sci] 2 Nov 20112\nferromagnetic semiconductor (Ga,Mn)As. Damping co-\ne\u000ecients of this material in the range of \u000b\u00180:004\u00000:04\nfor annealed samples have been reported.11{14The damp-\ning is anisotropically dependent on the magnetization\ndirection.11,12,14The few previous calculations of the\nGilbert damping constant in this material have indicated\nthat\u000b\u00180:003\u00000:04.11,15{17These theoretical works\nhave included the e\u000bects of disorder phenomenologically,\nfor instance, by applying the relaxation time approxima-\ntion. In contrast, Eq. (2) allows for studying the disor-\nder e\u000bects fully and non-perturbatively for the \frst time.\nIn agreement with Ref. 15, we show that spin-pumping\nfrom the localized d-electrons to the itinerant holes dom-\ninates the damping process in the clean limit. In the\ndi\u000busive regime, the breathing Fermi-surface e\u000bect is bal-\nanced by e\u000bects of the interband transitions, which cause\nthe damping to saturate. In determining the anisotropy\nof the Gilbert damping tensor, we \fnd that the shape of\nthe sample is typically more important than the e\u000bects\nof the strain and the cubic symmetry in the GaAs crys-\ntal.18This shape anisotropy of the Gilbert damping in\n(Ga,Mn)As has not been reported before and provides\na new direction for engineering the magnetization relax-\nation.\nII. MODEL\nThe kinetic-exchange e\u000bective Hamiltonian approach\ngives a reasonably good description of the electronic\nproperties of (Ga,Mn)As.19The model assumes that the\nelectronic states near the Fermi energy have the character\nof the host material GaAs and that the spins of the itin-\nerant quasiparticles interact with the localized magnetic\nMn impurities (with spin 5/2) via the isotropic Heisen-\nberg exchange interaction. If the s-d exchange interac-\ntion is modeled by a mean \feld, the e\u000bective Hamiltonian\ntakes the form:19,20\nH=HHoles +h(r)\u0001s; (5)\nwhereHHoles is the k\u0001pKohn-Luttinger Hamiltonian de-\nscribing the valence band structure of GaAs and h(r)\u0001s\nis a mean \feld description of the s-d exchange interac-\ntion between the itinerant holes and the local magnetic\nimpurities ( sis the spin operator). The exchange \feld\nhis antiparallel to the magnetization direction m. The\nexplicit form of HHoles that is needed for realistic model-\ning of the band structure of GaAs depends on the doping\nlevel of the system. Higher doping levels often require\nan eight-band model, but a six- or four-band model may\nbe su\u000ecient for lower doping levels. In the four-band\nmodel, the Hamiltonian is projected onto the subspace\nspanned by the four 3/2 spin states at the top of the\nGaAs valence band. The six-band model also includes\nthe spin-orbit split-o\u000b bands with spin 1/2. The spin-\norbit splitting of the spin 3/2 and 1/2 states in GaAs\nis 341 meV.21We consider a system with a Fermi level\nof 77 meV when measured from the lowest subband. Inthis limit, the following four-band model gives a su\u000ecient\ndescription:\nH=1\n2m\u0014\n(\r1+5\n2\r2)p2\u00002\r3(p\u0001J)2+h\u0001J\u0015\n+\n\r3\u0000\r2\nm(p2\nxJ2\nx+c:p:) +Hstrain +V(r): (6)\nHere, pis the momentum operator, Jiare the spin 3/2\nmatrices22and\r1,\r2and\r3are the Kohn-Luttinger pa-\nrameters.V(r) =P\niVi\u000e(r\u0000Ri) is the impurity scat-\ntering potential, where Riis the position of the impurity\niandViare the scattering strengths of the impurities23\nthat are randomly and uniformly distributed in the in-\nterval [\u0000V0=2;V0=2].Hstrain is a strain Hamiltonian and\narises because the (Ga,Mn)As system is grown on top\nof a substrate (such as GaAs).24The two \frst terms in\nEq. (6) have spherical symmetry, and the term propor-\ntional to\r3\u0000\r2represents the e\u000bects of the cubic sym-\nmetry of the GaAs crystal. Both this cubic symmetry\nterm25and the strain Hamiltonian24are small compared\nto the spherical portion of the Hamiltonian. A numeri-\ncal calculation shows that they give a correction to the\nGilbert damping on the order of 10%. However, the un-\ncertainty of the numerical results, due to issues such as\nthe sample-to-sample disorder \ructuations, is also about\n10%; therefore, we cannot conclude how these terms in-\n\ruence the anisotropy of the Gilbert damping. Instead,\nwe demonstrate that the shape of the system is the dom-\ninant factor in\ruencing the anisotropy of the damping.\nTherefore, we disregard the strain Hamiltonian Hstrain\nand the term proportional to \r3\u0000\r2in our investigation\nof the Gilbert damping.\nGaAs GaAs (Ga,Mn)As \nyx\nFIG. 1: We consider a (Ga,Mn)As system attached along the\n[010] direction to in\fnite ballistic GaAs leads. The scattering\nmatrix is calculated for the (Ga,Mn)As layer and one lattice\npoint into each of the leads. The magnetization is assumed to\nbe homogenous. In this paper, we denote the [100] direction\nas the x-axis, the [010] direction as the y-axis and the [001]\ndirection as the z-axis\nWe consider a discrete (Ga,Mn)As system with\ntransverse dimensions Lx2 f17;19;21gnm,Lz2\nf11;15;17gnm andLy= 50 nm and connected to in\fnite\nballistic GaAs leads, as illustrated in Fig. 1. The leads\nare modeled as being identical to the (Ga,Mn)As system,\nexcept for the magnetization and disorder. The lattice\nconstant is 1 nm, which is much less than the Fermi wave-\nlength\u0015F\u001810 nm. The Fermi energy is 0.077 eV when3\nmeasured from the lowest subband edge. The Kohn-\nLuttinger parameters are \r1= 7:0 and\r2=\r3= 2:5,\nimplying that we apply the spherical approximation for\nthe Luttinger Hamiltonian, as mentioned above.25We\nusejhj= 0:032 eV for the exchange-\feld strength. To\nestimate a typical saturation value of the magnetization,\nwe useMs= 10j\rj\u0016hx=a3\nGaAs withx= 0:05 as the doping\nlevel andaGaAs as the lattice constant for GaAs.26\nThe mean free path lfor the impurity strength V0is\ncalculated by \ftting the average transmission probability\nT=hGi=GshtoT(Ly) =l=(l+Ly),27whereGshis the\nSharvin conductance and hGiis the conductance for a\nsystem of length Ly.\nThe scattering matrix is calculated numerically us-\ning a stable transfer matrix method.28The disorder ef-\nfects are fully and non-perturbatively included by the\nensemble average h\u000bi=PNI\nn=1\u000bn=NI, whereNIis the\nnumber of di\u000berent impurity con\fgurations. All the\ncoe\u000ecients are averaged until an uncertainty \u000eh\u000bi=r\u0010\nh\u000b2i\u0000h\u000bi2\u0011\n=NIof less than 10% is achieved. The\nvertex corrections are exactly included in the scattering\nformalism.\nIII. RESULTS AND DISCUSSION\nWithout disorder, the Hamiltonian describing our sys-\ntem is rotationally symmetric around the axis parallel\ntoh. Let us brie\ry discuss how this in\ruences the\nparticular form of the Gilbert damping tensor ~ \u000b.29For\nclarity, we choose the coordinate axis such that the ex-\nchange \feld points along the z-axis. In this case, the\nHamiltonian is invariant under all rotations Rzaround\nthe z-axis. This symmetry requires the energy dissipa-\ntion _E/_mT~\u000b_m(_mTis the transposed of _m) of the\nmagnetic system to be invariant under the coordinate\ntransformations r0=Rzr(i.e., ( _m0)T~\u000b0_m0=_mT~\u000b_m\nwhere m0=Rzmand ~\u000b0is the Gilbert damping ten-\nsor in the rotated coordinate system). Because ~ \u000bonly\ndepends on the direction of m, which is unchanged\nunder the coordinate transformation Rz, ~\u000b= ~\u000b0and\nRT\nz~\u000bRz= ~\u000b. Thus, ~\u000bandRzhave common eigenvectors\b\nj\u0006i\u0011 (jxi\u0006ijyi)=p\n2;jzi\t\n, and the spectral decompo-\nsition of ~\u000bis ~\u000b=\u000b+j+ih+j+\u000b\u0000j\u0000ih\u0000j +\u000bzjzihzj.\nRepresenting the damping tensor in the fjxi;jyi;jzig\ncoordinate basis yields \u000b\u0011~\u000bxx= ~\u000byy= (\u000b++\u000b\u0000)=2,\n~\u000bzz=\u000bz, ~\u000byx= ~\u000bxy= 0, and\u000b+=\u000b\u0000. The last\nequality results from real tensor coe\u000ecients. However,\n\u000bzzcannot be determined uniquely from the energy dis-\nsipation formula _E/_mT~\u000b_mbecause _mis perpendicu-\nlar to the z-axis. Therefore, \u000bzzhas no physical signi\f-\ncance and the energy dissipation is governed by the single\nparameter\u000b. For an in\fnite system, this damping pa-\nrameter does not depend on the speci\fc direction of the\nmagnetization, i.e., it is isotropic because the symmetry\nof the Hamiltonian is not directly linked to the crystallo-\ngraphic axes of the underlying crystal lattice (when the\n0 0.5 1 1.5 200.2 0.4 0.6 0.8 1\nφ / πθ / π\n44.5 55.5 6x 10 −3 \n0 0.5 1 1.5 200.2 0.4 0.6 0.8 1\nφ / πθ / π\n4.5 55.5 6x 10 −3 a\nbFIG. 2: ( a) The dimensionless Gilbert damping parameter\n\u000bas a function of the magnetization direction for a system\nwhereLx= 17 nm,Ly= 50 nm and Lz= 17 nm. ( b) The\ndimensionless Gilbert damping parameter \u000bas a function of\nthe magnetization direction for a system where Lx= 21 nm,\nLy= 50 nm and Lz= 11 nm. Here, \u0012and\u001eare the polar and\nazimuth angles, respectively, that describe the local magne-\ntization direction m= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012). In both\nplots, the mean free path is l\u001822 nm\ncubic symmetry term in Eq. (6) is disregarded). For a\n\fnite system, the shape of the system induces anisotropy\nin the magnetization damping. This e\u000bect is illustrated\nin Fig. 2, which plots the e\u000bective damping in Eq. (4) as\na function of the magnetization directions for di\u000berent\nsystem shapes. When the cross-section of the conductor\nis deformed from a regular shape to the shape of a thin-\nner system, the anisotropy of the damping changes. The\nmagnetization damping varies from a minimum value of\naround 0.004 to a maximum value of 0.006, e.g., the\nanisotropy is around 50%. The relaxation process is\nlargest along the axis where the ballistic leads are con-\nnected, i.e., the y-axis. This shape anisotropy is about\nfour- to \fve-times stronger than the anisotropy induced\nby the strain and the cubic symmetry terms in the Hamil-\ntonian (6), which give corrections of about 10 percent.\nFor larger systems, we expect this shape e\u000bect to be-4\ncome less dominant. In these systems, the anisotropy\nof the bulk damping parameter, which is induced by the\nanisotropic terms in the Hamiltonian, should play a more\nsigni\fcant role. The determination of the system size\nwhen the strain and cubic anisotropy become comparable\nto the shape anisotropy e\u000bects is beyond the scope of this\npaper because the system size is restricted by the com-\nputing time. However, this question should be possible to\ninvestigate experimentally by measuring the anisotropy\nof the Gilbert damping as a function of the \flm thickness.\n0 1 2 3 4 52345678910 x 10 −3 \nLy/l ααmin\nαmax \nαmean \nFIG. 3: The e\u000bective dimensionless Gilbert damping (4) as a\nfunction of the disorder. Here, lis the mean free path and\nLyis the length of the ferromagnetic system in the transport\ndirection.\u000bminand\u000bmaxare the minimum and maximum val-\nues of the anisotropic Gilbert damping parameter and \u000bmean is\nthe e\u000bective damping parameter averaged over all the magne-\ntization directions. The system dimensions are Lx= 19 nm,\nLy= 50 nm and Lz= 15 nm.\nWe next investigate how the magnetization relaxation\nprocess depends on the disorder. Ref. 15 derives an ex-\npression that relates the Gilbert damping parameter to\nthe spin-\rip rate T2of the system: \u000b/T2(1 + (T2)2)\u00001.\nIn the low spin-\rip rate regime, this expression scales\nwithT2as\u000b/T\u00001\n2, while the damping parameter is\nproportional to \u000b/T2in the opposite limit . As ex-\nplained in Ref. 15, the low spin-\rip regime is dominated\nby the spin-pumping process in which angular momen-\ntum is transferred to the itinerant particles; the trans-\nferred spin is then relaxed with a rate proportional to\nT\u00001\n2. This process appears inside the ferromagnet itself,\ni.e., the spin is transferred from the magnetic system to\nthe itinerant particles in the ferromagnet, which are then\nrelaxed within the ferromagnet. Therefore, this relax-\nation mechanism is a bulk process and should not be\nconfused with the spin-pumping interface e\u000bect across\nthe normal metal jferromagnet interfaces reported in\nRef. 7. In (Ga, Mn)As, this bulk process corresponds to\nspin-pumping from the d-electrons of the magnetic Mn\nimpurities to the itinerant spin 3/2 holes in the valence\nband of the host compound GaAs. The transfer of spin\nto the holes is then relaxed by the impurity scatteringwithin the ferromagnet. By contrast, the opposite limit\nis dominated by the breathing Fermi-surface mechanism.\nIn this mechanism, the spins of the itinerant particles are\nnot able to follow the local magnetization direction adia-\nbatically and lag behind with a delay time of T2. In our\nsystem, which has a large spin-orbit coupling in the band\nstructure, we expect the spin-\rip rate to be proportional\nto the mean free path ( l/T2).30The e\u000bective dimen-\nsionless Gilbert damping (4) is plotted as a function of\ndisorder in Fig. 3. The damping ( \u000bmean) partly shows\nthe same behavior as that reported in Ref. 15. For clean\nsystems (i.e., those with a low spin-\rip rate regime), the\ndamping increases with disorder. In such a regime, the\ntransfer of angular momentum to the spin 3/2 holes is the\ndominant damping process, i.e., the bulk spin-pumping\nprocess dominates. \u000bmean starts to decrease for smaller\nmean free paths, implying that the main contribution\nto the damping comes from the breathing Fermi-surface\nprocess. Refs. 11,16,17 have reported that the Gilbert\ndamping may start to increase as a function of disorder in\ndirtier samples. The interband transitions become more\nimportant with decreasing quasi-particle life times and\nstart to dominate the intraband transitions (The intra-\nband transitions give rise to the breathing Fermi-surface\ne\u000bect). We do not observe an increasing behavior in the\nmore di\u000busive regime, but we \fnd that the damping sat-\nurates at a value of around 0.0046 (See Fig. 3). In this\nregime, we believe that the breathing Fermi-surface e\u000bect\nis balanced by the interband transitions. The damping\ndoes not vanish in the limit 1 =l= 0 due to scattering\nat the interface between the GaAs and (Ga,Mn)As lay-\ners in addition to spin-pumping into the adjacent leads\n(an interface spin-pumping e\u000bect, as explained above).\nFig. 3 shows that the shape anisotropy of the damp-\ning is reduced by disorder because the di\u000berence between\nthe maximum ( \u000bmax) and minimum ( \u000bmin) values of the\ndamping parameter decrease with disorder. We antici-\npate this result because disorder increases the bulk damp-\ning e\u000bect, which is expected to be isotropic for an in\fnite\nsystem.\nIV. SUMMARY\nIn this paper, we studied the magnetization damping\nin the ferromagnetic semiconductor (Ga,Mn)As. The\nGilbert damping was calculated numerically using a\nrecently developed scattering matrix theory of mag-\nnetization dissipation.8We conducted a detailed non-\nperturbative study of the e\u000bects of disorder and an inves-\ntigation of the damping anisotropy induced by the shape\nof the sample.\nOur analysis showed that the damping process is\nmainly governed by three relaxation mechanisms. In the\nclean limit with little disorder, we found that the magne-\ntization dissipation is dominated by spin-pumping from\nthe d-electrons to the itinerant holes. For shorter mean\nfree paths, the breathing Fermi-surface e\u000bect starts to5\ndominate, which causes the damping to decrease. In\nthe di\u000busive regime, the breathing Fermi-surface e\u000bect\nis balanced by the interband transitions and the e\u000bective\ndamping parameter saturates at a value on the order of\n0.005.\nFor the small samples considered in this study, we\nfound that the shape of the system was typically more\nimportant than the anisotropic terms in the Hamiltonian\nfor the directional dependency of the damping parame-\nter. This shape anisotropy has not been reported beforeand o\u000bers a new way of manipulating the magnetization\ndamping.\nV. ACKNOWLEDGMENTS\nThis work was partially supported by the European\nUnion FP7 Grant No. 251759 \\MACALO\".\n1For a review, see D. C. Ralph and M. Stiles, J. Magn.\nMagn. Mater. 320, 1190 (2008), and reference therein.\n2B. Heinrich, D. Fraitov\u0013 a, and V. Kambersky, Phys. Status\nSolidi 23, 501 (1967); V. Kambersky, Can. J. Phys. 48,\n2906 (1970); V. Korenman and R.E. Prange, Phys. Rev.\nB6, 2769 (1972); V.S. Lutovinov and M.Y. Reizer, Zh.\nEksp. Teor. Fiz. 77, 707 (1979) [Sov. Phys. JETP 50, 355\n(1979)]; V.L. Safonov and H.N. Bertram, Phys. Rev. B 61,\nR14893 (2000); J. Kunes and V. Kambersky, Phys. Rev. B\n65, 212411 (2002); V. Kambersky Phys. Rev. B 76, 134416\n(2007).\n3T.L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n4J.A.C. Bland and B. Heinrich, Ultrathin Magnetic Struc-\ntures III Fundamentals of Nanomagnetism (Springer Ver-\nlag, Heidelberg, 2004).\n5V. Kambersky, Czech. J. Phys. B 26, 1366 (1976).\n6K. Gilmore, Y.U. Idzerda, and M.D. Stiles, Phys. Rev.\nLett. 99, 027204 (2007).\n7Y. Tserkovnyak, A. Brataas and G.E.W. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n8A. Brataas, Y. Tserkovnyak, and G.E.W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008); Phys. Rev. B 84, 054416\n(2011).\n9A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak,\nand G. E. W. Bauer, Phys. Rev. Lett. 105, 236601\n(2010); Y. Liu, Z. Yuan, A. A. Starikov, and P. J. Kelly,\narXiv:1102.5305.\n10A similar analysis is presented in J. Seib, D. Steiauf, and\nM. F ahnle, Physical Review B 79, 092418 (2009).\n11J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J.K. Furdyna,\nW.A. Atkinson, and A.H. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n12Y.H. Matsuda, A. Oiwa, K. Tanaka, and H. Munekata,\nPhysica B 376-377 , 668 (2006).\n13A. Wirthmann et al. , Appl. Phys. Lett. 92, 232106 (2008).\n14Kh. Khazen et al. , Phys. Rev. B 78, 195210 (2008).\n15Y. Tserkovnyak, G.A. Fiete, and B.I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n16I. Garate and A.H. MacDonald, Phys. Rev. B 79, 064403(2009).\n17I. Garate and A.H. MacDonald, Phys. Rev. B 79, 064404\n(2009).\n18This shape anisotropy in the Gilbert damping should not\nbe confused with the shape anisotropy (in the anisotropy\n\feld) caused by surface dipoles in non-spherical systems.\n19T. Jungwirth, J. Sinova, J. Ma\u0014 sek, J. Ku\u0014 cera, and A. H.\nMacDonald, Rev. Mod. Phys. 78, 809864 (2006).\n20M. Abolfath, T. Jungwirth, J. Brum, and A. H. MacDon-\nald, Phys. Rev. B 63, 054418 (2001).\n21P.Y. Yu and M. Cardona, Fundamentals of Semicon-\nductors: Physics and Materials Properties , 3rd Edition\n(Springer Verlag, Berlin, 2005).\n22Note that in Eq. (6) the spin operator s(in the p-d ex-\nchange term) is represented in the basis consisting of the\nfour spin 3/2 states ( s=J=3). The factor 1 =3 is absorbed\nin the exchange \feld h.\n23In the discrete version of Eq. (6), as used in the numerical\ncalculation, we have one impurity at each lattice site.\n24A. Chernyshov, M. Overby, X. Liu, J.K. Furdyna, Y.\nLyanda-Geller, and L.P. Rokhinson, Nature Physics 5, 656\n(2009).\n25A. Baldereschi and N.O. Lipari, Phys. Rev. B 8, 2697\n(1973).\n26The prefactor of 10 comes from 4 Ga atoms per unit cell\ntimes spin 5/2 per substitutional Mn, which are assumed to\nbe fully polarized. The reduction of the net magnetization\ndue to the interstitial Mn ions and p holes are disregarded\nin our estimate.\n27S. Datta, Electronic Transport in Mesoscopic Systems\n(Cambridge University Press, Cambridge, England, 1995).\n28T. Usuki, M. Saito, M. Takatsu, R. A. Kiehl, and N.\nYokoyama, Phys. Rev. B 52, 8244 (1995).\n29D. Steiauf and M. F ahnle, Physical Review B 72, 064450\n(2005).\n30The spin relaxation time of holes in GaAs is on the scale of\nthe momentum relaxation time. See D.J. Hilton and C.L.\nTang, Phys. Rev. Lett. 89, 146601 (2002), and references\ntherein." }, { "title": "2111.07537v1.Convergence_Analysis_of_A_Second_order_Accurate__Linear_Numerical_Scheme_for_The_Landau_Lifshitz_Equation_with_Large_Damping_Parameters.pdf", "content": "arXiv:2111.07537v1 [math.NA] 15 Nov 2021CONVERGENCE ANALYSIS OF A SECOND-ORDER ACCURATE, LINEAR\nNUMERICAL SCHEME FOR THE LANDAU-LIFSHITZ EQUATION WITH\nLARGE DAMPING PARAMETERS\nYONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nAbstract. A second order accurate, linear numerical method is analyze d for the Landau-\nLifshitz equation with large damping parameters. This equa tion describes the dynamics of\nmagnetization, with a non-convexity constraint of unit len gth of the magnetization. The nu-\nmerical method is based on the second-order backward differe ntiation formula in time, combined\nwith an implicit treatment of the linear diffusion term and ex plicit extrapolation for the non-\nlinear terms. Afterward, a projection step is applied to nor malize the numerical solution at a\npoint-wise level. This numerical scheme has shown extensiv e advantages in the practical com-\nputations for the physical model with large damping paramet ers, which comes from the fact\nthat only a linear system with constant coefficients (indepen dent of both time and the updated\nmagnetization) needs to be solved at each time step, and has g reatly improved the numerical\nefficiency. Meanwhile, a theoretical analysis for this linea r numerical scheme has not been avail-\nable. In this paper, we provide a rigorous error estimate of t he numerical scheme, in the discrete\nℓ∞(0,T;ℓ2)∩ℓ2(0,T;H1\nh) norm, under suitable regularity assumptions and reasonab le ratio be-\ntween the time step-size and the spatial mesh-size. In parti cular, the projection operation is\nnonlinear, and a stability estimate for the projection step turns out to be highly challenging.\nSuch a stability estimate is derived in details, which will p lay an essential role in the convergence\nanalysis for the numerical scheme, if the damping parameter is greater than 3.\n1.Introduction\nThe Landau-Lifshitz (LL) equation [ 21], with quasilinearity and the constraint of unit length\nof magnetization, describes the evolution of the magnetiza tion in ferromagnetic materials with\napplications of information storage in the magnetic-based recording devices [ 24]. The nonlinear\nconservative term of the LL equation preserves the unit leng th of magnetization and drives the\nsystem. The remaining nonlinear part related to the harmoni c mapping of the LL equation is\ndissipated by a factor of damping parameters. Such a paramet er plays an important role for\nenergy evolution, which can be calculated [ 23].\nThere have been extensive numerical works for the LL equatio n [15,22,23,26]. One of the\nmost popular temporal discretization is the semi-implicit method [ 8,12,14], which turns out to\nremarkably relax restrictions of temporal step-size. For e xample, the linearly implicit backward\nEuler approach has been well studied in [ 1,2,8,14]. A combination of this numerical idea\nwith a high-order non-conforming finite element discretiza tion in space has been proposed and\nanalyzed in[ 2], inwhichaprojection isapplied toanapproximate tangent spacetothenormality\nconstraint. Moreover, a convergence analysis in both space and time has been established in\n[1], by evaluating the approximated error of time derivative t erm which is orthogonal to the\nmagnetization. Theerrorestimates for linearly implicit s chemes, basedon either backward Euler\nor Crank-Nicolson method, combined with finite element/fini te difference spatial discretization,\nhave been obtained in [ 3,4,14]. The backward differentiation formula (BDF)-based linearl y\nimplicit methods have been analyzed in [ 1,8], and a second-order accuracy have been rigorously\nproved under the same condition that temporal step-size pro portional to the spatial mesh-size\nin both space and time.\nDate: November 16, 2021.\n2010Mathematics Subject Classification. 35K61, 65M06, 65M12.\nKey words and phrases. Landau-Lifshitz equation, large damping parameters, line ar numerical scheme, second\norder accuracy, convergence analysis, stability estimate for the projection step.\n12 YONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nMeanwhile, it is noticed that, all these existing numerical methods lead to an asymmetric\nlinear system of equations with the coefficients dependent of the updated magnetization. An ef-\nficient numerical solver for such an asymmetric linear syste m is highly non-trivial, which usually\nresults in a very expensive computation cost in the three-di mensional simulation. Therefore,\na numerical scheme only involved with a constant coefficient l inear system, so that the coeffi-\ncients are independent of the updated magnetization, is hig hly desirable. In fact, such a linear\nnumerical scheme has been proposed and studied in a recent wo rk [7]. In more details, the\nsecond-order BDF stencil is used in the temporal discretiza tion, the perfect Laplace term (in\nthe harmonic mapping part) is treated implicitly, while the nonlinear terms are approximated\nby explicit extrapolation formulas. After an intermediate magnetization vector is obtained by\nthis linear algorithm, a projection of magnetization onto t he unit sphere is applied, to satisfy\nthe non-convex constraint of unit length. Of course, this nu merical approach leads to a linear\nsystem with constant coefficients independent the updated ma gnetization at each time step. Be-\ncause of this subtle fact, the linear numerical scheme has de monstrated great advantages in the\nsimulation of ferromagnetic materials for large damping pa rameters [ 7] . Furthermore, exten-\nsive simulation experiments have indicated that the propos ed linear numerical method preserves\nbetter stability property as the damping parameter takes la rge values, in comparison with all\nthe existing works [ 1–3,6,8], etc.\nOntheotherhand, atheoretical analysis oftheproposedlin ear numerical schemehasnot been\navailable, in spite of its extensive advantages in the pract ical computations for large dampingpa-\nrameters. Thekey theoretical difficulty is associated with t hefact that, afully explicit treatment\nof the nonlinear gyromagnetic term (by an extrapolation for mula) breaks its (energetic) conser-\nvative feature at the numerical level, so that a direct contr ol of this nonlinear term becomes a\nvery challenging issue. The only hopeful approach is to cont rol this term by the linear diffusion\nterm in theharmonicmappingpart, whilethefact that the non linear gyromagnetic term and the\nlinear diffusion term are updated by different temporal discret ization makes this estimate highly\nnon-trivial. In this article, we provide the convergence an alysis and the optimal rate error esti-\nmate for the proposed linear numerical scheme, in the discre teℓ∞(0,T;ℓ2)∩ℓ2(0,T;H1\nh) norm,\nif the damping parameter is greater than 3. To overcome the ab ove-mentioned difficulties, we\nbuild the stability estimate of the projection step, which w ill play a crucial role in the rigorous\nerror estimate for the original error function. In particul ar, a standard ℓ2stability estimate\nfor the projection step is not sufficient to recover the conver gence analysis, and an H1\nhstability\nestimate turns out to be necessary at the projection step, wh ich comes from the technique of\ncontrolling the nonlinear gyromagnetic term by the linear d iffusion term. In more details, the\na-prioriW1,∞\nhestimate for the numerical solution and a-prioriH1\nhestimate for the intermediate\nnumerical error function at the previous time step is needed in the error analysis. Meanwhile,\nthea-priori H1\nhestimate for the numerical error for the magnetization vect or can be controlled\nby a growth factor 1 + δacting on the H1\nhestimate of the intermediate magnetization error\nfunction, with δbeing arbitrary positive number. Such a W1,∞\nhbound for numerical solution\nandH1\nhestimate for the intermediate numerical error function can be recovered at the next time\nstep, with the help of the inverse inequality and a mild tempo ral constraint and large damping\nparameter (larger than three). In turn, an estimate for nume rical error function becomes a\nstraightforward consequence of an application of discrete Gronwall inequality, combined with\nthe fine estimate of a growth factor 1+ δacting on the H1\nhestimate of the intermediate error\nfunction.\nThe rest of this paper is organized as follows. In Section 2, we review the fully discrete\nnumerical schemes and state the main theoretical result of c onvergence. The detailed proof is\nprovided in Section 3. Finally, some concluding remarks are made in Section 4.CONVERGENCE ANALYSIS FOR THE LL EQUATION WITH LARGE DAMPING 3\n2.The mathematical model and the numerical scheme\n2.1.The Landau-Lifshitz equation. The LL equation is formulated as\n(2.1)\n\n∂tm(x,t) =−m×∆m+α∆m+α|∇m|2m,x∈Ω, t >0,\n∂nm(x,t)|∂Ω= 0, x∈∂Ω, t≥0,\nm(x,0) =m0(x), x∈Ω,\nwherexandtare the variables of space and time, respectively, Ω ⊂Rd(d= 1,2,3, withd\nbeing the spatial dimension) is a bounded domain and nis the unit outward normal vector\nalong∂Ω,m:=m(x,t) = (m1,m2,m3)T: Ω⊂Rd→S2represents the magnetization vector\nfield with |m|= 1,∀x∈Ω, andα >0 is the damping parameter. The notations ∂t,∇,\nand ∆ represent the temporal derivative, the gradient and th e Laplacian, respectively. The\nhomogeneous Neumann boundary condition is considered. The first term on the right hand side\nof (2.1) is the gyromagnetic term, and the remaining term related to αis the damping term.\nIn comparison with the ferromagnetic model [ 21], (2.1) only includes the exchange term which\nposes the main difficulties in numerical analysis, as done in t he literature [ 5,13,14]. To simplify\nthe presentation, we set Ω = [0 ,1]dand consider the 3D case in this paper, while the results\nhold for the 1D and 2D models.\n2.2.Finite difference discretization and the numerical method. We set the temporal\nstep-size as k >0, so that the time step instant becomes tn=nk(n≤/floorleftbigT\nk/floorrightbig\n, withTbeing the\nfinal time, ⌊·⌋being the floor operator). The spatial mesh-size is given by hx=1\nNx,hy=1\nNy,\nhz=1\nNz, withNx,NyandNzbeing the number of grid points of uniform partition along x,\nyandzdirections, respectively. We use the half grid points ( xi−1\n2,yj−1\n2,zℓ−1\n2) (also written as\n(ˆxi,ˆyj,ˆzℓ)), withxi−1\n2= (i−1\n2)hx,yj−1\n2= (j−1\n2)hyandzℓ−1\n2= (ℓ−1\n2)hz(i= 0,1,···,Nx+1;\nj= 0,1,···,Ny+1;ℓ= 0,1,···,Nz+1). The numerical domain becomes Ω h={(ˆxi,ˆyj,ˆzℓ)|i=\n0,1,···,Nx+ 1;j= 0,1,···,Ny+ 1;ℓ= 0,1,···,Nz+ 1}and the interior domain is Ω0\nh=\n{(ˆxi,ˆyj,ˆzℓ)|i= 1,···,Nx;j= 1,···,Ny;ℓ= 1,···,Nz}, and Ω h/Ω0\nhis the set of boundary\n(ghost) points. We introduce the notation of the discrete ve ctor grid function mh(x)∈R3\ndefined for x∈Ωhwithmi,j,ℓ=mh(ˆxi,,ˆyj,ˆzℓ) (similar notations for the scalar functions ),\nand the discrete homogeneous Neumann boundary condition re ads forix= 0,Nx,jy= 0,Ny,\nℓz= 0,Nzand 0≤i≤Nx+1,0≤j≤Ny+1,0≤ℓ≤Nz+1,\n(2.2) mix,j,ℓ=mix+1,j,ℓ,mi,jy,ℓ=mi,jy+1,ℓ,mi,j,ℓz=mi,j,ℓz+1.\nLetX={fh(x)∈R,x∈Ωh, fhsatisfies boundary condition ( 2.2)}be the scalar function space\nandX={mh(x)∈R3,x∈Ωh,mhsatisfies boundary condition ( 2.2)}be the vector-valued\nfunction space. The corresponding continuous version is de noted by Xe,Xe. The standard\nsecond-order centered difference approximation for the Lapl ace operator results in\n∆hmi,j,ℓ=δ2\nxmi,j,ℓ+δ2\nymi,j,ℓ+δ2\nzmi,j,ℓ, δ2\nxmi,j,ℓ=mi+1,j,ℓ−2mi,j,ℓ+mi−1,j,ℓ\nh2x,\nwhereδ2\ny,δ2\nzfor the approximation of ∂yy,∂zzcould be similarly defined. The forward finite\ndifference operators Dx,DyandDzare defined for fh∈X:\nDxfi,j,ℓ=fi+1,j,ℓ−fi,j,ℓ\nhx, Dyfi,j,ℓ=fi,j+1,ℓ−fi,j,ℓ\nhy, Dzfi,j,ℓ=fi,j,ℓ+1−fi,j,ℓ\nhz.\nThese finite difference operators could be applied to the scala r or vector grid functions in the\nsame way. The discrete gradient operator (forward) ∇hmhwithmh= (uh,vh,wh)T∈Xreads\nas\n∇hmi,j,ℓ=\nDxui,j,ℓDxvi,j,ℓDxwi,j,ℓ\nDyui,j,ℓDyvi,j,ℓDywi,j,ℓ\nDzui,j,ℓDzvi,j,ℓDzwi,j,ℓ\n.\nA semi-implicit numerical scheme has been proposed in [ 29], and used in the numerical sim-\nulation for small damping parameter models. In more details , semi-implicit approximations4 YONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nare applied to the two nonlinear terms, namely m×∆mandm×(m×∆m), in which ∆ m\nis treated implicitly, while the coefficient variables are ex plicitly updated via an extrapolation\nformula. The theoretical convergence analysis has been est ablished in a more recent work [ 8].\nHowever, this numerical scheme involves a large linear syst em with time-dependent coefficients,\nrelated to the updated magnetization at each time step, and t he symmetry is not available in\nthe linear system, due to the nonlinear structure. To overco me this subtle difficulty, which leads\nto significant computational costs (especially in the 3D cas e), we make use of the alternate PDE\nformulation ( 2.1), and treat the linear diffusion term α∆mimplicitly, while the two nonlinear\nterms, namely −m×∆mandα|∇m|2m, are discretized in a fully explicit way. Subsequently, a\npoint-wise projection is applied to the intermediate field, so that the numerical solution of mhas\na unit length at the point-wise level. In more details, the fo llowing numerical scheme has been\nstudied, namely Algorithm 2.1 : for given mh,˜mn\nh,mn+1\nh,˜mn+1\nh∈X, find˜mn+2\nh,mn+2\nh∈X\nby solving\n3\n2˜mn+2\nh(x)−2˜mn+1\nh(x)+1\n2˜mn\nh(x)\nk=−ˆmn+2\nh×∆hˆ˜mn+2\nh\n+α∆h˜mn+2\nh+α|Ah∇hˆmn+2\nh|2ˆmn+2\nh,x∈Ω0\nh,\nfollowed by the point-wise projection mn+2\nh=˜mn+2\nh\n|˜mn+2\nh|, where the extrapolation formula is\ndefined by ˆmn+2\nh= 2mn+1\nh−mn\nh,ˆ˜mn+2\nh= 2˜mn+1\nh−˜mn\nh, andAh∇h(second approxima-\ntion to the gradient operator) is an average gradient operat or defined for the gird function\nmh= (uh,vh,wh)T∈XasAh∇hmh=∇hAhmhandAhmh= (Axuh,Ayvh,Azwh):\nAxui,j,ℓ=ui,j,ℓ+ui−1,j,ℓ\n2,Ayvi,j,ℓ=vi,j,ℓ+vi,j−1,ℓ\n2,Azwi,j,ℓ=wi,j,ℓ+wi,j,ℓ−1\n2.\nInaddition, amodifiedversionisproposedby[ 7], namely Algorithm 2.2 : forgiven mh,mn+1\nh∈\nX, denote ˆmn+2\nh= 2mn+1\nh−mn\nh, and find mn+2\nh,˜mn+2\nh∈Xby solving\n3\n2˜mn+2\nh(x)−2mn+1\nh(x)+1\n2mn\nh(x)\nk=−ˆmn+2\nh×∆hˆmn+2\nh+α∆h˜mn+2\nh\n+α|Ah∇hˆmn+2\nh|2ˆmn+2\nh,x∈Ω0\nh, (2.3)\nmn+2\nh=˜mn+2\nh\n|˜mn+2\nh|. (2.4)\nRemark 2.1. The initial data is given by m0\nh=Phm0∈X, wherePh: [C(Ω)]3→Xis the\npoint-wise interpolation as\n(2.5) Phm0(x) =m0(x),x∈Ω0\nh.\nIn addition, the first-order semi-implicit projection schem e could be applied to obtain m1\nh, so\nthat the two-step numerical method could be jump started. Su ch a single-step first order algo-\nrithm will preserve the overall second-order accuracy in ti me; see the detailed analysis in the\nrelated works [19,20]for Cahn-Hilliard equation, in which a single step, first orde r semi-implicit\nalgorithm creates a second order accurate numerical soluti on in the first step.\nRemark 2.2. The primary difference between Algorithm 2.1 and Algorithm 2.2 is focused\non the temporal-derivative approximation, where Algorithm 2.1 uses the intermediate approx-\nimate magnetization, while Algorithm 2.2 takes the previou s projected values. Extensive numer-\nical experiments have demonstrated that, Algorithm 2.2 provides a much better stability than\nAlgorithm 2.1 in the simulation of the realistic ferromagnetic material w ith large damping pa-\nrameters, as reported in [7]. In this article, we present a theoretical justification of th e stability\nand convergence analysis for Algorithm 2.2 .CONVERGENCE ANALYSIS FOR THE LL EQUATION WITH LARGE DAMPING 5\n2.3.Main theoretical results. For simplicity of presentation, we make an assumption that\nNx=Ny=Nz=N(withd= 3) so that hx=hy=hz=h. An extension to the general case\nis straightforward. For the gird function fh,gh∈X, the discrete ℓ2inner product /an}⌊∇a⌋ketle{t·/an}⌊∇a⌋ket∇i}ht, discrete\n/⌊a∇d⌊l·/⌊a∇d⌊l2norm and discrete /⌊a∇d⌊l·/⌊a∇d⌊l∞norm are defined as\n/an}⌊∇a⌋ketle{tfh,gh/an}⌊∇a⌋ket∇i}ht=h3/summationdisplay\nx∈Ω0\nhfh(x)·gh(x),/⌊a∇d⌊lfh/⌊a∇d⌊l2=/radicalbig\n/an}⌊∇a⌋ketle{tfh,fh/an}⌊∇a⌋ket∇i}ht,/⌊a∇d⌊lfh/⌊a∇d⌊l∞= maxx∈Ω0\nh|fh(x)|.\nIn addition, the discrete H1\nh-norm is given by /⌊a∇d⌊lfh/⌊a∇d⌊l2\nH1\nh:=/⌊a∇d⌊lfh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l∇hfh/⌊a∇d⌊l2\n2, and the discrete ℓp\n(1< p <∞) norm is defined as /⌊a∇d⌊lfh/⌊a∇d⌊lp\np=h3/summationtext\nx∈Ω0\nh|fh(x)|p. Such norms induce the discrete\nspaces\nℓp={fh∈X| /⌊a∇d⌊lfh/⌊a∇d⌊lp<∞},0< p≤ ∞\nH1\nh={fh∈X| /⌊a∇d⌊lfh/⌊a∇d⌊lH1\nh<∞},\nW1,∞\nh={fh∈X| /⌊a∇d⌊lfh/⌊a∇d⌊l∞+/⌊a∇d⌊l∇hfh/⌊a∇d⌊l∞<∞},\nW1,4\nh={fh∈X| /⌊a∇d⌊lfh/⌊a∇d⌊l4+/⌊a∇d⌊l∇hfh/⌊a∇d⌊l4<∞},\nℓ∞(0,T;ℓ2) ={fn\nh∈X|max\nn/⌊a∇d⌊lfn\nh/⌊a∇d⌊l2<∞, n∈/bracketleftbigg\n0,/floorleftbiggT\nk/floorrightbigg/bracketrightbigg\n},\nℓ2(0,T;H1\nh) =\n\nfn\nh∈X|/parenleftBiggN/summationdisplay\nn=0/⌊a∇d⌊lfn\nh/⌊a∇d⌊l2\n2/parenrightBigg1\n2\n<∞, n∈/bracketleftbigg\n0,/floorleftbiggT\nk/floorrightbigg/bracketrightbigg\n\n.\nMeanwhile, we define the continuous spaces for the function f(x,t) = (f1,f2,f3) as below,\nC3([0,T];[C0(Ω)]3) :={f(x,t)∈Xe|d3\ndt3fi∈C0([0,T]),fi∈C0(Ω), i= 1,2,3},\nC2([0,T];[C2(¯Ω)]3) :={f(x,t)∈Xe|d2\ndt2fi∈C0([0,T]),d2\ndx2fi∈C0(¯Ω), i= 1,2,3},\nL∞([0,T];[C4(¯Ω)]3) :={f(x,t)∈Xe|ess supt∈[0,T]d4\ndx4fi∈C0(¯Ω), i= 1,2,3},\nwhereC0(Ω) is the space of continuous function.\nThe unique solvability of scheme ( 2.3)-(2.4) follows from the equivalent form of ( 2.3):\n(3\n2kI−α∆h)˜mn+2\nh(x) =qn+2\nh(x),x∈Ω0\nh,\nwhere˜mn+2\nh∈Xandqn+2\nh:=2mn+1\nh−1\n2mn\nh\nk−ˆmn+2\nh×∆hˆmn+2\nh+α|Ah∇hˆmn+1\nh|2ˆmn+2. The\nleft hand side corresponds to a positive-definite symmetric matrix, and the unique solvability of\nthe proposed scheme ( 2.3)-(2.4), as well as the Algorithm 2.1 , is obvious. With the fast discrete\nCosine transform, the above linear system can be very efficien tly solved.\nThe theoretical results concerning the convergence analys is is stated below.\nTheorem 2.1. Assume that the exact solution of ( 2.1) has the regularity me∈C3([0,T];[C0( ¯Ω)]3)∩\nC2([0,T];[C2(¯Ω)]3)∩L∞([0,T];[C4(¯Ω)]3). Denote mn\nh(n≥0) as the numerical solution ob-\ntained from ( 2.3)-(2.4) with the initial error satisfying /⌊a∇d⌊lPhme(·,tp)−mp\nh/⌊a∇d⌊l2+/⌊a∇d⌊l∇h(Phme(·,tp)−\nmp\nh)/⌊a∇d⌊l2=O(k2+h2), p= 0,1. In addition, we assume the technical assumption α >3, and\nC1h≤k≤ C2h, withC1,C2being the positive constants. Then the following convergenc e result\nholds for 2≤n≤/floorleftbigT\nk/floorrightbig\nash,k→0+:\n/⌊a∇d⌊lPhme(·,tn)−mn\nh/⌊a∇d⌊l2+/parenleftBig\nkn/summationdisplay\np=1/⌊a∇d⌊l∇h(Phme(·,tp)−˜mp\nh)/⌊a∇d⌊l2\n2/parenrightBig1\n2≤ C(k2+h2), (2.6)\nin which the constant C>0is independent of kandh.6 YONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\n2.4.A few preliminary estimates. In this section, some preliminary inequalities are derived ,\nwhich will be useful in the error analysis presented in the ne xt section. In addition, we have to\nbuild a stability estimate of the projection step in the nume rical algorithm.\nThe proof of the standard inverse inequality and discrete Gr onwall inequality can be obtained\nin existing textbooks and references; we just cite the resul ts here. In the sequel, for simplicity\nof notation, we will use the uniform constant Cto denote all the controllable constants.\nLemma 2.1. (Inverse inequality) [9–11]. For each vector-valued grid function eh∈X, we have\n/⌊a∇d⌊len\nh/⌊a∇d⌊l∞≤γh−1/2(/⌊a∇d⌊len\nh/⌊a∇d⌊l2+/⌊a∇d⌊l∇hen\nh/⌊a∇d⌊l2),/⌊a∇d⌊l∇hen\nh/⌊a∇d⌊l4≤γh−3/4/⌊a∇d⌊l∇hen\nh/⌊a∇d⌊l2, (2.7)\nin which constant γdepends on Ω, as well as the form of the discrete /⌊a∇d⌊l·/⌊a∇d⌊l2norm.\nThe following estimate will be utilized in the convergence a nalysis. A rough version has been\nprovided in a recent article [ 8]; here we give a further refined estimate.\nLemma 2.2 (Discrete gradient acting on cross product) .For grid functions fh,gh,Fh∈X,\nwe have for any δ >0\n/an}⌊∇a⌋ketle{tfh×∆hgh,Fh/an}⌊∇a⌋ket∇i}ht=/an}⌊∇a⌋ketle{tFh×fh,∆hgh/an}⌊∇a⌋ket∇i}ht, (2.8)\n/⌊a∇d⌊l∇h(fh×gh)/⌊a∇d⌊l2\n2≤(1+δ)/⌊a∇d⌊lfh/⌊a∇d⌊l2\n∞·/⌊a∇d⌊l∇hgh/⌊a∇d⌊l2\n2+Cδ/⌊a∇d⌊lgh/⌊a∇d⌊l2\n4·/⌊a∇d⌊l∇hfh/⌊a∇d⌊l2\n4. (2.9)\nProof.Equality ( 2.8) has been proved in [ 8], so that we only focus on the proof of ( 2.9).\nAt each numerical mesh cell, from (ˆ xi,ˆyj,ˆzℓ) to (ˆxi+1,ˆyj,ˆzℓ), the following expansion identity\nis valid:\n(2.10)Dx(fh×gh)i,j,ℓ=(Axfh)i,j,ℓ×(Dxgh)i,j,ℓ+(Axgh)i,j,ℓ×(Dxfh)i,j,ℓ,\nwith (Axfh)i,j,ℓ=1\n2((fh)i,j,ℓ+(fh)i+1,j,ℓ).\nIn turn, we get the following expansion, over each mesh cell:\n(2.11) Dx(fh×gh) = (Axfh)×(Dxgh)+(Dxfh)×(Axgh).\nSubsequently, a careful application of discrete H¨ older in equality reveals that\n/⌊a∇d⌊l(Axfh)×(Dxgh)/⌊a∇d⌊l2≤/⌊a∇d⌊lAxfh/⌊a∇d⌊l∞·/⌊a∇d⌊lDxgh/⌊a∇d⌊l2≤ /⌊a∇d⌊lfh/⌊a∇d⌊l∞·/⌊a∇d⌊lDxgh/⌊a∇d⌊l2, (2.12)\n/⌊a∇d⌊l(Dxfh)×(Axgh)/⌊a∇d⌊l2≤/⌊a∇d⌊lDxfh/⌊a∇d⌊l4·/⌊a∇d⌊lAxgh/⌊a∇d⌊l4≤ /⌊a∇d⌊lDxfh/⌊a∇d⌊l4·/⌊a∇d⌊lgh/⌊a∇d⌊l4, (2.13)\nin which the fact that /⌊a∇d⌊lAxfh/⌊a∇d⌊l∞≤ /⌊a∇d⌊lfh/⌊a∇d⌊l∞,/⌊a∇d⌊lAxgh/⌊a∇d⌊l4≤ /⌊a∇d⌊lgh/⌊a∇d⌊l4, has been applied. Then we get\n(2.14) /⌊a∇d⌊lDx(fh×gh)/⌊a∇d⌊l2≤ /⌊a∇d⌊lfh/⌊a∇d⌊l∞·/⌊a∇d⌊lDxgh/⌊a∇d⌊l2+/⌊a∇d⌊lDxfh/⌊a∇d⌊l4·/⌊a∇d⌊lgh/⌊a∇d⌊l4.\nThecorrespondingestimates in the yandzdirections can besimilarly derived, and the technical\ndetails are skipped for the sake of brevity. A combination of (2.14) and its counterparts in y\nandzdirections leads to\n(2.15)/⌊a∇d⌊l∇h(fh×gh)/⌊a∇d⌊l2\n2≤/⌊a∇d⌊lfh/⌊a∇d⌊l2\n∞·/⌊a∇d⌊l∇hgh/⌊a∇d⌊l2+3/⌊a∇d⌊l∇hfh/⌊a∇d⌊l2\n4·/⌊a∇d⌊lgh/⌊a∇d⌊l2\n4\n+6/⌊a∇d⌊lfh/⌊a∇d⌊l∞·/⌊a∇d⌊l∇hfh/⌊a∇d⌊l4·/⌊a∇d⌊lgh/⌊a∇d⌊l4\n≤(1+δ)/⌊a∇d⌊lfh/⌊a∇d⌊l2\n∞·/⌊a∇d⌊l∇hgh/⌊a∇d⌊l2+(3+9δ−1)/⌊a∇d⌊l∇hfh/⌊a∇d⌊l2\n4·/⌊a∇d⌊lgh/⌊a∇d⌊l2\n4,\nfor anyδ >0, in which the Cauchy inequality has been applied in the last step. Therefore, the\nnonlinear cross product estimate ( 2.9) has been derived. This finishes the proof of Lemma 2.2.\n/square\nThe following discrete Sobolev inequality has been derived in the existing works [ 17,18],\nfor the discrete grid function with periodic boundary condi tion; an extension to the discrete\nhomogeneous Neumann boundary condition can be made in a simi lar manner.\nLemma 2.3 (Discrete Sobolev inequality) .[17,18]For a grid function fh∈X, we have the\nfollowing discrete Sobolev inequality:\n/⌊a∇d⌊lfh/⌊a∇d⌊l4≤ C/⌊a∇d⌊lfh/⌊a∇d⌊l1\n4\n2·/⌊a∇d⌊lfh/⌊a∇d⌊l3\n4\nH1\nh≤ C(/⌊a∇d⌊lfh/⌊a∇d⌊l2+/⌊a∇d⌊lfh/⌊a∇d⌊l1\n4\n2·/⌊a∇d⌊l∇hfh/⌊a∇d⌊l3\n4\n2), (2.16)\nin which the positive constant Conly depends on the domain Ω.CONVERGENCE ANALYSIS FOR THE LL EQUATION WITH LARGE DAMPING 7\nThe following stability estimates of the point-wise projec tion (2.4) are crucial for the error\nanalysis, and the proof could be found in Appendix A.\nLemma 2.4. Assume the continuous vector function me∈[C(Ω)]3satisfies a regularity require-\nment/⌊a∇d⌊lme/⌊a∇d⌊lW1,∞≤C∗(withC∗being a positive constant) and the point-wise constraint |me|= 1.\nDenoting mh=Phme∈X, for any grid function ˜mh∈X, we define the projected grid func-\ntionmh∈Xasmh=˜mh\n|˜mh|, and introduce the error functions as eh(x) =mh(x)−mh(x),\n˜eh(x) =mh(x)−˜mh(x)(x∈Ωh). Under the a-priori assumptions on ˜ehor equivalently on\nthe profile ˜mh:\n(2.17) /⌊a∇d⌊l˜eh/⌊a∇d⌊l2≤2k15\n8,/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l2≤1\n2k11\n8,\nthe following estimates hold for sufficiently small kandhsatisfying C1h≤k≤C2h(withC1,C2\nbeing two positive constants)\n/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n2≥(1−k5\n4)/⌊a∇d⌊leh/⌊a∇d⌊l2\n2+(1−k1\n4)/⌊a∇d⌊l˜eh−eh/⌊a∇d⌊l2\n2, (2.18)\n/⌊a∇d⌊l∇heh/⌊a∇d⌊l2\n2≤(1+δ)/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l2\n2+Cδ/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n2,for anyδ >0. (2.19)\nIn addition, for the analysis of the BDF2 temporal stencil at the projection stage, a further\nrefined error estimate is needed.\nLemma 2.5. Consider m(1)\nh,m(2)\nh∈X(|m(q)\nh(x)|= 1,x∈Ωh,q= 1,2) with the W1,∞\nh\nbound/⌊a∇d⌊lm(q)\nh/⌊a∇d⌊l∞+/⌊a∇d⌊l∇hm(q)\nh/⌊a∇d⌊l∞≤C∗(q= 1,2). For any grid functions ˜m(1)\nh,˜m(2)\nh∈X, we\ndefine the projected grid functions m(q)\nh=˜m(q)\nh\n|˜m(q)\nh|, and introduce the error functions as e(q)\nh(x) =\nm(q)\nh(x)−m(q)\nh(x)∈X,˜e(q)\nh(x) =m(q)\nh(x)−˜m(q)\nh(x)∈X,q= 1,2. Under the a-priori\nassumptions for ˜e(q)\nh(q= 1,2):\n(2.20) /⌊a∇d⌊l˜e(q)\nh/⌊a∇d⌊l2≤2k15\n8,/⌊a∇d⌊l∇h˜e(q)/⌊a∇d⌊l2≤1\n2k11\n8, q= 1,2,\nand the assumptions for m(q)\nh(q= 1,2):\n(2.21) /⌊a∇d⌊lm(1)\nh−m(2)\nh/⌊a∇d⌊l∞≤1\n4k7\n8,\nthe following estimate is valid for sufficiently small kandhsatisfying C1h≤k≤C2h:\n(2.22)/vextendsingle/vextendsingle/vextendsingle/an}⌊∇a⌋ketle{t˜e(1)\nh−e(1)\nh,e(2)\nh/an}⌊∇a⌋ket∇i}ht/vextendsingle/vextendsingle/vextendsingle≤k5\n4/⌊a∇d⌊le(2)\nh/⌊a∇d⌊l2\n2+k1\n4/⌊a∇d⌊l˜e(1)\nh−e(1)\nh/⌊a∇d⌊l2\n2.\nWe leave the proof of Lemma 2.5to Appendix B. Lemmas 2.4and2.5essentially establish the\nstability of the projection step ( 2.4) under the assumptions that the previous numerical solutio n\nattnandtn+1are sufficiently close to the exact solution.\n3.The optimal rate convergence analysis: Proof of Theorem 2.1\nDenotem(x,t) =Phme(x,t)∈X(x∈Ωh) andmn\nh(x) =mh(x,tn) (n≥0). Around the\nboundary section z= 0, we set ˆ z0=−1\n2h, ˆz1=1\n2h, and we can extend the profile mto the\nnumerical “ghost” points, according to the extrapolation f ormula (2.2):\n(3.1) mi,j,0=mi,j,1,mi,j,Nz+1=mi,j,Nz,\nand the extrapolation for other boundaries can be formulate d in the same manner. The proof of\nsuch an extrapolation yields a higher order O(h5) approximation, instead of the standard O(h3)\naccuracy has been applied in [ 8]. Also see the related works [ 25,27,28] in the existing literature.\nPerforming a careful Taylor expansion for the exact solutio n around the boundary section\nz= 0, combined with the mesh point values: ˆ z0=−1\n2h, ˆz1=1\n2h, we get\nme(ˆxi,ˆyj,ˆz0) =me(ˆxi,ˆyj,ˆz1)−h∂zme(ˆxi,ˆyj,0)−h3\n24∂3\nzme(ˆxi,ˆyj,0)+O(h5)\n=me(ˆxi,ˆyj,ˆz1)−h3\n24∂3\nzme(ˆxi,ˆyj,0)+O(h5), (3.2)8 YONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nin which the homogenous boundary condition has been applied in the second step. It remains\nto determine ∂3\nzme(ˆxi,ˆyj,0), for which we use information from the rewritten PDE ( 2.1) and its\nderivatives. Applying ∂zto the first evolutionary equation in ( 2.1) along the boundary section\nΓz:z= 0 gives\n(m1)zt−2α(m1(∇m1·∇(m1)z+∇m2·∇(m2)z+∇m3·∇(m3)z))\n−α|∇me|2(m1)z−α((m1)zxx+(m1)zyy+∂3\nzm1)\n=(m3)z∆m2+m3((m2)zxx+(m2)zyy+∂3\nzm2)\n−(m2)z∆m3−m2((m3)zxx+(m3)zyy+∂3\nzm3),on Γz.(3.3)\nThe first, third terms, and the first two parts in the fourth ter m on the left-hand side of ( 3.3)\ndisappear, due to the homogeneous Neumann boundary conditi on form1. For the second term\non the left hand side, we observe that\n(3.4) ∇m1·∇(m1)z= (m1)x·(m1)zx+(m1)y·(m1)zy+(m1)z·(m1)zz= 0,on Γz,\nsince (m1)z= 0 on the boundary section. Similar derivations could be mad e to the two other\nterms on the left hand side:\n(3.5) ∇m2·∇(m2)z= 0,∇m3·∇(m3)z= 0,on Γz.\nMeanwhile, on the right hand side of ( 3.3), we see that the first and third terms, as well as the\nfirst two parts in the second and fourth terms, disappear, whi ch comes from the homogeneous\nNeumann boundary condition for m2andm3. Then we arrive at\n(3.6) α∂3\nzm1=−m3∂3\nzm2+m2∂3\nzm3,on Γz.\nSimilarly, we are able to derive the following equalities:\n(3.7)α∂3\nzm2=m1∂3\nzm3−m3∂3\nzm1,on Γz\nα∂3\nzm3=m2∂3\nzm1−m1∂3\nzm2,on Γz.\nIn turn, for any α >0, we observe that the matrix\nα m 3−m2\n−m3α m 1\nm2−m1α\nhas a positive\ndeterminant, so that the linear system ( 3.6)-(3.7) has only one trivial solution:\n(3.8) ∂3\nzm1=∂3\nzm2=∂3\nzm3= 0,on Γz.\nAs a result, an O(h5) consistency accuracy for the symmetric extrapolation is o btained:\n(3.9) me(ˆxi,ˆyj,ˆz0) =me(ˆxi,ˆyj,ˆz1)+O(h5),m(ˆxi,ˆyj,ˆz0) =m(ˆxi,ˆyj,ˆz1)+O(h5).\nIn other words, the extrapolation formula ( 3.1) is indeed O(h5) accurate.\nSubsequently, a detailed calculation of Taylor expansion, in both time and space, leads to the\nfollowing truncation error estimate:\n3\n2mn+2\nh(x)−2mn+1\nh(x)+1\n2mn\nh(x)\nk\n=−ˆmn+2\nh×∆hˆmn+2\nh+τn+2+α∆hmn+2\nh+α|Ah∇hˆmn+2\nh|2ˆmn+2\nh,x∈Ω0\nh,(3.10)\nwhereˆmn+2\nh= 2mn+1\nh−mn\nh∈X,τn+2∈Xand/⌊a∇d⌊lτn+2/⌊a∇d⌊l2≤ C(k2+h2). Introducing the\nnumerical error functions ˜en\nh=mn\nh−˜mn\nh∈X,en\nh=mn\nh−mn\nh∈X, and subtracting ( 2.3)-(2.4)\nfrom the consistency estimate ( 3.10), we have the error evolutionary equation at the interior\npointsx∈Ω0\nh, for 0≤n≤ ⌊T/k⌋−2:\n3\n2˜en+2\nh−2en+1\nh+1\n2en\nh\nk=−ˆmn+2\nh×∆hˆen+2\nh−/parenleftbig\n2en+1\nh−en\nh/parenrightbig\n×∆hˆmn+2\nh\n+α∆h˜en+2\nh+α|Ah∇hˆmn+2\nh|2ˆen+2\nh+τn+2\n+α/parenleftBig\nAh∇h(ˆmn+2\nh+ˆmn+2\nh)·Ah∇hˆen+2\nh/parenrightBig\nˆmn+2\nh(3.11)\nwithˆen+2\nh= 2en+1\nh−en\nh∈X.CONVERGENCE ANALYSIS FOR THE LL EQUATION WITH LARGE DAMPING 9\nBefore proceeding into the formal error estimate, we state t he bound for the exact solution\nmand the numerical solution mh. Since the exact solution me∈L∞([0,T];[C4(¯Ω)]3), the\nfollowing bound is available, for some positive constant C:\n/⌊a∇d⌊l∇r\nhmh(·,t)/⌊a∇d⌊l4,/⌊a∇d⌊l∇r\nhmh(·,t)/⌊a∇d⌊l∞≤ C, r= 0,1,2,3,0≤t≤T. (3.12)\nIn addition, we make the following a-priori assumption for the numerical error function:\n(3.13) /⌊a∇d⌊len\nh/⌊a∇d⌊l2≤k31\n16,/⌊a∇d⌊l˜en\nh/⌊a∇d⌊l2≤2k15\n8,/⌊a∇d⌊l∇h˜en\nh/⌊a∇d⌊l2≤1\n2k11\n8, n≤q+1.\nSuch an assumption will be recovered by the convergence anal ysis at time step tq+2. Based on\nthisa-priori assumption, we see that ( 2.17) is satisfied, so that we are able to apply Lemma 2.4\nand the estimate ( 2.19) to get\n/⌊a∇d⌊l∇hen\nh/⌊a∇d⌊l2≤5\n4/⌊a∇d⌊l∇h˜en\nh/⌊a∇d⌊l2+C/⌊a∇d⌊l˜en\nh/⌊a∇d⌊l2≤5\n8k11\n8+Ck15\n8≤k11\n8, n≤q+1. (3.14)\nIn turn, an application of inverse inequality implies the /⌊a∇d⌊l · /⌊a∇d⌊l∞and/⌊a∇d⌊l · /⌊a∇d⌊lW1,4\nhbounds of the\nnumerical error function en\nh(n≤q+1):\n(3.15)/⌊a∇d⌊len\nh/⌊a∇d⌊l∞≤C/⌊a∇d⌊len\nh/⌊a∇d⌊l2\nh3\n2≤C·k31\n16\nh3\n2≤Ck7\n16≤k3\n8,\n/⌊a∇d⌊l∇hen\nh/⌊a∇d⌊l4≤C/⌊a∇d⌊l∇hen\nh/⌊a∇d⌊l2\nh3\n4≤C·k11\n8\nh3\n4≤Ck5\n8≤k1\n2≤1\n3.\nSubsequently, the triangle inequality yields the desired W1,4\nhbound for the numerical solutions\nmn\nhand˜mn\nh(n≤q+1):\n/⌊a∇d⌊l∇hmn\nh/⌊a∇d⌊l4=/⌊a∇d⌊l∇hmn\nh−∇hen\nh/⌊a∇d⌊l4≤ /⌊a∇d⌊l∇hmn\nh/⌊a∇d⌊l4+/⌊a∇d⌊l∇hen\nh/⌊a∇d⌊l4≤ C+1\n3. (3.16)\nFurthermore, we need a sharper /⌊a∇d⌊l·/⌊a∇d⌊l∞bound for ˆmn+2\nh= 2mn+1\nh−mn\nh, which will be needed in\nthelatererroranalysis. Thefollowingextrapolationesti mateisvalid, duetothe C3([0,T];[C0(Ω)]3)\nregularity of the exact solution m(·,t):\n(3.17) mn+2\nh= 2mn+1\nh−mn\nh+O(k2).\nMeanwhile, since |m(x,t)| ≡1 (x∈Ω), we conclude that\n(3.18) /⌊a∇d⌊l2mn+1\nh−mn\nh/⌊a∇d⌊l∞≤1+Ck2, n≤q+1.\nIts combination with the a-priori assumption that /⌊a∇d⌊len\nh/⌊a∇d⌊l∞≤k+h, forn≤q+ 1, (as given\nby (3.13)), implies that\n(3.19)/⌊a∇d⌊lˆmn+2\nh/⌊a∇d⌊l∞=/⌊a∇d⌊l2mn+1\nh−mn\nh/⌊a∇d⌊l∞≤ /⌊a∇d⌊l2mn+1\nh−mn\nh/⌊a∇d⌊l∞+/⌊a∇d⌊l2en+1\nh−en\nh/⌊a∇d⌊l∞\n≤1+Ck2+3k3\n8≤α1:=/parenleftBig3+α\n6/parenrightBig1\n2,\nprovided that kandhare sufficiently small. In addition, we denote γ0:=α−3>0, so that\nα2\n1= 1+1\n6γ0.\nNext, weperformadiscrete ��2errorestimateat tq+2usingthemathematical induction. Taking\na discrete inner product with the numerical error equation ( 3.11) by˜en+2\nh∈X(n≤q+1) gives\nthat\nR.H.S.=/angbracketleftbig\n−/parenleftbig\n2mn+1\nh−mn\nh/parenrightbig\n×∆hˆen+2\nh,˜en+2\nh/angbracketrightbig\n−/angbracketleftbig/parenleftbig\n2en+1\nh−en\nh/parenrightbig\n×∆hˆmn+2\nh,˜en+2\nh/angbracketrightbig\n+/angbracketleftbig\nτn+2,˜en+2\nh/angbracketrightbig\n−α/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l2\n2+α/an}⌊∇a⌋ketle{t|Ah∇hˆmn+2\nh|2ˆen+2\nh,˜en+2\nh/an}⌊∇a⌋ket∇i}ht\n+α/angbracketleftBig/parenleftBig\nAh∇h(ˆmn+2\nh+ˆmn+2\nh)·Ah∇hˆen+2\nh/parenrightBig\nˆmn+2\nh,˜en+2\nh/angbracketrightBig\n=:˜I1+˜I2+˜I3−α/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l2\n2+˜I5+˜I6.(3.20)10 YONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nThen all the terms are accordingly analyzed. For the term ˜I1, a combination of the summation\nby parts formula by and Cauchy inequality results in\n˜I1=/angbracketleftbig\n−ˆmn+2\nh×∆hˆen+2\nh,˜en+2\nh/angbracketrightbig\n=/angbracketleftbig˜en+2\nh׈mn+2\nh,−∆hˆen+2\nh/angbracketrightbig\n=/angbracketleftBig\n∇h/bracketleftBig\n˜en+2\nh׈mn+2\nh/bracketrightBig\n,∇hˆen+2\nh/angbracketrightBig\n≤3\n2/vextenddouble/vextenddouble/vextenddouble∇h/parenleftBig\n˜en+2\nh׈mn+2\nh/parenrightBig/vextenddouble/vextenddouble/vextenddouble2\n2+1\n6/⌊a∇d⌊l∇hˆen+2\nh/⌊a∇d⌊l2\n2.(3.21)\nMeanwhile, an application of the cross product gradient est imate (2.9) implies that, for any\nδ >0, the following inequality is valid:\n(3.22)/vextenddouble/vextenddouble/vextenddouble∇h/parenleftBig\n˜en+2\nh׈mn+2\nh/parenrightBig/vextenddouble/vextenddouble/vextenddouble2\n2\n≤(1+δ)/⌊a∇d⌊lˆmn+2\nh/⌊a∇d⌊l2\n∞·/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l2\n2+Cδ/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n4·/⌊a∇d⌊l∇hˆmn+2\nh/⌊a∇d⌊l2\n4\n≤(1+δ)α2\n1/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l2\n2+Cδ/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n4,\nwhereCδis a positive constant dependent on δ, in which the a-priori bound estimates ( 3.15)\nand (3.19) have been applied. The term /⌊a∇d⌊l∇hˆen+2\nh/⌊a∇d⌊l2\n2can be analyzed as follows:\n(3.23)/⌊a∇d⌊l∇hˆen+2\nh/⌊a∇d⌊l2\n2=/⌊a∇d⌊l∇h(2en+1\nh−en\nh)/⌊a∇d⌊l2\n2≤6/⌊a∇d⌊l∇hen+1\nh/⌊a∇d⌊l2\n2+3/⌊a∇d⌊l∇hen\nh/⌊a∇d⌊l2\n2\n≤(1+δ)(6/⌊a∇d⌊l∇h˜en+1\nh/⌊a∇d⌊l2\n2+3/⌊a∇d⌊l∇h˜en\nh/⌊a∇d⌊l2\n2)+Cδ(/⌊a∇d⌊l˜en+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜en\nh/⌊a∇d⌊l2\n2),\nin which the estimate ( 2.19) (in Lemma 2.4) has been repeatedly applied, due to the a-priori\nassumption ( 3.13). Combining ( 3.22), (3.23) and (3.21), we get\n˜I1≤3\n2/vextenddouble/vextenddouble/vextenddouble∇h/parenleftBig\n˜en+2\nh׈mn+2\nh/parenrightBig/vextenddouble/vextenddouble/vextenddouble2\n2+1\n6/⌊a∇d⌊l∇hˆen+2\nh/⌊a∇d⌊l2\n2\n≤(1+δ)/parenleftBig3\n2α2\n1/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l∇h˜en+1\nh/⌊a∇d⌊l2\n2+1\n2/⌊a∇d⌊l∇h˜en\nh/⌊a∇d⌊l2\n2/parenrightBig\n+Cδ(/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n4+/⌊a∇d⌊l˜en+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜en\nh/⌊a∇d⌊l2\n2).(3.24)\nFor the term ˜I2, by the preliminary estimate ( 3.12) for the exact solution, we have\n˜I2=−/angbracketleftbig\nˆen+2\nh×∆hˆmn+2\nh,˜en+2\nh/angbracketrightbig\n≤1\n2/bracketleftbig\n/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊lˆen+2\nh/⌊a∇d⌊l2\n2·/⌊a∇d⌊l∆hˆmn+2\nh/⌊a∇d⌊l2\n∞/bracketrightbig\n≤C(/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len\nh/⌊a∇d⌊l2\n2).(3.25)\nFor the term ˜I3, an application of Cauchy inequality gives\n˜I3=/angbracketleftbig\nτn+2,˜en+2\nh/angbracketrightbig\n≤ C/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2+C(k4+h4). (3.26)\nIn terms of ˜I5, based on the W1,∞\nhbound (3.12) for the exact solution, an application of\ndiscrete H¨ older inequality gives\n/vextenddouble/vextenddouble/vextenddouble|Ah∇hˆmn+2\nh|2ˆen+2\nh/vextenddouble/vextenddouble/vextenddouble\n2≤/⌊a∇d⌊l∇hˆmn+2\nh/⌊a∇d⌊l2\n∞·/⌊a∇d⌊lˆen+2\nh/⌊a∇d⌊l2≤ C/⌊a∇d⌊lˆen+2\nh/⌊a∇d⌊l2, (3.27)\nand\n˜I5=α/an}⌊∇a⌋ketle{t|Ah∇hˆmn+2\nh|2ˆen+2\nh,˜en+2\nh/an}⌊∇a⌋ket∇i}ht ≤α/vextenddouble/vextenddouble/vextenddouble|Ah∇hˆmn+2\nh|2ˆen+2\nh/vextenddouble/vextenddouble/vextenddouble\n2·/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n≤Cα/⌊a∇d⌊lˆen+2\nh/⌊a∇d⌊l2·/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2≤ C(/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2).(3.28)\nFor the term ˜I6, an application of discrete H¨ older inequality gives\n˜I6=α/angbracketleftBig/parenleftBig\nAh∇h(ˆmn+2\nh+ˆmn+2\nh)·Ah∇hˆen+2\nh/parenrightBig\nˆmn+2\nh,˜en+2\nh/angbracketrightBig\n≤α/parenleftBig\n/⌊a∇d⌊l∇hˆmn+2\nh/⌊a∇d⌊l4+/⌊a∇d⌊l∇hˆmn+2\nh/⌊a∇d⌊l4/parenrightBig\n·/⌊a∇d⌊l∇hˆen+2\nh/⌊a∇d⌊l2·/⌊a∇d⌊lˆmn+2\nh/⌊a∇d⌊l∞·/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l4\n≤Cα/⌊a∇d⌊l∇hˆen+2\nh/⌊a∇d⌊l2·/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l4≤ Cα2γ−1\n0/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n4+γ0\n72/⌊a∇d⌊l∇hˆen+2\nh/⌊a∇d⌊l2\n2\n≤C/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n4+(1+δ)γ0\n24(2/⌊a∇d⌊l∇h˜en+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l∇h˜en\nh/⌊a∇d⌊l2\n2)+Cδ(/⌊a∇d⌊l˜en+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜en\nh/⌊a∇d⌊l2\n2),(3.29)CONVERGENCE ANALYSIS FOR THE LL EQUATION WITH LARGE DAMPING 11\nin which the W1,4\nhbounds ( 3.12), (3.16), for the exact and numerical solutions, as well as the\npreliminary error estimate ( 3.23), have been applied. On the other hand, the inner product of\nthe left hand side of ( 3.11) with˜en+2\nhturns out to be\nL.H.S.=1\n4k/parenleftbig\n/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2−/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l2˜en+2\nh−en+1\nh/⌊a∇d⌊l2\n2−/⌊a∇d⌊l2en+1\nh−en\nh/⌊a∇d⌊l2\n2\n+/⌊a∇d⌊l˜en+2\nh−2en+1\nh+en\nh/⌊a∇d⌊l2\n2/parenrightbig\n.\nIts combination with eqs. ( 3.24) to (3.26), (3.28) and (3.29) and (3.20) leads to\n1\n4k(/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2−/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l2˜en+2\nh−en+1\nh/⌊a∇d⌊l2\n2−/⌊a∇d⌊l2en+1\nh−en\nh/⌊a∇d⌊l2\n2)\n+α2/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l2\n2−(1+δ)(1+γ0\n12)/parenleftBig\n/⌊a∇d⌊l∇h˜en+1\nh/⌊a∇d⌊l2\n2+1\n2/⌊a∇d⌊l∇h˜en\nh/⌊a∇d⌊l2\n2/parenrightBig\n≤ Cδ/parenleftBign+2/summationdisplay\np=n/⌊a∇d⌊l˜ep\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n4/parenrightBig\n+C(k4+h4),(3.30)\nwhereα2= (α−3\n2α2\n1(1+δ)). Meanwhile, for the /⌊a∇d⌊l·/⌊a∇d⌊l4error estimate for ˜en+2\nh, an application\nof the discrete Sobolev inequality ( 2.16) (in Lemma 2.3) gives\n(3.31)Cδ/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n4≤Cδ(/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l1\n2\n2·/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l3\n2\n2)\n≤Cδ/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2+γ0\n12/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l2\n2,\nin which the Young’s inequality has been applied. Then we get\n(3.32)1\n4k/parenleftbig\n/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2−/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l2˜en+2\nh−en+1\nh/⌊a∇d⌊l2\n2−/⌊a∇d⌊l2en+1\nh−en\nh/⌊a∇d⌊l2\n2)\n+/parenleftBig\nα2−γ0\n12/parenrightBig\n/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l2\n2−(1+δ)(1+γ0\n12)/parenleftBig\n/⌊a∇d⌊l∇h˜en+1\nh/⌊a∇d⌊l2\n2+1\n2/⌊a∇d⌊l∇h˜en\nh/⌊a∇d⌊l2\n2/parenrightBig\n≤Cδ/parenleftBig\n/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜en+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜en\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len\nh/⌊a∇d⌊l2\n2/parenrightBig\n+C(k4+h4).\nIn particular, we observe that\n(3.33)/parenleftbig\nα−3\n2α2\n1(1+δ)−γ0\n12/parenrightBig\n−3\n2(1+δ)(1+γ0\n12)\n≥α−3α2\n1(1+δ)≥(3+γ0)−3(1+1\n6γ0)(1+δ)≥1\n4γ0,\nifδ >0 is chosen with (1+1\n6γ0)(1+δ)≤1+1\n4γ0.\nMoreover, an application of the a-priori assumption ( 3.13) into (3.32) yields\n(3.34)(1\n4k−Cδ)/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2+/parenleftBig3\n2(1+δ)(1+γ0\n12)+γ0\n4/parenrightBig\n/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l2\n2\n≤(1+δ)(1+γ0\n12)/parenleftBig\n/⌊a∇d⌊l∇h˜en+1\nh/⌊a∇d⌊l2\n2+1\n2/⌊a∇d⌊l∇h˜en\nh/⌊a∇d⌊l2\n2/parenrightBig\n+1\n4k/parenleftbig\n/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l2ˆen+2\nh/⌊a∇d⌊l2\n2)\n+Cδ/parenleftBig\n/⌊a∇d⌊l˜en+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜en\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len\nh/⌊a∇d⌊l2\n2/parenrightBig\n+C(k4+h4)\n≤3\n2(1+δ)(1+γ0\n12)·1\n4k11\n4+C(k23\n8+k15\n4+k4+h4)\n≤/parenleftBig3\n2(1+δ)(1+γ0\n12)+γ0\n8/parenrightBig\n·1\n4k11\n4,\nprovided that kandhare sufficiently small, and under the linear refinement requir ement,C1h≤\nk≤C2h. As a matter of fact, we can choose γ0>0 andδ >0 and make ksufficiently small so\nthat\n(3.35)3\n2(1+δ)(1+γ0\n12)+γ0\n8≤2,1\n4k−Cδ≥1\n6k.12 YONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nAs a result, by ( 3.34), we arrive at\n(3.36)1\n6k/⌊a∇d⌊l˜eq+2\nh/⌊a∇d⌊l2\n2≤1\n2k11\n4,/⌊a∇d⌊l∇h˜eq+2\nh/⌊a∇d⌊l2\n2≤1\n4k11\n4,\ni.e.,/⌊a∇d⌊l˜eq+2\nh/⌊a∇d⌊l2≤√\n3k15\n8≤2k15\n8,/⌊a∇d⌊l∇h˜eq+2\nh/⌊a∇d⌊l2≤1\n2k11\n8,\nso that the a-priori assumption is valid for ˜mq+2.\nWith the recovery of the a-priori estimate ( 3.36) at time step tq+2, we are able to apply\nestimates ( 2.18), (2.22) (in Lemmas 2.4and2.5), respectively:\n/⌊a∇d⌊l˜en+2\nh/⌊a∇d⌊l2\n2≥(1−k5\n4)/⌊a∇d⌊len+2\nh/⌊a∇d⌊l2\n2+(1−k1\n4)/⌊a∇d⌊l˜en+2\nh−en+2\nh/⌊a∇d⌊l2\n2, (3.37)/vextendsingle/vextendsingle/vextendsingle/an}⌊∇a⌋ketle{t˜en+2\nh−en+2\nh,ep\nh/an}⌊∇a⌋ket∇i}ht/vextendsingle/vextendsingle/vextendsingle≤k5\n4/⌊a∇d⌊lep\nh/⌊a∇d⌊l2\n2+k1\n4/⌊a∇d⌊l˜en+2\nh−en+2\nh/⌊a∇d⌊l2\n2, p=n+1,n+2. (3.38)\nMoreover, the following inequality becomes available for n≤q:\n(3.39)/⌊a∇d⌊l2˜en+2\nh−en+1\nh/⌊a∇d⌊l2\n2=/⌊a∇d⌊lˆen+3\nh/⌊a∇d⌊l2\n2+4/an}⌊∇a⌋ketle{t˜en+2\nh−en+2\nh,ˆen+3\nh/an}⌊∇a⌋ket∇i}ht+4/⌊a∇d⌊l˜en+2\nh−en+2\nh/⌊a∇d⌊l2\n2\n≥/⌊a∇d⌊l2en+2\nh−en+1\nh/⌊a∇d⌊l2\n2−8k5\n4/⌊a∇d⌊len+2\nh/⌊a∇d⌊l2\n2−8k1\n4/⌊a∇d⌊l˜en+2\nh−en+2\nh/⌊a∇d⌊l2\n2\n−4k5\n4/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2−4k1\n4/⌊a∇d⌊l˜en+2\nh−en+2\nh/⌊a∇d⌊l2\n2+4/⌊a∇d⌊l˜en+2\nh−en+2\nh/⌊a∇d⌊l2\n2\n≥/⌊a∇d⌊l2en+2\nh−en+1\nh/⌊a∇d⌊l2\n2−k(/⌊a∇d⌊len+2\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2)\n+(4−12k1\n4)/⌊a∇d⌊l˜en+2\nh−en+2\nh/⌊a∇d⌊l2\n2.\nGoing back ( 3.32), we arrive at the following estimate, for n≤q:\n(3.40)1\n4k/parenleftbig\n/⌊a∇d⌊len+2\nh/⌊a∇d⌊l2\n2−/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l2en+2\nh−en+1\nh/⌊a∇d⌊l2\n2−/⌊a∇d⌊l2en+1\nh−en\nh/⌊a∇d⌊l2\n2)\n+5−13k1\n4\n4k/⌊a∇d⌊l˜en+2\nh−en+2\nh/⌊a∇d⌊l2\n2+/parenleftBig\nα2−γ0\n12/parenrightBig\n/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l2\n2\n−(1+δ)(1+γ0\n12)/parenleftBig\n/⌊a∇d⌊l∇h˜en+1\nh/⌊a∇d⌊l2\n2+1\n2/⌊a∇d⌊l∇h˜en\nh/⌊a∇d⌊l2\n2/parenrightBig\n≤Cδn+2/summationdisplay\np=n(/⌊a∇d⌊l˜ep\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊lep\nh/⌊a∇d⌊l2\n2/parenrightBig\n+C(k4+h4).\nMeanwhile, for the terms /⌊a∇d⌊l˜ep\nh/⌊a∇d⌊l2\n2,p=n,n+1,n+2, an application of Cauchy inequality gives\n(3.41) /⌊a∇d⌊l˜ep\nh/⌊a∇d⌊l2\n2≤2(/⌊a∇d⌊lep\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜ep\nh−ep\nh/⌊a∇d⌊l2\n2), p=n,n+1,n+2.\nIts substitution into ( 3.40) leads to the following inequality for n≤q:\n(3.42)1\n4k/parenleftbig\n/⌊a∇d⌊len+2\nh/⌊a∇d⌊l2\n2−/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l2en+2\nh−en+1\nh/⌊a∇d⌊l2\n2−/⌊a∇d⌊l2en+1\nh−en\nh/⌊a∇d⌊l2\n2)\n+1\n4k/⌊a∇d⌊l˜en+2\nh−en+2\nh/⌊a∇d⌊l2\n2−Cδ(/⌊a∇d⌊l˜en+1\nh−en+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜en\nh−en\nh/⌊a∇d⌊l2\n2)\n+/parenleftBig\nα2−γ0\n12/parenrightBig\n/⌊a∇d⌊l∇h˜en+2\nh/⌊a∇d⌊l2\n2−(1+δ)(1+γ0\n12)/parenleftBig\n/⌊a∇d⌊l∇h˜en+1\nh/⌊a∇d⌊l2\n2+1\n2/⌊a∇d⌊l∇h˜en\nh/⌊a∇d⌊l2\n2/parenrightBig\n≤Cδ/parenleftBig\n/⌊a∇d⌊len+2\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len+1\nh/⌊a∇d⌊l2\n2+/⌊a∇d⌊len\nh/⌊a∇d⌊l2\n2/parenrightBig\n+C(k4+h4).\nIn turn, an application of discrete Gronwall inequality [ 16], combined with the fact ( 3.33),\nyields the desired convergence estimate at tq+2:\n(3.43)/⌊a∇d⌊len\nh/⌊a∇d⌊l2\n2+γ0kn/summationdisplay\np=0/⌊a∇d⌊l∇h˜ep\nh/⌊a∇d⌊l2\n2≤ CTeCT(k4+h4),∀n≤q+2≤/floorleftbiggT\nk/floorrightbigg\n,\ni.e.,/⌊a∇d⌊len\nh/⌊a∇d⌊l2+/parenleftBig\nγ0kn/summationdisplay\np=0/⌊a∇d⌊l∇h˜ep\nh/⌊a∇d⌊l2\n2/parenrightBig1\n2≤ C(k2+h2).CONVERGENCE ANALYSIS FOR THE LL EQUATION WITH LARGE DAMPING 13\nThe convergence estimate ( 2.6) has been proved at tq+2. In addition, we see that the a-priori\nassumption ( 3.13) has also been validated at the next time step tq+2, provided that kandhare\nsufficiently small. By mathematical induction, this complet es the proof of Theorem 2.1./square\nRemark 3.1. The condition α >3is a relatively strong constraint. In fact, such a condition\nis used in the estimate (3.24)for˜I1, since we need α >3to control these Laplace terms, due to\nthe explicit treatment of ∆hˆmn+2\nh. Meanwhile, such an inequality only stands for a theoretica l\ndifficulty, and the practical computations may not need that l arge value of α. In most practical\nsimulation examples, a value of α >1would be sufficient to ensure the numerical stability of the\nproposed numerical scheme ( 2.3)-(2.4).\nIn addition, the explicit treatment of the Laplace term, name ly∆hˆmn+2\nh= ∆h(2mn+1\nh−mn\nh),\nwill greatly improve the numerical efficiency, since only a co nstant-coefficient Poisson solver is\nneeded at each step. This crucial fact enables one to produce v ery robust simulation results at a\nmuch-reduced computational cost.\nRemark 3.2. In a recent work [7], a rough stability estimate for the projection step, namely ,\n/⌊a∇d⌊leh/⌊a∇d⌊l2≤2/⌊a∇d⌊l˜eh/⌊a∇d⌊l2+O(h2),/⌊a∇d⌊l∇heh/⌊a∇d⌊l2≤ C(/⌊a∇d⌊leh/⌊a∇d⌊l2+/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l2) +O(h2), has been proved. These\ninequalities are sufficient to establish the stability and co nvergence analysis for a semi-implicit\nnumerical scheme, if the BDF2 temporal stencil is formulate d as1\nk(3\n2˜mn+2\nh−2˜mn+1\nh+1\n2˜mn\nh).\nHowever, for the BDF2 temporal stencil formulated as1\nk(3\n2˜mn+2\nh−2mn+1\nh+1\n2mn\nh)(as given by\nAlgorithm 2.2 ), such a stability estimate are not sufficient to derive the st ability and conver-\ngence analysis, due to the singular coefficient1\nk. Instead, a much more refined stability estimate,\nas given by (2.18),(2.19)(in Lemma 2.4), is needed to pass through the convergence analysis.\nThe proof of these two refined inequalities has to be based on a m ore precise geometric analysis\nof the corresponding vectors, and the details will be presen ted in Appendix A.\nExtensive numerical experiments have demonstrated a bette r stability property of the temporal\nstencil in Algorithm 2.2 than that of Algorithm 2.1 , for physical models with large damping\nparameters. For the theoretical analysis of Algorithm 2.2 , the refined stability estimate (2.18),\n(2.19)has played a crucial role.\n4.Conclusions\nIn this paper, we have presented an optimal rate convergence analysis and error estimate\nfor a second-order accurate, linear numerical scheme to the LL equation. The second-order\nbackward differentiation formula is applied in the temporal d iscretization, the linear diffusion\nterm is treated implicitly, while the nonlinear terms are up dated by a fully explicit extrapolation\nformula. Afterward, a point-wise projection is applied to n ormalize the magnetization vector.\nIn turn, only a linear system independent of the updated magn etization needs to be solved\nat each time step, which has greatly improved the computatio nal efficiency, and many great\nadvantages of this numerical scheme have been reported in th e numerical simulation with large\ndamping parameters. The error estimate has been theoretica lly established in the discrete\nℓ∞(0,T;ℓ2)∩ℓ2(0,T;H1\nh) norm, under suitable regularity assumptions and reasonab le ratio\nbetween the time step-size and the spatial mesh-size. The ke y difficulty of the theoretical\nanalysis is associated with the fact that the projection ste p is highly nonlinear and non-convex.\nTo overcome this subtle difficulty, we build a stability estim ate for the projection step, which\nplays a crucial role in the derivation of the convergence ana lysis for the numerical scheme.\nAppendix A. Proof of lemma 2.4\nProof.First of all, an /⌊a∇d⌊l · /⌊a∇d⌊l∞bound for the numerical error ˜ehcan be derived, by the a-priori\nestimate ( 2.17):\n(4.1) /⌊a∇d⌊l˜eh/⌊a∇d⌊l∞≤γh−1\n2(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2+/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l2)≤γh−1\n2·k11\n8≤1\n4k3\n4,\nprovided that kandhare sufficiently small, and under the linear refinement requir ementC1h≤\nk≤C2h. Notice that the inverse inequality ( 2.7) has been applied in the first step. In turn, we14 YONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nobserve that the following bounds are available for the nume rical profile ˜mh:\n1−1\n4k3\n4≤ |˜mh| ≤1+1\n4k3\n4,at a point-wise level , (4.2)\n/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l4≤γh−3\n4/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l2≤γh−3\n4·1\n2k11\n8≤k1\n2, (4.3)\n/⌊a∇d⌊l∇h˜mh/⌊a∇d⌊l4≤ /⌊a∇d⌊l∇hmh/⌊a∇d⌊l4+/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l4≤C∗+1 :=M. (4.4)\nAgain, the inverse inequality ( 2.7) has been applied in the derivation.\nA careful calculation indicates that\n(4.5)eh=mh−mh=mh−˜mh+˜mh−˜mh\n|˜mh|=˜eh+˜mh\n|˜mh|(|˜mh|−1),so that\n˜eh=eh+˜eh,c,˜eh,c:=˜mh\n|˜mh|(1−|˜mh|).\nMeanwhile, at a fixed grid point (ˆ xi,ˆyj,ˆzℓ)∈Ω0\nh, we look at the triangle formed by the vectors:\n˜mh,˜ehandmh. In particular, we denote the angle between ˜mhandmhasθ. Since the lengths\nof these three vectors have the following estimates:\n1−1\n4k3\n4≤ |˜mh| ≤1+1\n4k3\n4,|˜eh| ≤1\n4k3\n4,|mh| ≡1,\na careful application of Sine law indicates that\n(4.6) 0 ≤sinθ≤1\n4k3\n4.\nAndalso, welook at thetriangle formedbythevectors: mh,ehandmh. Twosideshaveequal\nlengths: |mh|=|mh|= 1, andtheangle between mhandmhis exactly θ, sincemh=˜mh\n|˜mh|is in\nthe same direction as ˜mh. In turn, the angle between vectors mhandehis given by ϕ′=π\n2−θ\n2.\nBecause of this fact, the angle between vectors ˜mhandehis the same as ϕ′=π\n2−θ\n2,\nSubsequently, we denote the angle between vectors ˜ eh,candehasϕ. In fact, this angle has\nthe following representation\n(4.7) ϕ=ϕ′=π\n2−θ\n2,if|˜mh| ≤1;ϕ=π−ϕ′=π\n2+θ\n2,if|˜mh|>1.\nIn either case, the following estimate is valid:\n(4.8) |cosϕ|=|sinθ\n2| ≤sinθ≤1\n4k3\n4.\nConsequently, the definition of the point-wise inner produc t implies the following estimate\n(4.9)|˜eh|2=|eh+˜eh,c|2=|eh|2+|˜eh,c|2+2eh·˜eh,c\n=|eh|2+|˜eh,c|2+2|eh|·|˜eh,c|·cosϕ\n≥|eh|2+|˜eh,c|2−2|eh|·|˜eh,c|·1\n4k3\n4=|eh|2+|˜eh,c|2−1\n2k3\n4|eh|·|˜eh,c|\n≥|eh|2+|˜eh,c|2−(k5\n4|eh|2+1\n4k1\n4|˜eh,c|2)\n≥(1−k5\n4)|eh|2+(1−k1\n4)|˜eh−eh|2.\nNoticethatthisestimateisvalidatapoint-wiselevel, for anyfixedmeshpoint( i,j,ℓ). Therefore,\na summation in space leads to the first inequality ( 2.18).\nTo derive the second inequality ( 2.19), we will focus on the Dxpart in the discrete gradient;\nthe analysis for the DyandDzparts can be performed in a similar manner. We begin with theCONVERGENCE ANALYSIS FOR THE LL EQUATION WITH LARGE DAMPING 15\nfollowing expansion:\n(4.10)Dxeh=Dxmh−Dx˜mh\n|˜mh|=Dx/bracketleftbiggmh\n|˜mh|−˜mh\n|˜mh|/bracketrightbigg\n+Dx/bracketleftbigg\nmh−mh\n|˜mh|/bracketrightbigg\n=Dx˜eh\n|˜mh|−Dx/bracketleftbiggmh\n|˜mh|(1−|˜mh|)/bracketrightbigg\n=Dx˜eh\n|˜mh|−Dx/bracketleftbiggmh\n|˜mh|(mh+˜mh)·(mh−˜mh)\n1+|˜mh|/bracketrightbigg\n=Dx˜eh\n|˜mh|−Dx/bracketleftbigg\nmh(mh+˜mh)·˜eh\n|˜mh|+|˜mh|2/bracketrightbigg\n=Dx˜eh\n|˜mh|−Dx/bracketleftbigg\nmh(2mh−˜eh)·˜eh\n|˜mh|+|˜mh|2/bracketrightbigg\n=Dx˜eh\n|˜mh|−Dx/parenleftBig\nmh2mh·˜eh\n|˜mh|+|˜mh|2/parenrightBig\n+Dx/parenleftBig\nmh|˜eh|2\n|˜mh|+|˜mh|2/parenrightBig\n,\nin which the identity ˜mh=mh−˜ehhas been applied at the last two steps. Meanwhile, at each\nnumerical mesh cell, from ( i,j,ℓ) to (i+1,j,ℓ), the identity ( 2.10) is valid, so that the following\nexpansions are able to be made at the center location ( i+1/2,j,ℓ) over the numerical cell:\n(4.11)Dx˜eh\n|˜mh|−Dx/parenleftBig\nmh2mh·˜eh\n|˜mh|+|˜mh|2/parenrightBig\n=J1+J2+J3+J4+J5,\nJ1=Ax/parenleftBig1\n|˜mh|/parenrightBig\nDx˜eh−AxmhAx/parenleftBig2\n|˜mh|+|˜mh|2/parenrightBig\n(Axmh·Dx˜eh),\nJ2=−/parenleftBig\nA(2)\nx/parenleftBig1\n|˜mh|2/parenrightBig\nDx|˜mh|/parenrightBig\n˜eh, J3=−2(Dxmh)Ax/parenleftBigmh·˜eh\n|˜mh|+|˜mh|2/parenrightBig\n,\nJ4=(Axmh)Ax(mh·˜eh)A(2)\nx/parenleftBig2\n(|˜mh|+|˜mh|2)2/parenrightBig\n·(Dx|˜mh|+2Ax˜mh·Dx˜mh),\nJ5=−AxmhAx/parenleftBig2\n|˜mh|+|˜mh|2/parenrightBig\n(Dxmh·Ax˜eh),\nin which the nonlinear average operator A(2)\nxis introduced as\n(4.12) A(2)\nx/parenleftBig1\n(fh)2/parenrightBig\ni,j,ℓ=1\n(fh)i,j,ℓ(fh)i+1,j,ℓ,for scalar grid function fh.\nMoreover, by the point-wise a-priori estimate ( 4.2) for˜mh, the following bounds are available:\n(4.13)1\n1+1\n4k3\n4≤Ax/parenleftBig1\n|˜mh|/parenrightBig\n≤1\n1−1\n4k3\n4,\n1−3\n2k3\n4≤Ax/parenleftBig2\n|˜mh|+|˜mh|2/parenrightBig\n≤1+3\n2k3\n4,\n1\n(1+1\n4k3\n4)2≤A(2)\nx/parenleftBig1\n|˜mh|2/parenrightBig\n≤1\n(1−1\n4k3\n4)2,\n1\n2(1+1\n4k3\n4)4≤A(2)\nx/parenleftBig2\n(|˜mh|+|˜mh|2)2/parenrightBig\n≤1\n(1−1\n4k3\n4)4.\nFor the term J1, we make the following decomposition J1=J11+J12+J13to facilitate the\nanalysis\n(4.14)J11=Dx˜eh−Axmh(Axmh·Dx˜eh), J12=/parenleftBig\nAx/parenleftBig1\n|˜mh|/parenrightBig\n−1/parenrightBig\nDx˜eh,\nJ13=−Axmh/parenleftBig\nAx/parenleftBig2\n|˜mh|+|˜mh|2/parenrightBig\n−1/parenrightBig\n(Axmh·Dx˜eh).16 YONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nFor the quantity J11, we see that\n(4.15)|J11|2=|Dx˜eh|2+|Axmh|2(Axmh·Dx˜eh)2−2(Axmh·Dx˜eh)2\n≤|Dx˜eh|2−(Axmh·Dx˜eh)2,so that|J11| ≤ |Dx˜eh|,\nin which the last step comes from the fact that |Axmh|2≤1. For the quantity J12, thea-priori\nbounds ( 4.13) imply that\n(4.16) J12=/vextendsingle/vextendsingle/vextendsingleAx/parenleftBig1\n|˜mh|/parenrightBig\n−1/vextendsingle/vextendsingle/vextendsingle·|Dx˜eh| ≤1\n2k3\n4|Dx˜eh|.\nSimilarly, the quantity J13can be analyzed as\n(4.17)J13=|Axmh|·/vextendsingle/vextendsingle/vextendsingleAx/parenleftBig2\n|˜mh|+|˜mh|2/parenrightBig\n−1/vextendsingle/vextendsingle/vextendsingle·|Axmh|·|Dx˜eh|\n≤(1+Ch2)2(1\n(1−1\n4k3\n4)2|Dx˜eh| ≤(1+3\n4k3\n4)|Dx˜eh|.\nConsequently, a combination of ( 4.15)-(4.17) leads to\n(4.18) |J1| ≤ |J11|+|J12|+|J13| ≤(1+5\n4k3\n4)|Dx˜eh|,\nwhich turns out to be a point-wise inequality. In turn, a summ ation in space implies that\n(4.19) /⌊a∇d⌊lJ1/⌊a∇d⌊l2≤(1+5\n4k3\n4)/⌊a∇d⌊lDx˜eh/⌊a∇d⌊l2.\nFor theterm J2, wemake useof the a-prioribounds( 4.13), as well as thefact that |Dx|˜mh|| ≤\n|Dx˜mh|, so that an application of discrete H¨ older inequality give s\n(4.20) /⌊a∇d⌊lJ2/⌊a∇d⌊l2≤/vextenddouble/vextenddouble/vextenddoubleA(2)\nx/parenleftBig1\n|˜mh|2/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n∞·/⌊a∇d⌊lDx˜mh/⌊a∇d⌊l4·/⌊a∇d⌊l˜eh/⌊a∇d⌊l4≤(1+3\n4k3\n4)M/⌊a∇d⌊l˜eh/⌊a∇d⌊l4,\nin which the a-priori W1,4\nhbound (4.4) for the numerical profile ˜mhhas been applied. The\nother terms in the expansion ( 4.11) can be analyzed in a similar manner.\n(4.21)/⌊a∇d⌊lJ3/⌊a∇d⌊l2≤/⌊a∇d⌊lDxmh/⌊a∇d⌊l∞·max/parenleftBig2\n|˜mh|+|˜mh|2/parenrightBig\n·/⌊a∇d⌊lmh/⌊a∇d⌊l∞·/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n≤C∗(1+3\n4k3\n4)·1·/⌊a∇d⌊l˜eh/⌊a∇d⌊l2≤(1+3\n4k3\n4)M/⌊a∇d⌊l˜eh/⌊a∇d⌊l2,\n/⌊a∇d⌊lJ4/⌊a∇d⌊l2≤/⌊a∇d⌊lmh/⌊a∇d⌊l2\n∞·/⌊a∇d⌊l˜eh/⌊a∇d⌊l4·max/parenleftBig2\n(|˜mh|+|˜mh|2)2/parenrightBig\n·(1+2/⌊a∇d⌊l˜mh/⌊a∇d⌊l∞)/⌊a∇d⌊lDx˜mh/⌊a∇d⌊l4≤3\n2(1+3\n2k3\n4)M/⌊a∇d⌊l˜eh/⌊a∇d⌊l4,\n/⌊a∇d⌊lJ5/⌊a∇d⌊l2≤/��a∇d⌊lmh/⌊a∇d⌊l∞·max/parenleftBig2\n|˜mh|+|˜mh|2/parenrightBig\n·/⌊a∇d⌊lDxmh/⌊a∇d⌊l∞·/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n≤(1+3\n2k3\n4)C∗/⌊a∇d⌊l˜eh/⌊a∇d⌊l2≤(1+3\n2k3\n4)M/⌊a∇d⌊l˜eh/⌊a∇d⌊l2.\nTherefore, a substitution of ( 4.19), (4.20) and (4.21) into (4.11) leads to\n(4.22)/vextenddouble/vextenddouble/vextenddoubleDx˜eh\n|˜mh|−Dx/parenleftBig\nmh2mh·˜eh\n|˜mh|+|˜mh|2/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2\n≤(1+5\n4k3\n4)/⌊a∇d⌊lDx˜eh/⌊a∇d⌊l2+3M(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l4).CONVERGENCE ANALYSIS FOR THE LL EQUATION WITH LARGE DAMPING 17\nThe analysis for the last term on the right hand side of ( 4.10) can be similarly carried out;\nsome technical details are skipped for the sake of brevity.\n(4.23)/vextenddouble/vextenddouble/vextenddoubleDx/parenleftBig\nmh|˜eh|2\n|˜mh|+|˜mh|2/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2\n≤C/parenleftBig\n/⌊a∇d⌊lmh/⌊a∇d⌊l∞·max/parenleftBig2\n|˜mh|+|˜mh|2/parenrightBig\n·/⌊a∇d⌊l˜eh/⌊a∇d⌊l∞·/⌊a∇d⌊lDx˜eh/⌊a∇d⌊l2\n+/⌊a∇d⌊lDxmh/⌊a∇d⌊l∞·max/parenleftBig1\n|˜mh|+|˜mh|2/parenrightBig\n·/⌊a∇d⌊l˜eh/⌊a∇d⌊l∞·/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n+/⌊a∇d⌊lmh/⌊a∇d⌊l∞·max/parenleftBig1\n(|˜mh|+|˜mh|2)2/parenrightBig\n·(1+2/⌊a∇d⌊l˜mh/⌊a∇d⌊l∞)/⌊a∇d⌊lDx˜mh/⌊a∇d⌊l4\n·/⌊a∇d⌊l˜eh/⌊a∇d⌊l∞·/⌊a∇d⌊l˜eh/⌊a∇d⌊l4/parenrightBig\n≤Ck3\n4(/⌊a∇d⌊lDx˜eh/⌊a∇d⌊l2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l4)≤k5\n8(/⌊a∇d⌊lDx˜eh/⌊a∇d⌊l2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l4).\nFinally, a substitution of ( 4.22) and (4.23) into (4.10) yields\n(4.24)/⌊a∇d⌊lDxeh/⌊a∇d⌊l2≤(1+2k5\n8)/⌊a∇d⌊lDx˜eh/⌊a∇d⌊l2+(3M+1)(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l4),so that\n/⌊a∇d⌊lDxeh/⌊a∇d⌊l2\n2≤(1+5k5\n8)/⌊a∇d⌊lDx˜eh/⌊a∇d⌊l2\n2+(3M+1)2(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l4)2\n+4(3M+1)/⌊a∇d⌊lDx˜eh/⌊a∇d⌊l2(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l4)\n≤(1+5k5\n8)/⌊a∇d⌊lDx˜eh/⌊a∇d⌊l2\n2+2(3M+1)2(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n4)\n+δ\n4/⌊a∇d⌊lDx˜eh/⌊a∇d⌊l2\n2+16(3M+1)2δ−1(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l4)2\n≤(1+δ\n2)/⌊a∇d⌊lDx˜eh/⌊a∇d⌊l2\n2+(32δ−1+2)(3M+1)2(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n4),\nfor anyδ >0, provided that kis sufficiently small. Similar estimates can be derived for th e\ngradient in the yandzdirections; the technical details are skipped for the sake o f brevity.\n(4.25)/⌊a∇d⌊lDyeh/⌊a∇d⌊l2\n2≤(1+δ\n2)/⌊a∇d⌊lDy˜eh/⌊a∇d⌊l2\n2+(32δ−1+2)(3M+1)2(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n4),\n/⌊a∇d⌊lDzeh/⌊a∇d⌊l2\n2≤(1+δ\n2)/⌊a∇d⌊lDz˜eh/⌊a∇d⌊l2\n2+(32δ−1+2)(3M+1)2(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n4).\nThen we arrive at\n(4.26) /⌊a∇d⌊l∇heh/⌊a∇d⌊l2\n2≤(1+δ\n2)/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l2\n2+3(32δ−1+2)(3M+1)2(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n4).\nMeanwhile, by the discrete Sobolev inequality ( 2.16) (in Lemma 2.3), we get\n/⌊a∇d⌊l˜eh/⌊a∇d⌊l4≤ C(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l1\n4\n2·/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l3\n4\n2), (4.27)\nso that\n(4.28)3(32δ−1+2)(3M+1)2/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n4\n≤3(32δ−1+2)(3M+1)2C(/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n2+/⌊a∇d⌊l˜eh/⌊a∇d⌊l1\n2\n2·/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l3\n2\n2)\n≤Cδ/⌊a∇d⌊l˜eh/⌊a∇d⌊l2\n2+δ\n2/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l2\n2,∀δ >0,\nin which the Young’s inequality has been applied in the last s tep. Going back ( 4.26), we obtain\n(4.29) /⌊a∇d⌊l∇heh/⌊a∇d⌊l2≤(1+δ\n2)/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l2+δ\n2/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l2+Cδ/⌊a∇d⌊l˜eh/⌊a∇d⌊l2≤(1+δ)/⌊a∇d⌊l∇h˜eh/⌊a∇d⌊l2+Cδ/⌊a∇d⌊l˜eh/⌊a∇d⌊l2,\nprovided that kis sufficiently small. This finishes the proof of Lemma 2.4. /square18 YONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nAppendix B. Proof of lemma 2.5\nProof.Herem(1)\nhandm(2)\nhserve as theexact solution at different timesteps and ˜m(1)\nh,˜m(2)\nh∈X\nare the corresponding numerical solutions.\nThe/⌊a∇d⌊l · /⌊a∇d⌊l∞bounds for the error functions ˜e(q)\nhcan be derived, with the help of the a-priori\nestimate ( 2.20):\n(4.30)/⌊a∇d⌊l˜e(q)\nh/⌊a∇d⌊l∞≤γh−1\n2(/⌊a∇d⌊l˜e(q)\nh/⌊a∇d⌊l2+/⌊a∇d⌊l∇h˜e(q)\nh/⌊a∇d⌊l2)≤γh−1\n2·k11\n8≤1\n4k3\n4,\nso that 1 −1\n4k3\n4≤ |˜m(q)\nh| ≤1+1\n4k3\n4,at a point-wise level, q= 1,2.\nFor the numerical error functions at different time steps, the decomposition ( 4.5) is still valid:\n(4.31) ˜e(q)\nh=e(q)\nh+˜e(q)\nh,c,˜e(q)\nh,c:=˜m(q)\nh\n|˜m(q)\nh|(1−|˜m(q)\nh|).\nAt a fixed grid point (ˆ xi,ˆyj,ˆzk), we look at the triangle formed by the vectors: ˜m(q)\nh,˜e(q)\nh,m(q)\nh,\nand denote the angle between ˜m(q))\nhandm(q)\nhasθq. In turn, the estimate ( 4.6) is laid for each\nθq:\n(4.32)1−1\n4k3\n4≤ |˜m(q)\nh| ≤1+1\n4k3\n4,|˜eh| ≤1\n4k3\n4,|m(q)\nh| ≡1,\n0≤sinθj≤1\n4k3\n4, q= 1,2.\nSimilarly, in the triangle formed by the vectors: m(q)\nh,e(q)\nh,m(q)\nh, two sides have equal lengths:\n|m(q)\nh|=|m(q)\nh|= 1, and the angle between m(q)\nhandm(q)\nhis exactly θq. In turn, the angle\nbetween vectors m(q)\nhande(q)\nh, as well as the angle between vectors m(q)\nhande(q)\nhis given by\nϕ(q)=π\n2−θq\n2.\nMeanwhile, we denote the angle between m(1)\nhandm(2)\nhasθ∗. By the a-priori assumption\nand the fact that |m(1)\nh|=|m(2)\nh|= 1, we have an estimate for θ∗:\n(4.33) 2sinθ∗\n2≤1\n4k7\n8.\nSubsequently, we denote the angle between ˜m(1)\nhande(2)\nhasφ∗. By the above analyses, we see\nthat\n(4.34) φ∗=ϕ(2)±(θ1+θ∗) =π\n2−θ2\n2±(θ1+θ∗).\nIn either case, φ∗=π\n2−θ2\n2+(θ1+θ∗) orφ∗=π\n2−θ2\n2−(θ1+θ∗), the following estimate is valid:\n(4.35)|cosφ∗|=|sin(θ2\n2±(θ1+θ∗))| ≤sin(θ2\n2+θ1+θ∗)\n≤sinθ2\n2+sinθ1+sinθ∗≤sinθ2+sinθ1+sinθ∗\n≤1\n4k3\n4+1\n4k3\n4+1\n4k7\n8≤3\n4k3\n4,\ninwhich we have used thetriangular inequality, sin( s1+s2+s3)≤sins1+sins2+sins3, provided\nthats1,s2,s3>0 are sufficiently small. As a consequence, the definition of th e point-wise inner\nproduct implies the following estimate\n(4.36)/vextendsingle/vextendsingle/vextendsingle(˜e(1)\nh−e(1)\nh)·e(2)\nh/vextendsingle/vextendsingle/vextendsingle=|˜e(1)\nh−e(1)\nh|·|˜e(2)\nh|·|cosφ∗|\n≤|˜e(1)\nh−e(1)\nh|·|˜e(2)\nh|·3\n4k3\n4\n≤k1\n4|˜e(1)\nh−e(1)\nh|2+k4\n5|˜e(2)\nh|2.CONVERGENCE ANALYSIS FOR THE LL EQUATION WITH LARGE DAMPING 19\nAgain, this estimate is valid at a point-wise level for x∈Ω0\nh. 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Phys. 404(2020), 109104.\nLaboratory of Mathematics and Complex Systems and School of Mathematical Sciences, Beijing\nNormal University, Beijing, China.\nEmail address :yongyong.cai@bnu.edu.cn\nSchool of Mathematical Sciences, Soochow University, Suzh ou, China.\nEmail address :jingrunchen@suda.edu.cn (Corresponding author)\nMathematics Department, University of Massachusetts, Nor th Dartmouth, MA 02747, USA.\nEmail address :cwang1@umassd.edu\nDepartment of Mathematics, The Hong Kong University of Scie nce and Technology, Clear\nWater Bay, Kowloon, Hong Kong, China.\nEmail address :macjxie@ust.hk" }, { "title": "0812.1570v1.Landau_Damping_and_Alfven_Eigenmodes_of_Neutron_Star_Torsion_Oscillations.pdf", "content": "arXiv:0812.1570v1 [astro-ph] 8 Dec 2008Landau Damping and Alfven Eigenmodes of Neutron Star Torsio n Oscillations\nAndrei Gruzinov\nCCPP, Physics Department, New York University, 4 Washington P lace, New York, NY 10003\nABSTRACT\nTorsion oscillations of the neutron star crust are Landau da mped by the Alfven\ncontinuum in the bulk. For strong magnetic fields (in magneta rs), undamped Alfven\neigenmodes appear.\n1. Introduction and Conclusion\nIt is thought that Israel et al (2005) have detected torsion o scillations of the neutron star crust.\nHere we show that crustal modes are strongly affected by the Alf ven continuum in the bulk. For\nweek magnetic fields, there is Landau damping of crustal eige nmodes. At stronger magnetar-like\nmagnetic fields undamped Alfven eigenmodes appear. In this r egime, the Alfven waves in the bulk\ncontrol the torsion oscillations of the crust.\nIt might be that Israel et al (2005) have actually measured th e Alfven speed in the bulk\ncA, rather than the torsion speed in the crust ct. If so, the measured ν≈20Hz frequency gives\ncA≈2νR≈4×107cm/s (assuming that νis the fundamental Alfven eigenmode).\nWe calculate the Landau damping of the crustal torsion modes and Alfven eigenmode (un-\ndamped) frequencies. The thin-crust uniform-field constan t-density model is used, and we only\nconsider the antisymmetric axisymmetric modes. This is obv iously not good enough for serious\nneutron star seismology. One must include: (i) the non-axis ymmetric modes, (ii) the toroidal mag-\nnetic field, (iii) the realistic density profile. But the qual itative results – Landau damping and\nemergence of undamped Alfven eigenmodes – should be robust.\nThis paper is a formal mathematical development of the ideas of Levin (2007).\n2. Results\nWe use the following units and notations:\n•R= 1 is the stellar radius\n•cA= 1 is the Alfven speed in the bulk– 2 –\n•3Mis the crust to bulk mass ratio.\n•ctis the torsion sound speed in the crust (in units of cA).\nThe calculated frequencies ω(in units of cA/R) of the antisymmetric axisymmetric torsion\nmodes (l= 2,4,6,...,m= 0) are given in the table for various mass ratios Mand speed ratios ct.\nThe eigenmodes (and cuts, see §3 ) are also shown in the figure for M= 0.1,ct= 2.\n3. The eigenmode equation and the dispersion relation\nToroidal displacements of the crust launch Alfven waves int o the bulk. For an axisymmetric\nantisymmetric torsion mode, the toroidal displacement in t he bulk is ∝e−iωtsin(ωz), wherezis\nthe Cartesian coordinate along the magnetic field, with the e quatorial plane z= 0. Including the\ncorresponding force into the thin-crust eigenmode equatio n, we get\nc2\nt/parenleftbig\n(1−x2)f′′−4xf′/parenrightbig\n+ω2f=ωxcot(ωx)\nMf. (1)\nHerex≡cosθ,θis the spherical angle with θ= 0 alongz,fsinθis the toroidal displacement of\nthe crust, prime is the x-differentiation. The detailed deriv ation is given in the Appendix.\nFor the antisymmetric mode, the boundary condition at x= 0 isf= 0. The boundary\ncondition at x= 1 is given by the equation itself: −4c2\ntf′+ω2f=ωcot(ω)\nMf.\nTo obtain the dispersion relation in the form F(ω) = 0, choose some f(1) andf′(1) satisfying\nthex= 1 boundary condition. Then integrate (1) form x= 1 tox= 0 and put F(ω) =f(0).\nAccording to the Landau rule, one calculates F(ω) in the upper half-plane of complex ω. The\neigenmodes are given by the zeros of F(ω) on the real axis (undamped modes) and in the lower\nhalf plane (Landau damped modes). The value of Fin the lower half-plane should be obtained by\nthe analytic continuation from the upper half-plane.\nTable 1: Eigenfrequencies in units of cA/R.\nM= 0.1,ct= 1 2.10, 2.74, 3.03, 3.12, 5.05-0.001i, 5.75-0.002i, ...\nM= 0.1,ct= 3 3.06, 6.18-0.11i, 9.35-0.04i, ...\nM= 0.1,ct= 10 21.7-4.01i, 46.3-6.19i, ...\nM= 0.1,ct= 100 200-3.14i, 424-3.19i, ...\nM= 0.01,ct= 10 3.11, 6.26-0.005i, 9.41-0.006i, ...\nM= 0.01,ct= 30 103-76i, 198-144i, ...\nM= 0.01,ct= 100 219-55i, 490-102i, ...– 3 –\nThe analytic continuation of Fto the lower half-plane can be accomplished by complexifyin g\nxand integrating (1) from x= 1 tox= 0 along an arbitrary contour in the upper half-plane of x.\nThe shape of the x-contour determines the shape of the cuts in the lower half-plane of ω. We chose\nthe semicircle |x−0.5|= 0.5. This gives the cuts in the lower half-plane of ωrunning straight down\nfromω=πk.\nThe dispersion function Fwas calculated numerically. The eigenmodes, both the Landa u-\ndamped crustal modes and the undamped Alfven eigenmodes are given in the table. The real\nentries of the table (what we call Alfven eigenmodes), were c onfirmed by a straightforward real-\nnumbers integration of (1).\nDue to the cuts, the late-time asymptotic of the crustal moti on will also have algebraically\ndamped modes with frequencies ω=πk. The late time asymptotic is determined solely by the\nlocation of the tips of the cuts and is not affected by the choice of the x-contour. The zeros of F\nare also independent of the x-contour.\nFor generic parameters Mandct, numerical integration seems to be the only way. But there\nare limiting cases which can be treated analytically. These may serve to confirm that equation (1)\nactually makes sensible predictions and also to check the nu merical results:\n•ct≪1,M≫1 gives undampedcrustal modes ω2= (l2+l−2)c2\nt+1\nM,l= 2,4,6,.... This case\nisobvious–magneticfieldandthefluidbulkjustfollow thesl owandheavy crust, providingan\nadditional elasticity and therefore increasing the freque ncy. This case is unphysical, because\nthe crust is actually lighter than the bulk.\n•ct≫1,Mct≫1 gives Landau damped crustal modes ω=√\nl2+l−2ct−ial\nM,l= 2,4,6,...,\nalare calculable dimensionless numbers (of order unity for lo wl). This case is physical, it\noccurs for small enough magnetic fields.\nThe calculation is as follows. By the Euler’s formula,\ncot(ωx) =/summationdisplay\nk1\nωx−πk→/integraldisplay\ndk1\nωx−πk=−i. (2)\nNow (1) becomes\nc2\nt/parenleftbig\n(1−x2)f′′−4xf′/parenrightbig\n+ω2f=−iωx\nMf. (3)\nWe solved it using the first order perturbation theory in 1 /(Mct). Forl= 2, one calculates\na2= 5/16 = 0.313 – in agreement with the M= 0.1,ct= 100 entry of the table.\nI thank Yuri Levin for showing me the problem and for useful di scussions.\nThis work was supported by the David and Lucile Packard found ation.– 4 –\nA. Torsion oscillations of the elastic crust with the magnetized bulk\nConsider a neutron star with a thin crust. Choose the zaxis along the uniform magnetic field\nB. The toroidal displacement of the fluid bulk ξis described by the Alfven wave equation\n∂2\ntξ=c2\nA∂2\nzξ, (A1)\nwherec2\nA=B2\n4πρis the Alfven speed in the bulk of density ρ.\nThe toroidal displacement of the crust ψis described by the torsion wave equation with the\nmagnetic force from the bulk\n∂2\ntψ=c2\nt\nR2/parenleftbigg\n∂2\nθψ+cosθ\nsinθ∂θψ−1\nsin2θψ+2ψ/parenrightbigg\n+cosθTφz\nσ. (A2)\nHerectis the torsion sound speed in the crust, Ris the radius of the star, θis the polar angle with\nrespect toz,Tφzis the toroidal-vertical component of the Maxwell stress te nsor,σis the surface\ndensity of the crust.\nThe Maxwell stress is given by Tφz=−BBφ\n4π,Bφ=B∂zξ, which should be calculated at the\nboundary of the bulk, where ξ=ψ. We get:\n∂2\ntψ=c2\nt\nR2/parenleftbigg\n∂2\nθψ+cosθ\nsinθ∂θψ−1\nsin2θψ+2ψ/parenrightbigg\n−ρc2\nA\nσcosθ∂zξ|z=cosθ. (A3)\nAs a check, one confirms that this system conserves energy and angular momentum:\nE=1\n2/integraldisplay\nρdV/parenleftbig\n(∂tξ)2+c2\nA(∂zξ)2/parenrightbig\n+1\n2/integraldisplay\nσdA/parenleftbigg\n(∂tψ)2+c2\nt\nR2(sinθ∂θ(ψ\nsinθ))2/parenrightbigg\n(A4)\nL=/integraldisplay\nρdV rsinθ ∂tξ+/integraldisplay\nσdA Rsinθ ∂tψ, (A5)\nwhere/integraltext\ndVand/integraltext\ndAare volume and surface integrals and ris the radius.\nREFERENCES\nIsrael, G. L. et al, ApJL, 628, 53, 2005\nLevin, Y., Mon.Not.Roy.Astron.Soc, 377, 159, 2007\nThis preprint was prepared with the AAS L ATEX macros v5.2.– 5 –\n0 2 4 6 8 10-0.2-0.100.1\nFig. 1.—M= 0.1,ct= 2. Zeros and cuts of the dispersion function F(ω) in the complex ωplane." }, { "title": "1409.7452v2.An_ultimate_storage_ring_lattice_with_vertical_emittance_generated_by_damping_wigglers.pdf", "content": "arXiv:1409.7452v2 [physics.acc-ph] 7 Oct 2014An ultimate storage ring lattice with vertical emittance\ngenerated by damping wigglers\nXiaobiao Huang\nSLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025\nAbstract\nWediscuss theapproachofgeneratingroundbeamsforultimatest oragerings\nusing vertical damping wigglers (with horizontal magnetic field). The verti-\ncal damping wigglers provide damping and excite vertical emittance. This\neliminates the need to generate large linear coupling that is impractica l with\ntraditional off-axis injection. We use a PEP-X compatible lattice to de mon-\nstrate the approach. This lattice uses separate quadrupole and s extupole\nmagnets with realistic gradient strengths. Intrabeam scattering effects are\ncalculated. The horizontal and vertical emittances are 22.3 pm and 10.3 pm,\nrespectively, for a 200 mA, 4.5 GeV beam, with a vertical damping wigg ler\nof a total length of 90 meters, peak field of 1.5 T and wiggler period of 100\nmm.\nKeywords: ultimate storage ring, vertical emittance, damping wiggler\n1. Introduction\nInpresent daythirdgenerationlightsources, theverticalemitta nceisusu-\nally small compared to the horizontal emittance. It is typically a few p ercent\nof the latter or below without coupling correction and can reach pico -meter\nlevel with coupling correction. For ultimate storage rings (USR), it is not\nadvisable to maintain the same level vertical-to-horizontal emittan ce ratio.\nThis is because the horizontal emittance will already be diffraction-lim ited\nand hence there is no need to make the vertical emittance any smalle r. In\naddition, a smaller vertical emittance will cause significant emittance growth\ndue to intrabeam scattering (IBS) and also severe Touschek beam loss. Most\nEmail address: xiahuang@slac.stanford.edu (Xiaobiao Huang)\nPreprint submitted to Elsevier June 2, 2021USR designs to-date (such as PEP-X [3]) assume the vertical emitta nce to\nbe equal to the horizontal emittance, resulting in a round beam.\nVertical emittance in a storage ring can be generated with linear cou pling\nor vertical dispersion. A round beam can be achieved with 100% linear cou-\npling, in which case the horizontal and vertical emittances are 50% o f the\nnatural emittance. The reduction of horizontal emittance by a fa ctor of 2\nis a significant benefit of this approach. However, large coupling bet ween\nthe two transverse directions will cause injection difficulties for off- axis injec-\ntion. The injected beam, initially at a large horizontal offset, will take large\nvertical oscillation and likely get lost to small vertical apertures suc h as the\nsmall-gap insertion devices. Effectively, large coupling with small vert ical\napertures causes the dynamic aperture to decrease. This is expe rimentally\ndemonstrated on the SPEAR3 storage ring as is shown in Figure 1, wh ich\nshowsthattheinjectionefficiency dropstozeroatorbeforethec ouplingratio\nis increased to 26%. Large linear coupling may also reduce Touschek lif etime\nsince the horizontal oscillation of the Touschek particles will be coup led to\nthe vertical plane which usually has smaller apertures.\n00.050.10.150.20.250.30.3500.20.40.60.81\ncoupling ratioinjection efficiency\nFigure 1: Injection efficiency vs. coupling ratio at SPEAR3.\nThe second approach to generate vertical emittance is to create vertical\ndispersion inside dipole magnets. This will not cause severe injection a nd\nlifetime difficulties, but will lose the benefit of horizontal emittance re duc-\ntion. And since strong skew quadrupoles are needed to create larg e vertical\ndispersion, it may be inevitable to introduce large linear coupling.\nWe have studied a third approach which can mitigate the negative effe cts\nof both of the above approaches. In this approach we use vertica l damp-\ning wigglers (with horizontal magnetic field) to achieve both the redu ction\n2of horizontal emittance and the generation of vertical emittance . Damping\nwigglers are usually required for USRs because in USRs the dipole bend ing\nradius is large and hence the radiation energy loss from dipole magnet s is\ntoo small for sufficient damping, which is required for controlling collec tive\neffects such as intrabeam scattering and beam instabilities. Usually d amp-\ning wigglers have vertical magnet field that causes wiggling beam motio n on\nthe horizontal plane. The horizontal dispersion generated by the damping\nwiggler itself contributes to an increase of the horizontal emittanc e. The\nrelative emittance increase can be significant when the natural emit tance is\nsmall. Choosing to use small period damping wigglers alleviates the emit-\ntancegrowthproblem to someextent. But it putsa challenge to the damping\nwiggler design and increases the cost. A vertical damping wiggler doe s not\nincrease the horizontal emittance and in the same time generates t he desir-\nable vertical emittance. Therefore it is reasonable to use vertical damping\nwigglers for USRs. The idea of using vertical damping wigglers to gene rate\nvertical emittance has been independently proposed in Refs. [1, 2 ].\nIn this study we demonstrate this approach with a lattice that is com -\npatible with the PEP tunnel at SLAC National Accelerator Laborato ry. In\nsection 2 we use a simple model to calculate and compare the emittanc es\nwiththehorizontalandvertical dampingwiggler approaches. Inse ction3the\nPEP compatible lattice with vertical damping wigglers is presented. Em it-\ntance parameters with intrabeam scattering effects are given in se ction 4.\nThe conclusions are given in section 5.\n2. Theoretic calculation\nThe effects of vertical damping wigglers can be analytically estimated .\nSuppose the wiggler peak field is Bw, its length is Lw, and the horizontal\nfield is given by\nBx=Bwcoshkxcosks, B y= 0, (1)\nwherek= 2π/λwandλwis the wiggler period, then the vertical closed orbit\ninside the damping wiggler (DW) is [4]\nyco=1\nρwk2(1−cosks), y′\nco=1\nρwksinks, (2)\n3whereρw=Bρ/B wis the minimum bending radius. The vertical dispersion\ngenerated by the DW itself is\nDy=−1\nρwk2(1−cosks), D′\ny=−1\nρwksinks, (3)\nConsequently the radiation integral contributions are\nI2w=Lw\n2ρ2w, I 3w=4Lw\n3πρ3w,\nI4wy=3Lw\n8πρ4wk2, I5wy=4< βy> Lw\n15πρ5wk2, (4)\nwhere< βy>is the average vertical beta function across the DW. The\nemittances and momentum spread are given by\nσ2\nδ=γ2CqI3+I3w\nI2+I2w1\n2+Dx+Dy, (5)\nǫx=γ2CqI5\nI2+I2w1\n1−Dx, (6)\nǫy=γ2CqI5wy\nI2+I2w1\n1−Dy, (7)\nwhereI2−5are radiation integrals for the bare lattice and\nDx=I4\nI2+I2w,Dy=I4wy\nI2+I2w. (8)\nWe now consider a PEP-X compatible lattice at 4.5 GeV (see section 3).\nThe relevant radiation integrals without DWs are\nI2= 0.1026m−1, I3= 1.674×10−3m−2,\nI4=−0.1215m−1, I5= 3.092×10−7m−1. (9)\nAssuming the average beta function over the DW is 10 m, the emittan ces\nas a function of wiggler length is calculated and compared to the case with\na regular horizontal damping wiggler for various sets of peak magne tic field\nand wiggler period values. The results are shown in Figure 2. Clearly th e\nvertical DW provides damping of the horizontal emittance and in the mean-\ntime generates vertical emittance. The total emittance is only sligh tly larger\nthan the case with a regular horizontal DW. The difference is smaller f or\nsmaller wiggler periods.\n40 20 40 60 80 100010203040\nDW length (m)emittances (pm)\n \nB0 = 1.2 T, λ = 200 mmεx VDW\nεy VDW\nεx,y HDW\n0 20 40 60 80 100010203040\nDW length (m)total emittance (pm)\n \nB0 = 1.2 T, λ = 200 mmHDW\nVDW\n0 20 40 60 80 100010203040\nDW length (m)emittances (pm)\n \nB0 = 1.2 T, λ = 100 mmεx VDW\nεy VDW\nεx,y HDW\n0 20 40 60 80 100010203040\nDW length (m)total emittance (pm)\n \nB0 = 1.2 T, λ = 100 mmHDW\nVDW\n0 20 40 60 80 100010203040\nDW length (m)emittances (pm)\n \nB0 = 1.5 T, λ = 100 mmεx VDW\nεy VDW\nεx,y HDW\n0 20 40 60 80 100010203040\nDW length (m)total emittance (pm)\n \nB0 = 1.5 T, λ = 100 mmHDW\nVDW\nFigure 2: Comparison of the emittances of a PEP-X ring with vertical or horizontal\ndamping wigglers. Top row with Bw= 1.2 T and λw= 200 mm; middle row with\nBw= 1.2 T and λw= 100 mm; bottom tow with Bw= 1.5 T and λw= 100 mm. A 100%\ncoupling is assumed for the regular horizontal DW case.\n53. Application to a PEP-X compatible lattice\nWe have implemented the vertical DW approach for a PEP-X compatib le\nlattice with the design beam energy at 4.5 GeV. This lattice is similar to th e\nPEP-X USR design as it adopts the same MBA and fourth order achro mat\napproach [3]. The 2.2-km long PEP tunnel has a hexagonal geometry . There\nare six 120-m long straight sections which can be used to host long da mping\nwigglers. The lattice has 6 arcs, each consists of 8 MBA (with M= 7) cells.\nAn MBA cell is composed of 5 identical TME cells in the middle and two\nmatching cells at the ends. The MBA cell and the TME cell are shown in\nFigure 3. The TME and 7BA cell lengths are 3.12 m and 30.4 m, respec-\ntively. The TME dipole magnet is 1.12 m in length and its bending angle\nis 1.0475◦. This dipole is a combined-function magnet with a defocusing\nquadrupole component and the normalized gradient is −0.7989 m−2. The\nfocusing quadrupole (QF) is split into two halves to put the SF sextup ole in\nbetween. The length of each half is 0.18 m. The length of SF is 0.30 m. On e\nSD sextupole magnet is put at each end of the dipole. Its length is 0.21 m.\nThe matching dipole has no quadrupole gradient. Its length is 8% longe r\nthan the TME dipole. At each end of the MBA cell, outside of the match -\ning dipole, there is a quadrupole triplet. Three harmonic sextupoles a re put\nbetween these magnets. The minimum edge-to-edge distance for m agnets\nis 8 cm to accommodate coils and BPMs [5]. The quadrupole strength is\nbelow 51 T/m and the sextupole strength is below 7500 T/m2. With a bore\nradius of 12.5 mm, the pole tip magnetic field would be below 0.64 T for\nquadrupoles and below 0.59 T for sextupoles.\nThe insertion device straight sections between the MBA cells are 5 me ter\nlong. The horizontal and vertical beta functions at the centers o f these\nstraight sections are 0.8 m and 2.0 m, respectively. The horizontal b eta\nfunction is made very small to provide better matching of the electr on and\nphoton optics. But we keep the vertical beta at a level close to half of the\nID straight length to allow small gap insertion devices [6].\nThe 120-m long straight sections are filled with FODO cells. One of the\nlong straight sections houses the damping wigglers. The wiggler sect ions are\n4.06 meter long and are put between the quadrupoles of the FODO ce lls.\nThe optics functions are shown in Figure 4 for a FODO cell for the cas e with\nwiggler period at 200 mm and peak field at 1.2 T. Optics function for one\nhalf of the long straight section is as shown in Figure 5.\nThe ring lattice parameters for three wiggler settings are compare d in\n60.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0\ns (m)\nδE/ p0c = 0.\nTable name = TWISSTME\nWin32 version 8.51/15 08/03/13 13.40.34\n0.01.2.3.4.5.6.7.8.9.10.β(m)\n0.00.010.020.030.040.050.060.070.08\nDx(m) βxβyDx\n0.0 5. 10. 15. 20. 25. 30. 35. 40.\ns (m)\nδE/ p0c = 0.\nTable name = TWISSStandard cell.\nWin32 version 8.51/15 08/03/13 13.40.34\n0.02.55.07.510.012.515.017.520.022.525.0β(m)\n-0.010.00.010.020.030.040.050.060.070.08\nDx(m) βxβyDx\nFigure 3: The TME cell (left) and 7BA cell (right) for the PEP-X comp atible lattice.\n0.0 2. 4. 6. 8. 10.\ns (m)\nδE/ p0c = 0.\nTable name = TWISSFODO w/ wiggler\nWin32 version 8.51/15 08/03/13 13.40.34\n0.02.55.07.510.012.515.017.520.022.525.0β(m)\n-1.00-0.75-0.50-0.250.00.250.500.751.00\nDy(m) [*10**( - 3)]βxβyDy\nFigure 4: One FODO lattice period with damping wigglers.\n70.0 10. 20. 30. 40. 50. 60. 70. 80.\ns (m)\nδE/ p0c = 0.\nTable name = TWISSTME2FODOW\nWin32 version 8.51/15 08/03/13 13.40.34\n0.05.10.15.20.25.30.β(m)\n-0.010.00.010.020.030.040.050.060.070.08\nDx(m) βxβyDx\nFigure 5: Half of the DW straight section.\nTable 1. The parameters were calculated with MAD8 [7]. The results ag ree\nwith the prediction given in Figure 2. For the vertical DW sets of (1.2 T ,\n200 mm) and (1.5 T, 100 mm), the horizontal and vertical emittance s are\nnearly equal, with values down to 17/17 pm and 13/10 pm, respective ly.\nTable 1: Ring parameters with or without vertical damping wigglers\nparameters no DW VDW-1 VDW-2 VDW-3\nEnergy (GeV) 4.5\nCircumference (m) 2199.3\nνx,y 130.15/73.30\nαc 0.38×10−4\nVDW length (m) 0 90.8 89.4 89.4\nVDWBw(T) 1.2 1.2 1.5\nVDWλw(m) 0.2 0.1 0.1\nU0(MeV) 0.59 2.30 2.26 3.19\nǫx(pm) 36.5 16.5 16.7 12.7\nǫy(pm) 0 17.1 4.3 10.2\nσδ(×0.001) 0.77 1.05 1.05 1.10\ndamping time τx(ms) 47 21 22 17\ndamping time τy(ms) 112 29 29 21\ndamping time τs(ms) 176 17 18 12\n84. Intrabeam scattering calculation\nIntrabeam scattering (IBS) can significantly increase the emittan ce and\nenergy spread for very low emittance beams at high current. To ex amine\nhow the IBS effects may differ for the two approaches of generatin g vertical\nemittance, i.e., with full coupling or with vertical damping wiggler, we did\nIBS calculation for both cases for the PEP-X compatible lattice with t he\nhigh energy approximation model [3, 8]. Similar to PEP-X, we assume th e\nCoulomb log function is (l og) = 11. For the full coupling case, a horizontal\ndamping wiggler is put into the model to reduce emittance. The wiggler\nparameters are the same as the vertical damping wiggler. For the c ase cor-\nresponding to VDW-3 in Table 1 (i.e., with peak field 1.5 T, wiggler period\n100 mm and wiggler length 89.4 m), the emittances are 10.5 pm for both\nplanes. For the vertical wiggler case, a small linear coupling ratio of 0 .001\nis assumed. The vertical emittance is almost entirely generated with the\nvertical damping wiggler.\nThe emittances vs. beam current for the two cases with IBS effect s are\nshown in Figure 6. The bunch length is assumed to be σz= 2.7 mm, corre-\nsponding to an RF gap voltage of 6 MV with the 476.0 MHz rf system. Th e\ntotalnumber ofbunches isassumed tobe3300. Forthevertical D Wcase, the\nhorizontal emittance has a significant increase. But the vertical e mittance\ngrowth is very small. This is because the vertical dispersion is confine d to in-\nside the damping wiggler, which constitutes only a small fraction of th e ring,\nwhile the horizontal dispersion is present at all arc areas. In additio n, the\nvertical dispersion is much smaller than the horizontal dispersion wh ile the\nhorizontal and vertical emittances at zero current are nearly th e same. This\nis because the average bending field in the damping wiggler is much stro nger\nthan in the bending magnets. Overall the vertical IBS growth rate is much\nsmaller thanthehorizontal planebecause theIBS growthrateispr oportional\nto the dispersion invariant averaged over the ring circumference.\nThe distribution of the IBS growth rate for the vertical DW case is s hown\nin Figure 7 for the case with a 200 mA total current. For this case, t he\naverage IBS growth rates for x,y,pdirections are 26.1 s−1, 0.20 s−1and\n8.4 s−1, respectively. The corresponding emittances are ǫx= 22.3 pm and\nǫy= 10.3 pm and the momentum spread is σδ= 1.16×10−3. If harmonic\ncavities are used to lengthen the bunch to σz= 5 mm, the x,y,pIBS\ngrowth rates become 18.8 s−1, 0.13 s−1and 5.5 s−1, respectively, and the\nemittances and the momentum spread become ǫx= 18.4 pm,ǫy= 10.25 pm\n90 50 100 150 200 250051015202530\nI0 (mA)emittance (pm)\n \nεx,y HDW\nεx VDW\nεy VDW\nFigure 6: Emittance growth vs total beam current, assuming a unif orm current distribu-\ntion in 3300 bunches and a bunch length of 2.7 mm.\nandσδ= 1.13×10−3.\n0 500 1000 1500 2000 2500−10010203040506070\ns (m)δ (1/T), (s−1)\n \nδ (1/Tp)\nδ (1/Tx)\nδ (1/Ty)\nFigure 7: The distribution of local IBS growth rate for the three dim ensions for a beam\ncurrent of 200 mA in 3300 bunches.\n5. Conclusion\nFor ultimate storage rings that require off-axis injection, we propo se the\nuse of vertical damping wiggler (with horizontal magnetic field) to ge nerate\n10vertical emittance in order to obtain round beams. This approach is better\nthan generating round beams with 100% coupling because it does not couple\nthe large amplitude horizontal oscillation of the injected beam to the vertical\nplane and therefore the small vertical apertures in the ring does n ot pose\nsevere limitation to the dynamic aperture. It is shown that for damp ing\nwigglers with reasonably small wiggler period (e.g., λw= 100 mm), the total\nemittances of the two approaches are nearly equal.\nA PEP-X compatible lattice is designed to demonstrate this approach . It\nconsists of 6 arcs, each made up of 8 MBA cells. The beamenergy is as sumed\nto be 4.5 GeV. The quadrupoles and sextupoles are separate funct ion mag-\nnets, with strengths below 51 T/m and 7500 T/m2, respectively. The bare\nlattice horizontal emittance is 36.5 pm. The beta functions at the mid dle of\nstraights are0.8m horizontal and2.0 mvertical, which allows goodmat ching\nto the photon optics and supports the use of small gap insertion de vices.\nWhen a 90-m long vertical damping wiggler with peak field at 1.5 T and\nwiggler period at 100 mm is put into one of the long straight sections, t he\nhorizontal and vertical emittances are 13 pm and 10 pm, respectiv ely. The\nrms momentum spread is 1.1 ×10−3. Intrabeam scattering is calculated for\nthis case. For 200 mA beam current in 3300 bunches and a bunch leng th\nofσz= 2.7 mm, the horizontal and vertical emittance become 22.3 pm and\n10.3 pm, respectively. The rms momentum spread is 1 .16×10−3.\nAcknowledgment\nThe study is supported by DOE Contract No. DE-AC02-76SF00515 .\nReferences\n[1] X. Huang, “A PEP-X lattice with vertical emittance by damping wig-\nglers”, SSRL-AP-note 51, July 2013.\n[2] A. Bogomyagkov, et al, presentation at the 4th Low Emittance R ings\nWorkshop, September 2014.\n[3] Y. Cai, et al, Phys. Rev. ST Accel. Beams, 054002 (2012).\n[4] S. Y. Lee, Accelerator Physics , World Scientific (1999).\n11[5] A 7.5 cm minimum magnet separation is reserved in MAX-IV\ndesign. See Detailed Design Report on the MAX IV Facility,\nhttps://www.maxlab.lu.se/node/1136, August 2010, chapter 2.\n[6] T. Rabedeau, private communications (2013).\n[7] H. Grote, F.C. Iselin, The MAD Program User’s Manual , CERN/SL/90-\n13 (1990).\n[8] K. Bane, EPAC 2002, Paris, France (2002)\n12" }, { "title": "2401.09938v2.Real_space_nonlocal_Gilbert_damping_from_exchange_torque_correlation_applied_to_bulk_ferromagnets_and_their_surfaces.pdf", "content": "Real-space nonlocal Gilbert damping from exchange torque correlation applied to\nbulk ferromagnets and their surfaces\nBalázs Nagyfalusi,1,2,∗László Szunyogh,2,3,†and Krisztián Palotás1,2,‡\n1Institute for Solid State Physics and Optics, HUN-REN Wigner Research Center for Physics,\nKonkoly-Thege M. út 29-33, H-1121 Budapest, Hungary\n2Department of Theoretical Physics, Institute of Physics,\nBudapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary\n3HUN-REN-BME Condensed Matter Research Group,\nBudapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary\n(Dated: February 29, 2024)\nIn this work we present an ab initio scheme based on linear response theory of exchange torque\ncorrelation, implemented into the real-space Korringa-Kohn-Rostoker (RS-KKR) framework to cal-\nculate diagonal elements of the atomic-site-dependent intrinsic Gilbert damping tensor. The method\nis first applied to bcc iron and fcc cobalt bulk systems. Beside reproducing earlier results from the\nliterature for those bulk magnets, the effect of the lattice compression is also studied for Fe bulk,\nand significant changes for the Gilbert damping are found. Furthermore, (001)-oriented surfaces\nof Fe and Co are also investigated. It is found that the on-site Gilbert damping increases in the\nsurface atomic layer and decreases in the subsurface layer, and approaches the bulk value moving\nfurther inside the magnets. Realistic atomic relaxation of the surface layers enhances the identified\neffects. Thefirst-neighbordampingparametersareextremelysensitivetothesurfacerelaxation. De-\nspite their inhomogeneity caused by the surface, the transverse Gilbert damping tensor components\nremain largely insensitive to the magnetization direction.\nI. INTRODUCTION\nIt is highly demanded to understand and control\nthe dynamical processes governing the manipulation\nof various magnetic textures, such as atomic chains1,2,\nmagnetic skyrmions3,4or domain walls5, which can\nbe potentially used in future magnetic recording and\nlogic devices. These processes are often described by\nthe phenomenological Landau-Lifshitz-Gilbert (LLG)\nequation6,7,\n∂ ⃗ mi\n∂t=−γ ⃗ mi×⃗Beff\ni+α\nmi⃗ mi×∂ ⃗ mi\n∂t,(1)\nwhere ⃗ miis the magnetic moment at site i,mi=|⃗ mi|\nis its length, and γis the gyromagnetic ratio. The\nfirsttermonthe rhsofEq.(1)describestheprecession\nof⃗ miaround the effective magnetic field ⃗Beff\ni, while\nthe second term is the Gilbert damping due to the\nenergy dissipation to the lattice. Clearly, this latter\nterm causes the relaxation of the magnetization to its\nequilibrium value, which is controlled by the damping\nconstant αand plays a crucial role in the realization\nof high-speed spintronic devices.\nThe Gilbert damping constant αcan be deter-\nmined experimentally from the ferromagnetic reso-\nnance (FMR) spectroscopy where the damping pa-\nrameter is related to the line-width in the measured\nspectra8. FMR spectroscopy is a well-established\nmethod for bulk materials9,10, but especially in the\nlow temperature measurement it is controversial be-\ncause the intrinsic Gilbert damping needs to be sepa-\nrated from various extrinsic sources of the line-width,\ne.g., two-magnon scattering, eddy-current damping,\nradiative damping, spin-pumping, or the slow relaxer\nmechanism11–16. The comparison of experimental\nmeasurement to theoretical calculations is also made\ndifficult bythe sampleproperties likethe exactatomic\nstructure.From a theoretical perspective the ultimate goal is\nto develop a method to calculate the Gilbert damp-\ning parameters from the electronic structure of the\nmaterial. In the last decades there have been sev-\neral efforts to understand the damping process. The\nfirst successful method was developed by Kamberský\nwhorelatedthedampingprocesstothespin-orbitcou-\npling (SOC) in terms of the breathing Fermi surface\nmodel17, while he also proposed the spin-orbit torque\ncorrelation model18,19. Later on several other meth-\nods were introduced such as the spin-pumping20and\nlinear-response approaches11,21,22. A recent summary\nof these methods was published by Guimarães et al.23\nDue to the increased interest in noncollinear mag-\nnetism Fähnle et al.24suggested an inhomogeneous\ntensorial damping. The replacement of a scalar αby\na damping matrix αmeans that the damping field in\nEq.(1)isnolongerproportionaltothetimederivative\nof⃗ mi, it becomes a linear function of ∂ ⃗ mi/∂t. More-\nover, nonlocality of the damping process implies that\nthe damping field at site iexperiences ∂ ⃗ mj/∂tfor any\nsitej. The LLG equation (1) is then replaced by the\nset of equations25,\n∂ ⃗ mi\n∂t=⃗ mi×\n−γ⃗Beff\ni+X\njαij1\nmj∂ ⃗ mj\n∂t\n,(2)\nwhere the damping term is unfolded to pairwise con-\ntributions of strength αij. The appearance of non-\nlocal damping terms was evidenced for magnetic do-\nmain walls26,27by linking the Gilbert damping to the\ngradients of the magnetization. In NiFe, Co, and\nCoFeB thin films Li et al.28measured wave-number-\ndependent dissipation using perpendicular spin wave\nresonance, validating thus the idea of nonlocal damp-\ning terms. Different analytical expressions for αijare\nalready proposed22,25,29,30, and the nonlocal damp-\ning is found for bulk materials25,31as well as its ef-arXiv:2401.09938v2 [cond-mat.mtrl-sci] 28 Feb 20242\nfect on magnon properties of ferromagnets have been\ndiscussed32. Recent studies went further and, anal-\nogously to the higher order spin-spin interactions in\nspin models, introduced multi-body contributions to\nthe Gilbert damping33.\nThe calculation of the Gilbert damping prop-\nerties of materials has so far been mostly fo-\ncused on 3D bulk magnets, either in chemically\nhomogeneous11,19,23,25,34–36or heterogeneous (e.g.\nalloyed)11,22,31forms. There are a few studies avail-\nable reporting on the calculation of the Gilbert damp-\ning in 2D magnetic thin films12,23,37,38, or at surfaces\nand interfaces of 3D magnets31,35,37. The calculation\nof the Gilbert damping in 1D or 0D magnets is, due\nto our knowledge, not reported in the literature. Fol-\nlowing the trend of approaching the atomic scale for\nfunctional magnetic elements in future spintronic de-\nvices, the microscopic understanding of energy dissi-\npation through spin dynamics in magnets of reduced\ndimensions is inevitable and proper theoretical meth-\nods have to be developed.\nOur present work proposes a calculation tool for\nthediagonalelementsofthenon-localintrinsicGilbert\ndamping tensor covering the 3D to 0D range of mag-\nnetic materials on an equal footing, employing a real-\nspace embedding Green’s function technique39. For\nthis purpose, the linear response theory of the Gilbert\ndamping obtained by the exchange torque correlation\nis implemented in the real-space KKR method. As a\ndemonstration of the new method, elemental Fe and\nCo magnets in their 3D bulk form and their (001)-\noriented surfaces are studied in the present work. Go-\ning beyond comparisons with the available literature,\nnew aspects of the Gilbert damping in these materials\nare also reported.\nThe paper is organized as follows. In Sec. II the\ncalculation of the Gilbert damping parameters within\nthe linear response theory of exchange torque corre-\nlation using the real-space KKR formalism is given.\nSec. III reports our results on bulk bcc Fe and fcc Co\nmaterials and their (001)-oriented surfaces. We draw\nour conclusions in Sec. IV.\nII. METHOD\nA. Linear response theory within real-space\nKKR\nThe multiple-scattering of electrons in a finite clus-\nter consisting of NCatoms embedded into a 3D or 2D\ntranslation-invariant host medium is fully accounted\nfor by the equation39\nτC=τH\u0002\nI−(t−1\nH−t−1\nC)τH\u0003−1,(3)\nwhere τCand τHare the scattering path operator\nmatrices of the embedded atomic cluster and the host,\nrespectively, tCandtHare the corresponding single-\nsite scattering matrices, all in a combined atomic site\n(j, k∈ {1, ..., N C}) and angular momentum ( Λ,Λ′∈\n{1, ...,2(ℓmax+ 1)2}) representation: τ={τjk}=\n{τjk\nΛΛ′}andt={tj\nΛΛ′δjk}, where ℓmaxis the angularmomentum cutoff in describing the scattering events,\nand for simplicity we dropped the energy-dependence\nof the above matrices.\nFor calculating the diagonal Cartesian elements\nof the nonlocal Gilbert damping tensor connecting\natomic sites jandkwithin the finite magnetic atomic\ncluster, we use the formula derived by Ebert et al.22,\nαµµ\njk=2\nπmj\nsTr\u0010\nTj\nµ˜τjk\nCTk\nµ˜τkj\nC\u0011\n, (4)\nwhere µ∈ {x, y, z}, the trace is taken in the\nangular-momentum space and the formula has to\nbe evaluated at the Fermi energy ( EF). Here,\nmj\nsis the spin moment at the atomic site j,\n˜τjk\nC,ΛΛ′= (τjk\nC,ΛΛ′−(τkj\nC,Λ′Λ)∗)/2i, and Tj\nµis the\ntorque operator matrix which has to be calculated\nwithin the volume of atomic cell j,Ωj:Tj\nµ;ΛΛ′=R\nΩjd3rZj\nΛ(⃗ r)×βσµBxc(⃗ r)Zj\nΛ′(⃗ r),wherethenotationof\nthe energy-dependence is omitted again for simplicity.\nHere, βis a standard Dirac matrix entering the Dirac\nHamiltonian, σµare Pauli matrices, and Bxc(⃗ r)is the\nexchange-correlation field in the local spin density ap-\nproximation (LSDA), while Zj\nΛ(⃗ r)are right-hand side\nregular solutions of the single-site Dirac equation and\nthe superscript ×denotes complex conjugation re-\nstricted to the spinor spherical harmonics only22. We\nshould emphasize that Eq. (4) applies to the diagonal\n(µµ) elements of the Gilbert tensor only. To calcu-\nlate the off-diagonal tensor elements one needs to use,\ne.g., the more demanding Kubo-Bastin formula40,41.\nNote also that in noncollinear magnets the exchange\nfield Bxc(⃗ r)is sensitive to the spin noncollinearity42\nwhich influences the calculated torque operator ma-\ntrix elements, however, this aspect does not concern\nour present study including collinear magnetic states\nonly.\nNote that the nonlocal Gilbert damping is, in gen-\neral, not symmetric in the atomic site indices, αµµ\njk̸=\nαµµ\nkj, instead\nαµµ\nkj=mj\ns\nmksαµµ\njk(5)\nholds true. This is relevant in the present work for\nthe ferromagnetic surfaces. On the other hand, in\nferromagnetic bulk systems αµµ\njk=αµµ\nkjsince mj\ns=\nmk\ns=msfor any pair of atomic sites.\nIn practice, the Gilbert damping formula in Eq. (4)\nis not directly evaluated at the Fermi energy, but a\nsmall imaginary part ( η) of the complex energy is ap-\nplied, which is called broadening in the following, and\nits physical effect is related to the scattering rate in\nother damping theories19,25,37,43. Taking into account\nthe broadening η, the Gilbert damping reads\nαµµ\njk(η) =−1\n4h\n˜αµµ\njk(E+, E+) + ˜αµµ\njk(E−, E−)\n−˜αµµ\njk(E+, E−)−˜αµµ\njk(E−, E+)i\n,(6)\nwhere E+=EF+iηandE−=EF−iη, and the3\nindividual terms are\n˜αµµ\njk(E1, E2) =\n2\nπmj\nsTr\u0010\nTj\nµ(E1, E2)τjk\nC(E2)Tk\nµ(E2, E1)τkj\nC(E1)\u0011\n(7)\nwith E1,2∈ { E+, E−}, and the ex-\nplicitly energy-dependent torque opera-\ntor matrix elements are: Tj\nµ;ΛΛ′(E1, E2) =R\nΩjd3rZj×\nΛ(⃗ r, E1)βσµBxc(⃗ r)Zj\nΛ′(⃗ r, E2).\nB. Effective damping and computational\nparameters\nEq.(6)givesthebroadening-dependentspatiallydi-\nagonal elements of the site-nonlocal Gilbert damping\ntensor: αxx\njk(η),αyy\njk(η), and αzz\njk(η). Since no longitu-\ndinal variation of the spin moments is considered, the\ntwo transversal components perpendicular to the as-\nsumed uniform magnetization direction are physically\nmeaningful. Given the bulk bcc Fe and fcc Co sys-\ntems and their (001)-oriented surfaces with C4vsym-\nmetry under study in the present work, in the follow-\ning the scalar αrefers to the average of the xxand\nyyGilbert damping tensor components assuming a\nparallel magnetization with the surface normal z[001]-\ndirection: αjk= (αxx\njk+αyy\njk)/2 =αxx\njk=αyy\njk. From\nthe site-nonlocal spatial point of view in this work we\npresent results on the on-site (\" 00\"), first neighbor\n(denoted by \" 01\") and second neighbor (denoted by\n\"02\") Gilbert damping parameters, and an effective,\nso-called total Gilbert damping ( αtot), which can be\ndefined as the Fourier transform of αjkat⃗ q= 0. The\nFourier transform of the Gilbert damping reads\nα⃗ q=∞X\nj=0α0jexp(−i⃗ q(⃗ r0−⃗ rj))\n≈X\nr0j≤rmaxα0jexp(−i⃗ q(⃗ r0−⃗ rj)),(8)\nwhere r0j=|⃗ r0−⃗ rj|and the effective damping is\ndefined as\nαtot=α⃗ q=⃗0=∞X\nj=0α0j≈X\nr0j≤rmaxα0j.(9)\nSince we have a real-space implementation of the\nGilbert damping, the infinite summation for both\nquantities is replaced by an approximative summation\nfor neighboring atoms upto an rmaxcutoff distance\nmeasured from site \"0\". Moreover, note that for bulk\nsystems the effective damping αtotis directly related\nto the ⃗ q= 0mode of FMR experiments.\nThe accuracy of the calculations depends on many\nnumerical parameters such as the number of ⃗kpoints\nused in the Brillouin zone integration, the choice of\nthe angular momentum cutoff ℓmax, and the spatial\ncutoff rmaxused for calculating α⃗ qandαtot. Previ-\nous research25showed that the Gilbert damping heav-\nily depends on the broadening η, so we extended ourstudies to a wider range of η= 1meV to 1 eV. The\nsufficient k-point sampling was tested at the distance\nofrmax= 7a0(where a0is the corresponding 2D lat-\ntice constant) from the reference site with the broad-\nening set to 1mRy, and the number of ⃗kpoints was\nincreased up to the point, where the 5th digit of the\ndamping became stable. Maximally, 320400 ⃗kpoints\nwere used for the 2D layered calculation but the re-\nquested accuracy was reached with 45150 and 80600\n⃗kpoints for bulk bcc Fe and fcc Co systems, respec-\ntively.\nThe choice of ℓmaxwas tested through the whole η\nrange for bcc Fe, and it was based on the comparison\nof damping calculations with ℓmax= 2andℓmax= 3.\nThe maximal deviation for the on-site Gilbert damp-\ning was found at around η= 5mRy, but it was still\nless than 10%. The first and second neighbor Gilbert\ndamping parameters changed in a more significant\nway (by ≈50%) in the whole ηrange upon changing\nℓmax, yet the effective total damping was practically\nunchanged, suggesting that farther nonlocal damping\ncontributions compensate this effect. Since αtotis the\nmeasurable physical quantity we concluded that the\nlower angular momentum cutoff of ℓmax= 2is suffi-\ncient to be used further on.\nThe above choice of ℓmax= 2for the angular mo-\nmentumcutoff, themathematicalcriterionofpositive-\ndefinite αjk(which implies α⃗ q>0for all ⃗ qvectors),\nand the prescribed accuracy for the effective Gilbert\ndamping in the full considered η= 1meV to 1 eV\nrange set rmaxto 20 a0for both bcc Fe and fcc Co. It\nis worth mentioning that the consideration of lattice\nsymmetries made possible to decrease the number of\natomic sites in the summations for calculating α⃗ qand\nαtotby an order of magnitude.\nIII. RESULTS AND DISCUSSION\nOur newly implemented method was employed to\nstudy the Gilbert damping properties of Fe and Co\nferromagnetsintheirbulkand(001)-orientedsurfaces.\nIn these cases only unperturbed host atoms form\nthe atomic cluster, and the so-called self-embedding\nprocedure44is employed, where Eq. (3) reduces to\nτC=τHforthe3Dbulkmetalsand2Dlayeredmetal-\nvacuum interfaces.\nA. Bulk Fe and Co ferromagnets\nFirst we calculate and analyze the nonlocal and ef-\nfective dampings for bulk bcc Fe by choosing a 2D\nlattice constant of a0= 2.863Å. The magnitude of\nthe magnetic moments are obtained from the self-\nconsistent calculation. The spin and orbital moments\narems= 2.168µBandmo= 0.046µB, respectively.\nThe broadening is set to η= 68meV. The inset of Fig.\n1a) shows the typical function of the nonlocal Gilbert\ndamping α0jdepending on the normalized distance\nr0j/a0between atomic sites \"0\" and \" j\". In accor-\ndance with Ref. 25 the nonlocal Gilbert damping\nquickly decays to zero with the distance, and can be4\na)\n5 10 15 2005\n5 10 15 20−505\nr0j/a0α0j[×10−4]\nr0j/a0α0j·(r0j/a0)2[×10−4]\nb)\n5 10 15 20−202468\nrmax/a0αtot[×10−3]\nFIG. 1. a) Nonlocal Gilbert damping in bulk bcc Fe as a\nfunction of distance r0jbetween atomic sites \"0\" and \" j\"\nshown upto a distance of 20 a0(the 2D lattice constant is\na0= 2.863Å): the black squares are calculated α0jval-\nues times the normalized squared-distance along the [110]\ncrystallographic direction, and the red line is the corre-\nsponding fitted curve based on Eq. (10). The inset shows\nthe nonlocal Gilbert damping α0jvalues in the given dis-\ntance range. b) Convergence of the effective damping pa-\nrameter αtot, partial sums of α0jupto rmaxbased on Eq.\n(9), where rmaxis varied. The broadening is chosen to be\nη=68 meV.\nwell approximated with the following function:\nα(r)≈Asin (kr+ϕ0)\nr2exp(−βr).(10)\nTo test this assumption we assorted the atomic sites\nlying in the [110] crystallographic direction and fit-\nted Eq. (10) to the calculated data. In practice, the\nfit is made on the data set of α0j(r0j/a0)2, and is\nplotted in Fig. 1a). Although there are obvious out-\nliers in the beginning, the magnitude of the Gilbert\ndamping asymptotically follows the ∝exp(−βr)/r2\ndistance dependence. The physical reason for this de-\ncay is the appearance of two scattering path operators\n(Green’s functions) in the exchange torque correlation\nformula in Eq. (4) being broadened due to the finite\nimaginary part of the energy argument.\nIn our real-space implementation of the Gilbert\ndamping, an important parameter for the effective\ndamping calculation is the real-space cutoff rmaxin\nEq. (9). Fig. 1b) shows the evolution of the ef-\nfective (total) damping depending on the rmaxdis-\ntance, within which all nonlocal damping terms α0jare summed up according to Eq. (9). An oscillation\ncan similarly be detected as for the nonlocal damping\nitself in Fig. 1a), and this behavior was fitted with\na similar exponentially decaying oscillating function\nas reported in Eq. (10) in order to determine the ex-\npected total Gilbert damping αtotvalue in the asymp-\ntotic r→ ∞limit. In the total damping case it is\nfound that the spatial decay of the oscillation is much\nslower compared to the nonlocal damping case, which\nmakes the evaluation of αtotmore cumbersome. Our\ndetailed studies evidence that for different broaden-\ningηvalues the wavelength of the oscillation stays\nthe same but the spatial decay becomes slower as\nthe broadening is decreased (not shown). This slower\ndecay together with the fact that the effective (to-\ntal) damping value itself is also decreasing with the\ndecreasing broadening results that below the 10meV\nrange of ηthe amplitude of the oscillation at the dis-\ntance of 20 a0is much larger than its asymptotic limit.\nIn practice, since the total damping is calculated as\nther→ ∞limit of such a curve as shown in Fig. 1b),\nthis procedure brings an increased error for αtotbelow\nη= 10meV, and this error could only be reduced by\nincreasing the required number of atomic sites in the\nreal-space summation in Eq. (10).\nFig. 2 shows the dependence of the calculated on-\nsite, first- and second-neighbor and effective total\nGilbert damping parameters on the broadening η.\nThe left column shows on-site ( α00) and total ( αtot)\nwhile the right one the first ( α01) and second ( α02)\nneighbor Gilbert dampings. We find very good agree-\nment with the earlier reported results of Thonig et\nal.25, particularly that the on-site damping has the\nlargest contribution to the total damping being in the\nsame order of magnitude, while the first and second\nneighbors are smaller by an order of magnitude. The\nobtained dependence on ηis also similar to the one\npublished by Thonig et al.25:α00andαtotare in-\ncreasing with η, and α01andα02do not follow a\ncommon trend, and they are material-dependent, see,\ne.g., the opposite trend of α02with respect to ηfor\nFe and Co. The observed negative values of some of\nthesite-nonlocaldampingsarestillconsistentwiththe\npositive-definiteness of the full (infinite) αjkmatrix,\nwhich has also been discussed in Ref. 25.\nThe robustness of the results was tested against a\nsmall change of the lattice constant simulating the ef-\nfect of an external pressure for the Fe bulk. These re-\nsults are presented in the second row of Fig. 2, where\nthe lattice constant of Fe is set to a0= 2.789Å. In this\ncase the magnetic moments decrease to ms= 2.066µB\nandmo= 0.041µB. It can clearly be seen that the on-\nsite, first and second neighbor Gilbert dampings be-\ncome smaller upon the assumed 2.5% decrease of the\nlattice constant, but the total damping remains prac-\ntically unchanged in the studied ηrange. This sug-\ngests that the magnitudes of more distant non-local\ndamping contributions are increased.\nThe third row of Fig. 2 shows the selected damp-\ning results for fcc Co with a 2D lattice constant of\na0= 2.507Å. The spin and orbital moments are\nms= 1.654µBandmo= 0.078µB, respectively. The\nincreaseofthetotal, theon-site, andthefirst-neighbor5\n10−310−210−11000246810Fe -a0= 2.863˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Fe -a0= 2.863˚A\nη(eV)α[×10−4]α01\nα02\n10−310−210−11000246810Fe -a0= 2.789˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Fe -a0= 2.789˚A\nη(eV)α[×10−4]α01\nα02\n10−310−210−11000246810Co -a0= 2.507˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Co -a0= 2.507˚A\nη(eV)α[×10−4]α01\nα02\nFIG. 2. Left column: Local on-site ( α00, black square)\nand total ( αtot, red triangle) Gilbert damping as a func-\ntionofthebroadening ηforbccFe(001)with a0= 2.863Å,\nbcc Fe(001) with a0= 2.789Å, and fcc Co(001) with\na0= 2.507Å. Right column: Nonlocal first nearest neigh-\nbor (α01, black square) and second nearest neighbor ( α02,\nred triangle) Gilbert damping for the same systems.\ndampings with increasing ηis similar to the Fe case,\nand the on-site term dominates αtot. An obvious\ndifference is found for the second-neighbor damping,\nwhich behaves as an increasing function of ηfor Co\nunlike it is found for Fe.\nConcerning the calculated damping values, there is\na large variety of theoretical methods and calculation\nparameters, as well as experimental setups used in\nthe literature, which makes ambiguous to compare\nour results with others. Recently, Miranda et al.31\nreported a comparison of total and on-site damping\nvalues with the available theoretical and experimen-\ntal literature in their Table S1. For bcc Fe bulk they\nreported total damping values in the range of 1.3–\n4.2×10−3and for fcc Co bulk within the range of 3.2–\n11×10−3, and our results fit very well within theseranges around η≈100meV for Fe and for η >100\nmeV for Co. Moreover, we find that our calculated on-\nsite damping values for bcc Fe are larger ( >5×10−3)\nthan the reported values of Miranda et al.(1.6×10−3\nand 3.6 ×10−3), but for fcc Co the agreement with\ntheir reported total (3.2 ×10−3) and on-site damping\n(5.3×10−3) values is very good at our η= 136meV\nbroadening value.\n10−310−210−110010−510−410−310−2Fe\nη(eV)αtot\nαSOC=1\nαSOC=0\n10−310−210−110010−510−410−310−2Co\nη(eV)αtot\nαSOC=1\nαSOC=0\nFIG. 3. Effective (total) Gilbert damping for bcc Fe\n(left) and fcc Co (right) as a function of broadening ηon\na log-log scale. The error bars are estimated from the\nfitting procedure of Eq. (10). The red triangles show the\ncase with normal SOC ( αSOC=1), and the blue diamonds\nwhere SOC is switched off ( αSOC=0).\nNext, weinvestigatethespin-orbit-coupling-(SOC)-\noriginated contribution to the Gilbert damping. Our\nmethodmakesitinherentlypossibletoincludeaSOC-\nscaling factor in the calculations45. Fig. 3 shows the\nobtained total Gilbert damping as a function of the\nbroadening ηwith SOC switched on/off for bcc Fe\nand fcc Co. It can be seen that the effect of SOC\nis not dominant at larger ηvalues, but the SOC\nhas an important contribution at small broadening\nvalues ( η < 10−2eV), where the calculated total\nGilbert damping values begin to deviate from each\nother with/without SOC. As discussed in Ref. 23,\nwithout SOC the damping should go toward zero for\nzero broadening, which is supported by our results\nshown in Fig. 3.\nB. (001)-oriented surfaces of Fe and Co\nferromagnets\nIn the following, we turn to the investigation of the\nGilbertdampingparametersatthe(001)-orientedsur-\nfaces of bcc Fe and fcc Co. Both systems are treated\nas a semi-infinite ferromagnet interfaced with a semi-\ninfinite vacuum within the layered SKKR method46.\nIn the interface region 9 atomic layers of the ferromag-\nnet and 3 atomic layers of vacuum are taken, which is\nsandwiched between the two semi-infinite (ferromag-\nnet and vacuum) regions. Two types of surface atomic\ngeometries were calculated: (i) all atomic layers hav-\ning the bulk interlayer distance, and (ii) the surface\nand subsurface atomic layers of the ferromagnets have6\nTABLE I. Geometry relaxation at the surfaces of the fer-\nromagnets: change of interlayer distances relative to the\nbulk interlayer distance at the surfaces of bcc Fe(001) and\nfcc Co(001), obtained from VASP calculations. \"L1\" de-\nnotes the surface atomic layer, \"L2\" the subsurface atomic\nlayer, and \"L3\" the sub-subsurface atomic layer. All other\ninterlayer distances are unchanged in the geometry opti-\nmizations.\nL1-L2 L2-L3\nbcc Fe(001) -13.7% -7.7%\nfcc Co(001) -12.4% -6.4%\nbeenrelaxedintheout-of-planedirectionusingtheVi-\nenna Ab-initio Simulation Package (VASP)47within\nLSDA48. For the latter case the obtained relaxed\natomic geometries are given in Table I.\nFigure 4 shows the calculated layer-resolved on-\nsite and first-neighbor Gilbert damping values (with\nη= 0.68eV broadening) for the bcc Fe(001) and fcc\nCo(001) surfaces. It can generally be stated that the\nsurface effects are significant in the first 4 atomic lay-\ners of Fe and in the first 3 atomic layers of Co. We\nfind that the on-site damping ( α00) increases above\nthe bulk value in the surface atomic layer (layer 1:\nL1), and decreases below the bulk value in the sub-\nsurface atomic layer (L2) for both Fe and Co. This\nfinding is interesting since the spin magnetic moments\n(ms, shown in the insets of Fig. 4) are also consider-\nably increased compared to their bulk values in the\nsurface atomic layer (L1), and the spin moment enters\nthe denominator when calculating the damping in Eq.\n(4).α00increases again in L3 compared to its value in\nL2, thus it exhibits a nonmonotonic layer-dependence\nin the vicinity of the surface. The damping results ob-\ntained with the ideal bulk interlayer distances and the\nrelaxed surface geometry (\"R\") are also compared in\nFig. 4. It can be seen that the on-site damping is in-\ncreasedinthesurfaceatomiclayer(L1), anddecreased\nin the subsurface (L2) and sub-subsurface (L3) atomic\nlayers upon atomic relaxation (\"R\") for both Fe and\nCo. The first-neighbor dampings ( α01) are of two\ntypes for the bcc Fe(001) and three types for the fcc\nCo(001), see caption of Fig. 4. All damping values\nare approaching their corresponding bulk value mov-\ning closer to the semi-infinite bulk (toward L9). In\nabsolute terms, for both Fe and Co the maximal sur-\nface effect is about 10−3for the on-site damping, and\n2×10−4for the first-neighbor dampings. Given the\ndamping values, the maximal relative change is about\n15% for the on-site damping, and the first-neighbor\ndampings can vary by more than 100% (and can even\nchangesign)inthevicinityofthesurfaceatomiclayer.\nNote that Thonig and Henk35studied layer-resolved\n(effective) damping at the surface of fcc Co within the\nbreathing Fermi surface model combined with a tight-\nbinding electronic structure approach. Although they\nstudied a different quantity compared to us, they also\nreported an increased damping value in the surface\natomic layer, followed by an oscillatory decay toward\nbulk Co.\nSo far the presented Gilbert damping results cor-\nrespond to spin moments pointing to the crystallo-\n1 3 5 7 90.81Fe\n1 92.43\nlayermsms\nmR\ns\nlayerα00[×10−2]\nα00\nαR\n00\n1 3 5 7 90.81\n1 91.71.8\nlayermsms\nmR\nsCo\nlayerα00[×10−2]α00\nαR\n00\n1 3 5 7 9−2−101Fe\nlayerα01[×10−4]\nα01+αR\n01+\nα01−αR\n01−\n1 3 5 7 91234\nCo\nlayerα01[×10−4]α01+αR\n01+\nα01−αR\n01−\nα01 αR\n01FIG. 4. Evolution of the layer-resolved Gilbert damping\nfrom the surface atomic layer (L1) of bcc Fe(001) and fcc\nCo(001) toward the bulk (L9), depending also on the out-\nof-plane atomic relaxation \"R\". On-site ( α00) and first\nneighbor ( α01) Gilbert damping values are shown in the\ntop two and bottom two panels, respectively. The broad-\nening is η= 0.68eV. The empty symbols belong to the\ncalculations with the ideal bulk interlayer distances, and\nthe full symbols to the relaxed surface geometry, denoted\nwith index \"R\". Note that α01is calculated for nearest\nneighbors of atomic sites in the neighboring upper, lower,\nand the same atomic layer (for fcc Co only), and they are\nrespectively denoted by \" +\" (L-(L+1)), \" −\" (L-(L −1)),\nand no extra index (L-L). The insets in the top two panels\nshow the evolution of the magnitudes of the layer-resolved\nspin magnetic moments ms. The horizontal dashed line in\nall cases denotes the corresponding bulk value.\ngraphic [001] ( z) direction, and the transverse compo-\nnents of the damping αxxandαyyare equivalent due\ntothe C4vsymmetryofthe(001)-orientedsurfaces. In\norder to study the effect of a different orientation of\nall spin moments on the transverse components of the\ndamping, we also performed calculations with an ef-\nfective field pointing along the in-plane ( x) direction:\n[100] for bcc Fe and [110] for fcc Co. In this case, due\nto symmetry breaking of the surface one expects an\nanisotropy in the damping, i.e., that the transverse\ncomponents of the damping tensor, αyyandαzz, are\nnot equivalent any more. According to our calcula-7\ntions, however, the two transverse components of the\non-site ( αyy\n00andαzz\n00) and nearest-neighbor ( αyy\n01and\nαzz\n01) damping tensor, at the Fe surface differed by less\nthan 0.1 % and at the Co surface by less than 0.2 %,\ni.e., despite the presence of the surface the damping\ntensor remained highly isotropic. The change of the\ndamping with respect to the orientation of the spin\nmoments in zorxdirection (damping anisotropy)\nturned out to be very small as well: the relative dif-\nference in αyy\n00was 0.1 % and 0.3 %, while 0.5 % and\n0.1 % in αyy\n01for the Fe and the Co surfaces, respec-\ntively. For the farther neighbors, this difference was\nless by at least two orders of magnitude.\nIV. CONCLUSIONS\nWe implemented an ab initio scheme of calculat-\ning diagonal elements of the atomic-site-dependent\nGilbert damping tensor based on linear response the-\nory of exchange torque correlation into the real-space\nKorringa-Kohn-Rostoker (KKR) framework. To val-\nidate the method, damping properties of bcc Fe and\nfcc Co bulk ferromagnets are reproduced in good com-\nparison with the available literature. The lattice com-\npression is also studied for Fe bulk, and important\nchanges for the Gilbert damping are found, most pro-\nnounced for the site-nonlocal dampings. By investi-\ngating (001)-oriented surfaces of ferromagnetic Fe andCo, we point out substantial variations of the layer-\nresolved Gilbert damping in the vicinity of the sur-\nfaces depending on various investigated parameters.\nThe effect of such inhomogeneous dampings should be\nincluded into future spin dynamics simulations aim-\ning at an improved accuracy, e.g., for 2D surfaces and\ninterfaces. We anticipate that site-nonlocal damping\neffects become increasingly important when moving\ntoward physical systems with even more reduced di-\nmensions (1D).\nACKNOWLEDGMENTS\nThe authors acknowledge discussions with Danny\nThonig. Financial support of the National Research,\nDevelopment, and Innovation (NRDI) Office of Hun-\ngary under Project Nos. 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Fundamentally, these excitations are associated with an attractive interaction between\nelementary spin-excitations (i.e., magnons) and were predicted to occur in PMA materials in the\nabsence of damping [1, 2]. While damping, present in all magnetic materials, suppresses these\nexcitations, it is now possible to compensate damping by spin transfer torques through electrical\ncurrent \row in nanometer scale contacts to ferromagnetic thin \flms [3, 4]. A theory predicts the\nappearance of magnetic droplet solitons at a threshold current in nanocontacts [5] and, recently,\nexperimental signatures of droplet nucleation have been reported [6]. However, thus far, they\nhave been observed to be nearly reversible excitations, with only partially reversed magnetization\nand to be subject to instabilities that cause them to drift away from the nanocontacts (i.e., drift\ninstabilities) [6]. Here we show that magnetic droplet solitons can be stabilized in a spin transfer\nnanocontact. Further, they exhibit a strong hysteretic response to \felds and currents and a nearly\nfully reversed magnetization in the contact. These observations, in addition to their fundamental\ninterest, open up new applications for magnetic droplet solitons as multi-state high frequency\ncurrent and \feld tunable oscillators.\n1arXiv:1408.1902v1 [cond-mat.mtrl-sci] 8 Aug 2014Spin transfer torque nanooscillators (STNO) are nanometer scale electrical contacts to\nferromagnetic thin \flms that consist of a free magnetic layer (FL) and a \fxed spin-polarizing\nmagnetic layer [7{10]. The spin-transfer torque in such contacts can compensate the damp-\ning torque and excite spin-waves in the free layer, at a threshold dc current. When these\nspin-waves have a frequency less than the lowest propagating spin-wave modes in the fer-\nromagnetic \flm, the ferromagnetic resonance (FMR) frequency [11], they are localized in\nthe contact region. In PMA free layers this has been predicted to lead to dissipative droplet\nsolitons [5] (hereafter referred to as droplet solitons), which are related to the conserva-\ntive magnon droplets that were studied in uniaxial (easy axis type) ferromagnets with no\ndamping [1, 2]. In the nanocontact, energy dissipated due to damping (essentially friction)\nis compensated by energy input associated with spin-transfer torques, resulting in steady\nstate spin-precession. Droplet solitons are expected to be strongly localized in the contact\nregion, as well as to have spins precessing in-phase in the \flm plane [5]. In addition, for\nsu\u000ecient current the magnetization in the contact was predicted to be almost completely\nreversed relative to the \flm magnetization outside the contact. While precession frequencies\nbelow the FMR frequency were observed to appear at a threshold current in recent exper-\niments [6], there was no evidence for fully reversed spins in the contact. Here we report\nevidence for nearly full reversal of the magnetization in the contact region. Further, we\nshow that droplet solitons are stable, and exhibit a strong hysteretic response to currents\nand applied \felds. We also observe structure in their resistance versus \feld characteristics,\nsuggesting they experience a disordered pinning potential. These results are important to\nunderstanding and controlling their motion, nucleation and annihilation [12, 13].\nTo study droplet solitons we fabricated STNO with a free layer with PMA and an in-plane\nmagnetized polarizing layer (shown schematically in Fig. 1a) to measure their dc and high\nfrequency electrical characteristics. As the electrical signal in our STNO is associated with\nthe giant magnetoresistance e\u000bect, the in-plane magnetized polarizer allows us to detect\nin-plane precession of the magnetization of the free layer in the contact region, as discussed\nfurther below. Our layer structure consisted of a 4 nm thick Cobalt-Nickel (CoNi) multilayer,\n10 nm of Copper (Cu) followed by 10 nm of Permalloy (Ni 80Fe20, denoted Py) [14]. The\nCoNi multilayer is the free layer and has an easy magnetization axis perpendicular to the\nplane, while the Py layer serves as a polarizer and has its magnetization in the \flm plane.\nThe Cu layer decouples the magnetic layers while enabling spin-transport between the layers\n2dB a b\n1213141516 c\n051015\n024μ0H=0.4\tT\nPyCu\nCoNi\nSiO2applied I\nMCoNi \nMPyyz\nx\ndB \n01\n01dB dB \nμ0H=0.55\tT\nI(mA)f(GHz)\n20 22 24 26 28 3016171819\nSweep\tup\n23.52424.5\n30 32 34 3623.52424.5\nSweep\tdownSpacer\nPolarizerFLμ0H=0.8\tT\nI(mA)f(GHz)\nf(GHz) f(GHz)FIG. 1: Device schematic and dynamical properties. (a) Schematic of a magnetic droplet in\na STNO. An electrical current \rows through a nanocontact to a thin ferromagnetic layer (the free\nlayer, FL) towards a \fxed spin-polarizing layer. The external magnetic \feld is applied perpendicu-\nlar to the \flm plane (the z-direction in the \fgure). The droplet is a nearly reversed magnetization\nregion with spins precessing in the x\u0000yplane. (b & c) High frequency spectra as a function of\nthe current in applied \felds of (b) \u00160H= 0:4 and 0:55 T and (c) \u00160H= 0:8 T, showing both \feld\nswept-up and swept-down measurements. Hysteresis is found, as indicated by the vertical dashed\nlines.\n(i.e., the Cu layer is much thinner than its spin-di\u000busion length). We de\fned contacts to\nthe \flms that ranged from 70 to 200 nm in nominal diameter. Further, our sample layout\nenables electronic transport measurements from dc to microwave frequencies ( \u001850 GHz).\nWe characterized the magnetic properties of the layer stacks using ferromagnetic res-\nonance (FMR) spectroscopy. Most important for these studies, is that the CoNi has an\ne\u000bective perpendicular anisotropy \feld of \u00160HP= 0:25 T, which is de\fned to be the per-\npendicular anisotropy \feld minus the saturation magnetization, HP=HK\u0000Ms. We also\nstudied samples with smaller \u00160HP'0:1 T. We focus on the results for the STNO with\n\u00160HP= 0:25 T in this paper. (Results for the other series of STNO are presented in the\nSupplementary Materials section.)\nAs noted, the electrical response of our samples is associated with the giant magnetore-\nsistance e\u000bect. The device resistance depends on the relative magnetization alignment of\nthe free and polarizing layers, MR = ( R(H)\u0000RP)=RP=R0(1\u0000^ mFL\u0001^ mP)=2, where ^ mFL=P\nare unit vectors in the direction of the FL and polarizer magnetization respectively and\n3R0= (RAP\u0000RP)=RPis the fractional change in resistance between the device antiparallel\n(AP) and parallel (P) magnetization states. A magnetic \feld applied perpendicular to the\n\flm plane tilts the Py magnetic moments out of the \flm plane, mz;P=H=M sforH \u0016 0Ms= 1:1 T, the layer magnetizations align forming a\nP state. This results in the resistance of the STNO decreasing linearly with the applied \feld\nforH < M sand saturating when H > M s. On the other hand, precession of the in-plane\ncomponent of the FL magnetization leads to an oscillating resistance and thus a voltage\nresponse in the microwave range for \felds less than the saturation \feld of the polarizing\nlayer (i.e., when the polarizing layer has a component of magnetization in the \flm plane).\nFigure 1b shows measurements of the STNO high frequency response versus dc cur-\nrent at two di\u000berent applied perpendicular magnetic \felds at room temperature. At 0 :4\nT (mz;P'0:36) the STNO signal output frequency decreases with increasing bias current\n(i.e., there is a redshift of the signal) [15, 16]. While at 0 :55 T (mz;P'0:5) there is initially\na redshift of the oscillation frequency with increasing current, followed by an abrupt ( \u00183\nGHz) decrease of the signal frequency at a threshold current{that has been associated with\nthe creation of a droplet soliton [6]. The signal frequency after the jump is close to the Zee-\nman frequency, the prediction for a reversed droplet soliton, fZeeman =\r\u00160H=(2\u0019), where\r\nis the gyromagnetic ratio [1, 2]. Figure 1c shows that at 0 :8 T (mz;P'0:73) we only observe\nthe lower frequency (magnetic droplet) excitation and the frequency increases (blueshifts)\nwith increasing current (Fig. 1c). Most interestingly, at 0 :8 T the spectra are hysteretic \u0000the\nspectra depend on the \feld sweep direction. The droplet soliton nucleates at about 32 mA\nwith increasing current and is annihilated at about 31 mA with decreasing current. We\nnote also that with increasing \feld the magnetization of the Py saturates perpendicular to\nthe \flm plane\u0000in the same direction as the CoNi \u0000and thus the microwave signal vanishes.\nHowever, dc-magnetoresistance measurements still enable characterization of the droplet,\nbecause precession of FL changes its perpendicular component ( mz;FL) and this results in a\nvariation of the dc resistance (even when mz;P= 1).\nHence, to characterize magnetic droplet excitations \u0000both their onset and\nannihilation\u0000we measured the dc magnetoresistance (MR) of our devices at low temper-\nature (4:2 K) within a vector superconducting magnet. In Fig. 2a we show the MR for\n\felds applied perpendicular to the \flm plane; a resistance of zero corresponds to the \felds\n4ab\n15 20 25 30 350246810\nI(mA)\u0001MR(%)\n0.5 T0.6 T0.7 T0.8 T0.9 T1.0 T1.1 T1.2 T1.4 T\n1.3 T\n0 0.5 1 1.5 200.20.40.60.81\nc\u00010H(T)MR(%)\ndMeasured\nJumpMax MR\n0.6 0.8 1 1.2 1.40.20.40.60.81\u0001R (%)0.6 T0.9 T\n15 20 25 30 3500.51\u0001MR(%)\n\u00010H(T) I(mA)FIG. 2: Nucleation of droplet solitons with current. (a) Magnetoresistance of a 150 nm\ndiameter STNO versus \feld with I= 5 mA. Light blue and pink arrows represent the magnetic\nmoments for CoNi and Py; we outline the CoNi arrows that represent droplet soliton states. (b)\nMR as a function of the applied current for \felds ranging from \u00160H= 0:5 to 1.4 T. Curves are\nshifted vertically for clarity. (c) Expanded MR curves in (b) for the \felds \u00160H= 0:6 and 0.9 T,\nwith the onset of spin-wave excitations and the droplet state indicated. (d) The MR in the droplet\nsoliton state as a function of the applied \feld. The black curve is the expected maximum GMR of\nthe STNO as a function of the applied \feld, MR = R0H=M swhereR0is 0.9% and \u00160Msis 1.1 T.\nat which the magnetizations of the Py and CoNi align parallel, and the overall MR of the\nSTNO (R0= 0:9 %) corresponds to twice the value at zero \feld when the magnetizations of\nthe Py and CoNi are orthogonal. The dashed red curve in Fig. 2a illustrates the expected\nMR for a reversed CoNi magnetization (i.e., magnetization antialigned with the applied\n\feld). This curve is obtained by re\recting the measured MR about the horizontal line,\nMR(H= 0). Fig. 2b shows current swept MR measurements at a series of perpendicularly\napplied magnetic \felds. The signal to noise ratio increases at 4 :2 K and MR data can\n5now be used to detect both the onset of spin-waves excitations, corresponding to a small\n0:02\u00000:08% increase in MR, and the onset of droplet solitons, 0.85% change in resistance\n(see, for instance, the curves in Fig. 2c).\nThe resistance curves at constant applied \felds (Fig. 2b and c) show the onset of droplet\nsolitons when increasing the current and the annihilation of the soliton states when de-\ncreasing the current. There is hysteresis showing that there is an energy barrier separating\nSTNO states and indicating that the droplet soliton is stable for a range of applied currents.\nIn Fig. 2c we see that at low \felds ( \u00160H\u00180:6 and 0.9 T) there is \frst a small step in\nresistance that corresponds to the onset of a (small angle precession) spin-wave excitation\nand next a larger step in resistance that corresponds to the nucleation of the droplet soliton.\nWe have plotted the maximum change in the resistance step corresponding to the soliton\nexcitation in Fig. 2d and compared it with the maximum expected change in resistance\n(i.e., MR = R0H=M s). The overall change is almost the full MR, indicating the CoNi mag-\nnetization is reversed nearly completely in the nanocontact area. The overall droplet MR\nincreases with applied \feld because the relative alignment between Py and CoNi magnetiza-\ntion increases as the Py magnetization tilts with increasing applied \feld (see the schematic\nblue and pink arrows in Fig. 2d). The di\u000berence in resistance between that measured in the\ndroplet state and full magnetization reversal of the FL ranges from 20% at H= 0:6 T to just\n5% at \felds above 1 T; this di\u000berence may be due to precession in the droplet that decreases\nthe overall perpendicular component of the magnetization or by a small displacement of the\ndroplet from the contact center.\nNext we analyze the onset of droplet solitons as a function of applied \feld at constant\ncurrent. In Fig. 3a we show the MR as a function of the applied \feld for two \fxed applied\ncurrents,I= 25:5 mA andI= 32:5 mA. The MR curves clearly show both the onset and the\nannihilation of the droplet excitations as well as a remarkable hysteretic behavior, especially\nat large \felds. The MR curve for I= 32:5 mA (see Fig. 3a lower panel) shows the nucleation\nof a droplet soliton at a \feld of about 0 :3 T; at this \feld the magnetization of the CoNi\nlayer reverses and opposes the applied \feld. The resistance then increases with the applied\n\feld until it saturates when the Py magnetization saturates (i.e. when the magnetizations\nof CoNi and Py are antiparallel). At even larger \feld ( '3 T) there is a step decrease\nin resistance, which we associate with the droplet annihilation. When the applied \feld is\nreduced, the droplet nucleates at much lower \feld ( '1:4 T). The large \feld hysteresis is\n6a b\n0 0.2 0.4 0.600.20.40.6\nμ0Hx(T)MR(%)\nSoliton0stable\n&\nmz=10stable\n15 20 25 30 35123\nI(mA)μ0H(T)Annihilation0with0increasing0H\nAnnihilation0with0decreasing0I\nOnset0with0increasing0IOnset0with0decreasing0H\n0.5 1 1.5 2 2.5 300.20.40.60.8\nμ0H(T)MR(%)\n00.20.40.60.8MR(%)I=25.50mA\nI=32.50mA c\nI=260mA\nμ0H=2.00T\nμ0H=2.10T\nμ0H=2.20TFIG. 3: Hysteresis in the magnetoresistance (MR) data. (a) MR curves as a function of\nthe perpendicular applied \feld for currents of I= 25:5 mA andI= 32:5 mA. (b) Stability map of\ndroplet solitons: red circles show annihilation with increasing H, orange squares show annihilation\nwith decreasing I, blue circles show onset with decreasing H, and cyan squares show onset with\nincreasingI. (c) MR curves as a function of the in-plane \feld Hxfor an current of I= 26 mA. The\nthree curves correspond to a perpendicular applied \feld of 2.0 (blue) 2.1 (green), and 2.2 (red) T.\nconsistent with the current swept data at \fxed \feld, as we discuss further below.\nWe note that within the \feld range where droplet solitons are present there are additional\nand highly reproducible small steps in the resistance curves (Fig. 3a). These may originate\nfrom pinning of the soliton at di\u000berent sites within the contact. As the \feld increases,\nthe droplet soliton state becomes less energetically favorable and the soliton might shift\nto di\u000berent locations with slightly di\u000berent e\u000bective \felds caused by either current-induced\nOersted \felds or by small variations in the FL's magnetic anisotropy or magnetization. Such\nresistance states may also be due to changes in the droplet precession angles.\nWe next focus on the onset and annihilation of the soliton excitations when the Py layer is\nsaturated (\u00160H > 1:1 T) so the spin polarization of the current is constant with increasing\n\feld. There is hysteresis both in current sweeps at \fxed \feld and \feld sweeps at \fxed\ncurrent. In Fig. 3b we show the onset and annihilation conditions of the droplet soliton\n7found for both cases. The points marking the onset of the soliton fall on a straight line\n(both for decreasing \feld and for increasing current). Annihilation occurs at larger \felds\nand falls on a straight line as well{with a larger slope. The large area in between the two\nlines corresponds to the hysteretic region{the zone where the droplet soliton is present or\nabsent depending on the STNO \feld and current history. Similar measurements on a sample\nwith a smaller anisotropy ( HP\u00180:1 T) also showed the straight lines for both the onset\nand annihilation of the droplet soliton state with a smaller area hysteretic region, a width\nin applied \feld of about 0 :3 T (see the Supplementary Materials).\nWe also observed that droplet solitons can be annihilated (and thus rendered unstable)\nwith magnetic \felds applied in the \flm plane. We nucleated droplet solitons at a large\nperpendicular magnetic \feld and then applied an in-plane \feld until we annihilate the droplet\nsoliton. Figure 3c shows the MR as a function of the in-plane \feld at di\u000berent perpendicular\napplied \felds. We see that all the high-resistance states show an abrupt step down to the\nlow-resistance state with increasing in-plane \feld. Once we removed the in-plane \feld, we\nonly nucleated droplet solitons in cases for which the perpendicular \feld made the soliton\nstate stable and the no-soliton state unstable (i.e., perpendicular \felds and currents that\nare not in the hysteretic zone in Fig. 3b). Interestingly, the in-plane \feld values needed for\ndroplet annihilation depend on what pinning state the soliton was in before applying the\nin-plane \feld. Again, we observed the e\u000bect in samples with smaller anisotropy.\nWe now consider the basic physics of the soliton nucleation and annihilation. Droplet\nsolitons form in PMA thin \flms because spin torques can favor a layer magnetization an-\ntialigned with the magnetization of the polarizing layer (an AP state). Further, in PMA\n\flms (HP=Hk\u0000Ms>0) in a perpendicular applied \feld, any dynamical excitation\ntends to have a frequency that lies below the FMR frequency because the e\u000bective \feld\n(He\u000b=HPmz^ z) decreases with increasing precession angle (i.e. mz<1 and decreases with\nincreasing precession angle). Thus a soliton state is localized because its excitation frequency\nis below the frequency of propagating spin waves modes. As the applied perpendicular \feld\nis increased the current required to sustain the droplet also increases and eventually the\ndroplet annihilates. Hysteresis can result because the spin-torque required to maintain the\ndroplet state is less than that required to nucleate it.\nA quantitative understanding of droplet soliton hysteresis is possible through analysis\nof the Landau-Lifshitz-Gilbert-Slonczewski [3] equation describing the FL's magnetization\n8dynamics. In dimensionless parameters the LLGS equation reads:\n@m\n@t=m\u0002he\u000b\u0000\u000bm\u0002(m\u0002he\u000b) +\u001b0\u0011(mz)m\u0002(m\u0002mp); (1)\nwhere the precession (\frst term) and damping (second term) include the e\u000bective \feld\nhe\u000b=h0+hPmz^ z+r2m, the sum of the applied \feld, the e\u000bective perpendicular anisotropy\n\feld, and the exchange \feld. The second term's coe\u000ecient \u000bis the damping constant. The\n\felds are normalized to the saturation magnetization, Ms(e.g., h0=H0=Ms). The spin-\ntorque (third term in Eq. 5) includes the spin polarization direction of the applied current,\nmp, the torque asymmetry, \u0011(mz) (de\fned below), and \u001b0, which is proportional to the\ncurrent intensity. (Further details on the analysis is in the Supplementary Materials.)\nThe simplest analysis considers a macrospin and thus does not include the exchange\n\feld or allow for spatial variations in the magnetization. The dynamical equation for the\nperpendicular component of magnetization in an applied \feld perpendicular, h0=h0^ z, and\nwith a polarization also perpendicular to the \flm's plane, mp=^ z, is\n_mz=\u0000(1\u0000m2\nz)(\u001b0\u0011(mz)\u0000\u000b(h0+hPmz)): (2)\nThe state magnetized along the \feld direction, mz= 1, becomes unstable at a critical current\nand then it becomes stable again at a larger \feld. However, the reversed magnetization,\nmz=\u00001, has a di\u000berent stability threshold, leading to hysteresis. For certain parameters\nthere exists a third solution, 0 < m z<1 but it is unstable for the form of the spin-torque\nand spin-torque asymmetry that we consider.\nThe stability thresholds for the two solutions gives a state diagram, the range of current\nand \feld parameters in which there is bistability. For mz= 1 the threshold is\nh0=\u001b0\u0011(mz)=\u000b\u0000(hk\u00001) (3)\nand formz=\u00001 it is:\nh0=\u001b0\u0011(mz)=\u000b+ (hk\u00001): (4)\nIn the case with no spin-torque asymmetry, \u0011= 1, the state diagram in applied \feld versus\ncurrent (h0vs\u001b0) is two straight lines separating mz=\u00061 transitions. The width of the\nhysteresis in applied \feld is twice the e\u000bective anisotropy \feld, 2 hP. With an asymmetric\nspin torque [3] \u0011= 2\u00152=[(\u00152+ 1) + (\u00152\u00001)mz] withmz= 1 we have the same stability\ncondition but for mz=\u00001 the threshold slope changes. Thus an asymmetric spin torque\n9gives a \feld hysteresis that increases with the applied current, having a width in \feld of\ntwice the e\u000bective perpendicular anisotropy \feld plus a term linear in the applied current,\n2hP+\u001b0(\u00152\u00001).\nThe experimental data in Fig. 3b shows two straight threshold lines with di\u000berent slopes.\nFitting this data to a macrospin model gives an asymmetric spin torque \u00152= 1:8, with\na hysteresis extrapolated to zero current of 0 :7 T, which is larger than twice the layer\nmeasured e\u000bective perpendicular anisotropy \feld (0 :25 T). A state diagram for a STNO\nwith an e\u000bective perpendicular anisotropy \feld of \u00180:1 T was measured as well and could\nbe \ft assuming no spin-torque asymmetry ( \u0011= 1) and also showed a smaller \feld hysteresis\nof 0:3 T (see the Supplementary Materials).\nWe also considered a 2D model with exchange interactions and performed micromagnetic\nsimulations. Speci\fcally, we numerically solved the LLGS equation (Eq. 5) to study the\ndynamics of the FL's magnetization. Again, we considered the case when the permalloy\nlayer is saturated (i.e., the current polarization direction is constant, mp=^ z). We found\nthat droplet solitons are created at a critical current [5] and annihilate in a perpendicular\napplied \feld as well as with small in-plane magnetic \felds. Our results show that the\nhysteresis is the same as that of the macrospin model. The main di\u000berence is that the lines\nseparating the stability of the droplet soliton are shifted towards larger current densities{or\ntowards lower applied \felds{owing to the fact that the exchange induces di\u000busion in the\nsystem (see Supplementary Material).\nOur simulations also show how droplet solitons annihilate with in-plane \felds. In-plane\n\felds cause the droplet soliton to delocalize and eventually lose stability; this occurs when\nthe localized oscillations in the magnetization couple with the \flm's propagating spin-wave\nmodes. Our analysis and experiments also show that droplet solitons are stable at relatively\nsmall perpendicular \feld \felds. The macrospin model also predicts stable droplets at zero\napplied \feld, provided the polarizing layer is perpendicularly magnetized at zero \feld (i.e.\nhas a net PMA). We also found this in micromagnetic simulations with a perpendicularly\nmagnetized polarizer layer.\nIn summary we have demonstrated stable droplet solitons in nanocontacts to ferromag-\nnetic thin \flms. Our experimental results reveal a nearly complete reversal of the magnetiza-\ntion in the nanocontact and a large hysteresis between the onset and annihilation of soliton\nexcitations{both in \feld and current. This provides a new means for droplet solitons to\n10carry and store information, in addition to in their phase and amplitude [13]. Our modeling\nand micromagnetic simulations capture the main features of our experimental results and\nalso predict that droplet solitons can exist at zero applied \feld. We have also observed small\nand reproducible variations on the nanocontact resistance when varying the \feld suggesting\nthat the droplet solitons can be trapped at pinning sites or may have discrete precessional\nstates. The hysteresis and the observed discrete states could be useful in in an application\nthat requires a multistate oscillator.\nMethods\nThe layer stacks consist of Co and Ni, 6 \u0002(0.2Coj0.6Ni) capped with 0.2Co j5 Pt, sepa-\nrated by 10 Cu from a 10 Py layer, deposited by thermal and e-beam evaporation in an\nultra high vacuum chamber (thicknesses in nanometers). The Cu spacer layer was chosen to\nmagnetically decouple the in-plane magnetized Py layer from the out-of-plane magnetized\nCojNi multilayer; the Pt capping layer was used to further enhance the interface-induced\nperpendicular magnetic anisotropy of the Co jNi multilayer. Layer stacks were deposited on\noxides silicon wafers. The point contacts were de\fned by etching holes in a 50 nm thick\nsilicon dioxide layer deposited on top of the \flms. Electron-beam lithography was used to\nde\fned the point contacts with diameters ranging from 70 to 200 nm. Devices were pat-\nterned with a bottom electrode and a top electrode into structures suitable for both dc and\nmicrowave electrical measurements using optical lithography and etching techniques. Most\nof the nanofabrication process were carried out at the Center for Functional Nanomaterials\nat Brookhaven National Lab. The contacts were characterized by atomic force microscopy\nand scanning electron microscopy.\nOur layer stacks have been characterized before and after patterning with FMR spec-\ntroscopy (see Supplementary Materials) in order to determine the magnetic anisotropy of\nboth Py and CoNi. We used frequencies ranging from 1 to 40 GHz as a function of the ap-\nplied \feld at room temperature and a coplanar waveguide (CPW) to create the microwave\n\felds along with a network analyzer to record the absorption signal.\nFor the high frequency measurements our samples were contacted with picoprobes to\na current source and to a spectrum analyzer that recorded the signals in the presence of\napplied magnetic \felds. We used a broadband low noise 20 dB ampli\fer. The dc transport\n11measurements were carried out by contacting devices with wire bonds. Low temperature\ntransport measurements were conducted at 4.2 K in a three-dimensional vector supercon-\nducting magnet capable of producing bipolar \felds up to 8 T.\n12Supplementary materials\nStable Magnetic Droplet Solitons in Spin Transfer Nanocontacts\nI. ADDITIONAL MEASUREMENTS\nWe have measured samples with a similar layer stacks to the ones presented in the main\nmanuscript but with a slightly di\u000berent anisotropy \feld, \u00160HP(=\u00160(HK\u0000Ms)), of about 0.1\nT, determined using ferromagnetic resonance spectroscopy. While the nominal thicknesses\nof the Co and Ni layers were the same, we obtained slightly di\u000berent anisotropies in di\u000berent\ndeposition runs.\nWe have analyzed the stability of soliton excitations with applied magnetic \felds similar\nto the method presented in Fig. 3 of the main manuscript. Figure 4 shows a stability map of\n10 15 20 25 300.511.522.5\nI(mA)H(T)Annihilation with increasing H\nAnnihilation with decreasing I\nOnset with increasing IOnset with decreasing H\nFIG. 4: (a) Stability map of soliton excitations for a sample with an e\u000bective perpendicular\nanisotropy \feld \u00160HP, of about 0.1 T: red circles show annihilation with increasing H, orange\nsquares show annihilation with decreasing I, blue circles show onset with decreasing H, and cyan\nsquares show onset with increasing I.\n13soliton excitations as a function of the perpendicular applied \feld for a point contact with\na nominal radius of 50 nm. At \felds larger than 1 :1 T when the Py layer is saturated, we\ncan take the spin polarization of the current as a constant. The points separating the onset\nand annihilation of the droplet state fall on a nearly straight line and a hysteresis width in\n\feld of about 0 :3 T. We note here that this sample presents equal slopes for both onset and\nannihilation \feld versus current, suggesting the e\u000bect of the asymmetry in the spin torque\nis negligible (i.e. \u0011= 1).\nAt \felds below 1.1 T, the onset current (and the annihilation current) decreases with\nincreasing \feld (\u00181=H) because the Py layer is not saturated and the polarization of the\ncurrent and the spin-torque are mainly driven by the z-component of the magnetization of\nthe polarizing Py layer.\nII. MODELING AND MICROMAGNETICS\nTo understand the regions in applied \feld and current density where the droplet soliton\nstate is stable we discuss in more detail the argument used in the main manuscript, both\nthe macrospin model and micromagnetic simulations.\nWe again consider \frst the simpli\fed macrospin model that neglects spatial variations\nin the free layer magnetization. We start with the LLGS equation for the magnetization\ndynamics:\n@m\n@t=m\u0002he\u000b\u0000\u000bm\u0002(m\u0002he\u000b) +\u001bm\u0002(m\u0002mp); (5)\nwithhe\u000b=h0+hPmz^ z,\u000bis the damping, and \u001bis proportional to the current. Notice\nthere is no exchange and hP>0 is the e\u000bective perpendicular anisotropy, hP=hk\u00001. We\nhave normalized magnetization and time by the saturation magnetization, Ms, and Larmor\nfrequency,\r\u00160Ms, respectively.\nThe dynamics for the magnetization zcomponent is given by:\n_mz=\u0000(1\u0000m2\nz)(\u001b0\u0011(mz)\u0000\u000bhe\u000b); (6)\nwhere\u001b=\u001b0\u0011and\u0011the polarization spin torque asymmetry. The e\u000bective \feld, he\u000b=\nh0+hPmz, includes the anisotropy, hP>0, the external \feld in the z-direction,h0, and the\ndemagnetization \feld, \u0000mz.\n14We then perform a stability analysis of the zcomponent of the magnetization and see\nunder what conditions the constant solutions, mz, for Eq. 6, are either stable ( @mz(@tmz)<0)\nor unstable ( @mz(@tmz)>0). This reasoning has already been used in [17] and in [18] to\nexplain the switching of a nanopilar (in-plane and out-of-plane respectively).\nEquation 6 can have either 2 or 3 solutions depending on the parameters h0,\u001b,\u000bandhk:\nmz=\u00061\nmz=\u001b=\u000b\u0000h0\nhk\u00001if\u00001\u0014mz\u00141(7)\nHere, we note that the third solution is always unstable if there is perpendicular anisotropy\nand for the form of the spin-torque torque and spin-torque asymmetry we consider. We\nconsider a spin-transfer torque that depends on the angle between the free and polarizing\nlayer's magnetization [3, 19]:\n\u0011=2\u00152\n(\u00152+ 1) + (\u00152\u00001)mz: (8)\nSo\u0015= 1 is no asymmetry; and \u00156= 1 results in \u0011(mz= 1) = 1 and \u0011(mz=\u00001) =\u00152.\nOne can derive the condition for a macrospin state to lose stability for the two solutions\nthat correspond to the spins pointing along the applied \feld ( mz= 1) or in the opposite\ndirection (mz=\u00001).\nFormz= 1 we obtain\nh0=\u001b0\u0011(1)=\u000b\u0000(hk\u00001) (9)\nand formz=\u00001 we have\nh0=\u001b0\u0011(\u00001)=\u000b+ (hk\u00001): (10)\nWe can now see that the hysteresis for a \fxed current, \u001b0, is:\n\u0001h= (h0+ (hk\u00001)) (\u00152\u00001) + 2(hk\u00001): (11)\nIn Fig. 5 we plot the stability thresholds for mz=\u00061 using the parameters for CoNi\nmultilayered \flm having a saturation magnetization value, Ms, of 0.96 T. We plotted two\ndiagrams corresponding to: a)an e\u000bective perpendicular anisotropy \feld \u00160(Hk\u0000Ms) of\n0.15 T and no spin torque asymmetry, \u0015= 1, b)an e\u000bective perpendicular anisotropy \feld\n\u00160(Hk\u0000Ms) of 0.3 T and a spin torque asymmetry with \u00152= 1:8.\n150 0.02 0.04 0.0600.511.522.533.5\nσ0h0\n0 0.02 0.04 0.0600.511.522.5\nσ0h0α=0.03\nhk=1.15\nλ2=1α=0.03\nhk=1.3\nλ2=1.8\nmz=-1mz=1\nmz=-1mz=1\nmz=-1mz=1\n&mz=1\nmz=-1&FIG. 5: Stability thresholds versus applied current, \u001b0, and \feld,h0, separating stable and unstable\nregions for the solution mz=\u00061. The damping value is taken at \u000b= 0:03 and the anisotropy\nvalue, athk= 1:15 andhk= 1:3 that corresponds to the values for our measured samples.\nThis macrospin model has limitations because it neglects spatial variations in the mag-\nnetization. We have done 2 dimensional micromagnetic simulations to investigate droplet\nexcitations; onset, annihilation and hysteresis. We used the LLGS equation (Eq. 5) with an\ne\u000bective \feld that includes exchange:\nhe\u000b=h0+ (hk\u00001)mzz+r2m: (12)\nWe observe the formation of droplet solitons and their annihilation with both large out\nof plane \felds and also with in-plane \felds. Figure 6a and b show the \fnal state of a 50 nm\npoint contact with an applied current, \u001b0, of 0.1 under an applied \feld of h0=1.5. We have\nconsidered a \flm with an e\u000bective \feld of 0.15 T ( hk\u00191:15) and a damping, \u000bof 0:03.\nIn this case it does not matter what the initial state is because the droplet is stable and it\nalways forms no matter what the initial state is. To study the hysteresis in the micromagnetic\nsimulations we compared di\u000berent applied \felds, h0, and current densities, \u001b0, while keeping\nthe parameters hkand\u000bconstant. In all cases we simulated 2 events, i)initial condition\nis a nucleated droplet and ii)initial condition is magnetization pointing in the applied \feld\ndirection. We see hysteresis that has a size of twice the e\u000bective perpendicular anisotropy\nhk\u00001 and shows a linear dependence between the applied current, \u001band the applied \feld,\nh0, with a constant \u000b. In [5] it was shown, with some analytical treatment of the LLGS\n16α=0.03\nhk=1.15\nλ2=1\n0.02 0.04 0.06 0.08 0.100.511.522.5\nh0\nσ0\n−1000100\n−1000100−1−0.500.51\n−50 0 50−50\n0\n50a\nbc\nx(nm)y(nm)x(nm)y(nm)mzFIG. 6: Micromagnetic simulations (a) and (b) Final state for the magnetization, mzin center\nof the nanocontact for a 50 nm point contact. The \flm parameters are hk= 1:15,h0= 1:5,\u001b= 0:1\nand\u000b= 0:03. The simulation was run until a steady state was found corresponding to a reversed\nsoliton. (c) Curves in applied current, \u001b, and \feld,h, separating stable and unstable regions for the\nreversed solitons . The damping value is taken at \u000b= 0:03 and the anisotropy value, at hk= 0:15\nequation (eq. 5), the critical sustaining current depended linearly on the external h0with a\nfactor\u000b.\nWe have computed the stability diagram showing the \fnal state as a function of the\ninitial conditions. Figure 6c corresponds to a \flm with \u00160(Hk\u0000Ms) = 0:15 T of PMA and\nwe neglected spin torque asymmetry.\nThis modeling shows that one should expect annihilation of the reversed excitations and\nthat the hysteresis in the reversing is indeed intrinsic in the solution of the LLGS equation.\nIt also tells that the size of the hysteresis is 2 hP(i.e., it increases with the anisotropy). The\nmodel also predicts that if we were able to maintain the polarization of the current at zero\n\feld we could create a droplet just increasing the current, which could be achieved with a\nperpendicular magnetized polarizer.\n17Acknowledgements\nF.M. acknowledges support from EC, MC-IOF 253214, from Catalan Government\nCOFUND-FP7, and from MAT2011-23698. This research was supported by NSF-DMR-\n1309202 and in part by ARO-MURI, Grant No. W911NF-08-1-0317. Research carried out\nin part at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which\nis supported by the U.S. Department of Energy, O\u000ece of Basic Energy Sciences, under Con-\ntract No. DE-AC02-98CH10886.\n[1] A. Ivanov and A. M. Kosevich, Zh. Eksp. Teor. Fiz. 72, 2000 (1977).\n[2] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys. Rep. 194, 117 (1990).\n[3] J. C. Slonczewski, Journal of Magnetism and Magnetic Materials 1-2, L1 (1996).\n[4] J. C. Slonczewski, Journal of Magnetism and Magnetic Materials 195, L261 (1999).\n[5] M. A. Hoefer, T. J. Silva, and M. W. Keller, Phys. Rev. B 82, 054432 (2010).\n[6] S. M. Mohseni, S. R. Sani, J. Persson, T. N. Anh Nguyen, S. Chung, Y. Pogoryelov, P. K.\nMuduli, E. Iacocca, A. Eklund, R. K. Dumas, et al., Science 339, 1295 (2013).\n[7] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, T. V., and P. Wyder, Nature 406, 46\n(2000).\n[8] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman,\nand D. C. Ralph, Nature 425, 380 (2003).\n[9] W. Rippard, M. Pufall, S. Kaka, S. Russek, and T. Silva, Phys. Rev. Lett 92, 027201 (2004).\n[10] S. Bonetti, V. Tiberkevich, G. Consolo, G. Finocchio, P. Muduli, F. Manco\u000b, A. Slavin, and\nJ.\u0017Akerman, Phys. Rev. Lett. 105, 217204 (2010).\n[11] C. Kittel, Phys. Rev. 70, 965 (1946).\n[12] M. A. Hoefer, M. Sommacal, and T. J. Silva, Phys. Rev. B 85, 214433 (2012).\n[13] M. D. Maiden, L. D. Bookman, and M. A. Hoefer, Phys. Rev. B 89, 180409 (2014).\n[14] F. Maci\u0012 a, P. Warnicke, D. Bedau, M.-Y. Im, D. A. Fischer, P. Arena, and A. D. Kent, Jour.\nof Mag. Mag. Mat. 324, 3632 (2012).\n18[15] W. H. Rippard, A. M. Deac, M. R. Pufall, J. M. Shaw, M. W. Keller, S. E. Russek, G. E. W.\nBauer, and C. Serpico, Phys. Rev. B 81, 014426 (2010).\n[16] S. M. Mohseni, S. R. Sani, J. Persson, T. N. Anh Nguyen, S. Chung, Y. Pogoryelov, and\nJ. Akerman, Phys. Status Solidi RRL 5, 432 (2011).\n[17] S. Mangin and et al, Nature Materials 5, 210 (2006).\n[18] B. Ozyilmaz, A. D. e. Kent, and et al, Phys. Rev. Lett. 91, 067203 (2003).\n[19] D. C. Ralph and M. D. Stiles, Journal of Magnetism and Magnetic Materials 320, 1190 (2007).\n19" }, { "title": "1212.2073v1.Heat_induced_damping_modification_in_YIG_Pt_hetero_structures.pdf", "content": "arXiv:1212.2073v1 [cond-mat.mes-hall] 10 Dec 2012Heat-induced damping modification in YIG/Pt hetero-struct ures\nM. B. Jungfleisch,1,a)T. An,2,3K. Ando,2Y. Kajiwara,2,3K. Uchida,2,4V. I. Vasyuchka,1A. V. Chumak,1\nA. A. Serga,1E. Saitoh,2,3,5,6and B. Hillebrands1\n1)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern,\n67663 Kaiserslautern, Germany\n2)Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n3)CREST, Japan Science and Technology Agency, Tokyo 102-0076 , Japan\n4)PRESTO, Japan Science and Technology Agency, Saitama 332-0 012, Japan\n5)WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577,\nJapan\n6)Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195,\nJapan\n(Dated: 16 October 2018)\nWe experimentally demonstrate the manipulation of magnetization re laxation utilizing a temperature dif-\nference across the thickness of an yttrium iron garnet/platinum ( YIG/Pt) hetero-structure: the damping\nis either increased or decreased depending on the sign of the tempe rature gradient. This effect might be\nexplained by a thermally-induced spin torque on the magnetization pr ecession. The heat-induced variation\nof the damping is detected by microwave techniques as well as by a DC voltage caused by spin pumping into\nthe adjacent Pt layer and the subsequent conversion into a charg e current by the inverse spin Hall effect.\nDue to their interesting underlying physics and po-\ntential applications in magnon spintronics the spin Hall\neffect (SHE) and the inverse spin Hall effect (ISHE)\nattracted considerable attention in the last years.1,2\nMagnon spintronics is a new, emerging field in spin-\ntronics, that utilizes magnons (quanta of spin waves)\nas carriers of angular momentum. The combination\nof spin pumping and inverse spin Hall effect turned\nout to be a well suited technique for the detection of\nmagnons beyond the wavenumber limitations of most\nother methods.2The recent discoveryof the spin Seebeck\neffect (SSE) in magnetic insulators demonstrates the im-\nportance of heat currents in spintronics and opened up\nthe new field of spin caloritronics.3,4\nA key objective in the field of magnon spintronics\nis the control of magnetization relaxation and genera-\ntion of spin waves. To compensate spin-wave damp-\ning a common method is parametric amplification.2,5\nRecently, it was reported, that propagating spin waves\ncan also be amplified by injecting a spin current due\nto the SHE and the spin-transfer torque (STT) effect6\nand by the SSE.7Spin relaxation was manipulated by\nSHE and STT in Ni 81Fe198and by thermally-induced in-\nterfacial spin transfer in yttrium iron garnet/platinum\n(YIG/Pt) structures.9In all these experiments, magneti-\nzation dynamics is measured by using microwave tech-\nniques. However, two-magnon scattering can lead to\nthe excitation of secondary spin waves with much higher\nwavevectors as it has been shown in Refs.10,11. Even\nthough these secondary waves contribute significantly to\nspin pumping,12,13they cannot be detected by inductive\nantennas. Therefore, microwave measurements do not\nnecessarily give a thorough insight into magnetization\ndynamics.\na)Electronic mail: jungfleisch@physik.uni-kl.deIn this Letter, we report on the thermal manipula-\ntionofspin-waverelaxationmeasuredbyboth microwave\ntechniques as well as by spin pumping. The investi-\ngated sample consists of a YIG/GGG/YIG/Pt hetero-\nstructure. A temperature difference applied across the\nthickness of this structure leads to the longitudinal\nSSE: an imbalance between the magnon and electron\ntemperatures causes a spin current across the YIG/Pt\ninterface.3,4The generatedspin current transfers angular\nmomentum and, consequently, they might exert a torque\non the magnetization (see Fig. 1(b)). As a result, the\nFIG. 1. (Color online) (a) Schematic illustration of the exp er-\nimental setup. (b) Possible mechanism for the heat-induced\ndamping variation in the YIG film. Mdenotes the magneti-\nzation. (c) Typical example for a measurement of UISHEas a\nfunction of the applied magnetic field H.2\nFIG. 2. (Color online) (a) Typical example for the tempera-\nture difference ∆ Tacross the sample as a function of the ap-\nplied magnetic field H. ∆TPeltierdenotes the temperature dif-\nference applied by the Peltier element. At the ferromagneti c\nresonance field around 79 mT the magnetization precession\ncauses additional heating denoted as ∆ TMW. (b) Calculated\nspin Seebeck voltage USSEcomposed of UPeltier\nSSEgenerated by\n∆TPeltierandUMW\nSSEcreated by ∆ TMW.\nmagnetization precession can either be enhanced or sup-\npressed depending on the sign of the temperature differ-\nence and, thus, the direction of the spin current. This\nchange in the damping is equivalent to a variation of the\nferromagnetic resonance linewidth ∆ HFMRthat is mea-\nsuredbymicrowavereflectionaswellasbyspin pumping.\nA sketch of the experimental setup is shown in\nFig. 1(a). A 2.1 µm thick YIG film was grown by\nmeans of liquid phase epitaxy on both sides of a 500 µm\nthick gadolinium galliumgarnet(GGG) substrate. Using\nmolecular beam epitaxy, we then deposited a Pt layer of\n10 nm thickness on one side of the sample, fully covering\none of the YIG surfaces. As shown in Ref.14the Pt layer\nmight show ferromagnetic behavior on ferromagnetic in-\nsulators due to magnetic proximity effects. However, as\nshown in Ref.15, a possible contamination by the anoma-\nlous Nernst effect is negligibly small compared to the\nlongitudinal SSE contribution. A Peltier element, that\nis mounted on top of the Pt layer, generates a temper-\nature difference across the multilayer. In order to en-\nhance the temperature flow from the sample, the second\nYIG surface is covered with a sapphire substrate that\nis connected to a heat bath (sapphire is a good thermal\nconductor). The second YIG layer neither influences the\nmagnetic nor the electric measurements but it should be\nnoted that the temperature difference is applied across\nthe entire Pt/YIG/GGG/YIG sample stack. The mag-\nnetization precession is excited by a 500 µm wide copper\nmicrostrip antenna that is placed abovethe Pt layerwith\nan intervening isolation layer (see Fig. 1(a)). The tem-\nperature difference is monitored using an infrared cam-\nera, calibrated by two thermocouples.\nThe experiment is performed as follows: an exter-\nnal magnetic field His applied perpendicularly to the\nYIG waveguide in the YIG film plane. The magneti-\nzation precession is driven by the alternating magnetic\nfieldh(t) of a continuous microwave signal (see Fig. 1(a)\nand (b)) with a carrier frequency of 4 GHz and pow-\ners ofPMW= +14 dBm, +20 dBm, and +25 dBm.While sweeping the external magnetic field H, a tem-\nperature difference across the sample thickness is applied\nand recorded by the infrared camera. The electric volt-\nagedue to the ISHE, UISHE, andthe microwavereflection\nare measured simultaneously.\nIn the experiment, two different mechanisms con-\ntribute to the spin current: the spin Seebeck effect and\nthe spin pumping effect. The SSE originates from the\ndifference between the effective magnon temperature Tm\nand the effective electron temperature Teat the YIG/Pt\ninterface.3,4,16This temperature difference ∆ Tis created\nin two different ways in our experiment (see Fig. 2(a)):\nby the Peltier element (denoted by ∆ TPeltierin the fol-\nlowing) and by heating due to magnetization precession\nin resonance condition of the YIG film at HFMR(de-\nnoted by ∆ TMW). The second mechanism to create\na spin current is spin pumping by the externally ex-\ncited coherent magnetization precession.12Irrespective\nof its origin, the net spin current Jsinjected into the\nPt layer is transformed into a conventional charge cur-\nrentJc, perpendicular to both JsandH, by the ISHE\n(see Fig 1(c)).10,17As a result, charges accumulate at the\nedges of the Pt layer and a voltage UISHE=USP+USSE,\ncomposed of a spin pumping contribution USPand a\nSSE contribution USSE, can be measured (Fig. 1(c)).\nThe voltage USSE=UMW\nSSE+UPeltier\nSSEitself consists of\nthe voltage UMW\nSSEgenerated by heating ∆ TMWdue to\nmagnetization precession in resonance and UPeltier\nSSEgen-\nerated by the temperature difference ∆ TPeltier. In or-\nder to distinguish between the different contributions to\nUISHE=UPeltier\nSSE+UMW\nSSE+USPthe following procedure\nhas been used: for each temperature difference created\nwith the Peltier element, UISHEand ∆Twere recorded.\nFrom the off-resonance condition ( H > H FMR), we can\ndeduce a linear relation between USSE=UPeltier\nSSEand\n∆T. Using this linear SSE relation3,4,16enables us to re-\ncalculate the corresponding voltage UMW\nSSEatHFMRand,\nthus, the spin pumping voltage USP, respectively. In\nFig. 2 the evolution of the temperature difference as a\nfunction of the applied magnetic field Hand the corre-\nsponding SSE voltage USSEare shown for a microwave\npower of PMW= +25 dBm. Figure 2(a) clearly shows,\nFIG. 3. (Color online) (a) Typical spectra for 4 different tem -\nperaturedifferences at +25dBm. For positive (negative) tem -\nperature differences, the Pt is colder (hotter) than the YIG.\n(b) Measured resonance linewidth ∆ Has a function of the\ntemperature difference ∆ Tfor different microwave powers.3\nthat, in addition to the applied temperature difference of\n∆TPeltier≈4.6◦C, the temperature rises at HFMRby an\nadditional value of ∆ TMW≈0.6◦C. The corresponding\nvoltages UPeltier\nSSEandUMW\nSSEare illustrated in Fig. 2(b).\nFor ∆T= 0, the FMR driven spin pumping contribution\nUSPis dominant ( USP/(USP+USSE)≈99.9%).\nThe heat-induced spin current affects our measure-\nments in two different ways. On one side, it generates\na voltage USSEindependent of the absolute value of the\nexternally applied magnetic field H(crossing zero field\nresults in a change of the polarity of USSEaccording to\nthe SSE3,4,16), and on the other side, it most likely exerts\na torque on the magnetization, resulting in the manipu-\nlation of the relaxation damping (see Fig. 1(b)).\nFigure 1(c) shows a typical example for the measured\nISHE voltage UISHEwithout externally applied tempera-\nture difference. The voltagereachesits maximal absolute\nvalue at HFMR≈79 mT. In Fig. 3(a) the recalculated\nUSPdata is shown as a function of the external magnetic\nfieldHfor four different measured temperature differ-\nences ∆T. Heating and cooling the sample gives rise to a\nchange of the saturation magnetization MSresulting in\na resonance peak shift to higher or lower magnetic fields.\nAs it is obvious from Fig. 3(a), not only one but\nseveral modes contribute to the spin pumping voltage\nUSP. Therefore, the envelope of USPis fitted for each\ntemperature difference by a Gaussian function f(x) =\na·exp(−(x−b)2/(2c2)), where cdefines the linewidth\n∆Hwhichisameasureforthedamping α. Thelinewidth\n∆Hthat is determined in this way, does not necessarily\ncoincide with the real ferromagnetic resonance linewidth\n∆HFMRbut is proportional to it, i. e. ∆ H∝∆HFMR.\nThe linewidth ∆ Has a function of the temperature dif-\nference ∆ Tis shown in Fig. 3(b) for different microwave\npowersPMW. It is clearly visible that the total linewidth\n∆Hdecreases for one polarity of ∆ Tand increases for\nthe other. We also analyzed each mode separately and\nwe found that the qualitative behavior for each mode is\nthe same.\nAs it is visible from Fig. 3(b), the variation of the\nlinewidth ∆ Hper 1◦C temperature difference (slope in\nFig. 3(b)) is approximately the same for all microwave\npowers. For a temperature difference of ∆ T≈ ±4◦C,\nthe linewidth changes about 6%, independent of the mi-\ncrowave power. This independency is expected since the\ngenerated spin currents are thermally induced and, thus,\ndo not depend on the applied microwave power.3The\ndamping, i. e., the linewidth at ∆ T= 0 is larger for\nhigher microwave powers which is attributed to the on-\nset of non-linear effects.18–20\nNow we compare how the linewidth alters under the\ninfluence of a longitudinal temperature difference mea-\nsured by both spin pumping as well as microwave tech-\nniques. Since spin pumping is not sensitive to the spin-\nwave wavelength, the directly excited spin-wave modes\nas well as short-wavelength secondary waves contribute\nto the detected signal.10,11However, microwave reflec-\ntion mainly detects the primary excited uniform mode.FIG. 4. (Color online) Measured linewidth ∆ Has a func-\ntion of the measured temperature difference ∆ Tfor reflected\nmicrowave signal and spin pumping voltage USP.\nThe results are summarized in Fig. 4. For both mea-\nsurement techniques, the linewidth qualitatively behaves\nthe same in the investigated range of temperature dif-\nferences. Nevertheless, one can see that the slopes of\nthe two curves diverge leading to the assumption that\nthe uniform FMR mode is mostly effected by the heat-\ninduced damping modification. However, a quantitative\nstatement is not possible.\nAssuming, that the observed heat-induced damping\nvariationis dueto thermalspin currentsgeneratedbythe\nSSE, we calculate the variation of the magnetization re-\nlaxationin YIG/Pt hetero-structuresbased on the model\ndeveloped in Ref.8. We modify this model by substitut-\ning a heat-induced spin current for a SHE-generated spin\ncurrent: the charge current Jcis replaced by the tem-\nperature difference ∆ T. Thus, the generalized Landau-\nLifshitz-Gilbert (LLG) equation is expressed as\ndM\ndt=−γM×Heff+α0\nMsM×dM\ndt\n−γJSTT\nS\nM2sVFM×(M×σ),(1)\nwhereMis the magnetization, γthe gyromagnetic ra-\ntio,Heffthe effective magnetic field, Msthe saturation\nmagnetization, VFthe volume of the YIG layer, and\nσthe spin polarization vector. The Gilbert damping\nα=αF+∆αSPis the sum of the intrinsic damping con-\nstantαFof the isolated YIG layer and ∆ αSPis the ad-\nditional damping due to spin pumping in the adjacent\nPt layer.12JSTT\nSdescribes the heat-induced spin torque.\nBy introducing the injection and charge current conver-\nsion efficiency u= (e/¯h)(2πfMsdF/γ)v, wherevis the\nslope obtained by fitting our results (Fig. 3(b)), and by\nintroducing an additional temperature dependent damp-\ning parameter ∆ αSSE\nSTT, we obtain, in analogy to Ref.8,\nthe heat-induced spin torque\nJSTT\nS=AFv2πfMsdF\nγ∆T. (2)\nThespin-currentdensityisgivenby JS= 2e/(¯hAF)JSTT\nS.4\nPMW(dBm)JSTT\nS(×10−11Nm\n◦C∆T)JS(×109A\nm2◦C∆T)\n+14 1.74±0.15 3.70±0.32\n+20 2.11±0.18 4.49±0.39\n+25 2.01±0.08 4.27±0.16\nTABLE I. Comparison of spin torque JSTT\nSand spin current\ndensityJsfor different mircowave powers PMW.\nThe calculated spin torque JSTT\nSand the spin current\ndensityJSare summarized for different microwave pow-\nersin Table I. Forthese calculations MSis assumedto be\nconstant since ∆ Tleads to variations of only about 1%\nwhich cannotexplainthe observedbehavior. It shouldbe\nemphasizedthatourcalculatedheat-inducedspincurrent\ndensity per 1◦C is one to two orders of magnitude higher\nthan those generated by the SHE for the maximal DC\nvoltage pulses used in Ref.6ofU= 8 V (JS= 108A/m2,\nPt resistance ≈30 Ω, DC pulse length 300 ns, repetition\nrate 10 ms).\nTaking the large magnitude of the observed heat-\ninduced STT (linewidth change about 6%) for the com-\nparably small temperature difference across the actual\nYIG/Pt interface (less than 0.1◦C) into account, we\nmight consider influences on ∆ Hby other effects: (1)\nThechangeinelectricresistanceofthePtlayerduetothe\napplied temperature difference (less than approximately\n0.1%) cannot be the origin of the observed linewidth\nchange. (2)Inthepresentexperiment, phononspenetrat-\ning the entire sample stack (including the 500 µm thick\nGGG layer) are the main heat carriers.4,16Consequently,\nthey are the major cause for the thermally-induced spin\ncurrent.\nIn conclusion, the heat-induced damping modification\nin YIG/Pthetero-structureshasbeen shown. Themodu-\nlation ofthe relaxationcoefficient has been demonstrated\nby spin pumping as well as by microwave techniques.\nBoth techniques qualitatively show the same behavior.\nBesides that, our findings demonstrate, that every spin\npumping experiment, in which the coherent magnetiza-\ntion precession is driven by a microwave source, is ac-\ncompanied by heating. We have introduced a method\nto identify spin pumping from coherent magnons and\nSSE contribution from incoherent magnons to the ISHE\nvoltage, resulting in an increase of the ISHE voltage of\naround 0.1%. 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Lifshitz, Physikalische Zeitschrift d er\nSowjetunion 8, 153 (1935).\n22T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).\n23S. Takahashi and S. Maekawa, J. Magn. Magn. Mater. 310, 2067\n(2007)." }, { "title": "1003.2346v1.Damping_of_MHD_turbulence_in_partially_ionized_gas_and_the_observed_difference_of_velocities_of_neutrals_and_ions.pdf", "content": "arXiv:1003.2346v1 [astro-ph.GA] 11 Mar 2010Damping of MHD turbulence in partially ionized gas and the\nobserved difference of velocities of neutrals and ions\nD. Falceta-Gon¸ calves1, A. Lazarian2& M. Houde3\nABSTRACT\nTheoretical and observational studies on the turbulence of the in terstellar\nmedium developed fast in the past decades. The theory of superso nic magne-\ntized turbulence, as well as the understanding of projection effec ts of observed\nquantities, are still in progress. In this work we explore the charac terization\nof the turbulent cascade and its damping from observational spec tral line pro-\nfiles. We address the difference of ion and neutral velocities by clarif ying the\nnature of the turbulence damping in the partially ionized. We provide t heoret-\nical arguments in favor of the explanation of the larger Doppler bro adening of\nlines arising from neutral species compared to ions as arising from th e turbulence\ndamping of ions at larger scales. Also, we compute a number of MHD nu merical\nsimulations for different turbulent regimes and explicit turbulent dam ping, and\ncompare both the 3-dimensional distributions of velocity and the sy nthetic line\nprofile distributions. Fromthe numerical simulations, we place const raints on the\nprecision with which one can measure the 3D dispersion depending on t he tur-\nbulence sonic Mach number. We show that no universal correspond ence between\nthe 3D velocity dispersions measured in the turbulent volume and minim a of the\n2D velocity dispersions available through observations exist. For ins tance, for\nsubsonic turbulence the correspondence is poor at scales much sm aller than the\nturbulence injection scale, while for supersonic turbulence the cor respondence is\npoor for the scales comparable with the injection scale. We provide a physical\nexplanation of the existence of such a 2D-3D correspondence and discuss the\nuncertainties in evaluating the damping scale of ions that can be obta ined from\nobservations. However, we show that the statistics of velocity dis persion from\n1N´ ucleo de Astrof´ ısica Te´ orica, Universidade Cruzeiro do Sul - Ru a Galv˜ ao Bueno 868, CEP 01506-000,\nS˜ ao Paulo, Brazil\nemail: diego.goncalves@cruzeirodosul.edu.br\n2Astronomy Department, University of Wisconsin, Madison, 475 N. C harter St., WI 53711, USA\n3Department of Physics and Astronomy, the University of Western Ontario, London, Ontario, Canada,\nN6A 3K7– 2 –\nobserved line profiles can provide the spectral index and the energ y transfer rate\nof turbulence. Also, comparing two similar simulations with different vis cous\ncoefficients it was possible to constrain the turbulent cut-off scale. This may\nespecially prove useful since it is believed that ambipolar diffusion may b e one\nof the dominant dissipative mechanism in star-forming regions. In th is case, the\ndetermination of the ambipolar diffusion scale may be used as a complem entary\nmethod for the determination of magnetic field intensity in collapsing c ores. We\ndiscuss the implications of our findings in terms of a new approach to m agnetic\nfield measurement proposed by Li & Houde (2008).\nSubject headings: ISM: magnetic fields, ISM: kinematics and dynamics, tech-\nniques: radial velocities, methods: numerical, statistical\n1. Introduction\nThe interstellar medium (ISM) is known to be composed by a multi-phas e, turbulent\nand magnetized gas (see Brunt & Heyer 2002, Elmegreen & Scalo 200 4, Crutcher 2004,\nMcKee & Ostriker 2007). However, the relative importance of turb ulence and the magnetic\nfield in the ISM dynamics and in the formation of structures is still a ma tter of debate. More\nspecifically, typical molecular clouds present densities in the range o f 102−105cm−3, sizes\nL∼0.1−100pc, temperature T∼10−20 K, and lifetimes that are larger than the Jeans\ngravitational collapse timescale. The role of the magnetic field in prev enting the collapse\nis hotly debated in the literature (see Fiedge & Pudritz 2000, Falceta -Gon¸ calves, de Juli &\nJatenco-Pereira 2003, MacLow & Klessen 2004). Magnetic field can be removed from clouds\nin the presence of ambipolar diffusion arising from the differential drif t of neutrals and ions\n(Mestel &Spitzer 1986, Shu 1983)andreconnection diffusion which arisesfromfastmagnetic\nreconnection of turbulent magnetic field (Lazarian 2005). Nevert heless, the role of magnetic\nfields in the dynamics of ISM is difficult to underestimate.\nWe feel that a lot of the unresolved issues in the theory of star for mation are in part due\nto the fact that the amount of information on magnetic fields obtain able through presently\nused techniques is very limited. For molecular cloud the major ways of obtaining information\nabout magnetic fields amount to Zeeman broadening of spectral line s, which provides mea-\nsures of the field strength along the line of sight (see Crutcher 199 9) and the Chandrashekar-\nFermi(CF)method, whichuses thestatistics ofpolarizationvecto rs toprovidetheamplitude\nof the plane of the sky component of the field (see Hildebrand 2000) . However, since Zeeman\nmeasurements are restricted to rather strong magnetic fields (d ue to current observational\nsensitity) and therefore the measurements are restricted to de nse clouds and new measure-– 3 –\nments require a lot of observational time. At the same time, the CF m ethod relies on all\ngrains being perfectly aligned, which is known not to be the case in mole cular clouds (see\nLazarian 2007 for a review). The CR technique is also known to syste matically overestimate\nthe field intensity (Houde et al. 2009, Hildebrand et al. 2009), and to poorly map the mag-\nnetic field topology for super-Alfvenic turbulence (see Falceta-Go n¸ calves, Lazarian & Kowal\n2008).\nThe difficulty of the traditional techniques call for new approaches in measuring as-\ntrophysical magnetic fields. Recently, a number of such technique s has been proposed. For\ninstance, Yan&Lazarian(2006, 2007, 2008)discussed using ther adiativealignment ofatoms\nand ions having fine or hyperfine split of the ground of metastable lev els. The technique\nis based on the successful alignment of atoms in the laboratory con ditions, but it requires\nenvironments where radiative pumping dominates the collisional de-e xcitation of the levels.\nAnother new approach which we dwell upon in this paper is based on th e comparison\nof the ion-neutral spectral lines. Houde et al. (2000a, 2000b) ide ntified the differences of\nthe width of the lines of neutral atoms and ions as arising from their d ifferential interaction\nwith magnetic fields. It was assumed that because ions are forced in to gyromagnetic motions\nabout magnetic field lines that their spectral line profiles would thus r eveal the imprint of\nthe magnetic field on their dynamics.\nIn particular, as observations of HCN and HCO+in molecular clouds revealed signifi-\ncantly and systematically narrower ion lines, Houde et al. (2000a) pr oposed a simple expla-\nnation for these observations. The model was solely based on the s trong Lorentz interaction\nbetween the ion and the magnetic field lines, but also required the pre sence of turbulent mo-\ntions in the gas. More precisely, it was found that the observations of the narrower HCO+\nlines when compared to that from the coexistent HCN species could p otentially be explained\nif neutral particles stream pass magnetic field lines with the entraine d ions. Such a picture\ncould be a particular manifestation of the ambipolar diffusion phenome non.\nAlthough this model was successful in explaining the differences bet ween the velocities\nof ions and neutrals, the quantitative description of the model of d rift was oversimplified.\nFor example, it was neither possible to infer anything about the stre ngth of the magnetic\nfield nor was the ”amount” of ambipolar diffusion, which is at the root o f the observable\neffect described by the model, quantifiable in any obvious manner. Th e main reason for\nthese shortcomings resides in the way that turbulence and its inter play with the magnetic\nfield were treated in the analysis of Houde et al. (2000a); a more com plete and powerful\nmodel was required. The next step in the study of magnetized turb ulence and ambipolar\ndiffusion through the comparison of the coexistent ion/neutral sp ectral lines was taken by Li\n& Houde (2008) where a model of turbulence damping in partially ionize d gas was employed.– 4 –\nAssumption that the main damping mechanism is associated with ambipo lar diffusion, they\ndeduced proposed a way for evaluating the strength of the plane- of-the-sky component of\nthe magnetic field in molecular clouds.\nThemainideaisthat,inamagneticallydominatedscenario, cloudcollaps eandmagnetic\nenergy removal may be accelerated due to ambipolar diffusion of ions and neutral particles.\nAlthough as gravity becomes dominant the collapsing cloud continuou sly drags material to\nits core including the ions, which are frozen to the magnetic field lines, magnetic pressure\nslows down their infall, but not that of the neutrals. At this stage mo st of the matter, in\nneutral phase, continues to decouple from the ionic fluid and the fie ld lines leading to the\ndiffusion of magnetic energy and the collapse may develop further. T his ion-neutral drift,\nexcited by the ambipolar diffusion, is also responsible for damping the io nturbulent motions.\nThe increase in the net viscosity of the flow provides a cut-off in the t urbulent cells with\nturnover timescales lower than the period of collisions τi,n(see Lazarian, Vishniac & Cho\n[2004] for detailed review).\nSince the turbulent cascade is dramatically changed by the decouplin g of the ion and\nneutral fluids the observed velocity dispersion could reveal much o f the physics of collapse\nlength scales. The interpretation and reliability of this technique, ho wever, still need to\nbe corroborated with more detailed theoretical analysis, as well as numerical simulations of\nmagnetized turbulence.\nFor the past decade, because of its complicated, fully non-linear an d time-dependent na-\nture, magnetized turbulence has been mostly studied by numerical simulations (see Ostriker,\nStone & Gammie 2001, Cho & Lazarian 2005, Kowal, Lazarian & Beresn iak 2007). Sim-\nulations can uniquely provide the three-dimensional structure for the density, velocity and\nmagnetic fields, as well as two dimensional maps that can be compare d to observations (e.g.\ncolumn density, line profiles, polarization maps). Therefore, direct comparison of observed\nand synthetic maps may help reveal the magnetic topology and veloc ity structure.\nIn this paper we re-examine the assumptions made in this model and t est some of these\nassumptions using the MHD numerical simulations. In particular, we p rovide a number\nof numerical simulations of MHD turbulent flows, with different sonic a nd Alfvenic Mach\nnumbers. In §2, we describe the NIDR technique for the determination of the dam ping scales\nand magnetic field intensity from dispersion of velocities and the main t heoretical aspects\nof turbulence in partially ionized gases. In §3, we describe the numerical simulations and\npresent the results and the statistical analysis of the data. In §4, we discuss the systematic\nerrors intrinsic to the procedures involved, followed by the discuss ion of the results and\nsummary, in §5.– 5 –\n2. Turbulence in Partially Ionized Gas\n2.1. Challenge of interstellar turbulence\nIn 1941, Kolmogorov proposed the well-known theory for energy c ascade in incom-\npressible fluids. Under Kolmogorov’s approximation, turbulence evo lves from the largest to\nsmaller scales, up to the dissipation scales, as follows. Within the so-c alled inertial range,\ni.e the range of scales large enough for dissipation to be negligible but s till smaller than the\ninjection scales, the energy spectrum may be well described by,\nP(k)∼˙ǫα−1k−α, (1)\nwhere ˙ǫis the energy transfer rate between scales and α∼5/3. In this approximation,\nwithin the inertial range, the energy transfer rate is assumed to b e constant for all scales.\nTherefore, integrating Eq. 3 over kforα= 5/3, we obtain,\nσ2(k)∝/parenleftbigg3˙ǫ2/3\n2/parenrightbigg\nk−2/3. (2)\nHowever, reality is far more complicated. First, the ISM is threaded by magnetic fields,\nwhich may be strong enough to play a role on the dynamics of eddies an d change the\nscaling relations. Second, observations suggest that the ISM is, a t large scales, is highly\ncompressible. Third, many phases of the ISM (see Draine & Lazarian 1999 for typical\nparameters) are partially ionized.\nAttempts to include magnetic fields in the picture of turbulence includ e works by Irosh-\nnikov (1964) and Kraichnam (1965), which were done assuming that magnetized turbulence\nstays isotropic. Later studies proved that magnetic field introduc es anisotropy into turbu-\nlence (Shebalin, Matthaeus, & Montgomery 1983, Higdon 1984, Zan k & Matthaeus 1992,\nsee also book by Biskamp 2003).\nGoldreich & Sridhar (1995, henceforth GS95) proposed a model fo r magnetic incom-\npressible turbulence1based on the anisotropies in scaling relations, as eddies would evolve\ndifferently in directions parallel and perpendicular to the field lines: VA/Λ/bardbl∼vλ/λ, where\n1In the original treatment of GS95 the description of turbulence is lim ited to a situation of the velocity\nof injection at the injection scale VLbeing equal to VA. The generalization of the scalings when VL< VA\ncan be found in Lazarian & Vishniac (1999). The generalization for th eVL> VAis also straightforward (see\nLazarian 2006).– 6 –\nΛ/bardblis theparallel scale ofthe eddy and λis its perpendicular scale. These scales aremeasured\nin respect to the local2magnetic field. Combining this to the assumption of self-similarity\nin energy transfer rate, we get a Kolmogorov-like spectrum for pe rpendicular motions with\nα= 5/3 and, most importantly, the anisotropy in the eddies scales as Λ /bardbl∝λ2/3.\nIn spite of the intensive recent work on the incompressible turbulen ce (see Boldyrev\n2005, 2006, Beresnyak & Lazarian 2006, 2009), we feel that the GS95 is the model that can\nguide us in the research in the absence of a better alternative. The generalization of the\nGS95 for compressible motions are available (Lithwick & Goldreich 2001 , Cho & Lazarian\n2002, 2003) and they consider scalings of the fast and slow MHD mod es.\n2.2. Turbulence damping in partially ionized gas\nAs stated before, the turbulent cascade is expected to devel op down to\nscales where dissipation processes become dominant. The di ssipation scales are\nassociated to the viscous damping, which is responsible for the transfer of kinetic\ninto thermal energy of any eddy smaller than the viscous cuto ff scale.\nIn the ISM, e.g. in cold clouds, the gas is partially ionized a nd the coupling\nbetween neutrals, ions and magnetic fields gives rise to inte resting processes. As\nfar as damping is concerned one of the most interesting is the energy dissipation\nas the motions of ions and neutral particles decouple. While the issue of turbu-\nlence dissipation has been discussed extensively in the lit erature (see Minter &\nSpangler 1997), a generalization of the GS95 model of turbul ence for the case\nof the partially ionized gas was presented in Lazarian, Vish niac & Cho (2004)\n(LVC04). In their model, if the eddy turnover time ( τ) gets of order of the\nion-neutral collision rate ( t−1\nin) two fluids are strongly coupled. In this situation\na cascade cut-off is present.\nIn a strongly coupled fluid, using the scaling relation for th e inertial range\nvdamp∼Uinj(L−1\ninjldamp)1/3, the damping scale is given by (LVC04),\nldamp∼λmfp/parenleftbiggcn\nvl/parenrightbigg/parenleftbiggVA\nvl/parenrightbigg1/3\nfn, (3)\n2The latter issue does not formally allow to describe turbulence in the F ourier space, as the latter calls\nfor the description in respect to the global magnetic field.– 7 –\nwherefnis the neutral fraction, λmfpandcnthe mean free path and sound speed\nfor the neutrals, respectively, VAis the Alfv´ en speed, and subscript “inj” refers\nto the injection scale. Since the Alfvn speed depends on the m agnetic field,\nVA=B(4πρ)−1/2, Eq.(3) is rewritten as:\nB∼/parenleftbiggldamp\nλmfp/parenrightbigg3/parenleftbiggvl\ncn/parenrightbigg3\n(4πρ)1/2f−3\nnvl. (4)\nTherefore, for a given molecular cloud, if the decoupling of ions and neutrals\nis the main process responsible for the ion turbulence dampi ng Eq.(4) may be\na complementary estimation of B. The main advantage of this method is that\nBis the total magnetic field and not a component, parallel or pe rpendicular to\nthe LOS, as respectively obtained from Zeeman or CF-method f rom polarization\nmaps.\n2.3. Approach by Li & Houde 2008\nFrom the perspective of the turbulence above we can discuss the m odel adopted by Li &\nHoude (2008) for their study. The authors considered that damp ing of ion motions happen\nearlier than those by neutrals at sufficiently small scales. At large sc ales, ions and neutrals\nare well coupled through flux freezing and their power spectra sho uld be similar. At small\nscales the ion turbulence damps while the turbulence of neutral par ticles continues cascading\nto smaller scales. This difference may be detected in the velocity dispe rsions (σ) obtained\nfrom the integration of the velocity power spectrum over the wave numberk,\nσ2(k)∝/integraldisplay∞\nkP(k′)dk′\n∼bk−n, (5)\nconsidering P(k) a power-law spectrum function. Since the turbulence of ions is dam ped at\nthe diffusion/dissipation scale ( LD), while the turbulence of neutral particles may develop\nup to higher wavenumbers we may consider that the ions and neutra l particles present the\nsame distribution of velocities (well coupled) for L > L D. In this sense, the dispersion of\nneutral particles may be written as,\nσ2\nn(k)∝/integraldisplay∞\nkP(k′)dk′– 8 –\n=/integraldisplaykD\nkP(k′)dk′+/integraldisplay∞\nkDP(k′)dk′\n∼σ2\ni(k)+bk−n\nD, (6)\nwherekD≈L−1\nD. Eq.(5) may be directly compared to Eq. 2. In this case, we would obt ain\nn∼2/3 andbis related to the energy transfer rate ˙ ǫ. Therefore, once the fitting parameters\nof Eq. 1 are obtained from the observational data, it is possible to o btain the cascading\nconstants ˙ ǫandα.\nWith the dispersion of velocities for both ions and neutrals at differen t scaleskit is\npossible to calculate the damping scale kD. From Eq.(5), the dispersion of neutral particles\nprovides bandnconstants and, by combining neutral and ion dispersions, it is possib le to\ngetkD(Eq.6), i.e. the damping scale.\nFinally, as proposed by Li & Houde (2008) in a different context, it is p ossible to to\nevaluatemagneticfieldstrengthbyEq.(4). Wefeelthattheproce dureofmagneticfieldstudy\nrequires a separate discussion, due to its complexity, but in what fo llows we concentrate on\nthe interesting facts of observational determining of the charac teristics of turbulence and its\ndamping that are employed in the technique by Li & Houde (2008).\n2.4. Observational perspective\nEqs. (1) and (2) are based on the dispersion of a three-dimensiona l velocity field, i.e.\nsubvolumes with dimensions k−3. Observational maps of line profiles, on the other hand,\nprovide measurements of the velocity field integrated along the line o f sight (LOS) within the\nareaofthebeam, i.e. atotalvolume of k−2k−1\nmin(withk= 1/landkmin= 1/L, asLrepresents\nthe total depth of the structure observed - typically larger than l). Also, velocity dispersions\nare obtained from spectral line profiles, which are strongly depend ent on the column density,\ni.e. the distribution of matter along the LOS. These factors make th e comparison between\nobserved lines and theoretical distribution of velocity fields a hard t ask.\nFortunately, 3-dimensional numerical simulations of MHD turbulenc e may be useful in\nproviding both the volumetric properties of the plasma parameters as well as their synthetic\nmeasurements projected along given lines of sight, which may be com pared directly to ob-\nservations, such as the spectral line dispersion. In this sense, ba sed on the simulations of\nOstriker et al. (2001), Li & Houde (2008) stated that the actual dispersion of velocity is,\napproximately, the minimum value of the LOS dispersion, at each beam sizel, obtained in a\nlarge sample of measurements. However, Ostriker et al. (2001) pr esented a single simulation,\nexclusively for supersonic and sub-alfvenic turbulent regime, with lim ited resolution (2563).– 9 –\nThey also did not study increased viscosity, nor the correlation of m inima of the synthetic\ndispersion and the turbulent regimes and the distribution of gas alon g the LOS.\nIn the following sections we will describe the details regarding the est imation of line\ndispersions, but now comparing it with a larger set of numerical simula tions with different\nturbulent regimes and with finer numerical resolution. The idea is to d etermine whether\nthe technique is useful or not, and if there is any limitations with the d ifferent turbulent\nregimes. These tests are mandatory to ensure the applicability of t he NIDR method to ISM\nobservations.\n3. Numerical Simulations\nIn order to test the NIDR model, i.e to verify if the minimum dispersion o f the velocity\nmeasured along the line of sight for a given beamsize l×lis aproximately the actual value\ncalculated for a volume l3, we used a total of 12 3-D MHD numerical simulations, with 5123\nresolution, for 6 different turbulent regimes as described in Table 1, but repeated for viscous\nand inviscid models.\nThe simulations were performed solving the set of ideal MHD isotherm al equations, in\nconservative form, as follows:\n∂ρ\n∂t+∇·(ρv) = 0, (7)\n∂ρv\n∂t+∇·/bracketleftbigg\nρvv+/parenleftbigg\np+B2\n8π/parenrightbigg\nI−1\n4πBB/bracketrightbigg\n=f, (8)\n∂B\n∂t−∇×(v×B) = 0, (9)\n∇·B= 0, (10)\np=c2\nsρ, (11)\nwhereρ,vandpare the plasma density, velocity and pressure, respectively, B=∇×A\nis the magnetic field, Ais the vector potential and f=fturb+fviscrepresents the external\nsource terms, responsible for the turbulence injection and explicit viscosity. The code– 10 –\nsolves the set of MHD equations using a Godunov-type scheme, based on a\nsecond-order-accurate and the non-oscillatory spatial re construction (see Del\nZanna et al. 2003). The shock-capture method is based on the H arten-Lax-van\nLeer (1983) Riemann solver. The magnetic divergence-free i s assured by the\nuse of a constrained transport method for the induction equa tion and the non-\ncentered positioning of the magnetic field variables (see Lo ndrillo & Del Zanna\n2000). The code has been extensively tested and successfull y used in several\nworks (Falceta-Gon¸ calves, Lazarian & Kowal 2008; Le˜ ao et al. 2009; Burkhart\net al. 2009; Kowal et al. 2009; Falceta-Gon¸ calves et al. 201 0).\nThe turbulence is triggered by the injection of solenoidal perturba tions in Fourier space\nof the velocity field. Here, we solve the explicit viscous term as fvisc=−ρν∇2v, whereν\nrepresents the viscous coefficient and is set arbitrarily to simulate t he increased viscosity of\nthe ionic flows due to theambipolar diffusion. We run all the initial condit ions given inTable\n1 for both ν= 0 andν= 10−3, representing the neutral and ion particles fluids, respectively.\nEach simulations is initiated with an uniform density distribution, threa ded by an uniform\nmagnetic field. The simulations were run until the power spectrum is f ully developed. The\nsimulated box boundaries were set as periodic.\nInFig. 1we showtheresulting velocity power spectra offourofour models, representing\nthe four different turbulent regimes, i.e. (sub)supersonic and (su b)super-Alfv´ enic. Solid\nlines represent the non-viscous cases, and the dotted line the visc ous cases. The spectra are\nnormalized by a Kolmogorov power function P∝k−5/3. The inertial range of the scales\nis given by the horizontal part of the spectra. For the inviscid fluid, subsonic turbulence\npresents approximately flat spectra for 2 < k≤50. Supersonic turbulence, on the other\nhand, shows steeper power spectra within this range. Actually, as shown from numerical\nsimulations by Kritsuk et al. (2007) and Kowal & Lazarian (2007), sh ocks in supersonic\nflows are responsible for the filamentation of structures and the in crease in the energy flux\ncascade, resulting in a power spectrum slope ∼ −2.0. Fork >50, the power spectra show\na strong damping of the turbulence, resulting from the numerical v iscosity. For the viscous\nfluid, the damped region is broadened ( kcutoff∼20), due to the stronger viscosity.\n4. Relationship between 2D and 3D dispersion of velocities\n4.1. Comparing the synthetic to the 3-dimensional dispersi on of velocities\nTheoretically, as given by Eq.(6), the difference between the two sp ectra for each run\nmay be obtained from the observed dispersion of velocities. The nex t step then is to obtain– 11 –\nFig. 1.— Velocity power spectra of four of the models described in Tab le 1, separated by\nturbulent regime. The spectra are normalized with a Kolmogorov pow er function ( P∝\nk−5/3). Solid lines represent the non-viscous fluid, and the dotted line the viscous fluid.\nthe dispersion of velocity, for different scales l, from our simulations. However, as explained\npreviously, there are two different methods to obtain this paramet er. One represents the\nactualdispersion, calculatedwithinsubvolumes l3ofthecomputationalbox, whilethesecond\nrepresents the observational measurements and is the dispersio n of the velocity within the\nsubvolume l2L(assuming the gas is optically thin), where Lis the total depth of the box.\nIn order to match our calculations to observational measurement s we will use the density\nweighted velocity v∗=ρv(see Esquivel & Lazarian [2005]), which characterizes the line\nemission intensity proportional to the local density.\nTo obtain the actual dispersion of velocities as a function of the sca lel, we subdivide the\nbox inNvolumes of size l3. Then, we calculate the dispersion of v∗, normalized by the sound\nspeedcs, as the mean value of the local dispersions obtained for each subvo lume. For the\nsyntheticobservationaldispersion, wemustfirstlychooseagiven lineofsight(LOS).Here, for\nthe results shown in Fig. 2 we adopted x-direction. After, we subdiv ide the orthogonal plane\n(y-z) in squares of area l2, representing the beamsize. Finally, we calculate the dispersion of– 12 –\nv∗within each of the volumes l2L, normalized by the sound speed cs, for different values of\nl/L. The results of these calculations for each of the non-viscous mod els, is shown is Fig. 2.\nThe solid line and the triangles represent the average of the actual mean dispersion\nof velocities, while the crosses represent each of the synthetic ob served dispersion within\nl2L. Regarding the synthetic observational measurements, we see t hat increasing lresults\nin a decrease in the dispersion, i.e. range of values, of σv∗x. Also, since we use the density\nweighted velocity v∗, the mass distribution plays an important role in the calulation of\nσv∗. Denser regions will give a higher weight for their own local velocities a nd, therefore, if\nseveral uncorrelated denser regions are intercepted by the LOS ,σv∗will probably be larger.\nTherefore, we may understand the minimum value of σv∗x(l) as the dispersion obtained for\nthe given LOS that intercepts the lowest number of turbulent sub- structures. If a single\nturbulent structure could be observed, then σv∗x(l) would tend to the actual volumetric value\nif the overdense structure depth is ∼l. Also, as you increase l, the number of different\nstructures intercepting the line of sight increases, leading to large r values of the minimum\nobserved dispersion. Onthe other hand, themaximum observed dis persion is directly related\nto the LOS that intercepts most of the different turbulent struct ures. Since this number\nis unlikely to change, the maximum observed dispersion decreases wit hlsimply because\nof the larger number of points for statistics. As l→L,σv∗x(l) gets closer to the actual\nvolumetric dispersion. However, as noted in Fig. 2, the obtained values for l→L\nare slightly different. This is caused by the anisotropy in th e velocity field\nregarding the magnetic field, as the velocity components may be different along\nand perpendicular to B.\nDespite of this effect, the results presented in this work do n ot change when a\ndifferent orientation for the line of sight is chosen. Even th ough not shown in Fig.\n2, we have calculated the dispersion of velocity for LOS in y a nd z-directions.\nThe general trends shown in Fig. 2 are also observed, but a sli ght difference\nappears as l→L, exactly as explained above. This difference is expected to b e\nseen in sub-alfvenic cases because of the anisotropy in the v elocity distribution.\nIt is clear from Fig. 2 that the actual dispersion of velocities and a giv en observational\nline-widthmaybeverydifferent. Li&Houde(2008),basedonOstrik er etal.’swork, assumed\nthatifonechooses, fromalargenumberofobservationalmeasur ements alongdifferentLOS’s,\nthe minimum observational dispersion as the best estimation for the actual dispersion, the\nassociated error is minimized. Considering the broad range of obser ved dispersions obtained\nfrom the simulations for a given l, the minimum value should correspond to the actual\nvelocity dispersion. Actually, from Fig. 2 we see that the validity of su ch statement depends\nonland on the turbulent regime, though as a general result the scaling of the minimum– 13 –\nobserved dispersion follows the actual one.\nFor the subsonic models, the actual dispersion is lower than σv∗x(l), with increasing\ndifference as l/L→0. In these models, we see that for l >0.05Lthere is a convergence\nof the actual dispersion to the synthetic observational measure ments. At these scales the\nminimum value of σv∗x(l) is a good estimate of the velocity dispersion of the turbulence at\nthe given scale l. The difference between both values is less than a factor of 3 for all l’s,\nbeing of a few percent for l >0.03L. In this turbulent regime, mainly at the smaller scales,\nσv∗x(l) overestimates the true dispersion. For larger scales the associa ted error is very small\nand the two quantities give similar values.\nOn the other hand, for the highly supersonic models (M S∼7.0), the minimum value of\nσv∗x(l) underestimates the actual value, at most scales. Under this reg ime, the difference to\ntheactual dispersionisofafactor ∼2−4. Thebest matching between thetwo measurements\noccured for the marginally supersonic cases (M S∼1.5).\nAs a major result we found that the uncertainties associated to th e NIDR technique\ndependonthesonicMachnumberofthesystem, thoughtheassoc iatederrorsarenotextreme\nin any case. We see no major role of the Alfvenic Mach number on this t echnique.\n4.2. Minimum 2D velocity dispersion vs3D statistics\nWe have shown that the synthetic observed dispersion minima repre sent a fair approxi-\nmationfortheactual 3-dimensional dispersion ofvelocities forthe supersonic models, though\nit is slightly overestimated in subsonic cases, and underestimated in h ighly supersonic cases.\nWhat is the physical reason for that?\nOne of the most dramatic differences between subsonic and supers onic turbulence is\nthe mass density distribution. Subsonic turbulence is almost incompr essible, which means\nthat density fluctuations and contrast are small. Supersonic turb ulence, on the other hand,\npresent strong contrast and large fluctuation of density within th e volume. Strong shocks\nplay a major role on the formation of high density contrasts, and is t he main cause of high\ndensity sheets and filamentary structures in simulations. Strong s hocks also modify the\nturbulent energy cascade, opening the possibility for a more efficien t transfer of energy from\nlargeto small scales, resulting inasteeper energyspectra (typica lly withindex ∼ −2, instead\nof the Kolmogorov’s ∼ −5/3).\nObservationally, the determination of the velocity dispersion along t he line of sight is\nalways biased by the density distribution, i.e. a given line profile depend s on both the emis-– 14 –\nsion intensity and the Doppler shifts. For an optically thin gas, the em ission intensity is\ndirectly dependent on the density of the gas. Compared to subson ic, we expect the super-\nsonic turbulence to present a larger contrast of density in struct ures. Because of that, the\nsizes and the number of structures intercepting the line of sight ma y have a deep impact in\nthe determination of the observational dispersion of lines. In orde r to check this hypothesis\nwe calculated the number of structures, and their average sizes, along each of the synthetic\nlines of sight of the cube and determined the correlation with the velo city dispersion. The\nnumber and depths of the structures were obtained by an algor ithm that identi-\nfies peaks in density distributions. Basically, the algorit hm follows three steps.\nFirstly, for each line of sight with beamsize l2, it identifies the maximum peak\nof density and uses a threshold defined as the half value of thi s maximum of\ndensity. Secondly, it removes all cells with densities lowe r than the selected\nthreshold. The remaining data represents the dominant stru ctures within the\ngiven line of sight. Finally, the algorithm follows each lin e of sight detecting\nthe discontinuities, created by the use of a threshold, and c alculates the sizes of\nthese structures.\nIn Fig. 3 we show the correlation between the synthetic velocity disp ersion and the\nnumber of structures intercepted in all LOS, in x-direction, for ou r cube of the Model 3.\nFor all models the result is very similar, though not shown in these plot s. As noticed, the\nminimum dispersion corresponds to the LOSin which there isonly one st ructure intercepted,\ni.e. there is only one source that dominates the emission line. It makes complete sense if this\nsingle source is small in depth and, as a volume, we have the dispersion of a volume ∼l3. As\nwe increase the number of structures intercepted by the LOS, ea ch one contributes with a\ndifferent Doppler shift, resulting in a larger dispersion. However, as we can see from Fig. 3,\nthemaximum dispersion also corresponds toa single structure inthe LOS. The reasonisthat\nwe calculate the threshold for capturing clumps with the FWHM of the highest peak of the\ngiven LOS. If the LOS intercepts “voids”, which aretypically very lar gecompared to clumps,\nthe algorithm results in a number of structures equal onebut its depth, and consequently\nalso its velocity dispersion, is large. Taking into account observation al sensitivity, these low\ndensity regions are irrelevant. Furthermore, this picture is also us eful to understand the\nscaling relation of σ(l). As we increase l, increasing the beamsize, the number of structures\nin the LOS is higher resulting in the increase in velocity dispersion.\nIn Fig. 4, we compare the average number of structures intercep ting theLOS’s andtheir\naverage sizes as a function of the sonic Mach number. It is shown th at both the number\nof sources and their intrinsic sizes are inverselly correlated with the sonic Mach number.\nSubsonic models present lower contrast of densities, which corres ponds to larger overdense\nstructures. For MS= 0.7 we obtained a range of densities 0 .3< ρ/¯ρ <3. As discussed– 15 –\nabove, large structures correspond to integrated volumes ∼l2L, which deviates from the\nactual 3D velocity dispersion. The result is that the minimum synthet ic dispersion obtained\nfrom subsonic turbulence will overestimate the actual value. For s upersonic models, where\nclumps are systematically small, the dispersion minima correspond, fr om Fig. 3, to the\nsingle structures with volumes ∼l3, very close to the actual values. On the other hand, for\nl→L, the result is underestimated. Here, for MS= 1.5 we obtained a range of densities\n0.08< ρ/¯ρ <10, while for MS= 7.0 we obtained 0 .01< ρ/¯ρ <90.\n4.3. Turbulence dissipation scales\nIn order to obtain the dissipation scales of the ionized flows, we must apply Eqs. (5)\nand (6) and 2 to the simulated data. For the inviscid flows, we calculat ebandn(Eq.5)\nusing both the actual and minimum observed velocity dispersions. Th en, we use the data\nfrom the viscous simulation to calculate the difference σ2\nn−σ2\ni. Finally, from Eq.(6), knowing\nσ2\nn−σ2\ni,nandbit is possible to obtain kD. In Fig. 5 we show the data used to compute\nEqs.(5) and (6), i.e. boththe actual (lines) and the minimum observe d (symbols) dispersions\nof velocity for the viscous (stars) and inviscid fluids (squares). Th e fit parameters (Eq.5)\nfor the inviscid simulated data are shown in Table 2, where b acand nacwhere obtained for\nthe actual velocity dispersion and b obsand nobsfor the synthetic observed line widths. We\nsee thatnincreases with the sonic Mach number (M S), as explained below. In Table 2, we\nalso present the damping scale LD, obtained from Eq.(6). In the last column we show the\nratio between the actual and observational scales Lac\nD/Lobs\nD. The ratio of the obtained scales\nshowed a maximum difference of a factor of 5 between the actual va lue of the dispersion and\nthe one obtained from the synthetic observational maps. Also, co mpared to the expected\nvalues obtained visually from the spectra (Fig.1), there is a good cor respondence with Lobs\nD\ngiven in Table 2. This fact shows that the method indeed might be usef ul.\nRegarding the spectral index α(Eq.1), Table 2 gives α∼1.4−1.8 for the subsonic\nmodels, and α∼2.0−2.1 for the supersonic models. These parameters may be directly\ncompared to the values α∼1.7 andα∼2.0 expected for theoretical incompressible and\ncompressible turbulent spectra, respectively. Also, the increase ofbasMSincreases shows\nthat the energy transfer rate is larger for compressible models. F rom Table 2, we obtain an\naveraged value of ˙ ǫ∼0.6 for subsonic and 1.4 for supersonic models, in code units.– 16 –\nTable 1: Description of the simulations - Bis assumed in x-direction\nModelMSMA Description\n1 0.7 0.7 subsonic & sub-Alfvenic\n2 1.5 0.7 supersonic & sub-Alfvenic\n3 7.0 0.7 supersonic & sub-Alfvenic\n4 0.7 7.0 subsonic & super-Alfvenic\n5 1.5 7.0 supersonic & super-Alfvenic\n6 7.0 7.0 supersonic & super-Alfvenic\nsonic Mach number ( MS=∝angb∇acketleftv/cs∝angb∇acket∇ight)\nAlfvenic Mach number ( MA=∝angb∇acketleftv/VA∝angb∇acket∇ight)\nTable 2: Parameters of best fit and damping scales\nactual synthetic maps\nMSMAbacnac Lac\nD bobs nobs Lobs\nDLac\nD/Lobs\nD\n0.7 0.7 0.63(16) 0.39(2) 0.244(21) 1.01(9) 0.80(5) 0.088(13) 2.7\n0.7 7.0 1.38(12) 0.65(3) 0.246(21) 1.00(10) 0.76(4) 0.215(17) 1.1\n7.0 0.7 1.58(21) 1.01(4) 0.162(17) 1.58(13) 1.08(5) 0.318(23) 0.5\n7.0 7.0 1.82(18) 1.05(4) 0.041(6) 2.51(18) 0.95(6) 0.197(25) 0.2\nLac\nDandLobs\nDare placed in terms of the total size of the box, i.e. LD/L.– 17 –\n4.4. Accuracy of the method\nBefore describing a potential technique for future observations , we must stress out that\nthisworkmaybedividedintwoindependentparts. Thefirstisrelated tothecharacterization\nof the energy power spectrum of turbulence, including the energy transfer rate and cut-off\nlength, while the second is based on the use of this damping length whic h is used in the Li\n& Houde (2008) approach to determine the magnetic field.\nAsmentioned before, inmagnetized partiallyionized gases, several different mechanisms\nmay act in order to damp turbulent motions. The determination of th e magnetic field\nintensity from the damping scale is strictly dependent on the assump tion that the turbulent\ncut-off is due to the ambipolar diffusion, i.e. the ion-neutral diffusion s cale is larger than\nthe scales of any other dissipation mechanism. Unfortunately, this could not be tested in\nthe simulations since we did not include explicit two fluid equations to che ck the role of\nambipolar diffusion and other damping mechanisms in the turbulent spe ctra.\nInthesimulations, thecut-offisobtainedviaanexplicitandarbitrary viscouscoefficient.\nThe result is clear in the power spectra (Fig. 1), where the damping le ngth shifts to lower\nvalues of k. From those, we found out that the “observational” velocity dispe rsions for a\ngivenbeamsize l×l, inmost cases, donot coincidewiththeactualdispersion atscale l. Also,\nthe estimation from the minimum value of the observed dispersions ma y also be different\nfrom the expected measure. The associated errors depend on th e sonic Mach number and\non the scale itself, as shown in the previous section. However, the p arameters obtained for\nthe power-law of the synthetic observations and actual 3-dimens ional distributions showed\nto be quite similar. The theory behind this method is very simple, but st ill reasonable, as it\nassumes that the two fluids (ions and neutrals) would present the s ame cascade down to the\ndissipation scale, when they decouple, where the ion turbulence wou ld be sharply damped.\nWe believe that these errors may, eventually, be originated by the s hort inertial range of the\nsimulated data. We see from Fig. 1 that the turbulent damping range is broad, and not\nsharp compared to the inertial range, as assumed in this model. In t he real ISM, the power\nspectrum presents a constant slope within a much broader range, and the errors with real\ndata may be smaller.\n5. Discussion\nIn this work we studied the relationship between the actual 3-D disp ersion of velocities\nand the ones obtained from synthetic observational line profiles, i.e . the density weighted\nline profile widths, along different LOS’s. We study the possibility of the scaling relation of– 18 –\nthe turbulent velocity dispersion being determined from spectral lin e profiles. If correct, the\nobserved line profiles could allow us to determine in details the turbulen ce cascade and the\ndissipation lengths of turbulent eddies in the ISM. However, in order to check the validity\nof this approach, we performed a number of higher resolution turb ulent MHD simulations\nunder different turbulent regimes, i.e. for different sonic and Alfven ic Mach numbers, and\nfor different viscosity coefficients.\nBased on the simulations, we showed that the synthetic observed lin e width ( σv) is\nrelated to the number of dense structures intercepted by the LO S. Therefore, the actual\ndispersion at scale ltends to be similar to the line width obtained by a LOS within a\nbeam size l×lintercepting a single dense structure. It means that, the minimum o bserved\ndispersion may be the best estimative for the actual dispersion of v elocities, if a large number\nof LOS is considered, as assumed in the theoretical model.\nMoreover, by adopting a power-law for the spectrum function of σv, we were able to\nestimate the spectral index nand constant b(Eq. 5), given in Table 2. We see a good\ncorrespondence between the parameters obtained for the 3-D d ispersions and synthetic line\nprofiles. Furthermore, since nis associated with the turbulent spectral index α, andbwith\nthe energy tranfer rate between scales, line profiles may be usefu l in characterizing the\nturbulent cascade.\nAlso, we showed that the models under similar turbulent regime but wit h different\nviscosities will result in different dispersions of velocity, on both 3-D a nd synthetic line\nprofile measurements. The difference of σvfor the inviscid and viscous models, associated\nwith the parameters nandbpreviously obtained, gives an estimative of the damping scale\nLDof the viscous model (Eq. 6). The ratio between the damping scales obtained from the\n3-D and synthetic profile dispersions vary only by a factor ≤5.\nA good estimate of the cut-off scale of the ISM turbulence may bring light to much of\ntheuncertainties aboutthemechanisms thatareresponsible fort hedamping oftheturbulent\neddies. As we showed, if ambipolar diffusion is the main phenomenon res ponsible for the\ndissipation of turbulence, then it is possible to provide another meth od for the determination\nof magnetic field strength in dense cores, besides Zeeman splitting.\nWe believe that, i - the numerical resolution used in our models, and ii - the single\nfluid aproximation, with different viscosities to simulate neutrals and io ns separately, are the\nmain limitations of this work. The ISM may present more than 5 decade s of inertial range\nin its power spectrum while numerical simulations are, at best, limited t o 2-3. On the other\nhand, since the validity of the theoretical approximation presente d in this work depends on\nthe broadness of the inertial range, we expect this model to be ev en more accurate for finer– 19 –\nresolution, or for the ISM itself. Under the single fluid aproximation m ade in this work we\nfail in correlating the ambipolar diffusion with increased viscosity and, therefore it is not\npossible to test the estimation of Bfrom the damping scales as proposed above, though\nthe results related to the damping length and energy transfer rat es remain unchanged. In\na two-fluid simulation this is more likely to be fulfilled. We are currently imp lementing the\ntwo-fluid set of equations in the code, and will test this hypothesis in the future.\n6. Summary\nIn this work we presented an extensive analysis of the applicability of the NIDR method\nfor the determination of the turbulence damping scales and the mag netic field intensity, if\nambipolar diffusion is present, based on numerical simulations of visco us MHD turbulence.\nWe performed simulations with different characteristic sonic and Alfv enic Mach numbers,\nand different explicit viscous coefficients to account for the physica l damping mechanisms.\nAs main results we showed that:\n•the correspondence between the synthetic observational dispe rsion of velocities (i.e.\nfrom the 2D oserved maps) and the actual 3-dimensional dispersio n of velocities de-\npends on the turbulent regime;\n•for subsonic turbulence, the minimum inferred dispersion tends to o verestimate the\nactualdispersionofvelocitiesforsmallscales( l << L), butpresented goodconvergence\nat large scales ( l→L);\n•for supersonic turbulence, on the other hand, there is a converg ence at small scales\n(l << L), but the minimum inferred dispersion tends to underestimate the a ctual\ndispersion of velocities at large scales ( l→L);\n•even though not precisely matching, the actual velocity and the min imum velocity\ndispersion fromspectral lines were well fittedby a power-lawdistrib ution. Weobtained\nsimilar slopes and linear coefficients for both measurements, with α∼ −1.7 and−2.0\nfor subsonic and supersonic cases, respectivelly, as expected th eoretically;\n•the damping scales obtained from the fit for the both cases were sim ilar. The difference\nbetween the scales obtained from the two fits was less than a facto r of 5 for all models,\nindicating that the method may be robust and used for observation al data;\nThe work presented in this paper tests some of the key assumption s important process\nby technique of Li & Houde (2008). Evidently, more work is still requ ired in order to test the– 20 –\nfull range of applicability of the method (e.g., test cases of both mag netically and neutral\ndriven turbulence). The aforementioned implementation of two-flu id numerical simulations\nto better mimic ambipolar diffusion will be an important step in that direc tion.\nD.F.G. thank the financial support of the Brazilian agencies FAPESP ( No. 2009/10102-\n0) and CNPq (470159/2008-1), and the Center for Magnetic Self- Organization in Astrophys-\nical and Laboratory Plasmas (CMSO). A.L. acknowledges NSF grant AST 0808118 and the\nCMSO. M.H.’s research is funded through the NSERC Discovery Grant , Canada Research\nChair, Canada Foundation for Innovation, Ontario Innovation Tru st, and Western’s Aca-\ndemic Development Fund programs. 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Each triangle correspond to all LOS’s calculated from a cube\nof model 3.\n \n0246810 12 14 \n0 1 2 3 4 5 6 7 8\nSonic Mach number Avergage number of structures Parallel - visc 0.0 \nPerpendicular - visc 0.0 \nParallel - visc 1e-3 \nPerpendicular - visc 1e-3 \n010 20 30 40 50 60 \n0 1 2 3 4 5 6 7 8\nSonic Mach number Average size of structures Parallel - visc 0.0 \nPerpendicular - visc 0.0 \nParallel - visc 1e-3 \nPerpendicular - visc 1e-3 \nFig. 4.— Average number of structures within all LOS’s computed for all models as a\nfunction of the sonic Mach number.– 25 –\nFig. 5.— Actual (lines) and the minimum observed (symbols) dispersion s of velocity for the\nviscous (stars) and inviscid fluids (squares)." }, { "title": "2403.12615v1.Polarization_Dynamics_in_Paramagnet_of_Charged_Quark_Gluon_Plasma.pdf", "content": "Polarization Dynamics in Paramagnet of Charged\nQuark-Gluon Plasma\nLihua Dong∗1and Shu Lin†1\n1School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China\nMarch 20, 2024\nAbstract\nIt is commonly understood that the strong magnetic field produced in heavy ion collisions\nis short-lived. The electric conductivity of the quark-gluon plasma is unable to significantly\nextend the life time of magnetic field. We propose an alternative scenario to achieve this: with\nfinite baryon density and spin polarization by the initial magnetic field, the quark-gluon plasma\nbehaves as a paramagnet, which may continue to polarize quark after fading of initial magnetic\nfield. We confirm this picture by calculations in both quantum electrodynamics and quantum\nchromodynamics. In the former case, we find a splitting in the damping rates of probe fermion\nwith opposite spin component along the magnetic field with the splitting parametrically small\nthan the average damping rate. In the latter case, we find a similar splitting in the damping\nrates of probe quark with opposite spin components along the magnetic field. The splitting is\nparametrically comparable to the average damping rate, providing an efficient way of polarizing\nstrange quarks by the quark-gluon plasma paramagnet consisting of light quarks.\n∗donglh6@mail2.sysu.edu.cn\n†linshu8@mail.sysu.edu.cn\n1arXiv:2403.12615v1 [hep-ph] 19 Mar 20241 Introduction\nThe observations of spin polarization in Λ-hyperon in heavy ion collision experiments have revealed\nquark-gluon plasma (QGP) as spin polarized matter [1]. The polarization is attributed to vorticity\nof QGP coming from initial orbital angular momentum in off-central collisions [2]. Theories based\non spin-vorticity coupling have been developed in the past few years [3, 4, 5, 6, 7], giving satisfactory\nexplanation of global spin polarization [8, 9, 10, 11, 12]. However, the spin-vorticity coupling alone\npredicts an equal polarization for both Λ and anti-Λ, while experiments have found splitting of\npolarizations for Λ and anti-Λ, with the splitting more prominent at low energy collisions. Different\nmechanisms have been proposed to understand the splitting including spin-magnetic coupling [13,\n14, 15], mean-field effect [16], direct flow effect [17], helicity vortical effect [18] etc.\nWhile the mechanism of spin-magnetic coupling gives the correct sign of polarization splitting,\nit is generally expected that it cannot provide sufficient magnitude because the lifetime of the\nmagnetic field is short so that the remaining magnetic field at freezeout may be too weak. Indeed,\nrecent studies suggest magnetic field alone cannot explain the splitting at low energy [14, 15].\nThe evolution of magnetic field has been studied using evolution of in-medium electromagnetic\nfield [19, 20]. In order to have long life time for magnetic field, one needs to have large electric\nconductivity for QGP medium, which is not favored by lattice studies [21, 22, 23, 24]. Anisotropic\nconductivity in magnetized QGP has been considered in different approaches including lattice [25],\nholography [26, 27] and kinetic theories [28, 29, 30, 31, 32, 33, 34, 35]. However, the situation does\nnot improve significantly at phenomenologically relevant strength of magnetic field. Other methods\nof constraining the strength of magnetic field experimentally have been discussed in [36].\nMost previous studies have treated QGP as spinless fluid, which does not develop magne-\ntization under external magnetic field. Indeed this is true for charge neutral QGP, in which the\nspin polarization due to spin-magnetic coupling cancel among positive and negative charge carriers.\nHowever, the cancellation is incomplete in charged QGP, leading to nonvanishing magnetization.\nThis is most clearly seen in strong magnetic field limit, where the fermionic degrees of freedom are\ndominated by lowest Landau levels (LLL), see [37] for a recent review. The spin polarization of the\nLLL leads to net magnetization of charged QGP1. In particular, positively charged QGP relevant\nfor heavy ion phenomenology corresponds to a paramagnet.\nThe purpose of this paper is to suggest that the paramagnet of charged QGP can play the\nrole of magnetic field in dynamics of spin polarization. We shall propose the following picture: while\nthe magnetic field due to spectators in heavy ion collisions decays quickly, the strong magnetic field\n1QGP produced at low energy collisions has net baryon charge. It is also electrically charged for two-flavor QGP.\n2can convert the charged QGP consisting of light flavors into a paramagnet. The QGP paramagnet\ncontinues to polarize the strange quarks produced at later stage in QGP evolution, eventually giving\nrise to polarization of Λ hyperons [2]. The polarization is realized as a splitting of damping rates\nfor strange quark with opposite spin component along the magnetic field, which dynamically favors\nstrange quarks with negative spin component.\nThe paper is organized as follows: in Sec 2, we review photon self-energy in charged fluid\nconsisting of LLL states, and calculate the resummed photon propagator. We shall find an anti-\nsymmetric component unique to charged fluid, which is essential for polarization dynamics; in Sec 3,\nwe consider a probe fermion in the paramagnet and find a splitting in the damping rates of the\nprobe fermion with opposite spin component along the magnetic field. It provides a mechanism for\npolarizing the probe fermion; in Sec 4, we extend the analysis to probe quark in charged QGP. This\ncase is complicated by self-interaction of gluons, which gives rise to completely different dispersion\nof gluons. Nevertheless, we find the same mechanism exists for probe quark. We also discuss\nimplications for heavy ion phenomenology; Sec 5 is devoted to conclusion and discussion of future\ndirections.\nWe define ϵ0123= +1, Pµ= (p0,p),σµ= (1,σ) and ¯ σµ= (1,−σ).\n2 Photon in paramagnet\nIn this section, we study the dynamics of photon in charged magnetized plasma. The case for charge\nneutral plasma has been studied extensively in literature, see [37, 38, 39] and references therein. We\nshall focus on the difference in charged magnetized plasma. On general ground, charged magnetized\nplasma consisting of spin one half matter is also spin polarized with nonvanishing magnetization. It\nis known that medium with magnetization is gyrotropic [40], which is characterized by polarization\ntensor with purely imaginary off-diagonal components. It leads to splitting of right-handed and\nleft-handed electromagnetic waves. We shall see this is also true with the paramagnet. We will\nfirst present photon self-energy in charged magnetized plasma, which is then used to determined\nthe dispersion of electromagnetic waves. We will also calculate the resummed photon propagator\nto be used in Sec 3.\n2.1 Photon self-energy in charged magnetized plasma\nWe will use the real time formalism of finite temperature field theory in ra-basis [41]. The fields in\nra-basis are related to the counterpart on Schwinger-Keldysh contour by\nAr=1\n2(A1+A2), A a=A1−A2. (1)\n3The correlators in the ra-basis are defined as\nDµν\nra(x) =⟨Aµ\nr(x)Aν\na(0)⟩,\nDµν\nar(x) =⟨Aµ\na(x)Aν\nr(0)⟩,\nDµν\nrr(x) =⟨Aµ\nr(x)Aν\nr(0)⟩,\nDµν\naa(x) =⟨Aµ\na(x)Aν\na(0)⟩. (2)\nThe correlators in the Schwinger-Keldysh basis are given by\nDµν\n11(x) =θ(x0)⟨Aµ(x)Aν(0)⟩+θ(−x0)⟨Aν(0)Aµ(x)⟩,\nDµν\n22(x) =θ(−x0)⟨Aµ(x)Aν(0)⟩+θ(x0)⟨Aν(0)Aµ(x)⟩,\nDµν\n21(x) =⟨Aµ(x)Aν(0)⟩,\nDµν\n12(x) =⟨Aν(0)Aµ(x)⟩, (3)\ncorresponding to time-ordered, anti-time-ordered, greater and lesser correlators respectively. From\n(1), we can relate correlators in the two basis as\nDµν\nra(x) =1\n2(Dµν\n11(x)−Dµν\n22(x)−Dµν\n12(x) +Dµν\n21(x)),\nDµν\nar(x) =1\n2(Dµν\n11(x)−Dµν\n22(x) +Dµν\n12(x)−Dµν\n21(x)),\nDµν\nrr(x) =1\n4(Dµν\n11(x) +Dµν\n22(x) +Dµν\n12(x) +Dµν\n21(x)),\nDµν\naa(x) =Dµν\n11(x) +Dµν\n22(x)−Dµν\n12(x)−Dµν\n21(x). (4)\nUsing the explicit representations in (3) and θ(x0) +θ(−x0) = 1, we easily find\nDµν\nra(x) =θ(x0)⟨[Aµ(x), Aν(0)]⟩ ≡ − iDµν\nR(x),\nDµν\nar(x) =θ(−x0)⟨[Aµ(x), Aν(0)]⟩ ≡ − iDµν\nA(x),\nDµν\nrr(x) =1\n2{Aµ(x), Aν(0)},\nDµν\naa(x) = 0 . (5)\nwith Dµν\nR(x) and Dµν\nA(x) being retarded and advanced correlators respectively.\nIn Schwinger-Keldysh basis, the vertices can be obtained from the interaction terms in the\nLagrangian Jµ\n1(x)A1,µ(x)−Jµ\n2(x)A2,µ(x). We can convert the current density J1/2toJr/awith a\ndefinition parallel to (1) to arrive at the interaction terms Jµ\na(x)Ar,µ(x) +Jµ\nr(x)Aa,µ(x). Specific\nform of current density will be given in the next section. The photon self-energy in ra-basis is\n4simply the correlators of current density defined as follows\nΠµν\nar(x) =⟨Jµ\nr(x)Jν\na(0)⟩,\nΠµν\nra(x) =⟨Jµ\na(x)Jν\nr(0)⟩,\nΠµν\naa(x) =⟨Jµ\nr(x)Jν\nr(0)⟩. (6)\nNote that use the ralabeling from the Afields for Π as is done conventionally. In particular Π rr\ninstead of Π aavanishes identically.\nNow we focus on photon self-energy in charged magnetized plasma. The retarded self-energy\nis defined similar to (5) with A→J. The results for neutral magnetized plasma in LLL approxima-\ntion have been calculated using both field theory [42] and chiral kinetic theory [43]. The inclusion\nof anti-symmetric part in charged magnetized plasma has also been made using field theory [30]\nand chiral kinetic theory [44, 45], with the results in momentum space quoted below\nΠµν\nR=−e3B\n2π2q2\n3uµuν+q2\n0bµbν+q0q3u{µbν}\n(q0+iϵ)2−q2\n3+ie2µ\n2π2\u0010\nq0ϵµνρσ+u[µϵν]λρσqT\nλ\u0011\nuρbσ, (7)\nIn (7) Bis the magnetic field and µis the chemical potential for fermion number. For simplicity,\nwe consider medium consisting of a single species of fermion carrying positive electric charge. uµis\nfluid velocity and bµis the direction of the magnetic field. qµ\nT=bµ(q·b)+qµ−uµ(q·u) corresponds\nto spatial components of qperpendicular to bµ. The first term of (7) is symmetric in indices with\nthe pole coming from the chiral magnetic wave (CMW)) [46] in the LLL approximation. The second\nterm is anti-symmetric and purely imaginary. It comes from the Hall effect arising from the current\nalong the drift velocity in charged plasma [44, 45]. This can be confirmed in field theory [30] and in\nmagnetohydrodynamics [47]. If we work in local rest frame of the plasma, and point the magnetic\ninzdirection so that bµ= (0,0,0,1) when uµ= (1,0,0,0). Both [30] and [47] give Πxy\nR=ine\nBq0for\nqT= 0. In the LLL approximation, we can express the electric charge density nein terms of electric\nchemical potential µe=eµand susceptibility χ=eB\n2π2asne=µeχ=eµeB\n2π2, which agrees with (7).\nThe origin of the anti-symmetric component implies that (7) is valid on a time scale longer than\nthe relaxation time τRsuch that Hall current can establish.\nUsing (5) with A→J, we can determine the following correlator in ra-basis\nΠµν\nar(x) =−iΠµν\nR(x). (8)\n52.2 Electromagnetic wave in magnetized plasma\nWe proceed to find the polarization modes for photon by solving the Maxwell equations in the\nmagnetized plasma. We start with the Maxwell equations in coordinate space\n\u0000\n∂2ηµν−∂µ∂ν\u0001\nAν,r=jµ\nr=−iZ\nd4yΠµν\nar(x, y)Aν,r. (9)\nWorking in momentum space and taking the Coulomb gauge ∇ ·⃗A= 0 in local rest frame of the\nplasma, we can express the Maxwell equation as\nQ2Aµ−q0A0Qµ−Πµν\nRAν= 0. (10)\nThe polarization modes for photon can be obtained from the solutions of q2\n0. For pedagogical\npurpose, we first solve (10) for neutral plasma µ= 0, in which we obtain\nq2\n0=˜B+q2+O(B−1), A i=−A0qi\nTq0\n˜B, A3=A0q2\nTq0\n˜B,\nq2\n0=q2, A 0=A3= 0, qi\nTAi= 0. (11)\nwith ˜B=e3B/2π2andi= 1,2 labeling directions perpendicular to b. The first one is a gapped\nmode and the second one is lightlike2.\nTurning to the charged plasma, we can get three roots of q2\n0, corresponding to three polar-\nization modes of the photon as follows\nq2\n0=˜B+q2,\nq2\n0=1\n2\u0012\n˜µ2+q2\n⊥+ 2q2\n3−q\n4˜µ2q2\n3+\u0000\nq2\n⊥+ ˜µ2\u00012\u0013\n≡x2\n1,\nq2\n0=1\n2\u0012\n˜µ2+q2\n⊥+ 2q2\n3+q\n4˜µ2q2\n3+\u0000\nq2\n⊥+ ˜µ2\u00012\u0013\n≡x2\n2. (12)\nwith ˜ µ=e2µ/2π2andq2\n⊥=q2\n1+q2\n2. The first mode is the same gapped one as the neutral case. The\nsecond and third correspond to the space-like and the time-like low energy modes, respectively. The\norigin of the low energy modes is most clearly seen in the neutral limit where the two modes reduce\ntoq2\n0=q2\n3andq2\n0=q2respectively. The former corresponds to Landau damping, which arises\nfrom energy exchange between photon and LLL states. In the massless limit we consider, Landau\ndamping appears as a pole instead of a cut [48]. The latter corresponds to photon dispersion in\nvacuum. The effect of finite density medium is to shift the two poles. The actual propagating\nmodes are only the first and third ones in (12). One may expect to have three propagating modes\n2In the special case when qT= 0, the first mode disappears and the second mode becomes two degenerate ones,\nas photon does not feel the magnetic field. We are not interested in this trivial case.\n6rather than two due to collective motion in plasma [49]. Note that the self-energy (7) contains no\nexplicit temperature dependence, suggesting the medium is more like a Fermi sea rather than a\nplasma. It follows that the number of propagating modes matches that of the vacuum. We shall\nelaborate on this later.\nLet us take a close look at the low energy modes in the phenomenologically interesting limit\n˜µ≫q:x2\n1≈q2\n3q2\n˜µ2, x2\n2≈˜µ2. (13)\nIf we estimate the relaxation by its value in the absence of magnetic field τR∼1\ne4Tand take\n˜µ∼e2µ∼e2T, we find the mode x2no longer present in the low energy regime set by the Hall\neffect: q0∼τ−1\nR≪˜µ. This leaves only the mode x1in the low energy spectrum. To gain further\ninsights, we plug (12) into (10) to solve for Aµ. In the same limit ˜ µ≫q, we obtain\nq2\n0=x2\n1:A1\nA0=i(q2q+iq1|q3|)\nq2\n⊥q˜µ,A2\nA0=−i(q1q−iq2|q3|)\nq2\n⊥q˜µ,A3\nA0=q3\n|q3|q˜µ. (14)\nThe physical interpretation of this mode is most transparent if we focus on the regime q3≫q⊥,\nthat is a photon propagating almost along the magnetic field. We have thenA1\nA2≃ −ifrom (14).\nThis is analogous to one of the circular polarization in vacuum, but with the dispersion modified\nby the charged medium. This parity breaking mode will play an important role in polarizing probe\nfermions just as a paramagnet polarizing an ordinary metal.\n2.3 Resummed photon propagator\nIn the previous section, we have obtained the photon polarization modes by solving the Maxwell\nequations. These modes contain pole and Landau damping (also a pole in massless limit) contri-\nbutions to the spectral function of photon. In this section, we will derive the resummed photon\npropagator and extract the spectral function, from which we will find both pole and Landau damp-\ning contributions.\nWe start with the following bare photon propagators Dar\nµν(0),Dra\nµν(0)andDrr\nµν(0)in Coulomb\ngauge in thermal equilibrium\nDar\nµν(0)(Q) =i\n(q0−iϵ)2−q2\u0012\nPT\nµν+Q2uµuν\nq2\u0013\n,\nDra\nµν(0)(Q) =i\n(q0+iϵ)2−q2\u0012\nPT\nµν+Q2uµuν\nq2\u0013\n,\nDrr\nµν(0)(Q) = 2 π ϵ(q0)δ(Q2)\u00121\n2+fγ(q0)\u0013\u0012\nPT\nµν+Q2\nq2uµuν\u0013\n. (15)\nThe structures PT\nµνandQ2\nq2uµuνcorrespond to transverse and longitudinal components of the prop-\nagator respectively. The transverse projection operator PT\nµνis defined as PT\nµν=Pµν−PµαPνβQαQβ\n−Q2+(Q·u)2\n7with Pµν=uµuν−ηµνbeing the projection operator orthogonal to fluid velocity. In fluid’s rest\nframe, we have\nPT\n00=PT\n0i=PT\ni0= 0,\nPT\nij=δij−qiqj\nq2. (16)\nUsing the definitions (2), (6) and the couplings Jµ\na(x)Ar,µ(x) +Jµ\nr(x)Aa,µ(x), we may express the\npropagators up to first order in the self-energy as:\n\nDrrDra\nDar0\n\nµν=\nDrr\n(0)Dra\n(0)\nDar\n(0)0\n\nµν−\nDrr\n(0)Dra\n(0)\nDar\n(0)0\n\nµα\n0 Πra\nΠarΠaa\nαβ\nDrr\n(0)Dra\n(0)\nDar\n(0)0\n\nβν.\n(17)\nBy iteration, we deduce the resummed propagators satisfy the following equations\n\nDrrDra\nDar0\n\nµν=\nDrr\n(0)Dra\n(0)\nDar\n(0)0\n\nµν−\nDrr\n(0)Dra\n(0)\nDar\n(0)0\n\nµα\n0 Πra\nΠarΠaa\nαβ\nDrrDra\nDar0\n\nβν.\n(18)\nThe component form of the above reads\nDra\nµν=Dra\nµν(0)−Dra\nµα(0)Παβ\narDra\nβν,\nDar\nµν=Dar\nµν(0)−Dar\nµα(0)Παβ\nraDar\nβν,\nDrr\nµν=Drr\nµν(0)−Dra\nµα(0)Παβ\narDrr\nβν−\u0010\nDrr\nµα(0)Παβ\nra+Dra\nµα(0)Παβ\naa\u0011\nDar\nβν. (19)\nThe resummed propagators can be solved by inverting the following matrix equations\n\u0010\nδαµ+Dra\nαβ(0)Πβµ\nar\u0011\nDra\nµν=Dra\nαν(0),\n\u0010\nδαµ+Dar\nαβ(0)Πβµ\nra\u0011\nDar\nµν=Dar\nαν(0),\n\u0010\nδαµ+Dra\nαρ(0)Πρµ\nar\u0011\nDrr\nµν=\u0010\nDrr\nαν(0)−Drr\nαβ(0)Πβσ\nraDar\nσν−Dra\nαβ(0)Πβσ\naaDar\nσν\u0011\n. (20)\nWe first invert the first two equations to obtain Dra\nµν(Q) and Dar\nµν(Q), and then use the results to\ninvert the last equation to obtain Drr\nµν. Note that our knowledge about the self-energy from the\nLLL approximation should be viewed as leading terms in the limit B→ ∞ . It follows that we\nshould also keep only the leading terms in the resulting resummed propagators, which gives the\nfollowing results\nDra\nµν(Q) =\u00121\n(q0+iϵ)2−x2\n1+1\n(q0+iϵ)2−x2\n2\u0013Aµν(Q) +Sµν(Q)\u0000\nq2\n0−x2\n1\u0001\n+\u0000\nq2\n0−x2\n2\u0001,\nDar\nµν(Q) =Dra\nνµ(−Q),\nDrr\nµν(Q) =−2iπ ϵ(q0) (Sµν(Q) +Aµν(Q))\u00121\n2+fγ(q0)\u0013\u0012δ(q2\n0−x2\n1)\nq2\n0−x2\n2+δ(q2\n0−x2\n2)\nq2\n0−x2\n1\u0013\n. (21)\n8Here AµνandSµνare the anti-symmetric and symmetric tensors, defined respectively as\nAµν(Q) = −q0\nq2˜µ\u0010\nq0u[µϵνλρσ ]qλ\nTuρbσ−q2\n3ϵµνρσuρbσ+q3b[µϵνλρσ ]qλ\nTuρbσ\u0011\n,\nSµν(Q) = i\u0000\n−gµν\u0000\nq2\n0−q2\n3\u0001\n−q2\n3uµuν−q2\n0bµbν−b{µuν}q0q3\u0001\n+i\nq2\u0000\nu{µqν}q3\n0+b{µqν}q2\n0q3\u0001\n+i\nq4qµqν\u0000\nq2q2\n3−q2\n0\u0000\nq2+q2\n3\u0001\u0001\n. (22)\nClearly the low energy modes found in Sec 2.2 are present as poles of Dra\nµν(Q) and Dar\nµν(Q).\nThe gapped mode in Sec 2.2 is invisible after the limit B→ ∞ is taken in the resummed propagator.\nFrom the definition (2), it is easy to show that Drr\nµν(Q) is hermitian. This is indeed satisfied by\nthe corresponding expression in (21) with real symmetric components and purely imaginary anti-\nsymmetric components.\n3 Probe fermion in paramagnet\nWe consider a probe fermion interacting with the medium. We choose an unmagnetized probe\nfermion. This is motivated by heavy ion phenomenology: with the quick decay of the magnetic\nfield, the strange quarks produced at later stage are not spin polarized and can only interact\nwith the medium. We shall consider high density limit ˜ µ≫q. In this case the medium is like\na paramagnet, which is able to polarize probe fermion. We will corroborate the picture with\ncalculations of damping rates of probe fermions. For simplicity, we take the probe fermion to be\nmassless.\n3.1 Resummed fermion propagator and damping rate\nA probe fermion interacting with the medium will have a modified dispersion, with the damping\nrate given by imaginary part of the pole in the resummed retarded propagator. The procedure of\nderiving resummed propagator is similar to Sec 2.3. We start with the bare fermion propagators\ninra-basis.\nSar(0)(P) =i/P\n(p0−iϵ)2−p2,\nSra(0)(P) =i/P\n(p0+iϵ)2−p2,\nSrr(0)(P) =\u00121\n2−fe(p0)\u0013\n2πϵ(p0)/Pδ\u0000\nP2\u0001\n. (23)\nwith febeing the Fermi-Dirac distribution function when the fermion is in equilibrium. For the\nprobe fermion, we set fe= 0. The resummation equation for retarded propagator is analogous to\n9counterpart in (20)\nSra(P) =Sra(0)(P)−Sra(0)(P)Σar(P)Sra(P). (24)\nThe self-energy in (24) is defined by the Fourier transform of the following\nΣar(x) =⟨ηr¯ηa⟩, (25)\nwith η=e/Aψand ¯η=e¯ψ/Abeing the sources coupled to ¯ψandψrespectively. Inverting (24), we\nobtain the following resummed propagator\nSra=i\n/P+iΣar, (26)\nwhere we have dropped the iϵassuming the self-energy Σ aralready shifts the pole of p0from the\nreal axis. Since both medium and probe fermions are chiral, the self-energy also preserves the chiral\nsymmetry with the following decomposition\nΣar=Vµγµ+Aµγ5γµ. (27)\nThe decoupling of left and right-handed components is manifest in chiral representation of Dirac\nmatrices, with the following explicit denominator of (26)\n/P+iΣar=\n(Pµ+iVµ−iAµ)σµ\n(Pµ+iVµ+iAµ) ¯σµ\n. (28)\nThis allows us to treat left and right-handed components separately as\nSR\nra=i\n(Pµ+iVµ−iAµ)σµ=i(Pµ+iVµ−iAµ) ¯σµ\n(P+iV −iA)2,\nSL\nra=i\n(Pµ+iVµ+iAµ) ¯σµ=i(Pµ+iVµ+iAµ)σµ\n(P+iV+iA)2. (29)\nIt is clear that the effect of self-energy is to shift the momenta of left and right-handed components\nrespectively. The coefficient Aencodes the splitting between left and right-handed components. At\nfinite charge density, the medium is spin polarized. We suggest in Sec 2.2 that the Landau damping\nmode is parity-breaking. Thus we expect splitting between left and right-handed components.\nNow we present explicit calculation of the self-energy. Fig. 1 shows one of the self-energy\ndiagrams in ra-basis The corresponding self-energy contribution is given by3\nΣar(P) =e2Zd4Q\n(2π)4γµSra(0)(P−Q)γνDrr\nµν(Q). (30)\nThere is the other diagram from exchanging ra-labeling of photon and probe fermion in the loop\nin Fig. 1. Its contribution is suppressed because the probe fermion is not thermally populated.\n3With our definition (25), the interaction vertex is −einstead of −ie. The factor i2=−1 appears in the\nresummation equation (24).\n10μ νFigure 1: One-loop fermion self-energy Σ ar. Black solid circle represents the resummed photon\npropagator and the thick line indicates the probe fermion. The other diagram can be obtained\nby exchanging the ra-labeling of the photon and probe fermion in the loop. Its contribution is\nsuppressed because the probe fermion is not thermally populated.\n3.2 Damping in paramagnet\nNow we evaluate the self-energy in the high density limit ˜ µ≫q. We have shown in Sec 2.2 that\nonly the Landau damping mode survives in this limit. We then evaluate the integrals with (23) and\n(21) taking contribution from q2\n0=x2\n1only. We calculate separately anti-symmetric and symmetric\ncontributions to be denoted as ΣA\narand ΣS\nar. The anti-symmetric contribution reads\nΣA\nar(P) =e2\n(2π)3Zd4Q ϵ(q0)\n(P−Q)2+iϵ(p0−q0)\u00121\n2+fγ(q0)\u0013\u0012δ(q2\n0−x2\n1)\nq2\n0−x2\n2+δ(q2\n0−x2\n2)\nq2\n0−x2\n1\u0013\n×γµ(/P−/Q)γνAµν(Q), (31)\nWe first deal with γµ(/P−/Q)γνAµν(Q) by using the following relation\nγµγaγν=gµαγν−gµνγα+gανγµ−iϵµανβγ5γβ. (32)\nOnly the last anti-symmetric term contributes when contracted with Aµν, giving\nγµ(/P−/Q)γνAµν(Q) =−i(P−Q)αϵµανβγ5γβAµν(Q) =2i˜µ\nq2\u0000\nq2\n0f1+q0f2\u0001\n, (33)\nwith\nf1=\u0000\np⊥·q⊥−q2\u0001\nγ5γ3−p3γ5q⊥·γ⊥,\nf2=p0q2\n3γ5γ3+p0q3γ5q⊥·γ⊥+q3γ5γ0(q2−p·q), (34)\nbeing the coefficients of even and odd powers of q0. The perpendicular vectors are defined as\np⊥= (p1, p2) and similarly for q⊥andγ⊥.\nWe proceed by making several approximations: firstly the self-energy induces only a small\ncorrection to the dispersion, so for the purpose of finding damping rate of on-shell probe fermion we\n11may set P2= 0; secondly the Landau damping mode is nearly static, allowing us to approximate\n1\n2+fγ(Q)≃T\nq0; thirdly combining the on-shell condition and q0≪q≪p0, we approximate the\ndenominator of fermion propagator as\n1\n(P−Q)2+iϵ(p0−q0)≃1\n21\n(p·q+iϵp0), (35)\ndropping Q2≪2P·Qandq0p0. Then, (31) can be written as\nΣA\nar(P) =−ie2T\n(2π)3˜µZd3q\nq2(p·q+iϵp0)Zdq0\nq0δ\u0012\nq2\n0−q2\n3q2\n˜µ2\u0013\nϵ(q0)\u0000\nq2\n0f1+q0f2\u0001\n. (36)\nWe proceed with the integral of q0first. Since f1andf2are independent of q0, the integral receives\ncontribution from integrand even in q0as\nZdq0\nq0δ\u0012\nq2\n0−q2\n3q2\n˜µ2\u0013\nϵ(q0)\u0000\nq2\n0f1+q0f2\u0001\n=f1Z\ndq0δ\u0012\nq2\n0−q2\n3q2\n˜µ2\u0013\nq0ϵ(q0) =f1. (37)\nThe remaining integrals are evaluated using the residue theorem. The details can be found in\nAppendix. We quote the final results here. To be specific, we take p0>0 to arrive at the following\nresults\nΣA\nar(P) = −ic1γ5γ3+ic2γ5p⊥·γ⊥+c3γ5γ3, (38)\nwith\nc1=e2TqUV\n4π˜µ\u0012\n1−|p3|\np\u0013\n, c2=e2TqUV\n4π˜µ\u0012\n1−|p3|\np\u0013p3\np2\n⊥, c3=e2Tq2\nUV\n8π˜µ|p3|. (39)\nqUVis the ultraviolet cutoff of q⊥.\nThe calculation of the symmetric contribution proceeds similarly. We simply quote the final\nresult, collecting details in Appendix. For p0>0, we have\nΣS\nar=γ0d1+ip⊥·γ⊥d2+iγ3d3, (40)\nwith\nd1=e2T\n4πp0lnqUV\nqIR\np, d 2=e2T\n4πqUV\np2\n⊥\u0012\n1−|p3|\np\u0013\n, d 3=e2T\n4πqUV\npϵ(p3). (41)\nqIRis the infrared (IR) cutoff of q⊥.\nNow we can take Σ ar= ΣA\nar+ ΣS\narand compare with (27) and (29) to obtain damping\nrates of left and right-handed components respectively. We find it more instructive to obtain the\ncontributions to damping rate from ΣA\narand ΣS\narrespectively. In fact, if we keep linear order in\nΣA\nar∼ΣS\nar∼e2, the corresponding shifts of the poles from the vacuum counterpart are additive.\n12The imaginary part of the shift gives the damping rate. We first consider contribution from ΣA\nar.\nUsing (27) and (29), we easily find the poles given by\nL:p0≃p−c2p2\n⊥\np−c1p3\np−ic3p3\np,\nR:p0≃p+c2p2\n⊥\np+c1p3\np+ic3p3\np. (42)\nWe can see ΣA\narcauses the shifts of poles with opposite sign for both real and imaginary parts.\nThe real part corresponds to a chiral shift discussed in [50]. The imaginary part gives the following\ndamping rates\nΓL≃c3p3\np=e2Tq2\nUV\n8π˜µpϵ(p3),\nΓR≃ −c3p3\np=−e2Tq2\nUV\n8π˜µpϵ(p3). (43)\nThe cases with Γ <0 are unstable. These include right-handed component with p3>0 and left-\nhanded component p3<0. The implication is interesting: Due to spin-momentum locking, both\ncases have a positive spin component along the direction of the paramagnet. Interaction with the\nparamagnet tends to polarize the probe fermion by amplifying these modes. In contrast, left-handed\ncomponent with p3<0 and right-handed component with p3>0 have Γ >0. They both have a\nnegative spin component along the direction of the paramagnet and are damped out. This provides\na mechanism to polarize the probe fermion.\nNow we turn to the symmetric contribution. This contribution leads to identical shifts for\nthe left and right-handed components:\np0≃p+p2\n⊥d2\np+p3d3\np−id1=p+e2T\n4πqUV\np−id1. (44)\nThe corresponding damping rate is given by\nΓ =d1=e2T\n4πp0lnqUV\nqIR\np. (45)\nIt depends on both UV and IR cutoffs. While the UV cutoff is set by the boundary of low energy\nregime, the IR cutoff is fictitious. In fact, the logarithmic structure is reminiscent of the IR\ndivergence in damping rate of heavy fermion in thermal plasma [51].\nCombining the contributions from anti-symmetric and symmetric parts, we obtain a slightly\nmodified picture: probe fermion interacting with the medium will generically be damped. This is\nbecause the damping rate from the symmetric contribution is parametrically larger than the coun-\nterpart from the anti-symmetric contribution: d1≫c3|p3|\np. However, with the medium being like a\nparamagnet, modes with positive/negative spin component along the direction of the paramagnet\n13have smaller/larger damping rate, thus interaction tends to polarize the probe fermion. This occurs\nat a time scale t∼∆Γ−1∼˜µp\ne2Tq2\nUV. One may worry that at this time scale, the probe fermion\nhas been damped out completely because of the hierarchy d1≫c3|p3|\np. This can still have physical\nconsequence. If the probe fermion is continuously produced in the medium, the number density can\nmaintain despite of damping by the medium, but the polarization mechanism from the splitting of\ndamping rates always works. We will extend the analysis to QGP case in the next section, where\nwe will see the splitting of damping rates is significantly enhanced and parametrically similar to\nthe average damping rate, making the polarization dynamics more efficient.\n4 Probe quark in paramagnet of QGP\nNow we extend the analysis to probe quark in charged QGP. A new feature in this case is that gluon\nself-energy receives an additional contribution from gluon self-interaction, which is parametrically\nlarger than the counterpart from Hall effect. It follows that the dispersions we obtain from solving\nMaxwell equations no longer apply. We will identify low energy modes by finding the resummed\ngluon propagator and use it to calculate the splitting of damping rates for probe quark.\n4.1 Gluon propagator in charged QGP\nWe follow the procedure in Sec 2.3. The gluon bare propagator in Coulomb gauge is the same as\n(15) except for additional color structures\nDAB,ar\nµν(0)=δABDar\nµν(0),\nDAB,ra\nµν(0)=δABDra\nµν(0),\nDAB,rr\nµν(0)=δABDrr\nµν(0). (46)\nWe have used capital letters for color indices and the color structure is diagonal δAB. The gluon\nself-energy is given by\nΠµν,AB\nR=\u0014\n−g2eB\n2π2q2\n3uµuν+q2\n0bµbν+q0q3u{µbν}\n(q0+iϵ)2−q2\n3+ig2\n2π2µ\n2\u0010\nq0ϵµνρσ+u[µϵν]λρσqT\nλ\u0011\nuρbσ\n−Pµν\nTΠT−Pµν\nLΠL\u0015\nδAB, (47)\nwith Π T/Lbeing the transverse/longitudinal components from gluon loop. The explicit expressions\nin the hard thermal loop (HTL) regime are as follows\nΠT=m2\u0000\nx2+ (1−x2)xQ0(x)\u0001\n,\nΠL=−2m2(x2−1) (1−xQ0(x)), (48)\n14where m2=1\n6Ncg2T2is the thermal mass and Ncis the number of colors. The Legendre function\nQ0is defined as Q0(x) =1\n2ln|x+1\nx−1| −iπ\n2θ(1−x2). The symmetric components of (47) have been\nextensively discussed in [52]. The anti-symmetric component is obtained by a straightforward\ngeneralization of the calculations in [30] for a single species of quark carrying positive electric charge\nqf>0, with µbeing chemical potential for quark number density. The overall factor1\n2comes from\ncolor trace in the fundamental representation tr[ tAtB] =1\n2δAB. The physical interpretation is the\nchromo-Hall effect. Imagine applying a chromo-electric field in color direction Aperpendicular\nto the magnetic field. The quarks carrying both electric charge qfand effective chromo charge ¯ g\nwill develop a drift velocity vA=¯gEA\nqfBwhere the chromo-electric force and ordinary Lorentz force\nreaches a balance. This gives rise to a chromo current along the drift velocity\nJA= ¯gρvA=¯g2EA\nqfBχµ= ¯g2EAµ, (49)\nwhere we have used χ=qfB. To arrive at (47), we need to fix the effective chromo charge. This\nis most easily done in double line basis for color [53], in which the gluon color index is represented\nasA=ijand quark color indices are represented by iandj. The color matrices in fundamental\nrepresentation are given by\ntij\nkl=1√\n2\u0012\nδi\nkδj\nl−1\nNcδijδkl\u0013\n. (50)\nIt is most easily understood in the large Nclimit, in which the color indices of gluons and quarks are\nlocked. Naturally the corresponding quarks lead to chromo current in the same color direction as\nthe chromo-electric field with the effective charge ¯ g=1√\n2g, thus the factor1\n2is perfectly accounted\nfor.\nSince the color structure is trivial in both bare propagator and self-energy, we can simply\nignore it and then use (19) to obtain the resummed gluon propagator. We assume the following\nhierarchy: eB≫ΠT/L∼g2T2≫g2µq. We will first expand to leading order in B−1and then\nexpand to leading order in µ. The resulting resummed propagator contains both symmetric and\nanti-symmetric parts. The symmetric part exists in the absence of µand has been elaborated\nin [52]. This part does not lead to splitting of damping rate so we do not keep track of. The\nanti-symmetric part starts at O(µ), with the following explicit expression (suppressing the color\nstructure)\nDra,A\nµν(Q) =Q2q2\n(Q2−ΠT)\u0000\nq2Q2(q3\n0−q2\n3)−Q2q2\n3ΠT−q2\n0q2\n⊥ΠL\u0001Aµν\n2,\nDrr,A\nµν(Q) = 2 iIm\"\nQ2q2\n(Q2−ΠT)\u0000\nq2Q2(q3\n0−q2\n3)−Q2q2\n3ΠT−q2\n0q2\n⊥ΠL\u0001#\u00121\n2+fg(q0)\u0013Aµν\n2,(51)\n15where fgis the Bose-Einstein distribution for gluon. It is instructive to compare (51) with (21):\nwhile they share the same Lorentz structure, the corresponding spectral structures are entirely\ndifferent. In the photonic case, the spectrum is Landau damping poles and lightlike mode, both\nmodified by density. In the gluonic case, the spectrum contains two poles and Landau damping\ncut. The poles are located at\nQ2−ΠT= 0, q2Q2(q3\n0−q2\n3)−Q2q2\n3ΠT−q2\n0q2\n⊥ΠL= 0. (52)\nNot surprisingly they correspond to the transverse mode and mixed mode in HTL regime in the\nabsence of µ[52]. The location of the cut is at Q2<0, originating from fluctuations of on-shell\ngluons in the medium. Although the anti-symmetric part inherits most spectral features from the\nsymmetric part, there is one difference: the transverse mode and mixed mode are decoupled in the\nsymmetric part, but are coupled in the anti-symmetric part in the form of product in (51).\n4.2 Damping rate of probe quark\nNow we can proceed to calculate the splitting of damping rates from anti-symmetric part of gluon\npropagator. Similar to (30), we have for the quark self-energy\nΣar(P) =N2\nc−1\n2Ncg2Zd4Q\n(2π)4γµSra(0)(P−Q)γνDrr\nµν(Q), (53)\nwhere the overall factor comes from tAtA=N2\nc−1\n2Nc. Using (51) and (23), we obtain the following\nrepresentation\nΣA\nar(P) =N2\nc−1\n2Ncg2\n2Zd4Q\n(2π)4γµ i(/P−/Q)\n(P−Q)2+iϵ(p0−q0)γν\u00121\n2+fg(q0)\u0013\n×2iIm\"\n−Q2q2\n(Q2−ΠT)\u0000\nq2Q2(q3\n0−q2\n3)−Q2q2\n3ΠT−q2\n0q2\n⊥ΠL\u0001#\nAµν. (54)\nThe gamma matrices are evaluated in the same way as before\nγµ(/P−/Q)γνAµν(Q) =2i˜µ\nq2\u0000\nq2\n0f1+q0f2\u0001\n, (55)\nwith f1andf2taking the schematic forms cµγ5γµandcµbeing real functions of PandQ. We make\nthe following observation: the damping rate arises from purely imaginary shift of momentum. This\ncorresponds to real part of the coefficients of γ5γµin ΣA\nar. It is only possible when the iϵprescription\nis invoked in the integral. It amounts to keeping the real part of the following\nRei\n(P−Q)2+iϵ(p0−q0)=πδ((P−Q)2)ϵ(p0−q0). (56)\n16Taking P2= 0 as before and p0>0, we have the δ((P−Q)2)≃δ(2P·Q). The Dirac delta function\nis non-vanishing for spacelike qonly. It follows that the time-like poles do not contribute to the\nsplitting of damping rate, but only the Landau damping cut does, which significantly simplifies the\nintegration. The q0integral is performed with the Dirac delta function\nReΣA\nar(P) =N2\nc−1\n2Ncg2\n2Zd3q\n(2π)4π\n2pT\nq0\u00122i˜µ\nq2\u0013\u0000\nq2\n0f1+q0f2\u0001\n×2iIm\"\nQ2q2\n(Q2−ΠT)\u0000\nq2Q2(q3\n0−q2\n3)−Q2q2\n3ΠT−q2\n0q2\n⊥ΠL\u0001#\n|q0=ˆp·q. (57)\nWe can further simplify the integral by noting that Im\u0014\nQ2q2\n(Q2−ΠT)(q2Q2(q2\n0−q2\n3)−Q2q2\n3ΠT−q2\n0q2\n⊥ΠL)\u0015\nis odd\ninq0thus also odd under q→ −q. To have an integrand even under q→ −q, we can just keep\nthe following terms in f1andf2\nf1=−q2γ5γ3, f 2=q3q2γ5γ0. (58)\nWe then parameterize the quark self-energy as\nReΣA\nar(P)≡N2\nc−1\n2Ncg2\n2Zdq\n(2π)4πT˜µ\np\u0000\nh1γ5γ3+h2γ5γ0\u0001\n. (59)\nBy rotational invariance, h1andh2are even and odd functions of ˆ p3respectively. Their precise\nforms can only be obtained numerically. We use the following parameterization of q\nq=qcosαˆp+qsinαcosβˆb−cosγˆp\nsinγ+qsinαsinβˆb׈p\nsinγ. (60)\nWe have chosen ˆ pas the z-axis and the plane spanned by ˆ pandˆbas the z−xplane. γdenotes the\nangle between ˆ pandˆbwith cos γ= ˆp·ˆb. We have then ˆ p·q= cos αandd3q=q2dqdcosαdβ. The\nangular integration is performed numerically to obtain h1,2.\nTheq-dependence is of particular interest. It has been shown that the dynamical screening\ncrucial for damping rate is the same as the case without magnetic field in the IR limit [52]. It\nfollows that damping rate from symmetric contribution contains logarithmic divergence [54]. One\nmay expect similar logarithmic divergence in the splitting of damping rate from anti-symmetric\ncontribution. It turns out that this is not the case. Fig. 2 shows the q-dependence of h1,2for a\ngeneric cos γ. Both h1andh2are IR safe. In the UV h2decays more slowly than h1. Let us we\ndefine the q-integrated quantities as\nReΣA\nar(P) =H1γ5γ3−ϵ(p3)H2γ5γ0. (61)\nWe have taken into account the signs of h1andh2(note that the latter is an odd function of ˆ p3)\nsuch that both H1andH2are positive. For the range of cos γwe have explored, |h2|is larger than\n17●●●●●●●●●●●●●●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n■■■■■■■■■■■■■■■■■■■\n■■\n■\n■\n■\n0.10 1 10 100 1000 104q\nm10-610-40.011\n●mh1\n■-mh 2Figure 2: q-dependence of h1(disk) and h2(square) for ˆ p·ˆb= cos γ=1\n3. Both are finite in the IR\nand UV, with h2larger than h1in a wide range of q.\nh1in a wide range of q. It suggests the following relation: H2> H 1, which we assume to be true\ngenerically. This has interesting implication for the dispersions:\nL:p0≃p−ip3H1\np+iϵ(p3)H2,\nR:p0≃p+ip3H1\np−iϵ(p3)H2, (62)\nwith the following damping rate\nΓL≃ϵ(p3)\u0012\nH2−|p3|H1\np\u0013\n,\nΓR≃ −ϵ(p3)\u0012\nH2+p3H1\np\u0013\n. (63)\nClearly the damping rate is dominated by the H2contribution as H2>|p3|H1\np. We find (63) has\nthe same structure as (43) so that the previous reasoning applies: the right-handed mode with\np3>0 and left-handed mode with p3<0 are amplified with respect to their chiral partners. This\nis just the amplification of the mode with a positive spin component along the magnetic field, which\nprovides a mechanism to polarize the probe quark by the QGP paramagnet. Finally let us give a\nparametric estimate: Note thatR\ndqh1,2are dimensionless so they can only depend on cos γ. The\nsplitting of damping rate from the anti-symmetric contribution (59) can be estimated as g2Tµ\np. On\nthe other hand, the symmetric contribution to dynamical screening in the IR limit is independent\nof the magnetic field [52]. It is expected to lead to an average damping rate independent of the\nmagnetic field, for which we estimate as g2T. Assuming µ∼T∼p, we find that the splitting effect\ncan be significant in the context of heavy ion collisions.\nWe are ready to propose the following picture for polarization dynamics in heavy ion colli-\nsions: the initial strong magnetic field first polarizes the spin of light quarks in the QGP. At not\n18very high energy, the QGP carries finite baryon density. Due to mismatch of charges of up and\ndown quarks, the medium is also electrically charged, and thus can be treated as a paramagnet. The\ninitial magnetic field decays quickly so cannot affect the strange quarks produced in late stage of\nheavy ion collisions. Nevertheless, the spin polarized charged QGP serves as a paramagnet, which\ncan efficiently polarize the strange quarks. This is realized through the splitting in the damping\nrates for quarks with opposite spin component along the magnetic field.\n5 Conclusion and outlook\nWe have considered self-energies of photon/gluon in charged magnetized medium in strong magnetic\nfield limit. Finite charge density of the medium induces an anti-symmetric component in the self-\nenergies. We have found the anti-symmetric component leads to splitting of damping rates for\nprobe chiral fermion/quark with opposite spin component along the magnetic field. In the case\nof probe fermion, we have found the splitting of damping rates is parametrically smaller than the\naverage damping rate, while in the case of probe quark, due to self-interaction of gluon, the splitting\nof damping rate is significantly enhanced to be parametrically comparable to the average damping\nrate. Applying the results to heavy ion collisions, we propose the QGP consisting of light quarks\ncan be analogous to a paramagnet due to the interplay of finite magnetic field and baryon density.\nAfter decay of initial magnetic field, the paramagnet can continue to polarize the strange quarks\nproduced at late stage of heavy ion collisions. This provides a mechanism to effectively extend life\ntime of the magnetic field other than the electric conductivity.\nSeveral extensions of this work can be considered: although we have considered the strong\nmagnetic field limit, the mechanism of inducing splitting of damping rate in charged magnetized\nmedium is not necessarily restricted to the strong field limit. It is desirable to consider the weak\nfield limit, which might be more relevant for heavy ion phenomenology. It is more interesting to\nconsider the scenario with vorticity. Indeed an anti-symmetric component of self-energy for gluon\nis known in a vortical QGP [55]. It is expected to lead to splitting of damping rates for different\nspin states even for neutral QGP. For charged vortical QGP, it can be also viewed a paramagnet,\nmaking the damping rate dependent on both spin and charge of the probe. We leave these for\nfuture studies.\nAcknowledgments\nWe are grateful to Koichi Hattori, Aihong Tang and Pengfei Zhuang for helpful discussions. This\nwork is in part supported by NSFC under Grant Nos 12075328 and 11735007.\n19A Evaluation of the probe fermion self-energy\nIn this appendix, we evaluate the probe fermion self-energy which is necessary to determine the\ndamping rate in the main text. Let’s start from the anti-symmetric contribution of (36)\nΣA\nar(P) = −ie2T\n(2π)3˜µZd3q\nq2f1\np·q+iϵp0\n=−ie2Tγ5γ3\n(2π)3˜µZd3q\nq2p1q1+p2q2\np·q+iϵp0+ie2Tp3γ5\n(2π)3˜µZd3q\nq2q1γ1+q2γ2\np·q+iϵp0+ie2Tγ5γ3\n(2π)3˜µZd3q\np·q+iϵp0\n=ie2Tγ5γ3\n(2π)3˜µZ\nq⊥dq⊥dϕZdq3\np⊥q⊥cosϕ +p3q3+iϵp0\n−ie2Tγ5γ3\n(2π)3˜µp⊥Z\nq2\n⊥cosϕ dq ⊥dϕZdq3\n(q2\n⊥+q2\n3)(p⊥q⊥cosϕ +p3q3+iϵp0)\n+ie2Tp3γ5\n(2π)3˜µZ\nq⊥(q⊥·γ⊥)dq⊥dϕZdq3\n(q2\n⊥+q2\n3)(p⊥q⊥cosϕ +p3q3+iϵp0), (64)\nwith\nq⊥·γ⊥=q⊥\np⊥\u0000\n(p1cosϕ−p2sinϕ)γ1+ (p1sinϕ +p2cosϕ)γ2\u0001\n. (65)\nϕis the angle between p⊥andq⊥. We have used the cylindrical coordinates to calculate this\nintegral.\nNext, we will use the residue theorem to calculate the above integral. The sign of p0andp3\nwill affect the integral result. Therefore, we consider the following two cases which are related to\nour study: one case is p0>0 and p3>0, the other case is p0>0 and p3<0.\nAs for the first case ( p0>0 and p3>0), the integral results of q3are\nZdq3\np⊥q⊥cosϕ +p3q3+iϵp0=−iπ\np3,\nZdq3\n(q2\n⊥+q2\n3)(p⊥q⊥cosϕ +p3q3+iϵp0)=π\nq2\n⊥(ip3+p⊥cosϕ), (66)\nThen, (64) becomes\nΣA\nar(P) =ie2Tp3qUV\n8π2˜µp⊥γ5Z\ndϕ(p1cosϕ−p2sinϕ)γ1+ (p1sinϕ +p2cosϕ)γ2\nip3+p⊥cosϕ\n−ie2Tp⊥qUV\n8π2˜µγ5γ3Zdϕ cosϕ\nip3+p⊥cosϕ+e2Tq2\nUV\n8π˜µp3γ5γ3. (67)\nLetz=eiϕ,cosϕ =z2+1\n2zandsinϕ =z2−1\n2z, we can obtain\nΣA\nar(P) =e2Tp3qUV\n8π2˜µp⊥γ5I\n|z|=1dz\nz\u0012(z2+ 1)( p1γ1+p2γ2)\np⊥(z2+ 1) + 2 izp3+(z2−1)(p1γ2−p2γ1)\ni(p⊥(z2+ 1) + 2 izp3)\u0013\n−e2Tp⊥qUV\n8π2˜µγ5γ3I\n|z|=1dz\nzz2+ 1\np⊥(z2+ 1) + 2 izp3+e2Tq2\nUV\n8π˜µp3γ5γ3, (68)\n20The consequences ofH\ndzare\nI\n|z|=1dz\nz(z2+ 1)\np⊥(z2+ 1) + 2 izp3=2πi\np⊥\u0012\n1−p3\np\u0013\n,\nI\n|z|=1dz\nz(z2−1)\np⊥(z2+ 1) + 2 izp3= 0. (69)\nSubstituting (69) into (68) , we can get the final result\nΣA\nar(P) =−ie2TqUV\n4π˜µ\u0012\n1−p3\np\u0013\u0012\nγ5γ3−p3\np2\n⊥γ5\u0000\np1γ1+p2γ2\u0001\u0013\n+e2Tq2\nUV\n8π˜µp3γ5γ3. (70)\nWe can use a similar method to calculate the second case ( p0>0 and p3<0). The integral results\nofq3are\nZdq3\np⊥q⊥cosϕ +p3q3+iϵp0=iπ\np3,\nZdq3\n(q2\n⊥+q2\n3)(p⊥q⊥cosϕ +p3q3+iϵp0)=π\nq2\n⊥(p⊥cosϕ−ip3). (71)\nThen, (64) becomes\nΣA\nar(P) =ie2Tp3qUV\n8π2˜µp⊥γ5Z\ndϕ(p1cosϕ−p2sinϕ)γ1+ (p1sinϕ +p2cosϕ)γ2\np⊥cosϕ−ip3\n−ie2Tp⊥qUV\n8π2˜µγ5γ3Zdϕ cosϕ\np⊥cosϕ−ip3−e2Tq2\nUV\n8π˜µp3γ5γ3. (72)\nWe substitute z=eiϕ,cosϕ =z2+1\n2zandsinϕ =z2−1\n2zinto the above equation.\nΣA\nar(P) =e2Tp3qUV\n8π2˜µp⊥γ5I\n|z|=1dz\nz\u0012(z2+ 1)( p1γ1+p2γ2)\np⊥(z2+ 1)−2izp3+(z2−1)(p1γ2−p2γ1)\ni(p⊥(z2+ 1)−2izp3)\u0013\n−e2Tp⊥qUV\n8π2˜µγ5γ3I\n|z|=1dz\nzz2+ 1\np⊥(z2+ 1)−2izp3−e2Tq2\nUV\n8π˜µp3γ5γ3. (73)\nThe consequences ofH\ndzare\nI\n|z|=1dz\nzz2+ 1\np⊥(z2+ 1)−2izp3=2πi\np⊥\u0012\n1 +p3\np\u0013\n,\nI\n|z|=1dz\nzz2−1\n(p⊥(z2+ 1)−2izp3)= 0. (74)\nEventually, we get\nΣA\nar(P) =−ie2TqUV\n4π˜µ\u0012\n1 +p3\np\u0013\u0012\nγ5γ3−p3\np2\n⊥γ5\u0000\np1γ1+p2γ2\u0001\u0013\n−e2Tq2\nUV\n8π˜µp3γ5γ3. (75)\nCombining (70) and (75), we can back to the result of (38).\n21Let’s turn to the symmetric contribution. We start from the following expression\nΣS\nar(P) =e2\n(2π)3Zd4Q ϵ(q0)\n(P−Q)2+iϵ(p0−q0)\u00121\n2+fγ(q0)\u0013\u0012δ(q2\n0−x2\n1)\nq2\n0−x2\n2+δ(q2\n0−x2\n2)\nq2\n0−x2\n1\u0013\n×γµ(/P−/Q)γνSµν(Q), (76)\nWe first deal with γµ(/P−/Q)γνSµν(Q). The following result is obtained by considering only ∼q0\n0\nγµ(/P−/Q)γνSµν(Q) =−2iq2\n3\u0012\np0γ0+q·γ−(q·γ)(p·q)\nq2\u0013\n. (77)\nThen, (76) can be written as\nΣS\nar(P) =ie2T\n(2π)3˜µ2Zq2\n3\u0010\np0γ0+q·γ−(q·γ)(p·q)\nq2\u0011\n(p·q+iϵp0)d3qZdq0\nq0δ\u0012\nq2\n0−q2\n3q2\n˜µ2\u0013\nϵ(q0). (78)\nWe proceed with the integral of q0.\nZdq0\nq0δ\u0012\nq2\n0−q2\n3q2\n˜µ2\u0013\nϵ(q0) =˜µ2\nq2\n3q2. (79)\nBy using (79), we can simplify (78) and obtain\nΣS\nar(P) =ie2T\n(2π)3Zd3q\nq2(p·q+iϵp0)\u0012\np0γ0+q·γ−(q·γ)(p·q)\nq2\u0013\n. (80)\nIn cylindrical coordinates, the above equation can be rewritten as\nΣS\nar(P) =ie2Tγ0p0\n(2π)3Z\nq⊥dq⊥dϕZdq3\n(q2\n⊥+q2\n3)(p⊥q⊥cosϕ +p3q3+iϵp0)\n+ie2T\n(2π)3Z\nq⊥(q⊥·γ⊥)dq⊥dϕZdq3\n(q2\n⊥+q2\n3)(p⊥q⊥cosϕ +p3q3+iϵp0)\n+ie2Tγ3\n(2π)3Z\nq⊥dq⊥dϕZq3dq3\n(q2\n⊥+q2\n3)(p⊥q⊥cosϕ +p3q3+iϵp0)\n−ie2T\n(2π)3Z\nq⊥(q⊥·γ⊥)(q⊥·p⊥)dq⊥dϕZdq3\n(q2\n⊥+q2\n3)2(p⊥q⊥cosϕ +p3q3+iϵp0)\n−ie2Tp3\n(2π)3Z\nq⊥(q⊥·γ⊥)dq⊥dϕZq3dq3\n(q2\n⊥+q2\n3)2(p⊥q⊥cosϕ +p3q3+iϵp0)\n−ie2Tγ3\n(2π)3Z\nq⊥(q⊥·p⊥)dq⊥dϕZq3dq3\n(q2\n⊥+q2\n3)2(p⊥q⊥cosϕ +p3q3+iϵp0)\n−ie2Tγ3p3\n(2π)3Z\nq⊥dq⊥dϕZq2\n3dq3\n(q2\n⊥+q2\n3)2(p⊥q⊥cosϕ +p3q3+iϵp0). (81)\nWe still consider two cases. As for the case of p0>0 and p3>0, the integral results of q3are\nZq3dq3\n(q2\n⊥+q2\n3)(p⊥q⊥cosϕ +p3q3+iϵp0)=iπ\nq⊥(ip3+p⊥cosϕ),\nZdq3\n(q2\n⊥+q2\n3)2(p⊥q⊥cosϕ +p3q3+iϵp0)=π(2ip3+p⊥cosϕ)\n2q4\n⊥(ip3+p⊥cosϕ)2,\nZq3dq3\n(q2\n⊥+q2\n3)2(p⊥q⊥cosϕ +p3q3+iϵp0)=−p3π\n2q3\n⊥(ip3+p⊥cosϕ)2,\nZq2\n3dq3\n(q2\n⊥+q2\n3)2(p⊥q⊥cosϕ +p3q3+iϵp0)=πp⊥cosϕ\n2q2\n⊥(ip3+p⊥cosϕ)2, (82)\n22We plug (82) into (81) and calculate the integral of q⊥to get the following result.\nΣS\nar(P) =ie2Tγ0p0\n8π2lnqUV\nqIRZdϕ\nip3+p⊥cosϕ−e2Tγ3qUV\n8π2Zdϕ\nip3+p⊥cosϕ\n+ie2TqUV\n8π2p⊥Zdϕ\nip3+p⊥cosϕ\u0000\ncosϕ(γ1p1+γ2p2) +sinϕ(γ2p1−γ1p2)\u0001\n−ie2T\n16π2lnqUV\nqIRZ(2ip3+p⊥cosϕ)dϕ\n(ip3+p⊥cosϕ)2\u0000\n(cosϕ)2(γ1p1+γ2p2) +sinϕcosϕ (γ2p1−γ1p2)\u0001\n+ie2Tp2\n3\n16π2p⊥lnqUV\nqIRZdϕ\n(ip3+p⊥cosϕ)2\u0000\ncosϕ(γ1p1+γ2p2) +sinϕ(γ2p1−γ1p2)\u0001\n.(83)\nThen we can calculate the integral of ϕand write as\nZdϕ\nip3+p⊥cosϕ=−2iπ\np,\nZ\u0000\ncosϕ(γ1p1+γ2p2) +sinϕ(γ2p1−γ1p2)\u0001\ndϕ\nip3+p⊥cosϕ=2π\np⊥\u0012\n1−p3\np\u0013\n(p1γ1+p2γ2),\nZdϕ\n(ip3+p⊥cosϕ)2\u0000\ncosϕ(γ1p1+γ2p2) +sinϕ(γ2p1−γ1p2)\u0001\n=−2iπp⊥\np3(γ1p1+γ2p2),\nZ(2ip3+p⊥cosϕ)dϕ\n(ip3+p⊥cosϕ)2\u0000\n(cosϕ)2(γ1p1+γ2p2) +sinϕcosϕ (γ2p1−γ1p2)\u0001\n=−2iπp2\n3\np3(γ1p1+γ2p2),\n(84)\nSubstituting (84) into (83), we can obtain\nΣS\nar(P) =e2Tγ0p0\n4πplnqUV\nqIR+ie2Tγ3qUV\n4π2p+ie2TqUV\n4πp2\n⊥\u0012\n1−p3\np\u0013\n(γ1p1+γ2p2). (85)\nAs for the another case of p0>0 and p3<0, the integral results of q3change into\nZq3dq3\n(q2\n⊥+q2\n3)(p⊥q⊥cosϕ +p3q3+iϵp0)=−iπ\nq⊥(p⊥cosϕ−ip3),\nZdq3\n(q2\n⊥+q2\n3)2(p⊥q⊥cosϕ +p3q3+iϵp0)=π(p⊥cosϕ−2ip3)\n2q4\n⊥(p⊥cosϕ−ip3)2,\nZq3dq3\n(q2\n⊥+q2\n3)2(p⊥q⊥cosϕ +p3q3+iϵp0)=−p3π\n2q3\n⊥(p⊥cosϕ−ip3)2,\nZq2\n3dq3\n(q2\n⊥+q2\n3)2(p⊥q⊥cosϕ +p3q3+iϵp0)=πp⊥cosϕ\n2q2\n⊥(p⊥cosϕ−ip3)2. (86)\nAfter using the result of (86), (81) becomes\nΣS\nar(P) =ie2Tγ0p0\n8π2lnqUV\nqIRZdϕ\np⊥cosϕ−ip3+e2Tγ3qUV\n8π2Zdϕ\np⊥cosϕ−ip3\n+ie2TqUV\n8π2p⊥Zdϕ\np⊥cosϕ−ip3\u0000\ncosϕ(γ1p1+γ2p2) +sinϕ(γ2p1−γ1p2)\u0001\n−ie2T\n16π2lnqUV\nqIRZ(p⊥cosϕ−2ip3)dϕ\n(p⊥cosϕ−ip3)2\u0000\n(cosϕ)2(γ1p1+γ2p2) +sinϕcosϕ (γ2p1−γ1p2)\u0001\n+ie2Tp2\n3\n16π2p⊥lnqUV\nqIRZdϕ\n(p⊥cosϕ−ip3)2\u0000\ncosϕ(γ1p1+γ2p2) +sinϕ(γ2p1−γ1p2)\u0001\n.(87)\n23We take advantage of the same method as before to integrate ϕto get the following result.\nZdϕ\np⊥cosϕ−ip3=−2iπ\np,\nZ\u0000\ncosϕ(γ1p1+γ2p2) +sinϕ(γ2p1−γ1p2)\u0001\ndϕ\np⊥cosϕ−ip3=2π\np⊥\u0012\n1 +p3\np\u0013\n(p1γ1+p2γ2),\nZdϕ\n(p⊥cosϕ−ip3)2\u0000\ncosϕ(γ1p1+γ2p2) +sinϕ(γ2p1−γ1p2)\u0001\n=−2iπp⊥\np3(γ1p1+γ2p2),\nZ(p⊥cosϕ−2ip3)dϕ\n(p⊥cosϕ−ip3)2\u0000\n(cosϕ)2(γ1p1+γ2p2) +sinϕcosϕ (γ2p1−γ1p2)\u0001\n=−2iπp2\n3\np3(γ1p1+γ2p2),\n(88)\nIn the end, we can obtain\nΣS\nar(P) =e2Tγ0p0\n4πplnqUV\nqIR−ie2Tγ3qUV\n4π2p+ie2TqUV\n4πp2\n⊥\u0012\n1 +p3\np\u0013\n(γ1p1+γ2p2). 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Lett.\nB, 818:136386, 2021.\n28" }, { "title": "2304.05789v1.Micromagnetics_simulations_and_phase_transitions_of_ferromagnetics_with_Dzyaloshinskii_Moriya_interaction.pdf", "content": "MICROMAGNETICS SIMULATIONS AND PHASE TRANSITIONS OF\nFERROMAGNETICS WITH DZYALOSHINSKII-MORIYA INTERACTION\nPANCHI LI, SHUTING GU, JIN LAN, JINGRUN CHEN, WEIQING REN, AND RUI DU\nAbstract. Magnetic skyrmions widely exist in a diverse range of magnetic systems, including chiral magnets\nwith a non-centrosymmetric structure characterized by Dzyaloshinkii-Moriya interaction (DMI). In this study, we\npropose a generalized semi-implicit backward di\u000berentiation formula projection method, enabling the simulations\nof the Landau-Lifshitz (LL) equation in chiral magnets in a typical time step-size of 1 ps, markedly exceeding the\nlimit subjected by existing numerical methods of typically 0 :1 ps. Using micromagnetics simulations, we show\nthat the LL equation with DMI reveals an intriguing dynamic instability in magnetization con\fgurations as the\ndamping varies. Both the isolated skyrmionium and skyrmionium clusters can be consequently produced using\na simple initialization strategy and a speci\fc damping parameter. Assisted by the string method, the transition\npath between skyrmion and skyrmionium, along with the escape of a skyrmion from the skyrmion clusters, are\nthen thoroughly examined. The numerical methods developed in this work not only provide a reliable paradigm\nto investigate the skyrmion-based textures and their transition paths, but also facilitate the understandings for\nmagnetization dynamics in complex magnetic systems.\n1.Introduction\nMagnetic skyrmions are whirling structures of magnetization that have been observed in diverse types of\nmagnetic systems [1, 2]. Due to the topological protection, skyrmions maintain the extremely stability under\nexternal perturbations, and are thus frequently treated as particle-like objects [3, 4]. In addition, the individual\nskyrmion possesses small size down to the nanometer range, and high mobility under electric currents [5, 6],\nthus is a promising candidate for future ultradense information storages and logic techniques [7, 8].\nThe formation of skyrmions is facilitated by the Dzyaloshinkii-Moriya interaction (DMI) [9, 10], but for a\nrange of moderate strength. When the DMI strength is weak, the ground state is the homogeneous ferro-\nmagnetic domain of uniform magnetization; in another limit of strong DMI, the spin spiral state forms. For\nthe medium strength of DMI, skyrmion as well as skyrmionium, a similar texture yet with a trivial topology,\nare spontaneously engendered [5, 11, 12, 13, 14]. These magnetic textures, depicting the inhomogeneous dis-\ntribution of magnetizations, represent local minima of magnetic free energy in chiral magnets with nonzero\nDMI. Gradient descent methods [15] therefore can be applied to search these minima states. In dynamics, the\nLandau-Lifshitz (LL) equation [16, 17] guides the evolution of magnetization.\nMicromagnetics simulations is an important tool to study the magnetization dynamics, where the LL equa-\ntion is solved numerically. There are vast literatures on numerical methods for the LL equation without DMI\n(see recent reviews [18, 19] and the references therein). During the past two decades, semi-implicit projection\nmethods [20, 21, 22, 23] and the tangent plane scheme [24, 25] have been developed for micromagnetics sim-\nulations to achieve a suitable trade-o\u000b between e\u000eciency and numerical stability. However, while the DMI\nis crucial in the generation and transition of skyrmion (and skyrmionium), the incorporation of DM \feld to\nmicromagnetics simulations is rarely studied. The prominent obstacle in the numerical modeling is the chiral\nboundary conditions (a nonhomogeneous Neumann boundary condition), due to the curling nature of DM \feld.\nBeside inhomogeneity, the DM \feld also imposes stringent constraint for the temporal step-size, typically in\n0:1 ps in existing methods [26, 27].\nTo address these challenges posed by the DMI, we develop a generalized semi-implicit backward di\u000berentia-\ntion formula (BDF) projection scheme with the second-order accuracy in both space and time to numerically\nsolve the LL equation with DMI. A time step-size of 1 :0 ps is permissible in our method, that substantially\nreduces the computational expenses by one order of magnitude. Dynamic instability of the LL dynamics is ob-\nserved that distinct stable magnetization con\fgurations stabilize contingent on the damping parameter, from\nthe initialization e3. Diverse skyrmion textures such as isolated skyrmion, isolated skyrmionium and their\nclusters are therefore generated, and minimal energy path (MEP) between these textures are determined with\nthe assistance of the string method. In addition, the protection by the chiral boundary and the propelling by\nlocal magnetic \feld are also demonstrated.\nThe remaining sections of this paper are structured as follows. In Section 2, we present the second-order\nsemi-implicit method for the LL equation, along with the harmonic map heat \row technique to investigate the\nmagnetic free energy with DMI, and the string method to locate the transition path. Skyrmion-based textures\nunder di\u000berent circumstances are then generated in Section 3, and the phase transitions associated with these\ntextures are visualized in Section 4. The concluding remarks are given in Section 5.\nDate : April 13, 2023.\n1arXiv:2304.05789v1 [math.NA] 12 Apr 20232 PANCHI LI, SHUTING GU, JIN LAN, JINGRUN CHEN, WEIQING REN, AND RUI DU\n2.Models and numerical methods\nMagnetic skyrmions were initially discovered in 2009 [7] and have since been extensively studied due to their\nunique features, including extraordinary metastability, and their small physical size (usually below \u0018100 nm),\nwhich results in high mobility at low-current densities [8, 15, 28]. On the atomic scale, DMI has the form\n(2.1) EDM=\u0000X\nhijidij\u0001(mi\u0002mj);\nwhere dijdenotes the DMI vector between atomic indices iandj, having a direction dependent on the system's\ntype [29], mirepresents the atomic moment with a unit length, and the summation is over all atomic indices\nthat have \fnite neighbor interactions, hiji[30, 31]. The total spin Hamiltonian comprises the Heisenberg\nexchange interaction, the DMI, and the interaction with an applied magnetic \feld H, represented as:\n(2.2) E=\u0000JX\nhijimi\u0001mj\u0000X\nhijidij\u0001(mi\u0002mj)\u0000\u00160X\niH\u0001mi;\nwhereJis the exchange coupling constant, and \u00160denotes the vacuum permeability. The Heisenberg model\n(2.2) outlined in [15, 32] disregards the anisotropy term and the stray \feld.\n2.1.The continuum model. In the continuum model, the magnetization is represented as a vector \feld\nthat is dependent on the spatial variable, M=M(x) wherex2\n. For non-centrosymmetric bulk materials\nthat have chirality, the continuous equivalent of (2.2) is given by:\n(2.3)F[M] =Z\n\nA\nM2sjrMj2dx+D\nM2sZ\n\n(r\u0002M)\u0001Mdx\n\u0000\u00160Z\n\nH\u0001Mdx+\u00160\n2Z\nR3jrUj2dx+Z\n\n\b\u0012M\nMs\u0013\ndx;\nwhereAis the exchange constant, Dis the DMI constant, and \n \u001aRd(d= 1;2;3) is the region occupied\nby the magnetic body. Below the Curie temperature, the saturation magnetization Msis a constant and it\nsatis\fesjM(x)j=Ms. The fourth term is the dipolar energy that is de\fned by the Newtonian potential\nN(x) =\u00001\n4\u00191\njxjin the form\n(2.4) U(x) =Z\n\nrN(x\u0000x0)\u0001M(x0)dx0:\nDenote the stray \feld by Hs=\u0000rU(x), then the dipolar energy can be rewritten as \u0000\u00160\n2R\n\nHs\u0001Mdx.\nThe anisotropy energy \b( M(x)=Ms) : \n!R+is a smooth function. For a uniaxial ferromagnet with the\neasy-axis direction e1= (1;0;0)T, the anisotropy energy has the form \b( M(x)=Ms) =Ku(M2\n2+M2\n3)=M2\ns=\nKu(M2\ns\u0000M2\n1)=M2\nswithKuthe anisotropy constant. A magnetic skyrmion induced by the DMI has a topology\nnumber (or skyrmion number), which is de\fned by [3, 5]\n(2.5) Q=1\n4\u0019M3sZ\nM\u0001\u0012@M\n@x\u0002@M\n@y\u0013\ndxdy=\u00061:\nIn a skyrmion lattice, the topology number is proportional to the accumulated isolated skyrmion.\nIn the dynamic case, the magnetization M=M(x;t) follows the phenomenological Landau-Lifshitz-\nGilbert (LLG) equation (an equivalent of the LL equation)\n(2.6)@M\n@t=\u0000\rM\u0002H+\u000b\nMsM\u0002@M\n@t;\nwhere\ris the gyromagnetic parameter, \u000bis the dimensionless damping parameter, and His the e\u000bective \feld\ncalculated by the variation of the energy functional (2.3),\n(2.7) H=2A\nM2s\u0001M\u00002Ku\nM2s(M2e2+M3e3) +\u00160H+\u00160Hs\u00002D\nM2sr\u0002M:\nAs a result of calculus of variations, the non-homogeneous Neumann boundary condition, also known as the\nchiral boundary, is derived\n(2.8)@M\n@\u0017\f\f\f\n@\n=\u0000D\n2AM\u0002\u0017\nwith\u0017being the unit outward normal vector. This is important for the energy dissipation law of the magne-\ntization dynamics stated in the following theorem.3\nTheorem 2.1. LetM2L1([0;T]; [H1(\u0016\n)]3)\\C1([0;T]; [C1(\u0016\n)]3)be the solution of (2.6) -(2.8) , then the\nfollowing energy dissipation law holds\n(2.9)dF[M]\ndt\u00140\nif the external magnetic \feld is independent of time.\nProof. For vectorsv;w2H1(\n), it holds that r\u0001(v\u0002w) =w\u0001(r\u0002v)\u0000v\u0001(r\u0002w). Taking the volume\nintegral and applying the divergence theorem, we have\n(2.10)Z\n\nr\u0001(v\u0002w)dx=Z\n\nw\u0001(r\u0002v)\u0000v\u0001(r\u0002w)dx=Z\n@\n(v\u0002w)\u0001\u0017dS:\nThen, we get\n(2.11)d\ndtZ\n\nM\u0001(r\u0002M)dx= 2Z\n\n@M\n@t\u0001(r\u0002M)dx\u0000Z\n@\n\u0010\nM\u0002@M\n@t\u0011\n\u0001\u0017dS:\nDue to the nonhomogeneous boundary condition, we have\nZ\n\n@M\n@t\u0001\u0001Mdx=Z\n@\n@M\n@t\u0001rM\u0001\u0017dS\u00001\n2d\ndtZ\n\njrMj2dx\n=\u0000D\n2AZ\n@\n@M\n@t\u0001(M\u0002\u0017)dS\u00001\n2d\ndtZ\n\njrMj2dx\n=D\n2AZ\n@\n\u0010\nM\u0002@M\n@t\u0011\n\u0001\u0017dS\u00001\n2d\ndtZ\n\njrMj2dx: (2.12)\nTaking inner product with (2.6) by \rH\u0000\u000b\nMs@M\n@t, we arrive at\n0\u0014\u000b\n\rMsZ\n\n\u0010@M\n@t\u00112\ndx=Z\n\n@M\n@t\u0001Hdx\n=2A\nM2sZ\n\n@M\n@t\u0001\u0001Mdx\u00002Ku\nM2sZ\n\n@M\n@t\u0001(M2e2+M3e3)dx+\n\u00160Z\n\n@M\n@t\u0001Hsdx+\u00160Z\n\n@M\n@t\u0001Hdx\u00002D\nM2sZ\n\n@M\n@t\u0001(r\u0002M)dx\n=D\nM2sZ\n@\n\u0010\nM\u0002@M\n@t\u0011\n\u0001\u0017dx\u0000A\nM2sd\ndtZ\n\njrMj2dx\u0000Ku\nM2sd\ndtZ\n\n(M2\n2+M2\n3)dx+\n\u00160d\ndtZ\n\nM\u0001(H+Hs)dx\u0000D\nM2sd\ndtZ\n\nM\u0001(r\u0002M)dx\u0000D\nM2sZ\n@\n\u0010\nM\u0002@M\n@t\u0011\n\u0001\u0017dx\n=\u0000dF\ndt;\nwhere we use the fact that the external magnetic \feld is independent of time. This completes the proof. \u0003\nSkyrmion-based patterns induced by the DMI exhibit superior mobility in response to current \felds. To\naccount for the magnetic interactions with an external current, we include the spin transfer torque (STT)\nsupplied by the spin-polarized current in the LLG model, as described in [33]:\n(2.13)@M\n@t=\u0000\rM\u0002H+\u000b\nMsM\u0002@M\n@t\u0000b\nM2sM\u0002(M\u0002(j\u0001r)M)\u0000b\u0018\nMsM\u0002(j\u0001r)M;\nHerein,Pdenotes the polarization rate, jrepresents the current density vector, b=P\u0016B=(eMs(1+\u00182)), where\n\u0016Bis the Bohr magneton, eis the elementary charge, and \u0018is a dimensionless parameter that characterizes\nthe degree of non-adiabaticity.\nDenote\n^H=\u00002Ku\nM2s(M2e2+M3e3) +\u00160H+\u00160Hs+b\n\rM2sM\u0002(j\u0001r)M+b\u0018\n\rMs(j\u0001r)M;\nthen (2.13) can be rewritten as the LL form\n(2.14)@M\n@t=\u0000\r\n1 +\u000b2M\u0002\u00102A\nM2s\u0001M\u00002D\nM2sr\u0002M+^H\u0011\n\u0000\n\r\u000b\n1 +\u000b2M\u0002\u0010\nM\u0002\u00102A\nM2s\u0001M\u00002D\nM2sr\u0002M+^H\u0011\u0011\n:4 PANCHI LI, SHUTING GU, JIN LAN, JINGRUN CHEN, WEIQING REN, AND RUI DU\nDe\fne the dimensionless variables m=M=Ms,he=H=Ms,hs=Hs=Msand spatial rescaling x!Lx\nwithLbeing the length of ferromagnetic body. The dimensionless magnetic free energy I[m] satisfying\nF[M] =\u00160M2\nsI[m] is\n(2.15) I[m] =Z\n\n\u000f\n2jrmj2+\u0014\n2(r\u0002m)\u0001m\u0000he\u0001m\u00001\n2hs\u0001m+ \b (m) dx\nwith\u000f= 2A=(\u00160M2\nsL2) and\u0014= 2D=(\u00160M2\nsL). Meanwhile, we take the time rescaling t!(1+\u000b2)(\u00160\rMs)\u00001t\nand get the dimensionless LL equation\n(2.16)@m\n@t=\u0000m\u0002\u0010\n\u000f\u0001m\u0000\u0014r\u0002m+^h\u0011\n\u0000\u000bm\u0002\u0010\nm\u0002\u0010\n\u000f\u0001m\u0000\u0014r\u0002m+^h\u0011\u0011\n;\nwhere ^h=\u0000q(m2e2+m3e3) +he+hs+b\n\u00160\rM2sm\u0002(j\u0001r)m+b\u0018\n\u00160\rM2s(j\u0001r)mwithq= 2Ku=(\u00160M2\ns). Note\nthat^H=Ms^h.\nIn the absence of spin-polarized current, i.e. j=0, we have the following LL equation with boundary and\ninitial conditions\n(2.17)8\n>>>><\n>>>>:@m\n@t=\u0000m\u0002h\u0000\u000bm\u0002(m\u0002h) in [0 ;T]\u0002\n;\n@m\n@\u0017=\u0000\u0014bm\u0002\u0017 on [0;T]\u0002@\n;\nm(0) = m0with\f\fm0\f\f= 1 inft= 0g\u0002\n;\nwhere\n(2.18) h=\u0000\u000eI\n\u000em=\u000f\u0001m\u0000\u0014r\u0002m\u0000q(m2e2+m3e3) +he+hs:\nHere\u0014b=DL=(2A) is proportional to L, implying the sti\u000bness of the boundary.\nA stable or metastable state, such as skyrmion and skyrmionium, given by the LL equation satis\fes\n(2.19) m=ch\nwithcbeing a constant for jmj= 1 in a point-wise sense. Due to the energy dispassion of the LL equation,\nthe convergent procedure should stop at the local minimizer of the energy functional (2.3) with\n(2.20)8\n<\n:\u000eI\n\u000em= 0;\ns.t.jmj= 1:\nGradient decent methods, such as the nonlinear conjugate gradient method [15], therefore can be applied\nto search the minima of the magnetic free energy. Alternatively, the stable magnetization con\fguration can\nbe obtained by simulating the dynamics driven by the harmonic map heat \row equation. In this case, the\nminimization problem is formulated as\n(2.21) infn\nI(m)j@m\n@\u0017=\u0000\u0014bm\u0002\u0017on@\n;jm(x)j= 1;8x2\no\n:\nUsing the Lagrange multiplier method with ^\u0015being the Lagrange multiplier, we get\n(2.22) L(m;^\u0015) =I(m) +^\u0015\n2Z\n\n((m)2\u00001)dx:\nAt stationary points, it holds\n\u000eL\n\u000em=\u0000h+^\u0015m= 0;\n(m)2\u00001 = 0:\nSo we have ^\u0015= (m;h). Therefore, the harmonic map heat \row reads as\n(2.23)@m\n@t=\u0000\u000eL\n\u000em=h\u0000(m;h)m\nsubject to the constraint jmj= 1 and the nonhomogeneous boundary condition. Thus, we also consider the\nharmonic map heat \row system\n(2.24)8\n>>>><\n>>>>:@m\n@t=\u0000m\u0002(m\u0002h); in [0;T]\u0002\n;\n@m\n@\u0017=\u0000\u0014bm\u0002\u0017; on [0;T]\u0002@\n;\nm(0) = m0; inft= 0g\u0002\n;\nwherejm0j= 1.5\n2.2.Numerical methods. FeGe is a representative chiral ferromagnet with strong spin-orbit coupling. Here\nwe list its physical parameters in Table 1. It is clear that \u0014and\u0014dare two leading parameters due to the DMI.\nTable 1. Physical parameters of FeGe ( L= 80 nm).\nParameter Value (unit) Dimensionless quantity\nKu 0 (J=m3) q= 0\nA 8:78\u000210\u000012(J=m)\u000f\u00191:48\u000210\u00002\nMs 3:84\u0002105(A=m)\nD 1:58\u000210\u00003(J=m2)\u0014\u00190:21,\u0014d\u00197:20\nThe \frst parameter results a strong curl \feld in the LL equation and the second one leads to a sti\u000b boundary\ncondition, both of which will be examined later.\nIn the framework of \fnite di\u000berence method, we construct unknowns on half grid points as m(xi;yj;zk) =\nm((i\u00001\n2)hx;(j\u00001\n2)hy;(k\u00001\n2)hz). Herehx= 1=nx,hy= 1=ny,hz= 1=nzandh=hx=hy=hzholds for\nuniform spatial meshes. The indexes i;j;k are valued with i= 1;\u0001\u0001\u0001;nx,j= 1;\u0001\u0001\u0001;nyandk= 1;\u0001\u0001\u0001;nz. For\nthe sake of clarity, the approximations of the boundary condition and operator ronx-direction are depicted\nbelow.\n⋯⋯⋯𝑥0𝑥1𝑥2𝑥𝑛𝑥+1 𝑥𝑛𝑥 𝑥𝑛𝑥−1 ⋯⋯⋯\n0 1\nFigure 1. Grids along the x-direction with two ghost points x0andxnx+1.\nLet\u0017= (1;0;0)T, the boundary condition at x= 1 is discretized as\nm(xnx+1;yj;zk)\u0000m(xnx;yj;zk)\nhx=\u0000\u0014bm(xnx+1;yj;zk) +m(xnx;yj;zk)\n2\u0002\u0017;\ni.e.8\n>>>>><\n>>>>>:m1(xnx+1;yj;zk) =m1(xnx;yj;zk);\nm2(xnx+1;yj;zk) =1\u0000k2\nbx\n1 +k2\nbxm2(xnx;yj;zk)\u00002kbx\n1 +k2\nbxm3(xnx;yj;zk);\nm3(xnx+1;yj;zk) =2kbx\n1 +k2\nbxm2(xnx;yj;zk) +1\u0000k2\nbx\n1 +k2\nbxm3(xnx;yj;zk);\nwherekbx=\u0014bhx=2. Similarly, discretization of the boundary condition at x= 0 yields\n8\n>>>>><\n>>>>>:m1(x0;yj;zk) =m1(x1;yj;zk);\nm2(x0;yj;zk) =1\u0000k2\nbx\n1 +k2\nbxm2(x1;yj;zk) +2kbx\n1 +k2\nbxm3(x1;yj;zk);\nm3(x0;yj;zk) =\u00002kbx\n1 +k2\nbxm2(x1;yj;zk) +1\u0000k2\nbx\n1 +k2\nbxm3(x1;yj;zk):\nBoundary conditions along yandzdirections are discretized in a similar way.\nThe operatorrxis discretized as\nrxm(x1;yj;zk)\u0019m(x1;yj;zk)\u0000m(x0;yj;zk)\nhx;\nrxm(xi;yj;zk)\u0019m(xi+1;yj;zk)\u0000m(xi\u00001;yj;zk)\n2hx;\nrxm(xnx;yj;zk)\u0019m(xnx+1;yj;zk)\u0000m(xnx;yj;zk)\nhx;\nwherei= 2;\u0001\u0001\u0001;nx\u00001,j= 1;\u0001\u0001\u0001;nyandk= 1;\u0001\u0001\u0001;nz.ryandrzare discretized similarly.\nRemark 1.In thin \flms, the DMI arises from robust spin-orbit couplings at the edges, whereby magnetization\nis absent beyond the sample. In addition, the exchange interactions are non-symmetric and intrinsically\ndirectional at the boundaries. To numerically capture such features, the introduction of ghost points outside\nthe material is commonly employed to approximate spatial derivatives within the vicinity of the boundaries.\nThe resulting discretization of the roperator is achieved as described above.6 PANCHI LI, SHUTING GU, JIN LAN, JINGRUN CHEN, WEIQING REN, AND RUI DU\nThe Laplacian operator is descritized as\n\u0001m(xi;yj;zk)\u0019m(xi\u00001;yj;zk)\u00002m(xi;yj;zk) +m(xi+1;yj;zk)\nh2x+\nm(xi;yj\u00001;zk)\u00002m(xi;yj;zk) +m(xi;yj+1;zk)\nh2y+\nm(xi;yj;zk\u00001)\u00002m(xi;yj;zk) +m(xi;yj;zk+1)\nh2z:\nRegarding time-stepping, the standard second-order backward di\u000berentiation formula (BDF2) is employed\n3mn+1\u00004mn+mn\u00001\n2k=\u0000mn+1\u0002(\u000f\u0001mn+1\u0000\u0014r\u0002mn+1+^hn+1)\n\u0000\u000bmn+1\u0002(mn+1\u0002(\u000f\u0001mn+1\u0000\u0014r\u0002mn+1+^hn+1)):\nThe prevalent feature of this approach involves the utilization of an implicit methodology, thereby necessitating\nnonlinear solvers at each time step. Notably, a semi-implicit scheme incorporating a projection step has been\ndeveloped with the motivation of ensuring maintenance of jmj= 1 [23]. This scheme treats the DMI term\nas implicit, primarily due to its dominance in the e\u000bective \feld. The ensuing semi-implicit BDF2 projection\nscheme exhibits the following features. Given its second-order time accuracy, simulations in micromagnetics\nemploying this scheme may adopt a step-size \u0001 t= 1 ps. Conversely, earlier approaches were limited in their\nability to utilize sub-picosecond time step-sizes.\nAlgorithm 2.1. Set^ mn+1= 2mn\u0000mn\u00001and~hn+1= 2^hn\u0000^hn\u00001.\n(i)Compute ~ mn+1such that\n3~ mn+1\u00004mn+mn\u00001\n2\u0001t=\u0000^ mn+1\u0002(\u000f\u0001~ mn+1\u0000\u0014r\u0002~ mn+1+~hn+1)\n\u0000\u000b^ mn+1\u0002(^ mn+1\u0002(\u000f\u0001~ mn+1\u0000\u0014r\u0002~ mn+1+~hn+1)) (2.25)\n(ii)Projection ontoS2:\n(2.26) mn+1=1\nj~ mn+1j~ mn+1\nFor the harmonic map heat \row, a similar algorithm is proposed.\nAlgorithm 2.2. Set^ mn+1= 2mn\u0000mn\u00001and~hn+1= 2^hn\u0000^hn\u00001.\n(i)Compute ~ mn+1such that\n3~ mn+1\u00004mn+mn\u00001\n2\u0001t=\u0000^ mn+1\u0002(^ mn+1\u0002(\u000f\u0001~ mn+1\u0000\u0014r\u0002~ mn+1+~hn+1)):\n(ii)Projection ontoS2:\nmn+1=1\nj~ mn+1j~ mn+1:\nThe semi-implicit BDF1 approach is employed as a precursor to the BDF2 scheme, and allows for the\ncalculation of m1. This initial step does not impart any alteration to the overall second-order accuracy\nexhibited by the numerical scheme. Additionally, it is worth mentioning that if the relative change in energy\nbetween two consecutive time steps is less than 1 :0\u000210\u00009in the simulation, a steady state is considered to\nhave been attained.\nIn order to search the minimum energy transition paths of skyrmion-based magnetic textures, here we\nfurther introduce the string method [34, 35]. By de\fnition, a curve \rconnecting two local minima satis\fes\n(2.27) ( rI)?(\r) = 0;\nwhere (rI)?is the component of rInormal to\r. Then\r:=f'(a);a2[0;1]gde\fnes a MEP from one local\nminima to the other. After an initial parametrization of the curve is picked and usually the equal arc-length\nparametrization is used, the curve evolve to the MEP following the equation\n(2.28) 't=\u0000(rI('))?+\u0015\u001c;\nwhere (rI('))?=rI(')\u0000(rI(');\u001c)\u001c,\u001cis the unit tangent vector along 'with\u001c='a=j'aj, and\u0015is the\nLagrange multiplier uniquely determined by the choice of parametrization. Let \u0016\u0015=\u0015+ (rI(');\u001c), then the\n(2.28) can be rewritten as\n(2.29) 't=\u0000rI(') +\u0016\u0015\u001c:7\nIn order to \fnd the MEP, the time-splitting method is applied to solve (2.29). Details are given in Algorithm\n2.3. A convergent string means that all the images satisfy\n(2.30) ( rI('i))?= 0:\nIn our simulations, this is replaced by the stopping criterion ( TOL = 1:0e-06) as\n(2.31) max\nijjI('i;tn)\u0000I('i;tn+1)jj1\u0014TOL:\nAlgorithm 2.3. Choose an initial string \r0with inclusion of N+ 1images such that \r0:=f'i='(ai);ai2\n[0;1];i= 0;\u0001\u0001\u0001;Ng.\n\u000fStep 1: Evolve the images on the string following the gradient \row\n(2.32) @t'i=\u0000rI('i):\nFrom current images f'n\nig,f'\u0003\nigare obtained by one time stepping. Here an image is denoted by\nthe magnetization con\fguration m(x);x2\n, and the harmonic map heat \row (2.24) is solved by\nAlgorithm 2.2.\n\u000fStep 2: Compute the parametrization fa\u0003\nigby\ns0= 0;si=si\u00001+\f\f'\u0003\ni\u0000'\u0003\ni\u00001\f\f;i= 1;\u0001\u0001\u0001;N;\nand then the updated mesh fa\u0003\nigis normalized by a\u0003\ni=si=sN.\n\u000fStep 3: Parametrization of the string by equal arc-length and projection. The images f^'n+1\nigare\nobtained by cubic spline interpolation at uniform grid points fai=i=Ng, and the new images f'n+1\nig\nare obtained after the projection step.\n\u000fStep 4: Go back to Step 1 and iterate until convergence.\nRemark 2.The approach to the string method utilized in this present work di\u000bers from the standard version\noutlined in [35, 36]. This variance essentially stems from the novel conservation requirement on the length of\nmagnetization. Accordingly, Step 3 in the method is crafted to incorporate a projection step. Furthermore,\ntreating images as tensors characterized by 'i2R3\u0003nx\u0003ny\u0003nzmakes it possible to consider the existence of a\nreversible mapping between 'andm, which is captured by mappings L:'!mandL\u00001:m!'in the\nimplemented technique.\n3.Micromagnetics simulations\nThis section initiates with the utilization of Algorithm 2.1 and Algorithm 2.2 in generating stable magnetic\ntextures. The simulation carried out considers a FeGe sample having a consistent spatial mesh size of 2 \u00022\u0002\n2 nm3.\nSpeci\fcally, this simulation focuses on a sample with dimensions 80 \u000280\u00026 nm3. One observation of\nnotable interest is the onset of a dynamic instability in the LL equation, attributed to the presence of the DMI.\nStarting from an initial uniform state of m0= (0;0;1)T, the system relaxes into di\u000berent stable con\fgurations\nin response to variations in the damping parameter. As depicted in Fig. 2, the LL equation readily generates\n(a)Q=\n\u00001\n(b)Q= 0\n (c)Q= 0\n (d)Q= 1\nFigure 2. Isolated skyrmions and isolated skyrmioniums by means of the LL equation with\ndi\u000berent damping parameters \u000b= 0:05;0:07;0:2;0:6. The color of background represents the\ncomponent m3and arrows represent the in-plane components m1andm2.\nskyrmions ( Q=\u00061) and skyrmioniums ( Q= 0) with diverse damping parameters \u000b. We also document\nthe energies and spatially averaged magnetization of the four con\fgurations accounting for skyrmions and\nskyrmioniums in Table 2. For varied \u000bvalues within the interval (0 ;1], the system reliably converges towards\none of the four con\fgurations.\nThe publication [14] reveals that a skyrmionium is a composite structure composed of two topological\nmagnetic skyrmions possessing Q= 1 andQ=\u00001, and its motion is swifter when driven by an external8 PANCHI LI, SHUTING GU, JIN LAN, JINGRUN CHEN, WEIQING REN, AND RUI DU\nTable 2. Energy and spatially averaged magnetization hmi= (hm1i;hm2i;hm3i)Tof isolated\nskyrmions and isolated skyrmioniums.\nLabel energy(10\u000018J)hmxihmyihmzi \u000b\nFig. 2(A) -3.8989 0.0047 0.0038 -0.1204 0.05/0.06\nFig. 2(B) -3.0936 -0.0035 0.0036 -0.2627 0.07/0.08/0.09/0.1\nFig. 2(C) -3.0938 0.0188 -0.0038 0.2629 0.2/0.3/0.4/0.5\nFig. 2(D) -3.8986 0.0021 0.0042 0.1216 0.01/\u0001\u0001\u0001/0.04/0.6/\u0001\u0001\u0001/1.0\nout-of-plane current than that of a skyrmion. In order to gain a better understanding of their fundamental\ndi\u000berences, the energy density distribution is visualized. Let L(m) denote the energy density distribution in\nthe absence of the Dzyaloshinskii-Moriya interaction (DMI), and D(m) denote the energy density distribution\ncorresponding to the DMI. The combination of these two, T(m) =L(m) +D(m), represents the total energy\ndensity distribution. As illustrated in Fig. 3, the di\u000berence between the skyrmion and skyrmionium is mainly\nattributable toD(m), as their energy distributions without the DMI appear similar along the axes passing\nthrough the center. However, the energy density distribution of the DMI is almost entirely opposite.\n(a)L(m)\n (b)D(m)\n (c)T(m)\n(d)L(m)\n (e)D(m)\n (f)T(m)\nFigure 3. The energy density distribution along the centered slice of the material in the xy-\nplane. Top row: energy density distribution of the skyrmion with Q= 1. Bottom row: energy\ndensity distribution of the skyrmionium.\nIn the context of the harmonic map heat \row equation, the relaxation of the system is aimed toward\nachieving a single skyrmion con\fguration. As portrayed in Fig. 4, a meticulous scrutiny of the quantity hm3i\nindicates a swift formation of the skyrmion, followed by a prolonged period of relaxation that is required to\nsatisfy the stability criterion.\nIsolated skyrmion and skyrmionium structures have been successfully generated in a controllable manner.\nSubsequently, we have expanded our e\u000borts towards generating skyrmion clusters in a ferromagnetic sample\nwith dimensions of 200 \u0002200\u00026 nm3. Initiated from a con\fguration where m0= (0;0;1)T, the system\nundergoes relaxation to yield varying clusters as we adjust the damping parameter \u000b. A skyrmion cluster is\ncharacterized based on the number of skyrmions it comprises and the nature of their interconnected structures.\nFig. 5 depicts representative skyrmion clusters generated by employing the LL equation.\nThe mutual interactions between individual skyrmions can lead to the formation of skyrmion lattices and\nclusters. As illustrated in Fig. 5, the local structure of skyrmion lattices can be realized by the presence\nof skyrmion clusters. For instance, by adopting a speci\fc initialization scheme, the square skyrmion lattice\nstructure can be easily generated as a periodic replica of the square skyrmion cluster, as shown in either\nFig. 5(D) or Fig. 6. The initial magnetization exhibits a rectangular shape with in-plane dimensions of\n40 nm\u000240 nm, a con\fguration which yields skyrmions having a diameter of 40 nm. For the skyrmion lattice\ngenerated via the LL equation, the stable energy and spatially averaged magnetization are \u00001:7066e-17 J and\nhmi= (\u00000:58e-04;0:39e-03;0:23)T, respectively. Meanwhile, the skyrmion lattice produced by means of the\nharmonic map heat \row method possesses stable energy and averaged magnetization values of \u00001:6971e-17 J\nandhmi= (0:28e-03;0:04;0:23)T, respectively.9\n10 ps\n 20 ps\n 45 ps\n 100 ps\n 350 ps\n(a)Snapshots of relaxation driven by the harmonic map heat \row.\n10 ps\n 20 ps\n 45 ps\n 100 ps\n 350 ps\n(b)Snapshots of relaxation driven by the LL equation.\n0 1 2 3 4\ntime (s) 10-10-4-2024energy (J)10-18\nLL dynamics\nharmonic map heat flow\n(c)Energy evolution.\n0 0.5 1 1.5 2 2.5 3 3.5 4\ntime (s)10-1001020m110-3\n0 0.5 1 1.5 2 2.5 3 3.5 4\ntime (s)10-1001020m210-3\n0 0.5 1 1.5 2 2.5 3 3.5 4\ntime (s)10-1000.51m3 (d)hmievolution.\nFigure 4. Comparison of the relaxation driven by the LL equation and the harmonic map\nheat \row. The same initialization m0= (0;0;1)Tis used and the same skyrmion with Q= 1\nis reached. The damping parameter \u000b= 0:6 is used in the LL equation.\n(a)\u000b=\n0:04.\n(b)\u000b=\n0:05.\n(c)\u000b=\n0:09.\n(d)\u000b=\n0:5.\nFigure 5. Representative skyrmion clusters formed in the 200 \u0002200\u00026 nm3ferromagnet.\nThe emergence of isolated skyrmioniums and skyrmionium clusters is a subject that has received little\nattention in the existing literature. In this study, we leverage the dynamic instability exhibited by the LL\nequation to create isolated skyrmioniums and skyrmionium clusters with precise damping parameter settings.\nWhen the ferromagnetic material attains an Lvalue of 80 nm, both isolated skyrmions and skyrmioniums\ncan be observed, with the latter's radius being considerably larger than that of the former, speci\fcally in\nan unsaturated phase. Consequently, we broaden our investigation by considering a sample of dimensions\n500\u0002500\u00026 nm3and adopt two di\u000berent initialization strategies to generate skyrmionium clusters.10 PANCHI LI, SHUTING GU, JIN LAN, JINGRUN CHEN, WEIQING REN, AND RUI DU\nFigure 6. Left: Initial magnetization con\fguration. Middle and right: Stable magnetization\ncon\fgurations formed by the LL equation and harmonic map heat \row, respectively. The back\nground color represents the magnetization component m3. The magnetization within the blue\nblocks is (0 ;0;\u00001)T, while the remianing area is (0 ;0;1)T.\nOur \frst strategy involves placing nine rectangles with m3=\u00001 within the initial magnetization con\fgura-\ntion and adjusting the inter-block distance. The dimensions of the blocks are \fxed at 100 nm \u0002100 nm, while\nthe damping parameter is set to \u000b= 0:2. When the blocks are spaced at 25 nm, four skyrmions emerge near\nthe corners along with a set of nine skyrmioniums arranged in the shape of a \rower, as displayed in Fig. 7(B).\nAs the inter-block distance increases to 66 nm, four skyrmions are generated at diverse locations in a regular\nskyrmionium lattice. It is important to note that the damping parameter plays a crucial role in the formation\nof skyrmionium clusters, unlike skyrmion clusters and skyrmion lattices. Furthermore, in contrast to skyrmion\nclusters and lattices, individual skyrmions consistently coexist with skyrmioniums in the skyrmionium clusters.\n(a)\n (b)\n (c)\nFigure 7. Initial con\fguration (A) and skyrmionium clusters (B) and (C). The sample size is\n500 nm\u0002500 nm\u00026 nm and the damping parameter is \u000b= 0:2 in the LL equation.\nNext, we reduce the size of the blocks in the initial con\fguration to 50 nm \u000250 nm and set \u000b= 0:1 in\nthe LL equation. As illustrated in Fig. 8, a skyrmion lattice comprising of 29 skyrmions and an additional\nisolated skyrmion located at a corner is produced. However, we observe a defective lattice in this case due to\nthe existence of three distinct types of skyrmion clusters. Speci\fcally, the skyrmion clusters are categorized\ninto three types: (1) a skyrmion is encircled by \fve neighboring skyrmions; (2) a skyrmion is surrounded by\nsix neighboring skyrmions; and (3) a skyrmion is surrounded by seven neighboring skyrmions. The occurrence\nof the \frst two types of clusters is also observed in a ferromagnetic material with dimensions of 200 nm \u0002\n200 nm\u00026 nm. When the isolated skyrmions coalesce into an interconnected structure, the energy density at\nthe junctions of any two skyrmions exhibits a notably higher magnitude when compared to other locations.\nComprehending the phase transition between magnetic textures holds signi\fcant importance in the \feld of\nspintronics. Pertaining to skyrmion-based textures, experimental observations have revealed various transitions\nsuch as those between skyrmion clusters facilitated by a magnetic \feld [37], transitions between skyrmion\nlattice structures induced by a magnetic \feld [38], transitions between skyrmioniums driven by spin-polarized\ncurrent [14], and the formation of skyrmions via ultrafast laser pulses [39]. The local order in magnetization is\ndisrupted and then re-established to generate skyrmion and skyrmionium structures as demonstrated in [39].\nMotivated by these experimental \fndings, we aim to investigate the generation of skyrmion and skyrmionium\nstructures in a skyrmion lattice con\fguration during the re-stabilization process through simulations.\nIn our simulations, we perturbe the magnetization order of the stable magnetic texture depicted in Fig. 8 by\nlocally revaluating it randomly, in accordance with the LL equation. Speci\fcally, in Fig. 9, the magnetization11\n(a) Initializa-\ntion\n(b) Equilib-\nrium\n(c)T(m)\nFigure 8. A skyrmion lattice over a 500 nm \u0002500 nm\u00026 nm ferromagnet. Here an isolated\nskyrmion is presented near the bottom-right corner, which resulted from the defectiveness of\nthe sample. From the initial magnetization con\fguration (A), the system reaches (B) with\nspatial energy density distribution (C), following the LL dynamics with \u000b= 0:1.\n(a)Magnetization within the centered circle with radius 100 nm is randomly reval-\nued and re-stabilized.\n(b)Magnetization within the centered circle with radius 120 nm is randomly reval-\nued and re-stabilized.\nFigure 9. Generation of skyrmioniums and skyrmions in a skyrmion lattice under pertuba-\ntions. The initial magnetization con\fguration is a skyrmion lattice with the magnetization\nover a centered circle randomly pertubated, which is fed into the LL dynamics as the initial\ncondition. A new stable magnetization con\fguration is obtained by following the LL equation.\nFirst column: initial con\fguration. Second column: stable magnetization con\fguration. Third\ncolumn: energy density distribution of the stable magnetization con\fgurations.\norder within a centered circular domain was randomly revalued while the radius of the circle was adjusted.\nThe results demonstrate that this perturbation spontaneously induces the formation of either a skyrmionium\nor several skyrmions, thus initiating the transition of the skyrmion structure. A damping value of \u000b= 0:1 was\nemployed, as depicted in Fig. 9 where the generation of the skyrmionium is observed as a consequence of the\naforementioned perturbation.12 PANCHI LI, SHUTING GU, JIN LAN, JINGRUN CHEN, WEIQING REN, AND RUI DU\n4.Transition paths\nThe collapse of an isolated skyrmion, its subsequent escape through a boundary, and division into two\nidentical skyrmions, have been observed. The \frst two transitions shed light on the connection between isolated\nskyrmions and the classical magnetic saturation state, while the last transition doubles the system's topological\nnumber. This study aims to identify transition paths pertaining to changes in the topological number of both\nisolated skyrmions and skyrmion clusters. To this end, the string method, previously described in literature,\nhas been employed to identify transition paths between magnetic textures with di\u000bering topological numbers.\nSubsequently, micromagnetic simulations have been conducted to realize the phase transition corresponding\nto the identi\fed path.\nThe primary objective of this study is to investigate the transition between a skyrmion and a skyrmionium.\nBy considering an isolated skyrmionium with a topological number Q= 0 and an isolated skyrmion with Q= 1\nas the two initial endpoints, a transition path can be established through the implementation of the string\nmethod. Such a path is illustrated in Fig. 10, where stable and saddle points are highlighted. Speci\fcally, the\ntransition from the skyrmionium to skyrmion with Q=\u00001 is accompanied by the escape of a nucleus located\nat the core of the skyrmionium from the protective ring pattern. It is worth noting that the nucleus of the\nskyrmionium can be regarded as a reversed skyrmion and can be easily controlled by an in-plane current, while\nthe external ring cannot be as readily removed due to the pronounced boundary protection. Hence, this study\nprovides valuable insights into the transition from an isolated skyrmionium to an isolated skyrmion, whereby\nthe topological number is e\u000bectively erased as a result of the removal of the nucleus skyrmion.\n0 0.2 0.4 0.6 0.8 1-3.8875-3.7057-3.5239-3.3421-3.1603-2.9785\nABC\nDE\nFG\nH\nB\n C\n D\n E\n F\n G\n H\nFigure 10. Phase transition between an isolated skyrmionium Q= 0 and an isolated skyrmion\nQ= 1. (A) The transition path between the skyrmionium and the skyrmion. The red points\ndenote the local minima and saddle points on the path. (B)-(H) are the magnetization con\fg-\nurations corresponding to minima and saddle points on the MEP.\nDuring the transition path between a skyrmion with a topological number of Q=\u00001 and one with Q= 1, a\nmetastable state characterized by a skyrmion junction with a topological number of Q= 0 is encountered. The\nchange in topological number follows the sequence \u00061!0!\u00071 along this transition path, as demonstrated\nin Fig. 11. In order to realize the transition from a skyrmion with Q= 1 to one with Q=\u00001 in the LL\nequation, an in-plane current is applied, with a chosen damping parameter of \u000b= 0:6. The simulation is\nconducted in two stages. Firstly, from 0 \u0018850 ps, a current with u=\u0000bJ=\u0000150 m=s and\f= 0:5 along the\ndirection\u0000e1is applied, and the system reaches the skyrmion junction. Subsequently, from 1 :5 ns to 2:1 ns,\na current with u=\u000050 m=s and\f= 0:4 is applied along the direction \u0000e2, and the system relaxes to the\nskyrmion with Q=\u00001. The simulation results demonstrate that the energy barrier during the transition from\na skyrmion to a skyrmion junction is higher than that from the skyrmion junction to the skyrmion, owing to\nthe superior stability of isolated skyrmions.\nThis study then proceeds to investigate the transition between skyrmion clusters with a change in topological\nnumber. The transition process is illustrated in Fig. 12, where a skyrmion escapes through a boundary, leading13\n0 1000 2000 3000\nTime (ps)-4-3.8-3.6-3.4 Energy (J)10-18\n0 ps\n 250 ps\n 750 ps\n 1:5 ns\n 2 ns\n 2:5 ns\n 3 ns\nFigure 11. Energy evolution when a spin-polarized current is applied and 7 representative\nsnapshots of magnetization con\fguration are visualized. Top row: energy evolution driven by\nthe LL equation. Black real lines represent the dynamics when the current is removed, while\nred and blue dashed lines represent the dynamics when the current is applied with di\u000berent\ndirections and strengthes. Bottom row: 7 snapshots at di\u000berent times, corresponding to the\npentagrams during the energy evolution.\nto a transition between two skyrmion clusters and a subsequent reduction in the system's topological number.\nIt is worth noting that although there are alternative mechanisms for inducing this transition, such as the\ncollapse of a skyrmion or the merger of two skyrmions into one, the transition path illustrated in Fig. 12\nrepresents the MEP in this particular case.\n0 0.2 0.4 0.6 0.8 1-1.9-1.85-1.810-17\na\nb\nc\nd\nFigure 12. Phase transition between skyrmion clusters and a skyrmion escapes from a bound-\nary.\nThe uniform application of an external \feld (i.e., magnetic or current \feld) leads to the simultaneous\nmovement of all skyrmions within the cluster. Hence, it becomes challenging to induce transitions between\nskyrmion clusters using this approach. As a viable alternative, we propose the use of a local magnetic \feld\nto manipulate individual skyrmions. To demonstrate this, we consider the use of a magnetic \feld to pull the14 PANCHI LI, SHUTING GU, JIN LAN, JINGRUN CHEN, WEIQING REN, AND RUI DU\ncentral skyrmion from the cluster, causing it to eventually escape through a boundary. In this demonstration,\nthe magnetic \feld strength is chosen to be \u00002:5e3T, which proves adequate for overcoming boundary sti\u000bness,\nbut is considered too strong for inducing desired skyrmion movements. In addition, skyrmions can be attracted\n0 ps\n 1 ns\n 2:5 ns\n 3 ns\n 12:5 ns\nFigure 13. A local out-of-plane magnetic \feld applied over the green square domain drives\none skyrmion across the boundary. The magnetic \feld with magnitude \u00002:5e3T moves along\nthee1direction with velocity 40 m =s.\nor repelled by an out-of-plane magnetic \feld. For instance, when a local \feld is applied over an in-plane domain\nof 40\u000240 nm2during a time interval of 0 \u0018200 ps, two neighboring skyrmions are observed to move towards\neach other, eventually resulting in their merger.\n0 ps\n 40 ps\n 200 ps\n 400 ps\n 5:8 ns\nFigure 14. Mergence of two skyrmions when an out-of-plane magnetic \feld \u00000:5ezT is applied\nover the green square domain within the time period [0 ;200 ps].\n5.Conclusion\nThis study proposes a generalized, second-order accurate, semi-implicit projection scheme for solving the\nLandau-Lifshitz (LL) equation with the Dzyaloshinskii-Moriya interaction (DMI), which enables the use of\nlarger step-sizes for micromagnetics simulations. It is observed that the LL system exhibits a dynamic insta-\nbility, and that various stable magnetization con\fgurations can be generated by means of simple initializa-\ntion as the damping parameter varies, including isolated skyrmions, isolated skyrmionium, skyrmion clusters,\nskyrmionium clusters, and combinations thereof, in a controlled manner. The string method is employed to\nidentify minimal energy paths connecting di\u000berent stable magnetization con\fgurations. In particular, the\ntransition between a skyrmion with Q= 1 and one with Q=\u00001 involves a local minimizer characterized by a\nskyrmion junction with Q= 0. Moreover, for skyrmion clusters, a transition path is determined that involves\na skyrmion escaping through the boundary. The proposed method o\u000bers a dependable strategy for studying\nskyrmion textures and their transition paths, which can greatly enhance our understanding of magnetization\ndynamics for spintronics applications.\nAcknowledgments\nP. Li thanks for the helpful discussion of Zhiwei Sun, and acknowledges the program of China Scholarships\nCouncil No. 202106920036. S. Gu acknowledges the support of NSFC 11901211 and the Natural Science\nFoundation of Top Talent of SZTU GDRC202137. J. Lan acknowledges the support of NSFC (Grant No.\n11904260) and Natural Science Foundation of Tianjin (Grant No. 20JCQNJC02020). J. Chen acknowledges\nthe support of NSFC (Grant No. 11971021). R. Du was supported by NSFC (Grant No. 12271360).\nReferences\n[1] C. Back, V. Cros, H. Ebert, K. Everschor-Sitte, A. Fert, M. Garst, T. Ma, S. Mankovsky, T. L. Monchesky, M. Mostovoy,\nN. Nagaosa, S. S. P. Parkin, C. P\reiderer, N. Reyren, A. Rosch, Y. Taguchi, Y. Tokura, K. von Bergmann, and J. Zang. The\n2020 skyrmionics roadmap. J.Phys. D:Appl. Phys., 53(36):363001, jun 2020.\n[2] K. Wang, V. Bheemarasetty, J. Duan, S. Zhou, and G. Xiao. Fundamental physics and applications of skyrmions: A review.\nJ.Magn. Magn. Mater., 563:169905, 2022.\n[3] S. 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Lett., 110:177205, Apr 2013.16 PANCHI LI, SHUTING GU, JIN LAN, JINGRUN CHEN, WEIQING REN, AND RUI DU\nEmail address :LiPanchi1994@163.com\nSchool of Mathematical Sciences, Soochow University, Suzhou, 215006, China\nEmail address :gst1988@126.com\nCollege of Big Data and Internet, Shenzhen Technology University, Shenzhen 518118, China\nEmail address :lanjin@tju.edu.cn\nCenter for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, 92 Weijin\nRoad, Tianjin 300072, China.\nEmail address :jingrunchen@ustc.edu.cn\nSchool of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China\nSuzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou, Jiangsu\n215123, China\nEmail address :matrw@nus.edu.sg\nDepartment of Mathematics, National University of Singapore, 119076, Singapore\nEmail address :durui@suda.edu.cn\nSchool of Mathematical Sciences, Soochow University, Suzhou, 215006, China.\nMathematical Center for Interdisciplinary Research, Soochow University, Suzhou, 215006, China." }, { "title": "1712.01239v4.DAMPE_Electron_Positron_Excess_in_Leptophilic__Z___model.pdf", "content": "arXiv:1712.01239v4 [hep-ph] 29 May 2018Prepared for submission to JHEP\nDAMPE Electron-Positron Excess in Leptophilic Z′\nmodel\nKarim Ghorbaniaand Parsa Hossein Ghorbanib\naPhysics Department, Faculty of Sciences, Arak University, Arak 38156-8-8349, Iran\nbInstitute for Research in Fundamental Sciences (IPM), Scho ol of Particles and Accelerators, P.O.\nBox 19395-5531, Tehran, Iran\nE-mail:karim1.ghorbani@gmail.com ,parsaghorbani@gmail.com\nAbstract: Recently the DArk Matter Particle Explorer (DAMPE) has repo rted an excess\nintheelectron-positron fluxofthecosmicrayswhichisinte rpretedasadarkmatter particle\nwith the mass about 1 .5 TeV. We come up with a leptophilic Z′scenario including a Dirac\nfermion dark matter candidate which beside explaining the o bserved DAMPE excess, is\nabletopassvariousexperimental/observational constrai nts includingtherelicdensity value\nfrom the WMAP/Planck, the invisible Higgs decay bound at the LHC, the LEP bounds\nin electron-positron scattering, the muon anomalous magne tic moment constraint, Fermi-\nLAT data, and finally the direct detection experiment limits from the XENON1t/LUX. By\ncomputing the electron-positron flux produced from a dark ma tter with the mass about\n1.5 TeV we show that the model predicts the peak observed by the D AMPE.\nKeywords: Cosmology of Theories beyond the SM, Dark MatterContents\n1 Introduction 1\n2 Model 3\n3 Relic Density and Invisible Higgs Decay 5\n4 Muon Anomalous Magnetic Moment 7\n5 LEP constraint 8\n6 Direct Detection 10\n7 Neutrino Trident Production and τDecay 12\n8 DAMPE Excess 12\n9 Constraints from Fermi-LAT 14\n10 Conclusion 15\nA Dark Matter Annihilation Cross Sections 15\n1 Introduction\nOne of the signals of a new physics could be the observation of any excess in the energy\nspectra of the cosmic rays. The search for such an excess in th e electron and positron spec-\ntra have been already in progress by different particle detect ors in the space; the PAMELA\nsatellite experiment observed an abundance of the positron in the cosmic radiation energy\nrange of 15 −100 GeV [ 1], also a positron fraction in primary cosmic rays of 0 .5−350\nGeV [2] and 0.5−500 GeV [ 3] and the measurement of electron plus positron flux in the\nprimary cosmic rays from 0 .5 GeV to 1 TeV [ 4] reported by the Alpha Magnetic Spectrom-\neter (AMS02). The motivation of the current paper is however the recent report of the\nfirst results of the DArk Matter Particle Explorer (DAMPE) wi th unprecedentedly high\nenergy resolution and low background in the measurement of t he cosmic ray electrons and\npositrons (CREs) in 25 GeV to 4 .6 TeV energy range [ 5]. At energy about 1 .4 TeV a peak\nassociated to a monoenergetic electron source is observed. This excess is interpreted by a\ndark matter particle with the mass around 1 .5 TeV annihilating into electron and positron\nin a nearby subhalo in the Milky Galaxy about 0 .1−0.3 kpc distant from the solar system.\nThe dark matter annihilation cross section times velocity i s estimated to be in the range\n– 1 –∼10−26−10−24cm3/sfor the aforementioned dark matter mass. For an interpretat ion of\nthe DAMPE data see [ 6].\nThere are already several papers that have tried to explain t his excess using different\nmodels. In [ 7] a vector-like fermion DM with a new U(1) gauge boson which on ly couples to\nthe firsttwo lepton generation is used to explain theDAMPE da ta. Inthis direction, model\nindependent analysis performed with fermion DM in [ 8] and with scalar and fermionic DM\nin [9]. There are also studies within the simplified models with a Z′gauge bosons couples\nonly to the first family of leptons (electrophilic interacti on) or to the other families as well\n[10–13]. There is another study in [ 14] where electron flavored fermion DM can interact\nwith the first generation lepton doublet via an inert scalar d oublet or with right-handed\nelectron via a charged scalar singlet. In addition, the exce ss is studied in Hidden Valley\nmodel with lepton portal DM [ 15], radiative Dirac seesaw model [ 16] and gauged Le−Lµ\nmodel [17]. It is also studied that the DM particles annihilate to two i ntermediate scalar\nparticles and then the scalars decay to DM fermions [ 18]. In [19] it is shown that a DM\ncandidate with cascade decay can explain the DAMPE TeV elect ron-positron spectrum.\nThere are detailed analysis on the morphology of CRE flux cons idering properties of the\nprimary electron sources [ 20–22].\nMeanwhile, it should benoted that there may exist some possi ble exotic sources for the\nexcess or it may originate from some standard sources like pu lsars or supernova remnants.\nIn this work we interpret the excess due to the DM annihilatio n in a nearby halo.\nTo explain the DAMPE excess, we come up with a leptophilic Z′dark matter scenario\nthat contains a Dirac fermion which plays the role of the dark matter candidate. Besides,\nin the dark sector we introduce a U(1)′gauge symmetry and a complex scalar that to-\ngether with the Dirac fermion are charged under this U(1)′gauge symmetry. The dark\nsector communicates with the standard model sector through two portals. One portal is\nthrough the mixing of the complex scalar with the standard mo del Higgs particle and the\nother portal comes from the interaction of the U(1)′gauge boson, Z′, merely with the\nleptons in the standard model, hence being a leptophilic Z′portal. One of the distinctive\ncharacteristics of our two-portal model is that the DM-nucl eon elastic scattering begins\nat one loop level. Therefore there is a large region in the par ameter space which evades\ndirect detection. Thus, indirect detection searches becom e very important tools to probe\nthe viable parameter space of the present model.\nIn addition to the constraints from the relic density as well as the direct and indirect\nbounds on the dark matter model, we examine the model if it is c onsistent also with the\nnew observed DAMPE bump in the electron and positron flux in th e cosmic rays.\nThe paper have the following parts. In the next section we ela borate the setup of\nour leptophilic dark matter scenario. In section 3the dark matter relic density and the\ninvisible Higgs decay are computed and compared with bounds from the WMAP/Planck\nand the LHC. Next we take into account the muon magnetic anoma ly and shrink the viable\nspace of parameters. Constraints from the LEP is discussed i n section 5. In section 6we\nconstrain more the model with limits from the direct detecti on experiments specially the\nrecent XENON1t and LUX experiments. Discussions on the neut rino trident production\nandτdecay is given in section 7. We also find a viable space of parameter consistent with\n– 2 –the excess observed by the DAMPE in section 8. The Fermi-LAT constraint is discussed\nin section 9. Finally we conclude in section 10.\n2 Model\nWe explore a leptophilic two-portal dark matter scenario. T hat is, a fermionic candidate of\ndark matter connected to the standard model particles throu gh vector and Higgs portals.\nThe vector in the dark sector interacts with all the lepton fla vors in the SM but with no\ninteraction with the quarks. The Lagrangian of the model can be written in three parts,\nL=LSM+LDM+Lint, (2.1)\nwhere the dark matter Lagrangian consists of a Dirac fermion playing the role of the dark\nmatter and a complex scalar field both charged under U(1)′,\nLDM=−1\n4F′\nµνF′µν+¯ψ/parenleftbig\niγµD′\nµ−mψ/parenrightbig\nψ\n+/parenleftbig\nD′\nµϕ/parenrightbig/parenleftbig\nD′µϕ/parenrightbig∗−m2(ϕϕ∗)−1\n4λs(ϕϕ∗)2.(2.2)\nwhere theU(1)′field strength is denoted by F′\nµν, theψis the Dirac fermion and ϕstands\nfor the complex scalar. The dark sector covariant derivativ e is defined as,\nD′\nµ=∂µ−ig′zZ′\nµ. (2.3)\nwhich acts on the fields in the dark sector as well as the lepton s in the SM with g′being\nthe strength of its coupling and zthe charge of the field acting on.\nHere we study a leptophilic model in which the U(1)′gauge boson, Z′, interacts only\nwith the leptons in the SM but also with a hypothetical right- handed neutrino for the\nreason that will be discussed latter on. It is therefore nece ssary to modify the covariant\nderivative in the SM to include a term for the new coupling,\nDSM\nµ→D′SM\nµ=DSM\nµ−ig′zZ′\nµ, (2.4)\nwhereg′is theU(1)′coupling in the dark sector and zis the dark charge of the leptons\nthat the covariant derivative acts on.\nThe interaction Lagrangian then reads,\nLint=−λ′(ϕϕ∗)/parenleftBig\nHH†/parenrightBig\n+g′zELZ′\nµ¯ELγµEL+g′zeRZ′\nµ¯eRγµeR\n+g′zνRZ′\nµ¯νRγµνR,(2.5)\nwhereELandeRare respectively the three families of left-handed lepton d oublets and\nright-handed lepton singlets including the right handed ne utrinos. Notice that we have\nconsidered universal charges for all families of the lepton s, i.e. we have taken the zeLto\nbe the lepton U(1)′charge for each family of left-handed lepton doublet and zeRto be the\nU(1)′charge foreR,µRandτR.\n– 3 –Having introduced a new U(1)′coupled to the dark matter and the chiral fermions\nin the SM, one must be careful about the triangle anomalies. I n order to remove such\nanomalies we choose the charges in eq. ( 2.5) to take the following two leptophilic choices,\nA)zµ/negationslash= 0,zνµ/negationslash= 0\nzeL= 2a, zµL=−a, zτL=−a\nzνeL= 2a, zνµL=−a, zντL=−a\nzeR=−2a, zµR=a, zτR=a\nzνeR=−2a, zνµR=a, zντR=a(2.6)\nB)zµ= 0,zνµ= 0\nzeL=a, zµL= 0, zτL=−a\nzνeL=a, zνµL= 0, zντL=−a\nzeR=−a, zµR= 0, zτR=a\nzνeR=−a, zνµR= 0, zντR=a(2.7)\nwhereais a real number. In the choice Ait is assumed that the charge of the lepton µ\nand that of its neutrino νµare non-zero while in the choice Bwe setzµ=zνµ= 0. We\nwill clarify latter on the reasoning for these choices. The e xistence of the right-handed\nneutrinos are crucial; without them the triangle anomalies can not be fixed. The charge of\nthe dark matter Dirac fermion, zψ, suffices to have opposite values for its left-handed and\nright-handed components, i.e. zψL=−zψR. Fixinga= 1 and substituting the anomaly-\nfree charges in eq. ( 2.6) and eq. ( 2.7) into eq. ( 2.5) we obtain two interaction Lagrangians,\nA)\nLint=−λ′(ϕϕ∗)/parenleftBig\nHH†/parenrightBig\n−2g′Z′\nα¯eγαγ5e−2g′Z′\nα¯νeγαγ5νe\n+g′Z′\nα¯µγαγ5µ+g′Z′\nα¯νµγαγ5νµ\n+g′Z′\nα¯τγαγ5τ+g′Z′\nα¯ντγαγ5ντ,(2.8)\nB)\nLint=−λ′(ϕϕ∗)/parenleftBig\nHH†/parenrightBig\n−g′Z′\nα¯eγαγ5e−g′Z′\nα¯νeγαγ5νe\n+g′Z′\nα¯τγαγ5τ+g′Z′\nα¯ντγαγ5ντ.(2.9)\nAs seen in eqs. ( 2.8) and (2.9) the vector boson Z′couples to the leptons axially. Note\nthat in eqs. ( 2.8) and (2.9) the fields e,µandτare all Dirac fermions. Let us turn back\nto scalars in the SM and in the dark sector. The Higgs potentia l as usual is composed of a\n– 4 –quadratic and a quartic term which guarantees a non-zero vevfor the Higgs field fixed by\nthe experiment to be vh= 246 GeV. After the electroweak symmetry breaking we denote\nthe Higgs doublet as H†= (0vh+h) wherehis the fluctuation around the vevbeing a\nsinglet real scalar. The complex scalar field, ϕin eq. (2.2) has two degrees of freedom\nout of which only one takes non-zero expectation value. The c omponent that takes zero\nexpectation value goes for the longitudinal part of the Z′dark gauge boson. Therefore,\nϕ→vs+s, (2.10)\nwithsbeing a real scalar which mixes with the SM Higgs and vsthe vacuum expectation\nvalue of the scalar ϕ. The Higgs portal interaction term in eq. ( 2.5) together with the\nscalar potential in eq. ( 2.2) and the Higgs potential at the vevof the scalars, leads to a\nnon-diagonal mass matrix for the field space of hands. We diagonalize the mass matrix\nby rotating in the handsspace by the mixing angle θ(see [23] for more details). After\ndiagonalizing the mass matrix we end up with the physical mas ses that we denote by mh,\nms. We are keeping the same notations for the scalar fields handsafter the mixing.\nThe couplings of the model are λ′in eq. (2.5), the Higgs quartic coupling λhin the Higgs\npotential, andthescalar quarticcoupling λsineq. (2.2). Thesecouplingsareall expressible\nin terms of the physical masses, mh,msand the mixing angle θ,\nλh=m2\nssin2θ+m2\nhcos2θ\n2v2\nh,\nλs=m2\nscos2θ+m2\nhsin2θ\nv2s/2−v2\nh\nv2sλ′,\nλ′=m2\nh−m2\ns\n2√\n2vhvssin2θ.(2.11)\nThe vacuum stability conditions on the potential already gi ve rise to the following con-\nstraints on the couplings,\nλh>0,\nλsvs2>λ′v2\nh,\nvs2(λhλs−2λ′2)>v2\nhλ′λh.(2.12)\nThe free parameters of the model can then be assigned as mψ,ms,θ,vsandg′.\n3 Relic Density and Invisible Higgs Decay\nThe fermionic DM candidate in the present model is a weakly in teracting massive particle\n(WIMP). Thebasicverticestobuildthediagramsrelevantfo rtheannihilationprocessesare\nas follows. The DM has a vector-type interaction with Z′, i.e., via the vertex Z′\nµ¯ψγµψ, and\nthe new gauge boson has axial-vector interactions with the S M leptons, i.e., via the vertex\nZ′\nµ¯lγ5γµl. Moreover, there are two types of vertices for the Z′coupled to the SM Higgs and\nthe new scalar, i.e., Z′µZ′\nµhandZ′µZ′\nµs. It is therefore possible to have DM annihilation\nin s-channel via Z′exchange, ¯ψψ→¯ee,¯µµ,¯ττ,¯νlνl,Z′h,Z′s(for model B,¯ψψ→¯µµ,\n– 5 –100101102\n 10 100mψ [GeV]\nsin(θ) = 0.05vs = 600 GeVModel AmS [GeV]\nmZ´ [GeV]WMAP/Planck regions\nExcluded by Higgs invisible decay\n 0 500 1000 1500 2000 2500 3000\n100101102\n 10 100mψ [GeV]\nsin(θ) = 0.1vs = 600 GeVModel AmS [GeV]\nmZ´ [GeV]WMAP/Planck regions\nExcluded by Higgs invisible decay\n 0 500 1000 1500 2000 2500 3000\nFigure 1 . The plots show the mass of the Z′against the mediator mass in model Afor two mixing\nanglesleft)sinθ= 0.05 andright)sinθ= 0.1. The gray points are excluded by the invisible Higgs\ndecay bound. The scan is done over the parameters with 10−30.\nOn the other hand, the anisotropy of surface spins is given by Néel’s model [ 25], according to which anisotropy only\noccurs at sites with reduced coordination. The correspondi ng anisotropy constant is denoted by ks=Ks/J >0.\nTherefore, we write\nHan,i=\n\n−kc(si·ez)2, i ∈core\n+1\n2kszi/summationdisplay\nj∈nn(si·uij)2, i∈surface.(4)\nuijis a unit vector connecting the nearest neighbors (nn) iandj.\nPhysical parameters : A word is in order regarding the orders of magnitude of these parameters. For cobalt, the\nlattice parameter is a= 0.3554nm and the magnetic moment per atom is µa=n0µB, withn0being the number of Bohr\nmagnetons per atom ( n0≃1.7) andµB= 9.274×10−24J/T the Bohr magneton. Hence, µa≃1.58×10−23J/T. The\n(bulk) exchange coupling is J≃8mev or1.2834×10−21J/atom, which yields τs∼70fs. Next, the magneto-crystalline\nanisotropy constant is roughly Kc≃3×10−24J/atom and the surface anisotropy constant is Ks≃5.22×10−23\nJoule/atom. As such, kc≡Kc/J≃0.00234 andks≡Ks/J≃0.04. The latter value is within the range of those\ninferred from several experimental studies. Indeed, one ma y findKs/J≃0.1for cobalt [ 26],Ks/J≃0.06for iron\n[27], andKs/J≃0.04for maghemite particles [ 28]. For later reference, we note that for a nanomagnet with the\natomic magnetic moment µaand magneto-crystalline anisotropy Kcgiven above, the Stoner-Wohlfarth switching\nfieldHSW= 2Kc/µaevaluates to 0.38T.\nAnother word is in order regarding the notations for the anis otropy constants. The capital letter Kstands for\nthe macroscopic constant usually given in J/m3for the volume and J/m2for the surface. It can also be converted\nintoJ/atom upon using the unit cell of the underlying lattice. The lower case letter kis the dimensionless anisotropy\nconstant that results from a normalization with respect to t he largest energy in the system, namely the exchange\ncoupling J,i.e.k≡K/J, for core and surface. In the figures presented below, instea d of using the dimensionless\nconstant k, we use the anisotropy constant in units of meVwhich should be understood as meV/atom.\nThe dynamics of the atomic spins miis described with a set of ( 2N) coupled damped Landau-Lifshitz equations\n(LLE)\ndsi\ndτ=−si×heff,i−αsi×(si×heff,i), (5)\nwhereheff,i≡(µaHeff)/Jis the (dimensionless) effective field acting on the atomic sp in at site i, given by heff,i=\n−δH/δsi;αis the dimensionless damping parameter ( ∼0.01).\nWe use the (second-order) Heun algorithm to solve the set of e quations ( 5) starting from a given initial state (a\nconfiguration of Nspinssi) and a set of parameters kc,ks,hdcand rf field characteristics, i.e.amplitude hrf=µaH0\nrf/J\nand frequency ̟=ωτs. The calculations will be performed for two anisotropy confi gurations:\n1. Model TUA (textured uniaxial anisotropy): all spins, core and surfac e, have the same uniaxial anisotropy in\nthezdirection, but the corresponding constant may be different f or core and surface spins.\n2. Model NSA (Néel surface anisotropy): the spins in the core are attribu ted the uniaxial anisotropy easy axis\naccording to the first line in Eq. ( 4) with constant kcand easy axis ez, whereas those on the surface have their\nanisotropy given by the second line with constant ks.\nIn this study we vary the amplitude hrfand frequency ̟of the rf field and determine their optimal values for which\nthe net magnetic moment switches, for a given and fixed DC magn etic field.\nNumerical procedure : we recall that the main objective here is to achieve magneti zation reversal under an rf field,\nin addition to a DC magnetic field whose intensity is supposed ly smaller than the critical value required for switching3\naccording to the macrospin Stoner-Wohlfarth model. In addi tion, we intend to demonstrate that the intensity and\nfrequency of the rf field that are usually required by the swit ching of a macrospin, are further optimized in nanomagnets\nwhich exhibit spin misalignment induced by surface anisotr opy. Accordingly, we prepare the system in an initial state,\nnamely a spin configuration with all spins aligned in a given d irection, say in the +zdirection and apply a DC magnetic\nfield in the −zdirection. In fact, to avoid having a vanishing vector produ ctsi×Hdcat the initial time, we set Hdc\nat an angle slightly different from π(e.g.179◦). While maintaining the rf magnetic field off, we set the dampi ng\nparameter to some (relatively) large value ( ∼1) and let the spins evolve in time according to Eq. ( 5) until the\nequilibrium state is reached. Since the DC field is smaller th an the switching field, the spins remain in the local\nminimum defined by the anisotropy energy. Next, we reset the d amping parameter to α∼0.02and switch on the rf\nfield. Then, we let the spins evolve in time, using again Eq. ( 5), and record the net magnetic moment m(t)defined\nas\nm=1\nNN/summationdisplay\ni=1mi. (6)\n0 10 20 30 40 50 60 70050100150200250\nm(0).m(t) ≤ − 0.5\nm(0).m(t) ≤ − 0.75\nm(0).m(t) ≤ − 0.9 Hrf (mT)\n ω (GHz)NSA: Kc = Ks = 0.008meV\nHdc = 8mT\nFigure 1: Switching phase diagram for different switching co nditions. Each symbol represents a point of the diagram for w hich\nswitching is achieved. N= 113.\nWe stop this time evolution when two criteria are satisfied: F irst we require that m(0)·m(t)≤ε, withεa negative\nnumber. This condition is required to ensure switching. Acc ordingly, in Fig. 1, we show the ̟-hrfphase diagram for\nthe NSA model with different values of ε(|ε|<1). We see that indeed the phase diagram strongly depends on th e\nswitching criterion. More explicitly, for a given hrf, a higher |ε|requires a higher frequency ̟since more energy has\nto be pumped into the system from the rf field. On the other hand , at a given frequency ̟, the larger |ε|the smaller\nis the critical hrf. This can be understood by the fact that a small value of |ε|requires the system to be maintained\nhalfway from the more stable (deeper) minimum and this requi res a stronger field. In all subsequent results of the\npresent study, we adopted the most stringent criterion, i.e.with the largest value of |ε|. At finite temperature, this\nwould insure the highest thermal stability for the magnetic state of the system. In fact, a second criterion is necessary\nin order to ensure that the final magnetic moment remains in th e minimum reached after switching. For this, we\nrequire that the fluctuations of the net magnetic moment deca y to a small value, i.e./bardblδm(t)/bardbl/lessorsimilar10−3.\nNow we discuss the effect of the damping parameter αin Eq. ( 5). The present study is concerned with the\nmagnetization dynamics at very low temperatures. As such, t he magnetization switching can only be achieved by\ngoing over the energy barrier. In this case, the role of the da mping term in Eq. ( 5) is to drive the net magnetic\nmoment of the nanomagnet towards the effective field and thus t o push the system into its global energy minimum. In\nparticular, it does not affect the time trajectory of the net m agnetic moment during its switching process. However,\nthe value of αdoes have a bearing on the computing time needed for the syste m to reach equilibrium, i.e.its global\nminimum, but does not affect the global switching diagram as f ar as the values of hrfand̟are concerned. This\nis indeed confirmed by the results shown in Fig. 2. Consequently, this study and all subsequent results have b een\nobtained for a fixed value of α, stated after Eq. ( 5).4\n0 10 20 30 40 50 60 70050100150200250\n α = 0.025\n α = 0.02\n α = 0.015\n α = 0.01 Hrf (mT)\n ω (GHz)N = 113\nKc = 0.008meV, Ks = 10Kc\nFigure 2: Switching phase diagram for different values of the damping parameter.\nIn general, it is a rather involved task to deal with the dynam ics within a many-spin approach, especially the\ncalculation of relaxation rates, magnetization reversal, AC susceptibility and so on. Indeed, it is a formidable task\nto perform a detailed analysis of the various critical point s (minima, maxima, saddle points) of the energy which are\nrequired for the study of the relaxation processes [ 29–32]. In the particular case of the present study of the switchin g\n̟-hrfphase diagram, it is not so easy to analyze the various switch ing mechanisms. As such, it is desirable to replace\nthe many-spin problem by some effective macroscopic model. A ccordingly, it has been shown [ 33–36] that, under some\nconditions which are quite plausible for today’s state-of- the-art grown nanomagnets, namely of weak surface disorder ,\none can map this atomistic approach onto a macroscopic model for the net magnetic moment mdefined in Eq. ( 6),\nevolving in the effective potential (H.O.T. = higher-order t erms)\nEeff=−k2m2\nz+k4/parenleftbig\nm4\nx+m4\ny+m4\nz/parenrightbig\n+H.O.T. (7)\nWe remark in passing that there is a rich literature of works [ 7,24,37–39] that study, in the macrospin approach, the\nswitching of a nanomagnet assisted by the circularly polari zed microwave field in Eq. ( 3) and, in some cases, analytical\nanalysis is performed in order to determine the various stab le states of the system and their optimal switching paths.\nIn the present situation, the effective model in Eq. ( 7), as compared to the macrospin studied in the cited works, ta kes\ninto account the internal structure and boundary effects in t he nanomagnet, and these lead to the quartic term in\nthe energy potential in Eq. ( 7). Because of the latter, highly nonlinear contributions ar e brought in and it is thereby\ndifficult, if not impossible, to carry out similar analytical investigations even at equilibrium.\nIn the sequel, we will refer to Eqs. ( 7,6) as the effective-one-spin problem (EOSP) while the many-sp in nanomagnet,\ndescribed by the set of equations ( 1) and ( 4), will be referred to as the many-spin problem (MSP). We emph asize\nthat the two leading terms in Eq. ( 7) have coefficients k2andk4whose value and sign are functions of the atomistic\nparameters ( J,Kc,Ks,etc) and of the size and shape of the nanomagnet [ 33,35,36,40]. Moreover, we note that\nboth the core and surface may contribute to k2andk4. For example, when the anisotropy in the core is uniaxial,\nk2≃kcNc/N, whereNcis the number of atoms in the core, see Ref. 34. In fact, even in this case the quartic term\nappears and is a pure surface contribution. Regarding the co efficient of the quartic contribution, for a sphere we\nhaveK4=κK2\ns/zJ(ork4=κk2\ns/z) whereκis a surface integral [ 33] and for a cube, k4=/parenleftbig\n1−0.7/N1/3/parenrightbig4k2\ns/z\n[40]. The details of the conditions under which this effective mo del is applicable are discussed in Ref. 33and may be\nsummarized as follows: the spin misalignment (or canting) s hould not be too strong. Therefore, the effective model\ncan be built for a nanomagnet whose surface anisotropy const antKs, as compared to the spin-spin exchange coupling\nJ, is small ( ks/lessorsimilar1). On the other hand, the nanomagnet size should not be too lar ge for it to be considered as a\nsingle magnetic domain, for a given underlying material.\nIn Fig. 3we plot the time evolution of the net magnetic moment of a nano magnet of size 11×11×11with the\nparameters indicated in the legend, for both the NSA (a) and T UA (b) and . In both situations, we see that for the\nsurface anisotropy constant considered, the effective mode l (EOSP) recovers very well the behavior of the many-spin5\n0 2,5 5 7,5 10 12,5 15-1-0,500,51\nEOSP\nMSP\nNSA: Kc = Ks = 0.008meV\nHrf = 80mT, ω = 20 GHz\nHdc = 8mT\nt (ns)m(0).m(t)\nN = 113\na)\n0 2,5 5 7,5 10-1-0,500,51\nEOSP\nMSP\nTUA: Kc = Ks = 0.008meV\nHrf = 80mT, ω = 20 GHz\nHdc = 8mT\nt (ns)m(0).m(t)\nN = 113\nb)\nFigure 3: Magnetization temporal evolution for a nanomagne t ofN= 113spins. (Upper panel) EOSP compared to MSP+NSA\nand (lower panel), EOSP compared to MSP and TUA.\nnanomagnet (MSP). Hence, in the sequel all results will be pr esented for the EOSP model, since the corresponding\ncalculation spares us much CPU time while producing the same results, as long as the validity conditions for EOSP\nare met, which will be the case for all subsequent calculatio ns [See discussion below]. Note that when the size and\nthe other parameters of the MSP system are varied, the coeffici entsk2andk4of the EOSP model ( 7) are accordingly\nadjusted.\nII. MORE RESULTS AND DISCUSSION\nA. Temporal evolution\nIn Fig. 4we plot the results for a nanomagnet of size N= 11×11×11. The upper panel is the ( ω,Hrf) phase\ndiagram for TUA and NSA. In both models, the value of the aniso tropy constant is the same for all spins within\nthe nanomagnet. The physical parameters used are given in th e caption and legends. The result in Fig. 4(a) shows\nthat the NSA model with a nonuniform distribution of the anis otropy easy axes allows for switching in a region with\nsmaller amplitude and frequency (left lower corner of the di agram) than the TUA model. On the other hand, the\nlatter allows for a wider domain of switching field intensiti es, though of higher values. Next, the graph in Fig. 4(b)\nshows that, for a given amplitude and frequency of hrf, switching is obtained with NSA and not with TUA. Again,\nthis means that the misalignment induced by the Néel surface anisotropy favors switching. This result, however,\ndepends on frequency as is shown by the graphs in Fig. 4(b) and (c). Now, comparing the graphs in Figs. 4(b) and\n(d), we see that switching is achieved within TUA but at the ex pense of a higher rf field amplitude. Under this field,\nand higher frequencies, switching is achieved in both model s, as witnessed by the graph in Fig. 4(e).\nThe general explanation of these observations resides in th e anisotropy configuration within the two models, NSA\nand TUA, for a box-shaped nanomagnet. For the TUA model, the a nisotropy is textured with all easy axes pointing\nin thez(vertical) direction. So, on the horizontal facets, NSA and TUA have the same easy axis[ 34]. However, on\nthe other four facets and along the edges, NSA has a perpendic ular effective anisotropy direction. As such, in the\nTUA model the effective anisotropy field is stronger (more rig id) and thereby the magnetization switching requires\nhigher external fields and/or higher frequencies.\nB. Effect of surface anisotropy ( Ks)\nThe first three phase diagrams in Fig. 5are obtained for TUA and NSA models with the same (fixed) aniso tropy\nconstant in the core and increasing anisotropy constant at t he surface. Comparing the three diagrams, we first\nobserve that the TUA model renders a phase diagram that shift s to the right along the diagonal, i.e.towards higher\namplitudes and frequencies of the rf field. This is due to the f act that as Ksincreases it makes the system with textured\nanisotropy ( i.e.TUA) more rigid. On the other hand, the NSA seems to be less sen sitive to the increase of Ks, apart6\nfrom a small shift to lower amplitudes of hrf. In fact, once the latter has reached the sufficient value for i nducing\na nonuniform spin configuration, the switching mechanism re mains the same. However, as Ksis further increased,\nthe phase diagram shows more “holes” which means that switch ing is no longer possible for all field amplitudes and\nfrequencies. This is shown by the last phase diagram in Fig. 5. In fact, when surface anisotropy is strong enough (but\nnot too strong) as to induce spin misalignment, several gene rations of spin waves are created with different amplitudes\nand frequencies [ 9,10,41,42], but only those modes that are at resonance or whose frequen cies match the rf field\nfrequency do participate in the switching mechanism [ 38,39]. On the other hand, the results in the lower right panel\nof Fig. 5show that the effective model (EOSP) reaches the limits of its validity regarding the parameters Ks. Indeed,\nin Refs. [ 34,36], it was checked that the validity limit on ks=Ks/Jis about 0.25for a simple cubic lattice and 0.35\nfor a face-centered cubic lattice.\nSumming up, it is clearly demonstrated that surface effects, as exemplified by the NSA model, open up switching\nchannels for relatively low rf fields intensities and freque ncies, as compared to the textured-anisotropy model TUA.\nThis is seen in all phase diagrams in Fig. 5and in subsequent ones. Indeed, they all exhibit a switching area that\nstretches to the left lower corner, though they remain narro wer at higher intensities and frequencies.\nIn general, in a nanomagnet with nonuniform anisotropy, tra nsverse spin waves are excited which are not necessarily\nspin waves in an equilibrium state. These transverse spin flu ctuations trigger spin motion which may grow up into a\ncoherent dynamics and ultimately induce magnetization swi tching[ 9,10,41,42].\nC. Size effect\nIn Fig. 6we plot the ̟-hrfphase diagram for both the NSA and TUA nanomagnets of varying size (cube\nside),N= 9,11,15,25,50,100which corresponds to a ratio of surface-to-total number of s pins equal to Nst=\n64%,45%,35%,22%,12%,6%, respectively. The anisotropy constants are chosen so that Ks/Kc= 10.\nFirst, we see that, in what regards the TUA model, as the size i ncreases ( Nstdecreases) the phase diagram\nextends towards lower amplitudes and frequencies of the rf fi eld. For example, for the size N= 7, corresponding to\nNst= 64% ,i.e.with a large surface contribution (in number of spins and ani sotropy strength), the phase diagram of\nTUA “disappears” which means that switching in this case wou ld require much higher values of the rf field amplitude\nand frequency. On the other hand, upon increasing the size, t he surface contribution decreases in terms of spins\nnumber, and thereby the core effective anisotropy becomes re latively stronger than that on the surface and, as such,\nlow-frequency and small-field dynamics takes over. This is i llustrated in Fig. 6by the results for size N ≥11.\nThe situation for the NSA model is somewhat reversed. First, for all these sizes and the corresponding surface\ncontributions, magnetization switching is achieved for al l rf fields (amplitude and frequency) in the explored ranges.\nMoreover, the NSA phase diagram widens in the lower left corn er of the diagram, i.e.for lower amplitudes and\nfrequencies of the rf field when Nstdecreases. This means that, as was shown by the results in Fig .5, when the\ncontribution of surface anisotropy dominates, there are on ly some specific modes that contribute to switching.\nAs can be seen in Fig. 6, at very large system sizes ( e.g.50,100), the surface contribution becomes negligible and\nthe two models tend to (asymptotically) yield the same switc hing diagram, since the black diagram widens and the\nblue one shrinks. This is mainly due to the fact that as Nst→0, the effective anisotropy in the two models becomes\nthe same.\nIII. CONCLUSION\nWe have studied the effect of surface anisotropy, and the spin misalignment it induces, on the magnetization\nswitching in a many-spin nanomagnet subjected to an rf magne tic field, on top of a DC magnetic field. Our study is\nbased on the numerical solution of the damped Landau-Lifshi tz equation for exchange-coupled atomic spins.\nWe have demonstrated that surface effects, as exemplified by N éel’s model for surface atoms, open up switching\nchannels with relatively low rf fields intensities and frequ encies, as compared to the model with parallel easy axes,\nor the so-called macrospin model. In general, these favorab le switching conditions depend on the size and shape of\nthe nanomagnet and, of course, on the underlying magnetic pr operties and energy parameters. 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Lett. 99, 227207 (2007).\n[40] D. A. Garanin, Phys. Rev. B 98, 054427 (2018).\n[41] V. I. Yukalov, Phys. Rev. B 71, 184432 (2005) .\n[42] V. I. Yukalov, V. K. Henner, and P. V. Kharebov, Phys. Rev. B 77, 134427 (2008) .9\n0 10 20 30 40 50 60 70050100150200250\nNSA: Kc = Ks = 0.008 meV\nTUA: Kc = Ks = 0.008 meV Hrf (mT)\n ω (GHz)Hdc = 8 mT\n(a)\n0 2,5 5 7,5 10-1-0,500,51\nt (ns)m(0).m(t)\nHrf = 30 mT, ω = 5 GHz\nHdc = 8 mTTUA Kc = Ks= 0.008 meV\nNSA Kc = Ks = 0.008 meV\n(b)\n0 5 10 15-1-0,500,51\nt (ns)m(0).m(t)\nHrf = 30 mT, ω = 15 GHz\nHdc = 8 mTNSA Kc = Ks = 0.008 meV\nTUA Kc = Ks = 0.008 meV\n(c)\n0 2,5 5 7,5 10-1-0,500,51\nNSA Kc = Ks = 0.008 meV\nTUA Kc = Ks = 0.008 meV\nt (ns)m(0).m(t)\nHrf = 70 mT, ω = 5 GHz\nHdc = 8 mT\n(d)\n0 2,5 5 7,5 10-1-0,500,51\nNSA Kc = Ks = 0.008 meV\nTUA Kc = Ks = 0.008 meV\nt (ns)m(0).m(t)\nHrf = 70 mT, ω = 15 GHz\nHdc = 8 mT\n(e)\nFigure 4: Upper panel: phase diagram for TUA and NSA models. M iddle panel: time evolution for Hrf= 30mT andω= 5 GHz\n(left) and ω= 15 GHz (right). Lower panel: time evolution for Hrf= 70mT andω= 5 GHz (left) and ω= 15 GHz (right). In\nall cases Hdc= 8mT .N= 113.10\n0 10 20 30 40 50 60 70050100150200250\nTUA: Kc = 0.008meV, Ks = 0.025meV\nNSA: Kc = 0.008meV, Ks = 0.025meV Hrf (mT)\n ω (GHz)0 10 20 30 40 50 60 70050100150200250\nTUA: Kc = 0.008meV, Ks = 0.05meV\nNSA: Kc = 0.008meV, Ks = 0.05meV Hrf (mT)\n ω (GHz)\n0 10 20 30 40 50 60 70050100150200250\nTUA: Kc = 0.008meV, Ks = 0.08meV\nNSA: Kc = 0.008meV, Ks = 0.08meV Hrf (mT)\n ω (GHz)0 10 20 30 40 50 60 70050100150200250\nNSA: Kc = 0.008meV, Ks = 0.32meV Hrf (mT)\n ω (GHz)\nFigure 5: Phase diagram for TUA and NSA with increasing (down wards) surface anisotropy constant Ks.Hdc= 8mT .\nN= 113.11\n0 10 20 30 40 50 60 70050100150200\nNSA: Kc = 0.008meV, Ks = 10Kc Hrf (mT)\n ω (GHz)N = 73\n0 10 20 30 40 50 60 70050100150200250\nTUA: Kc = 0.008meV, Ks = 10Kc\nNSA: Kc = 0.008meV, Ks = 10Kc Hrf (mT)\n ω (GHz)N = 153\n0 10 20 30 40 50 60 70050100150200250\nTUA: Kc = 0.008meV, Ks = 10Kc\nNSA: Kc = 0.008meV, Ks = 10Kc Hrf (mT)\n ω (GHz)N = 253\n0 10 20 30 40 50 60 70050100150200250\nTUA: Kc = 0.008meV, Ks = 10Kc\nNSA: Kc = 0.008meV, Ks = 10Kc Hrf (mT)\n ω (GHz)N = 503\n0 10 20 30 40 50 60 70050100150200250\nTUA: Kc = 0.008meV, Ks = 10Kc\nNSA: Kc = 0.008meV, Ks = 10Kc Hrf (mT)\n ω (GHz)N = 1003\nFigure 6: Phase diagram for the model NSA with varying size N= 73,113,153,253,503,1003." }, { "title": "1806.00658v1.Ultra_low_damping_insulating_magnetic_thin_films_get_perpendicular.pdf", "content": "1 \n Ultra -low damping insulating magnetic thin films get perpendicular \n \nLucile Soumah1, Nathan Beaulieu2, Lilia Qassym3, Cécile Carrétero1, Eric Jacquet1, Richard Lebourgeois3, \nJamal Ben Youssef2, Paolo Bortolotti1, Vincent Cros1, Abdelmadjid Anane1* \n \n1 Unité Mixte de Physique CNRS , Thales, Univ. Paris -Sud, Université Paris Saclay, 91767 Palaiseau, France \n2LABSTICC, UMR 6285 CNRS, Université de Bretagne Occidentale, 29238 Brest, France \n3 Thales Research and Technology, Thales 91767 Palaiseau , France \n* Email : madjid.anane@u -psud.fr \n \nA magnetic material combining low losses and large Perpendicular Magnetic Anisotropy (PMA) is still a \nmissing brick in the magnonic and spintronic field s. We report here on the growth of ultrathin Bismuth \ndoped Y 3Fe5O12 (BiYIG ) films on Gd 3Ga 5O12 (GGG) and substituted GGG (sGGG) (111) oriented \nsubstrates. A fine tuning of the PMA is obtained using both epitaxial strain and growth induced \nanisotrop ies. Both spontaneously in -plane and out -of-plane magnetized thin films can be elaborated . \nFerromagnetic Resonance (FMR) measurement s demonstrate the high dynamic quality of these BiYIG \nultrathin films , PMA films with Gilbert damping values as low as 3 10-4 and FMR linewidth of 0.3 mT at \n8 GHz are achieved even for films that do not exceed 30 nm in thickness . Moreover, w e measure \nInverse Spin Hall Effect (ISHE) on Pt/BiYIG stack s showing that the magnetic insulator ’s surface is \ntransparent to spin current making it appealing for spintronic applications . \n \n 2 \n Introduction. \nSpintronic s exploit s the electron’s spin in ferromagnetic transition metal s for data storage and data \nprocessing. Interestingly, as spintronics codes information in the angular momentum degre es of \nfreedom , charge transport and therefore the use of conducting materials is not a requirement, opening \nthus electronics to insulators . In magnetic insulators (MI), pure spin currents are described using \nexcitation states of the ferromagnetic background named magnons (or spin waves). Excitation, \npropagation and detection of magnons are at the confluent of the emerging concepts of magnonics \n1,2, caloritronics3 and spin -orbitronics4. Magnons, and their classical counterpart , the spin waves (SWs) \ncan carry information over distances as large as millimeters in high quality thick YIG films, with \nfrequencies extending from the GHz to the THz regime5–7. The main figure of merit for magnonic \nmaterials is the Gilbert damping 1,5,8 which has to be as small as possible. This makes the number of \nrelevant materials for SW propagation quite limited and none of them has yet been found to possess a \nlarge enough perpendicular magnetic anisotropy (PMA ) to induce spontaneous out -of-plane \nmagnetization . We report here on the Pulsed Laser Deposition (PLD) growth of ultra -low loss MI \nnanometers -thick films with large PMA : Bi substituted Yttrium Iron Garnet ( BixY3-xFe5O12 or BiYIG ) where \ntunability of the PMA is achieved through epitaxial strain and Bi doping level. The peak -to-peak FMR \nlinewidth (that characterize the losses) can be as low as 𝜇0𝛥𝐻pp=0.3 mT at 8 GHz for 30 nm thick \nfilms. This material thus opens new perspectives for both spintronic s and magnonic s fields as the SW \ndispersion relation can now be easily tuned through magnetic anisotropy without the need of a large \nbias magnetic field. Moreover, energy efficient data storage devices based on magn etic textures existing \nin PMA materials like magnetic bubble s, chiral domain walls and magnetic skyrmions would benefit from \nsuch a low loss material for efficient operation9. \nThe study of micron -thick YIG films grown by liquid phase epitaxy (LPE) was among the hottest topics in \nmagnetism few decades ago . At this time, it has been already noticed that unlike rare earths (Thulium, \nTerbium, Dysprosium …) substitutions, Bi substitution does not overwhelmingly increase the magnetic \nlosses10,11 even though it induces high uniaxial magnetic anisotropy12–14 . Very recently, ultra -thin MI \nfilms showing PMA have been the subject of an increasing interes t 15,16: Tm 3Fe5O12 or BaFe 12O19 \n(respectively a garnet and an hexaferrite) have been used to demonstrate spin -orbit -torque \nmagnetization reversal using a Pt over -layer as a source of spin current 4,17,18. However, their large \nmagnetic losses prohibit their use as a spin -wave medium (reported va lue o f 𝜇0𝛥𝐻pp of TIG is 16.7 mT at \n9.5 GHz)19. Hence, whether it is possible to fabricate ultra -low loss thin films with a large PMA that can \nbe used for both magnonics and spintronics applications remains to be demonstrated . Not only l ow 3 \n losses are important for long range spin wave propagation but they are also necessary for spin transfer \ntorque oscillators (STNOs) as the threshold current scales with the Gilbert damping20. \nIn the quest for the optimal material platform , we explore here the growth of Bi doped YIG ultra -thin \nfilms using PLD with different substitution; BixY3-xIG (x= 0.7, 1 and 1.5) and having a thickness ranging \nbetween 8 and 50 nm. We demonstrate fine tuning of the magnetic anisotropy using epitaxial strain and \nmeasure ultra low Gilbert damping values ( 𝛼=3∗10−4) on ultrathin films with PMA . \nResults \nStructural and magnetic characterization s \nThe two substrates that are used are Gallium Gadolinium Garnet (GGG) which is best lattice matched to \npristine YIG and substituted GGG (sGGG) which is traditionally used to accommodate substituted YIG \nfilms for photonics applications . The difference between Bi and Y ionic radii ( rBi = 113 pm and rY = 102 \npm)21 leads to a linear increase of the BixY3-xIG bulk lattice parameter with Bi content (Fig. 1 -(a) and Fig. \n1-(b)). In Fig. 1, we present the (2−) X-ray diffraction patterns (Fig.1 -(c) and 1 -(d)) and reciprocal \nspace maps (RSM) (Fig.1 -(e) and 1 -(f)) of BiYIG on sGGG(111) and GGG(111) substrates respectively . The \npresence of ( 222) family peaks in the diffraction spectra shown in Fig. 1 -(b) and 1 -(c) is a signature of the \nfilms’ epitaxial quality and the presence of Laue fringes attest s the coherent crystal structure existing \nover the whole thickness. As expected, all films on GGG are under compressive strain, whereas films \ngrown on sGGG exhibit a transition from a tensile (for x= 0.7 and 1) towards a compressive ( x= 1.5) \nstrain . Reciprocal Space Mapping of these BiYIG samples shown in Fig.1 -(e) and 1 -(f) evidences the \npseudomorphic nature of the growth for all films , which confirms the good epitaxy. \nThe static magnetic properties of the films have been characterized using SQUID ma gnetometry, Faraday \nrotation measurements and Kerr microscopy. As the Bi doping has the effect of enhancing the magneto -\noptical response 22–24, we measure on average a large Faraday rotation coefficients reaching up to 𝜃F =\n−3 °.𝑚−1 @ 632 nm for x= 1 Bi doping level and 15 nm film thickness . Chern et al .25 performed PLD \ngrowth of BixY3-xIG on GGG and reported an increase of 𝜃F\n𝑥= −1.9 °.𝜇𝑚−1 per Bi substitution x @ 632 \nnm. The Faraday rotation coefficients we find are slightly larger and m ay be due to the much lower \nthickness of our films as 𝜃F is also dependent on the film thickness26. The saturation magnetization ( Ms) \nremains constant for all Bi content (see Table 1) within the 10% experimental errors . We observe a clear \ncorrelation between the strain and the shape of the in -plane and out -of-plane hysteresis loop s reflecting \nchanges in the magnetic anisotropy. Wh ile films under compressive strain exh ibit in -plane anisotropy, \nthose under tensile strain show a large out -of-plane anisotrop y that can eventually lead to an out -of-\nplane easy axis for x= 0.7 and x= 1 grown on sGGG. The transition can be either induced by ch anging the 4 \n substrate (Fig.2 -(a)) or the Bi content ( Fig. 2-(b)) since both act on the misfit strain. We ascribe the \nanisotropy change in our films to a combination of magneto -elastic anisotropy and growth induced \nanisotropy, this later term being the domin ant one (see Supplementary Note 1). \nIn Fig. 2 -(c), we show the magnetic domains structures at remanance observed using polar Kerr \nmicroscopy for Bi1Y2IG films after demagnetization : µm-wide maze -like magnetic domains demonstrate s \nunambiguously that the magnetic easy axis is perpendicular to the film surface . We observe a decrease \nof the domain width (Dwidth) when the film thickness ( tfilm) increases as expected from magnetostatic \nenergy considerations. In fact, a s Dwidth is severa l order s of magnitude larger than tfilm, a domain wall \nenergy of σDW 0.7 and 0.65 mJ.m-2 (for x= 0.7 and 1 Bi doping) can inferred using the Kaplan and \nGerhing model27 (the fitting procedure is detailed in the Supplementary Note 2). \n \nDynamical characterization and spin transparency . \nThe most striking feature of these large PMA films is their extremely low magne tic losses that we \ncharacterize using Ferromagnetic Resonance (FMR) measurements. First of all, we quantify by in-plane \nFMR the anisotropy field HKU deduced from the effective magnetization ( Meff): HKU = M S – Meff (the \nprocedure to derive Meff from in plan e FMR is presented in Supplementary Note 3 ). HKU values for BiYIG \nfilms with different doping levels grown on various substrates are summarized in Table 1. As expected \nfrom out-of-plane hysteresis curves , we observe different signs for HKU. For spontaneously out -of-plane \nmagnetized samples , HKU is positive and large enough to fully compensate the demagnetizing field while \nit is negative for in -plane magnetized films. From these results , one can expect that fine tuning of the Bi \ncontent allow s fine tuning of the effective magnetization and consequently of the FMR resonan ce \nconditions. We measure magnetic losses on a 30nm thick Bi1Y2IG//sGGG film under tensile strain with \nPMA (Fig. 3 -(a)). We use the FMR absorption line shape by extracting the peak -to-peak linewidt h (𝛥𝐻pp) \nat different out-of-plane angle for a 30nm thick perpendicularly magnetized Bi 1Y2IG//sGGG film at 8 GHz \n(Fig. 3-(b)). This yields an optimal value of 𝜇0𝛥𝐻pp as low as 0.3 mT (Fig. 3-(c)) for 27° out-of-plane polar \nangle. We stress here that state of the art PLD grown YIG//GGG films exhibit similar values for 𝛥𝐻pp at \nsuch resonant conditions28. This angular dependence of 𝛥𝐻pp that shows pronounced variations at \nspecific angle is characteristic of a two magnons scattering relaxation process with few \ninhomogenei ties29. The value of this angle is sample dependent as it is related to the distribution of the \nmagnetic inhomogeneities . The dominance in our films of those two i ntrinsic relaxation processes \n(Gilbert damping and two magnons scattering) confirms the high films quality . We also derived the \ndamping value of th is film (Fig. 3-(d)) by selecting the lowest linewidth (corresponding to a specific out of 5 \n plane angle) at each frequency, the spread of the out of plane angle is ±3.5 ° around 30.5 °. The obtained \nGilbert damping value is α = 3.10-4 and the peak -to-peak extrinsic linew idth 𝜇0𝛥𝐻0 =0.23 mT a re \ncomparable to the one obtained for the best PLD grown YIG//GGG nanometer thick films28 (α =2.10-4). \nFor x= 0.7 Bi doping, the smallest observed FMR linewidth is 0.5 mT at 8 G Hz. \nThe low magnetic losses of BiYIG films could open new perspectives for magnetization dynamics control \nusing spin-orbit torques20,30,31. For such phenomenon interface transparency to spin curr ent is then the \ncritical parameter which is defined using the effective spin -mixing conductance ( 𝐺↑↓). We use spin \npumping experiments to estimate the increase of the Gilbert damping due to Pt deposition on Bi1Y2IG \nfilms. The spin mixing conductance can thereafter be calculated using 𝐺↑↓=4𝜋𝑀s𝑡film\n𝑔eff𝜇B(𝛥𝛼) where 𝑀s and \n𝑡film are the BiYIG magnetization saturation and thickness, 𝑔eff is the effective Landé factor ( 𝑔eff=2), \n𝜇B is the Bohr magneton and 𝛥𝛼 is the increase in the Gilbert damping constant induced by the Pt top \nlayer. We obtain 𝐺↑↓=3.9 1018m−2 which is comparable to what is obtained on PLD grown YIG//GGG \nsystems 28,32,33. Consequently , the doping in Bi should not alter the spin orbit -torque efficiency and spin \ntorque devices made out of BiYIG will be as energy efficient as their YIG counterpart . To further confirm \nthat spin current cross es the Pt/BiYIG interface , we measure Inverse Spin Hall Effect (ISHE) in Pt for a Pt/ \nBi1.5Y1.5IG(20nm)/ /sGGG in -plane magnetized film (to fulfill the ISHE geometry requirements the \nmagnetization needs to be in -plane and perp endicular to the measured voltage ). We measure a \ncharacteristic voltage peak due to ISHE that reverses its sign when the static in-plane magnetic field is \nreversed (Fig. 4). We emphasize here that the amplitude of the s ignal is similar to that of Pt/ YIG//GGG in \nthe same experimental conditions. \nConclusion \nIn summary, this new material platform will be highly beneficial for magnon -spintronics and related \nresearch fields like caloritronics. In many aspects , ultra -thin BiYIG films offer new leverages for fine \ntuning of the magnetic properties with no drawbacks compared to the reference materials of th ese \nfields: YIG. BiYIG with its higher Faraday rotation coefficient (almost two orders of magnitude more than \nthat of YIG) will increase the sensitivity of light based detection technics that can be used (Brillouin light \nspectroscopy (BLS) or time resolved Kerr microscopy34). Innovative scheme s for on -chip magnon -light \ncoupler could be now developed bridging the field of magnonics to th e one of photonics. From a \npractical point of view , the design of future active devices will be much more flexible as it is possible to \neasily engineer the spin waves dispersion relation through magnetic anis otropy tuning without the need \nof large bias magnetic fields. For instance, working in the forward volume waves configuration comes 6 \n now cost free, whereas in sta ndard in -plane magnetized media one has to overcome the demagnetizing \nfield. As the development of PMA tunnel junctions was key in developing today scalable MRAM \ntechnology , likewise, we believe that P MA in nanometer -thick low loss insulator s paves the path to new \napproaches where the magnonic medium material could also be used to store information locally \ncombining therefore the memory and computational functions, a most desirable feature for the brain -\ninspired neuromorphic paradigm . \n 7 \n Methods \nPulsed Laser Deposition (PLD) growth \nThe PLD growth of BiYIG films is realized using stoichiometric BiYIG target. The laser used is a frequency \ntripled Nd:YAG laser ( λ =355nm), of a 2.5Hz repetition rate and a fluency E varying from 0.95 to 1.43 \nJ.cm-2 depending upon the Bi doping in the target. The distance between target and substrate is fixed at \n44mm. Pri or to the deposition the substrate is annealed at 700°C under 0.4 mbar of O 2. For the growth, \nthe pressure is set at 0.25 mbar O 2 pressure. The optimum growth temperature varies with the Bi \ncontent from 400 to 550°C. At the end of the growth, the sample is cooled down under 300 mbar of O 2. \n \nStructural characterization \nAn Empyrean diffractometer with Kα 1 monochromator is used for measurement in Bragg -Brentano \nreflection mode to derive the (111) interatomic plan distance. Reciprocal Space Mapping is performed on \nthe same diffractometer and we used the diffraction along the (642) plane direction which allow to gain \ninformation on the in-plane epitaxy relation along [20 -2] direc tion. \n \nMagnetic characterization \nA quantum design SQUID magnetometer was used to measure the films’ magnetic moment ( Ms) by \nperforming hysteresis curves along the easy magnetic direction at room temperature. The linear \ncontribution of the paramagnetic (sGG G or GGG) substrate is linearly subtracted. \nKerr microscope (Evico Magnetics) is used in the polar mode to measure out-of-plane hysteresis curves \nat room temperature. The same microscope is also used to image the magnetic domains structure after \na demagn etization procedure. The spatial resolution of the system is 300 nm. \nA broadband FMR setup with a motorized rotation stage was used. Frequencies from 1 to 20GHz have \nbeen explored. The FMR is measured as the derivative of microwave power absorption via a low \nfrequency modulation of the DC magnetic field. Resonance spectra were recorded with the applied static \nmagnetic field oriented in different geometries (in plane or tilted of an angle 𝜃 out of the strip line \nplane). For o ut of plane magnetized samples the Gilbert damping parameter has been obtained by \nstudying the angular linewidth dependence. The procedure assumes that close to the minimum \nlinewidth (Fig 3a) most of the linewidth angular dependence is dominated by the inhomogeneous \nbroadening, thus opt imizing the angle for each frequency within few degrees allows to estimate better 8 \n the intrinsic contribution. To do so we varied the out of plane angle of the static field from 2 7° to 34 ° for \neach frequency and w e select the lowest value of 𝛥𝐻pp. \nFor Inverse spin Hall effect measurements, the same FMR setup was used, however here the modulation \nis no longer applied to the magnetic field but to the RF power at a frequency of 5kHz. A Stanford \nResearch SR860 lock -in was used a signal demodulator. \nData availability : \nThe data that support the findings of this study are available within the article or from the corresponding \nauthor upon reasonable request . \nAcknowledgements: \n We acknowledge J. Sampaio for preliminary Faraday rotation measurements and N. Rey ren and A. \nBarthélémy for fruitful discussions. This research was supported by the ANR Grant ISOLYIG (ref 15 -CE08 -\n0030 -01). LS is partially supported by G.I.E III -V Lab. France. \n \nAuthor Contributions : \nLS performed the growth, all the measurements, the da ta analysis and wrote the manuscript with AA . NB \nand JBY conducted the quantitative Faraday Rotation measurements and participated in the FMR data \nanalysis. LQ and fabricated the PLD targets . RL supervised the target fabrication and participated in the \ndesign of the study . EJ participated in the optimization of the film growth conditions. CC supervised the \nstructural characterization experiments. AA conceived the study and w as in charge of overall direction . \nPB and VC contributed to the design and implem entation of the research . 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Lett. 117, 1–5 (2016). \n 11 \n Figures Captions : \n \nFigure 1 -Structural properties of ultra -thin BiYIG films . \n(a) and (b) : Evolution of the target cubic lattice parameter of BixY3-xIG, the dashed line represents the \nsubstrate (sGGG and GGG respectively) lattice parameter and allow s to infer the expected tensile or \ncompressive strain arising for each substrate/target combination. \n(c) and ( d): 2𝜃−𝜔 X-Ray diffraction scan along the (111) out-of-plane direction for BixY3-xIG films gr own \non sGGG (111) and GGG (111) respectively. From the film and substrate diffraction peak position , we can \nconclude about the nature of the strain. Compressive strain is observed for 1.5 doped films grown on \nsGGG substrate and for all films grown on GGG w hereas tensile strain occurs for films with x= 0.7 and x= \n1 Bi content grown on sGGG. \n(e) and (f) : RSM along the evidence the (642) oblique plan showing pseudomorphic growth in films: both \nsubstrate and film the diffraction peak are aligned along the qx\\\\[20-2] direction. The relative position of \nthe diffraction peak of the film (up or down) along qx is related to the out-of-plane misfit between the \nsubstrate and the film (tensile or compressive). \nFigure 2 -Static magnetic properties . \n(a) Out-of plane Kerr hysteresis loop performed in the polar mode for Bi0.7Y2.3IG films grown on the two \nsubstrates: GGG and sGGG \n(b) Same measurement for BixY3-xIG grown on sGGG with the three different Bi doping ( x= 0.7, 1 and 1.5) . \nBi0.7Y2.3IG//GGG is in -plane magnetized whereas perpendicular magnetic anisotropy (PMA) occurs for x= \n0.7 and x= 1 films grown on sGGG: square shaped loops with low saturation field ( µ0Hsat about 2.5 mT) are \nobserved. Those two films are experiencing tensile strain . Whereas the inset shows that the Bi1.5Y1.5IG \nfilm saturates at a much higher field wi th a curve characteristic of in -plane easy magnetization direction. \nNote that for Bi1.5Y1.5IG//sGGG µ0Hsat ≈290mT >µ0Ms≈162mT which points toward a negative uniaxial \nanisotropy term ( µ0HKU) of 128mT which is coherent with the values obtained from in plane FMR \nmeasurement . \n(c) Magnetic domains structure imaged on Bi 1Y2IG//sGGG films of three different thicknesses at \nreman ant state after demagnetization . The scale bar, displayed in blue , equal s 20 µm . Periods of the \nmagnetic domains structure ( Dwidth) are derived using 2D Fast Fourier Transform . We obtained Dwidth =3.1, \n1.6 and 0.4 µm for tBi1Y2IG= 32, 47 and 52 nm respectively. We note a decrease of Dwidth with increasing \ntBi1Y2IG that is coherent with the Kaplan and Gehring model valide in the case Dwidth>>tBiYIG. 12 \n Figure 3-Dynamical properties of BiYIG films with PMA. \n(a) Sketch of the epitaxial configuration for Bi1Y2IG films , films are grown under tensi le strain giving rise \nto tetragonal distortion of the unit cell. \n(b) Out-of-plane angular depend ence of the peak -to-peak FMR linewidth ( 𝛥𝐻pp) at 8 GHz on a 30 nm \nthick Bi1Y2IG//sGGG with PMA (the continuous line is a guide for the eye) . The geometry of the \nmeasurement is shown in top right of the graph. The wide disparity of the value for the peak to peak \nlinewidth 𝛥𝐻pp is attributed to the two magnons scattering process and inhomogeneties in the sample . \n(c) FMR absorption linewidth o f 0.3 mT for the same film at measured at 𝜃=27°. (d) Frequency \ndependence of the FMR linewidth . The calculated Gilbert damping parameter and the extrinsic linewidth \nare displayed on the graph . \nFigure 4- Inverse Spin Hall Effect of BiYIG films with in plane magnetic anisotropy. \nInverse Spin Hall Effect (ISHE) voltage vs magnetic field measured on the Pt / Bi1.5Y1.5IG//sGGG sample in \nthe FMR resonant condition at 6 GHz proving the interface transparency to spin current. The rf excitation \nfield is about 10-3 mT which corresponds to a linear regime of excitation. Bi1.5Y1.5IG//sGGG present s an in-\nplane easy magnetization axis due to a growth under compressive strain. \n \n 13 \n Table 1 - Summary of the magnetic properties of BixY3-xIG films on GGG and sGGG \nsubstrates. \nThe saturation magnetization is roughly unchanged. The effective magnetization Meff obtained through \nbroad -Band FMR measurements allow to deduce the out -of-plane anisotropy fields HKU (HKU =Ms-Meff) \nconfirming the dramatic changes of the out-of-plane magnetic anisotropy variations observed in the \nhysteresis curves . \n \nBi doping Substrate µ0MS(mT) µ0Meff(mT) µ0HKU(mT) \n0 GGG 157 200 -43 \n0.7 sGGG 180 -151 331 \n0.7 GGG 172 214 -42 \n1 sGGG 172 -29 201 \n1 GGG 160 189 -29 \n1.5 sGGG 162 278 -116 14 \n Figure 1 \n (f) \n0.5 1.0 1.51.2351.2401.2451.2501.255 Cubic Lattice parameter in nmBi content \n sGGG\n45 50 55103106109\n Intensity (cps)\n2 angle(°)0.711.5\n0.711.5\n1.70 1.724.254.304.35qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+01 1E+05\n1.74 1.754.234.274.30qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n1.73 1.764.214.254.29qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n0.7 1 1.5\nsGGGBi0.7Y2.3IG\nsGGGBi1Y2IG\nBi1.5Y1.5IGsGGG\n45 50 55102105108\n Intensity (cps)\n2 angle (°)\n0.5 1.0 1.51.2351.2401.2451.2501.255 Cubic Lattice parameter in nmBi content \n \nGGG0.711.5\n0.7511.5\n1.75 1.774.204.304.40qz//[444] (rlu)*10\nqx//[2-20] (rlu)*101E+01 1E+05\n1.750 1.7754.284.324.35qz//[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n0.7\nGGG\nBi0.7Y2.3IGGGG\nBi1Y2IG1sGGG\nGGG(a)\n(b)(c)\n(d)(e)\n(f)15 \n Figure 2 \n \n \n \n16 \n Figure 3 \n \n \n \n17 \n Figure 4 \n-130 -120 -110 110 120 130-4000400800V (nV)\nµ0H (mT)2 \n Supplementary Notes 1- Derivation of the magneto -elastic anisotropy \n \nThe out -of-plan e anisotropy constant KU is ascribed to be a result of, at least , two contributions : a \nmagneto -elastic anisotropy term induced by strain (𝐾MO) and a term that is due to preferential \noccupation of Bi atoms of non equivalent dodecahedral sites of the cubic u nit cell. This last term is \nknown as the growth induced anisotropy term 𝐾GROWTH . From X -ray characterizations and f rom the \nknown properties of the thick BiYIG LPE grown films, it is possible to calculate the expected values of \n𝐾MO in each doping/substra te combination. We thereafter deduce 𝐾GROWTH from the relation \n𝐾U=𝐾MO+ 𝐾GROWTH . KMO is directly proportional to the misfit between the film and the substrate: \n𝐾MO=3\n2∙𝐸\n1−𝜇∙𝑎film −𝑎substrate\n𝑎film∙𝜆111 \nWhere E, µ and λ111 are respectively the Young modulus, the Poisson coefficient, and the \nmagnetostrictive constant along the (111) direction. Those constant are well established for the bulk1: E= \n2.055.1011 J.m-3, µ= 0.29 . The magnetostriction coefficient λ111 for the thin film case is slightly higher \nthan that of the bulk and depends upon the Bi rate x: 𝜆111(𝑥)=−2.819 ∙10−6(1+0.75𝑥) 2. The two \nlattice par ameter entering in the equation: afilm and asubstrate correspond to the lattice parameter of the \nrelaxed film structure and of the subs trate. Under an elastic deformation afilm can be derived with the \nPoisson coefficient: \n𝑎film =𝑎substrate −[1−𝜇111\n1+𝜇111]𝛥𝑎⊥ with ∆𝑎⊥=4√3𝑑444film−𝑎substrate \nAll values for the different target/substrate combinations are displayed in the Table S-1. We note here \nthat a negative (positive) misfit corresponding to a tensile (compressive) strain will favor an out-of-plane \n(in-plane ) magnetic anisotropy which is coherent with what is observed in our samples. To estimate the \ncontribution to the magnetic energy of the magneto elastic anisotropy term we compare it to the \ndemagnetizing field 𝜇0𝑀s that favors in-plane magnetic anisotropy in thin films. Interestingly the \nmagneto elastic field ( 𝜇0𝐻MO) arising from 𝐾MO (𝜇0𝐻MO=2𝐾MO\n𝑀s) never exceed 30% o f the \ndemagnetizing fields and therefore cannot alon e be responsible of the observed PMA. \n \nStudies on µm-thick BiYIG films grown by LPE showed that PMA in BiYIG arises due to the growth \ninduced anisotropy term 𝐾GROWTH , this term is positive 3 for the case of Bi substitution . We have \ninferred 𝜇0𝐻GROWTH values for all films using 𝐾U constants measured by FMR. The results are \nsummarized in Supplementary Figure 1. One can clearly see that 𝐾GROWTH is strongly substrate \ndependent and therefore does not depend sol ely on the Bi content. We conclude that strain play s a role \nin Bi3+ ion ordering within the unit cell . \n \n 3 \n \n \nSupplementary Figure 1- Summary of the inferred values of the effective magnetic \nanisotropy out -of-plan fields. \n \nHorizontal dash lines represent the magnitude of the demagnetization field µ0Ms. When µ0HKU is larger \nthan µ0Ms (dot line) films have a PMA, they are in -plane magnetized otherwise . \n0200400µ0HKuvs Bi content \n HKu = Hstrain+ Hgrowth\nµ0Ms \n1.50.7µ0HKu (mT) µ0 Hstrain\n µ0 Hgrowth 1\nµ0Ms\n-1000100200\n0.7 µ0HKu (mT) µ0Hstrain\n µ0Hgrowth\n14 \n Supplementary Table 1 - Summary of the films’ calculated magneto -elastic \nanisotropy constant ( KMO) and the corresponding anisotropy field HMO \n \n \n Bi content substrate afilm(Å) Δ a┴/afilm KMO (J.m-3) µ0 HMO(mT) µ0 Hdemag (mT) \n0.7 sGGG 12.45 0.6 5818 81 179 \n0.7 GGG 12.41 -0.4 -4 223 -61 172 \n1.0 sGGG 12.47 0.3 3 958 57 172 \n1.0 GGG 12.42 -0.6 -6 500 -102 160 \n1.5 sGGG 12.53 -0.6 -8 041 -124 157 5 \n Supplementary Notes 2- Derivation of the domain wall energy \n \nTo derive the characteristic domain wall energy σDW for the maze shape like magnetic domains , we use \nthe Kaplan et al. model4. This model applies in our case as the ratio of the film thickness ( tfilm) to the \nmagnetic domain width ( Dwidth) is small (𝑡film\n𝐷width~ 0.01). The domain wall width and the film thickness are \nthen expected to be linked by: \n𝐷width =𝑡filme−π1.33𝑒𝜋𝐷0\n2𝑡film where 𝐷0=2𝜎w\n𝜇0𝑀s2 is the dipolar length. \nHence we expect a linear dependence of ln(𝐷width\n𝑡film) vs 1\n𝑡film : \nln(𝐷width\n𝑡film)=π𝐷0\n2∙ 1\n𝑡film+Cst. (1) \nThe magnetic domain width of Bi xY3-xIG//sGGG ( x = 0.7 and 1) for several thicknesses are extracted from \n2D Fourier Transform of the Kerr microscopy images at remanence. In Supplementary Fig ure 2, we plot \nln(𝐷width\n𝑡film) vs 1\n𝑡film which follow s the expected linear dependence of Equation (1) . We infer from the zero \nintercept an estimat ion of the dipolar length D0 of BiYIG films doped at 0.7 and 1 in Bi : D0 x=0.7= 16.5 µm \nand D0 x=1= 18.9µm. The corresponding domain wall energy are respectively 0.7 mJ .m-2 and 0.65 mJ .m-2. \nEven if the small difference in domain wall energy between the two Bi content may not be significant \nregarding the statistical fitting errors, it correlates to the decrease of the out-of-plane anisotropy ( KU) \nwith increasing the Bi content. 6 \n Supplementary Figure 2- Evolution of the domain width vs film thickness \n \nln(𝐷width\n𝑡film) vs 1\n𝑡film for Bi xY3-xIG//sGGG films doped at 0.7 (a) and 1 (b) in Bi. Dots correspond to the \nexperimental values. The dashed line is the linear fit that allows to extract the D0 parameter. \n \n25 50 752468\n ln(Dwidth/tfilm)\ntfilm(µm-1)50 1002468\n ln(Dwidth/tfilm)\n/tfilm(µm-1)Bi0.7Y2.3IG//sGGG\nD0=16.5µm\nBi1Y2IG//sGGG\nD0=18.9µm\n(a) \n(b) 7 \n \nSupplementary Notes 3 - Damping and effective magnetic field derivation \n \nFrom In Plane frequency dependent of FMR we can derive the effective magnetization (𝑀eff) using the \nKittel law: \n𝑓res=𝜇0𝛾√𝐻res(𝐻res+𝑀-eff) \nWhere γ is the gyromagnetic ratio of the BiYIG (assumed to be same as the one of the YIG) : γ=28 GHz.T-1. \n𝐻res and 𝑓res are respectively the FMR resonant field and frequency . The uniaxial magnetic anisotropy \ncan thereafter be derived using the saturation magnetizatio n form squid magnetometry using : \nMeff=Ms-HKU. The Gilbert damping ( α) and the inhomogeneous linewidth ( ΔH 0) which are the two \nparamaters defining the magnetic relaxation are obtained from the evolution of the peak to peak \nlinewidth ( ΔHpp) vs the resonant frequency ( fres): \n𝛥𝐻pp=𝛥𝐻0+2\n√3𝛼𝑓res\n𝜇0𝛾 (2) \nThe first term is frequency independent and often attributed magnetic inhomogeneity’s (anisotropy, \nmagnetization). \n 8 \n Supplementary Figure 3 - µ0ΔHpp vs fres on 18 nm thick Bi15.Y1.5IG//sGGG \n \nThe l inewidth frequency dependence from 5 to 19 GHz for in plane magnetized Bi 1.5Y1.5IG//sGGG sample \nallow to ex tract the damping and the inhomogeneous linewidth parameter using the Equation (2) . \n \n \n0 5 10 150.00.51.01.5 \n Hpp(mT)\nfres (GHz)H0=0.5 mT\n=1.9*10-39 \n Supplementary References: \n \n1. Hansen, P., Klages, C. ‐P. & Witter, K. Magnetic and magneto ‐optic properties of praseodymium ‐ \nand bismuth ‐substituted yttrium iron garnet films. J. Appl. Phys. 60, 721–727 (1986). \n2. Ben Youssef, J., , Legall, H. & . U. P. et M. C. Characterisation and physical study of bismuth \nsubstituted thin garnet films grown by liquid phase epitaxy (LPE). (1989). \n3. Fratello, V. J., Slusky, S. E. G., Brandle, C. D. & Norelli, M. P. Growth -induced anisotropy in \nbismuth: Rare -earth iron garnets. J. Appl. Phys. 60, 2488 –2497 (1986). \n4. Kaplan, B. & Gehring, G. A. The domain structure in ultrathin magnetic films. J. Magn. Magn. \nMater. 128, 111–116 (1993). \n " }, { "title": "1805.03799v1.Dust_modification_of_the_plasma_conductivity_in_the_mesosphere.pdf", "content": "arXiv:1805.03799v1 [physics.plasm-ph] 10 May 2018Dust modification of the plasma conductivity in the\nmesosphere\nB.P. Pandeya,∗, S.V. Vladimirovb,1\naDepartment of Physics & Astronomy, Macquarie University, S ydney 2109, NSW,\nAustralia\nbMetamaterials Laboratory, National Research University o f Information Technology,\nMechanics & Optics, St. Petersburg 199034, Russia and Joint Institute of High\nTemperatures, Russian Academy of Sciences, Izhorskaya 13/ 2, 125412 Moscow, Russia,\nSchool of Physics, The University of Sydney, Sydney 2006, NS W, Australia\nAbstract\nRelativetransverse drift(withrespect totheambient magneticfie ld) between\nthe weakly magnetized electrons and the unmagnetized ions at the lo wer alti-\ntude (<80km) and between the weakly magnetized ions and unmagnetized\ndust at the higher altitude ( <90km) gives rise to the finite Hall conduc-\ntivity in the Earth´ s mesosphere. If, on the other hand, the numb er of free\nelectrons is sparse in the mesosphere and most of the negative cha rge resides\non the weakly magnetized, fine, nanometre sized dust powder and p ositive\ncharge on the more massive, micron sized, unmagnetized dust, the sign of\nthe Hall conductivity due to their relative transverse drift will be op posite\nto the previous case. Thus the sign of the Hall effect not only depen ds on\nthe direction of the local magnetic field but also on the nature of the charge\ncarrier in the partially ionized dusty medium.\nAs the Hall and the Ohm diffusion are comparable below 80km, the low\nfrequency ( ∼10−4−10−5s−1) long wavelength ( ∼103−104km) waves will\nbe damped at this altitude with the damping rate typically of the order\nof few minutes. Therefore, the ultra–low frequency magnetohyd rodynamic\nwaves can not originate below 80km in the mesosphere. However, ab ove\n80km since Hall effect dominates Ohm diffusion the mesosphere can ho st\nthe ultra–low frequency waves which can propagate across the ion osphere\n∗Corresponding author\nEmail addresses: birendra.pandey@mq.edu.au (B.P. Pandey),\nsergey.vladimirov@sydney.edu.au (S.V. Vladimirov)\nPreprint submitted to JATP November 6, 2018with little or, no damping.\nKeywords: Mesosphere, D region, Plasma conductivity, Waves\nPACS:52.27.Lw, 94.20.de, 94.05.Bf\n1. Introduction\nThe Earth´ sionosphereconsists ofhorizontally stratifiedlayers o fpartially\nionized gas immersed in its magnetic field. The various altitude range of the\nionosphere is divided for convenience as D, E and F layer, with the D re gion\nspanning between 60 −90km, the E region between ∼90−150km and the\nF region between 150 −400 km [1]. The temperature from the ground up to\n∼15km altitude (troposphere) decreases with height. The tempera ture rises\nin the stratosphere ( ∈[15,50]km) before decreasing again in the mesosphere\nwhere it has the lowest value at about ∼80−90km. The temperature in\nthe mesosphere is about ∼190K though 160K or even lower temperature is\nalso possible at occasions. Surprisingly, the mesopause temperatu re in the\npolar regions is higher in winter than in summer.\nThe altitude profiles of the dominant neutrals as well as the ionized co m-\nponents and their variations above about ∼100km is quite well known.\nHowever, the lower ionosphere ( ∼60−100km), owing to the limited ex-\nperimental data base, to a large extent, is still poorly understood . The\n60−150km altitude region is too high for balloons and too low for satellite\nobservations posing considerable observational challenge, yet un derstanding\nof this region is crucial to the behaviour of radio transmission, the in itiation\nof sprites above thunderstorm etc. Owing to the D-region´ s impac t on the\nglobal climate change, this region has started receiving renewed at tention [2].\nWhile the magnetic field shields Earth from the solar wind, meteoroids\nand dust freely penetrate the atmosphere. Meteors are observ ed at all alti-\ntudebetween70 −400kmwithsmallmeteorsevaporatingbetween70 −120km\n[3]. Meteoric smoke particles ( ∼nm in size; 1nm = 10−9m) form from the\nrecondensing of the ablated meteoroid material at 80 −90km [4, 5]. The po-\nlar mesospheric summer echoes (PMSE) and noctilucent clouds (NLC s) also\ncalled polar mesospheric clouds (PMCs) are also observed at this altit ude.\nThere is a strong correlation between the observations of NLCs an d PMSEs\nsuggesting that they might have a common origin. The PMSE refers t o the\nstrong radar echoes observed at 50 −1.3GHz due to electron scattering at\nBragg scale whereas the NLC which is also observed in the similar range of\n2frequencies refers to the formation of water ice particles [6, 7, 8]. Typical\ndensity of these ice particles can vary between ∼10 to 103cm−3and their\nradius can vary between ∼10−100nm [8, 9]. It is quite plausible that the\nsize of the dust particles and the condensation of the water vapou r in the\nNLCs are interlinked. For example, the presence of large dust can n oticeably\nreduce the concentration of water vapour in the upper atmosphe re and this\nin turn can decrease the particle sizes [10].\nThe measurements during PMSE conditions shows a pronounced dep le-\ntion of the electron number density. In fact electron density can d ecrease\nby an order of magnitude where strong PMSE are observed. The de pletion\nof electron density, usually called electron bite–outs , is a generic feature of\nthe PMSE [11, 12]. It would appear that the electron deficit at this alt itude\noccurs only during PMSEs. However, the electron depletion is much m ore\ngeneric in the D-region owing to the availability of meteoric smoke part icles.\nIn fact, there is a distinct deficit of electrons between 80 and 90km . This\ndeficit canbeexplained by thepresence ofnegatively chargedmete oric smoke\nparticles [15].\nClearly, the plasma composition in the altitude region between 60 −\n100km is chemically complex. Whereas toward the E-region, the domin ant\nions are molecular (NO+and O+\n2), below a marked transition height clus-\nter ions (H+(H2O)nand NO+(H2O)n) constitute the bulk of the population\n[13, 14]. The presence of charged and neutral dust (mesospheric smoke par-\nticles, ice particles) are also important constituent of the partially io nized\ngas. After forming at higher altitude ( ∼90km), the largest of dust particles\nsettles to the lower altitude where they are visible as NLC whereas sm aller\ndust isvisiblethrough strong radar echoes [8, 9, 16, 17]. Estimates for the\nnumber density of small dust particles in the ionosphere ( ∼a few tens of\nnm to sub-visual in size) appears generally to be larger than about 1 02cm−3\nunder PMSE conditions [18].\nAs noted above the dust in the D-region is either neutral or carry e lec-\ntronic charge. The ratio of charged to neutral dust particles is ab out 5 to\n10 percent of the total dust number density [19, 20] implying that t he small\ndust particles are predominantly neutral. However, large ( >20nm) ice crys-\ntals are often negatively charged in the NLC. The presence of large dust can\nsignificantly augment the electron recombination rate causing the e lectron\nbiteouts. The average charge on the dust is negative owing to the la rge mo-\nbility of electrons. The typical charge Zon the dust will be −1efor particles\nwith radius a∈[1,10]nm,−2efor 30nm particles and −3efor 50nm parti-\n3cles [9]. Here eis the electron charge. We note that on the one hand, charged\nand neutral grains couple to the electromagnetic field via collisions wit h the\nelectrons and ions and on the other hand, charge fluctuation modifi es this\nfield [21, 22, 23]. Thus, dust-plasma coupling is responsible for some o f the\nnovel collective features in a dusty medium [24, 25, 26, 27]. The collisio n be-\ntween the plasma, neutrals and dust grains not only causes the diss ipation of\nthe high frequency waves but can also help the dissipationless propa gation of\nthe low frequency fluctuations. For example, if various collision freq uencies\nare higher than the dynamical frequency of interest then collision w ill move\nthe bulk medium (which is a sum of the plasma, dust and neutral partic les)\ntogether. In such a scenario collision facilitates undamped propaga tion of\nthe wave [28, 29, 30, 31]. However, in the opposite limit, when the collis ion\nfrequencies are smaller than the dynamical frequency of interest , collision\ncauses damping of the waves.\nThe 80−120kmregion of Earth´ s ionosphere is weakly ionised and weakly\nmagnetized with the neutral number density ( > nn= 1014cm−3at 80km)\nfar exceeding the ion number density ( ni= 103cm−3). The sub–visible small\ndust (∼nm) number density could be similar to the ion number density.\nAdding to this complexity is the role of the ambient magnetic field. The\ndynamical processes in the ionosphere are strongly controlled by t he cou-\npling of the largely neutral D and lower E region to the magnetic field. T his\ncoupling is facilitated by the frequent collisions between the neutrals and the\ndusty plasma particles which transmits the Lorentz force to the ne utrals. It\nis pertinent to recall here that in the past the D-layer was consider ed either\na purely neutral layer [32] or, the role of dust in the mesospheric pla sma\ndynamics was completely ignored [33]. But as is clear from the above de -\nscription, the abundance of tiny charged grains may well exceed th e electron\nabundance in the D-region and thus the plasma transport propert ies which\nneglects the presence of grains at this altitude is incomplete.\nThe role of dust in modifying the plasma conductivity is well known in\nthe space [34, 35] and astrophysical environment [29, 36]. The pre sence of\ndust not only affects the ionization structure of the plasma but also the gas\nphase abundances [36, 37]. Further, owing to the large mass and siz e distri-\nbution, the grains can couple directly as well as indirectly to the magn etic\nfield. Thus the charged grains not only modifies the ambipolar time-sc ale,\nbut may also give rise to the Hall diffusion [36]. By ambipolar diffusion here\nwe imply the diffusion of the magnetic field against the sea of neutrals d ue\nto the relative slippage of the frozen–in ions against the neutrals [38 ]. This\n4is different from the ambipolar diffusion of the plasma particles against the\nelectric field [39]. As we shall see, the mesosphere can be described in the\nframework of magnetohydrodynamics (with Ohm, ambipolar and Hall diffu-\nsion operating at various scale heights). This approach is different f rom the\nusualelectrodynamicsapproachinwhichthemagneticfieldisassume dstatic.\nAs has been noted by Parker (2007), in the magnetohydrodynamic approach,\nthe bulk velocity of the plasma fluid and the magnetic field is the primary\nvariable whereas in the electrodynamic approach, electric field and c urrent is\nthe primary variable. Both paradigm may sometimes arrive at the sam e con-\nclusion [41]. In the present work, we shall adopt the magnetohydro dynamic\napproach and first explore relative importance of the various diffus ivities\nbefore investigating the wave propagation in the dusty mesospher ic layer.\nWe shall provide an expression for the generalized Ohm´ s law in the ne xt\nsection where the relative importance of Ohm, Hall and ambipolar diffu sion\nis also discussed . In section III we describe the effect of Hall and Oh m\ndiffusion on the propagation of low frequency waves. It is shown tha t the\nlong wavelength waves may suffer significant damping if most of the ele ctrons\nhave been moped by the dust. In section IV discussion of the result along\nwith a brief summary is presented.\n2. Formulation\nWe shall assume a weakly ionized medium consisting of electrons, ions,\ncharged grains and neutral particles. In order to better elucidat e the role of\nthe charged dust, we shall assume that the grains have same size a nd ignore\ntheir size distribution. Likewise, for simplicity the difference between the\nmolecular and cluster ions will be neglected. Since the D-region is weak ly\nionized the inertia and the thermal pressure terms in the plasma mom entum\nequations can be neglected.\nThe most accurate way of calculating the transport properties of a gas is\nto employ Chapman-Enskog method which was applied to the ionosphe re by\nCowling [42]. This method is analytically involved and thus we shall adopt\na much simpler free–path method similar to the one employed by Baker and\nMartyn [43] for the ionosphere. Free-path theory assumes that the gas is\nnot accelerating, i.e. it makes no distinction between the values of th e mean\nvelocity at the beginning of the free pathand at the instant conside red. Thus\nthe charged particles drift through the sea of neutrals due to inst ant Lorentz\n5force, i.e.\n0 =−qjnj/parenleftbigg\nE′+vj×B\nc/parenrightbigg\n−ρjνjnvj. (1)\nHereqjnj(E′+vj×B/c) is the Lorentz force and E′=E+vn×B/cis the\nelectric field in the neutral frame with EandBas the electric and magnetic\nfields respectively, nj,vjis the number density and velocity, qjis the charge\non the plasma particles and dust and cis the speed of light. The last term\non the right hand in the above equation is the collision momentum excha nge\nterm in the neutral frame of reference. The electron-neutral a nd ion-neutral\ncollision frequencies are [1]\nνen= 5.4×10−10nnT1/2\ne,\nνin= 2.6×10−9nnA−1/2. (2)\nHereAis the mean neutral molecular mass in atomic mass units. Assuming\nA= 30 above collision frequency can be written in the following form [44]\nνen= 7.64×104nn+13T1/2\n+200,\nνin= 4.7×103nn+13A−1/2\n30. (3)\nHereA30=A/30,nn+13=nn/1013cm−3andT+200=T/200K. Since the\ndust-neutral collision rate is [45]\n< σv > dn= 7.1×10−9T1\n2\n+200a2\n−7cm3s−1, (4)\nwitha−7=a/10−7cm, the dust-neutral collision frequency becomes\nνdn=/parenleftbiggmn\nmd/parenrightbigg\nnn< σv > dn≃237nn+13., (5)\nHerewehaveassumed mn= 20mpwithmp= 1.67×10−24gandmd= 10−20g\nfor the dust mass density ρd= 2gm/cm−3.\nWe shall define the plasma Hall parameter\nβj=/parenleftbiggωcj\nνjn/parenrightbigg\n, (6)\nas the ratio of the cyclotron ωcj=qjB/mjcto the collision νjnfrequencies.\nThe value of this parameter is a measure of how well the magnetic field is\ncoupled to the neutral matter.\n6Height[Km]75 80 85 90log10(β)\n-6-4-202\nβe\nβi\nβd\nFigure 1: The variation of Hall βwhich is a ratio of the cyclotron to the plasma-neutral\ncollision frequencies is shown against the altitude in the above figure. The neutral number\ndensity for the summer mesosphere has been taken from [46].\nSinceωce= 5.3×106B0.3s−1,ωci= 102B0.3s−1andωcd= 0.5B0.3s−1for\na nm–sized grain with Z=−1, various Hall beta become\nβe≃70T−1\n2\n+200n−1\nn+13B0.3,\nβi≃2×10−2A1\n2\n30n−1\nn+13B0.3,\nβd≃2×10−3a−2\n−7T−1\n2\n+2n−1\nn+13B0.3, (7)\nwhereB0.3=B/0.3G. Clearly depending on the value of Hall βthe charged\ngrains may couple directly or indirectly to the ambient magnetic field. I n\nFig. (1) the variation of Hall β, Eq. (7) is plotted against the altitude. We\nsee from the figure that the electrons are magnetized above 78km altitude\nwhereas ions and nanometre sized or, larger grains are uncoupled t o the\nmagnetic field (which we shall also call unmagnetized for short) in the entire\nmesosphere. Therefore, the ions and dust grains can only indirect ly couple\nto the magnetic field since in this case βd≪βi<1.\nThe current Jand the electric field Eare related via conductivity tensor\nσ≈\nJ=σ/bardblE′\n/bardbl+σPE′\n⊥+σHE′×b, (8)\nwhereb=B/|B|and the parallel, Pedersen and Hall conductivities can be\n7written as [37]\nσ/bardbl=/parenleftBigec\nB/parenrightBig/summationdisplay\njnj|Zj|βj,\nσP=/parenleftBigec\nB/parenrightBig/summationdisplay\njnj|Zj|βj\n1+β2\nj,\nσH=−/parenleftBigec\nB/parenrightBig/summationdisplay\njnjZj\n1+β2\nj, (9)\nwhereZj=±1 is the sign of the charged particle. The Ohm ( ηO), Hall(ηH)\nand ambipolar ( ηA) diffusivities are [37]\nηO=/parenleftbiggc2\n4π/parenrightbigg1\nσ/bardbl, ηA=c2\n4π/parenleftbiggσP\nσ2\n⊥−1\nσ/bardbl/parenrightbigg\n,\nηH=−/parenleftbiggc2\n4π/parenrightbiggσH\nσ2\n⊥, (10)\nwhereσ⊥=/radicalbig\nσ2\nP+σ2\nHand the Pedersen diffusivity is\nηP=ηO+ηA. (11)\nConductivity σhas the unit 1 /sin Eq. (9). Similarly, the diffusivity has the\nunit cm2/s in Eq. (10). In Fig. 2(a) we plot various conductivities closely\ncorresponding to the PMSE altitude (78 −80km) where electron bite out\noccurs. The data from ECOMA/MASS campaign [Fig. (5)] of [47] have\nbeen used for the above plot. The corresponding diffusivities in Fig. [2 (b)]\nsuggestthattheHallandOhmdiffusionarecomparablebelow80kman dHall\ndominate beyond this altitude. The ambipolar diffusion is several orde rs of\nmagnitude smaller than boththe Ohmand Hall diffusion. Thus the Pede rsen\ndiffusivity at this altitude is solely due to Ohm diffusion. Clearly, Hall is the\ndominant diffusion in the electron bite-out region. Although we have u sed\nECOMA/MASS campaign data the dominance of Hall over other diffusio n\nis a generic feature of the upper D-region.\nHow important is the presence of grains in the D-region for the Hall d if-\nfusion? We know that the electron and ion densities are nearly equal after\nabout∼90km and the dust has very little role in affecting the plasma con-\nductionproperties abovethis altitude. However, belowthis altitude , thedust\n8Height[Km]75 80 85 90log10(σ)\n3456\nσO\nσH\nσP(a)\nHeight[Km]75 80 85 90log10(η)\n11131517\nηOηH\nηA(b)\nFigure 2: The plasma conductivities (1 /s) [panel (a)] and corresponding diffusivities\n(cm2/s) [bottom panel (b)] is plotted against the PMSE altitude where the electron bite\nout occurs. Data is taken from ECOMA/MASS campaign [Fig. 5 [47]]. The subscript\nH,O,A andPin the figure stands for Hall, Ohm, ambipolar and Pedersen.\ngrains play an important role in the magnetic diffusion. In order to delin -\neate the role of dust grains, we rewrite the Hall and Pedersen cond uctivities\n[Eq. (9)] as\nσH/parenleftbigecne\nB/parenrightbig=1\n1+β2\ne−/parenleftbiggni\nne/parenrightbigg1\n1+β2\ni+/parenleftbiggng\nne/parenrightbigg1\n1+β2\ng,\nσP/parenleftbigecne\nB/parenrightbig=βe\n1+β2\ne+/parenleftbiggni\nne/parenrightbiggβi\n1+β2\ni+/parenleftbiggng\nne/parenrightbiggβg\n1+β2\ng.\n(12)\nIn the lower D-layer ( ≪75km), the collision frequencies are so high (in\ncomparison with therespective cyclotron frequencies) that the e lectrons, ions\nand dust grains are unmagnetized, i.e. βg≪βi≪βe≪1. In the absence\nof dust, the conductivity is isotropic since σ/bardbl≈σPandσH≈0. However, in\nthe presence of grains this isotropy is lost since\nσ/bardbl≈σP≈/parenleftBigec\nB/parenrightBig\nneβe,\nσH≈/parenleftBigecni\nB/parenrightBig\nβ2\ni/bracketleftBig\n1−/parenleftbiggng\nni/parenrightbigg/parenleftbiggβg\nβi/parenrightbigg2/bracketrightBig\n, (13)\nwhere in the above expression for the Hall conductivity σHwe have assumed\n9Height[Km]75 80 85 90log10(σH)\n3456\n(a)electron+ion+dust\nion+dust\nHeight[Km]75 80 85 90log10(σP)\n345\n(b) only electron\nFigure 3: The Hall conductivity (1 /s) at PMSE altitude with and without the electron\nterm in Eq. (12) is plotted in the top panel (a). The Pedersen condu ctivity due only to\nthe electron term is plotted in panel (b).\nβg≪βi<1< βeand/parenleftbiggni\nne/parenrightbigg\n>1\nβ2\ni. (14)\nMaking use of (7), Eq. (14) gives ni/ne>2.5×(103−104) at 80−90kms\naltitude. Thisinequalitymaybeeasilysatisfiedintheelectronbiteoutr egion\nwhere most of the electrons are mopped by the dust grain.\nIn Fig. (3) we plot the Hall conductivity in the top panel with and with-\nout the electron term in Eq. (12). It is clear from the figure that be yond\n78−80km altitude in the PMSE region, the Hall conductivity is determined\nby the relative transverse drift between the weakly magnetized ion s (βi<1)\nand unmagnetized dust ( βg≪1). Below ∼78−80km, the Hall conductivity\nis due to the relative drift between the unmagnetized ions and weakly mag-\nnetized electrons. Note that at this altitude since βe<1 the electron-neutral\ncollision easily dissipates this relative drift (current). Thus both the Hall and\nOhm diffusion are comparable below ∼80km in Fig. (2).\nClearly the origin of the Hall effect in the lower and upper D-region is\nquite different. Whereas it is the relative transverse electron-ion d rift in\nthe lower D-region, it is the ion-charged dust drift in the upper D-re gion\nthat leads to the Hall conductivity. When the electron is also strong ly un-\nmagnetized in the lower D-region [ βe≪1, not shown in Fig. (1)], the Hall\nconductivity disappears. The Pedersen conductivity [bottom pane l Fig. (3)]\nis determined mainly by the electron term in Eq. (12) since there is ver y little\n10difference with the σPcurve in the top panel of Fig. (2).\nInverting generalized Ohms law Eq. (8) in terms of electric field Egives\nc2\n4πE′=ηOJ/bardbl+ηHJ×b+ηPJ⊥. (15)\nTaking curl of Eq. (15) and using of Maxwell´ s equation c∇×E=−∂tB, we\nget following induction equation\n∂B\n∂t=∇×/bracketleftbigg\n(v×B)−4πηO\ncJ+4πηA\nc(J×b)×b\n−4πηH\nc(J×b)/bracketrightbigg\n. (16)\nHerevis the bulk fluid velocity [44].\nAs the size distribution of the dust in the mesosphere is skewed towa rds\nnanometre (nm) or, smaller particles, it is quite plausible that the fine , nm\nsized dust may mop up all the electrons in the plasma leaving behind the\nlarger dust ( ∼few 10s of nanometre) to collect the more sluggish positive\nions. This may happen especially when the fraction ionization (ratio of the\nelectronandneutralnumber densities) inthepartiallyionizedgasisv ery low,\ni.e. when the plasma is weakly ionized. The observation of the nighttime\npolar mesosphere over Kiruna, Sweden confirms such an expectat ion [48].\nThus when the fine dust particles ( <50amu) mops up most of the electrons\nand the positive charge resides on the micron and sub-micron sized g rains\n(See Fig. 5 of [48]), it is the relative transverse drift (with respect t o the\nmagnetic field) of the negatively charged and magnetized fine dust p owder\nagainst the almost stationary unmagnetized or, weakly magnetized positive\ndust that will give rise to the Hall effect. Since the direction of the Ha ll\nelectric field in this case will be opposite to the case discussed above w here\nit was the plasma particles that was drifting against the negatively ch arged\ndust, the sign of the Hall term in the induction equation (16) will beco me\npositive. Such a reversal of sign of the Hall term is well known in the d usty\nplasmas [31] and has recently been investigated in the plume region of the\nSaturn´ s moon Enceladus [49].\nIt is pertinent to ask at this stage, what is the spatial scale over wh ich\nthe magnetic field couples to the plasma in the D-region. In order to a nswer\nthis question we need to compare the advective and diffusive terms in the\ninduction equation. Thus from Eq. (16) we get L∼ηH/v. Assuming v∼\n11vA∼0.1−1km/s forηH∼1014cm2/s [Fig. (2)] the characteristic length\nscaleLturns out to be about ∼103−104km. Thus we conclude that\nthe Hall diffusion of the magnetic field is important only for the ultra-lo w\nfrequency waves corresponding to the planetary waves of similar w avelength\n[50]. For the shorter wavelengths, magnetic diffusion in Eq. (16) sho uld be\nsafely neglected. However, the ideal MHD framework of the D-reg ion plasma\n(which is partially ionized) is not the same as the ideal MHD of fully ionized\nplasmas [51].\nAs can be seen from Eqs. (10) and (13), the ratio of the ambipolar a nd\nHall diffusion coefficient is\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleηA\nηH/vextendsingle/vextendsingle/vextendsingle/vextendsingle∼/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggσP\nσ2\n⊥−1\nσ/bardbl/parenrightbiggσ2\n⊥\nσH/vextendsingle/vextendsingle/vextendsingle/vextendsingle≈0, (17)\nand thus the ambipolar diffusion in the induction Eq. (16) is unimportan t.\nWe shall note yet again that here we are talking about the ambipolar d iffu-\nsion of the magnetic field against the sea of neutrals. This is different than\nthe ambipolar diffusion of the electrons against the electric field. The pres-\nence of dust particles may significantly reduce the electron density , resulting\nthe slowing down of electron diffusion which may facilitate the survival of\nplasma structures over long (10s of minutes to hours) lifetime [52]. H owever,\nlike the usual MHD framework, the present, dusty magnetohydro dynamics\nframework does not dwell upon the electric diffusion.\nThe ratio of the Hall and Ohm diffusion coefficient is\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleηH\nηO/vextendsingle/vextendsingle/vextendsingle/vextendsingle∼/vextendsingle/vextendsingle/vextendsingle/vextendsingleσH\nσ/bardbl/vextendsingle/vextendsingle/vextendsingle/vextendsingle∼(1+P)β2\ni\nβe/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−/parenleftbiggng\nni/parenrightbigg/parenleftbiggβg\nβi/parenrightbigg2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (18)\nwhere\nP=Zng\nne, (19)\nis Havnes parameter. In the electron bit-out regions where most o f the elec-\ntrons has been mopped by the dust, we see that ng∼ni(owing to the\nquasi-neutrality) and thus, it is not guaranteed that the Hall will do minate\nOhm diffusion. Clearly, both the Ohm and Hall diffusion may operate on\nthe same footing in the lower mesosphere conforming what we alread y know\nfrom Fig. (2). As we shall see in the next section whereas Hall will cau se the\nlow frequency, long wavelength left and right circularly polarized cyc lotron\nand whistler waves the Ohm will cause the damping of such waves.\n123. Waves in the mesosphere\nDefining the bulk fluid mass density and velocity as ρ≈ρn,v≈vnand\nafter assuming a barotropic relation P=c2\nsρwherecsis the sound speed we\nlinearize the following equations\n∂ρ\n∂t+∇·(ρv) = 0, (20)\nρdv\ndt=−∇P+J×B\nc, (21)\ntogetherwiththeinductionEq.(16)aroundahomogenousbackgr oundB0, ρ0, P0\nand get the following equations\nωδρ−ρk·δv= 0,\nωδv=c2\ns/parenleftbiggk·δv\nω/parenrightbigg\nk+/bracketleftBig\n(b·δb)k−(k·b)δb/bracketrightBig\nv2\nA,(22)\nwherevA=B/√4πρis the Alfv´ enspeed in the bulk fluid and δb=δB/|B|.\nWe define ¯ ω2=ω2−k2c2\ns, dot equation (22) with k, and use k·δb= 0 to\nwrite\nk·δv=/parenleftBigωA\n¯ω/parenrightBig2\nω(b·δb). (23)\nHereωA=kvAis the Alfv´ enfrequency. Making use of Eq. (23) Eq. (22) can\nbe written as\nωδv=/bracketleftbigg/parenleftBigω\n¯ω/parenrightBig2\n(b·δb)k−(k·b)δb/bracketrightbigg\nv2\nA. (24)\nThe linearized induction equation after setting ηA= 0 can be written as\n/parenleftbig\nω−ik2ηO/parenrightbig\nδb= [(k·δv)b−(k·b)δv]\n−iηH(k·b)k×δb. (25)\nDefining ˆk·b= cosθ,(ˆk=k/|k|) and diffusivities DO=k2ηO,DH=k2ηH\nand eliminating δvandk·δvfrom the above equation yields\n/bracketleftbig\nω2−iDOω−ω2\nAcos2θ/bracketrightbig\nδb=/parenleftBigω\n¯ω/parenrightBig2\nω2\nA\n×(δb·b)/parenleftBig\nb−ˆkcosθ/parenrightBig\n−iDHωcosθ/parenleftBig\nˆk×δb/parenrightBig\n. (26)\n13After some straightforward algebra we get following dispersion rela tion\n/bracketleftbigg\nω2/parenleftbigg\n1−ω2\nA\n¯ω2sin2θ/parenrightbigg\n−iDOω−ω2\nAcos2θ/bracketrightbigg\n×/parenleftbig\nω2−iDOω−ω2\nAcos2θ/parenrightbig\n−D2\nHω2cos2θ= 0. (27)\nThis dispersion relation acquires a familiar form [25, 31] when wave is pr op-\nagating along the ambient magnetic field ( θ= 0)\nω2−ik2ηOω−ω2\nA=±ηHk2ω. (28)\nIn the Ohm limit the solution is\nω=ik2η0\n2±ωA/parenleftbigg\n1−k2η2\nO\n4v2\nA/parenrightbigg1/2\n, (29)\nwhich for small magnetic diffusion describes an Alfv´ enwave propaga ting at\nfrequency ω≈ωA, and which is experiencing damping at a rate k2ηO/2.\nThe short wavelength fluctuations when the magnetic field diffusion s peed is\nof the order or, larger than twice the Alfv´ enspeed, i.e. kηO>2vAdo not\nsurvive as there is no oscillatory solution, only damping. When θ=π/2, the\nHall term drops out from Eq. (27). The damped magnetosonic mode is the\nsolution of the dispersion relation.\nEq. (27) in the Hall limit becomes\n/parenleftbig\nω2−ω2\nAcos2θ/parenrightbig2−/parenleftbiggω2−ω2\nAcos2θ\nω2−k2c2\ns/parenrightbigg\nω2ω2\nAsin2θ\n=/parenleftbig\nk2ηH/parenrightbig2ω2cos2θ. (30)\nFor the waves propagating transverse to the ambient field, i.e. θ=π/2, the\ndispersion relation (30) gives\nω2=k2/parenleftbig\nc2\ns+v2\nA/parenrightbig\n, (31)\nwhich is the usual magnetosonic branch. For the waves propagatin g along\nthe background magnetic field ( θ= 0) above dispersion relation (30) gives\nω=/parenleftBigωW\n2/parenrightBig/parenleftBigg\n1±/radicalBigg\n1±4ω2\nA\nω2\nW/parenrightBigg\n. (32)\n14where\nωW=k2ηH, (33)\nis the whistler frequency. The positive and negative sign inside the sq uare\nbracket in (32) corresponds to the left and circularly polarized wav es owing\nto the handedness of the Hall effect.\nNote that the Eq. (32) is the nonlinear dispersion relation for the ex -\nactly parallel (say along axis z), circularly polarized Alfv´ enwaves, b=\nexpi(ωτ−kξ) [28, 29]. Here τandξare stretched variables and b=\nBx+iBy. The weakly nonlinear and weakly dispersive waves propagat-\ning exactly along or, slightly oblique to the magnetic field satisfies comp lex\nderivative nonlinear Schrodinger (DNLS) equation. For exactly par allel case,\nthe DNLS equation admits localized envelop soliton [53]\n|b|2= 2/parenleftbiggV\nvA/parenrightbigg/bracketleftBig√\n2cosh/parenleftbiggξ−V τ\nL/parenrightbigg\n−1/bracketrightBig−1\n, (34)\nwhereVrepresents the velocity in the comoving frame, L=ηH/2Vis the\nwidth and 2 V/vAis the maximum amplitude of the soliton. Here we have\nassumed right circularly polarized waves. For typical Alfv´ enspeed ∼0.1−\n1km and ηH∼1014−1015cm2/s [Fig. 2(b)] the typical width of the soliton\nturns out to be ∼103−105km. However, the presence of Ohmic dissipation\nin the mesosphere may give shock-like structure due to dissipation o f the\nwave energy.\n4. Discussion and summary\nIn a partiallyionized medium in thepresence ofmagnetic field the plasma\nconductivity becomes anisotropic. The three orthogonal compon ents of this\nanisotropic tensor are (i) ambipolar, when the magnetic field is froze n in\nthe electron fluid and slips through the neutrals, (ii) Hall– when the pla sma\nparticles are partially coupled or, uncoupled to the magnetic field and , (iii)\nOhm, when the plasma particles cannot drift with the neutrals owing t o fre-\nquent collisions. In the mesosphere however since only the electron fluid is\nfrozeninthemagneticfieldabove80kmandtheionsareeither unmag netized\n(in which case they move with the neutrals owing to frequent collisions ) or,\nweakly magnetized (i.e. the relative slip between the ions and the neut rals\nis small) the ambipolar diffusion is negligible. Thus only the Hall and Ohm\nis important in the mesosphere. However, the origin of Hall effect is q uite\n15different below and above 80km. Whereas below 80km altitude the Hall\neffect is due to the relative transverse drift between the weakly ma gnetized\nelectrons and unmagnetized ions (the dust provides a stationary n eutraliz-\ning background), above this altitude, the Hall effect is due to the re lative\ndrift between the weakly magnetized ions and unmagnetized dust. N ote that\nthe distribution of the dust in the mesospheric layer determines the sign of\nthe Hall effect. If the negative charge largely resides on the fine, n anometre\nsized dust grains and positive charge resides on the more massive (f ew 10s of\nnanometre) grains [48], then the sign of the Hall electric field will be op posite\nto the previous case as in this case the transverse drift of the mag netized neg-\native fine dust against the more massive unmagnetized positive dust causes\nthe Hall effect. The sign change of the Hall term in the induction equa -\ntion (16) may affect the polarization of the waves in the mesospheric plasma\nlayers.\nAs the Hall and Ohm diffusion are comparable below 80km, the low fre-\nquency (ω∼10−4−10−5s−1) fluctuations will be damped with the damping\nratek2ηO∼few minutes. Therefore, the ultra–low frequency waves will\nbe heavily damped below 80km in the mesosphere. However, above 80 km\nthe mesosphere can host ultra–low frequency waves as Hall domina tes Ohm.\nAs a result the long wavelength, low frequency circularly polarized wh istlers\ncan propagate across the ionosphere with little or, no damping at all. Since\nthese low frequency waves have very large wavelength they may co uple to the\nmagnetosphere. For example, the fundamental frequencies at w hich magne-\ntosphere resonates is 1.3 and 1.9mHz [54]. Equating this frequency w ith\nthe whistler k2ηHgivesλ∼103−104km. Therefore, it is quite likely that\nthe generation of the low frequency waves in the lower E-region or, upper\nmesosphere is responsible for the mesospheric resonant tuning.\nThe present magnetohydrodynamic formulation of the weakly ionize d\nplasma is complementary to the usual electrodynamic formulation. W hereas\nin the present magnetohydrodynamics framework the bulk plasma fl ow ve-\nlocity (v) and magnetic field ( B) are the primary variable, it is the current\n(J) and electric field ( E) that are often used to describe the ionosphere dy-\nnamics. Therefore, the present framework does not deal with th e diffusion\nof the electron against the electric field, which also is called the ambipo lar\ndiffusion. We have seen that for the mesospheric parameters, amb ipolar dif-\nfusion of the magnetic field is unimportant. The effect of dust on the electron\ndiffusion in the presence of externalelectric field cannot be addressed in this\nframework. Importantly since the electric field is assumed to have it s origin\n16in theexternalevents (e.g. solar wind [55] and references therein), it cannot\npenetrate the quasi–neutral plasma except in a very thin boundar y layer of\nthe order of the Debye Length. Therefore, the limited penetratio n of such\nan electric field in the plasma will not be able to set the convective motio n\nand drive currents [56].\nFollowing is the summary of the results.\n1. The Ohm and Hall is the main components of the plasma conductivity\nin the mesosphere.\n2. Whereas Ohm and Hall are comparable in the lower mesosphere, Ha ll\ndominates Ohm above 80km.\n3. The physical mechanism of Hall effect differs in the lower and upper\nmesosphere. The relative transverse (with respect to the ambien t magnetic\nfield) drift between the weakly magnetized electrons and unmagnet ized ions\ncauses theHallinthelower mesosphere. Intheuppermesosphere therelative\ndrift between the weakly magnetized ions and unmagnetized nano me tre or,\nlarger dust causes Hall effect.\n4. If the negative change in the mesosphere is carried, mainly by the fine\ndust particles and the positive charge is carried by the more massive almost\nimmobile dust, the Hall effect in this case can be caused by the relative\ntransverse drift of the magnetized fine dust against the positively charged\ndust. The sign of the Hall effect in this case will be opposite of the pre vious\ncase. Thus the sign of the Hall effect not only depends on the direct ion of\nthe of the local magnetic field but also on the nature of the charge c arrier in\nthe plasma.\n5. The mesosphere can host low frequency long wavelength waves t hat may\npropagate undamped through the ionosphere.\nACKNOWLEDGEMENTS TheworkofSVVwaspartiallysupported\nby the Government of Russian Federation (Grant 08-08). 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The coupling is treated at the adiabatic level of the medium's response, which mediates\na \frst-order in frequency dissipative interaction along with an instantaneous Heisenberg exchange.\nThe resultant damped spin precession yields exceptional points (EPs) in the coupled spin dynamics,\nwhich should be experimentally accessible with the existing magnetic heterostructures. In partic-\nular, we show that an EP is naturally approached in an antiferromagnetic dimer by controlling\nlocal damping, while the same is achieved by tuning the dissipative coupling between spins in the\nferromagnetic case. Extending our treatment to one-dimensional spin chains, we show how EPs\ncan emerge within the magnonic Brillouin zone by tuning the dissipative properties. The critical\npoint, at which an EP pair emerges out of the Brillouin zone center, realizes a gapless Weyl point in\nthe magnon spectrum. Tuning damping beyond this critical point produces synchronization (level\nattraction) of magnon modes over a \fnite range of momenta, both in ferro- and antiferromagnetic\ncases. We thus establish that damped magnons can generically yield singular points in their band\nstructure, close to which their kinematic properties, such as group velocity, become extremely sen-\nsitive to the control parameters.\nI. INTRODUCTION\nOver the past several decades, it became clear that\na class of unconventional degeneracies are abundant in\ndiverse physical systems undergoing non-Hermitian dy-\nnamics [1]. This generally concerns a Schr odinger-type\nevolution of a complex-valued vector \feld v(t):\nid\ndtv=Hv; (1)\nin terms of a non-Hermitian matrix-valued \\Hamilto-\nnian\"H(w), parametrized by a set of complex-valued\nsystem parameters w. The dynamics governed by Eq. (1)\nbecome extremely sensitive to the values of wclose to\npoints w0whereHis not diagonalizable. In fact, the\nstandard diagonalization of Hatw0would yield a branch\npoint singularity dubbed exceptional point (EP) [2]. The\nde\fning property of w0is the degeneration of two or more\neigenvectors at an eigenfrequency degeneracy point. For\nexample, a spin-1 =2 raising operator ^ \u001b+= ^\u001bx+i^\u001by, ex-\npressed in terms of two 2 \u00022 Pauli matrices ^ \u001bi, has a sole\neigenvector, with zero eigenvalue, corresponding to the\nspin-up state. As such, ^ \u001b+cannot be diagonalized by a\nsimilarity transformation.\nThe eigenenergy solutions \u0015ofH(w) realize, as a func-\ntion of w, multivalued functions with branch cuts termi-\nnating at branch points. Smoothly varying one of the\ncomplex-valued parameters w, the eigenenergies can be\nrepresented as single-valued functions on a multisheet\nRiemann surface, with branch points marking coales-\ncence of two or more energy levels. These EPs thus\nprovide genuine singularities, which can manifest promi-\nnently in microwave [3] and optical [4] response proper-\nties and hybrid dynamical systems [5{8], scattering prob-\nlems and sensing [9], open and quasi-Hermitian quantum\nsystems [10], etc. [1], particularly in regard to their topo-\nlogical encircling aspects [11]. The emergence of EPs\nin quasiparticle band structures, furthermore, tremen-dously enriches their topological classi\fcation in crys-\ntalline materials [12].\nIn this paper, we argue that EPs are also common-\nplace in pure spin dynamics, based on several generic\nexamples, even in the absence of external driving, such\nas spin-transfer torque [13]. In particular, an isotropic\nantiferromagnetic spin pair harbors an EP already in its\nsinglet-like ground state. We show how this EP gets in-\nherited by extended antiferromagnetic dynamics, mani-\nfesting in Weyl singularities and synchronization (level\nattraction) within magnonic band structure, which are\ntunable by the dissipative properties of the environment.\nWhile somewhat less natural, similar EPs can also be\nengendered by ferromagnetic systems.\nThe paper is structured as follows: In Sec. II, a gen-\neral model for dissipatively coupled spin dynamics is for-\nmulated, based on exchange and spin-pumping mediated\ninteractions [14]. In Sec. III, we specialize to the ferro-\nmagnetic case, \frst revealing EPs in simple two-spin dy-\nnamics and then extending the treatment to the magnon\nband structure in a spin chain. In Sec. IV, a similar pro-\ngram is carried out for the antiferromagnetic case, before\nsummarizing the paper in Sec. V.\nII. GENERAL TWO-SPIN DYNAMICS\nA. Reactive and dissipative coupling\nConsider an isotropic system composed of two classical\nspins described by the Hamiltonian\nH=\u0000b1\u0001S1\u0000b2\u0001S2\u0000JS1\u0001S2: (2)\nb{parametrize individual Zeeman splittings and J\nHeisenberg exchange between the spins. This Hamilto-\nnian, according to the classical spin algebra fSa;Sbg=\n\u000fabcSc(withf:::gstanding for the Poisson bracket andarXiv:1911.01619v2 [cond-mat.mes-hall] 26 Dec 20192\n\u000fabcdenoting the Levi-Civita symbol), describes a cou-\npled Larmor precession of the spins:\n_S{\u0011fS{;Hg=S{\u0002(b{+JS~{): (3)\nHere, ~{= 2;1 for{= 1;2, respectively. A possible phys-\nical realization of such a system can be provided by a\nmagnetic bilayer coupled through a normal-metal spacer\n[15].\nIn addition to a RKKY-type exchange J, we generally\nalso need to add a dissipative coupling mediated by spin\npumping through the spacer [15], which enters the equa-\ntions of motion as a nonlocal Gilbert damping [16]. For\nsmall-angle dynamics, relative to a common equilibrium\norientation (supposing b{are collinear), the full coupled\nequations become:\n(1 +g{S{\u0002)_S{+G(s{\u0002_s{\u0000s~{\u0002_s~{) =S{\u0002(b{+JS~{):(4)\nGparametrizes the strength of the spin pumping across\nthe spacer (which is related to the spin-mixing conduc-\ntance [14, 17]), driven by the orientational dynamics of\ns{\u0011S{=S{. We also included local Gilbert damping [16]\ng{in each of the magnetic layers, which parametrizes the\nquality factor Q{= 1=2g{S{of intrinsic magnetic dynam-\nics. See Fig. 1 for a schematic. Note that the equations\nof motion (4) would need to be revised in the general\nlarge-angle case [14, 15], in order to preserve the magni-\ntude of S{. In the following, we will see that the nonlocal\ndamping/Gis essential in establishing EPs in the fer-\nromagnetic case, while it will turn out to be unimportant\nfor the antiferromagnetic case.\nS1\nAAACBnicbVDLSsNAFL2pr1pfVZduBovgqiQq6MJFwY3LirYWmlAm00k7dDIJMxMhhOz9Arf6Be7Erb/hB/gfTtosbOuBC4dz7uUejh9zprRtf1uVldW19Y3qZm1re2d3r75/0FVRIgntkIhHsudjRTkTtKOZ5rQXS4pDn9NHf3JT+I9PVCoWiQedxtQL8UiwgBGsjeS6IdZjP8ju84EzqDfspj0FWiZOSRpQoj2o/7jDiCQhFZpwrFTfsWPtZVhqRjjNa26iaIzJBI9o31CBQ6q8bJo5RydGGaIgkmaERlP170WGQ6XS0DebRUa16BXif14/0cGVlzERJ5oKMnsUJBzpCBUFoCGTlGieGoKJZCYrImMsMdGmprkvaZFM5aYXZ7GFZdI9azrnTefuotG6LhuqwhEcwyk4cAktuIU2dIBADC/wCm/Ws/VufVifs9WKVd4cwhysr1/i0ZoaS2\nAAACBnicbVDLSsNAFL2pr1pfVZduBovgqiRV0IWLghuXFe0DmlAm00k7dDIJMxMhhO79Arf6Be7Erb/hB/gfTtosbOuBC4dz7uUejh9zprRtf1ultfWNza3ydmVnd2//oHp41FFRIgltk4hHsudjRTkTtK2Z5rQXS4pDn9OuP7nN/e4TlYpF4lGnMfVCPBIsYARrI7luiPXYD7KH6aAxqNbsuj0DWiVOQWpQoDWo/rjDiCQhFZpwrFTfsWPtZVhqRjidVtxE0RiTCR7RvqECh1R52SzzFJ0ZZYiCSJoRGs3UvxcZDpVKQ99s5hnVspeL/3n9RAfXXsZEnGgqyPxRkHCkI5QXgIZMUqJ5aggmkpmsiIyxxESbmha+pHkyNTW9OMstrJJOo+5c1J37y1rzpmioDCdwCufgwBU04Q5a0AYCMbzAK7xZz9a79WF9zldLVnFzDAuwvn4B5GiaGw==\nb1\nAAACBnicbVDLSsNAFL2pr1pfVZduBovgqiQq6MJFwY3LCvYBTSiT6aQdOpmEmYkQQvZ+gVv9Anfi1t/wA/wPJ20WtvXAhcM593IPx485U9q2v63K2vrG5lZ1u7azu7d/UD886qookYR2SMQj2fexopwJ2tFMc9qPJcWhz2nPn94Vfu+JSsUi8ajTmHohHgsWMIK1kVw3xHriB5mfD51hvWE37RnQKnFK0oAS7WH9xx1FJAmp0IRjpQaOHWsvw1IzwmlecxNFY0ymeEwHhgocUuVls8w5OjPKCAWRNCM0mql/LzIcKpWGvtksMqplrxD/8waJDm68jIk40VSQ+aMg4UhHqCgAjZikRPPUEEwkM1kRmWCJiTY1LXxJi2QqN704yy2sku5F07lsOg9XjdZt2VAVTuAUzsGBa2jBPbShAwRieIFXeLOerXfrw/qcr1as8uYYFmB9/QL615opb2\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GAAAB+3icbVDLSsNAFL2pr1pfVZdugkVwVRIVdOGi4EKXLdgHtKFMpjft0MkkzEyEEPoFbvUL3IlbP8YP8D+ctFnY6oGBwzn3cs8cP+ZMacf5skpr6xubW+Xtys7u3v5B9fCoo6JEUmzTiEey5xOFnAlsa6Y59mKJJPQ5dv3pXe53n1AqFolHncbohWQsWMAo0UZq3Q+rNafuzGH/JW5BalCgOax+D0YRTUIUmnKiVN91Yu1lRGpGOc4qg0RhTOiUjLFvqCAhKi+bB53ZZ0YZ2UEkzRPanqu/NzISKpWGvpkMiZ6oVS8X//P6iQ5uvIyJONEo6OJQkHBbR3b+a3vEJFLNU0MIlcxktemESEK16WbpSponUzPTi7vawl/Suai7l3W3dVVr3BYNleEETuEcXLiGBjxAE9pAAeEZXuDVmllv1rv1sRgtWcXOMSzB+vwBOOGVSg==\ng1\nAAAB/XicbVDLSsNAFL2pr1pfVZduBovgqiRa0IWLghuXFe0D2lAm00k6dDIJMxMhhOIXuNUvcCdu/RY/wP9w0mZhWw8MHM65l3vmeDFnStv2t1VaW9/Y3CpvV3Z29/YPqodHHRUlktA2iXgkex5WlDNB25ppTnuxpDj0OO16k9vc7z5RqVgkHnUaUzfEgWA+I1gb6SEYOsNqza7bM6BV4hSkBgVaw+rPYBSRJKRCE46V6jt2rN0MS80Ip9PKIFE0xmSCA9o3VOCQKjebRZ2iM6OMkB9J84RGM/XvRoZDpdLQM5Mh1mO17OXif14/0f61mzERJ5oKMj/kJxzpCOX/RiMmKdE8NQQTyUxWRMZYYqJNOwtX0jyZmppenOUWVknnou5c1p37Rq15UzRUhhM4hXNw4AqacActaAOBAF7gFd6sZ+vd+rA+56Mlq9g5hgVYX7+b3ZYOg2\nAAAB/XicbVDLSsNAFL2pr1pfVZduBovgqiRV0IWLghuXFe0D2lAm00k6dDIJMxMhhOIXuNUvcCdu/RY/wP9w0mZhWw8MHM65l3vmeDFnStv2t1VaW9/Y3CpvV3Z29/YPqodHHRUlktA2iXgkex5WlDNB25ppTnuxpDj0OO16k9vc7z5RqVgkHnUaUzfEgWA+I1gb6SEYNobVml23Z0CrxClIDQq0htWfwSgiSUiFJhwr1XfsWLsZlpoRTqeVQaJojMkEB7RvqMAhVW42izpFZ0YZIT+S5gmNZurfjQyHSqWhZyZDrMdq2cvF/7x+ov1rN2MiTjQVZH7ITzjSEcr/jUZMUqJ5aggmkpmsiIyxxESbdhaupHkyNTW9OMstrJJOo+5c1J37y1rzpmioDCdwCufgwBU04Q5a0AYCAbzAK7xZz9a79WF9zkdLVrFzDAuwvn4BnXSWDw==\nFIG. 1. A general two-spin system coupled through a dissi-\npative environment. The spins S{precess in their respective\ninternal \felds b{and couple through a Heisenberg exchange J.\nIn addition, individual viscous (Gilbert) damping g{is com-\nplemented with a mutual spin-pumping-mediated viscosity G\nin the coupled dynamics.\nB. Energy dissipation and external pumping\nNote that the dissipative coupling /Ga\u000bects only the\nout-of-phase precession of the two (macro)spin orienta-\ntions. This could be thought of as a viscosity associated\nwith relative spin dynamics [15]. Calculating energy dis-\nsipation according to the equation of motion (4),\nP\u0011\u0000 _H=H1\u0001_S1+H2\u0001_S2; (5)where H{\u0011\u0000@S{His the e\u000bective \feld conjugate to spin\n{, we \fnd\nP=g1_S2\n1+g2_S2\n2+G(_s1\u0007_s2)2; (6)\nin the case of the dynamics near the parallel (antipar-\nallel) con\fguration. This dissipation is guaranteed to\nbe positive-semide\fnite in the physical situation where\ng1;g2;G\u00150 [17]. The above equations thus describe\nthe dynamics of a stable system near its thermodynamic\nequilibrium.\nFormally,Pis positive-semide\fnite i\u000b g{+G=S2\n{\u00150\nandjGj=S1S2\u0014p\n(g1+G=S2\n1)(g2+G=S2\n2). Macro-\nscopic magnetic spins can be locally (thermoelectrically)\npumped out of their natural thermodynamic equilibrium\nby subjecting them to spin-transfer [18] or spin Seebeck\n[19] torques, which can shift the e\u000bective local damping\ng{into negative values [20] and possibly even invalidate\nthe stability requirement P\u00150. We can exploit this\nin practice for expanding the parameter space of experi-\nmentally tunable coe\u000ecients that govern our dynamical\nsystem.\nIII. FERROMAGNETIC ALIGNMENT\nTo be more speci\fc, we now orient the Zeeman terms\nalong thezaxis,b{=b{z, and linearize spin dynamics\nin terms of small deviations from the initially parallel\ncon\fguration, S{\u0019S{z. Owing to the axial symmetry,\nwe switch to the natural circular coordinates: S{\u0011(Sx\n{+\niSy\n{)=p2S{, which obey canonical algebra (in the case of\nsmall-angle dynamics):\nifS;S\u0003g\u00191; (7)\nfor each site {(with the inter-site Poisson brackets van-\nishing). We thus see that the quantized S!p\n~aobey\nbosonic statistics: [ a;ay]!ifS;S\u0003g= 1.a, which\nis proportional to the spin-raising operator, thus consti-\ntutes the magnon \feld.\nA. Equation of motion\nThe linearized dynamics following from Eq. (4) are de-\nscribed by\n(1 +i\u000b{)_S{\u0000i\u000b0_S~{=\u0000i!{S{+i!0S~{; (8)\nwhere!{\u0011b{+JS~{,\u000b{\u0011g{S{+G=S{and we de-\nnoted by the primed coe\u000ecients, !0\u0011JpS1S2and\n\u000b0\u0011G=pS1S2, the reactive and dissipative inter-spin\ncouplings, respectively. Equation (8) can \fnally be re-\ncast in the matrix (Schr odinger-like) form:\n(1 +i^d)id\ndtS=^hS; (9)3\nwhere\nS=\u0012\nS1\nS2\u0013\n;^h=!++!\u0000^\u001bz\u0000!0^\u001bx (10)\nis the e\u000bective 2\u00022 magnon Hamiltonian, and\n^d=\u000b++\u000b\u0000^\u001bz\u0000\u000b0^\u001bx (11)\nis the damping tensor, parametrized by the Pauli matri-\nces ^\u001ba.!\u0006\u0011(!1\u0006!2)=2 (and similarly for \u000b\u0006) are the\n(anti)symmetrized frequencies. The dynamics is \fnally\ncast as\nid\ndtS=^HS; (12)\nin terms of the non-Hermitian \\Hamiltonian\"\n^H\u0011(1 +i^d)\u00001^h: (13)\nLet us simplify ^Hby assuming smallness of the damp-\ning parameters, \u000b\u001c1 (implying large quality factors for\nthe resonant modes), as well as of the interspin detuning\nand coupling, !\u0000;!0\u001c!+. This leads to\n^H\u0019(1\u0000i\u000b\u0000^\u001bz+i\u000b0^\u001bx)(1 +\r\u0000^\u001bz\u0000\r0^\u001bx)~!+\n\u0019[1 + (\r\u0000\u0000i\u000b\u0000)^\u001bz\u0000(\r0\u0000i\u000b0)^\u001bx] ~!+\n\u0011(1 + ^H)~!+\u0019~!++^H!+;(14)\nwhere\r\u0000\u0011!\u0000=!+\u001c1 and\r0\u0011!0=!+\u001c1 are\nthe normalized detuning and coupling, respectively, and\n~!+\u0011!+=(1 +i\u000b+) is the complex-valued normal fre-\nquency of the unperturbed symmetric mode. By diago-\nnalizing the normalized and shifted Hamiltonian\n^H= (\r\u0000\u0000i\u000b\u0000)^\u001bz\u0000(\r0\u0000i\u000b0)^\u001bx; (15)\nwe can \fnally decompose the linearized dynamics into\ndamped modes of the form /e\u0000i(1+\u0015)~!+t, where\u0015is one\nof the two (complex-valued) eigenvalues of ^H. For conve-\nnience, we summarize the key quantities parametrizing\nthe coupled dynamics in Table I.\nTABLE I. Parameters of the collinear two-spin system.\n\r\u0000normalized frequency asymmetry, !\u0000=!+\n\u000b\u0000damping asymmetry\n\r0normalized exchange coupling, !0=!+\n\u000b0dissipative coupling\nB. Exceptional points\nThe diagonalization of the matrix (15) breaks down\nat the exceptional points, where the eigenvectors asso-\nciated with degenerate eigenvalues coalesce [21], rulingout a diagonalizing similarity transformation. The two\neigenvalues are given by\n\u0015\u0006=\u0006p\n(\r\u0000\u0000i\u000b\u0000)2+ (\r0\u0000i\u000b0)2; (16)\nwhere we are making the convention for the square root\nto evaluate the principal value. An EP occurs when \u0015\u0006=\n0, while the individual terms under the square root are\nnot both zero. In this case, ^H6= 0, while ^H2= 0, which\ncon\frms that indeed the matrix cannot be diagonalized.\nA ready example of this is provided by a matrix /^\u001ba+\ni^\u001bb, in terms of two distinct Pauli matrices ^ \u001baand ^\u001bb, as\nwe have already mentioned.\nExpanding the (complex-valued) energy eigenvalues\n(16) near such an EP, as a function of some complex-\nvalued parameter that parametrizes ^H, would generically\nde\fne a square-root singularity. The trivial degeneracy\nof^H= 0 signals a diabolic point , on the other hand, which\nwould harbor a Berry-curvature monopole [22] associated\nwith a Weyl singularity.\n1. The reactive and dissipative scenarios\nEven if the two-spin system is symmetric, S1=S2, we\nmay still introduce possible asymmetries in the individ-\nual resonant frequencies and the associated broadenings.\nThe EP condition (16) translates into\n\r\u0000\u0000i\u000b\u0000=\u0006i(\r0\u0000i\u000b0)6= 0: (17)\nThe EPs can then be realized for physical real-valued\nparameters when \u000b\u0000=\u0007\r0and\r\u0000=\u0006\u000b0. Two ba-\nsic practical scenarios can then be envisioned: (1) The\ncoupling between the spins is purely reactive, \r06= 0,\nwhile\u000b0= 0, in which case the resonances need to be\ntuned,\r\u0000= 0, resulting in two EPs when \u000b\u0000=\u0006\r0; and\n(2) The coupling between the spins is purely dissipative,\n\u000b06= 0, while\r0= 0, in which case the local dissipation\nneeds to be symmetric, \u000b\u0000= 0, resulting in two EPs\nwhen\r\u0000=\u0006\u000b0. The latter scenario is especially attrac-\ntive, as it naturally occurs in a magnetic bilayer system\nwith a di\u000busive normal-metal spacer [14, 15]. Resonant\ntuning across the EP has been realized in Ref. [15], not\nmaking the connection with the EP perspective, however,\nat the time.\n2. Topology of the exceptional points\nLet us look more closely into the frequency eigenvalues\n(16) in the vicinity of these exceptional points. As an\nexample, suppose we have a magnetic bilayer coupled\nthrough a purely dissipative coupling, i.e., \r0= 0. Let\nus take the dissipative coupling \u000b0to be \fxed, while the\nlocal resonance conditions and damping, \r\u0000and\u000b\u0000, are\nallowed to be tuned by the control of local \felds and\nthermoelectric pumping [20]. The frequency eigenvalues4\nare then given by\n\u0015\u0006=\u0006p\n(\r\u0000\u0000i\u000b\u0000)2\u0000\u000b02: (18)\nWe can rewrite these eigenfrequencies as \u0015\u0006=\n\u0006p\nz2\u0000\u000b02, in terms of the fully-tunable complex-valued\nparameter z\u0011\r\u0000\u0000i\u000b\u0000[which couples to the z-\ncomponent Pauli matrix in the original Hamiltonian\n(15)]. If\u000b\u0000= 0,zis swept along the real axis by vary-\ning\r\u0000, passing through two EP at z=\u0006\u000b0. The corre-\nsponding frequency eigenvalues are plotted in Fig. 2. For\nj\r\u0000=\u000b0j<1, they are purely imaginary, with the anti-\nsymmetric mode damped relative to the symmetric one.\nAmusingly, both modes here are perfectly synchronized,\nin regard to their real frequency components. This is\ndubbed level attraction [7, 8], in contrast to the usual\nlevel repulsion for a hybridized Hermitian system. At\nj\r\u0000=\u000b0j>1, the modes bifurcate, with a vanishing rela-\ntive damping. The EP transition from the purely real to\nthe purely imaginary eigenvalues \u0015\u0006atj\r\u0000=\u000b0j= 1, as\ndepicted in Fig. 2, is related to the spontaneous breaking\nof aPTsymmetry [10].\nIt may be useful to recall, that these frequency eigen-\nvalues\u0015\u0006are normalized and shifted by a complex-\nvalued ~!+[cf. Eq. (14)]. The corresponding physical\neigenfrequencies must both have negative imaginary com-\nponents, according to the overall thermodynamic stabil-\nity of the system [so long as dissipation Pin Eq. (6) is\npositive, which should be true in and near equilibrium].\n\u0000±/↵0\nAAACD3icbVDLSsNAFJ3UV62vaJdugkV0VRMVdFnoxmUFWwtNCDeTSTt0JgkzEyGEfoRf4Fa/wJ249RP8AP/DaZuFbT1w4XDOudzLCVJGpbLtb6Oytr6xuVXdru3s7u0fmIdHPZlkApMuTlgi+gFIwmhMuooqRvqpIMADRh6DcXvqPz4RIWkSP6g8JR6HYUwjikFpyTfrLtPhEHw35RcusHQEZ77ZsJv2DNYqcUrSQCU6vvnjhgnOOIkVZiDlwLFT5RUgFMWMTGpuJkkKeAxDMtA0Bk6kV8yen1inWgmtKBF6YmXN1L8bBXApcx7oJAc1ksveVPzPG2QquvUKGqeZIjGeH4oyZqnEmjZhhVQQrFiuCWBB9a8WHoEArHRfC1fy6WdyontxlltYJb3LpnPVdO6vG6122VAVHaMTdI4cdINa6A51UBdhlKMX9IrejGfj3fgwPufRilHu1NECjK9fV0Wc/A==\u0000\u0000/↵0\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1AAAB+3icbVDLSsNAFL2pr1pfVZduBovgqiQq6LLQjcsW7APaUCbTm3bo5MHMRCghX+BWv8CduPVj/AD/w0mbhW09MHA4517umePFgitt299WaWt7Z3evvF85ODw6PqmennVVlEiGHRaJSPY9qlDwEDuaa4H9WCINPIE9b9bM/d4zSsWj8EnPY3QDOgm5zxnVRmo7o2rNrtsLkE3iFKQGBVqj6s9wHLEkwFAzQZUaOHas3ZRKzZnArDJMFMaUzegEB4aGNEDlpougGbkyypj4kTQv1GSh/t1IaaDUPPDMZED1VK17ufifN0i0/+CmPIwTjSFbHvITQXRE8l+TMZfItJgbQpnkJithUyop06ablSvzPJnKTC/OegubpHtTd27rTvuu1mgWDZXhAi7hGhy4hwY8Qgs6wADhBV7hzcqsd+vD+lyOlqxi5xxWYH39AhgClTs=\u00001\nAAAB/HicbVDLSsNAFL2pr1pfUZduBovgxpKooMtCNy6r2Ae0oUymk3boZBJmJkII9Qvc6he4E7f+ix/gfzhps7CtBwYO59zLPXP8mDOlHefbKq2tb2xulbcrO7t7+wf24VFbRYkktEUiHsmujxXlTNCWZprTbiwpDn1OO/6kkfudJyoVi8SjTmPqhXgkWMAI1kZ6uHAHdtWpOTOgVeIWpAoFmgP7pz+MSBJSoQnHSvVcJ9ZehqVmhNNppZ8oGmMywSPaM1TgkCovmyWdojOjDFEQSfOERjP170aGQ6XS0DeTIdZjtezl4n9eL9HBrZcxESeaCjI/FCQc6Qjl30ZDJinRPDUEE8lMVkTGWGKiTTkLV9I8mZqaXtzlFlZJ+7LmXtXc++tqvVE0VIYTOIVzcOEG6nAHTWgBgQBe4BXerGfr3fqwPuejJavYOYYFWF+/hLGVcg==\u00001\nAAAB/HicbVDLSsNAFL2pr1pfUZduBovgxpKooMtCNy6r2Ae0oUymk3boZBJmJkII9Qvc6he4E7f+ix/gfzhps7CtBwYO59zLPXP8mDOlHefbKq2tb2xulbcrO7t7+wf24VFbRYkktEUiHsmujxXlTNCWZprTbiwpDn1OO/6kkfudJyoVi8SjTmPqhXgkWMAI1kZ6uHAHdtWpOTOgVeIWpAoFmgP7pz+MSBJSoQnHSvVcJ9ZehqVmhNNppZ8oGmMywSPaM1TgkCovmyWdojOjDFEQSfOERjP170aGQ6XS0DeTIdZjtezl4n9eL9HBrZcxESeaCjI/FCQc6Qjl30ZDJinRPDUEE8lMVkTGWGKiTTkLV9I8mZqaXtzlFlZJ+7LmXtXc++tqvVE0VIYTOIVzcOEG6nAHTWgBgQBe4BXerGfr3fqwPuejJavYOYYFWF+/hLGVcg==0AAAB+3icbVDLSsNAFL2pr1pfVZduBovgqiQq6LLQjcsW7APaUCbTm3bo5MHMRCghX+BWv8CduPVj/AD/w0mbhW09MHA4517umePFgitt299WaWt7Z3evvF85ODw6PqmennVVlEiGHRaJSPY9qlDwEDuaa4H9WCINPIE9b9bM/d4zSsWj8EnPY3QDOgm5zxnVRmrbo2rNrtsLkE3iFKQGBVqj6s9wHLEkwFAzQZUaOHas3ZRKzZnArDJMFMaUzegEB4aGNEDlpougGbkyypj4kTQv1GSh/t1IaaDUPPDMZED1VK17ufifN0i0/+CmPIwTjSFbHvITQXRE8l+TMZfItJgbQpnkJithUyop06ablSvzPJnKTC/OegubpHtTd27rTvuu1mgWDZXhAi7hGhy4hwY8Qgs6wADhBV7hzcqsd+vD+lyOlqxi5xxWYH39AhZrlTo=ReAAACDnicbVDLSsNAFJ34rPUV69LNYBFclUQFXRbcuKxiH9CEMpnetEMnD2YmYggBP8EvcKtf4E7c+gt+gP/hJO3Cth4YOJxz557L8WLOpLKsb2NldW19Y7OyVd3e2d3bNw9qHRklgkKbRjwSPY9I4CyEtmKKQy8WQAKPQ9ebXBd+9wGEZFF4r9IY3ICMQuYzSpSWBmbNKXdkHk8gd0SA72Bg1q2GVQIvE3tG6miG1sD8cYYRTQIIFeVEyr5txcrNiFCMcsirTiIhJnRCRtDXNCQBSDcrc3N8opUh9iOhX6hwqf79kZFAyjTw9GRA1FgueoX4n9dPlH/lZiyMEwUhnQb5CccqwkUReMgEUMVTTQgVTN+K6ZgIQpWuay4lLS6Tue7FXmxhmXTOGvZ5w769qDebT9OGKugIHaNTZKNL1EQ3qIXaiKJH9IJe0ZvxbLwbH8bndHTFmLV6iOZgfP0Ch8adjw==ImAAACGHicbVDLSgMxFM34rPU16krcBIvgqsyooMuC4GNXwT6gM5RMetuGJpkhyRTLUPQ73LvVX3Anbt35B36G02kXtvVA4HDOzb2HE0ScaeM439bC4tLyympuLb++sbm1be/sVnUYKwoVGvJQ1QOigTMJFcMMh3qkgIiAQy3oXY78Wh+UZqG8N4MIfEE6krUZJSaVmva+l+1IrkIF2lwrADn0lMC3omkXnKKTAc8Td0IKaIJy0/7xWiGNBUhDOdG64TqR8ROiDKMchnkv1hAR2iMdaLT6LNKSCNB+8pAlGOKj1G/hdqjSJw3O1L+fEiK0HoggnRTEdPWsNxL/8xqxaV/4CZNRbEDS8aF2zLEJ8agS3GIKqOGDlBCqWBoX0y5RhJq0uKkrg1EyPUyrcWeLmCfVk6J7WnTvzgql0tO4pBw6QIfoGLnoHJXQDSqjCqLoEb2gV/RmPVvv1of1OR5dsCbF7qEpWF+/k3Kh8w==\nFIG. 2. Complex-valued eigenfrequencies (18), when \u000b\u0000= 0.\nThe two EPs at j\r\u0000=\u000b0j= 1 engender singularities, at which\nthe eigenfrequencies switch from being purely imaginary (at\nsmaller local frequency asymmetries \r\u0000) to purely real (at\nlarger asymmetries \r\u0000). Such eigenfrequency structure has\nbeen observed in a symmetric iron-based magnetic bilayer [15]\n(see also Ref. [14], for a more detailed analysis), as well as\nhybrid microwave-cavity based systems [6{8].\nClose to either of the two EPs, z!z0=\u0006\u000b0, we\nobtain a square-root singularity in the full complex plane,\nz2C:\n\u0015\u0006!\u0006p\n2z0(z\u0000z0) (when\u000b06= 0): (19)\nWhen the dissipative coupling vanishes, \u000b0!0, on the\nother hand, the two EPs merge, resulting in a single Weylpoint in the spectrum (at ^H!0):\n\u0015\u0006!\u0006z(when\u000b0= 0): (20)\nThe latter is of course just a trivial scenario of decou-\npled circular precession, near the degeneracy point. For\na \fnite spin pumping, \u000b06= 0, the eigenfrequencies /pz\nconstitute a double-valued function in the original com-\nplex plane z, while being single-valued on the Riemann\nsurface, which consists of two sheets emanating from the\nEP (the branch point) and stitched up along the branch\ncut (that is typically chosen along the negative real axis\nofz).\n\u0000\u0000\n(null)(null)(null)(null)↵\u0000\n(null)(null)(null)(null)\n122✏\n(null)(null)(null)(null)\nt(null)(null)(null)(null)Re\u0000±\n(null)(null)(null)(null)2p✏\n(null)(null)(null)(null)\nt(null)(null)(null)(null)2p✏\n(null)(null)(null)(null)Im\u0000±\n(null)(null)(null)(null)(1)\n(2)↵0\n(null)(null)(null)(null)\u0000↵0\n(null)(null)(null)(null)\nFIG. 3. The evolution of the eigenvalues upon the passage of\nan exceptional point z0!\u000b0to the right (1) or left (2). The\nsolid lines in the insets show the respective values of the real\nand imaginary parts of the frequency eigenvalues (in units ofp\n2\u000b0). The dashed lines are, respectively, the imaginary and\nreal parts.\nIfzpasses just to the right of the EP (path 1 in Fig. 3):\nz=z0+\u000f\u0000it, where\u000f>0 is a small shift controlled by\n\fxing\r\u0000!z0+\u000fand\u000b\u0000!tis continuously varied,\nthe eigenvalues\n\u0015\u0006/\u0006p\n\u000f\u0000it (21)\nhave their two real parts anticrossing and two imaginary\nparts crossing, at t!0. When the passage is performed\nto the left of the EP (path 2 in Fig. 3), \u000f <0, the real\nparts cross while the imaginary parts anticross. At the\nWeyl point, \u000f= 0, both parts cross, of course. This cross-\ning/anticrossing behavior is generic for basic topological\nreasons [21].\nC. Spin chain\nThe EPs discussed above can also be approached in\nmomentum space, by considering a dimerized chain of\ntwo-spin composites. As a starting point to that end,\nwe analyze a chain of Nidentical spins, with periodic5\nboundary conditions, exchange-coupled as in Eq. (2):\nH=\u0000NX\n|=1b|\u0001S|\u0000JX\nh||0iS|\u0001S|0; (22)\nwhere the double sum runs over the nearest neighbors\n(theNth spin being a neighbor to the 1st one). The\nlinearized dynamics (8) then obey\n(1 +ig|S)_S|\u0000i\u000b0(_S|\u00001+_S|+1\u00002_S|)=2\n=\u0000ib|S|+i!0(S|\u00001+S|+1\u00002S|)=2;\n(23)\nscaling, for convenience, the de\fnitions of the coupling\nparameters \u000b0and!0up by a factor of 2. If g|\u0011\u000b=S\nandb|\u0011b, the solutions are plane waves Sj/ei(kj\u0000!t),\nwith the dispersion\n!=b+ 2!0sin2k\n2\n1 +i\u0000\n\u000b+ 2\u000b0sin2k\n2\u0001: (24)\nIt has a \fnite positive curvature in the (real part of the)\nfrequency as well as in the e\u000bective Gilbert damping (or\nthe inverse quality factor), as k!0 [14]. The wave\nnumberkruns over the Brillouin zone ( \u0000\u0019;\u0019]. Assum-\ning the linearization of the spin dynamics is performed\nwith respect to a stable state (i.e., energy minimum of\nan isolated spin chain), the eigenfrequency (24) exhibits\nno singularities.\n……\n!0\nAAACAXicbVDLSgMxFM34rPVVdekmWERXZUYFXbgouHFZwT6gHUomvdOGJpMhyQjD0JVf4Fa/wJ249Uv8AP/DTDsL23ogcDjnXu7JCWLOtHHdb2dldW19Y7O0Vd7e2d3brxwctrRMFIUmlVyqTkA0cBZB0zDDoRMrICLg0A7Gd7nffgKlmYweTRqDL8gwYiGjxFip3ZMChuSsX6m6NXcKvEy8glRRgUa/8tMbSJoIiAzlROuu58bGz4gyjHKYlHuJhpjQMRlC19KICNB+No07wadWGeBQKvsig6fq342MCK1TEdhJQcxIL3q5+J/XTUx442csihMDEZ0dChOOjcT53/GAKaCGp5YQqpjNiumIKEKNbWjuSpon0xPbi7fYwjJpXdS8y5r3cFWt3xYNldAxOkHnyEPXqI7uUQM1EUVj9IJe0Zvz7Lw7H87nbHTFKXaO0Bycr1+7HZfL↵0\nAAACAXicbVDLSsNAFL3xWeur6tJNsIiuSqKCLlwU3LisYB/QhnIznbRDJ5MwMxFC6MovcKtf4E7c+iV+gP/hpM3Cth4YOJxzL/fM8WPOlHacb2tldW19Y7O0Vd7e2d3brxwctlSUSEKbJOKR7PioKGeCNjXTnHZiSTH0OW3747vcbz9RqVgkHnUaUy/EoWABI6iN1O4hj0d41q9UnZozhb1M3IJUoUCjX/npDSKShFRowlGpruvE2stQakY4nZR7iaIxkjEOaddQgSFVXjaNO7FPjTKwg0iaJ7Q9Vf9uZBgqlYa+mQxRj9Sil4v/ed1EBzdexkScaCrI7FCQcFtHdv53e8AkJZqnhiCRzGS1yQglEm0amruS5snUxPTiLrawTFoXNfey5j5cVeu3RUMlOIYTOAcXrqEO99CAJhAYwwu8wpv1bL1bH9bnbHTFKnaOYA7W1y+2MZfI!1\nAAACAnicbVDLSgMxFL1TX7W+qi7dBIvgqsyooAsXBTcuK9gHtEPJpJk2NMkMSUYYhu78Arf6Be7ErT/iB/gfZtpZ2NYDgcM593JPThBzpo3rfjultfWNza3ydmVnd2//oHp41NZRoghtkYhHqhtgTTmTtGWY4bQbK4pFwGknmNzlfueJKs0i+WjSmPoCjyQLGcHGSt1+JOgID7xBtebW3RnQKvEKUoMCzUH1pz+MSCKoNIRjrXueGxs/w8owwum00k80jTGZ4BHtWSqxoNrPZnmn6MwqQxRGyj5p0Ez9u5FhoXUqAjspsBnrZS8X//N6iQlv/IzJODFUkvmhMOHIRCj/PBoyRYnhqSWYKGazIjLGChNjK1q4kubJ9NT24i23sEraF3Xvsu49XNUat0VDZTiBUzgHD66hAffQhBYQ4PACr/DmPDvvzofzOR8tOcXOMSzA+foFieKYPg==!2\nAAACAnicbVDLSgMxFM34rPVVdekmWARXZaYKunBRcOOygn1AO5RMeqcNTTJDkhGGoTu/wK1+gTtx64/4Af6HmXYWtvVA4HDOvdyTE8ScaeO6387a+sbm1nZpp7y7t39wWDk6busoURRaNOKR6gZEA2cSWoYZDt1YAREBh04wucv9zhMozSL5aNIYfEFGkoWMEmOlbj8SMCKD+qBSdWvuDHiVeAWpogLNQeWnP4xoIkAayonWPc+NjZ8RZRjlMC33Ew0xoRMygp6lkgjQfjbLO8XnVhniMFL2SYNn6t+NjAitUxHYSUHMWC97ufif10tMeONnTMaJAUnnh8KEYxPh/PN4yBRQw1NLCFXMZsV0TBShxla0cCXNk+mp7cVbbmGVtOs177LmPVxVG7dFQyV0is7QBfLQNWqge9RELUQRRy/oFb05z8678+F8zkfXnGLnBC3A+foFi3mYPw==↵1\nAAACAnicbVC7SgNBFL0bXzG+opY2g0GwCrsqaGERsLGMYB6QLOHuZDYZMvtgZlZYlnR+ga1+gZ3Y+iN+gP/hbLKFSTwwcDjnXu6Z48WCK23b31ZpbX1jc6u8XdnZ3ds/qB4etVWUSMpaNBKR7HqomOAha2muBevGkmHgCdbxJne533liUvEofNRpzNwARyH3OUVtpG4fRTzGgTOo1uy6PQNZJU5BalCgOaj+9IcRTQIWaipQqZ5jx9rNUGpOBZtW+oliMdIJjljP0BADptxslndKzowyJH4kzQs1mal/NzIMlEoDz0wGqMdq2cvF/7xeov0bN+NhnGgW0vkhPxFERyT/PBlyyagWqSFIJTdZCR2jRKpNRQtX0jyZmppenOUWVkn7ou5c1p2Hq1rjtmioDCdwCufgwDU04B6a0AIKAl7gFd6sZ+vd+rA+56Mlq9g5hgVYX7+E85g7↵2\nAAACAnicbVDLSsNAFL3xWeur6tJNsAiuSlIFXbgouHFZwT6gDeVmOmmHTiZhZiKE0J1f4Fa/wJ249Uf8AP/DSZuFbT0wcDjnXu6Z48ecKe0439ba+sbm1nZpp7y7t39wWDk6bqsokYS2SMQj2fVRUc4EbWmmOe3GkmLoc9rxJ3e533miUrFIPOo0pl6II8ECRlAbqdtHHo9xUB9Uqk7NmcFeJW5BqlCgOaj89IcRSUIqNOGoVM91Yu1lKDUjnE7L/UTRGMkER7RnqMCQKi+b5Z3a50YZ2kEkzRPanql/NzIMlUpD30yGqMdq2cvF/7xeooMbL2MiTjQVZH4oSLitIzv/vD1kkhLNU0OQSGay2mSMEok2FS1cSfNkamp6cZdbWCXtes29rLkPV9XGbdFQCU7hDC7AhWtowD00oQUEOLzAK7xZz9a79WF9zkfXrGLnBBZgff0ChoqYPA==unit cellAAACB3icbVC7SgNBFJ2NrxhfUUubwSBYhV0VtLAI2FhGMA9IljA7uZsMmZldZ2aFsMTeL7DVL7ATWz/DD/A/nE22MIkHLhzOuZd7OEHMmTau++0UVlbX1jeKm6Wt7Z3dvfL+QVNHiaLQoBGPVDsgGjiT0DDMcGjHCogIOLSC0U3mtx5BaRbJezOOwRdkIFnIKDFW8rtK4EQy80SB81654lbdKfAy8XJSQTnqvfJPtx/RRIA0lBOtO54bGz8lyjDKYVLqJhpiQkdkAB1LJRGg/XQaeoJPrNLHYaTsSIOn6t+LlAitxyKwm4KYoV70MvE/r5OY8MpPmYwTA5LOHoUJxybCWQO4zxRQw8eWEKqYzYrpkChCje1p7ss4S6YnthdvsYVl0jyreudV7+6iUrvOGyqiI3SMTpGHLlEN3aI6aiCKHtALekVvzrPz7nw4n7PVgpPfHKI5OF+/8JGatA==\nFIG. 4. Spin waves in a ferromagnetically-ordered spin chain.\nThe neighboring sites are interacting via reactive, !0, and\ndissipative, \u000b0, exchange couplings. A unit cell is composed\nof a nonidentical spin pair, with the individual frequencies\nand damping parametrized by !{and\u000b{, respectively.\nTo allow for a dimerization of spin dynamics (a term we\nuse broadly to account for a two-spin unit cell), we next\nsuppose the onsite \feld and damping are alternating in\nmagnitude between neighboring sites, as in the preceding\ntwo-spin model. See Fig. 4 for a schematic. We will now\nlook for solutions of the form\nS{|(t) =S{(t)eik|; (25)\nwhere{= 1;2 is the sublattice index and |is the unit-cell\nindex labeling repeated site pairs. The ensuing dynam-\nics of S{(t) are then governed by a k-dependent 2\u00022Hamiltonian:\n^H=\u0014\n1 +i\u0012\n\u000b++\u000b\u0000^\u001bz\u0000\u000b0cosk\n2^\u001bk\nx\u0013\u0015\u00001\n\u0002\u0012\n!++!\u0000^\u001bz\u0000!0cosk\n2^\u001bk\nx\u0013\n:\nThis is fully analogous to Eqs. (10)-(13) but with \u000b0!\n\u000b0cosk\n2and!0!!0cosk\n2now modulated by the factor\nof cosk\n2and rotating the Pauli matrix,\n^\u001bx!^\u001bk\nx\u0011cosk\n2^\u001bx+ sink\n2^\u001by; (26)\nby the angle k=2 around the zaxis.\nFocusing on a purely dissipative coupling by setting\n!0!0, as before, we \fnd frequency eigenvalues [normal-\nized by ~!+\u0011!+=(1 +i\u000b+)] in the form (18):\n\u0015\u0006=\u0006r\n(\r\u0000\u0000i\u000b\u0000)2\u0000\u000b02cos2k\n2; (27)\nwith\r\u0000and\u000b\u0000now denoting the frequency and damp-\ning asymmetries on the adjacent lattice sites. The role\nof the spin-pumping coupling \u000b0is maximized at k= 0,\nwhere the antisymmetric mode gets damped relative to\nthe symmetric one, and gets diminished towards the Bril-\nlouin zone boundaries.\nAs before, we can reach the EP singularity by set-\nting\u000b\u0000!0 and varying \r\u0000and/ork. Physically, this\ncorresponds to a homogeneous spin chain, apart from a\nstaggering of the applied magnetic \feld (parametrized\nby\r\u0000). Suppose all the parameters of the system are\n\fxed, withj\u000b0=\r\u0000j>1, so that there are two real-valued\nmomenta,\nk\u0006=\u00062 cos\u00001\f\f\f\r\u0000\n\u000b0\f\f\f; (28)\nat which the EPs shown in Fig. 2 are traversed as we\nmove within the Brillouin zone. For k2(k\u0000;k+), we are\ne\u000bectively inside the circle shown in Fig. 2, while out-\nside otherwise. In Fig. 5, we plot the corresponding (real\npart of the) eigenfrequencies, \u0015\u0006=\r\u0000, for\u000b0=\r\u0000varied\nbetween 0 and 5. \u000b0=\r\u0000!1 corresponds to the critical\npoint, at which the two EPs merge into a single Weyl\npoint and subsequently disappear from the real momen-\ntum axis at smaller dissipative coupling \u000b0.\nNote that at \u000b0=\r\u0000<1, a \fnite dissipative coupling\n\u000b0endows dynamics with a dispersion, despite the ab-\nsence of any Heisenberg exchange J. For\u000b0=\r\u0000>1,\non the other hand, the two modes get synchronized at\nRe\u0015\u0006= 0 (with one mode damped relative to the other)\nfork\u0000< k < k +. These two qualitatively distinct\nregimes of the coupled dynamics are separated by a Weyl\npoint emerging at \u000b0=\r\u0000= 1. We expect these charac-\nteristic dispersions, synchronization, and Weyl criticality\nto provide practical experimental handles to explore the\nconsequences of the emergence of the EPs in our spin6\n⇡AAAB/XicbVDLSgMxFL3js9ZX1aWbYBFclRkVdFnoxmVF+4B2KJk004ZmMiHJCMNQ/AK3+gXuxK3f4gf4H2baWdjWA4HDOfdyT04gOdPGdb+dtfWNza3t0k55d2//4LBydNzWcaIIbZGYx6obYE05E7RlmOG0KxXFUcBpJ5g0cr/zRJVmsXg0qaR+hEeChYxgY6WHvmSDStWtuTOgVeIVpAoFmoPKT38YkySiwhCOte55rjR+hpVhhNNpuZ9oKjGZ4BHtWSpwRLWfzaJO0blVhiiMlX3CoJn6dyPDkdZpFNjJCJuxXvZy8T+vl5jw1s+YkImhgswPhQlHJkb5v9GQKUoMTy3BRDGbFZExVpgY287ClTRPpqe2F2+5hVXSvqx5VzXv/rpabxQNleAUzuACPLiBOtxBE1pAYAQv8ApvzrPz7nw4n/PRNafYOYEFOF+/AJSWUw==\u0000⇡\nAAAB/nicbVDLSgMxFL3js9ZX1aWbYBHcWGZU0GWhG5cV7APaoWTSTBuaZIYkIwxDwS9wq1/gTtz6K36A/2GmnYVtPRA4nHMv9+QEMWfauO63s7a+sbm1Xdop7+7tHxxWjo7bOkoUoS0S8Uh1A6wpZ5K2DDOcdmNFsQg47QSTRu53nqjSLJKPJo2pL/BIspARbHLpsh+zQaXq1twZ0CrxClKFAs1B5ac/jEgiqDSEY617nhsbP8PKMMLptNxPNI0xmeAR7VkqsaDaz2ZZp+jcKkMURso+adBM/buRYaF1KgI7KbAZ62UvF//zeokJ7/yMyTgxVJL5oTDhyEQo/zgaMkWJ4aklmChmsyIyxgoTY+tZuJLmyfTU9uItt7BK2lc177rmPdxU642ioRKcwhlcgAe3UId7aEILCIzhBV7hzXl23p0P53M+uuYUOyewAOfrF22hloo=k+\nAAAB/XicbVDLSsNAFL2pr1pfUZduBosgCCVRQZeFblxWtA9oQ5lMJ+3QySTMTIQQil/gVr/Anbj1W/wA/8NJm4VtPTBwOOde7pnjx5wp7TjfVmltfWNzq7xd2dnd2z+wD4/aKkokoS0S8Uh2fawoZ4K2NNOcdmNJcehz2vEnjdzvPFGpWCQedRpTL8QjwQJGsDbSw2RwMbCrTs2ZAa0StyBVKNAc2D/9YUSSkApNOFaq5zqx9jIsNSOcTiv9RNEYkwke0Z6hAodUedks6hSdGWWIgkiaJzSaqX83MhwqlYa+mQyxHqtlLxf/83qJDm69jIk40VSQ+aEg4UhHKP83GjJJieapIZhIZrIiMsYSE23aWbiS5snU1PTiLrewStqXNfeq5t5fV+uNoqEynMApnIMLN1CHO2hCCwiM4AVe4c16tt6tD+tzPlqyip1jWID19Qua0pYTk\u0000\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Re\u0000\nAAACC3icbVDLSsNAFJ3UV62PRl26GSyCq5KooMtCNy6r2Ac0oUwm03bozCTMTIQQ8gl+gVv9Anfi1o/wA/wPJ20WtvXAhcM593AvJ4gZVdpxvq3KxubW9k51t7a3f3BYt4+OeypKJCZdHLFIDgKkCKOCdDXVjAxiSRAPGOkHs3bh95+IVDQSjzqNic/RRNAxxUgbaWTXM09y+EByj5lQiEZ2w2k6c8B14pakAUp0RvaPF0Y44URozJBSQ9eJtZ8hqSlmJK95iSIxwjM0IUNDBeJE+dn88RyeGyWE40iaERrO1b+JDHGlUh6YTY70VK16hfifN0z0+NbPqIgTTQReHBonDOoIFi3AkEqCNUsNQVhS8yvEUyQR1qarpStp8ZnKTS/uagvrpHfZdK+a7v11o9UuG6qCU3AGLoALbkAL3IEO6AIMEvACXsGb9Wy9Wx/W52K1YpWZE7AE6+sXorSbjg==\u0000\u0000\nAAACAnicbVDLSgMxFL3js9ZX1aWbYBHcWGZU0GWhG5cV7APaoWTSTBuaZIYkIwxDd36BW/0Cd+LWH/ED/A8z7Sxs64HA4Zx7uScniDnTxnW/nbX1jc2t7dJOeXdv/+CwcnTc1lGiCG2RiEeqG2BNOZO0ZZjhtBsrikXAaSeYNHK/80SVZpF8NGlMfYFHkoWMYGOlbn+EhcCDy0Gl6tbcGdAq8QpShQLNQeWnP4xIIqg0hGOte54bGz/DyjDC6bTcTzSNMZngEe1ZKrGg2s9meafo3CpDFEbKPmnQTP27kWGhdSoCOymwGetlLxf/83qJCe/8jMk4MVSS+aEw4chEKP88GjJFieGpJZgoZrMiMsYKE2MrWriS5sn01PbiLbewStpXNe+65j3cVOuNoqESnMIZXIAHt1CHe2hCCwhweIFXeHOenXfnw/mcj645xc4JLMD5+gV73Zg7\n↵0\nAAACAXicbVDLSsNAFL3xWeur6tLNYBFdlUQFXRa6cVnBPqAN5WY6aYdOHsxMhBC68gvc6he4E7d+iR/gfzhps7CtBwYO59zLPXO8WHClbfvbWlvf2NzaLu2Ud/f2Dw4rR8dtFSWSshaNRCS7HiomeMhammvBurFkGHiCdbxJI/c7T0wqHoWPOo2ZG+Ao5D6nqI3U6aOIx3gxqFTtmj0DWSVOQapQoDmo/PSHEU0CFmoqUKmeY8fazVBqTgWblvuJYjHSCY5Yz9AQA6bcbBZ3Ss6NMiR+JM0LNZmpfzcyDJRKA89MBqjHatnLxf+8XqL9OzfjYZxoFtL5IT8RREck/zsZcsmoFqkhSCU3WQkdo0SqTUMLV9I8mZqaXpzlFlZJ+6rmXNech5tqvVE0VIJTOINLcOAW6nAPTWgBhQm8wCu8Wc/Wu/Vhfc5H16xi5wQWYH39ArhMl88=\nFIG. 5. The (positive real part of the) eigenfrequencies (27),\nwithkswept over the Brillouin zone. Here, we set \u000b\u0000!0 and\nvary\u000b0=\r\u0000! f0;0:6;0:9;1;1:1;1:5g. The EPsk\u0006, Eqs. (28),\nare marked for \u000b0=\r\u0000= 1:5. Im\u0015= 0 when Re \u00156= 0 and\nvice versa.\nchain. It is intriguing, in particular, how the group ve-\nlocity changes in a steplike fashion around the EPs k\u0006.\nWe remind, in the closing of this section, that these\ndynamics are associated with an overall decaying enve-\nlope function governed by the complex-valued frequency\n~!+\u0011!+=(1+i\u000b+), which would ensure stability in equi-\nlibrium. If the spin chain is thermoelectrically pumped\n[20], in order to e\u000bectively tune \u000b+to zero, the symmet-\nric mode would go unstable within the ( k\u0000;k+) interval\ndelineated by the EPs, if \u000b0>j\r\u0000j.\nIV. ANTIFERROMAGNETIC ALIGNMENT\nA. Two spins\nReturning to our two-spin dynamics, Eq. (4), let us\nnow consider an antiferromagnetic (AF) state, J <0, of\ntwo equal spins. We will suppose S{\u0019(\u00001){Sz, for sites\n{= 1;2. The local \felds are b{= [b+ (\u00001){K]z=\n(\u00001){b{z,b{\u0011(\u00001){b+K, in terms of an easy-axis\nanisotropy K\u00150 and a collinearly applied uniform \feld\nb. The canonical transverse coordinates are now conve-\nniently de\fned as S{\u0011(\u00001){(Sx\n{+iSy\n{)=p\n2S, obeying\nifS{;S\u0003\n{g\u0019(\u00001){. Linearizing equations of motion (4)\nin terms of these coordinates, we get\n[(\u00001){+i\u000b{]_S{\u0000i\u000b0_S~{=\u0000i!{S{+i!0S~{; (29)\nwhere, as before, !{\u0011b{+!0,\u000b{\u0011g{S+\u000b0, and we\ndenoted by the primed coe\u000ecients !0\u0011jJjSand\u000b0\u0011\nG=S. Adhering to the form of Eq. (9), this system of two\nequations can be written in terms of\n^d=\u0000g\u0000S\u0000(g+S+\u000b0)^\u001bz+i\u000b0^\u001by (30)\nand\n^h=b\u0000(K+!0)^\u001bz+i!0^\u001by; (31)whereg\u0006\u0011(g1\u0006g2)=2. The net e\u000bective non-Hermitian\nHamiltonian (13) is thus\n^H\u0019[1 +ig\u0000S+i(g+S+\u000b0)^\u001bz+\u000b0^\u001by]\n\u0002[b\u0000(K+!0)^\u001bz+i!0^\u001by];(32)\nsupposing, as before, that all the dimensionless damping\nparameters are small.\nSettingg\u0000!0,g+S+\u000b0!\u000b,b!0, andK+!0!\u0014,\nwe get\n^H\u0019(1 +i\u000b^\u001bz+\u000b0^\u001by) (i!0^\u001by\u0000\u0014^\u001bz)\n=\u0000i(\u000b\u0014\u0000\u000b0!0) +i(\u000b!0\u0000\u000b0\u0014)^\u001bx+ (i!0^\u001by\u0000\u0014^\u001bz);\n(33)\nwith constraints \u000b\u0015\u000b0and\u0014\u0015!0. This Hamiltonian\ndescribes a two-site antiferromagnet, with e\u000bective local\ndamping\u000b, local Larmor frequency \u0014, dissipative cou-\npling\u000b0, and exchange coupling !0. Dropping the con-\nstant piece,/(\u000b\u0014\u0000\u000b0!0)\u00150, which governs an overall\ndecay, we get\nH2=\u00142\u0000!02\u0000(\u000b!0\u0000\u000b0\u0014)2: (34)\nSetting this to zero, in order to locate the EP, we thus\nrequire\n\u000b!0\u0000\u000b0\u0014=\u0006p\n\u00142\u0000!02; (35)\nwith the expression under the square root being positive\n(for a stable AF con\fguration with K\u00150). Writing\n\u000b=a+\u000b0, where a\u00150 is the intrinsic local damp-\ning (excluding spin pumping), the EP condition can be\nrewritten as\na!0=\u000b0K\u0006p\nK(K+ 2!0): (36)\nSupposing, as before, that \u000b0\u001c1 and also !0\u001dK\n(strong exchange), we \fnally get\na\u0019r\n2K\n!0\u001c1; (37)\nwhich is consistent with the smallness of a. This value\nof damping corresponds to the quality factor of the AF\nresonance\u00181, however. In the spirit of these approxima-\ntions, we thus end up with the full frequency eigenvalues\nfollowing from Hamiltonian (33):\n\u0015\u0006\u0019\u0000ia!0\u0006p\n2(1\u0000a!0=!)!; (38)\nnear the EP point a\u0019!=!0, where!\u0011p\n2K!0is the\nintrinsic AF resonance frequency.\nB. AF vs F cases\nIt is amusing to remark that two spins interacting by\na pure antiferromagnetic exchange, in the absence of any\ndamping and additional \felds, naturally realize an EP:\n^H0\nAF=!0(i^\u001by\u0000^\u001bz); (39)7\naccording to Eqs. (30) and (31), after setting to zero all\nterms but!0. This is in stark contrast to the analogous\nferromagnetic case, where\n^H0\nF=!0(1\u0000^\u001bx); (40)\naccording to Eq. (10). ^H02\nAF= 0, while ( ^H0\nF=!0\u00001)2= 1\n(having subtracted the constant part), suggests that the\nAF dynamics is more peculiar. Indeed, decomposing the\nsmall-angle dynamics into the symmetric and antisym-\nmetric components: S\u0006= (S1\u0006S2)=2, we get\nAF : _S\u0000= 0;_S+= 2i!0S\u0000;\nF : _S+= 0;_S\u0000=\u00002i!0S\u0000:(41)\nBoth cases exhibit a zero mode, S+, corresponding to\na reorientation of the overall order parameter (N\u0013 eel in\nthe AF and magnetic in the F cases). The distortion of\nthis order, i.e., S\u00006= 0, triggers its small-angle preces-\nsion with frequency 2 !0in the F case, while resulting in\nan apparently unbounded growth of S+in the AF case.\nThe antiferromagnetic EP point thus results in a break-\ndown of the linearized treatment. We of course know the\ncorresponding outcome in the full spin dynamics: The\nN\u0013 eel order parameter precesses in the plane perpendicu-\nlar to the distortion S\u0000, which parametrizes relative spin\ncanting, with the frequency /!0S\u0000. This simple exam-\nple illustrates how an EP takes the coupled dynamics out\nof the linearized perturbative treatment, necessitating a\nfall back on a more complete description.\nC. Spin chain\nViewing the above two-spin system as a unit cell of an\nin\fnite homogeneous spin chain (which is thus naturally\ndimerized), we are looking for solutions of the form (25),\nwhere{= 1;2 is the sublattice index, as before, and |\nlabels the unit cells. See Fig. 6 for a schematic. The\nresultant equations of motion [cf. Eq. (29)] are\n[(\u00001){+i\u000b{]_S{\u0000i\u000b0_S~{1 +e(\u00001){ik\n2=\n\u0000i!{S{+i!0S~{1 +e(\u00001){ik\n2;(42)\nscaling, for convenience, the de\fnitions of the coupling\nparameters \u000b0and!0up by a factor of 2. The analogs\nof Eqs. (30) and (31) become (having set g\u0000!0,g+S+\n\u000b0!\u000b,b!0, andK+!0!\u0014):\n^d=\u0000\u000b^\u001bz+i\u000b0cosk\n2^\u001bk\ny (43)\nand\n^h=\u0000\u0014^\u001bz+i!0cosk\n2^\u001bk\ny; (44)\nwhere\n^\u001bk\ny\u0011cosk\n2^\u001by\u0000sink\n2^\u001bx (45)is the Pauli matrix ^ \u001byrotated by angle k=2 around the\nzaxis. These equations reduce to Eqs. (30) and (31) in\nthe limit of k= 0.\n……!0\nAAACAXicbVDLSgMxFM34rPVVdekmWERXZUYFXbgouHFZwT6gHUomvdOGJpMhyQjD0JVf4Fa/wJ249Uv8AP/DTDsL23ogcDjnXu7JCWLOtHHdb2dldW19Y7O0Vd7e2d3brxwctrRMFIUmlVyqTkA0cBZB0zDDoRMrICLg0A7Gd7nffgKlmYweTRqDL8gwYiGjxFip3ZMChuSsX6m6NXcKvEy8glRRgUa/8tMbSJoIiAzlROuu58bGz4gyjHKYlHuJhpjQMRlC19KICNB+No07wadWGeBQKvsig6fq342MCK1TEdhJQcxIL3q5+J/XTUx442csihMDEZ0dChOOjcT53/GAKaCGp5YQqpjNiumIKEKNbWjuSpon0xPbi7fYwjJpXdS8y5r3cFWt3xYNldAxOkHnyEPXqI7uUQM1EUVj9IJe0Zvz7Lw7H87nbHTFKXaO0Bycr1+7HZfL↵0\nAAACAXicbVDLSsNAFL3xWeur6tJNsIiuSqKCLlwU3LisYB/QhnIznbRDJ5MwMxFC6MovcKtf4E7c+iV+gP/hpM3Cth4YOJxzL/fM8WPOlHacb2tldW19Y7O0Vd7e2d3brxwctlSUSEKbJOKR7PioKGeCNjXTnHZiSTH0OW3747vcbz9RqVgkHnUaUy/EoWABI6iN1O4hj0d41q9UnZozhb1M3IJUoUCjX/npDSKShFRowlGpruvE2stQakY4nZR7iaIxkjEOaddQgSFVXjaNO7FPjTKwg0iaJ7Q9Vf9uZBgqlYa+mQxRj9Sil4v/ed1EBzdexkScaCrI7FCQcFtHdv53e8AkJZqnhiCRzGS1yQglEm0amruS5snUxPTiLrawTFoXNfey5j5cVeu3RUMlOIYTOAcXrqEO99CAJhAYwwu8wpv1bL1bH9bnbHTFKnaOYA7W1y+2MZfI!1\nAAACAnicbVDLSgMxFL1TX7W+qi7dBIvgqsyooAsXBTcuK9gHtEPJpJk2NMkMSUYYhu78Arf6Be7ErT/iB/gfZtpZ2NYDgcM593JPThBzpo3rfjultfWNza3ydmVnd2//oHp41NZRoghtkYhHqhtgTTmTtGWY4bQbK4pFwGknmNzlfueJKs0i+WjSmPoCjyQLGcHGSt1+JOgID7xBtebW3RnQKvEKUoMCzUH1pz+MSCKoNIRjrXueGxs/w8owwum00k80jTGZ4BHtWSqxoNrPZnmn6MwqQxRGyj5p0Ez9u5FhoXUqAjspsBnrZS8X//N6iQlv/IzJODFUkvmhMOHIRCj/PBoyRYnhqSWYKGazIjLGChNjK1q4kubJ9NT24i23sEraF3Xvsu49XNUat0VDZTiBUzgHD66hAffQhBYQ4PACr/DmPDvvzofzOR8tOcXOMSzA+foFieKYPg==!2\nAAACAnicbVDLSgMxFM34rPVVdekmWARXZaYKunBRcOOygn1AO5RMeqcNTTJDkhGGoTu/wK1+gTtx64/4Af6HmXYWtvVA4HDOvdyTE8ScaeO6387a+sbm1nZpp7y7t39wWDk6busoURRaNOKR6gZEA2cSWoYZDt1YAREBh04wucv9zhMozSL5aNIYfEFGkoWMEmOlbj8SMCKD+qBSdWvuDHiVeAWpogLNQeWnP4xoIkAayonWPc+NjZ8RZRjlMC33Ew0xoRMygp6lkgjQfjbLO8XnVhniMFL2SYNn6t+NjAitUxHYSUHMWC97ufif10tMeONnTMaJAUnnh8KEYxPh/PN4yBRQw1NLCFXMZsV0TBShxla0cCXNk+mp7cVbbmGVtOs177LmPVxVG7dFQyV0is7QBfLQNWqge9RELUQRRy/oFb05z8678+F8zkfXnGLnBC3A+foFi3mYPw==↵1\nAAACAnicbVC7SgNBFL0bXzG+opY2g0GwCrsqaGERsLGMYB6QLOHuZDYZMvtgZlZYlnR+ga1+gZ3Y+iN+gP/hbLKFSTwwcDjnXu6Z48WCK23b31ZpbX1jc6u8XdnZ3ds/qB4etVWUSMpaNBKR7HqomOAha2muBevGkmHgCdbxJne533liUvEofNRpzNwARyH3OUVtpG4fRTzGgTOo1uy6PQNZJU5BalCgOaj+9IcRTQIWaipQqZ5jx9rNUGpOBZtW+oliMdIJjljP0BADptxslndKzowyJH4kzQs1mal/NzIMlEoDz0wGqMdq2cvF/7xeov0bN+NhnGgW0vkhPxFERyT/PBlyyagWqSFIJTdZCR2jRKpNRQtX0jyZmppenOUWVkn7ou5c1p2Hq1rjtmioDCdwCufgwDU04B6a0AIKAl7gFd6sZ+vd+rA+56Mlq9g5hgVYX7+E85g7↵2\nAAACAnicbVDLSsNAFL3xWeur6tJNsAiuSlIFXbgouHFZwT6gDeVmOmmHTiZhZiKE0J1f4Fa/wJ249Uf8AP/DSZuFbT0wcDjnXu6Z48ecKe0439ba+sbm1nZpp7y7t39wWDk6bqsokYS2SMQj2fVRUc4EbWmmOe3GkmLoc9rxJ3e533miUrFIPOo0pl6II8ECRlAbqdtHHo9xUB9Uqk7NmcFeJW5BqlCgOaj89IcRSUIqNOGoVM91Yu1lKDUjnE7L/UTRGMkER7RnqMCQKi+b5Z3a50YZ2kEkzRPanql/NzIMlUpD30yGqMdq2cvF/7xeooMbL2MiTjQVZH4oSLitIzv/vD1kkhLNU0OQSGay2mSMEok2FS1cSfNkamp6cZdbWCXtes29rLkPV9XGbdFQCU7hDC7AhWtowD00oQUEOLzAK7xZz9a79WF9zkfXrGLnBBZgff0ChoqYPA==unit cellAAACB3icbVC7SgNBFJ2NrxhfUUubwSBYhV0VtLAI2FhGMA9IljA7uZsMmZldZ2aFsMTeL7DVL7ATWz/DD/A/nE22MIkHLhzOuZd7OEHMmTau++0UVlbX1jeKm6Wt7Z3dvfL+QVNHiaLQoBGPVDsgGjiT0DDMcGjHCogIOLSC0U3mtx5BaRbJezOOwRdkIFnIKDFW8rtK4EQy80SB81654lbdKfAy8XJSQTnqvfJPtx/RRIA0lBOtO54bGz8lyjDKYVLqJhpiQkdkAB1LJRGg/XQaeoJPrNLHYaTsSIOn6t+LlAitxyKwm4KYoV70MvE/r5OY8MpPmYwTA5LOHoUJxybCWQO4zxRQw8eWEKqYzYrpkChCje1p7ss4S6YnthdvsYVl0jyreudV7+6iUrvOGyqiI3SMTpGHLlEN3aI6aiCKHtALekVvzrPz7nw4n7PVgpPfHKI5OF+/8JGatA==\nFIG. 6. Spin waves in an antiferromagnetically-ordered spin\nchain. The neighboring sites are interacting via reactive, !0,\nand dissipative, \u000b0, exchange coupling. A unit cell is com-\nposed of an antiferromagnetic dimer, with the individual fre-\nquencies and damping parametrized by !{and\u000b{, respec-\ntively.\nThe spectrum and the subsequent EP analysis can thus\nbe obtained from Eq. (34), after scaling \u000b0and!0by\ncosk\n2, which results in the k-dependent eigenfrequencies\n\u0015\u0006\u0019\u0000ia!0\u0006s\n!2+!02\u0012\nsin2k\n2\u0000acos2k\n2\u0013\n:(46)\nWe are omitting here terms /\u000b0, which can be shown\nto be unimportant when the intrinsic damping ais ap-\nproaching the EP point. The EP is located at\nacosk\n2\u0019r\n(!=!0)2+ sin2k\n2: (47)\nFora\u001c1, we thus need to focus on k!0, which gives\na\u0019p\n(!=!0)2+ (k=2)2\u0019!k=!0; (48)\nsupposing, as before, that K\u001c!0and thus!\u001c!0.\n!k\u0011q\n!2+!02sin2k\n2is the intrinsic undamped disper-\nsion. Note that in the absence of damping, a!0, the\nEP is reached at k!0, requiring the absence of any\nanisotropy, !!0. This reproduces the elementary anti-\nferromagnetic EP discussed in Sec. IV B.\nClose to the EP, the frequency eigenvalues (46) are\ngiven by\n\u0015\u0006\u0019\u0000ia!0\u0006p\n2(1\u0000a!0=!k)!k: (49)\nThis leads to a qualitatively similar behavior in the k-\ndependent eigenfrequency dispersions, as already dis-\ncussed for the ferromagnetic case (cf. Fig. 5, along with\nthe associated discussion). In particular, for aAAAB/XicbVDLSgMxFL3js9ZX1aWbYBFclRkVdFnoxmVF+4B2KJk004ZmMiHJCMNQ/AK3+gXuxK3f4gf4H2baWdjWA4HDOfdyT04gOdPGdb+dtfWNza3t0k55d2//4LBydNzWcaIIbZGYx6obYE05E7RlmOG0KxXFUcBpJ5g0cr/zRJVmsXg0qaR+hEeChYxgY6WHvmSDStWtuTOgVeIVpAoFmoPKT38YkySiwhCOte55rjR+hpVhhNNpuZ9oKjGZ4BHtWSpwRLWfzaJO0blVhiiMlX3CoJn6dyPDkdZpFNjJCJuxXvZy8T+vl5jw1s+YkImhgswPhQlHJkb5v9GQKUoMTy3BRDGbFZExVpgY287ClTRPpqe2F2+5hVXSvqx5VzXv/rpabxQNleAUzuACPLiBOtxBE1pAYAQv8ApvzrPz7nw4n/PRNafYOYEFOF+/AJSWUw==\u0000⇡\nAAAB/nicbVDLSgMxFL3js9ZX1aWbYBHcWGZU0GWhG5cV7APaoWTSTBuaZIYkIwxDwS9wq1/gTtz6K36A/2GmnYVtPRA4nHMv9+QEMWfauO63s7a+sbm1Xdop7+7tHxxWjo7bOkoUoS0S8Uh1A6wpZ5K2DDOcdmNFsQg47QSTRu53nqjSLJKPJo2pL/BIspARbHLpsh+zQaXq1twZ0CrxClKFAs1B5ac/jEgiqDSEY617nhsbP8PKMMLptNxPNI0xmeAR7VkqsaDaz2ZZp+jcKkMURso+adBM/buRYaF1KgI7KbAZ62UvF//zeokJ7/yMyTgxVJL5oTDhyEQo/zgaMkWJ4aklmChmsyIyxgoTY+tZuJLmyfTU9uItt7BK2lc177rmPdxU642ioRKcwhlcgAe3UId7aEILCIzhBV7hzXl23p0P53M+uuYUOyewAOfrF22hloo=k+\nAAAB/XicbVDLSsNAFL2pr1pfUZduBosgCCVRQZeFblxWtA9oQ5lMJ+3QySTMTIQQil/gVr/Anbj1W/wA/8NJm4VtPTBwOOde7pnjx5wp7TjfVmltfWNzq7xd2dnd2z+wD4/aKkokoS0S8Uh2fawoZ4K2NNOcdmNJcehz2vEnjdzvPFGpWCQedRpTL8QjwQJGsDbSw2RwMbCrTs2ZAa0StyBVKNAc2D/9YUSSkApNOFaq5zqx9jIsNSOcTiv9RNEYkwke0Z6hAodUedks6hSdGWWIgkiaJzSaqX83MhwqlYa+mQyxHqtlLxf/83qJDm69jIk40VSQ+aEg4UhHKP83GjJJieapIZhIZrIiMsYSE23aWbiS5snU1PTiLrewStqXNfeq5t5fV+uNoqEynMApnIMLN1CHO2hCCwiM4AVe4c16tt6tD+tzPlqyip1jWID19Qua0pYTk\u0000\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!0\nAAACAXicbVDLSgMxFL3js9ZX1aWbYBFdlRkVdFnoxmUF+4B2KJk004YmmSHJCMPQlV/gVr/Anbj1S/wA/8NMOwvbeiBwOOde7skJYs60cd1vZ219Y3Nru7RT3t3bPzisHB23dZQoQlsk4pHqBlhTziRtGWY47caKYhFw2gkmjdzvPFGlWSQfTRpTX+CRZCEj2Fip048EHeGLQaXq1twZ0CrxClKFAs1B5ac/jEgiqDSEY617nhsbP8PKMMLptNxPNI0xmeAR7VkqsaDaz2Zxp+jcKkMURso+adBM/buRYaF1KgI7KbAZ62UvF//zeokJ7/yMyTgxVJL5oTDhyEQo/zsaMkWJ4aklmChmsyIyxgoTYxtauJLmyfTU9uItt7BK2lc177rmPdxU642ioRKcwhlcgge3UId7aEILCEzgBV7hzXl23p0P53M+uuYUOyewAOfrF704l9I=Re\u0000\nAAACC3icbVDLSsNAFJ3UV62PRl26GSyCq5KooMtCNy6r2Ac0oUwm03bozCTMTIQQ8gl+gVv9Anfi1o/wA/wPJ20WtvXAhcM593AvJ4gZVdpxvq3KxubW9k51t7a3f3BYt4+OeypKJCZdHLFIDgKkCKOCdDXVjAxiSRAPGOkHs3bh95+IVDQSjzqNic/RRNAxxUgbaWTXM09y+EByj5lQiEZ2w2k6c8B14pakAUp0RvaPF0Y44URozJBSQ9eJtZ8hqSlmJK95iSIxwjM0IUNDBeJE+dn88RyeGyWE40iaERrO1b+JDHGlUh6YTY70VK16hfifN0z0+NbPqIgTTQReHBonDOoIFi3AkEqCNUsNQVhS8yvEUyQR1qarpStp8ZnKTS/uagvrpHfZdK+a7v11o9UuG6qCU3AGLoALbkAL3IEO6AIMEvACXsGb9Wy9Wx/W52K1YpWZE7AE6+sXorSbjg==\n!\nAAACAHicbVDLSgMxFL3js9ZX1aWbYBFclRkVdFnoxmUF+4B2KJk008YmmSHJCMPQjV/gVr/Anbj1T/wA/8NMOwvbeiBwOOde7skJYs60cd1vZ219Y3Nru7RT3t3bPzisHB23dZQoQlsk4pHqBlhTziRtGWY47caKYhFw2gkmjdzvPFGlWSQfTBpTX+CRZCEj2Fip3Y8EHeFBperW3BnQKvEKUoUCzUHlpz+MSCKoNIRjrXueGxs/w8owwum03E80jTGZ4BHtWSqxoNrPZmmn6NwqQxRGyj5p0Ez9u5FhoXUqAjspsBnrZS8X//N6iQlv/YzJODFUkvmhMOHIRCj/OhoyRYnhqSWYKGazIjLGChNjC1q4kubJ9NT24i23sEralzXvqubdX1frjaKhEpzCGVyABzdQhztoQgsIPMILvMKb8+y8Ox/O53x0zSl2TmABztcvV9GXoQ==\naAAACBnicbVDLSsNAFL2pr1pfVZduBovgqiQq6LLQjcsK9gFNKDfTSTt0MgkzEyGE7v0Ct/oF7sStv+EH+B8mbRe29cCFwzn3cg/HjwXXxra/rdLG5tb2Tnm3srd/cHhUPT7p6ChRlLVpJCLV81EzwSVrG24E68WKYegL1vUnzcLvPjGleSQfTRozL8SR5AGnaHLJdUM040DhJMPpoFqz6/YMZJ04C1KDBVqD6o87jGgSMmmoQK37jh0bL0NlOBVsWnETzWKkExyxfk4lhkx72SzzlFzkypAEkcpHGjJT/15kGGqdhn6+WWTUq14h/uf1ExPceRmXcWKYpPNHQSKIiUhRABlyxagRaU6QKp5nJXSMCqnJa1r6khbJdNGLs9rCOulc1Z3ruvNwU2s0Fw2V4QzO4RIcuIUG3EML2kAhhhd4hTfr2Xq3PqzP+WrJWtycwhKsr190Q5p7\nFIG. 7. The (positive real part of the) eigenfrequencies (46),\nwithkswept over the Brillouin zone. Here, we set !=!0= 0:3\nand increase afrom 0 to 0:5, in increments of 0 :1 (with the 0 :3\ncorresponding to the gapless dispersion). The EPs k\u0006, which\nsolve Eq. (49), are marked for a= 0:5. Note how increasing\nthe ordinary Gilbert damping acloses the anisotropy gap !\nin the intrinsic AF dispersion.\ndynamics. A stronger damping, a>p\n2K=!0, however,\nresults in the synchronized (zero-frequency damped) dy-\nnamics within the two exceptional points k\u0006(correspond-\ning to!k=!0=a), without any dispersion. Finally, a\nWeyl point with a closed gap is obtained at a=p\n2K=!0.\nIn this critical case (corresponding to the AF resonance\nquality factor\u00181), an ordinary Gilbert damping closes\nthe gap opened in the undamped AF dynamics by an\neasy-axis anisotropy, restoring the linear Goldstone-mode\ndispersion. We plot the positive real part of the disper-\nsion (46) in Fig. 7.\nV. SUMMARY AND OUTLOOK\nWhen embedded into a dissipative environment, cou-\npled spin dynamics can yield exceptional points that have\na drastic e\u000bect on their spectral properties. We focused\nour discussion on ferromagnetic and antiferromagnetic\ndimers and spin chains. In the antiferromagnetic case,\nthe EPs can be experimentally accessed by simply tun-\ning the overall damping of the system. The ferromagneticcase requires a nonlocal dissipative coupling, which can\nbe realized by internal magnetic spin pumping into itin-\nerant degrees of freedom. In both cases, the EP emerges\nin the magnon band structure as a linearly dispersing\nWeyl point, which acts as a precursor to a \rat magnon\ndispersion that extends over a \fnite range of momenta.\nIn the strongly-damped regimes, the EP singularities,\ntherefore, separate the dispersing and nondispersing re-\ngions of the magnon band structure. These examples\nshow how controlling dissipation can dramatically modify\nkinetic properties of magnons, such as a steplike change\nin their group velocities, in the vicinity of an EP. The re-\nsultant thermodynamic response and kinetic coe\u000ecients\nof the magnon gas, such as its spin conductivity and spin\ndi\u000busion length [23], could then be invoked to exhibit this\nphysics, in addition to coherent microwave probes.\nReducing the spins and crossing over to the quantum\nregime of the coupled dynamics, at low temperatures, it\ncan be interesting to explore how the EPs evolve into\nthe quantum limit of magnetic \ructuations. It may be\nintriguing, for example, to look for the associated features\nin the entanglement properties between individual spins\n[24] or magnetic sublattices [25].\nBraiding around the classical EPs in the parameter\nspace taps into their Riemannian topological aspects,\nmanifested, for example, in characteristic phase changes\nand non-Abelian braiding representation in the eigenvec-\ntor evolution [11], along with the associated topologi-\ncal energy transfer [5]. When applying such braiding\nto the evolution of an open quantum spin system, one\nmay be compelled to investigate the possibility of robust\nfeatures inherited from the topology in the underlying\nclassical counterpart, in regard, for example, to quantum\ngates and information processing tasks. In the future\nworks, it could also be interesting to study the braiding\nof magnons in momentum space (or mixed parameter-\nmomentum space), both in the classical and quantum\nregimes.\nACKNOWLEDGMENTS\nWe thank Can-Ming Hu and Gerrit E. W. Bauer for\ndrawing attention to this problem and Benedetta Flebus\nand Mostafa Tanhayi Ahari for helpful discussions. The\nwork was supported by NSF under Grant No. DMR-\n1742928.\n[1] W. D. Heiss, J. Phys. A: Math. Theor. 45, 444016 (2012).\n[2] T. Kato, Perturbation Theory of Linear Operators\n(Springer, Berlin, 1966).\n[3] C. Dembowski, H.-D. Gr af, H. L. Harney, A. Heine,\nW. D. Heiss, H. Rehfeld, and A. Richter, Phys. Rev.\nLett. 86, 787 (2001); J. Doppler, A. A. Mailybaev,\nJ. B ohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Mil-\nburn, P. Rabl, N. Moiseyev, and S. Rotter, Nature 537,76 (2016).\n[4] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia,\nR. El-Ganainy, D. N. Christodoulides, and M. Kha-\njavikhan, Nature 548, 187 (2017); M.-A. Miri and\nA. Al\u0013 u, Science 363, eaar7709 (2019).\n[5] H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Nature\n537, 80 (2016).\n[6] D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q.9\nYou, Nature Comm. 8, 1368 (2017).\n[7] N. R. Bernier, L. D. T\u0013 oth, A. K. Feofanov, and T. J.\nKippenberg, Phys. Rev. A 98, 023841 (2018).\n[8] M. Harder, Y. Yang, B. M. Yao, C. H. Yu, J. W. Rao,\nY. S. Gui, R. L. Stamps, and C.-M. Hu, Phys. Rev. Lett.\n121, 137203 (2018).\n[9] W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and\nL. Yang, Nature 548, 192 (2017).\n[10] C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).\n[11] M. Berry, Ann. New York Acad. Sci. 755, 303 (1995);\nA. A. Mailybaev, O. N. Kirillov, and A. P. Seyranian,\nPhys. Rev. A 72, 014104 (2005); Q. Zhong, M. Kha-\njavikhan, D. N. Christodoulides, and R. El-Ganainy,\nNature Comm. 9, 4808 (2018); X.-L. Zhang, S. Wang,\nB. Hou, and C. T. Chan, Phys. Rev. X 8, 021066 (2018).\n[12] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys.\nRev. X 9, 041015 (2019).\n[13] A. Galda and V. M. Vinokur, Phys. Rev. B 94, 020408(R)\n(2016); 97, 201411(R) (2018); arXiv:1903.09729.\n[14] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[15] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban, and G. E. W. Bauer, Phys. Rev. Lett.90, 187601 (2003); Y. Tserkovnyak, A. Brataas, and\nG. E. W. Bauer, Phys. Rev. B 67, 140404(R) (2003).\n[16] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[17] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[18] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996); L. Berger, Phys. Rev. B 54, 9353 (1996).\n[19] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature\nMater. 11, 391 (2012).\n[20] S. A. Bender, R. A. Duine, and Y. Tserkovnyak, Phys.\nRev. Lett. 108, 246601 (2012); S. A. Bender, R. A.\nDuine, A. Brataas, and Y. Tserkovnyak, Phys. Rev. B\n90, 094409 (2014).\n[21] W. D. Heiss, Phys. Rev. E 61, 929 (2000).\n[22] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984).\n[23] L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef,\nand B. J. van Wees, Nature Phys. 11, 1022 (2015).\n[24] J. Zou, S. K. Kim, and Y. Tserkovnyak,\narXiv:1909.09653.\n[25] A. Kamra, E. Thingstad, G. Rastelli, R. A. Duine,\nA. Brataas, W. Belzig, and A. Sudb\u001c, Phys. Rev. B\n100, 174407 (2019)." }, { "title": "1703.03198v3.Material_developments_and_domain_wall_based_nanosecond_scale_switching_process_in_perpendicularly_magnetized_STT_MRAM_cells.pdf", "content": "Material developments and domain wall based nanosecond-scale switching process in\nperpendicularly magnetized STT-MRAM cells\nThibaut Devolder\u0003and Joo-V on Kim\nCentre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud,\nUniversit ´e Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France\nJ. Swerts, S. Couet, S. Rao, W. Kim, S. Mertens, and G. Kar\nIMEC, Kapeldreef 75, B-3001 Leuven, Belgium\nV . Nikitin\nSAMSUNG Electronics Corporation, 601 McCarthy Blvd Milpitas, CA 95035, USA\nWe investigate the Gilbert damping and the magnetization switching of perpendicularly magnetized FeCoB-\nbased free layers embedded in magnetic tunnel junctions adequate for spin-torque operated magnetic memories.\nWe first study the influence of the boron content in MgO / FeCoB /Ta systems alloys on their Gilbert damping pa-\nrameter after crystallization annealing. Increasing the boron content from 20 to 30% increases the crystallization\ntemperature, thereby postponing the onset of elemental diffusion within the free layer. This reduction of the in-\nterdiffusion of the Ta atoms helps maintaining the Gilbert damping at a low level of 0.009 without any penalty on\nthe anisotropy and the magneto-transport properties up to the 400\u000eC annealing required in CMOS back-end of\nline processing. In addition, we show that dual MgO free layers of composition MgO/FeCoB/Ta/FeCoB/MgO\nhave a substantially lower damping than their MgO/FeCoB/Ta counterparts, reaching damping parameters as\nlow as 0.0039 for a 3 ˚A thick Tantalum spacer. This confirms that the dominant channel of damping is the\npresence of Ta impurities within the FeCoB alloy. On optimized tunnel junctions, we then study the duration of\nthe switching events induced by spin-transfer-torque. We focus on the sub-threshold thermally activated switch-\ning in optimal applied field conditions. From the electrical signatures of the switching, we infer that once the\nnucleation has occurred, the reversal proceeds by a domain wall sweeping though the device at a few 10 m/s.\nThe smaller the device, the faster its switching. We present an analytical model to account for our findings. The\ndomain wall velocity is predicted to scale linearly with the current for devices much larger than the wall width.\nThe wall velocity depends on the Bloch domain wall width, such that the devices with the lowest exchange\nstiffness will be the ones that host the domain walls with the slowest mobilities.\nI. INTRODUCTION\nTunnel magnetoresistance (TMR) and spin transfer torque\n(STT) – the fact that spin-polarized currents manipu-\nlate the magnetization of nanoscale magnets and in par-\nticular magnetic tunnel junction (MTJ) nanopillars – are\nthe basic phenomena underpinning an emerging technol-\nogy called Spin-Transfer-Torque Magnetic Random Access\nMemory (STT-MRAM)1, which combines high endurance,\nlow power requirement2,3, CMOS back-end-of-line (BEOL)\ncompatibility4and potentially large capacity5.\nThe core of an STT-MRAM stack is a magnetic tunnel\njunction composed6of an FeCoB/MgO/FeCoB central block.\nOne of the FeCoB layer is pinned to a high anisotropy syn-\nthetic ferrimagnet to create a fixed reference layer (RL) sys-\ntem while the second FeCoB acts as a free layer (FL). Histor-\nically, the FL is capped with (or deposited on) an amorphous\nmetal such as Ta4,7and more recently capped with a second\nMgO layer to benefit from a second interface anisotropy7–9\nin the so-called ’dual MgO’ configuration. So far, it is un-\nclear whether this benefit of anisotropy can be obtained with-\nout sacrificing the other important properties of the free layer,\nin particular the Gilbert damping.\nIn this paper, we will first tailor the Boron content inside\nthe FeCoB alloy to improve the properties of Ta / FeCoB /\nMgO ’single MgO’ free layers and their resilience to thermal\nannealing. The idea is to postpone the FeCoB crystalliza-tion till the very last stage of the BEOL annealing. Indeed\nmaintaining the amorphous state of FeCoB allows to mini-\nmize the interdiffusion of materials –in our case: tantalum–\nwithin the stack. This interdiffusion is otherwise detrimental\nto the Gilbert damping.\nWe then turn to dual MgO systems comprising a Ta spacer\nlayer in the midst of the FL. This spacer is empirically needed\nto allow proper crystallization and to effectively get perpen-\ndicular magnetic anisotropy (PMA)8,10–14. Unfortunately, the\npresence of heavy elements inside the FeCoB free layer is ex-\npected to alter its damping and to induce some loss of mag-\nnetic moment usually referred as the formation of magneti-\ncally dead layers. We study to what extend the Ta spacer in\nthe dual MgO free layers affects the damping and how this\ndamping compares with the one that can be obtained with sin-\ngle MgO free layers. Once optimized, damping factors as low\nas 0.0039 can be obtained a dual MgO free layer.\nBesides the material issues, the success of STT-MRAM\nalso relies on the capacity to engineer devices in accordance\nwith industry roadmaps concerning speed and miniaturiza-\ntion. To achieve fast switching and design devices accordingly\noptimized, one needs to elucidate the physical mechanism by\nwhich the magnetization switches by STT. Several categories\nof switching modes – macrospin15, domain-wall based16,\nbased on sub-volume nucleation17or based on the spin-wave\namplification18– have been proposed, but single-shot time-\nresolved experimental characterization of the switching patharXiv:1703.03198v3 [cond-mat.mtrl-sci] 4 Sep 20172\nare still scarce19–21. Here we study the nanosecond-scale spin-\ntorque-induced switching in perpendicularly magnetized tun-\nnel junctions with sizes from 50 to 300 nm. Our time-resolved\nexperiments argue for a reversal that happens by the motion\nof a single domain wall, which sweeps through the sample\nat a velocity set by the applied voltage. As a result, the\nswitching duration is proportional to the device length. We\nmodel our finding assuming a single wall moving in a uni-\nform material as a result of spin torque. The wall moves with\na time-averaged velocity that scales with the product of the\nwall width and the ferromagnetic resonance linewidth, such\nthat the devices with the lowest nucleation current densities\nwill be the ones that host the domain walls with the lowest\nmobilities.\nThe paper is split in first a material science part, followed\nby a study of the magnetization reversal dynamics. After a de-\nscription of the samples and the caracterization methods, sec-\ntion II C describes how to choose the optimal Boron content\nin an FeCoB-based free layer for STT-MRAM applications.\nSection II D discusses the benefits of ’dual MgO’ free layers\nwhen compared to ’single MgO’ free layers. Moving to the\nmagnetization switching section, the part III A gathers the de-\nscription of the main properties of the samples and the experi-\nmental methods used to characterize the STT-induced switch-\ning speed. Section III B describes the electrical signatures of\nthe switching mechanism at the nanosecond scale. The latter\nis modeled in section III C in an analytical framework meant\nto clarify the factors that govern the switching speed when the\nreversal involves domain wall motion.\nII. ADVANCED FREE LAYER DESIGNS\nA. Model systems under investigation\nOur objective is to study advanced free layer designs in\nfull STT-MRAM stacks. The stacks were deposited by phys-\nical vapor deposition in a Canon-Anelva EC7800 300 mm\ncluster tool. The MgO tunnel barriers were deposited by\nRF-magnetron sputtering. In dual MgO systems, the top\nMgO layer was fabricated by oxidation of a thin metallic Mg\nfilm. All stacks were post-deposition annealed in a TEL-MSL\nMRT5000 batch furnace in a 1 T perpendicular magnetic field\nfor 30 minutes. Further annealing at 400\u000eC were done in a\nrapid thermal annealing furnace in a N 2atmosphere for a pe-\nriod of 10 minutes.\nWe will focus on several kinds of free layers embod-\nied in state-of-the art bottom-pinned Magnetic Tunnel Junc-\ntions (MTJ) with various reference systems comprising ei-\nther [Co/Ni] and [Co/Pt] based hard layers22,23. Although we\nshall focus here on FLs deposited on [Co/Ni] based synthetic\nantiferromagnet (SAF) reference layers, we have conducted\nthe free layer development also on [Co/Pt] based reference\nlayers. While specific reference layer optimization leads to\nslightly different baseline TMR properties, we have found that\nthe free layer performances were not impacted provided the\nSAF structure is stable with the concerned heat treatment (not\nshown).The first category of samples are the so-called ’single-\nMgO’ free layers. We shall focus on samples with a free\nlayer consists of a 1.4 nm thick Fe 60Co20B20or a 1.6 nm\nthick Fe 52:5Co17:5B30layer sandwiched between the MgO\ntunnel oxide and a Ta (2 nm) metal cap. Note that these\nso-called ”boron 20%” and ”boron 30%” samples have dif-\nferent boron contents but have the same number of Fe+Co\natoms. A sacrificial4Mg layer is deposed before the Ta cap\nto avoid Ta and FeCoB mixing during the deposition, and\navoid the otherwise resulting formation of a dead layer. The\nMg thickness is calibrated so that the Mg is fully sputtered\naway upon cap deposition. This advanced capping method has\nproven to provide improved TMR ratios and lower RA prod-\nucts thanks to an improved surface roughness and a higher\nmagnetic moment4.\nThe second category of free layers are the so-called ’dual\nMgO’ free layers in which the FeCoB layer is sandwiched\nby the MgO tunnel oxide and an MgO cap which concur to\nimprove the magnetic anisotropy. The exact free layer com-\npositions are MgO (1.0 nm) / Fe 60Co20B20(1.1 nm) / spacer\n/ Fe 60Co20B20(0.9 nm) / MgO (0.5 nm). We study shall two\nspacers: a Mg/Ta(3 ˚A) spacer and a Mg/Ta(4 ˚A) spacer, both\ncomprising a sacrificial Mg layer.\nB. Experimental methods used for material quality assessment\nWe studied our samples by current-in-plane tunneling\n(CIPT), vibrating sample magnetometry (VSM) and Vector\nNetwork Ferromagnetic resonance (VNA-FMR)24in out-of-\nplane applied fields. CIPT was performed to extract the tun-\nnel magneto-resistance (TMR) and the resistance-area product\n(RA) of the junction. VSM measurements of the free layer\nminor loops have been used to extract the areal moments. We\nthen use VNA-FMR to identify selectively the properties of\neach subsystem. Our experimental method is explained in\nFig. 1, which gathers some VNAFMR spectra recorded on\noptimized MTJs. The first panel records the permeability of\na single MgO MTJ in the ffield-frequencygparameter space.\nWe systematically investigated a sufficiently large parameter\nspace to detect 4 different modes whose spectral characters\ncan be used to index them22. Three of the modes belong to the\nreference system that comprises 3 magnetic blocks coupled\nby interlayer exchange coupling through Ru and Ta spacers\nas usually done22,23; the properties of these 3 modes are inde-\npendent from the nature of the free layer. While we are not\npresently interested in analyzing the modes of the fixed sys-\ntem – thorough analyses can be found in ref.22,23– we empha-\nsize that it is necessary to detect all modes to unambiguously\nidentify the one belonging to the free layer, in order to study\nit separately. The free layer modes are the ones having V-\nshaped frequency versus field curves [Fig. 1(a)], whose slope\nchanges at the free layer coercivity. in each sample, the free\nlayer modes showed an asymmetric Lorentzian dispersion for\nthe real part of the permeability and a symmetric Lorentzian\ndispersion for the imaginary part [see the examples Fig. 1(b,\nc)]. As we found no signature of the two-layer nature of the\ndual MgO free layers, we modeled each free layer as a sin-3\nSingle MgOfree layerModes of the reference layers\nDual MgOTa spacer\u0000f2f=0.006\u0000f2f=0.016Contrastx 10Permeabilitymap\n↵=12@\u0000f@f=0.0039\nFIG. 1. (Color online). Examples of MTJ dynamical properties to\nillustrate the method of analysis. (a) Microwave permeability versus\nincreasing out-of-plane field and frequency for an MTJ with a sin-\ngle MgO free layer after an annealing of 300\u000eC. Note that the scale\nof the permeability was increased by a factor of 10 above 58 GHz\nfor a better contrast. The apparent vertical bars are the eigenmode\nfrequency jumps at the different switching fields of the MTJ. (b)\nReal and imaginary parts of the experimental (symbols) and modeled\n(lines) permeability for an out-of-plane field of 1.54 T for the same\nMTJ. The model is for an effective linewidth \u0001f=(2f) = 0:016,\nwhich includes both the Gilbert damping and a contribution from the\nsample inhomogeneity. (c) Same but for a dual MgO free layer based\non a 3 ˚A Ta spacer, modeled with \u0001f=(2f) = 0:006. (d) Cross\nsymbols: FMR half frequency linewidth versus FMR frequency for\na dual MgO free layer based on a 3 ˚A Ta spacer. The line is a guide\nto the eye corresponding to a Gilbert damping of 0.0039.\nglemacrospin, disregarding whether it was a single MgO or a\ndual MgO free layer.\nFMR frequency versus field fits [see one example in\nfig. 2(c)] were used to get the effective anisotropy fields\nHk\u0000Msof all free layers25. The curve slopes are \r0, where\n\r0= 230 kHz.m/A is the gyromagnetic factor \rmultiplied\nby the vacuum permeability \u00160. It was consistent with a spec-\ntroscopic splitting Land ´e factor ofg\u00192:08. Damping analy-\nsis was conducted as follows: the free layer composition can\nyield noticeable differences in the FMR linewidths [see for in-\nstance Fig. 1(b) and (c)]. To understand these differences, we\nsystematically separated the Gilbert damping contribution to\nthe linewidth from the contribution of the sample’s inhomo-\ngeneity using standard VNA-FMR modeling25. This is doneby plotting the half FMR linewidth \u0001f=2versus FMR fre-\nquencyfFMR [see one example in Fig. 1(d)]. The Gilbert\ndamping is the curve slope and the line broadening arising\nfrom the inhomogeneity of the effective field within the free\nlayer is the zero frequency intercept1\n2\r0\u0001fjf=0) of the curve.\nC. Boron content and Gilbert damping upon annealing of\nsingle MgO free layers\nDesigning advanced free layer in STT-MRAM stacks re-\nquires to minimize the Gilbert damping of the used raw ma-\nterial. In Ta/FeCoB/MgO ’single MgO’ free layers made of\namorphous FeCoB alloys or made of FeCoB that has been\njust crystallized, a damping of 0.008 to 0.011 can be found\ntypically19,25. (Note that lower values can be obtained but for\nthicknesses and anisotropies that are not adequate for spin-\ntorque application26). The damping of Ta/FeCoB/MgO sys-\ntems generally degrades substantially when further annealing\nthe already crystallized state27. Let us emphasize than even in\nthe best cases26, the damping of FeCoB based free layers are\nstill very substantially above the values of 0.002 or slightly\nless than can be obtained on FeCo of Fe bcc perfect single\ncrystals28,29.\nThere are thus potentially opportunities to improve the\ndamping of free layers by material engineering. We illustrate\nthis in fig. 2 in which we show that a simple increase of the\nBoron content is efficient to maintain the damping unaffected,\neven upon annealing at 400\u000eC in a single MgO free layer. In-\ndeed starting from Ta/FeCoB/MgO ’single MgO’ free layers\nsharing the same damping of 0.009 after annealing at 300\u000eC\n(not shown), an additional 100\u000eC yields\u000b= 0:015for the\nfree layer with 20% of boron, while the boron 30% free lay-\ners keep a damping of \u000b= 0:009[see fig. 2(d)]. Meanwhile\nthe anisotropies of these two free layers remain perpendicu-\nlar [fig. 2(c)] with \u00160(Hk\u0000Ms)being 0.27 and 0.17 T, re-\nspectively, after annealing at 400\u000eC. Let us comment on this\ndifference of damping.\nTwo mechanisms can yield to extra damping: spin-\npumping30and spin-flip impurity scattering of the conduc-\ntion electrons by a spin-orbit process31. Tantalum is known\nto be a poor spin-sink material as this early transition metal\nhas practically no delectrons and therefore its spin-pumping\ncontribution to the damping of an adjacent magnetic layer is\nweak30. We expect a spin pumping contribution to the damp-\ning of Ta (2 nm) / FeCoB (1.4 nm) / MgO ’single MgO’\nfree layers that compares with for instance that measured by\nMizukami et al. on Ta (3 nm) / Fe 20Ni80(3 nm) which was\nundetectable32since below 0.0001; we therefore expect that\nthe spin-pumping contribution to the total free layer damping\nis too negligible to account for the differences observed be-\ntween a free layer and the corresponding perfect single crys-\ntals. The main remaining contribution to the damping is the\nmagnon scattering by the paramagnetic impurities within the\nFeCoB material33. Indeed the Ta atoms within an FeCoB layer\nare paramagnetic impurities that contribute to the damping ac-\ncording to their concentration like any paramagnetic dopant;\nhowever the effect with Ta is particularly large34as Fe and Co4\n/s50/s50 /s50/s52 /s50/s54/s48 /s49\n/s49/s48/s50/s48/s51/s48\n/s49/s48 /s50/s48 /s51/s48/s49/s48 /s50/s48 /s51/s48\n/s48/s46/s51/s48/s46/s54\n/s40/s100/s41/s66/s111/s114/s111/s110/s32/s51/s48/s37\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41/s40/s99/s41\n/s40/s98/s41/s40/s97/s41/s66/s111/s114/s111/s110/s32/s50/s48/s37/s84/s114/s97/s110/s115/s118/s101/s114/s115/s101/s32/s112/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121\n/s32/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s32/s66/s32/s40/s84/s41\n/s72/s97/s108/s102/s32/s108/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s71/s72/s122/s41\n/s32/s84\n/s97/s110/s110/s101/s97/s108/s32/s61/s32/s52/s48/s48/s176/s67\nFIG. 2. (Color online). Properties of single MgO free layers after\nannealing at 400\u000eC. (a) and (b): Real part (narrow lines) and imagi-\nnary part (bold lines) of the free layer permeability in a field of 0.7 T.\nThe lines are macrospin fits. (c) Ferromagnetic resonance frequency\nversus field curves. (d) Half linewidth versus FMR frequencies. The\nlines have slopes of \u000b= 0:009(red, B30%) and \u000b= 0:015(green,\nB20%)\natoms in direct contact with Ta atoms loose part of their mo-\nment and get an extra paramagnetic character, an effect usu-\nally referred as a ”magnetically dead layer”. Qualitatively, the\nTa atoms in the inner structure of the free layer degrade its\ndamping.\nAs the cap of Ta / FeCoB / MgO ’single MgO’ free lay-\ners contain many Ta atoms available for intermixing, a strong\ndegradation of the damping can be obtained in single MgO\nsystems when interdiffusion occurs. To prevent interdiffu-\nsion, we used the following strategy. Amorphous materials\n(including the glassy metals like FeCoB) are known to be ef-\nficient diffusion barriers, as they exhibit atom mobilities that\nare much smaller than their crystalline counterparts. To avoid\nthe diffusion of Ta atoms to the inner part of the FeCoB free\nlayer, a straightforward way is to maintain the FeCoB in an\namorphous state as long as possible during the annealing.\nIn metal-metalloid glasses, the crystallization temperature in-\ncreases with the metalloid content. In our FeCoB free lay-\ners, we find crystallization temperatures of 200, 300, 340 and\n375\u000eC for boron contents of respectively 10%, 20%, 25% and\n30%. Increasing the boron content in FeCo alloys is a way\nto conveniently increase the crystallization temperature and\nthus preserve a low damping. However since to obtain large\nTMR requires the FeCoB to be crystalline35,36, one should en-\ngineer the boron content such that the crystallization tempera-\nture matches with that used in the CMOS final BEOL anneal-\ning of 400\u000eC. In practice, we have found that this situation\nis better approached with a boron content of 30% than 0% to\n25%.D. Gilbert damping in single MgO and dual MgO free layers\nIn our search to further improve the free layers for STT-\nMRAM applications, we have compared the damping of op-\ntimized ’single MgO’ and optimized ’dual MgO’ free layers.\nFor a fair comparison, we first compare samples made from\nFeCoB with the same boron content of 20% and the same\n300\u000eC annealing treatement. From Fig. 1(b) and (c), there\nis a striking improvement of the FMR linewidths when pass-\ning from a single MgO to a dual MgO free layer. To discuss\nthis difference in linewidth, we have separated the Gilbert\ndamping contribution to the linewidth from the contribution\nof the sample’s inhomogeneity. We find that dual MgO sys-\ntems have systematically a substantially lower damping than\nsingle MgO free layers which confirms the trends indepen-\ndently observed by other authors9. Damping values as low\nas low as 0.0039\u00060:005were obtained in Ta 3 ˚A-spacer dual\nMgO stacks [Fig. 1(d)] after 300\u000eC annealing. Samples with\na thicker Ta spacer exhibit an increased damping (not shown).\nThis trend –lower damping in dual MgO systems –is main-\ntained after 400\u000eC annealing; for that annealing temperature,\nthe best damping are obtained for a slightly different internal\nconfiguration of the dual MgO free layer. Indeed a damping of\n0.0048 was obtained (not shown) in MgO / Fe 52:5Co17:5B30\n(1.4 nm) / Ta (0.2 nm) / Fe 52:5Co17:5B30(0.8 nm). This\nshould be compared with that the corresponding single MgO\nfree layer which had a damping of 0.009 for the same an-\nnealing condition [Fig. 2(d)]. This finding is consistent with\nthe results obtained on the single MgO free layer if we as-\nsume that the Ta impurities within an FeCoB layer contribute\nto the damping according to their concentration. Somehow,\nthe number of Tantalum atoms in the initial structure of the\nfree layer sets an upper bound for the maximum degradation\nof the damping upon its interdiffusion that can occur during\nthe annealing. Notably, the single MgO free layers contain\nmuch more Ta atoms (i.e. 2 nm compared to 0.2 to 0.4 nm)\navailable for intermixing: not only the initial number of Ta\nimpurities within the FeCoB layer directly after deposition is\nlarger in the case of single MgO free layer, but in addition a\nmuch stronger degradation of the damping can be obtained in\nsingle MgO systems when interdiffusion occurs, in line with\nour experimental findings. This interpretation – the dominant\nsource of damping is the Ta content – is further strengthened\nby the fact that the thickness of the Ta spacer strongly impacts\nthe damping in dual MgO free layers.\nLet us now study the spin-torque induced switching process\nin nanopillars processed from optimized MTJs.\nIII. SPIN-TORQUE INDUCED SWITCHING PROCESS\nA. Sample and methods for the switching experiments\nIn this section we use two kinds of perpendicularly magne-\ntized MTJ: a ’single MgO’ and a ’dual MgO’ free layer whose\nproperties are detailed respectively in ref.19and20. Note that\nthe devices are made from stacks that do not include all the\nlatest material improvement described in the previous sec-5\ntions and underwent only moderate annealing processes of\n300\u000eC. The ’single MgO’ free layer samples include a 1.4 nm\nFeCoB 20%free layer and a Co/Pt based reference synthetic\nantiferromagnet. Its most significant properties include19an\nareal moment of Mst\u00191:54mA, a damping of 0.01, an ef-\nfective anisotropy field of 0.38 T, a TMR of 150% . The ’dual\nMgO’ devices are made from tunnel junctions with a 2.2 nm\nthick FeCoB-based free layer and a hard reference system also\nbased on a well compensated [Co/Pt]-based synthetic antifer-\nromagnet. The perpendicular anisotropy of the (much thicker)\nfree layer is ensured by a dual MgO encapsulation and an iron-\nrich composition. After annealing, the free layer has an areal\nmoment of Mst\u00191:8mA and an effective perpendicular\nanisotropy field 0.33 T. Before pattering, standard ferromag-\nnetic resonance measurements indicated a Gilbert damping\nparameter of the free layer being \u000b= 0:008. Depending on\nthe size of the patterned device, the tunnel magnetoresistance\n(TMR) is 220 to 250%.\nBoth types of MTJs were etched into pillars of various size\nand shapes, including circles from sub-50 nm diameters to 250\nnm and elongated rectangles with aspect ratio of 2 and foot-\nprint up to 150\u0002300 nm. The MTJs are inserted in series\nbetween coplanar electrodes [Fig. 3(a)] using a device integra-\ntion scheme that minimizes the parasitic parallel capacitance\nso as to ensure an electrical bandwidth in the GHz range. The\njunction properties19,20are such that the quasi-static switching\nthresholds are typically 500 mV . Spin-wave spectroscopy ex-\nperiments similar to ref.37indicated that the main difference\nbetween the two sample series lies in the FL intralayer ex-\nchange stiffness. It is A= 8\u00009pJ/m in the 2.2 nm thick\ndual MgO free layers of the samples of ref.20and more usual\n(\u001920pJ/m) in the 1.4 nm thick ’single MgO’ free layers of\nthe samples of ref.19.\nFor switching experiments, the sample were characterized\nin a set-up whose essential features are described in Fig. 3(a):\na slow triangular voltage ramp is applied to the sample in se-\nries with a 50 \n oscilloscope. As the device impedance is\nmuch larger than the input impedance of the oscilloscope, we\ncan consider that the switching happens at an applied voltage\nthat is constant during the switching. We capture the elec-\ntrical signature of magnetization switching by measuring the\ncurrent delivered to the input of the oscilloscope [Fig. 3(b)].\nWhen averaging several switching events [as conducted in\nFig. 3(b)], the stochasticity of the switching voltage induces\nsome rounding of the electrical signature of the transition.\nHowever, the single shot switching events can also be cap-\ntured (Fig. 4-5). In that case, we define the time origins in\nthe switching as the time at which a perceivable change of the\nresistance suddenly happens (see the convention in Fig. 5).\nThis will be referred hereafter as the ”nucleation” instant.\nThis measurement procedure – slow voltage ramp and time-\nresolved current – entails that the studied reversal regime is\nthe sub-threshold thermally activated reversal switching. This\nsub-threshold thermally-activated switching regime is not di-\nrectly relevant to understand the switching dynamics in mem-\nory devices in which the switching will be forced by short\npulses of substantially higher voltage21. However elucidat-\ning the sub-threshold switching dynamics is of direct inter-\n50 ΩMTJVoltagebias\nOscilloscope50 Ω(a)(b)FIG. 3. (Color online). (a) Sketch of the experimental set-up. Mea-\nsurement procedure: the device is biased with a triangular kHz-rate\nvoltage (green) and the current (red) is monitored by a fast oscillo-\nscope connected in series. (b) The switching transitions are seen as\nabrupt changes of the current (red) followed by a change of the cur-\nrent slope. The resistance (blue) can be computed from the voltage-\nto-current ratio when the current is sufficiently non-zero. In this fig-\nure, the displayed currents and resistances are the averages over 1000\nevents for a 250 nm device with a dual MgO free layer of thickness\n2.2 nm and a weak exchange stiffness.\nest for the quantitative understanding of read disturb errors\nthat may happen at applied voltages much below the writing\npulses. Note finally that sending directly the current to the os-\ncilloscope has a drawback: the current decreases as the MTJ\narea such that the signal-to-noise ratio of our measurement\ndegrades substantially for small device areas (Fig. 5). As a\nresult, the comfortable signal-to-noise ratio allows for a very\nprecise determination of the onset of the reversal in large de-\nvices, but the precision degrades substantially to circa 500 ps\nfor the smallest (40 nm) investigated devices.\nB. Switching results\nIn samples whose (i) reference layers are sufficiently fixed\nto ensure the absence of back-hopping19and (ii) in which the\nstray field from the reference layer is rather uniform20, opti-\nmized compensation of the stray field of the reference layers\nleads to a STT-induced switching with a simple and abrupt\nelectrical signature [Fig. 4(a)]. If examined with a better time\nresolution, the switching event [Fig. 4(b)] appears to induce\na monotonic ramp-like evolution of the device conductance.\nFor a given MTJ stack, the switching voltage is practically in-\ndependent from the device size and shape in our interval of\ninvestigated sizes (not shown). This finding is consistent with\nthe consensual conclusion that the switching energy barrier\nis almost independent from the device area38,39for device ar-\neas above 50 nm. In spite of this quasi-independence of the\nswitching voltage and the device size, the switching duration\nwas found to strongly depend on device size (Fig. 5); we have\nfound that smaller devices switch faster, and the trend is that\nthe switching duration correlates linearly with the longest di-\nmension of the device. This is shown in Fig. 5: 40 nm devices\nswitch in typically 2 to 3 ns whereas devices that are 6 times6\n33PAP\nFIG. 4. (Color online). Single-shot time-resolved absolute value\nof the current during a spin-torque induced switching for parallel to\nantiparallel switching for a circular device of diameter 250 nm made\nwith a weak exchange stiffness, dual MgO 2.2 nm thick free layer.\n(a) Two microsecond long time trace, illustrating that the switching\nis complete, free of back-hopping phenomena, and occurs between\ntwo microwave quiet states. (b) 30 ns long time trace illustrating\nthe regular monotonic change of the device conductance during the\nswitching.\nlarger switch in 10 to 15 ns.\nSuch a reversal path can be interpreted this way: once a do-\nmain is nucleated at one edge of the device, the domain wall\nsweeps irreversibly through the system at a velocity set by\nthe applied voltage [sketch in Fig. 4(b)]. The average domain\nwall speed is then about 20 nm/ns for the low-exchange-free-\nlayers of ref.20. The other devices (not shown but described in\nref.19) based on a ’single MgO’ free layer with a more bulk-\nlike exchange switch with a substantially higher apparent do-\nmain wall velocity, reaching 40 m/s.\nC. Switching Model: domain wall-based dynamics\nTo model the switching, we assume that there is a domain\nwall (DW) which lies at a position qand moves along the\nlongest axis xof the device. The domain wall is assumed\nto be straight along the ydirection, as sketched in Fig. 4(b).\nWe describe the wall in the so-called 1D model40: the wall is\nassumed to be a rigid object of fixed width \u0019\u000epresenting a\ntilt\u001eof its magnetization in the device plane; by convention\n\u001e= 0is for a wall magnetization along x, i.e. a N ´eel wall.\n-10-5051015202501002000510m\nodelduration (ns)switchingd\nevice diam. (nm) \n90 nm \n60 nm 150 nm4\n0 nm80 nmCurrent (norm.)T\nime after nucleation (ns)250 nm FIG. 5. (Color online). Single-shot time-resolved conductance traces\nfor parallel to antiparallel switching events occurring at at -0.5 V\nfor circular devices of various diameters. The curves are for the de-\nvices whose dual MgO free layer has a thickness of 2.2 nm and has a\nweak exchange stiffness. The curves have been vertically offset and\nvertically normalized to ease the comparison. The time origins and\nswitching durations are chosen at the perceivable onset and end of\nthe conductance change: they are defined by fitting the experimental\nconductance traces by 3 segments (see the sketch labelled ”model”).\nInset: duration of the switching events versus free layer diameter\n(symbols) and linear fit thereof with an inverse slope of 20 m/s.\nThe local current density at the domain wall position is\nwrittenj. The wall is subjected to an out-of-plane field Hz\nassumed to vary slowly in space at the scale of the DW width.\njis assumed to transfer p\u00191spin per electron to the DW by\na pure Slonczewski-like STT. We define\n\u001b=~\n2e\r0\n\u00160MSt(1)\nas the spin-transfer efficiency in unit such that \u001bjis a fre-\nquency. With typical FeCoB parameters, i.e. magnetization\nMs2[1:1;1:4]MA/m and free layer thickness t2[1:4;2:2]\nnm, we have \u001b2[0:018;0:036] Hz / (A/m2) where the low-\nest value corresponds to the largest areal moment Mst. With\nswitching current density of the order of 4\u00021010A/m2, this\nyields\u001bjdcbetween 0.72 and 1.4 GHz.\nFollowing ref.41, the wall position qand and wall tilt \u001eare7\nlinked by the two differential equations:\n_\u001e+\u000b\n\u000e_q=\r0Hz; (2)\n_q\n\u000e\u0000\u000b_\u001e=\u001bjdc+\r0HDW\n2sin(2\u001e) (3)\nin which\u0019\u000eis the width of a Bloch domain wall in an ultra-\nthin film, with \u000e2= 2A=(\u00160MsHeff\nk)whereAis the exchange\nstiffness. A wall parameter \u000e= 12 nm will be assumed for\nthe normal exchange 1.4 nm free layer from various estimates\nincluding ref.37for the exchange stiffness and ref.22for the\nanisotropy of the free layer. The domain wall stiffness field42\nHDWis the in-plane field that one would need to apply to have\nthe wall transformed from a Bloch wall to a N ´eel wall. As it\nexpresses the in-plane demagnetization field within the wall,\nit depends on the wall width \u0019\u000eand on the wall length when\nthe finite size of the device constrains the wall dimensions.\nUsing42, the domain wall stiffness field can be estimated\nto be at the most 20 mT in our devices. In circular devices,\nthe domain wall has to elongate upon its propagation38such\nthat the domain wall stiffness field HDWdepends in princi-\nple on the DW position. It should be maximal when the wall\nis along the diameter of the free layer. However we will see\nthatHDWis not the main determinant of the dynamics. In-\ndeed in the absence of stray field and current, the Walker field\nHWalker is proportional to the domain wall stiffness field times\nthe damping parameter, i.e. HWalker =\u000bH DW=2. As the sam-\nples required for STT switching are typically made of low\ndamped materials with \u000b < 0:01, the Walker field is very\nsmall and likely to be smaller than the stray fields emanating\nfrom either the reference layers or the applied field. This very\nsmall Walker field has implications: in practice as soon as\nthere is some field of some applied current, any domain wall\nin the free layer is bound to move in the Walker regime and\nto make the back-and-forth oscillatory movements that are in-\nherent to this regime. The DW oscillates at a generally fast\n(GHz) frequency43such that only the time-averaged velocity\nmatters to define how much it effectively advances.\nTo see quantitatively the effect of a constant current on the\ndomain wall dynamics, we assume that the sample is invari-\nant along the domain wall propagation direction (x) (like in\nan hypothetical stripe-shaped sample). Solving numerically\nEq. 2 and 3, we find that the Walker regime is maintained for\njdc6= 0(not shown). Two points are worth noticing:\nThe time-averaged domain wall velocity h_qivaries linearly\nwith the applied current density. When in the Walker regime,\nthe current effect can be understood from Eq. 3. Indeed the\nsin(2\u001e)term essentially averages out in a time integration as\n\u001eis periodic, and the term \u000b_\u001eis neligible, such that the time-\naveraged wall velocity reduces to:\nh_qi\u0019\u000e\u001bj dc (4)\nFor\u000e= 12 nm and\u001bjin the range of 1.4 GHz at the switching\nvoltage for the bulk-like exchange stiffness sample with free\nlayer thcikness 1.4 nm, the previous equation would predict atime-averaged domain wall velocity of 17 m/s (or nm/ns) dur-\ning the switching. More compact domain walls are expected\nfor the samples with a weaker exchange stiffness; the twice\nlower\u001bj\u00190:72GHz related to the larger thickness would\nreinforce this trend to a much a lower domain wall velocity (9\nm/s for our material parameters estimates). This expectation\ncompares qualitatively well with our experimental findings of\nslower walls in weakly exchanged materials (Fig. 5).\nWe wish to emphasize that Eq. 4 can be misleading regard-\ning the role of damping. Indeed a too quick look at Eq. 4 could\nlet people wrongly conclude the domain wall velocity is es-\nsentially set by the areal moment Mstand that the wall veloc-\nity under STT from a current perpendicular to the plane (CPP\ncurrent) is independent from the damping factor (see Eq. 1).\nHowever this is not the case as the switching current jdcis\na sweep-rate-dependent and temperature-determined fraction\n\u00112[1\n2;1]of the zero temperature instability current jc0of a\nmacrospin in the parallel state, which reads15,44:\njc0=\u000b4e\n~1 +p2\np\u00160MstHeff\nk\n2(5)\nwherep\u00191is an effective spin polarization.\nUsing Eq. 1, 4 and 5, the time-averaged wall velocity at the\npractical switching voltage is:\nh_qi\u0019\u000b\u000e\r 0Heff\nk\u0011 (6)\nThis expression indicates that the samples performing best\nin term of switching current (minimal damping and easy nu-\ncleation thanks to a small exchange) will host domain walls\nthat are inherently slow when pushed by the CPP current in\nthe Walker regime. The domain wall speed scales with the\ndomain wall width, which may be the reason why the low\nexchange stiffness samples host domain walls that are experi-\nmentally slower.\nTo summarize, once nucleated at the instability of the uni-\nformly magnetized state at jdc=\u0011jc0, the domain wall flows\nin a Walker regime through the device. The switching dura-\ntion varies thus simply with the inverse current:\n\u001cswitch =L\n\u000e\u001bj dc\u0019L\n\u000e\u00021\n\u000b\r0Heff\nk\u00021\n\u0011(7)\nLet us comment on this equation which is the main con-\nclusion of this section. The underlying simplifications are:\n(i) a rigid wall (ii) that does not sense the sample’s edges\n(iii) that moves at a speed equal to its average velocity in the\nWalker regime (iv) at a switching voltage that is independent\nfrom the sample geometry. Under these assumptions, the du-\nration of the switching scales with the length Lof the sam-\nple, as observed experimentally. It also scales with the inverse\nof the zero-field ferromagnetic resonance linewith 2\u000b\r0Heff\nk.\nThe practical switching voltage is below the zero temperature\nmacrospin switching voltage by a factor \u0011, which gathers the\neffect of the thermal activation and of the sweeping rate of the\napplied voltage45.\u0011\u00191=2for quasi-static experiments like\nreported here and \u0011!1for experiments in which the voltage\nrise timeVmax=_Vis short enough compared to the switching\nduration (Eq. 7).8\nIV . SUMMARY AND CONCLUSION\nIn summary, we have investigated the Gilbert damping of\nadvanced free layer designs: they comprise FeCoB alloys with\nvariable B contents from 20 to 30% and are organized in the\nsingle MgO or dual MgO free layer configuration fully em-\nbedded in functional STT-MRAM magnetic tunnel junctions.\nIncreasing the boron content increases the cristallization tem-\nperature, thereby postponing the onset of elemental diffusion\nwithin the free layer. This reduction of the interdiffusion of\nthe Ta atoms helps maintaining the Gilbert damping at a low\nlevel without any penalty on the anisotropy and the transport\nproperties. Thereby, increasing the Boron content to at least\n30% is beneficial for the thermal robustness of the MTJ up\nto the 400\u000erequired in CMOS back-end of line processing.\nIn addition, we have shown that dual MgO free layers have a\nsubstantially lower damping than their single MgO counter-\nparts, and that the damping increases as the thickness of the\nTa spacer within dual MgO free layers. This indicates that\nthe dominant source of extra damping is the presence of Ta\nimpurities within the FeCoB alloy. Using optimized MTJs,\nwe have studied the duration of the switching events as in-\nduced by spin-transfer-torque. Our experimental procedure –\ntime-resolving the switching with a high bandwidth but dur-\ning slow voltage sweep – ensures that we are investigating\nonly sub-threshold thermally activated switching events. In\noptimal conditions, the switching induces a ramp-like mono-\ntonic evolution of the device conductance that we interpret\nas the sweeping of a domain wall through the device. The\nswitching duration is roughly proportional to the device size:\nthe smaller the device, the faster it switches. We studied twoMTJ stacks and found domain wall velocities from 20 to 40\nm/s. A simple analytical model using a rigid wall approxima-\ntion can account for our main experimental findings. The do-\nmain wall velocity is predicted to scale linearly with the cur-\nrent for device sizes much larger than the domain wall widths.\nThe domain wall velocity depends on the material parame-\nters, such that the samples with the thinnest domain walls will\nbe the ones that host the domain walls with the lowest mo-\nbilities. Schematically, material optimization for low current\nSTT-induced switching (i.e. in practice: fast nucleation be-\ncause of low exchange stiffness Aand low damping \u000b) will\ncome together with slow STT-induced domain wall motion at\nleast in the range of device sizes in which the STT-induced re-\nversal proceeds through domain wall motion. If working with\nSTT-MRAM memory cells made in the same range of device\nsizes, read disturb should be minimal (if not absent) provided\nthat the voltage pulse used to read the free layer magnetiza-\ntion state has a duration much shorter than the time needed\nfor a domain wall to sweep through the device at that voltage\n(Eq. 7).\nACKNOWLEDGMENT\nThis work is supported in part by IMEC’s Industrial Affil-\niation Program on STT-MRAM device, in part by the Sam-\nsung Global MRAM Innovation Program and in part by\na public grant overseen by the French National Research\nAgency (ANR) as part of the Investissements dAvenir pro-\ngram (Labex NanoSaclay, reference: ANR-10-LABX-0035).\nT. D. would like to thank Andr ´e Thiaville, Paul Bouquin and\nFelipe Garcia-Sanchez for useful discussions.\n\u0003thibaut.devolder@u-psud.fr\n1A. V . Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii,\nR. S. Beach, A. Ong, X. Tang, A. Driskill-Smith, W. H.\nButler, P. B. Visscher, D. Lottis, E. Chen, V . 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Available:\nhttp://link.aps.org/doi/10.1103/PhysRevLett.92.088302" }, { "title": "2012.15426v1.Damping_of_slow_surface_kink_modes_in_solar_photospheric_waveguides_modeled_by_one_dimensional_inhomogeneities.pdf", "content": "arXiv:2012.15426v1 [astro-ph.SR] 31 Dec 2020Draft version January 1, 2021\nTypeset using L ATEXpreprint style in AASTeX61\nDAMPING OF SLOW SURFACE KINK MODES IN SOLAR PHOTOSPHERIC WAV EGUIDES\nMODELED BY ONE-DIMENSIONAL INHOMOGENEITIES\nShao-Xia Chen,1Bo Li,1Tom Van Doorsselaere,2Marcel Goossens,2Hui Yu,1and\nMicha¨el Geeraerts2\n1Shandong Provincial Key Laboratory of Optical Astronomy an d Solar-Terrestrial Environment, Institute of Space\nSciences, Shandong University, Weihai 264209, China\n2Centre for mathematical Plasma Astrophysics (CmPA), KU Leu ven, Celestijnenlaan 200B bus 2400, B-3001 Leuven,\nBelgium\nSubmitted to ApJ\nABSTRACT\nGiven the recent interest in magnetohydrodynamic (MHD) waves in p ores and sunspot umbrae, we\nexaminethedampingofslowsurfacekinkmodes(SSKMs) bymodelings olarphotosphericwaveguides\nwith a cylindrical inhomogeneity comprising a uniform interior, a unifor m exterior, and a continuous\ntransition layer (TL) in between. Performing an eigen-mode analysis in linear, resistive, gravity-free\nMHD, our approach is idealized in that, among other things, our equilib rium is structured only\nin the radial direction. We can nonetheless address two damping mec hanisms simultaneously, one\nbeing the Ohmic resistivity, and the other being the resonant absor ption of SSKMs in the cusp and\nAlfv´ en continua. We find that the relative importance of the two me chanisms depends sensitively\non the magnetic Reynolds number ( Rm). Resonant absorption is the sole damping mechanism for\nrealistically large values of Rm, and the cusp resonance in general dominates the Alfv´ en one unle ss\nthe axial wavenumbers are at the lower end of the observationally r elevant range. We also find that\nthe thin-boundary approximation holds only when the TL-width-to- radius ratios are much smaller\nthan nominally expected. The Ohmic resistivity is far more important f or realistically small Rm.\nEven in this case, SSKMs are only marginally damped, with damping-time -to-period-ratios reaching\n∼10 in the parameter range we examine.\nKeywords: magnetohydrodynamics (MHD) — Sun: magnetic fields — Sun: photo-\nsphere — waves\nCorresponding author: Bo Li\nbbl@sdu.edu.cn2 Chen et al.\n1.INTRODUCTION\nLow-frequency waves and oscillations in the magnetohydrodynamic (MHD) regime often prove\nkey in the gas-magnetic field interactions throughout the solar atm osphere (see e.g., the textbook\nbyRoberts 2019 ). When generated by, say, sub-photospheric convective motion s, these waves\nmay be sufficiently energetic to play an active role in atmospheric heat ing, thereby helping shape\nsuch atmospheric structures as coronal loops (see e.g., Parnell & De Moortel 2012 ;Arregui 2015 ;\nCranmer & Winebarger 2019 , for recent reviews). Even when not that energetic, these wave s\ncan passively encode some rich information on their host magnetic st ructures in their observa-\ntional signatures. Conversely, this information can be deciphered with increasingly refined theo-\nries of MHD waves in an inhomogeneous medium, constituting the field o f “coronal seismology” or\nmore generally “solar atmospheric seismology” (SAS, Roberts et al. 1984 ; see also recent reviews\nby e.g., Roberts 2000 ,Nakariakov & Verwichte 2005 ,Nakariakov & Kolotkov 2020 ). Whichever\nthe application, MHD waves have been customarily classified with a sch eme largely established\nfor a canonical configuration where a magnetic structure is seen a s a field-aligned plasma cylinder\nwith circular cross-section, and the equilibrium quantities are struc tured only in the radial direc-\ntion (Wentzel 1979 ,Spruit 1982 ,Edwin & Roberts 1983 , hereafter ER83; see also Rosenberg 1970 ,\nZajtsev & Stepanov 1975 ,Cally 1986 ). We restrict ourselves to collective waves, “collective” in the\nsense thatthey involve coherent motionsofthefluidparcels inthes tructure itself andpossibly itssur-\nroundings. Collective waves (“modes” hereafter) may be classified as sausage ( m= 0), kink ( m= 1),\nand fluting modes ( m≥2) by their azimuthal wavenumber m. They may be independently grouped\ninto slow and fast modes according to their axial phase speeds. Alte rnatively, the spatial dependence\nof the associated perturbations in the surroundings enables the n otions of trapped and leaky modes\nas a measure for classification. Likewise, the notions of surface an d body modes are enabled by an\nexamination of the spatial behavior within the magnetic structure. One therefore ends up with an\nextensive list of combinations like fast kink body modes and slow sausa ge surface modes1. In the\nstructured solar corona, there have been ample observations fo r modes pertaining to a substantial\nfraction of these combinations (see the reviews by e.g., Banerjee et al. 2007 ;Nakariakov et al. 2016 ;\nWang 2016 ). In particular, the ubiquity of radial fundamental kink modes has enabled one to seis-\nmologically construct the spatial distributions of the magnetic field s trength in a substantial portion\nof some active region (AR, Anfinogentov & Nakariakov 2019 ) or even across several ARs ( Yang et al.\n2020).\nThere is a long history of observational studies on MHD waves in the lo wer solar atmosphere as well\n(see e.g., the review by Khomenko & Collados2015 onwaves in sunspots, andthat by Jess et al. 2015\non waves in the chromosphere). However, it seems that the classifi cation of these waves with the\nER83 scheme was made possible only relatively recently. Take the quie t chromosphere for instance.\nCyclic transverse displacements, the defining characteristic of ra dial fundamental kink modes, have\nbeen abundantly observed in slender fibrils imaged in both Ca II 3969 ˚A with SUNRISE/SuFI\n(Jafarzadeh et al. 2017 ) and H αwith ROSA ( Morton et al. 2012 ). Likewise, the latter study also in-\ndicated the omnipresence of breathing-like motions of the H αfibrils, thereby suggesting the ubiquity\n1Some issue arises when one applies this classification scheme to the ab undantly observed kink modes that are\ntrapped in thin coronal loops and that possess axial phase speeds comparable to the Alfv´ en speed. Customarily called\n“fast kink body modes”, they turn out to be essentially Alfv´ enic su rface waves with mixed properties (for details, see\nGoossens et al. 2009 ,2012). In particular, they survive in incompressible MHD, making them diffic ult to be classified\nas “fast”. We therefore deem it more appropriate to call them “th e radial fundamental kink mode”.Waves in photospheric waveguides 3\nof fast sausage modes. In both studies, the width of chromosphe ric fibrils reaches down to ∼150 km,\nwhich highlights the importance of high spatial resolution in enabling su ch classifications as kink\nor sausage modes. The same can be said for waves in chromosphere s above sunspot umbrae. For\ninstance, the long-known spiral wave patterns (SWPs) with period icities of ∼3 mins (e.g., Su et al.\n2016, and references therein) were recently shown to be compatible wit h the superposition of slow\nbody modes with m= 0 and m= 1 (m= 0 and m= 2) in the case of one-armed (two-armed)\nSWPs (Kang et al. 2019 ). Likewise, clear signatures of slow kink body modes were identified in the\ntemporallyandspatiallyfilteredH αimagesequencesacquiredwithROSAmountedontheDunnSolar\nTelescope (DST, Jess et al. 2017 ). Moving to the photosphere in either sunspot umbrae or pores, w e\nnote that slow sausage modes tend to be ubiquitous (e.g., Dorotoviˇ c et al. 2008 ;Fujimura & Tsuneta\n2009;Morton et al. 2011 ;Dorotoviˇ c et al. 2014 ;Moreels et al. 2015a ;Grant et al. 2015 ;Freij et al.\n2016). Inaddition,surfacemodestendtobefavoredoverbodymodes inporeswithlargersizesand/or\nstrongermagneticfields( Keys et al.2018 ,Gilchrist-Millar et al.2020 ; seealso Stangalini et al.2018 ).\nThe waves in the lower solar atmosphere, be them kink or sausage mo des, were suggested to reach\nchromospheric levels with a Poynting flux of up to ∼27 kW m−2(e.g.,Fujimura & Tsuneta 2009 ), a\nvaluesufficiently largeto meet thechromospheric heating requireme nt. Equally useful aretheir appli-\ncations to SAS as initiated by Fujimura & Tsuneta (2009) andMoreels et al. (2015a). Observations\naside, theoretical studies have proven crucial for establishing th e phase-differences between rele-\nvant perturbations and therefore for aiding mode identification ( Moreels & Van Doorsselaere 2013 ;\nMoreels et al. 2013 ). Likewise, expressions for the energy and energy flux densities a re inevitable for\nexamining the energetics ( Moreels et al. 2015b ).\nThisstudy isintended toexamine thedamping ofslowsurface kinkmod es(SSKMs) inphotospheric\nwaveguides representative of pores and sunspot umbrae, there by partially addressing the fate of\nSSKMs generated by, say, sub-photospheric convective motions . Our motivation is twofold. Firstly,\nin contrast to the apparent omnipresence of slow surface sausag e modes (SSSMs), photospheric\nkink modes were implicated only in the Hinode/SP measurements repor ted byFujimura & Tsuneta\n(2009) to our knowledge, and the kink mode therein is more likely to be a fast surface one. We\ntake this as encouraging rather than discouraging for a theoretic al study on SSKMs, the reason\nprimarily related to their theoretical properties and their likely excit ation. Theoretically, SSKMs\ntend to be indistinguishable from SSSMs in terms of eigen-frequencie s (see e.g., Figure 3 in ER83).\nWhile the azimuthal variations of the two modes are distinctively differ ent, the random broadband\nmotions that excite SSSMs are likely to excite SSKMs as well. Secondly, SSSMs in pores were\nobservationally demonstrated to be heavily damped over photosph eric heights ( Grant et al. 2015 ;\nGilchrist-Millar et al. 2020 ). So far this heavy damping has only been partially addressed. Work ing\nin the thin-boundary (TB) approximation, the largely analytical stu dy byYu et al. (2017a) indicated\nthatthe resonant absorption ofSSSMs inthe cusp continuum may r esult inadamping-time-to-period\nratio (τ/P) of∼10. This was taken further by Sadeghi & Karami (2019) who incorporated magnetic\ntwist in the equilibrium. As such SSSMs further resonantly couple to t he Alfv´ en continuum, thereby\nleading to some stronger damping. Note that the Alfv´ en and cusp c ontinua arise when the piece-\nwise constant model is replaced by a model in which the equilibrium quan tities vary in a continuous\nmanner in a transitional layer (TL) from their internal to their exte rnal values. By construction, the\nTB results apply only when the TL width ( l) is far below the cylinder radius ( R). However, in the\nabove-referenced TB studies, strong damping tends to occur wh enl/Ris substantial, and one may4 Chen et al.\nquestion whether the TB damping rates hold. Indeed, focusing on t he damping of SSSMs due to\nthe cusp resonance, Chen et al. (2018, hereafter C18) showed with resistive MHD computations that\nthe numerically derived damping rates are in general well below the TB values found by Yu et al.\n(2017a). C18 further showed that in general Ohmic resistivity tends to be more important than\nresonant absorption for damping SSSMs, a result that was furthe r corroborated by the analytical\nstudy in Geeraerts et al. (2020, G20 from here on). Compared with SSSMs, SSKMs are theoretically\nappealing in that they can resonantly couple to the m= 1 Alfv´ en waves even in the absence of\nmagnetic twist ( Yu et al. 2017b , Y17 hereafter). In Y17, the damping of SSKMs was solely due to\nresonant absorption and the cusp resonance was shown to domina te for large axial wavenumbers,\nresulting in values of down to ∼10 forτ/P. Now two questions arise, given that Y17 adopted the\nTB approximation, and given the C18 results regarding SSSMs. First ly, how well do the Y17 results\napply as far as the damping due to resonant absorption is concerne d? Secondly, what is the role of\nOhmic resistivity in the damping of SSKMs?\nSome remarks prove necessary at this point. We will adopt resistive , gravity-free, MHD as our the-\noretical framework, in accordance with which we will additionally take our equilibrium configuration\nto be structured only transversely. Both call into question how we ll our results apply to photospheric\nstructures in reality, the structuring of which is way more complicat ed than assumed here. We leave\nthe discussions on this aspect till the end of this manuscript. It suffi ces to note here that by “photo-\nspheric modes” we refer to the modes supported by some idealized m agnetic structures, for which the\nordering of the internal and external characteristic speeds non etheless captures some realistic features\nof pores and sunspot umbrae. Bearing the caveats in mind, we list tw o features that make our study\nconsiderably new relative to available ones on photospheric modes. F irstly, so far the damping of\nSSKMs was examined only by Y17 to our knowledge. Relative to Y17, ou r study is new in that we\nwill simultaneously account for the effects of the Ohmic resistivity an d resonant absorption, whereas\nY17 considered only resonant absorption. Secondly, we will analyze the wave damping from the\nperspective of energy balance, whereas all the afore-referenc ed studies approached the wave damping\nexclusively from the perspective of eigen-frequencies. As we will ac knowledge where more appropri-\nate, this is not to say that wave damping has not been examined from the energetics perspective. It\nis just that “photospheric modes” have not been examined this way as far as we know.\nThis manuscript is organized as follows. Section 2details the specification of our equilibrium\nconfiguration, and how we formulate and numerically solve the pertin ent eigenvalue problem. The\nnumerical results are then described in Section 3. Section 4summarizes the present study, ending\nwith some concluding remarks.\n2.MODEL DESCRIPTION\n2.1.Equilibrium Configuration\nWe adopt resistive, gravity-free MHD throughout, for which the p rimitive variables are the mass\ndensity ( ρ), velocity ( v), magnetic field ( B), and thermal pressure ( p). We denote the equilibrium\nquantities with the subscript 0, and assume that no equilibrium flow is p resent (v0= 0). Working in\na cylindrical coordinate system ( r,θ,z), we assume that all equilibrium quantities depend only on r.\nThe equilibrium magnetic field ( B0) is taken to be in the z-direction. As such B0(r) andp0(r) are\nrelated by the transverse force balance condition\np0(r)+B0(r)2\n2µ0= const, (1)Waves in photospheric waveguides 5\nwhereµ0is the magnetic permeability in free space. The adiabatic sound speed cs, Alfv´ en speed vA,\nand cusp (tube) speed cTare then defined by\nc2\ns(r) =γp0\nρ0, v2\nA(r) =B2\n0\nµ0ρ0, c2\nT(r) =c2\nsv2\nA\nc2\ns+v2\nA, (2)\nwithγ= 5/3 being the adiabatic index.\nWe model a photospheric waveguide as a field-aligned cylinder with mea n radiusR. Following C18,\nwe realize this by assuming that c2\nsandc2\nTtake the following form\nE(r) =\n\nEi, 0< r < r i=R−l/2 ;\nEi+Ee\n2−Ei−Ee\n2sinπ(r−R)\nl,ri≤r≤re=R+l/2 ;\nEe, r > r e,(3)\nwhereErepresents c2\nsandc2\nT. In addition, the subscripts i and e refer to the equilibrium values at\nthe cylinder axis and in the ambient medium, respectively. As such the equilibrium configuration\ncomprises a uniform interior ( r < r i), a uniform exterior ( r > r e) and a transition layer (TL)\ncontinuouslyconnectingthetwo. Thisequilibriumisfullyspecifiedbyas etofdimensional parameters\n[ρi,vAi,R] together with a set of dimensionless parameters [ csi/vAi,cse/vAi,vAe/vAi,l/R]. As in C18,\nwe fix [csi,cse,vAe] at [0.5,0.75,0.25]vAi, which is identical to ER83 and also in agreement with the\ngenerally accepted values for sunspot umbrae and pores (e.g., Jess et al. 2017 ;Yu et al. 2017b ). The\ninternal and external cusp speeds then read [ cTi,cTe] = [0.4472,0.2372]vAi. The ratio of the TL\nwidth to the mean radius ( l/R), on the other hand, is allowed to vary. The spatial distributions fo r\nboth the characteristic speeds and primitive variables are displayed in Figure 1, wherel/Ris taken\nto be 0.2 for illustration purposes. Among the dimensional parameters, vAiandRare relevant for our\npurposes. Wetake vAi= 12km s−1inview of Figure4 in Grant et al. (2015), who inturnconstructed\nthe values with the “hot” umbral model from Maltby et al. (1986). With pores and sunspot umbrae\nin mind, we assume that Rranges from 500 km (the lower limit for pores, see Sobotka 2003 ) to\n104km (for large sunspot umbrae, see Figure 1 in Stangalini et al. 2018 ).\n2.2.Formulation of the Eigenvalue Problem and Method of Solutio n\nLetthesubscript1denotesmall-amplitudeperturbationstotheeq uilibrium. Thelinearizedresistive\nMHD equations then read\nρ0∂v1\n∂t=−∇p1+(∇×B0)×B1\nµ0+(∇×B1)×B0\nµ0, (4)\n∂B1\n∂t=∇×/parenleftbigg\nv1×B0−η\nµ0∇×B1/parenrightbigg\n, (5)\n∂p1\n∂t=−v1·∇p0−γp0∇·v1+2(γ−1)η\nµ2\n0(∇×B1)·(∇×B0). (6)\nHereηis the Ohmic resistivity, taken to be constant for simplicity. We adopt an eigen-value-problem\n(EVP) perspective by Fourier-analyzing any perturbation as\ng1(r,θ,z;t) = Re{˜g(r)exp[−i(ωt−mθ−kz)]}, (7)6 Chen et al.\nwhereωis the angular frequency, and k(m) represents the axial (azimuthal) wavenumber. With\ntilde we denote the Fourier amplitude. We take kas real-valued, but allow ωto be complex-valued.\nIf some quantity is complex, then we denote its real and imaginary pa rts with subscripts R and I,\nrespectively. It follows that the period P= 2π/ωR, and the damping time τ= 1/|ωI|. Instabilities\nare not of interest, hence ωI<0. The axial phase speed is defined by vph=ωR/k.\nWe proceed with normalizing Equations ( 4) to (6). Among the necessary normalizing constants,\nwe take R,vAi, andρias independent ones, following from which are the normalizing consta nts\nfor time ( R/vAi), the magnetic field ( Bi≡/radicalbig\nµ0ρiv2\nAi) and thermal pressure ( ρiv2\nAi=B2\ni/µ0). The\nOhmic resistivity ends up in the magnetic Reynolds number Rm=µ0RvAi/η. In component form,\nthe resulting dimensionless equations then read\nω˜vr=−B0\nρ0/parenleftBigg\nk˜Br+id˜Bz\ndr/parenrightBigg\n−i˜Bz\nρ0dB0\ndr−i\nρ0d˜p\ndr, (8)\nω˜vθ=−B0\nρ0/parenleftBigg\nk˜Bθ−m˜Bz\nr/parenrightBigg\n+m˜p\nrρ0, (9)\nω˜vz=i˜Br\nρ0dB0\ndr+k˜p\nρ0, (10)\nω˜Br=−kB0˜vr+1\nRm/parenleftBigg\nid2˜Br\ndr2+i\nrd˜Br\ndr−im2˜Br\nr2−ik2˜Br+2m˜Bθ\nr2−i˜Br\nr2/parenrightBigg\n, (11)\nω˜Bθ=−kB0˜vθ+1\nRm/parenleftBigg\nid2˜Bθ\ndr2+i\nrd˜Bθ\ndr−im2˜Bθ\nr2−ik2˜Bθ−2m˜Br\nr2−i˜Bθ\nr2/parenrightBigg\n, (12)\nω˜Bz=−i˜vrdB0\ndr−B0/parenleftbigg\nid˜vr\ndr+i˜vr\nr−m˜vθ\nr/parenrightbigg\n+1\nRm/parenleftBigg\nid2˜Bz\ndr2+i\nrd˜Bz\ndr−im2˜Bz\nr2−ik2˜Bz/parenrightBigg\n,(13)\nω˜p=−c2\nsρ0/parenleftbigg\nid˜vr\ndr+i˜vr\nr−m˜vθ\nr−k˜vz/parenrightbigg\n−i˜vrdp0\ndr+2(γ−1)\nRmdB0\ndr/parenleftBigg\nk˜Br+id˜Bz\ndr/parenrightBigg\n.(14)\nWe specialize to the case where m= 1, given that SSKMs are of interest. Equations ( 8) to (14)\nconstitute a standard EVP when supplemented with appropriate bo undary conditions (BCs). The\nBCs at the cylinder axis ( r= 0) are that d˜ vr/dr= d˜vθ/dr= d˜Br/dr= d˜Bθ/dr= ˜vz=˜Bz= ˜p= 0.\nWith trapped modes in mind, we require that all variables vanish when rapproaches infinity. From\nhere onward, we see the physical variables as dimensional again for the ease of description.\nThe EVP is formulated and solved with the general-purpose finite-ele ment code PDE2D ( Sewell\n1988), which was first introduced to the solar context by Terradas et al. (2006) to our knowledge.\nThe computational domain spans from 0 to some outer boundary rM. A nonuniform grid is adopted,\namong which a considerable number of grid points are deployed in the T L to resolve the possible\noscillatory behavior therein. The total number of grid points typica lly ranges from 5 ×104up to\n2.5×105, among which up to 50% are deployed in the TL in some cases. Overall t he strategy is that\ndeploying more grid points in the TL leads to no discernible difference to our results, by which weWaves in photospheric waveguides 7\nmean both the eigen-frequencies and eigen-functions. The outer boundary is placed at a sufficiently\nlarge distance of rM= 50Rto ensure that further increasing rMdoes not introduce any appreciable\nchange to our results. Our numerical results are remarkably accu rate, as will be seen from the\nenergetics examinations. Taking [ l/R,kR,R m] as free parameters, one may formally express ωas\nωR\nvAi=G/parenleftbigg\nkR,l\nR,Rm/parenrightbigg\n. (15)\nWithl/Rdifficult to constrain observationally, we allow l/Rto vary as broadly as the code permits.\nEvaluating Rmnecessarily involves the electric resistivity, for which purpose we qu ote the values for\nelectric conductivity listed in Table 2 of Kovitya & Cram (1983) for a model sunspot umbra in the\ntemperature range of 3500 to ∼6400 K. Given the range of Rwe have discussed and the internal\nAlfv´ en speed we have fixed, Rmthen ranges from 104to 6.6×107.\nThe range of kRis difficult to constrain, given that a concrete identification of SSKMs has yet\nto appear. We nonetheless assume that kR/greaterorsimilar0.3 with the limited knowledge on the axial wave-\nlength (λ= 2π/k) of photospheric modes. This adopted range of kRis inferred primarily from\nthe measurements of photospheric slow sausage modes. Gatherin g the measurements for pores and\nsunspot umbrae of various sizes, one finds that the period tends t o range between ∼1.5 and 20 mins\n(Fujimura & Tsuneta 2009 ;Morton et al. 2011 ;Moreels et al. 2015a ;Grant et al. 2015 ;Freij et al.\n2016;Keys et al. 2018 ;Gilchrist-Millar et al. 2020 ). Boldly assuming that this periodicity applies to\nSSKMs in our equilibrium, we find that λranges from ∼500 to∼6500 km by approximating the\naxial phase speed vphwithcTi. We gather further information from the spectropolarimetric mea sure-\nments with Fe I 6173 ˚A and Ca II 8542 ˚A for a large sunspot as reported by Stangalini et al. (2018).\nThe phase delay between the oscillatory signals in these two lines was f ound to peak at δφ≈0.5 rad\n(Figure 3 therein). As suggested by the same study, the differenc e between the formation heights of\nthe two lines ( d) is∼800 km. Equating 2 πd/λtoδφleads to a λ≈104km. We therefore deem λto\nrange from 500 to 104km, from which we then deduce that kR/greaterorsimilar0.3. This deduction is admittedly\nidealized. To name but one caveat, in general one should equal 2 πd/λto 2nπ+δφ(n= 0,1,2,···).\nLettingn= 1, one finds a λof∼740 km, which is drastically different from the case where n= 0.\nThis subtlety is representative of multi-wavelength studies on wave s in the lower solar atmosphere.\nIt is necessary to address some technical details regarding Equat ion (15), for which purpose we\nrecall that only the parameters in the parentheses are seen as ad justable. The ER83 results were\nestablished inideal MHD( η≡0)andforasteptransverse distribution( l= 0). Inthiscase, anSSKM\ncan be identified without ambiguity for a given kR. However, care needs to be exercised in resistive\nMHD (η∝ne}ationslash= 0), in which case a dense set of solutions arise (see e.g., Figure 2 in Geeraerts et al. 2020\neven though SSSMs were examined therein). We would like to find a uniq ue solution out of this dense\nset, “unique” in the sense that it falls back to a standard SSKM as in E R83. As in C18, for a target\ncombination [ kR,l/R,R m], this is done by first fixing [ kR,l/R] and then looking for thesolution lying\non the curve that becomes independent on RmwhenRmis sufficiently large. This procedure relies\non two facts, one being that the Rm-independent eigen-frequencies pertain to resonantly damped\ncollective modes, and the other being that resonantly damped mode s are physically connected to\ntheir ER83 counterparts (see the review by Goossens et al. 2011 ; see also Poedts & Kerner 1991 , and\nVan Doorsselaere et al. 2004 ).\n2.3.Resonantly Damped SSKMs in the Thin-Boundary Limit8 Chen et al.\nWhenRmis irrelevant in Equation ( 15), the functional dependence therein can be established\nwith a largely analytical approach in the TB limit ( l/R≪1). In this case, the eigen-frequencies\nfor resonantly damped SSKMs are governed by the dispersion relat ion (DR) derived by Y17, which\nreads\nρi(ω2−k2v2\nAi)−ρe(ω2−k2v2\nAe)µi\nµeQ1\n+iπρiρe(ω2−k2v2\nAi)(ω2−k2v2\nAe)G1\nµe/bracketleftBigg\nk2\nρc|∆c|/parenleftbiggc2\nsc\nc2sc+v2\nAc/parenrightbigg2\n+1\nρA|∆A|r2\nA/bracketrightBigg\n= 0.(16)\nOut of the many symbols, relevant here are the definitions for ∆ cand ∆ A,\n∆c=d(ω2−k2c2\nT)\ndr/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=rc,∆A=d(ω2−k2v2\nA)\ndr/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=rA, (17)\nwhich were first introduced by Sakurai et al. (1991). Here the subscript c pertains to the cusp\nresonance, which takes place at rcwhereωR=kcT. Likewise, by the subscript A we refer to the\nAlfv´ en resonance, which takes place at rAwhereωR=kvA. We note that the terms in the square\nparentheses derive from the jumps in the radial Lagrangian displac ement ([˜ξ]) across the resonant\nlocations, and these jumps are responsible for SSKMs to be resona ntly damped ( Goossens et al.\n2011). As such, the importance of the cusp resonance relative to the A lfv´ en one can be measured by\nthe ratio ( X) between the two terms, which equals [ ˜ξ]c/[˜ξ]Ain the TB limit (see Equations 19 and\n20 in Y17).\nEquation( 16)isalsonecessary tosolvegiventhatwewouldliketocomparethedam pingratesinthe\nTB limit with those found with the resistive approach. We start with an initial guess for ω, thereby\nfindingtemporaryvaluesfor rcandrA. Thetermsinthesquareparentheses arethenevaluated, which\nenables us to solve Equation ( 16) to update ω. With this updated ωas a further initial guess, this\nprocess is iterated until convergence is met, thereby yielding a uniq ue eigen-frequency that depends\nonly on [ kR,l/R].\n2.4.Energetics of Linear Waves in Resistive MHD\nIt turns out necessary to examine the wave damping from the pers pective of energy balance. For\nthis purpose, we dot Equations ( 4) and (5) withv1andB1/µ0, respectively. Likewise, we multiply\nEquation ( 6) byp1/(γp0). Adding up the resulting equations then leads to a conservation law\n∂ǫ\n∂t=−∇·f−sideal−sres, (18)\nwhere\nǫ=1\n2ρ0v2\n1+B2\n1\n2µ0+p2\n1\n2γp0, (19)\nf=ptotv1−1\nµ0(v1·B1)B0+η\nµ0j1×B1 (20)\nsideal=j0·(v1×B1)+1\nγp0p1v1·∇p0, (21)\nsres=η/bracketleftbigg\nj2\n1−2(γ−1)\nγp1j1·j0\np0/bracketrightbigg\n. (22)Waves in photospheric waveguides 9\nFurthermore, ptotis the Eulerian perturbation of total pressure, and j0(j1) is the equilibrium (per-\nturbed) electric current density,\nptot=p1+B0·B1\nµ0,j0=∇×B0\nµ0,j1=∇×B1\nµ0. (23)\nIntegrating Equation ( 18) over some fixed volume V, one finds that\nd\ndt/integraldisplay\nVǫdV=−/contintegraldisplay\n∂VdA·f−/integraldisplay\nV(sideal+sres)dV , (24)\nwhere∂Vis the surface that encloses V.\nSome remarks on Equation ( 18) are needed. Firstly, within the framework of resistive MHD, Equa-\ntion (18) governs the energetics of arbitrary small-amplitude wave-like per turbations in an arbitrary\nnonuniform medium, provided that an equilibrium flow ( v0) is absent and hence the equilibrium is\nin force balance as governed by\n−∇p0(x)+j0(x)×B0(x) = 0. (25)\nEvidently, Equation( 25)simplifiestoEquation( 1)foraradiallystructuredequilibrium. Secondly, the\nideal version ( η= 0) of Equation ( 18) was first derived by Bazer & Hurley (1963) to our knowledge.\nWell known is that ǫandfrepresent the wave energy and energy flux densities, respective ly. Less\nwell known is the sidealterm, for which Leroy(1985) offered a physical interpretation where both\nterms on the right-hand side (RHS) of Equation ( 21) are connected to the rate of work done on the\nperturbed flow field ( v1) by some perturbations to the Amp´ ere force density. The first t erm results\ndirectly from j0×B1, while the second term derives from the perturbation to the mass d ensity of a\nfluid parcel that experiences the unperturbed Amp´ ere force ( j0×B0, see Equation 25). Evidently,\nsidealis relevant, at least in principle, for wave energetics unless B0is potential. Thirdly, introducing\na finiteηresults in both the sresterm and an additional Poyning flux associated with the conductive\nelectric field (the last term on the RHS of Equation 20). The first term on the RHS of Equation ( 22)\nevidently represents the Joule dissipation rate, while the second te rm can be loosely interpreted\nas resulting from the interaction between the perturbed ( j1) and unperturbed ( j0) electric current\ndensities. While a conservation law in the full form of Equation ( 18) is not available as far as we\nknow, we tend not to claim to have derived it because introducing ηis rather straightforward. On\ntop of that, energy balance can be expressed in multiple ways or pos sibly even in an infinite number\nof ways (see e.g., Bogdan et al. 2003 , Section 4). It suffices to note only one example, where one\ndots Equations ( 4) and (5) withv1andB1/µ0, respectively. Combining the resulting equations and\nintegrating over V, oneends upwith analternative conservationlaw asderived in Poedts et al. (1989,\nEquation 4.3).\nOur way for examining the wave energetics differs from relevant pre vious ones in three aspects.\nFirstly, we take Equation ( 18) (or equivalently Equation 24) as our starting point. Compared with\nthe formulation in, say, Poedts et al. (1989), Equation ( 18) has the advantage that ǫandfhave\nclearer physical interpretations. For instance, involving Equation (6) in the derivation, we have a\nclearly defined internal energy term in ǫ. Likewise, fthen involves explicitly the well-known acoustic\nfluxp1v1. Secondly, our examination of the wave energetics is intended to ex pand our understanding\nof the damping rates, which are found simultaneously with the eigen- functions. This is different10 Chen et al.\nfrom driven problems as examined in, say, Poedts et al. (1989), where waves with real frequencies\nare driven into the system and the rate of energy absorption in the Alfv´ en continuum is evaluated by\nworkingwiththeresistiveeigen-functions(seealso Goossens et al.2011 ,formoreondrivenproblems).\nThirdly, our way is also different from the limited number of available stu dies where the philosophy is\nsimilar(Arregui et al.2011 ,Soler et al.2013 ; seealso Wright & Thompson 1994 ). Basicallythepoint\nis that these studies were exclusively conducted by assuming a vanis hing gas pressure because the\nfocus was on the resonant absorption of coronal kink modes in the Alfv´ en continuum. It follows from\nthis assumption that the equilibrium magnetic field B0is potential and hence sidealis not involved.\nIn our equilibrium setup, B0is necessarily non-potential and hence sidealneeds to be evaluated. On\ntop of that, we will examine both resonantly and resistively damped m odes.\n3.NUMERICAL RESULTS\n3.1.Overview of Different Damping Regimes\nNow we are in a position to examine the damping of SSKMs. For this purp ose we start with\nFigure2, which shows how the dispersive properties of SSKMs depend on the magnetic Reynolds\nnumber ( Rm) for a number of combinations of [ l/R,kR]. Presented in Figure 2a is the damping-\ntime-to-period ratio τ/P, which derives from the real ( ωR, Figure2b) and imaginary ( ωI, Figure2c)\nparts of the eigen-frequency. The area shaded yellow pertains to the values of Rmbetween 104and\n6.6×107, a range likely to apply to pores and sunspot umbrae. Note that Figu re2is a pictorial\nrepresentation of Equation ( 15), which in turn represents the numerical solutions to the EVP in\nresistive MHD as laid out in Equations ( 8) to (14). As such, the damping of SSKMs is expected\nto experience the combined effects due to Ohmic resistivity and reso nant absorption. Indeed, three\nregimes show up for all pairs of [ l/R,kR]. When Rmis sufficiently large, all curves in Figure 2a level\noff, a defining signature for resonant absorption to be solely respo nsible for damping collective modes\n(Poedts & Kerner 1991 ). We therefore label this regime as the “resonant regime”. Moving away\nfrom this regime toward smaller Rm, one sees that τ/Ptends to decrease until a knee is reached,\nbeyond which τ/Ppossesses a simple linear Rm-dependence. We call this last regime the “Ohmic\nregime” because the damping of SSKMs can be solely attributed to Oh mic resistivity. To show this,\nwe note that τ/P∝ωR/|ωI|. We note further that the axial phase speeds ( vph=ωR/k, Figure2b)\nof SSKMs differ little from cTifor all combinations of [ l/R,kR,R m] examined in this study, meaning\nin particular that vphand hence ωRare essentially constants for a given [ l/R,kR] in the present\ncontext. On the other hand, for a given [ l/R,kR], one intuitively expects that |ωI| ∝η∝1/Rm\nfor weak damping, which is indeed the case as demonstrated by Figur e2c. The end result is then\nτ/P∝Rm. The range of Rmthat lies between the two regimes is to be named the “intermediate\nregime”, where Ohmic resistivity and resonant absorption both play a role for damping SSKMs. This\nintermediate regime is not as clearly separated from the resonant r egime as it is from the Ohmic one.\nFor definitiveness, we nonetheless regard the resonant regime to start at an Rmbeyond which τ/P\nvaries by no more than 1 .5% over a decade or so.\nOverall, the point to draw from Figure 2is that in general the importance of Ohmic resistivity\nrelative to resonant absorption for damping SSKMs needs to be ass essed on a case-by-case basis,\ngiven the broad coverage of the shaded area. That said, the damp ing rates can be order- or even\norders-of-magnitude larger than the ones deriving from resonan t absorption when Ohmic resistivity\ndominates, as happens for waveguides with, say, small sizes and/o r weak magnetic fields (see theWaves in photospheric waveguides 11\ndefinition for Rm). This is true despite that Rmconsistently exceeds 104, a value that seems to\nbe sufficiently large for one to propose resonant absorption as the primary damping mechanism by\ndrawing experience from pertinent coronal studies. Indeed, whe n coronal radial fundamental kink\nmodes are of interest, the Alfv´ en resonance tends to become th e sole damping mechanism when Rm/greaterorsimilar\n3×103−105(see e.g., Figure 2 in Terradas et al. 2006 ; Figure 9 in Guo et al. 2016 ). These values are\nmany orders-of-magnitude smaller than typically accepted corona l values for Rm, making resonant\nabsorption a leading mechanism to account for the damping of the ab undantly measured coronal\nkink modes ( Goossens et al. 1992 ;Ruderman & Roberts 2002 ;Goossens et al. 2002 ). What Figure 2\nhighlights is that in general Ohmic resistivity needs to be incorporate d in studies of collective modes\nin photospheric waveguides. This point has already been raised by bo th C18 and Geeraerts et al.\n(2020) who examined photospheric SSSMs, and derives from the consider able difference between the\nphysical conditions of photospheric waveguides and the coronal o nes. As an example, we note that\nthe waveguides in this study are under-dense ( ρi/ρe<1, Figure 1b), whereas bright coronal loops\nare consistently over-dense ( ρi/ρe>1).\nTheRm-dependence of the behavior of SSKMs can also be examined from th e perspective of energy\nbalance. We start by rewriting the Fourier ansatz (Equation 7) as\ng1(r,θ,z;t) = Re{˜g(r)exp[−i(ωRt−kz−θ)]}exp(ωIt),\nsuch that the oscillatory ( ωR) and damping ( ωI) time-dependencies are explicitly separated. One\nthen finds that the product of any two first-order quantities g1andh1, when averaged over one axial\nwavelength ( λ= 2π/k), evaluates to\n∝an}b∇acketle{tg1(r,θ,z;t)h1(r,θ,z;t)∝an}b∇acket∇i}ht ≡1\nλ/integraldisplayλ\n0g1(r,θ,z;t)h1(r,θ,z;t)dz=g1h1exp(2ωIt),(26)\nwhere\ng1h1=1\n2Re/braceleftBig\n˜g∗(r)˜h(r)/bracerightBig\n=1\n2Re/braceleftBig\n˜g(r)˜h∗(r)/bracerightBig\n. (27)\nThe asterisks represent the complex conjugate, and g1h1is seen to depend only on r. Now take a\nvolumeVto be the space bounded by two transverse planes that are an axia l wavelength apart. One\nreadilyfindsthatthenetwaveenergyfluxleavingthisvolumevanishe s, giventhatonlytrappedmodes\nare of interest (see the surface integral in Equation 24). After some further algebra, Equation ( 24)\nbecomes\n−2ωIE=Sideal+Sres, (28)\nwhere\nE=/integraldisplay∞\n0¯ǫ(r′)r′dr′, Sideal=/integraldisplay∞\n0¯sideal(r′)r′dr′, Sres=/integraldisplay∞\n0¯sres(r′)r′dr′. (29)\nThe barred quantities in the integrand can be readily evaluated by ap plying Equation ( 27) to the\nproducts of first-order quantities involved in Equations ( 19) to (22). Physically speaking, Equa-\ntion (28) means that the wave damping is realized in two channels, one throug h the Ohmic resistivity\nand the other through the mutual interaction between the waves and the equilibria.12 Chen et al.\nFigure3examines the Rm-dependence of some key terms in Equation ( 28) for the same set of\ncombinations [ l/R,kR] as in Figure 2. The shaded area is also inherited from there. Let us start\nwith Figure 3a, where ( Sideal+Sres)/Eis plotted in conjunction with −2ωI(the black diamonds).\nNote that the latter is essentially a direct output from our code, wh ereas the former needs to be\nevaluated with the eigen-functions. With Equation ( 28), one expects that the diamonds should lie\non the curves, which is seen to be indeed the case. In fact, a close a greement between −2ωIand\n(Sideal+Sres)/Eis found for all values of Rmat a given [ l/R,kR], with the diamonds representing\nonly a small subset in order not to obscure the curves. On the one h and, this agreement validates\nthe remarkable numerical accuracy of our computations. On the o ther hand, the curves leave out\nthe explicit k-dependence and hence are less crowded than in Figure 2c, making the three regimes of\nwave damping more evident. Now move on to Figure 3b, where Sideal/Sresis plotted to evaluate the\nrelative importance of the two terms. This evaluation is informative in that as far as we know, the\nSidealterm has not been explicitly examined, at least not in the context of p hotospheric modes. To\nproceed, we note that the Sresterm is always positive for all the computations because it is by far\ndominated by the Joule dissipation rate (see Equation 22). In contrast, the Sidealterm can be either\npositive or negative, thereby serving as a sink or source in physical terms. One further sees that Sres\ndominates SidealwhenRmis either sufficiently large or sufficiently small, or equivalently when the\nmodes are very deep in either the Ohmic or the resonant regime. Whe nSideal/Sresis appreciable,\none sees that its magnitude tends to be stronger for smaller values ofl/Rat a given kR(see the red\ncurves with different linestyles). Likewise, this magnitude tends to b e stronger for smaller kRwhen\nl/Ris given (see the solid curves with different colors). These details asid e, we conclude that Sideal\nneeds to be considered for examining the energetics of SSKMs in our model equilibria, given that\n|Sideal/Sres|may reach ∼28% in the case where [ l/R,kR] = [0.1,0.7].\nSome further understanding of the different damping regimes can b e gained from Figure 4, where\n¯sresis plotted as a function of rfor a fixed pair of [ l/R,kR] = [0.1,0.7]. Here the shaded area\nrepresents the transition layer (TL). A number of values for Rmare adopted, and the corresponding\n¯sresis rescaled such that its maximum attains unity. Above all, one sees th at ¯sresalmost vanishes\nidentically in the exterior. The black curve in Figure 4a pertains to lg Rm= 4.3, and is representative\nof the Ohmic regime. Most notable in this case is that ¯ sresis extended over both the interior and the\nTL.Onthecontrary, pertainingtoan Rmfairlydeepintheresonantregime, theredcurveinFigure 4b\nis characterized by the concentration of ¯ sresin two very narrow layers. Evidently, the inner and outer\nlayers correspond to the cusp and Alfv´ en resonances, respect ively. Given that the axial phase speeds\nof resonantly damped SSKMs are close to the internal tube speed ( cTi, see Figure 2b), one would\nhave expected the locations of the resonances in view of the radial profiles of cTandvA(Figure1).\nExamining the rest of the curves in Figure 4, one sees that ¯ sresbecomes increasingly concentrated\naround the resonances as Rmincreases. This behavior also offers a qualitative explanation for the\nratherinvolved behavior of Sideal/Sresshown inFigure 3b. Forthispurpose wenotethatby definition,\nSideal(Sres) collects all contributions from sideal(sres) over the entire range of r. However, sidealdoes\nnot vanish only in the TL. If then follows that Sideal/Sresis determined by two factors, one being how\nwidesidealis distributed relative to sresand the other being how sidealcompares with sresat specific\nlocations. It turns out that at sufficiently small Rm, the former factor is more important, leading to\nsmall values of Sideal/Sresdespite that sidealcan be comparable to sreslocally. When Rmis sufficientlyWaves in photospheric waveguides 13\nlarge, the latter factor is more important, once again resulting in sm all values of Sideal/Sreseven\nthoughsidealcan be distributed in a more extended manner.\nOne may have noticed two peculiarities in Figure 2a, one being that some curves do not extend\nto the largest Rmin the plot, the other being that some curves are absent for some c ombinations\nof [l/R,kR]. Both result from some limitations of our numerical method. Recall t hat, given a pair\nof [l/R,kR], we first look for the resonant regime and then vary Rmtoward the target value such\nthat a unique eigen-frequency can be found. Two technical issues arise. When Rmis extremely\nlarge, sometimes the computational resources required for reso lving the extremely fine scales where\nresonant absorption occurs may be too demanding. This explains th e first peculiarity. Another issue\nis that sometimes our code does not converge to a unique solution wh en we reduce Rmfrom the\nresonant regime. This happens when kRis large for a given l/Ror whenl/Ris large for a given kR.\nNote that this is not to say that a proper resonant regime cannot b e found in these situations. It is\njust that for a given pair of [ l/R,kR], theτ/Pprofile spans only a narrow range of Rmsuch that it\nis not straightforward to address the importance of the Ohmic res istivity for damping the SSKMs.\nThese technical issues notwithstanding, we are allowed to further examine resonant absorption by\ncapitalizing on the computability of the resonant regime for those pa irs of [l/R,kR] that cover a\nmuch broader range than presented in Figure 2. This examination is important in its own right,\nbecause resonant absorption is indeed the sole damping mechanism w henRmis sufficiently large.\n3.2.Resonantly Damped SSKMs\nFigure5presents the l/R-dependence of (a) the damping-time-to-period-ratios ( τ/P), (b) the axial\nphase speeds ( ωR/k), and (c) the damping rates ( ωI) in units of kvAifor resonantly damped SSKMs\nforanumber of axialwavenumbers ( kR). Inadditiontothevalues computedwith resistive MHD(the\nsolid curves), Figure 5also presents the TB expectations (dashed) found by solving Equa tion (16).\nOn top of that, we have also plotted the results found by solving Equ ation (16) with the second term\nin the square parentheses neglected (the dotted curves). For t he ease of description, let “full TB” and\n“cusp TB” refer to the dashed and dotted curves, respectively. One may question why the “cusp TB”\nresults are shown together with the “full TB” ones. We will address this question shortly, because\nit seems better to start with a comparison between the resistive an d “full TB” results. Examining\nFigure5a, one sees that the smaller the value of l/R, the smaller the difference between the solid and\ndashed curves, a qualitative behavior implied in the TB approximation b y construction. What needs\nto be examined is how small an l/Rshould be for the “full TB” results to well agree with the resistive\nresults for a given kR. We quantify this by first defining ǫ≡ |(τ/P)res/(τ/P)TB−1|to measure the\ndifference in the values for τ/Pfound with the two approaches. We further define ( l/R)TBto be\nthe critical value only below which ǫ≤30%. One finds that in general ( l/R)TBdecreases with\nkR, reading 0 .074 (0.026) for a kRof 0.3 (4.3). This k-dependence of ( l/R)TBwas also found for\nphotospheric SSSMs in C18, and can be explained by the same intuitive reasoning therein. After all,\nwith the axial wavelength ( λ= 2π/k) also a relevant lengthscale, the TL width ( l) should be much\nsmaller than both Randλfor the TB limit to hold. Somehow surprising is that ( l/R)TBis so small.\nIndeed,ǫevaluates to 2 .6 for anl/Ras small as 0 .1 whenkR= 4.3. For this pair of [ l/R,kR], one\nsees that the substantial value for ǫresults from the differences in the l/R-dependence of the resistive\nand “full TB” profiles. While the τ/Pprofile in the TB limit decreases monotonically with l/Rin\nthe plotted range, its resistive counterpart possesses a non-mo notonic behavior as characterized by\nthe appearance of a minimal τ/P. On the other hand, one sees from Figure 5b that the axial phase14 Chen et al.\nspeeds (ωR/k) are consistently smaller than cTiby only a small amount. It then follows that the\nbehavior of τ/P, theexistence ofaminimal τ/Pinparticular, derives fromthe l/R-dependence ofthe\nimaginary part of the eigen-frequencies ( ωI, Figure5c). We note that the existence of a minimal τ/P\nholds for all the resistive computations we have conducted. We not e further that this non-monotonic\nl/R-dependence has been seen for the resonantly damped photosph eric SSSMs (C18) and coronal\nradial fundamental kink modes alike ( Soler et al. 2013 , S13 hereafter). This S13 study approached\nthe resonant absorption of coronal kink modes in the Alfv´ en cont inuum from the ideal quasi-mode\nperspective with the Frobenius method. Furthermore, the non-m onotonic l/R-dependence of the\ndamping rates for a particular transverse profile (Figure 6 in S13) w as interpreted from energetics\nconsiderations by evaluating the energy flux injected into the reso nance and the integrated wave\nenergy. In what follows, we analyze the l/R-dependence of the damping rates of SSKMs from the\nenergetics perspective as well.\nWe start by considering the entire volume bounded by two transver se planes that are separated by\nan axial wavelength. We then exclude two cylindrical shells, the inner shell bounded by r−\ncandr+\nc,\nand the outer one bounded by r−\nAandr+\nA. LetVdenote the resulting volume. The subscripts c and\nA are adopted because these shells will be associated with the cusp a nd Alfv´ en resonances shortly.\nFor now it suffices to assume that r−\nc< r+\nc< r−\nA< r+\nA, and that the Ohmic resistivity is negligible\nfor determining the eigen-functions in V. Equation ( 24) then becomes\n−2ωIˆE=ˆFc+ˆFA+ˆSideal, (30)\nwhere\nˆE=/parenleftBigg/integraldisplayr−\nc\n0+/integraldisplayr−\nA\nr+\nc+/integraldisplay∞\nr+\nA/parenrightBigg\n¯ǫ(r′)r′dr′, (31)\nˆSideal=/parenleftBigg/integraldisplayr−\nc\n0+/integraldisplayr−\nA\nr+\nc+/integraldisplay∞\nr+\nA/parenrightBigg\n¯sideal(r′)r′dr′, (32)\nˆFc=r−\nc¯fr(r−\nc)−r+\nc¯fr(r+\nc)≡/braceleftbig\nr¯fr/bracerightbig\nc, (33)\nˆFA=r−\nA¯fr(r−\nA)−r+\nA¯fr(r+\nA)≡/braceleftbig\nr¯fr/bracerightbig\nA. (34)\nNote that we have introduced the brace operator {q}to evaluate the difference of some quantity q\nacross a shell. In addition, ¯fris found by applying the bar operation (Equation 27) to the radial\ncomponent of the wave energy flux density ( f, Equation 20). In the absence of the Ohmic resistivity,\nfrsimplifies to ptotv1rand consequently ¯fr= Re(˜p∗\ntot˜vr)/2. Evidently, Equation ( 30) means that\nthe wave energy in the volume Vis lost via both net fluxes into the shells ( ˆFcandˆFA) and the\ninteractions with the equilibrium ( ˆSideal).\nWith Equation ( 30) we recall some features of the ideal MHD quasi-mode approach su ch that\nsome intricacies with our resistive MHD approach can be appreciated . As far as resonantly damped\nmodes are concerned, the ideal quasi-mode approach has been ex clusively conducted by assuming\na vanishing gas pressure (S13, and Soler 2017 )2. It then follows that only the Alfv´ en resonance\n2Here by “ideal” we specifically refer to the quasi-mode approach whe re dissipative effects are discarded from\nthe outset. To our knowledge, this approach has been realized only through the Frobenius method formulated in\nWright & Thompson (1994) and first practiced in the solar context by S13. Some approaches can be regarded as\nnearly ideal, with the formulation developed in Sakurai et al. (1991) as a much-employed prototype. Restricting\nourselves to Equation ( 16), we mean by “nearly ideal” that the outcomes from these approac hes do not explicitly\ninvolve dissipative effects but dissipative MHD is nonetheless involved in the mathematical procedure.Waves in photospheric waveguides 15\narises, and sidealvanishes identically. On top of that, the Ohmic resistivity is conceptu ally irrelevant\nin the first place, and resonant absorption can be visualized as takin g place in a resonant layer that\nis infinitely thin ( r−\nA=r+\nA). In view of Equation ( 30), thatωIdoes not vanish in S13 is then due\nto the discontinuity in ¯fracross the Alfv´ en resonance, which in turn derives from the disco ntinuities\nin ˜ptotand ˜vr(see Figure 3 therein). Without concrete ideal MHD computations, we refrain from\ndiscussing further how resonantly damped SSKMs behave when the y are eventually found this way.\nRather, it suffices to note that the ideal eigen-frequencies are ex pected to agree with their resistive\ncounterparts but this is not true regarding the eigen-functions. This latter aspect has some bearings\non how we evaluate the terms in Equation ( 30) with our resistive eigen-functions. The point is, if\nneglecting the subscripts c and A, then one finds that Equation ( 30) holds for an arbitrary Vas long\nas resistivity can be neglected therein. Evidently, we need to assoc iate the two cylindrical shells with\nthe resonances for the evaluation to make physical sense. Howev er, some intricacies arise regarding\nhow to determine the borders of the shells, known as dissipative laye rs (DLs, e.g., Goossens et al.\n2011). For the much-studied Alfv´ en resonance, the DL width ( δA) is known to take the larger one\nout of the resistive ( δA,res) and non-stationarity scales ( δA,NS). Here δA,resdepends on the Ohmic\nresistivity as η1/3(Sakurai et al. 1991 ), whereas δA,NSis independent of ηbut proportional to |ωI|\n(Ruderman et al. 1995 ;Tirry & Goossens 1996 ). Regarding the DL width pertaining to the cusp\nresonance ( δc), a resistive spatial scale ( δc,res) is also relevant and possesses the same η-dependence\n(Sakurai et al. 1991 ). Likewise, the cusp version ( δc,NS) of the non-stationarity scale was shown to\nbe formally the same as δA,NS(Tirry et al. 1998 , see the discussions in connection to Equations 17\nto 19 therein). It then follows that the DL widths ( δcandδA) in the present context will be Rm-\nindependent when Rmbecomes sufficiently large. In this case, no Rm-dependence will show up for the\nhatted terms in Equation ( 30), making their evaluation more definitive. However, our code does n ot\nallowRmto increase indefinitely. This means in particular that δAnever reaches the non-stationarity\nscale in our computations, and the hatted terms in Equation ( 30) are in general dependent on Rm.\nSo how do we evaluate the hatted terms, supposing a fixed pair of [ l/R,kR]? One possible way is\nto draw experience from the analytical study by Goossens et al. (1995), who showed that\nr±\nA≈rA±5δA, (35)\nfortheAlfv´ enresonance, providedthat δAisdeterminedby δA,res. However, Equation( 35)isnolonger\nvalid when δAis determined by δA,NS(e.g.,Tirry & Goossens 1996 , Figure 1). We therefore choose an\nempirical procedure to determine r±\nA,cfor a given Rm. The idea is simply that by definition, resistivity\nis important only in the DLs, namely the two shells in the present conte xt. Basically, we compare\nthe radial profile of some eigen-function against that obtained at a substantially larger R′\nm=ζRRm,\nand locate the shells by looking for where the relative difference betw een the two profiles exceeds\nsome critical value ζcri. We consistently choose |˜vr|as the eigen-function, and adopt a ζR= 2. In\naddition, we take ζcrito be 4×10−4and 10−5to determine the cusp and Alfv´ en DLs, respectively.\nEmploying other reasonable values for ζRorζcrileads only to insignificant differences to our results.\nFigure6displays the radial profiles of Re˜ vrfor a number of Rmas labeled, with [ l/R,kR] fixed at\n[0.2,2]. Here two different intervals are distinguished to show the details in the DLs pertaining to\nthe cusp (Figure 6a) and Alfv´ en (Figure 6b) resonances. Each profile is rescaled such that ˜ ptot= 1\natr= 1.25R, otherwise the comparison between different profiles will not make s ense. Note that\nthe details for rescaling the eigen-functions are not essential. Rat her, with Figure 6a we show that16 Chen et al.\nthe width of the cusp DL ( δc) decreases when Rmincreases, accompanied by the appearance of more\nand more oscillations. This behavior is in line with what was pointed out by Ruderman et al. (1995)\nand numerically demonstrated by S13 for coronal kink modes reson antly absorbed in the Alfv´ en\ncontinuum (see also Andries 2003 , Figure 2.3). If Rmfurther increases, both studies showed that\nthe width of the Alfv´ en DL ( δA) will eventually be determined by δA,NS. In other words, eventually\nthe Alfv´ en DL width will become a constant, and the effect of increa singRmis to introduce an\nincreasing number of oscillations into the DL. We can discern some simila r signature for the cusp\nresonance here in that δcis found to depend on Rmonly weakly when lg Rm/greaterorsimilar9. With Figure 6b\nas an illustration, we note that the Alfv´ en DLs in all of our computat ions consistently narrow when\nRmincreases, showing no evidence that non-stationarity ultimately de cidesδAfor the SSKMs in the\nRm-range that we examine.\nSome remarks are necessary in this context. Our first set of rema rks is connected with Figure 6,\nfrom which one sees that the oscillations in the DL(s) can be very str ong when Rmis large. This\ndoes not invalidate the examination of resonantly damped modes fro m the perspective of dissipative\neigen-modes. Rather, this means that if resonant absorption is ex amined from the initial-value-\nproblem (IVP) perspective, then the perturbation energy is incre asingly transferred from global\ncoherent motions to localized motions around resonant surfaces. For coronal radial fundamental\nkink modes, this was established analytically by Ruderman & Roberts (2002) and has been shown\nby a considerable number of numerical studies (e.g., Terradas et al. 2006 ;Soler & Terradas 2015 ;\nEbrahimi et al. 2020 ;Guo et al. 2020 , to name only a few). To our knowledge, however, a similar\nstudy on photospheric modes from the IVP perspective has yet to appear. Our second set of remarks\npertainstoFigure 2, wherethedampingofSSKMsisseentoundergoacontinuous trans itionfromthe\nOhmictotheresonant regime. Wenotethatasimilar problemwasaddr essed forsurfaceAlfv´ enwaves\nin a one-dimensional slab equilibrium by Ruderman & Goossens (1996), even though viscosity was\nadoptedasthedissipativefactortherein. Thetransitionwasshow ntobeorganizedbyadimensionless\nparameter ( Rg) thatinvolves therelative variationofthelocalAlfv´ enspeed, the ratiooftheTL width\nto the axial wavelength, and the viscous Reynolds number. Our Figu re2plots the damping rates\nas a function of Rmfor a given pair of [ l/R,kR], showing that different damping regimes can indeed\nbe told apart. However, the range of Rmpertaining to, say, the intermediate regime is different\nfor different combinations of l/RandkR. If some Rgcan be established for photospheric modes in\nlight ofRuderman & Goossens (1996), then one expects a unified range of Rgfor the intermediate\nregime for different computations, leading to a substantially simpler w ay for discriminating different\ndamping regimes.\nWe are finally ready to examine the l/R-dependence of the damping rates with the aid of Equa-\ntion (30). It suffices to examine a fixed axial wavenumber of kR= 2. Figure 7presents, with circles\nof different colors, the l/R-dependencies of ˆFA/ˆE,ˆFc/ˆE, andˆSideal/ˆE, as well as the sum of the\nthree ratios for a number of Rm. Note that the range of Rmin general depends on l/R. This is due\nto two reasons, one being that SSKMs at different l/Rin general enter into the resonant regime at\ndifferent Rm, and the other being the technical difficulty for us to conduct comp utations when Rm\nis too large. With the exception of the sum, the symbols representin g a given quantity for a given\nRmare connected by a solid line with the same color coding. At any examine dl/R, one sees that\nthe circles of different colors representing the sum cannot be distin guished from one another. This\nis understandable because they evaluate to −2ωI, which is Rm-independent. In fact, replotting theWaves in photospheric waveguides 17\npertinent set of values for ωIwith the dashed curve, we see the expected behavior for the sums to\nagree with −2ωI. On the other hand, one sees that ˆFA/ˆEincreases monotonically with l/Rfor all the\nexamined values of Rm. Put another way, the wave damping due to the Alfv´ en resonance is stronger\nfor a larger l/R. Two reasons are therefore responsible for the apparently coun ter-intuitive behavior\nfor|ωI|to decrease with l/Rwhenl/R/greaterorsimilar0.12. Primarily, it happens because the wave damping due\nto the cusp resonance behaves this way. To a lesser but non-neglig ible extent, it happens because\nˆSideal/ˆEconsistently weakens with l/Rin this range of l/R.\nAn important product of Figure 7is thatˆFcconsistently exceeds ˆFA, meaning that the cusp\nresonance is consistently more important than the Alfv´ en one for damping the SSKMs with this\nparticular kR. What about the relative importance for other values of kR? Going back to Figure 5,\none can most readily evaluate this for the TB results, given the availa bility of an explicit dispersion\nrelation (Equation 16). In this case, one is allowed to isolate, say, the cusp resonance by keeping\nonly the first term in the square parentheses, as we have done whe n producing the “cusp TB” curves.\nRecall that the ratio of the first to the second term is denoted by X. To proceed, let ωc\nIandωA\nI\ndenotethedamping ratesdue tothecusp andAlfv´ enresonances alone, respectively. Likewise, let ωTB\nI\ndenote the damping rate when both resonances are accounted fo r. It then follows that ωTB\nI≈ωc\nI+ωA\nI\nandωc\nI/ωA\nI≈X, when the damping is sufficiently weak. Now let Ydenote the ratio of the “cusp TB”\nvalue for τ/Pto the “full TB” one. It further follows that Y≈1+1/Xbecauseτ/P∝ωR/|ωI|. In\nother words, one expects that Y <2 when the cusp resonance is stronger ( X >1) andY≈1 when\nit dominates ( X≫1). With this preparation, let us examine Figure 5a and focus on where the TB\nlimit holds ( l/R≤(l/R)TB). An inspection of the green curves pertinent to kR= 0.7 indicates that\nYhappens to be ≈2, corresponding to the situation where the two resonances are e qually important.\nThe cusp (Alfv´ en) resonance becomes increasingly important whe nkRincreases (decreases) fromthis\ncritical value. Note that the transition of the relative importance o f the resonances consistently takes\nplace atkR≈0.7 whenl/R≤(l/R)TB. For now it suffices to consider a given l/R. The existence of\na critical kRis not surprising in view of Equation ( 16), which suggests that\nX=k2Λ, (36)\nwhere\nΛ =r2\nAρA\nρc|∆A|\n|∆c|/parenleftbiggc2\nsc\nc2sc+v2\nAc/parenrightbigg2\n=r2\nAρA\nρc|dv2\nA/dr|r=rA\n|dc2\nT/dr|r=rc/parenleftbiggc2\nsc\nc2sc+v2\nAc/parenrightbigg2\n. (37)\nWe arrive at the second equality by replacing ∆ Aand ∆ cwith Equation ( 17). If one assumes that\nrA≈rc≈R, then Equation ( 36) simplifies to Equation (14) in Soler et al. (2009), where the effect\nof the cusp continuum was examined for damping the transverse os cillations in prominence threads.\nGiven the appearance of k2inX, one would expect the cusp resonance to dominate for large kR,\nan expectation that indeed holds. Less trivial is at which kRthe cusp resonance starts to dominate.\nWhenkvaries, it turns out that the k-dependence of Λ derives essentially from the k-dependence of\n(dc2\nT/dr)r=rc. This subtlety results from the extremely weak dispersion of SSKMs (see Figure 5b).\nWithωR/kclose tocTi, neither rcnorrAis sensitive to k, and hence the insensitivity of nearly all\nthe terms involved in Λ, with (d c2\nT/dr)r=rcbeing the only exception. However weak its k-dependence\nis,ωR/knonetheless increases toward cTiwith decreasing k. Therefore the cusp resonance point ( rc)\nmoves toward the inner boundary of the transition layer (see Figur e1a), thereby leading to some18 Chen et al.\nsubstantial decrease in |dc2\nT/dr|r=rc(see Equation 3)3. When it comes to the k-dependence of X,\nthek-dependence of (d c2\nT/dr)r=rcand hence that of Λ somehow counteract the k2term. While X\nremains largely determined by k2, this subtlety means that the critical kR≈0.7 is specific to the\nequilibrium configuration adopted in this study. The insensitivity of th e critical kRtol/R, on the\nother hand, simply follows from the l/R-insensitivity of ωR/kfor a given k(see Figure 5b).\nWhen the TB approximation is not that satisfactory ( l/R >(l/R)TB), one needs to employ the\nresistive computations to examine the relative importance of the re sonances. While isolating one\nresonance is no longer possible, such an examination remains possible because the importance of\nthe cusp relative to the Alfv´ en resonance is quantified by ˆFc/ˆFAin view of Equation ( 30). At this\npoint, we need to clarify how Xin Equation ( 36) relates to ˆFc/ˆFAto avoid possible confusion. For\nthis purpose, we recall that the governing equations on the primitiv e variables were sufficient for\nSakurai et al. (1991) to develop the mathematical prototype that enables the derivat ion of the TB\ndispersion relation ( 16) whereXis involved. We recall further that Equation ( 30) is valid regardless\nof the TB approximation. It is just that ˆFc/ˆFAcan be explicitly expressed by Equation ( 36) when\nthe TB limit holds. To see this, we use Equations ( 33) and (34) to arrive at\nˆFc\nˆFA={rRe(˜p∗\ntot˜vr)}c\n{rRe(˜p∗\ntot˜vr)}A, (38)\nwhere the braces are recalled to evaluate the difference of some qu antity across a dissipation layer\n(DL) of finite width. In the TB limit, the DLs are thin as well and ˜ ptotis continuous across a DL (see\nGoossens et al. 2011 , for details). We can then assume that r−\nc≈r+\nc≈rc,r−\nA≈r+\nA≈rA, and replace\nthe braces with a pair of square parentheses. Further assuming t hatr˜ptotis the same at the two\nresonances, one finds that Equation ( 38) evaluates to [˜ vr]r=rc/[˜vr]r=rAor equivalently [ ˜ξr]r=rc/[˜ξr]r=rA\nbecause ˜vr=−iω˜ξ. The general expression ( 38) therefore falls back to Equation ( 36). In other\nwords, the relative importance ˆFc/ˆFAis fully determined by the jumps in the radial speed in the TB\nlimit. When the TB limit is not that good an approximation, the relative imp ortance can be roughly\nmeasured by how {˜vr}ccompares with {˜vr}Aif{r˜ptot}cis not that different from {r˜ptot}A.\nFigure8presents the radial profiles of the Fourier amplitudes of some relev ant perturbations for an\nequilibrium profile with a TL-width-to-radius ratio ( l/R) of 0.2. The vertical dash-dotted lines mark\nthe boundaries of the TL. The top row represents the Eulerian per turbation to total pressure (˜ ptot),\nwhile the rest pertain to the radial (˜ vr), azimuthal (˜ vθ), and axial (˜ vz) speeds. The eigen-functions\nare normalized such that ˜ ptot= 1 where |˜ptot|attains its maximum. For any eigen-function, we use\nthe black and red curves to plot its real and imaginary parts, respe ctively. Two axial wavenumbers\n(kR) are examined, one being 0 .3 (the left column) and the other being 2 (right). Note that the\nresistive solutions for different values of kRpertain to different magnetic Reynolds numbers ( Rm).\nTo be specific, for each kR, we choose an Rmthat is barely inside the resonant regime. The reason\nfor doing this is that when Rmis too large, ˜ vrin the DLs may be too oscillatory for us to estimate\nthe relative importance of the resonances. Actually the oscillatory behavior can be seen in the lower\ntwo rows in Figure 8, from which the cusp (Alfv´ en) resonance can be readily told by the strong\noscillations in ˜ vz(˜vθ). The so-called 1 /s-singularities are also clearly visible. The presence of both\nresonances is further seen in the plots for ˜ vr, where both the jumps across resonances (e.g., the black\n3It can be readily shown that d c2\nT/dr∝δr≡(r−ri) for a given l/Rwhen 0<|δr| ≪l. Takel/R= 0.03 for\ninstance. We find that δrreads [2.4,1.4,0.7]×10−3RforkR= [2,0.7,0.3]. These values of δrare consistently far\nbelowl, let alone ri. However, the decrease of |dc2\nT/dr|r=rcwith decreasing kRis substantial.Waves in photospheric waveguides 19\ncurves) and logarithmic singularities (e.g., the red curves) are evide nt4. In contrast, the curves for\n˜ptotareessentially continuous, as expected in the case of weak damping . Comparing the two columns,\none sees a change in the behavior of the magnitude of {˜vr}crelative to {˜vr}A. While {˜vr}cis less\nstrong than {˜vr}AforkR= 0.3, the opposite happens for kR= 2. One then expects that the cusp\nresonance is less important in the former case, but is stronger in th e latter.\nEquation ( 30) can be used to perform a more quantitative evaluation of the relat ive importance\nof the resonances as a function of the axial wavenumber kR. For this purpose, we adopt a fixed\nl/R= 0.2 and plot the ratios of the relevant terms in Figure 9in a format nearly identical to\nFigure7. One sees that the sum of the ratios, ( ˆFc+ˆFA+ˆSideal)/ˆE, agrees with the dashed curve\nrepresenting the values of −2ωI, meaning that the computed eigen-functions are remarkably accu rate.\nMore importantly, one sees that the cusp resonance surpasses t he Alfv´ en one at kR∼1 in terms of\nthe importance for damping the SSKMs. That the cusp resonance d ominates for large kRis nothing\nunexpected. What Figures 5and9suggest is that the cusp resonance may not be the dominant\nresonance throughout the entire kRrange that is likely to be observationally relevant.\nThe relative importance of the two resonances aside, one may ques tion how efficient the resonant\ndamping of SSKMs can be. For this purpose, we recall that for each kR, there exists an l/Rthat\noptimizes the resonant damping (see Figure 5a; see also Yu & Van Doorsselaere 2019 for a relevant\nstudy on coronal kink waves). Let ( l/R)mindenote this l/R, and (τ/P)mindenote the value that the\ndamping-time-to-period ratio attains therein. Figure 10presents both ( l/R)min(the red curve) and\n(τ/P)min(green)asafunctionof kR. Oneseesthat( l/R)mindecreases with kR, reachingvaluesbelow\n0.02 when kR/greaterorsimilar16. This inspired us to examine the TB expectations at a given ( l/R)min, and the\ncorresponding results are shown by the blue curve labeled ( τ/P)TB. Despite that ( l/R)mineventually\ntends to be small, the TB expectations are seen to consistently ove restimate the resonant damping\nrather considerably. In other words, at a given kR, the TB limit holds only for l/Rthat is even\nsmaller than ( l/R)min(see Figure 5a). One further sees that when kRincreases, the resistive value\nfor (τ/P)minrapidly decreases at first but levels off when kRexceeds, say, 10. The asymptotic value\nthat (τ/P)minattains at kR= 30 is merely ∼70. From this we conclude that, in the observationally\nrelevant range of kR, resonant absorption does not play an important role for damping t he SSKMs\nin our equilibrium setup.\n4.SUMMARY\nMotivatedbythe considerable interest insolar photospheric collect ive modes, we have examined the\ndamping of slow surface kink modes (SSKMs) in photospheric wavegu ides with equilibrium quantities\nrepresentative of pores and sunspot umbrae. We adopted a cylind rical equilibrium configuration\ncomprising a uniform interior, a uniform exterior, and a transition lay er (TL) that continuously\nconnects the two. By formulating the relevant eigenvalue problem in the framework of resistive,\nlinear MHD, we were allowed to simultaneously account for two mechan isms for damping SSKMs in\naself-consistent manner, onebeing the Ohmicresistivity, andtheo ther being the resonant absorption\nof SSKMs in the cusp and Alfv´ en continua. We find that the diversity of the geometric and physical\nparameters of photospheric waveguides, wrapped up in the magne tic Reynolds number ( Rm), means\nthat the relative importance of the two mechanisms needs to be ass essed on a case-by-case basis. At\n4These singular profiles are only apparent because the resistivity is n onetheless finite (see the review by\nGoossens et al. 2011 , for details).20 Chen et al.\nrealistically small values of Rm, the Ohmic resistivity plays a far more important role, resulting in\ndamping-time-to-period-ratios τ/P∼10 in the parameter range we examine. In contrast, resonant\nabsorptionistheonlydampingmechanism forsomerealisticallylarge Rm. Inthislattercase, thecusp\nresonance in general dominates the Alfv´ en one except for very lo ng axial wavelengths. Regardless, in\nthe observationally relevant wavelength range, the resonant dam ping rate turns out to be negligible,\nleading to values for τ/Pto consistently exceed ∼70.\nWhen SSKMs are damped solely by resonant absorption, we have com pared the damping rates\nobtained by the resistive approach with those found by solving the e xplicit dispersion relation valid\nin the thin-boundary (TB) limit. In general we find that TB values agr ee with the resistive ones only\nwhen the transition-layer-width-to-radius ratios ( l/R) are much smaller than the nominal values\nfor the TB limit to apply. This is particularly true for short axial wavele ngths, which are more\nobservationally relevant. From the practical point, this means tha t when the resonant damping of\nSSKMs is of interest, one needs to adopt approaches such as the r esistive one, despite that the TB\nlimit proves much less computationally expensive.\nBefore closing, let us name a limited number of intricacies that are likely inherent in theoreti-\ncal studies on photospheric modes by contrasting our findings with representative coronal studies\non radial fundamental kink modes. One, the range of Rmwhere resonant absorption is expected\nto dominate for damping collective modes. Drawing experience from t he coronal studies by, say,\nTerradas et al. (2006) andGuo et al. (2016), one is inclined to propose that photospheric SSKMs\nare damped primarily by resonant absorption given that Rmtends to exceed ∼104. However, this\nis not the case. Two, the range of applicability of the TB limit regarding the resonant damping\nof collective modes. While the TB limit is constructed to hold only when l/Ris small, the range\nof its applicability is drastically different in photospheric cases from th at in coronal cases. In coro-\nnal studies, the TB expectations for the resonant damping rates of kink modes tend to be a good\napproximation well beyond the nominal range of applicability (e.g., Van Doorsselaere et al. 2004 ;\nSoler et al. 2014 ). In particular, Figure 1 in the latter study indicates that the true values for τ/P\ndeviate from the TB ones by no larger than 80% even when l/Rreaches two, the maximum allowed\nby the equilibrium configuration. And this deviation is /lessorsimilar20% when l/R/lessorsimilar0.2. This is not the case\nfor photospheric SSKMs, for which the relative deviation can readily exceed unity even for l/Ras\nsmall as, say, 0 .1. We take these two intricacies as strengthening rather than wea kening the need\nto further the theoretical examinations on photospheric modes. For instance, the second intricacy\nmeans that other approaches like the one outlined for coronal kink modes by Soler et al. (2013) can\nreadily find photospheric applications. Likewise, the first intricacy m eans that the present study on\nthe non-ideal mechanisms for damping SSKMs needs to be taken fur ther. Drawing experience from\nthe study by Geeraerts et al. (2020) on the damping of photospheric slow surface sausage modes\n(SSSMs) , we note that the Ohmic resistivity can be so efficient that S SSMs are damped within a\nfraction of the wave period even if Rmremains/greaterorsimilar104(Figure 3 therein). While this was found for\nphotospheric waveguides with piece-wise constant transverse dis tributions of the equilibrium quan-\ntities, there seems to be no reason to believe that photospheric wa veguides cannot be transversely\nstructured this way.\nSomeremarksonthecaveatsofourapproacharealsonecessary , forwhichpurposewestartbynoting\nthatinrealitythesolarphotosphereisstructuredinawaymuchmor ecomplicatedthanmodeledhere.\nIn fact, this complication is to such anextent that we have to restr ict the following discussions to onlyWaves in photospheric waveguides 21\none additional factor that is expected to be important in the wave d ynamics. This factor, the effect of\ngravity, was neglected in our study from the outset. As a consequ ence, essentially only three spatial\nscales were relevant, namely the waveguide radius ( R), the TL width ( l), and the axial wavelength\n(λ= 2π/k). Introducing gravity considerably complicates the problem, some aspects of which were\nsummarized by Roberts(2019, Chapters 9 to 11 and references therein). For our purpose, it s uffices\nto assume that the gravitational acceleration is aligned with the wav eguide axis. A number of\nadditional spatial scales become relevant even if we assume that ne ither the equilibrium temperature\nnor the mean molecular weight varies much over photospheric height s. These scales characterize\nhow rapid the equilibrium pressure and magnetic field strength vary. LetHdenote the shortest one\namong these scales. Given the axial stratification, the axial wavele ngth (λ) cannot be unambiguously\ndefined, but nonetheless can be understood as the spatial scale c haracterizing the axial variation of\nthe perturbations. For our results to apply, the condition λ≪Hshould be satisfied but proves quite\ndemanding for the majority of our computations in view of the small v alues ofHat photospheric\ntemperatures. It then follows that, in principle our results apply on ly to ER83-like photospheric\nwaveguides, by which we mean the idealized configurations where the equilibrium quantities are\nstructured only transversely and the involved characteristic spe eds follow some canonical ordering.\nNote that here we distinguish between the ER83-like and ER83 equilibr ia by one key difference,\nnamely the transverse structuring is allowed to be continuous in the former as realized by a non-\nvanishing lin this study. With this distinction we note that all the referenced ob servational studies\non photospheric modes have adopted the ER83 framework to aid th e mode identification and/or the\nevaluation of energetics. Focusing on wave damping, we believe that our computations contributed\nto the better understanding of photospheric collective modes, “p hotospheric modes” in the sense as\npracticed in available observational and theoretical efforts. In ad dition, our results can be of help for\nfurther theoretical/numerical studies that address additional r ealistic effects on photospheric modes.\nFor instance, if one adopts a theoretical framework that extend s ours by accounting for gravity,\nthen our results can be used to validate the pertinent computation s by specializing to the cases\nwhereλ≪H. We believe that such a validation is warranted, given that such furt her theoretical\ndevelopments are necessarily more involved than presented here. That said, we concede that it is\ndifficult toassess howwell ourresults carryover totheextremely c omplicatedphotospheric structures\n(see also Roberts 2019 , the introduction sections in Chapters 9 and 10).\nWith the above-mentioned caveats in mind, let us discuss whether ou r results can help constrain\nthe likely damping mechanism(s) for an identified SSKM. We start by no ting that, in the case of weak\ndamping, the damping-time-to-period ratios τ/Pfor temporally damped modes are nearly identical\nto the damping-length-to-wavelength ratios LD/λfor spatially damped ones if the wave dispersion is\nweak(seee.g., Terradas et al.2010 ,Equation40). InviewoftheextremelyweakdispersionofSSKMs,\nthis means that the following discussions apply to SSKMs that experie nce either temporal or spatial\ndamping. For the ease of discussion, let us slightly generalize Equatio n (15) by counting all possible\nphysical quantities that may influence the dispersive properties of SSKMs in our framework. Some\nstraightforward dimensional analysis yields that the complex-value d frequency ωcan be expressed\nas\nωR\nvAi=H/parenleftbiggcsi\nvAi,cse\nvAi,vAe\nvAi,l\nR,Rm;kR/parenrightbigg\n, (39)22 Chen et al.\nwhere the function His dictated by the eigen-value problem. Note that the arguments on the RHS\nof Equation ( 39) are all dimensionless, and are independent from one another. One may further\nformally express τ/P(or equivalently LD/λ) as\nτ\nP=L/parenleftbiggcsi\nvAi,cse\nvAi,vAe\nvAi,l\nR,Rm;kR/parenrightbigg\n. (40)\nTo proceed, let us assume that the set of [ csi/vAi,cse/vAi,vAe/vAi] is measurable. Our Figure 10then\nindicates that τ/Pcannot be lower than the asymptotic value ( τ/P)min,asympthat the green curve\nattains at large kR, provided that resonant absorption is the sole damping mechanism. Conversely,\nthe wave damping cannot be attributed to resonant absorption if τ/Pis observed to be smaller than\n(τ/P)min,asymp, which depends only on [ csi/vAi,cse/vAi,vAe/vAi]. In this case, one is allowed to rule\nout resonant absorption even without knowing any quantity in the s et [l/R,R m,kR]. The situation\nis much more involved when the observed τ/Pexceeds ( τ/P)min,asymp, and one needs to survey the\ncombinations of [ l/R,R m,kR] (see Figure 2). However, it remains possible to say a few words on\nthe likely damping mechanism(s). For this purpose, we further assu me thatl/RandkRare known.\nNote that it is non-trivial to evaluate the electric resistivity ( η) in a photospheric medium (e.g.,\nWang 1993 ;Khomenko & Collados 2012 ;Ni et al. 2016 ). Nonetheless, one can rather safely assume\nthat the resulting Rmlies in the range pertaining to the shaded area in Figure 2, given that this\nrange encompasses the rather extreme values of the waveguide r adius (R). With a known set of\n[csi/vAi,cse/vAi,vAe/vAi,l/R;kR], one may construct the Rm-dependence of τ/P, thereby finding the\nrange of τ/Pallowed by the range of Rm. Now two situations arise. If the observed τ/Plies in\nthis range, then one is allowed to constrain the possible Rm. If the measured τ/Pis outside this\nrange, then the wave damping can be attributed to neither the Ohm ic resistivity nor the resonant\nabsorption. To name but one additional mechanism, we note that we have adopted the values for\nthe electric conductivity from Kovitya & Cram (1983). As such, only the Ohmic resistivity due to\nelectron-neutral and electron-ion collisions was addressed. In re ality, the Cowling resistivity due to\nion-neutral collisions may be substantially stronger (e.g., Wang 1993 , Figure 1). A detailed study on\nthe effects of the Cowling resistivity on photospheric modes is certa inly warranted, but is nonetheless\nleft for a future work.\nWe thank both referees for their comments, which helped improve t he manuscript substantially.\nThis research was supported by the National Natural Science Fou ndation of China (BL: 41674172,\n11761141002, 41974200; SXC:41604145, HY:41704165). 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Thes e profiles are generated by specifying the\ntransverse distributions of the adiabatic sound ( cs) and cusp ( cT) speeds, for which a continuous transition\nlayer (TL) connects a uniform interior to a uniform exterior (see Equation 3). This TL is located between\nri=R−l/2 andre=R+l/2, where Ris the mean waveguide radius and lthe TL width. For illustration\npurposes, here l/Ris arbitrarily chosen to be 0 .2.Waves in photospheric waveguides 27\nFigure 2. Dependence on the magnetic Reynolds number ( Rm) of the dispersive properties of slow surface\nkink modes in a photospheric waveguide. Plotted are (a) the d amping-time-to-period ratio τ/P, (b) the\naxial phase speed ωR/k, and (c) the damping rate ωIin units of kvAi. A number of combinations of the\ntransition layer width ( l/R) and axial wavenumber ( kR) are examined as labeled. The area shaded yellow\ncorresponds to the range of Rmthat is likely to be relevant for pores and sunspot umbrae. Se e the text for\ndetails.28 Chen et al.\nFigure 3. Dependence on the magnetic Reynolds number ( Rm) of the terms involved in the energy balance\n(see Equation 28and the associated definitions). A number of combinations of the transition layer width\n(l/R) and axial wavenumber ( kR) are examined as labeled. The area shaded yellow correspond s to the range\nofRmthat is likely to be relevant for pores and sunspot umbrae. Th e diamonds represent −2ωI, taken from\na small subset of the values presented in Figure 2c. See the text for details.Waves in photospheric waveguides 29\nFigure 4. Radial distributions of the ¯ sresterm for a number of representative values of the magnetic\nReynolds number ( Rm) at a given pair of [ l/R,kR] = [0.1,0.7]. Here ¯ sresis evaluated by applying the\nbar operation (Equation 27) to the definition of sres(Equation 22). The shaded area represents where the\nequilibrium quantities are nonuniform. See the text for det ails.30 Chen et al.\nFigure 5. Dispersive properties of resonantly damped slow surface ki nk modes, shown by the dependence\non the dimensionless layer width ( l/R) of (a) the damping-time-to-period ratios ( τ/P), (b) the axial phase\nspeeds (ωR/k), and (c) the damping rates ( ωI) in units of kvAi. A number of axial wavenumbers ( kR) are\nexamined as represented by the different colors. The results f rom self-consistent resistive computations are\ngiven by the solid curves. For each pair of [ l/R,kR], the explicit dispersion relation (Equation 16) in the\nthin-boundary limit is also solved. The dashed curves repre sent the computations where both the cusp and\nAlfv´ en resonances are accounted for, whereas the dotted cu rves take into account only the cusp resonance.Waves in photospheric waveguides 31\nFigure 6. Radial distributions of the real part of the Fourier amplitu de of the radial flow speed (Re˜ vr) in\nthe intervals embracing (a) the cusp, and (b) the Alfv´ en dis sipative layers. A number of magnetic Reynolds\nnumbers ( Rm) are examined as labeled, whereas [ l/R,kR] is fixed at [0 .2,2]. The eigenfunctions are rescaled\nsuch that ˜ ptot= 1 atr= 1.25R.32 Chen et al.\nFigure 7. Dependenceonthetransitionlayer width( l/R)oftheratiosofthehattedtermsinEquation( 30).\nThe axial wavenumber is fixed at kR= 2. For a given l/R, a number of magnetic Reynolds numbers ( Rm)\nare examined to evaluate the hatted terms as represented by t he symbols of different colors. With solid\ncurves we connect the symbols representing ˆFc,ˆFA, andˆSidealfor the same Rmbut different l/R. The\ndashed curve represents −2ωI, the values of which are direct outputs of our code. See the te xt for details.Waves in photospheric waveguides 33\nFigure 8. Radial profilesof somerelevant eigen-functions foundinse lf-consistent resistive computations for\nresonantly damped slow surface kink modes. Two different valu es for the axial wavenumber are examined,\none corresponding to kR= 0.3 (the left column) and the other to kR= 2 (right). The vertical dash-\ndotted lines mark the boundaries of the transition layer, wh ich corresponds to a layer-width-to-radius-ratio\nofl/R= 0.2. The black (red) curves correspond to the real (imaginary) parts. The top row pertains to\nthe Eulerian perturbation to total pressure (˜ ptot), while the rest correspond to the radial (˜ vr), azimuthal\n(˜vθ), and axial (˜ vz) speeds. The eigen-functions are normalized such that ˜ ptot= 1 where its modulus attains\nmaximum. Different values for the magnetic Reynolds number ( Rm) are adopted in the two columns, see\nthe text for details.34 Chen et al.\nFigure 9. Similar to Figure 7except that the dependence on the axial wavenumber ( kR) is examined.\nHere the dimensionless transition layer width is fixed at l/R= 0.2.Waves in photospheric waveguides 35\nFigure 10. Dependenceontheaxial wavenumber( k)oftheoptimaldamping-time-to-periodratio( τ/P)min\nand thetransition-layer-width-to-radius ratio ( l/R)minwheretheoptimal dampingis reached. Thevalues for\n(τ/P)minand (l/R)minare derived from self-consistent resistive computations. At an (l/R)minthus derived,\nthe explicit dispersionrelation in the TB limit (Equation 16) is also solved, yielding ( τ/P)TBfor comparison.\nSee the text for details." }, { "title": "1805.08733v3.Uniqueness_of_the_Cauchy_datum_for_the_tempered_in_time_response_and_conductivity_operator_of_a_plasma.pdf", "content": "arXiv:1805.08733v3 [math.AP] 21 Dec 2022UNIQUENESS OF THE CAUCHY DATUM FOR THE\nTEMPERED-IN-TIME RESPONSE AND CONDUCTIVITY\nOPERATOR OF A PLASMA\nOLIVIER LAFITTE1,2AND OMAR MAJ3,4\nAbstract. We study the linear Vlasov equation with a given electric fiel d\nE∈ S, where Sis the space of Schwartz functions. The associated damped\npartial differential equation has a unique tempered solutio n, which fixes the\nneeded Cauchy datum. This tempered solution then converges to the causal\nsolution of the linear Vlasov equation when the damping para meter goes to\nzero. This result allows us to define the plasma conductivity operator σ, which\ngives the current density j=σ(E) induced by the electric field E. We prove\nthatσis continuous from Sto its dual S′. We can treat rigorously the case\nof uniform non-magnetized non-relativistic plasma (linea r Landau damping)\nand the case of uniform magnetized relativistic plasma (cyc lotron damping).\nIn both cases, we demonstrate that the main part of the conduc tivity operator\nis a pseudo-differential operator and we give its expression rigorously. This\nmatches the formal results widely used in the theoretical ph ysics community.\nContents\n1. Introduction 2\n1.1. Known results on existence and uniqueness 3\n1.2. Framework of this paper 5\n1.3. Main results 1: the non-relativistic, one-dimensional case 7\n1.4. Main results 2: relativistic, three-dimensional case 8\n1.5. Concluding remarks and structure of the paper 10\n2. Characterization of the response operator: a simple case stud y 11\n3. Uniform isotropic plasmas in one spatial dimension 12\n3.1. Solution of the linearized Vlasov equation 13\n3.2. Current density and conductivity operator 16\n3.3. Proofs for the results of section 1.3 20\n4. PDE of the form −(u1∂u2−u2∂u1)ϕ(θ,u)−ia(θ,u)ϕ(θ,u) =ψ(θ,u) 20\n4.1. Equation with a polynomial source term 21\n4.2. Equation with source term in Sand uniqueness in S′. 23\n5. Response of a uniform magnetized plasma 30\n5.1. Notation 30\n5.2. The roots of a0−nand limε→0+(1/sinπaε) 32\n5.3. Solution of the linear Vlasov equation for the magnetized case 36\n5.4. Current density and conductivity operator 40\n5.5. Proof of the results of section 1.4 44\n6. Stationary and non-stationary phase results for some integra ls 45\nAppendix A. Notation and basic definitions 53\nAppendix B. Proofs for the case study of section 2 54\n12 O. LAFITTE AND O. MAJ\nAppendix C. Causal solutions of linear kinetic equations 57\nAppendix D. The Hilbert transform and its action on symbols 60\nAppendix E. A useful linear algebra result 63\nAcknowledgments 66\nReferences 66\n1.Introduction\nA plasma is a collection of a sufficiently large number of electrically charg ed\nparticles of various species (electrons, protons, and ions of differ ent elements), sub-\nject to electromagnetic fields. In kinetic theory, the configuratio n of a plasma is\nspecified by a family of functions fs:R×R3×R3→R+, labeled by the index of\nparticle species sand defined so that fs(t,x,p) gives the density of particles of the\nspeciessat the time t, positionxand relativistic momentum p.\nThe equations governing the evolution of the distribution functions {fs}s, to-\ngether with the electric field E:R×R3→R3and the magnetic field B:R×R3→\nR3, are given by the relativistic Vlasov-Maxwell-Landau system, which w rites\n(1)\n\n∂tfs+vs·∇xfs+qs/parenleftbig\nE+vs×B/c/parenrightbig\n·∇pfs=Cs({fs′}s′),\n∂tE−c∇×B=−4π/summationdisplay\nsqs/integraldisplay\nR3vs(p)fs(t,x,p)dp,\n∂tB=−c∇×E,\n∇·B= 0,\n∇·E= 4π/summationdisplay\nsqs/integraldisplay\nR3fs(t,x,p)dp,\nwhere the relativistic velocity vsis defined by vs(p) =p/[msγs(p)], withγs(p) =/parenleftbig\n1+p2/m2\nsc2/parenrightbig1/2,Csis the relativistic Landaucollisionoperator[6, 13] and depends\non{fs′}s′,cis the speed of light, msis the mass of particles of the species s, andqs\nis their electric charge (c.g.s. units are used throughout the paper ).\nHowever, a variety of reduced models are also in use for modeling plas mas and\ngasesinspecialcases. Forinstance, atmoderateenergiestheno n-relativisticversion\nis used, which follows from (1) by setting γs(p) = 1, so that vs(p) =p/ms, and by\nreplacingCswith the non-relativistic Landau collision operator. When the collision\noperatorCscan be neglected, one recovers the Vlasov-Maxwell system both in the\nrelativisticandnon-relativisticversions. The Vlasov-Maxwellsyste mcanbe further\nreduced, when all effects of the magnetic field can be neglected; th en Maxwell’s\nequations are replaced by the Poisson equation for the electrosta tic potential φ,\nE=−∇φ, and the Vlasov-Maxwell system reduces to the Vlasov-Poisson sy stem.\nFor electrically neutral particles (a gas), qs= 0, the electromagnetic part of the\nsystem can be dropped and the collision operator Csis given by the Boltzmann\noperator. This givesthe Boltzmannequation[16]. Onecanalsoreplac ethe collision\noperator by simpler models, such as the BGK (Bhatnagar, Gross an d Krook [8])\noperator, leading to the BGK kinetic model [16].\nThe cases mentioned above are just some of the most common kinet ic models for\nplasmas and gases but other “combinations” of self-consistent fo rces and collisionTEMPERED-IN-TIME RESPONSE OF A PLASMA 3\noperators are also considered. The literature on kinetic models is va st and has\nmany applications. We shall not attempt to give a review here.\nIn this paper we are particularly interested in applications to the stu dy of high-\nfrequency electromagnetic waves in high-temperature plasmas. F or such problems,\nrelativistic effects have to be accounted for (at least for the elect rons), but collisions\ncanbeneglected,i.e. Cs= 0,sincethetimescaleofinterestismuchshorterthanthe\ncollision time. In addition, the wave is a small perturbation of the elect romagnetic\nfields of the plasma so that a formal linearization of the Vlasov equat ion can be\nphysically justified. As a result, the linearized relativistic Vlasov-Max wellsystem is\nthe physically appropriate model for such applications. Simpler mode ls such as the\nlinearized non-relativistic Vlasov-Maxwell or the linearized Vlasov-Po isson system,\nevenin reduced dimension d<3, can be interestingas well, for fundamental plasma\ntheory.\nIt is howeverworth starting from an overview of the mathematical results for the\nnon-linearmodels, inordertounderstandtheexpectedregularity ofthedistribution\nfunctions and the electromagnetic field. The mathematical works o n such a large\nclass of models focus in particular on the associated Cauchy problem [25, 21, 41,\nand references therein].\n1.1.Known results on existence and uniqueness. For the non-relativistic\nVlasov-Maxwell system, i.e. equation (1) with γ= 1 andCs= 0, Wollman [56]\nobtained a localexistence and uniqueness result: for a single-spec iesnon-relativistic\nplasma and with initial data in Hs,s≥5, and with compactly supported initial\nparticle distribution, there is a time T >0 depending on the initial conditions such\nthat a unique solution f∈C/parenleftbig\n[0,T],Hs(R6)/parenrightbig\n∩C1/parenleftbig\n[0,T],Hs−1(R6)/parenrightbig\nexists. Later\nAsano and Ukai [3], Degond [19], and Schaeffer [53] independently pro ved similar\nresults on the local existence and uniqueness of a solution ( fs,E,B) with lifespan\nindependent of the speed of light c. The fact that the lifespan is independent of c\nallowed them to take the limit c→ ∞and to show convergence to a solution of the\nVlasov-Poisson system. E.g., Degond showed the local existence an d uniqueness of\nthe solution in Sobolev spaces Hswiths≥3; specifically, if the initial data are\ninH3, and the initial distribution function is non-negative, f0\ns≥0, and compactly\nsupported in velocity, Degond has shown that there exist a time T >0, depending\non the initial data but not on the speed of light c, and a solution ( fs,E,B) in\nL∞(0,T;H3(R6)×H3(R3)×H3(R3)/parenrightbig\n. Wollman [57] improved his earlier result:\nagain for a single-species non-relativistic plasma, he showed local ex istence of a C1\nsolutionwithinitialdata f0∈C1\n0(R6)andE0,B0∈H3(R3); specificallythereis T >\n0 depending on the initial condition such that there exists a solution f∈C1([0,T]×\nR6) which is unique as an element of L1/parenleftbig\n0,T;H3(R6)/parenrightbig\n∩C/parenleftbig\n[0,T],H2(R6)/parenrightbig\n.\nFor the relativistic Vlasov-Maxwell system with multiple species Glasse y and\nStrauss [27] proved existence of a unique global solution in C1(R×R3×R3) with\ninitial data f0\ns∈C1\n0(R3×R3),E0,B0∈C2(R3) under the a priori assumption\nthat a solution fsis compactly supported in momenta and the radius of the sup-\nport is bounded by a continuous function of time. A simpler proof was given by\nBouchut, Golse and Pallard [12], and a variant of this result has been proposed by\nKlainerman and Staffilani [36]. Existence and uniqueness of a global C1solution\nfort∈[0,+∞), (x,p)∈R6has been shown by Glassey and Schaeffer [26] with\ncompactly supported initial data satisfying appropriate conditions that require, in\nparticular, the initial distribution function and electromagnetic field s to be small4 O. LAFITTE AND O. MAJ\ninC1andC2norms respectively. We note also the results on global existence wit h\nsmall data for the relativistic Vlasov-Maxwell system obtained by Ho rst [32]. The\nkey observation is that the decay of /ba∇dblE(t)/ba∇dbl2\nL2+/ba∇dblB(t)/ba∇dbl2\nL2fort→+∞completely\ndetermine the electromagnetic field, without initial conditions. Hors t makes use of\na fixed-point argument to show global existence of solutions: given the electromag-\nnetic fields ( E,B) in a suitable class of functions, he constructs the characteristic\nflowforthekineticequation,fromwhichhecomputesthechargean delectriccurrent\ndensities that generate new electromagnetic fields ( E′,B′). This defines an operator\nQ: (E,B)/ma√sto→(E′,B′) which is a contraction if the initial distribution f0and its\nderivatives are small enough. In this construction, the field ( E′,B′) is obtained as\nthe solution of Maxwell’s equations with the condition /ba∇dblE(t)/ba∇dbl2\nL2+/ba∇dblB(t)/ba∇dbl2\nL2→0 for\nt→+∞, [32, definition 2.6 and 3.5].\nGlobal existence of weak solution without small data assumption is du e to\nDi Perna and Lions [22]: for the non-relativistic case and with one par ticle species,\ngiven initial data f0∈L1∩L2(R3×R3) andE0,B0∈L2(R3) with the conditions\nf0≥0 and/integraldisplay\nR3×R3|v|2f0dxdv<+∞,\nthey prove existence of\nf∈L∞/parenleftbig\n0,+∞;L1(R3×R3)/parenrightbig\n,E,B∈L∞/parenleftbig\n0,+∞;L2(R3)/parenrightbig\n,\nthat satisfythe non-relativisticVlasov-Maxwellsystem in the sens e ofdistributions.\nTheconditionsonthedataarethenaturalonessince f(t,·,·)isaphase-spacedensity\nof particles (and thus must be non-negative and in L1) and the quantity\nE=1\n2m/integraldisplay\nR3×R3|v|2fdxdv+1\n8π/parenleftbig\n/ba∇dblE/ba∇dbl2\nL2+/ba∇dblB/ba∇dbl2\nL2/parenrightbig\n,\nisthetotalenergyofthesystem. DiPernaandLionshaveshownt hatforsuchglobal\nweak solutions, one has E(t)≤ E(0), that is, they are finite-energy solutions. The\nkey idea of this proof is the use of renormalized solutions [21, 41], and this idea has\nbeen applied to the Boltzmann equation [23] as well. More recently, th e relativistic\nversion has been addressed by Rein [51], while time-periodic weak solu tions in\nbounded (in space) domains have been considered by Bostan [11].\nAs for the Vlasov-Poisson system the study of the Cauchy problem developed\nalong the same lines, moving from local-in-time classical solutions up to global\nweak solutions [26, 21, and references therein]. Particularly, the w orks by Asano\nand Ukai, Degond, and Schaeffer cited above give results on the exis tence and\nuniqueness of local solutions the Vlasov-Poisson system. There ar e however earlier\nresults on local solutions [30, 31] and on global weak solutions [2, 3 3]. Existence of\na globalC1solutions with small initial data has been established by Bardos and\nDegond [5]. The first results on global classical solutions is due to Pfa ffelmoser\n[47], where “classical” here means that the characteristics system for the Vlasov\nequation has a unique classical solution and fis constant along the characteristics.\nAll these results are for the fully nonlinear problem. In this work, ho wever, we\naddress the linearized problem and we focus specifically on the assoc iated linear\nkinetic equation. Wollman [57, section 3] reports the classical result s on the exis-\ntence ofC1solutions for such linear problems, the proof of which is based on the\nstandard method of characteristics. Specifically, if the electric an d magnetic fields\nE,Bare inC/parenleftbig\n[0,T];H3(R3)/parenrightbig\nwithH3-norm bounded uniformly in time, the linearTEMPERED-IN-TIME RESPONSE OF A PLASMA 5\nnon-relativistic equation with initial datum f0∈C1\n0(R3×R3) has a unique classical\nsolution f∈C1([0,T]×R3×R3). HereTis the life-span of the fields and does not\ndepend on the initial distribution.\n1.2.Framework of this paper. In this paper, we consider a given stationary\nconfiguration {Fs,0(x,p)}sof the particle distribution functions, with zero electric\nfieldE0= 0, and a constant magnetic field B0. Then we address the linearized\nsystem around the stationary solution ( {Fs,0}s,E0,B0). The associated unknowns\nare the linear perturbations ( {fs}s,E,B), wherefsis the perturbed distribution\nfunction for the particle species s, whileEandBare the perturbations of electric\nand magnetic field, respectively.\nSince we consider a linearized problem, the solution for fsis not necessarily non-\nnegative, but Fs,0+fs≥0 (essentially, if Fs,0+fsfails to be non-negative, we no\nlonger are in the linear regime). We need to keep the assumption on th e existence\nof the velocity moments of fsand theL1-in-xbehavior (or L1\nlocin the idealized case\nof a plasma with an infinite number of particles, e.g., a uniform plasma ov er the\nwhole space).\nWe could expect to need and to be able to consider given arbitrary init ial data.\nOur aim howeveris characterizingand determining what is referredt o as thedielec-\ntric response of the plasma in the physics literature [14, 54, 38, 10]. The underlying\nphysical idea is that the application of a small-amplitude electromagne tic perturba-\ntion determines a small change in the distribution functions fs, which represents the\nresponse of the plasma to the imposed electromagnetic disturbanc e. It is similar\nto the construction of the operator Qin Horst’s fixed-point argument mentioned\nabove. In addition however, physical reasoning suggests that th e response of the\nplasma should be uniquely determined by and depend continuously on t his imposed\nperturbation, hence there should be no need to prescribe a Cauch y datum.\nThe evolution of a small perturbation fsinduced by an externally imposed small\nelectric field disturbance Eis governed by the linear relativistic Vlasov equation\n[9, 14, 35, 54],\n(2)∂tfs+vs·∇xfs+qs/parenleftbig\nvs×B0/c/parenrightbig\n·∇pfs=−qs/parenleftbig\nE+vs×B/c/parenrightbig\n·∇pFs,0,\nwhere the electric field Eof the disturbance is given, e.g., E∈[S(R4)]3, and the\nmagnetic field depends linearly on the electric field via the Faraday law,\n(3) ∂tB+c∇×E= 0.\nSinceEis given, this is a system of partial differential equations for ( fs,B) which\nreads/braceleftigg\n∂tfs+vs·∇xfs+qs(vs×B0/c)·∇pfs=−qs/parenleftbig\nE+vs×B/c/parenrightbig\n·∇pFs,0,\n∂tB=−c∇×E.\nFrom this system, by introducing the new unknown gs=∂tfs, one deduces the\ndecoupled system\n(4)/braceleftigg\n∂tgs+vs·∇xgs+qs(vs×B0/c)·∇pgs=−qs/parenleftbig\n∂tE−vs×∇×E/parenrightbig\n·∇pFs,0,\n∂tB=−c∇×E.\nThe associated homogeneous equation for gsis\nH(gs):=∂tgs+vs·∇xgs+qs/parenleftbig\nvs×B0/c/parenrightbig\n·∇pgs= 0.6 O. LAFITTE AND O. MAJ\nWe shall see that if we impose a control of the growth at t→ ±∞, the modified\nequation H(gs,ν) +νgs,ν= 0 for any ν >0 has the unique solution gs,ν= 0. This\nis similar to scattering theory [39], in which the scattered field is deter mined by\nconditions at infinity. The introduction of the damping term νgs,νis analogous\nto the limiting absorption principle [39, 24, 52] for elliptic equations of t he form\nAu= (λ+iε)u+fforε→0+(resp.ε→0−), where one constructs two resol-\nvents for the elliptic operator A. A classical application of the limiting absorption\nprinciple consists in selecting the unique outgoing-wave solution of th e Helmholtz\nequation. Other examples include the solution of elliptic equations, as well as the\nextensionoftheresolventoftheoperator −∆+V[24,45,15, andreferencestherein].\nRecently the idea of using scattering theory for the linearized Vlaso v-Maxwell sys-\ntem has been developed by Despr´ es in order to prove linear Landau damping for\ninhomogeneous equilibrium distributions [20]. In this paper, however, we shall not\ntake advantage of scattering theory, but rather focus on the s olution selected by\nthe growth conditions at infinity in time and the limiting absorption princ iple.\nThe main idea here is to apply the limiting absorption principle to either th e\ninhomogeneous problem (2) or (4). We shall see that for ν >0, there is a unique\ntempered solution, which has a limit for ν→0+, and the limit itself is a tempered\nsolution of either (2) or (4) without damping. We will find that the limit a mounts\nexactly to the solution which is referred to as the causal solution in the physics\nliterature and which describes the response of the plasma to the imp osedE.\nFinding the response of a system is classical. For example, in the mode ling of\nelectric circuits with capacitance Cinductance Land resistance R, the electric\nchargeq:t/ma√sto→q(t) satisfies the ordinary differential equation\nLC¨q+RC˙q+q=CU0cos(ωt),\nwhich leads to the response\nq(t) = Re/bracketleftigCU0eiωt\n1−LCω2+iωRC/bracketrightig\n,\neven though one has infinitely many solutions (depending on Cauchy d ata).\nReturning to the linearized Vlasov equation, one can also envisage th e use of\nother dissipation mechanisms such as the Fokker-Planck operator ν∇v·(∇vf−vf)\nor the collision operators mentioned above, but this is not addresse d here.\nHaving identified the unique solution of the linearized Vlasov equation w hich\ndescribes the response of the plasma to the imposed electric field dis turbance, the\ncorresponding unique perturbation of the electric current densit y is defined by\n(5) j(t,x):=/summationdisplay\nsqs/integraldisplay\nR3vs(p)fs(t,x,p)dp,\nwhich requires that the solution fsis at least in L1(the relativistic velocity is\nbounded by the speed of light, |vs(p)|< c). We shall generalize such integrals to\nthe case of distributional solutions and see that, since fsdepends linearly on E, the\ninduced current density jis given by the action of a linear operator on the electric\nfieldE, namely,\n(6) j=σ(E).\nThis is referred to as the linear constitutive relation of the plasma an d the operator\nσis the conductivity operator. Equation (6) is also referred to as (g eneralized)\nOhm’s law.TEMPERED-IN-TIME RESPONSE OF A PLASMA 7\nA precise mathematical analysis of the response of a plasma is import ant be-\ncause it is the basis for the construction of constitutive relations f or linear plasma\nwaves, the simplest of which being the Ohm’s law. Together with Maxwe ll’s equa-\ntions, it determines the linear wave equation describing plasma waves [10]. The\nsame problem has been considered by Omnes for a bounded plasma [46 ]. More\nrecently, Cheverry and Fontaine [18, 17] have addressed the ch aracteristic variety\n(or dispersion relation) for the linearized Maxwell-Vlasov system usin g asymptotic\nmethods, but here we focus on the properties of the plasma const itutive relation as\nan operator.\nWe carry out this ideas for the one-dimensional non-relativistic linea rized Vlasov\nequation without background magnetic field (non-magnetized) and for the three-\ndimensional relativistic linearized Vlasov equation with uniform backgr ound and\nwith constant magnetic field.\n1.3.Main results 1: the non-relativistic, one-dimensional cas e.We con-\nsider first the case of a non-magnetized, non-relativistic plasma in o ne dimension in\nspace and velocity. We also restrict ourselves to the case of a single particle species\n(and thus drop the index s). This last simplification does not imply any further\nloss of generality as the current density is the sum of the currents carried by the\nindividual species. With background distribution F0∈ S(R) depending on v∈R\nonly, andE∈ S(R2), we consider the linear kinetic equation\n(7) Lf:=∂tf+v∂xf=−q\nmEF′\ns,0,\nwhich is a reduced version of equation (2). The damped form is\n(8) Lνfν:=∂tfν+νfν+v∂xfν=−q\nmEF′\n0.\nIffis a generic distribution (not necessarily a solution of (7)) with finite fi rst\nvelocity moment, i.e., with v/ma√sto→vf(t,x,v) inL1for all (t,x), we define the operator\n(9) J(f)(t,x):=q/integraldisplay\nRvf(t,x,v)dv,\nwhich gives the associated current density, cf. equation (5), in th e one-dimensional,\nnon-relativistic case.\nTheorem 1.1. LetF0∈ S(R)be given.\n(i)Ifν >0, for anyE∈ S(R2)equation (8) has a solution fν∈ S(R3)which\nis unique as an element of S′(R3), andjν=J(fν)∈ S(R2).\n(ii)Forν→0+,fνandjνhave pointwise limits f∈C∞\nb(R3)andj∈C∞\nb(R2),\nrespectively; in addition, fν→f, andjν→jalso in the topology of S′.\n(iii)The limitfis a solution of equation (7) and j=J(f).\nBy the limiting absorption principle of theorem 1.1, to each E∈ S(R2) we thus\ncan associate a unique f, and thus a unique current density j. The conductivity\noperator is then defined as the map\n(10) σ:S(R2)∋E/ma√sto→j∈C∞\nb(R2)⊂ S′(R2).\nFor anyχ∈C∞\n0(R) withχ= 1 in a neighborhood of zero, we also introduce the\nfollowing Fourier multiplier (cf. appendix A for definitions and notation s)\n(11) F/parenleftbig\nσ1−χ(E)/parenrightbig\n(ω,k) =/parenleftbig\n1−χ(k)/parenrightbig\nˆσph(ω,k)ˆE(ω,k),8 O. LAFITTE AND O. MAJ\nwhereFdenotes the Fourier transform and ˆ σphis the physical conductivity tensor\n(explicit formula given in equation (33)).\nTheorem 1.2. The mapσdefined by equation (10) is linear and continuous from\nS(R2)→ S′(R2)and for every χ∈C∞\n0(R)withχ= 1near zero,σ(E) =σ1−χ(E)\nfor allEsatisfying ˆE(ω,k) = 0ifk∈suppχ.\nRemark 1.\n(1) The limit festablished in theorem 1.1 coincides with the causal solution\nof (7), which is reviewed in appendix C.\n(2) Expressions for the solutions fν,f, their Fourier transforms, the associated\ncurrents, and the operators σνandσare given in section 3.\n(3) The operator σχ:=σ−σ1−χis well defined and σχ(E) = 0 when ˆE(ω,k) =\n0 forksmall. Expressions of both σandσχare available (see proposi-\ntion 3.5).\n(4) Theorem 1.1 can be straightforwardly generalized to the case o f a non-\nmagnetized non-relativistic plasma with a spatially non-homogeneous equi-\nlibrium distribution of the form F0(x,v) =n0(x)˜F0(v), for which the veloc-\nity distribution is the same at any point in space. For such equilibria, σ1−χ\nis a pseudo-differential operator. The expression of the symbol is obtained\nin section 3.2, remark 7.\n1.4.Main results 2: relativistic, three-dimensional case. The second case\nunder consideration is the relativistic Vlasov equation with a uniform b ackground\nmagneticfield, thatis, B0inequation(2)istakenconstantandnon-zero. Wechoose\nB0=|B0|e/bardbldirected along the third axis of a Cartesian frame {e1,e2,e3=e/bardbl}. It\nis natural to normalize the relativistic momentum ptomsc, and thus to introduce\nnormalized momentum variables\n(12) u:=p/(msc)u⊥:= (u2\n1+u2\n2)1/2, u /bardbl:=u3.\nWe define the relativistic cyclotron frequency for the considered s pecies,\n(13) Ω s(u):=1\nγ(u)qs|B0|\nmsc= sgn(qs)ωc,s\nγ(u)>0,\nwhich has the sign of the charge qsand depends on u2throughγ(u) = (1+u2)1/2,\nwhereas the classical cyclotron frequency ωc,s:=|qsB0|/(msc)>0 is a positive\nconstant.\nThe background distribution functions are taken uniform and gyrotropic , i.e.,\nFs,0is constant in time tand spacexand depends only on u/bardbl,u⊥, namely,\n(14) Fs,0(t,x,p) =ns,0\n(msc)3Gs(u/bardbl,u⊥),\nwherens,0>0 is the constant background particle density and Gsis such that\nu/ma√sto→Gs(u3,(u2\n1+u2\n2)1/2/parenrightbig\nbelongs to S(R3). Such property is usually satisfied\nby the background distribution functions of practical interest. T he momentum\ndistribution Gshas unit norm in L1(R×R+,2πu⊥du/bardbldu⊥).\nInsteadofaddressingequation(2)directly,weconsiderthekinet icequationin(4)\nforgs=∂tfs, which amounts to\n(15) Vsgs=−qs/parenleftbig\n∂tE−vs×∇×E/parenrightbig\n·∇pFs,0,TEMPERED-IN-TIME RESPONSE OF A PLASMA 9\nwhere\n(16) Vs=∂t+vs·∇x+sgn(qs)ωc,s\nγ(u×e/bardbl)·∇u,\nwithvs=p/(msγs) =cu/γ(u) being the relativistic velocity as a function of the\nnormalized momentum.\nWe add to equation (15) a damping term, with the idea of applying the lim it-\ning absorption principle. Because of the u-dependent relativistic factor ωc,s/γthe\ndamping coefficient will be multiplied by γafter Fourier transform in time and\nspace of equation (15). It is therefore convenient to allow the dam ping coefficient\nνsto depend on the species sand on momentum ufrom the beginning, subject to\nthe conditions\n(17)\n\nνs∈C∞(R3),\nthere exists ν0>0 :γ(u)νs(u)≥ν0,\n(u1∂u2−u2∂u1)νs(u) = 0,\nand there exists m∈R:|∂α\nuνs(u)| ≤Cα(1+u2)m,∀u∈R3,∀α∈N3\n0,\nwhereCα∈Rare constants depending only on the order αof the derivatives. For\nexample, if νs∈C∞\nb(R3) is a function of u2\n1+u2\n2andu3only, then conditions (17)\nare fulfilled with m= 0. We shall see that the dissipation-less limit is independent\nof the choice of this damping function.\nForε >0 and for any function νs, satisfying conditions (17), we consider the\nregularized equation\n(18) Vs,εgs,ε=−qs/parenleftbig\n∂tE−v×∇×E/parenrightbig\n·∇pFs,0,\nwhereVs,ε=Vs+ενs.\nIfgsare generic distributions with finite first velocity moment, i.e., gs(t,x,·)∈\nL1(R3) for every ( t,x), we define\n(19) K({gs})(t,x):=/summationdisplay\nsqs(msc)3/integraldisplay\nR3vs(u)gs(t,x,u)du,\nwhich gives the time-derivative of the current (5) when gs=∂tfs. For this model,\nthe limiting absorption principle parallels theorem 1.2.\nTheorem 1.3. LetFs,0∈ S(R3)be given uniform gyrotropic distribution functions\nand letνsbe any function satisfying conditions (17).\n(i)Ifε >0, for anyE∈[S(R4)]3equation (18) has a solution gs,ε∈ S(R7)\nwhich is unique as an element of S′(R7),∂tjε=K({gs,ε})∈[S(R4)]3.\n(ii)Forε→0+,gs,εand∂tjεhave limits gsand∂tjinS′, independent of νs.\n(iii)The limitgsbelongs toC∞\nb(R7)and is a classical solution of equation (15);\nin addition gsbelongs to the domain of Kand∂tj=K({gs})∈C∞\nb(R4).\nTherefore we can define the map\n(20) ς: [S(R4)]3∋E/ma√sto→∂tj∈[S′(R4)]3.\nWe shall see that it can be represented by a Fourier multiplier if we exc lude the\nhyperplane ω= 0 in Fourier space. For any cut-off function χ∈C∞\n0(R) withχ= 1\nin a neighborhood of zero, we define the Fourier multiplier ς1−χby\n(21) F(ς1−χ(E))(ω,k) =/parenleftbig\n1−χ(ω)/parenrightbig\nˆς0(ω,k)ˆE(ω,k),10 O. LAFITTE AND O. MAJ\nwhere ˆς0(ω,k) is the limit established in proposition 5.10. An expression for ˆ ς0is\ngiven in equation (81), and proposition 5.10 establishes that ˆ ς0is continuous for\nω/\\e}atio\\slash= 0 andC∞whereω2/\\e}atio\\slash=c2k2+n2ω2\nc,sfor alln∈Zand all species s.\nTheorem 1.4. The mapςdefined in (20) is continuous and for any χ∈C∞\n0(R)\nwithχ= 1in a neighborhood of zero, ς(E) =ς1−χ(E)ifˆE(ω,k) = 0forω∈suppχ.\nRemark 2.\n(1) The hypothesis that νsis in the kernel of the operator u1∂u2−u2∂u1ex-\npresses the fact that νsmust be gyrotropic.\n(2) Thesolution gsistheuniquecausalsolutionof(15)(defined inappendixC).\n(3) We also have pointwise convergence of gs,ε→gs.\n(4) Existenceanduniquenessofthesolution gs,εisestablishedviaFouriertrans-\nform, while the causal solution is obtained by integration along the ch arac-\nteristics, cf. appendix C. Hence the proof of theorem 1.3 (iii) estab lishes a\nlink between the formulations in Fourier and physical variables.\n(5) An explicit expression of the linear operator ςvalid without restriction of\nthe support of Eis given below in proposition 5.9.\n1.5.Concluding remarks and structure of the paper. Theorems 1.2 and 1.4\nin particular show that the response of a uniform plasma to oscillator y electromag-\nnetic disturbances can be expressed by a Fourier multiplier. Althoug h limited to\na simple plasma equilibrium, these results support the physics theorie s that rely\non the pseudo-differential form of the conductivity operator [37 , 7, 44, 42, 43, 10].\nMore precisely, even though the response of a plasma is rigorously n ot a pseudo-\ndifferential operator, it can be written as the sum of a pseudo-diffe rential operator\nplus a remainder which vanishes if the spectrum of the electric-field d isturbance is\nsupported away from ω= 0 (ork= 0 in the simpler case of theorem 1.1); this is\ntypically the case in the envisaged applications, since the frequency of the pertur-\nbation is set by an external source and it is tuned to resonate with t he cyclotron\nmotion of a particle species.\nIn order to illustrate these specific applications at least qualitatively , we return\nto the linearized Vlasov-Maxwell system, that is,\n(22)\n\n∂tfs+vs(p)·∇xfs+qs(vs×B0/c)·∇pfs=−qs(E+vs×B/c)·∇pFs,0,\n∂tE−c∇×B=−4π/summationdisplay\nsqs/integraldisplay\nR3vsfsdp,\n∂tB+c∇×E= 0,\nforfs,E, andB. The two equations for the electromagnetic field ( E,B) imply\n(formally at least, by taking the time-derivative of the Amp` ere-Ma xwell law)\n∂2\ntE+c2∇×/parenleftbig\n∇×E/parenrightbig\n+4π∂tj= 0,\nwhere∂tj=K({∂tfs}). This equation depends on the time-derivative of the in-\nduced current ∂tjrather than on jalone, and the map (20) give ∂tj=ς(E) when\nE∈ S(R4). However, if the solution is highly oscillatory (high-frequency wave s),\nˆE(ω,k) = 0 near ω= 0 and we can replace ςby the Fourier multiplier (21) with\nthe low-frequency cut-off, obtaining\n(23) D(i∂t,−i∂x)E:=∂2\ntE+c2∇×/parenleftbig\n∇×E/parenrightbig\n+4πς1−χ(E) = 0,TEMPERED-IN-TIME RESPONSE OF A PLASMA 11\nwhich is a constant-coefficients pseudo-differential equation for t he electric field\nonly. Theorem 1.4 implies that the operator D(i∂t,−i∂x) : [S(R4)]3→[S′(R4)]3\nis continuous and we have established regularity results for its symb ol in proposi-\ntion5.10. Thesymbol, inparticular,ispolynomiallyboundedandthusth eoperator\nextends to\nD(i∂t,−i∂x) : [H(s)(R4)]3→[H(s−m)(R4)]3, s∈R,\nwhereH(s)(R4) is the space of w∈ S′(R4) such that (1 + (ω\nˆω)2+ (ck\nˆω)2)s\n2ˆw∈\nL2(R4), with the normalization frequency chosen by ˆ ω:= max s|ωc,s|and with\nm= max{2M,2}where the integer Mbeing the degree of the polynomial bound\nfor ˆς0established in proposition 5.10 in section 5.4.\nThe semi-classical methods commonly used to find approximate solut ions of the\nwave equation (23) (cf. the work of Prater et al. [48] for an overv iew of com-\nputational tools) are valid under strong assumptions on the symbo l of the oper-\natorD. These assumptions are smoothness of the symbol and the fact t hat its\nanti-Hermitian part is small in a certain sense (weakly non-Hermitian o perators)\n[37, 7, 44, 42, 43, 10]. In this paper we analyze the construction of the operator\nin detail. Theorem 1.4 shows that the full operator ςdefined in (21) has an addi-\ntional contribution that accounts for the low-frequency respon se of the plasma. For\nthe high-frequency part (operator (21)), smoothness of the s ymbol is established\nalmost everywhere in Fourier space, cf. proposition 5.10. As for th e assumption of\nweak anti-Hermitian part, this is always violated near cyclotron reso nances and the\napplication of standard computational methods is justified by heur istic arguments\nonly. Propagation near a resonance has been addressed in the phy sics literature\n[55, 4] but a satisfactory theory is not available. Then the only rigor ous approach\nto the problem would be the direct numerical computation of the solu tion of the\nlinearized Vlasov-Maxwell system, which is computationally too expen sive in real-\nistic cases. The precise characterization of response operators may help to improve\nthe available methods toward including resonances.\nIn section 2, a simple case study is presented in order to illustrate th e ideas. The\nrestofthe paper isdedicated to the proofs. In section 3the case ofa non-relativistic\nisotropic plasma in one dimension is addressed, while section 5 is dedicat ed to rela-\ntivistic uniform magnetized plasmas. An overviewof notations and st andarddefini-\ntions together with technical results can be found in the appendice s. In appendix C\nin particular a precise definition of causal solution is given for advect ion equations\nassociated to global-in-time flows.\n2.Characterization of the response operator: a simple case st udy\nIn this section, we study a simple model which however contains the e ssential\nelements of the full problem. The aim of these simple considerations is showing\nhow the limiting absorption principle determines the causal solution of a hyperbolic\nequation. All proofsarestraightforwardandreportedinappend ix Bforthe reader’s\nconvenience.\nGivenv∈ S(R1+d), we consider the equation\n∂tu(t,x) =v(t,x), u(0,·) =u0∈C∞\nb(Rd),\nwhereC∞\nbis the space of smooth bounded functions with bounded derivatives , cf.\nappendix A for the precise definition.12 O. LAFITTE AND O. MAJ\nProposition 2.1. Forv∈ S(R1+d), there exists a unique solution uinC∞\nb(R1+d)\nof the equation ∂tu=vsuch that limt→−∞u(t,x) = 0pointwise in x, and\nu(t,x) =/integraldisplayt\n−∞v(s,x)ds.\nThe mapv/ma√sto→uis a continuous linear operator both from S(R1+d)→ S′(R1+d)\nand from S(R1+d)→L∞(R1+d).\nIf we think of uas the response to a localized perturbation v, causality requires\nthatu→0 fort→ −∞since the perturbationdecreasesfasterthan anypolynomial\nwhent→ −∞. Hence, the solution given in proposition 2.1 is referred to as the\ncausal solution. Since v∈ S(R1+d),\n/integraldisplay0\n−∞v(s,·)ds\nis finite. The causality principle selects a unique initial condition u0given by\nu0(x) =/integraldisplay0\n−∞v(s,x)ds,\nthereby allowing us to define the linear continuous operator v/ma√sto→u, which we view\nas the response of the operator ∂tfor a perturbation v. We also note that the limit\nfort→+∞of the solution gives the time integral of the perturbation/integraltext\nRv(s,x)ds.\nThe following simple result provides us with a characterization of the c ausal\nsolution, that is used in this paper for more general problems.\nProposition 2.2. The damped problem ∂tuν+νuν=vinS′(R1+d)withν >0\nandv∈ S(R1+d)has a unique solution uν∈ S′(R1+d). It belongs to S(R1+d)and\nit is given by\nuν(t,x) =/integraldisplayt\n−∞e−ν·(t−t′)v(t′,x)dt′.\nFurthermore, uν→uinS′(R1+d)asν→0+, whereuis the causal solution\nobtained in proposition 2.1.\nRemark 3.The integral in the definition of uνis absolutely convergent as t−t′≥0\non the domain of integration and v(·,x)∈L1(R). The fact that uν∈ S(R1+d) is\nproven by showing that the Fourier transform ˆ uνis inS(R1+d).\nProposition 2.2 establishes that the admissible solution uof the model without\ndissipation (in the sense that u→0 whent→ −∞) is the limit of the unique\nsolutionuνinS′of the model with dissipation.\n3.Uniform isotropic plasmas in one spatial dimension:\nthe standard linear Landau damping\nHere we consider in detail the case of a non-magnetized non-relativ istic plasma\nin one spatial dimension and for a single particle species. Coupled to th e Poisson\nequation this is the textbook example for linear Landau damping. Equ ation (2)\nreduces to\n(24) ∂tf(t,x,v)+v∂xf(t,x,v) =−(q/m)E(t,x)F′\n0(v),TEMPERED-IN-TIME RESPONSE OF A PLASMA 13\nwhereF0∈ S(R) is the equilibrium distribution function, and E∈ S(R2) is the\nelectricfield perturbation. We dropthe speciesindex ssincewe considerone species\nonly. In this case the linearized Vlasov operator is the free-transp ort operator\nL=∂t+v∂x.\nViewed as an operator from S′(R3) into itself, Lhas a non-trivial null space given\nby all tempered distributions with partial Fourier transform in ( t,x) of the form\nˆf= 2πκ∗(δ⊗h)forh∈ S′(R2)andwithκ: (ω,k,v)/ma√sto→(ω−kv,k,v)beingavolume-\npreserving diffeomorphism of R3, andκ∗denotes the pull-back of distributions.\nUsually, this is formally written as ˆf(ω,k,v) = 2πδ(ω−kv)ˆh(k,v), in the physics\nliterature. In particular, (24) has infinitely many solutions in S′.\nWe consider a damped version of the advection operator and then p ass to the\nlimit to recover a solution of the original problem (limiting absorption pr inciple).\nMore precisely, for a given F0∈ S(R), we first prove the existence and uniqueness\nof the solution fνinS′(R3) of (8). We find that the unique solution fν∈ S′(R3)\nis an element of S(R3) and we thus define a map from S(R2) toS(R3). We\nfirst calculate the Fourier transform of fνin (t,x) (proposition 3.1), from which\nwe deduce the function itself. We obtain from this unique solution the damped\ncurrentjν(t,x):=/integraltext\nRvfν(t,x,v)dvthrough its Fourier transform, and the limits\nwhenν→0+, respectively in S′(R3) and in S′(R2) offν(proposition 3.2) and of\njν(proposition 3.3). This shows that lim ν→0+jν=σ(E) whereσis an operator\n(called the conductivity operator), for which we give expressions. Indeed, equality\n(34) below gives its pointwise limit as a Fourier multiplier and its global exp ression\nis described in proposition 3.5, which rewrites using the plasma physics language\nas proposition 3.6 on the object called conductivity.\n3.1.Solution of the linearized Vlasov equation. In this section we prove ex-\nistence and uniqueness of the solution of the damped problem (8), w hich reads\nLνfν=−(q/m)EF′\n0,\nwhereLν=L+ν=∂t+v∂x+ν. Let us also define the function\nfν,∗(x,v):=−(q/m)F′\n0(v)/integraldisplay0\n−∞eνsE/parenleftbig\ns,x+vs/parenrightbig\nds,\nthe integral being absolutely convergent.\nProposition 3.1. For anyν >0,E∈ S(R2)andF0∈ S(R), equation (8) has a\nsolutionfν∈ S(R3), given by its Fourier transform\n(25) ˆfν=−i(q/m)ˆEF′\n0\nω−kv+iν.\nThis is the unique solution in S′(R3)and the unique (classical) solution of the\nCauchy problem\nLνfν(t,x,v) =−(q/m)E(t,x)F′\n0(v), fν(0,x,v) =fν,∗(x,v),\nwith initial condition given at t= 0.\nRemark 4.In proposition 3.1, we distinguish between the equation in S′and the\nequation stated for C1(R2) functions.14 O. LAFITTE AND O. MAJ\nProof.ForF0∈ S(R),E∈ S(R2) andν >0, after partial Fourier transform in\n(t,x) we observe that necessarily a solution in S′of the considered equation is given\nby (25). Since the function ( ω,k,v)/ma√sto→(ω−kv+iν)���1belongs to C∞and has\npolynomially bounded derivatives, from equation (25) we check that ˆfν∈ S(R3),\nand thusfν∈ S(R3). However the sequence {fν}νis not uniformly bounded in S.\nThe inverse partial Fourier transform gives\n(26)fν(t,x,v) =−(q/m)F′\n0(v)/integraldisplayt\n−∞e−ν·(t−s)E/parenleftbig\ns,x−v·(t−s)/parenrightbig\nds,\nand we note that\nX(s,t,x,v) =x−v·(t−s), V(s,t,x,v) =v,\nis the solution of the equations for the characteristic curves of Lνintegrated back-\nward in time from ( t,x,v). Therefore (26) is the classical solution as claimed. /square\nRemark5.Thisresultshowsthattherequirement fν∈ S′(R3)leadstotheselection\nofaspecificinitialcondition fν,∗fortheCauchyproblem,thusuniquelydetermining\nthe solution. Conversely, the solution of the Cauchy problem with init ial condition\nfν,∗is an element of Sand thus of S′.\nAs an alternative way to illustrate how the condition fν∈ S′(R3) leads to the\nselection of a specific Cauchy datum, one can consider for fν,0∈L2(R2) the initial\nvalue problem in C1/parenleftbig\nR,L2(R2)/parenrightbig\n,\n/braceleftigg\n∂tfν(t,x,v)+v∂xfν(t,x,v)+νfν(t,x,v) =−(q/m)E(t,x)F′\n0(v),\nfν(0,x,v) =fν,0(x,v).\nPerforming the Fourier transform in space, we obtain an ordinary d ifferential equa-\ntion almost everywhere in the ( k,v)-space,\n/braceleftigg\n∂t˜fν(t,k,v)+ikv˜fν(t,k,v)+ν˜fν(t,k,v) =−(q/m)˜E(t,k)F′\n0(v),\n˜fν(0,x,v) =˜fν,0(k,v),\nthe solution of which is\n˜fν(t,k,v) =e−(ν+ikv)t/bracketleftig\n˜fν,0(k,v)−(q/m)F′\n0(v)/integraldisplayt\n0e(ν+ikv)s˜E(s,k)ds/bracketrightig\n.\nWe see that fν,˜fν∈C∞/parenleftbig\nR,L2(R2)/parenrightbig\n. The integral factor has a finite limit for\nt→ −∞,\n˜fν,∗(k,v) = (q/m)F′\n0(v) lim\nt→−∞/integraldisplayt\n0e(ν+ikv)s˜E(s,k)ds\n=−(q/m)F′\n0(v)/integraldisplay0\n−∞e(ν+ikv)s˜E(s,k)ds,\nand˜fν,∗∈L2(R2). We showthat if ˜fν,0/\\e}atio\\slash=˜fν,∗, then the solution ˜fν, orequivalently\nfν, is not tempered in time for any k,vfixed. With this aim we write\n˜fν(t,k,v) =e−(ν+ikv)t/bracketleftbig˜f(0)\nν(k,v)−˜fν,∗(k,v)/bracketrightbig\n−qF′\n0(v)\nm/integraldisplayt\n−∞e−(ν+ikv)(t−s)˜E(s,k)ds,\nandone observesthat, forevery k,v, the secondterm on the right-handside belongs\ntoC∞\nb(R) and thus to S′(R). As for the the first term on the right-hand side, for\nanyk,v, andν >0 the function t/ma√sto→e−νt+ikvtis not tempered, since there existsTEMPERED-IN-TIME RESPONSE OF A PLASMA 15\na test function ϕ(t) =e−ν√\n1+t2−ikvtinS(R) such that/integraltexte−νt+ikvtϕ(t)dt= +∞.\nHence the first term cannotbe the partialFouriertransformofa distribution, which\nwas the hypotheses that allowed us to write the equation in Fourier s pace.\nIt follows that we have ˜fν∈ S′if and only if the initial condition satisfies\n˜f(0)\nν(k,v) =˜fν,∗(k,v) almost everywhere, that is,\n˜fν,0(k,v) =˜fν,∗(k,v) =−(q/m)F′\n0(v)/integraldisplay0\n−∞e(ν+ikv)s˜E(s,k)ds.\nThe corresponding solution amounts to\n˜fν(t,k,v) =−(q/m)F′\n0(v)/integraldisplayt\n−∞e−(ν+ikv)(t−s)˜E(s,k)ds,\nand this is the unique solution in S′(R3). Upon inserting the full Fourier transform\nofE(t,x), one can check that this gives (26).\nWe apply now the limiting absorption principle, that is we consider the limit of\nthe distribution function fνforν→0+.\nProposition 3.2. For anyE∈ S(R2)andF0∈ S(R), the solution fνdefined\nin (26) has a pointwise limit for ν→0+given by\nf(t,x,v) =−(q/m)F′\n0(v)/integraldisplayt\n−∞E/parenleftbig\ns,x−v·(t−s)/parenrightbig\nds,\nwhich is in C∞\nb(R3), withf(t,x,·)∈ S(R), and solves L0f=−(q/m)EF′\n0.\nProof.We observe that for ν >0 ands1/2, then 1/(1+s2)mis integrable and by the\ndominated convergence theorem, for any ( t,x,v)∈R3,\nfν(t,x,v)ν→0+\n− −−− →f(t,x,v):=−(q/m)F′\n0(v)/integraldisplayt\n−∞E/parenleftbig\ns,x−v·(t−s)/parenrightbig\nds.\nWe observe that the pointwise limit is the causal solution of linear adve ction equa-\ntionL0f=−(q/m)F′\n0Ein the sense of appendix C and the characteristic flow sat-\nisfies the hypothesis of proposition C.1. Hence proposition C.2 gives f∈C∞\nb(R3).\nSincef(t,x,v) is proportional to F′\n0(v), we havef(t,x,·)∈ S(R). /square\nRemark 6.We can deduce other properties of the solution f. Sincef(t,x,·) is\nrapidly decreasing, we also have f(t,x,·)∈L1(R) as it should be (in view of its\nmeaning as particle density). Continuity implies that f(t,·,·) is inL1(K×R) for\nevery compact K⊂R, and this is physically appropriate for such an idealized\nmodel, which, being spatially uniform, has an infinite number of particle s: only the\nnumber of particles /ba∇dblf(t,·,·)/ba∇dblL1(K×R)in a compact spatial domain Khas to be\nfinite.\nThe pointwise limit obtained in proposition 3.2 is referred to as the resp onse of\nthe plasma to the perturbation E.16 O. LAFITTE AND O. MAJ\n3.2.Current density and conductivity operator. We can now compute the\nelectric current density via equation (5), namely,\n(27) jν(t,x) =q/integraldisplay\nRvfν(t,x,v)dv,\nandjν∈ S(R2). The map E/ma√sto→jν=σν(E) defines a linear continuous operator\nσν:S(R2)→ S(R2) which is given by the Fourier multiplier\n(28) ˆν(ω,k) = ˆσν(ω,k)ˆE(ω,k),ˆσν(ω,k) =−iq2\nm/integraldisplay\nRvF′\n0(v)\nω−kv+iνdv.\nThe continuity of σνin particular follows from the estimate\n/vextendsingle/vextendsingle∂α\nω∂β\nkˆσν(ω,k)/vextendsingle/vextendsingle≤C\nνα+β+1/integraldisplay\nR/vextendsingle/vextendsinglevβ+1F′\n0(v)/vextendsingle/vextendsingledv,\nfor any non-negative integers α,β, where the constant Cdepends only on q2/m,\nα, andβ. We observe that this estimate is not uniform in νas expected, since the\nsequencefνis not uniformly bounded in S.\nFor the slightly more general case of non-homogeneous equilibria of the form\nF0(x,v) =n0(x)˜F0(v) withn0∈C∞\nb, one can define ˜ σν:S(R2)→ S(R2) as the\nFourier multiplier with symbol\n(29) /hatwide˜σν(ω,k) =−iq2\nm/integraldisplay\nRv˜F′\n0(v)\nω−kv+iνdv,\nand obtain the induced current\n(30) ˆ ν(ω,k) =/hatwide˜σν(ω,k)/hatwidestn0E(ω,k).\nIn this case the conductivity operator is\njν=σν(E):= ˜σν(n0E),\nand it amounts to the pseudo-differential operator\n(31)jν(t,x) =1\n(2π)2/integraldisplay\ne−iω(t−t′)+ik(x−x′)n0(x′)/hatwide˜σν(ω,k)E(t′,x′)dt′dx′dωdk,\nwhere the integral is in the sense of oscillatory integrals and the sym bol of the\noperator is ˆ σν(x′,ω,k) =n0(x′)/hatwide˜σν(ω,k).\nBy using the dominated convergence theorem we have that the limit o f the\ncurrent density jνis equal to the current carried by the limit distribution function\nf. Specifically we have\nProposition 3.3. WithE∈ S(R2)andF0∈ S(R), the function defined by\nj(t,x) =−q2\nm/integraldisplay\nDtvF′\n0(v)E/parenleftbig\ns,x−v·(t−s)/parenrightbig\ndsdv,\nwithDt= (−∞,t]×R, belongs to C∞\nb(R2), hence to S′. The mapσ:E/ma√sto→j=σ(E)\nis a linear continuous operator from S(R2)→ S′(R2).\nProof.In the domain Dt= (−∞,t]×R, we change variables to s′=t−s,v′=v\nfor (s,v)∈Dt, and thus ( s′,v′)∈[0,+∞)×R. Then\nj(t,x) =−q2\nm/integraldisplay+∞\n0/integraldisplay\nRvF′\n0(v)E(t−s,x−vs)dvds.TEMPERED-IN-TIME RESPONSE OF A PLASMA 17\nThe integrand E(t,x,s,v) =vF′\n0(v)E(t−s,x−vs) is such that\n∂α\nt∂β\nxE(t,x,s,v) =vF′\n0(v)∂α\nt∂β\nxE(t−s,x−vs)≤|vF′\n0(v)|/parenleftbig\n1+(t−s)2/parenrightbigm/ba∇dblE/ba∇dblα+β+2m.\nFor anyt∈R, this upper bound belongs to L1(R2). By the dominated convergence\ntheorem, we deduce that j∈C∞\nb(R2) and|∂α\nt∂β\nxj(t,x)| ≤Cm/ba∇dblE/ba∇dblα+β+2m.\nAsC∞\nb(R2)⊂ S′(R2) and/ba∇dblj/ba∇dblL∞(R2)≤C/ba∇dblE/ba∇dbl2m, one has that the map E/ma√sto→j\nfromS → S′is continuous. /square\nProposition 3.4. Whenν→0+, the sequence jνdefined in equation (28) con-\nverges tojboth pointwise in R2and inS′(R2).\nProof.WithE(t,x,s,v) =vF′\n0(v)E/parenleftbig\ns,x−v·(t−s)/parenrightbig\nasintheproofofproposition3.3,\nwe have\njν(t,x)−j(t,x) =q2\nm/integraldisplay\nDt/parenleftbig\n1−e−ν(t−s)/parenrightbig\nE(t,x,s,v)dsdv.\nSincee−ν(t−s)≤1 for (s,v)∈Dt, we have\n/parenleftbig\n1−e−ν(t−s)/parenrightbig/vextendsingle/vextendsingleE(t,x,s,v)/vextendsingle/vextendsingle≤/ba∇dblF0/ba∇dbl2m2+2/ba∇dblE/ba∇dbl2m1\n(1+s2)m1(1+v2)m2\nandform1,m2>1/2the bound is in L1. The dominatedconvergencetheoremthen\nyields pointwise convergence: lim/parenleftbig\njν(t,x)−j(t,x)/parenrightbig\n= 0 for all ( t,x)∈R2. The\nsame estimate also gives convergence in S′(R2): for any test function φ∈ S(R2),\n/a\\}b∇acketle{tjν−j,φ/a\\}b∇acket∇i}ht=q2\nm/integraldisplay\nR2/integraldisplay\nDt/parenleftbig\n1−e−ν(t−s)/parenrightbig\nE(t,x,s,v)dsdvφ(t,x)dtdx,\nand the integrand is bounded by /ba∇dblF0/ba∇dbl2m2+2/ba∇dblE/ba∇dbl2m1(1+s2)−m1(1+v2)−m2|φ(t,x)|\nwhich is integrable. Again the dominated convergence theorem allows us to pass\nto the limit in the integral and obtain /a\\}b∇acketle{tjν−j,φ/a\\}b∇acket∇i}ht →0. /square\nFor an explicit calculation of the conductivity operator we consider t he limit\nν→0+in Fourier space. As a tempered distribution, ˆ νacts onψ∈ S(R2) by\n(32) /a\\}b∇acketle{tˆν,ψ/a\\}b∇acket∇i}ht=−i(q2/m)/integraldisplay\nR2×RvF′\n0(v)ˆE(ω,k)ψ(ω,k)\nω−kv+iνdωdkdv.\nWe now want to pass to the limit for ν→0+. We use the Hilbert transform\n(appendix D).\nLetG(v) =vF′\n0(v)/n0wheren0>0 is the uniform background plasma density\nand letωp=/radicalbig\n4πq2n0/mbe the plasma frequency of the considered species. For\n(ω,k)∈R2,k/\\e}atio\\slash= 0, let\n(33) ˆ σph(ω,k):=−iω2\np\n4π1\nk/bracketleftig\nπH(G)(ω/k)−iπG(ω/k)/bracketrightig\n,\nwhereH(G) is the Hilbert transform of G. The tensor ˆ σphis the same as the one\nobtained formallyin the physics literature. We have4 πiωˆσph(ω,k) =ω2\npH(ω/k) for\nk/\\e}atio\\slash= 0, where the function H∈C∞(R) is given by H(z) =z[πH(G)(z)−iπG(z)].\nThen we have\n(34) lim\nν→0+ˆσν= ˆσphpointwise in ( ω,k)∈R2, k/\\e}atio\\slash= 0,18 O. LAFITTE AND O. MAJ\nsince ˆσν(ω,k) =−i(ω2\np/4π)Aν(ω,k), whereAνis defined in appendix D, and equa-\ntion (34) follows from lemma D.2.\nLet us introduce λ>0, a cut-off function χ∈C∞\n0(R),χ(z) = 1 for |z| ≤1/2 and\nsuppχ⊂(−1,1), and letχλ(k) =χ(λk). The real number λcan be interpreted as\na scale-length in the frequency domain. Then we define two operato rsσλ,1−χ,σλ,χ:\nS(R2)→ S′(R2) given by\n(35)σλ,1−χ(E):=F−1/parenleftbig\n(1−χλ)ˆσphˆE/parenrightbig\n, σλ,χ(E):=σ−σλ,1−χ.\nWe note that σλ,1−χis continuous since it is the composition of continuous opera-\ntions, and we have shown in proposition 3.3 that σis continuous, therefore σλ,χis\ncontinuous.\nProposition 3.5. The operators σandσλ,χdefined in proposition 3.3 and equa-\ntion (35), respectively, are such that\n/a\\}b∇acketle{tF/parenleftbig\nσλ,χ(E)/parenrightbig\n,ˆψ/a\\}b∇acket∇i}ht=iω2\np\n4π/integraldisplay\nR2χλ(k)G(v)/bracketleftig\nπH/parenleftbigˆEˆψ(·,k)/parenrightbig\n(kv)+iπˆEˆψ(kv,k)/bracketrightig\ndkdv,\n/a\\}b∇acketle{tF/parenleftbig\nσ(E)/parenrightbig\n,ˆψ/a\\}b∇acket∇i}ht=iω2\np\n4π/integraldisplay\nR2G(v)/bracketleftig\nπH/parenleftbigˆEˆψ(·,k)/parenrightbig\n(kv)+iπˆEˆψ(kv,k)/bracketrightig\ndkdv,\nfor allE,ψ∈ S(R2).\nProof.LetG(v) =vF′\n0(v)/n0,ω2\np= 4πq2n0/m, and for any φ∈ S(R2), letAνand\nBνbe the integrals defined in appendix D. Let\nIν\nλ,1−χ(φ):=−iω2\np\n4π/integraldisplay\nR3/parenleftbig\n1−χλ(k)/parenrightbig\nG(v)φ(ω,k)\nω−kv+iνdωdkdv\n=−iω2\np\n4π/integraldisplay\nR2/parenleftbig\n1−χλ(k)/parenrightbig\nφ(ω,k)Aν(ω,k)dωdk,\nIν\nλ,χ(φ):=−iω2\np\n4π/integraldisplay\nR3χλ(k)G(v)φ(ω,k)\nω−kv+iνdωdkdv\n=−iω2\np\n4π/integraldisplay\nR2χλ(k)G(v)Bν(v,k)dkdv.\nSincek/\\e}atio\\slash= 0in thesupportof1 −χλ, the integrandshaveapointwiselimit as ν→0+\ncomputed in lemma D.2. In addition, lemma D.2 shows that the integrand s are\nbounded by a function in L1uniformly in ν. The dominated convergence theorem\napplies and we can pass to the limit ν→0+in the integrals, obtaining\nIν\nλ,1−χ(φ)→ −iω2\np\n4π/integraldisplay\nR2/parenleftbig\n1−χλ(k)/parenrightbig\nφ(ω,k)1\nk/bracketleftig\nπH(G)(ω/k)−iπG(ω/k)/bracketrightig\ndωdk,\nIν\nλ,χ(φ)→ −iω2\np\n4π/integraldisplay\nR2χλ(k)G(v)/bracketleftig\n−πH/parenleftbig\nφ(·,k)/parenrightbig\n(kv)−iπφ(kv,k)/bracketrightig\ndkdv.\nParticularly, Iν\nλ,1−χ(φ)→/integraltext /parenleftbig\n1−χλ(k)/parenrightbig\nˆσph(ω,k)φ(ω,k)dωdk. Then we have\n/a\\}b∇acketle{tˆν,ˆψ/a\\}b∇acket∇i}ht=Iν\nλ,1−χ(ˆEˆψ)+Iν\nλ,χ(ˆEˆψ)ν→0+\n− −−− →/angbracketleftbig\nF/parenleftbig\nσλ,1−χ(E)/parenrightbig\n,ˆψ/angbracketrightbig\n+iω2\np\n4π/integraldisplay\nR2χλ(k)G(v)/bracketleftig\nπH/parenleftbigˆEˆψ(·,k)/parenrightbig\n(kv)+iπˆEˆψ(kv,k)/bracketrightig\ndkdv.TEMPERED-IN-TIME RESPONSE OF A PLASMA 19\nOn the other hand we know from proposition 3.4 that as ν→0+,jν→j=σ(E)\ninS′, hence\n/a\\}b∇acketle{tˆν,ˆψ/a\\}b∇acket∇i}htν→0+\n− −−− →/angbracketleftbig\nF/parenleftbig\nσ(E)/parenrightbig\n,ˆψ/angbracketrightbig\n,\nand by definition σ(E) =σλ,1−χ(E) +σλ,χ(E). Uniqueness of the limit gives the\nclaimed expression for σλ,χ.\nThe claimed expression for σ(E) follows analogously on noting that\n/a\\}b∇acketle{tˆν,ˆψ/a\\}b∇acket∇i}ht=Iν(ˆEˆψ):=−iω2\np\n4π/integraldisplay\nR2G(v)Bν(v,k)dkdv,\nwhere the function Bνis now computed with φ=ˆEˆψ, i.e.,\nBν(ω,k) =/integraldisplay\nRˆE(ω,k)ˆψ(ω,k)\nω−kv+iνdω.\nAsν→0+the right-hand side converges to /a\\}b∇acketle{tF/parenleftbig\nσ(E)/parenrightbig\n,ˆψ/a\\}b∇acket∇i}ht, while the limit of the\nleft-hand side is dealt with as in the case of Iν\nλ,χ. /square\nThe operator σλ,χdoes not play any role when the electric field perturbation is\nsupported away from k= 0, i.e., for non-constant fields. More precisely we have\nthe following result, which expresses the usual Ohm’s law for a unifor m plasma.\nCorollary 3.6. IfE∈ S(R2)is such that ˆE(ω,k) = 0for|k| ≤1/λ, then\nˆ∈C∞(R2)andˆ(ω,k) = ˆσph(ω,k)ˆE(ω,k).\nProof.By hypothesis χλ(k)ˆE(ω,k) = 0 for all ( ω,k)∈R2, hence,σλ,χ(E) = 0; this\nfollows directly from the expression given in proposition 3.5 since, in pa rticular,\nχλ(k)H/parenleftbigˆEˆψ(·,k)/parenrightbig\n(kv) =H/parenleftbig\nχλˆEˆψ(·,k)/parenrightbig\n(kv). Then\nF/parenleftbig\nσ(E)/parenrightbig\n=F/parenleftbig\nσλ,1−χ(E)/parenrightbig\n= (1−χλ)ˆσphˆE= ˆσphˆE,\nsince (1 −χλ)ˆE=ˆE. The fact that ˆ is inC∞follows from the properties of\nthe Hilbert transform summarized in proposition D.1, that imply in part icular,\nH(G)∈H∞(R). /square\nRemark 7.In the case of non-homogeneous equilibria of the form F0(x,v) =\nn0(x)˜F0(v), the statement of corollary 3.6 remains true with GandˆEreplaced\nbyv˜F0(v) and/hatwidestn0E, respectively. Particularly, one has\nF/parenleftbig\nσλ,1−χ(E)/parenrightbig\n(ω,k) =/parenleftbig\n1−χλ(k)/parenrightbig/hatwide˜σph(ω,k)/hatwidestn0E(ω,k),\nwhere/hatwide˜σphis obtained from (29) in analogy with ˆ σph. Then, the operator\nσλ,1−χ(E)(t,x) =1\n(2π)2/integraldisplay\ne−iω(t−t′)+ik(x−x′)\n×[n0(x′)/parenleftbig\n1−χλ(k)/parenrightbig/hatwide˜σph(ω,k)]E(t′,x′)dt′dx′dωdk\nis pseudo-differential with symbol n0(x′)/parenleftbig\n1−χλ(k)/parenrightbig/hatwide˜σph(ω,k), recoveringan expres-\nsion similar to (31).20 O. LAFITTE AND O. MAJ\n3.3.Proof of the main results for the non-magnetized non-relati vistic\none-dimensional case (section 1.3). We collect at last the partial results of\nthis section and give the proofs of the two theorems stated in sect ion 1.3.\nProof of theorem 1.1. (i) The fact that fνbelongs to S(R3) and is the unique so-\nlution of equation (8) in S′(R3) is proven in proposition 3.1. The current density\njνand the operator σνare given in equation (27) and (28) and related comments.\n(ii) and (iii) Pointwise convergence fν→fis established in proposition 3.2 and\nan expression for the solution fis given there. In proposition 3.2, it is also proven\nthatf(t,x,·)∈ S(R) for every ( t,x)∈R2. As for the convergence of fν→fin the\ntopology of S′, proposition 3.2 establishes pointwise (but not uniform) convergen ce\nfν→fwith limitf∈C∞\nb; in addition, for any integer m≥1/2, we have\n/vextendsingle/vextendsinglefν(t,x,v)/vextendsingle/vextendsingle≤ /ba∇dblqF′\n0/m/ba∇dbl0·/ba∇dblE/ba∇dbl2m/integraldisplay+∞\n−∞ds\n(1+s2)m,\nuniformly in ν∈[0,+∞). Therefore for every ϕ∈ S(R3), the function ( fν−f)ϕ\nsatisfies/vextendsingle/vextendsinglefν(t,x,v)−f(t,x,v)/vextendsingle/vextendsingle/vextendsingle/vextendsingleϕ(t,x,v)/vextendsingle/vextendsingle≤C/vextendsingle/vextendsingleϕ(t,x,v)/vextendsingle/vextendsingle.\nAs|ϕ| ∈L1, (fν−f)ϕsatisfiesthehypothesisofthedominatedconvergencetheorem\nand\n/a\\}b∇acketle{tfν−f,φ/a\\}b∇acket∇i}ht=/integraldisplay\nR3(fν−f)φdtdxdv →0,for allφ∈ S(R3).\nIn proposition 3.3, it is shown that j=J(f)∈C∞\nb(R2), and proposition 3.4\nestablishes the limit jν→jboth pointwise and in S′(R2). /square\nProof of theorem 1.2. Proposition 3.3 also establishes the continuity of the oper-\natorσ:E/ma√sto→j. The relation to the physical conductivity operator is proven in\ncorollary 3.6. /square\n4.Study of a PDE with parameters of the form\n−(u1∂u2−u2∂u1)ϕ(θ,u)−ia(θ,u)ϕ(θ,u) =ψ(θ,u)\nIn this section we establish existence and uniqueness results for a p artial differ-\nential equation with parameters that arises in the study of the rela tivistic, linear\nVlasov equation with uniform magnetic field B0, addressed below in section 5.\nSpecifically the equation is\n(36) −(u1∂u2−u2∂u1)ϕ(θ,u)−ia(θ,u)ϕ(θ,u) =ψ(θ,u),\nwherea,ψ∈C∞(Rl×R3) are given complex-valued functions of u∈R3and\ndepend on parameters θ∈Rl. The operator −(u1∂u2−u2∂u1) originates from the\nLorentz force term q(v×B0)·∇pwithq >0 andB0constant and directed along\nthe third axis. Eventually, the parameters θwill be related to the Fourier variables\n(τ,ξ), andatoaεdefined in equation (56) of section 5 below. Therefore, we assume\nthatasatisfies a condition similar to (17), that is,\n(37a)a∈C∞(Rl×R3,C),(u1∂u2−u2∂u1)a(θ,u) = 0,/vextendsingle/vextendsingleIma(θ,u)/vextendsingle/vextendsingle≥η>0,\nfor a given constant η >0. As we need to control the growth of derivatives of the\nsolution at infinity, we shall also assume that\n(37b) |∂α\nθ∂β\nua(θ,u))| ≤Cα,β(1+θ2+u2)m,∀α∈Nl\n0,∀β∈N3\n0,TEMPERED-IN-TIME RESPONSE OF A PLASMA 21\nuniformly in ( θ,u)∈Rl×R3for a given m∈Rand with constants Cα,β>0\ndepending on the multi-indices.\nFirst we establish the uniqueness of the solution under rather gene ral conditions.\nLemma 4.1. Leta∈C∞(Rl×R3)satisfy condition (37a), Θ⊆Rlbe an open\nset, and let ϕ∈L2\nloc(Θ×R3)be a function with weak derivatives ∂u1ϕ,∂u2ϕ∈\nL2\nloc(Θ×R3)and such that\n−(u1∂u2−u2∂u1)ϕ−iaϕ= 0,a.e. inΘ×R3.\nThen,ϕ= 0a.e. inΘ×R3.\nProof.For almost all ( θ,u3)∈Θ×R, the function ˜ ϕ(u1,u2):=ϕ(θ,u1,u2,u3)\nbelongs toH1/parenleftbig\nBr(0)/parenrightbig\nfor everyr>0, whereBr(0)⊂R2is the open ball of radius\nrand centered in zero in R2. From the equation we deduce\n−(u1∂u2−u2∂u1)|˜ϕ|2+2Im(a)|˜ϕ|2= 0.\nThe first term amounts to the divergence of the vector field ( −u2,u1)|˜ϕ|2which is\ntangent to ∂Br(0), hence Gauss theorem for the divergence, which holds for H1\nfunctions, gives\n0 =/integraldisplay\nBr(0)(u1∂u2−u2∂u1)|˜ϕ|2du1du2= 2/integraldisplay\nBr(0)Ima|˜ϕ|2du1du2,\nfor every radius r>0. We can now conclude upon accounting for hypotheses (37a).\nIf Ima≥η>0, we have\n0≤η/integraldisplay\nBr(0)|˜ϕ|2du1du2≤/integraldisplay\nBr(0)Ima|˜ϕ|2du1du2= 0.\nIf instead −Ima≥η>0,\n0≤η/integraldisplay\nBr(0)|˜ϕ|2du1du2≤ −/integraldisplay\nBr(0)Ima|˜ϕ|2du1du2= 0.\nIn both cases we deduce /integraldisplay\nBr(0)|˜ϕ|2du1du2= 0,\nand thus ˜ϕ= 0 a.e. in Br(0) for allrand for almost all ( θ,u3)∈Θ×R. It follows\nthatϕ= 0 a.e. in Θ ×R3. /square\nIn the remaining part of this section, we first give an existence resu lt for the\ncase in which the source term ψis a polynomial in ( u1,u2); this is based on an\nalgebraic argument. Then, we prove the existence of a smooth solu tionϕ∈C∞\nwhenψ∈C∞and of a solution ϕ∈ Swhenψ∈ S. The latter implies uniqueness\nof the solution in S′.\n4.1.Equation with a polynomial source term. Let the source term in equa-\ntion (36) be a polynomial of the form\n(38) ψ(θ,u) =/summationdisplay\n0≤m+n≤LYm,n(θ,u3)um\n1un\n2,Ym,n∈C∞,\nand let us consider for z∈C\\Zthe equation\n(39) −(u1∂u2−u2∂u1)˜ϕ(z;θ,u)−iz˜ϕ(z;θ,u) =ψ(θ,u).22 O. LAFITTE AND O. MAJ\nWe can search for solutions of the form\n(40) ˜ ϕ(z;θ,u) =/summationdisplay\n0≤m+n≤LXm,n(z;θ,u3)um\n1un\n2,\nthat is, a polynomial with at most the same degree as the source ter m. Substitution\ninto (39) yields\n−/summationdisplay\nm=1/summationdisplay\nn=0(n+1)Xm−1,n+1um\n1un\n2+/summationdisplay\nm=0/summationdisplay\nn=1(m+1)Xm+1,n−1um\n1un\n2\n−iz/summationdisplay\nm=0/summationdisplay\nn=0Xm,num\n1un\n2=/summationdisplay\nm=0/summationdisplay\nn=0Ym,num\n1un\n2,\nwhere the sums are all finite since m+n≤L. We observe that only the coefficients\nXm,nwithm+n=ℓforℓ= 0,1,2,...are coupled. For every integer 0 ≤ℓ≤L,\nwe define\nxℓ= (Xℓ−j,j)ℓ\nj=0, yℓ= (−iYℓ−j,j)ℓ\nj=0,\nthen the linear equation for the coefficients splits into ( ℓ+1)-dimensional blocks of\nthe form\n(41) ( Aℓ−z)xℓ=yℓ,0≤ℓ≤L,\nwhere the matrix Aℓ∈C(ℓ+1)×(ℓ+1)is defined and given in appendix E. For each ℓ,\nequation (41) has a unique solution when zis not an eigenvalue of the matrix Aℓ.\nLemma E.1 shows that the spectrum of Aℓis given by {2s−ℓ:s= 0,1,...,ℓ}\nand it is contained in the set of relative integers Zfor anyℓ. Hence, if z∈C\\Z\nequation (41) has a unique solution for all ℓ.\nLemma 4.2. Ifz∈C\\Zandψis given by (38), equation (39) has a solution\nwhich is of the form (40) with Xm,n(·;θ,u3)analytic in C\\Z, andXm,n(z;·)∈\nC∞(Rl×R).\nProof.Lemma E.1 establishes that the matrix Aℓis diagonalizablewith eigenvalues\n2s−ℓ,s= 0,...,ℓ. We denote by SandT=S−1the matrices (explicitly given in\nthe proof of lemma E.1) such that TAℓSis diagonal. Then, for z∈C\\Z,Aℓ−z\nis invertible and equation (41) has a unique solution xℓ= (Xm,n)m+n=ℓgiven by\nXℓ−j,j(z;θ,u3) =ℓ/summationdisplay\nr,s=0SjrTrs\n2r−ℓ−zYℓ−s,s(θ,u3),\nwhich is analytic in z∈C\\Z, andC∞in (θ,u3). /square\nWe can now use ˜ ϕto construct the unique solution of (36).\nProposition 4.3. Leta∈C∞(Rl×R3,C)satisfy the condition (37a), ψbe given\nin the form (38), ˜ϕbe the solution established in lemma 4.2, and let\nϕ(θ,u):= ˜ϕ/parenleftbig\na(θ,u);θ,u).\nThenϕ∈C∞(Rl×R3)is the unique solution of (36).\nProof.The fact that ϕ∈C∞is a solution follows by equation (39) and assump-\ntion (37a) which in particular implies\n−(u1∂u2−u2∂u1)ϕ(θ,u) =−(u1∂u2−u2∂u1)˜ϕ(z;θ,u)|z=a(θ,u).\nUniqueness has been proven in lemma 4.1. /squareTEMPERED-IN-TIME RESPONSE OF A PLASMA 23\n4.2.Equation with source term in Sand uniqueness in S′.Ifϕ∈C1is\na solution of (36) and ( u⊥,φ)∈R+×[0,2π] are polar coordinates defined by\nu1=u⊥cosφ,u2=−u⊥sinφ, then the functions\nU(r,φ) =ϕ(θ,u⊥cosφ,−u⊥sinφ,u3),\nV(r,φ) =ψ(θ,u⊥cosφ,−u⊥sinφ,u3),\nwith parameter r= (θ,u⊥,u3) satisfy\n(42) ∂φU(r,φ)−i˜a(r)U(r,φ) =V(r,φ), U(r,0) =U(r,2π),\nwith\n˜a(r) =a(θ,u⊥cosφ,−u⊥sinφ,u3),\nwhich is independent of φbecause of condition (37a).\nRemark8.The choice ofthe angle φ, in the clockwise direction, is unusual for polar\ncoordinates. This is motivated by the fact that, with this definition, φincreases in\nthe direction of gyration of a positively charged particle under the L orentz force.\nFor smooth solutions ϕ∈C∞of (36), we find that the derivatives ∂α\nθ∂β\nuϕwith\nthe sameorderofdifferentiationin ( u1,u2) arerelatedto the solutionofanordinary\ndifferential equation analogous to (42). In fact, differentiating eq uation (36) yields\n∂α\nθ∂β\nu[−(u1∂u2−u2∂u1)ϕ] =−(u1∂u2−u2∂u1)(∂α\nθ∂β\nuϕ)\n−β1∂α\nϑ∂β1−1\nu1∂β2+1\nu2∂β3\nu3ϕ+β2∂α\nϑ∂β1+1\nu1∂β2−1\nu2∂β3\nu3ϕ,\nfor any multi-index α∈Nl\n0andβ= (β1,β2,β3)∈N3\n0. This can be shown either\ndirectly using the identities\n∂β1\nu1∂β2\nu2(u1∂u2ϕ) =u1∂β1\nu1∂β2+1\nu2ϕ+β1∂β1−1\nu1∂β2+1\nu2ϕ,\n∂β1\nu1∂β2\nu2(u2∂u1ϕ) =u2∂β1+1\nu1∂β2\nu2ϕ+β2∂β1+1\nu1∂β2−1\nu2ϕ,\nor by induction over β1andβ2. Therefore, the ( ℓ+1)-dimensional complex-vector-\nvalued function defined by\n(43)ϕα,β,ℓ(θ,u) =/parenleftbig\n∂α\nθ∂β\nuϕ(θ,u)/parenrightbig\nβ1+β2=ℓ=/parenleftbig\n∂α\nθ∂ℓ−j\nu1∂j\nu2∂β3\nu3ϕ(θ,u)/parenrightbigℓ\nj=0,\nsatisfies the system of partial differential equations\n(44) −(u1∂u2−u2∂u1)ϕα,β,ℓ−i(a+tAℓ)ϕα,β,ℓ=ψα,β,ℓ,\nwhereAℓare the same matrices introduced in equation (41) and studied in app en-\ndix E, and the right-hand side is the ( ℓ+1)-dimensional-vector-valued function\n(45)ψα,β,ℓ=/parenleftig\n∂α\nθ∂β\nuψ+i/summationdisplay\nα′<α/summationdisplay\nβ′<β/parenleftbiggα\nα′/parenrightbigg/parenleftbiggβ\nβ′/parenrightbigg\n(∂α−α′\nθ∂β−β′\nua)(∂α′\nθ∂β′\nuϕ)/parenrightigℓ\nj=0,\nwithβ= (β1,β2,β3),β1=ℓ−jandβ2=j,j= 0,...,ℓ. For (u1,u2)/\\e}atio\\slash= (0,0), let\nUα,β,ℓ(r,φ) =ϕα,β,ℓ(θ,u⊥cosφ,−u⊥sinφ,u3),\nVα,β,ℓ(r,φ) =ψαβ,ℓ(θ,u⊥cosφ,−u⊥sinφ,u3).\nThen, ifϕ∈C∞is a solution of (36), necessarily it must hold that\n(46)/braceleftigg\n∂φUα,β,ℓ(r,φ)−i/parenleftbig\n˜a(r)+tAℓ/parenrightbig\nUα,β,ℓ(r,φ) =Vα,β,ℓ(r,φ),\nUα,β,ℓ(r,0) =Uα,β,ℓ(r,2π).24 O. LAFITTE AND O. MAJ\nForℓ= 0 the matrix Aℓreduces toA0= 0, hence equation (42) is a special case\nof (46) obtained for α=β= 0 andℓ= 0. In general, equation (46) is a system\nofℓ+1 first-order ordinary differential equations on [0 ,2π] with periodic boundary\nconditions, for which we have the following result.\nProposition 4.4. Forℓ∈N0, letMℓbe a(ℓ+1)×(ℓ+1)diagonalizable, complex\nmatrix with eigenvalues λℓ,j∈C\\Z,j= 0,1,...,ℓ. Then, for any function\nVℓ∈C∞([0,2π],Cℓ+1)satisfyingVℓ(0) =Vℓ(2π), there exists a unique solution\nUℓ∈C∞([0,2π],Cℓ+1)of\nU′\nℓ(φ)−iMℓUℓ(φ) =Vℓ(φ), Uℓ(0) =Uℓ(2π),\ngiven in Fourier series by\nUℓ(φ) =/summationdisplay\nn∈Z[i(Mℓ−n)−1ˆVℓ,n]e+inφ,ˆVℓ,n=1\n2π/integraldisplay2π\n0Vℓ(φ)e−inφdφ,\nand ifImλℓ,j/\\e}atio\\slash= 0, the solution satisfies, for all φ∈[0,2π],\n|Uℓ(φ)|∞≤κℓ\nλℓ,mmax\nφ′∈[0,2π]|Vℓ(φ′)|∞,\nwhere, forz= (z0,z1,...,zℓ)∈Cℓ+1,|z|∞:= maxj|zj|is theL∞norm in Cℓ+1,\nκℓis a constant depending only on Mℓ, andλℓ,m= minj|Imλℓ,j|.\nProof.SinceVℓ∈C∞([0,2π],Cℓ+1), for anyµ∈N0, the Fourier coefficients satisfy\n|n|µ|ˆVℓ,n|∞≤maxφ′|∂µ\nφVℓ(φ′)|∞, hence the corresponding Fourier series converges\ninCk([0,2π],Cℓ+1) for everyk∈N0. Dini’s test implies that the sum of the Fourier\nseries is equal to Vℓ, that is,Vℓcan be represented by a Fourier series.\nAnalogously a function Uℓ∈C1([0,2π],Cℓ+1) can be represented by a Fourier\nseries, with convergence in C1([0,2π],Cℓ+1) and it is a solution if and only if the\nFourier coefficients ˆUℓ,nsatisfy\n(Mℓ−n)ˆUℓ,n=iˆVℓ,n.\nAs it was assumed that Mℓis diagonalizable, that is, there exists a non-singular\ncomplex matrix Sℓsuch thatS−1\nℓMℓSℓ= Λℓwhere Λℓ= diag(λℓ,0,λℓ,1,...,λℓ,ℓ) is\nthe diagonal matrix of eigenvalues. Then Mℓ−n=Sℓ(Λℓ−n)S−1\nℓis non-singular\nfor alln∈Z, if and only if λℓ,j/\\e}atio\\slash∈Z. Since this is the case, the Fourier coefficients\nof the solution are uniquely determined and given by ˆUℓ,n=i(Mℓ−n)−1ˆVℓ,n. The\nnorm of the Fourier coefficients can be readily estimated by\n|ˆUℓ,n|∞≤ |Sℓ|∞·|(Λℓ−n)−1|∞·|S−1\nℓ|∞·|ˆVℓ,n|∞≤κℓ\nδℓ|ˆVℓ,n|∞,\nwhereκℓ=|Sℓ|∞|S−1\nℓ|∞is the conditionnumber ofthe matrix Sℓand thus depends\nonly onMℓ, whileδℓ= minj,n|λℓ,j−n|>0 measures the distance of the eigenvalues\nfromZ. Since for n/\\e}atio\\slash= 0,|ˆVℓ,n|∞=O(|n|−µ) for allµ∈N0, the Fourier series of\nUℓconverges in Ck([0,2π],Cℓ+1) for every k∈N0, hence the sum Uℓbelongs to\nC∞([0,2π],Cℓ+1) and it is the unique classical solution of the problem.\nWe obtain an equivalent representation of the classical solution. In factUℓmust\nnecessarily satisfy/parenleftbig\ne−iMℓφUℓ(φ)/parenrightbig′=e−iMℓφVℓ(φ),TEMPERED-IN-TIME RESPONSE OF A PLASMA 25\nand the general solution of this equation, with arbitrary initial cond itionUℓ(0), is\ne−iMℓφUℓ(φ) =Uℓ(0)+/integraldisplayφ\n0e−iMℓφ′Vℓ(φ′)dφ′.\nThen the periodic boundary condition Uℓ(0) =Uℓ(2π) amounts to\n/parenleftbig\ne−2πiMℓ−1/parenrightbig\nUℓ(0) =/integraldisplay2π\n0e−iMℓφ′Vℓ(φ′)dφ′.\nThe matrix on the left-hand side is diagonalizable with eigenvalues e−2πiλℓ,j−1;\nforλℓ,j∈C\\Zthe eigenvalues are all non-zero, the matrix is invertible, and the\nintegration constant Uℓ(0) is uniquely determined. At last one finds that there is a\nunique periodic solution given by\n(47)Uℓ(φ) = [e−2πiMℓ−1]−1/integraldisplay2π\n0e+iMℓ(φ−φ′)Vℓ(φ′)dφ′+/integraldisplayφ\n0e+iMℓ(φ−φ′)Vℓ(φ′)dφ′.\nEquation (47) shows that Uℓ∈C∞([0,2π],Cℓ+1). The components of the vector\nS−1\nℓUℓare given by\n(48)/parenleftbig\nS−1\nℓUℓ(φ)/parenrightbig\nj=1\ne−2πiλℓ,j−1/integraldisplay2π\n0e+iλℓ,j(φ−φ′)/parenleftbig\nS−1\nℓVℓ(φ′)/parenrightbig\njdφ′\n+/integraldisplayφ\n0e+iλℓ,j(φ−φ′)/parenleftbig\nS−1\nℓVℓ(φ′)/parenrightbig\njdφ′.\nTherefore,\n/vextendsingle/vextendsingle/parenleftbig\nS−1\nℓUℓ(φ)/parenrightbig\nj/vextendsingle/vextendsingle≤max\nφ′/vextendsingle/vextendsingle/parenleftbig\nS−1\nℓVℓ(φ′)/parenrightbig\nj/vextendsingle/vextendsingle/bracketleftigg/vextendsingle/vextendsingle/vextendsingle1\ne−2πiλℓ,j−1/vextendsingle/vextendsingle/vextendsingle/integraldisplay2π\n0e−Imλℓ,j(φ−φ′)dφ′\n+/integraldisplayφ\n0e−Imλℓ,j(φ−φ′)dφ′/bracketrightigg\n.\nFor the factor in square brackets, we use\n/vextendsingle/vextendsingle1−e−2πiλℓ,j/vextendsingle/vextendsingle≥/vextendsingle/vextendsingle1−|e−2πiλℓ,j|/vextendsingle/vextendsingle=/vextendsingle/vextendsingle1−e2πImλℓ,j/vextendsingle/vextendsingle,\nso that/vextendsingle/vextendsingle/vextendsingle1\ne−2πiλℓ,j−1/vextendsingle/vextendsingle/vextendsingle≤1\n|e2πImλℓ,j−1|.\nFor anyφ1>0 andy/\\e}atio\\slash= 0 we have\n/integraldisplayφ1\n0e−y(φ−φ′)dφ′=e−yφ\ny/parenleftbig\neyφ1−1/parenrightbig\n=e−yφ\n|y|/vextendsingle/vextendsingleeyφ1−1/vextendsingle/vextendsingle,\nand the two needed integrals are obtained for φ1= 2πandφ1=φ. Hence,\n/vextendsingle/vextendsingle/parenleftbig\nS−1\nℓUℓ(φ)/parenrightbig\nj/vextendsingle/vextendsingle≤1\n|Imλℓ,j|/bracketleftig\ne−Imλℓ,jφ+/vextendsingle/vextendsingle1−e−Imλℓ,jφ/vextendsingle/vextendsingle/bracketrightig\nmax\nφ′/vextendsingle/vextendsingle/parenleftbig\nS−1\nℓVℓ(φ′)/parenrightbig\nj/vextendsingle/vextendsingle.\nIf Imλℓ,j>0, the term in square brackets is equal to one and we obtain\n(49)/vextendsingle/vextendsingle/parenleftbig\nS−1\nℓUℓ(φ)/parenrightbig\nj/vextendsingle/vextendsingle≤1\n|Imλℓ,j|max\nφ′/vextendsingle/vextendsingle/parenleftbig\nS−1\nℓVℓ(φ′)/parenrightbig\nj/vextendsingle/vextendsingle.26 O. LAFITTE AND O. MAJ\nIfImλℓ,j<0, weconsider/parenleftbig\nS−1\nℓUℓ(2π−φ)/parenrightbig\njand uponchangingintegrationvariable\nin (48) we obtain\n/parenleftbig\nS−1\nℓUℓ(2π−φ)/parenrightbig\nj=−/bracketleftigg\n1\ne2πiλℓ,j−1/integraldisplay2π\n0e−iλℓ,j(φ−φ′)/parenleftbig\nS−1\nℓVℓ(2π−φ′)/parenrightbig\njdφ′\n+/integraldisplayφ\n0e−iλℓ,j(φ−φ′)/parenleftbig\nS−1\nℓVℓ(2π−φ′)/parenrightbig\njdφ′/bracketrightigg\n.\nThe factor in square brackets has the same form as the right-han d side of (48) with\nλℓ,jreplaced by −λℓ,jand now Im( −λℓ,j)>0. Hence we obtain\n/vextendsingle/vextendsingle/parenleftbig\nS−1\nℓUℓ(2π−φ)/parenrightbig\nj/vextendsingle/vextendsingle≤1\n|Imλℓ,j|max\nφ′/vextendsingle/vextendsingle/parenleftbig\nS−1\nℓVℓ(φ′)/parenrightbig\nj/vextendsingle/vextendsingle.\nSinceφis arbitrary, this is equivalent to (49) for Im λℓ,j<0. Since |Imλℓ,j| ≥\nλℓ,m>0, taking the maximum over jin (49) yields\n/vextendsingle/vextendsingleS−1\nℓUℓ(φ)/vextendsingle/vextendsingle\n∞≤1\nλℓ,mmax\njmax\nφ′/vextendsingle/vextendsingle/parenleftbig\nS−1\nℓVℓ(φ′)/parenrightbig\nj/vextendsingle/vextendsingle=1\nλℓ,mmax\nφ′/vextendsingle/vextendsingleS−1\nℓVℓ(φ′)/vextendsingle/vextendsingle\n∞.\nThen/vextendsingle/vextendsingleUℓ(φ)/vextendsingle/vextendsingle\n∞≤/vextendsingle/vextendsingleSℓ/vextendsingle/vextendsingle\n∞/vextendsingle/vextendsingleS−1\nℓUℓ(φ)/vextendsingle/vextendsingle\n∞≤κl\nλℓ,mmax\nφ′/vextendsingle/vextendsingleVℓ(φ′)/vextendsingle/vextendsingle\n∞,\nwhich is the claimed estimate. /square\nAs a corollary we obtain the solution of the problem stated in proposit ion 4.4\nwith generic parameters r∈ O ⊆ Rm, whereOis an open set.\nCorollary 4.5. Form∈N,O ⊆Rm, andℓ∈N0, letMℓ∈C∞(O,R(ℓ+1)×(ℓ+1))\nandVℓ∈C∞(O ×[0,2π],Cℓ+1)be such that\n1)for anyr∈ O, there is a non-singular matrix Sℓ(r)for which Λℓ(r) =\nSℓ(r)−1Mℓ(r)Sℓ(r)is diagonal with eigenvalues λℓ,j(r)∈C\\Rsatisfying\n|Imλℓ,j(r)| ≥η>0forj= 0,...ℓ,r∈ Oand for a given η>0,\n2)V(r,0) =V(r,2π)forr∈ O, and\n3)Sℓ,S−1\nℓ, andλℓ,jare of class C∞(O).\nThen, the problem\n∂φUℓ(r,φ)−iMℓ(r)Uℓ(r,φ) =Vℓ(r,φ), Uℓ(r,0) =Uℓ(r,2π),\nhas a unique solution Uℓ∈C∞(O ×[0,2π],Cℓ+1), and, for all φ∈[0,2π],\n|Uℓ(r,φ)| ≤κℓ\nηmax\nφ′∈[0,2π]|Vℓ(r,φ′)|.\nProof.A function Uℓis a solution if and only if, for any r∈ O,Uℓ(r,·) solves the\nordinary differential equation in proposition 4.4. Assumptions 1) and 2) imply that\nall the hypotheses of proposition 4.4 are verified and we note that λℓ,m(r)≥η\nuniformly for r∈ O. Therefore, for any r∈ O, there is a unique solution Uℓ(r,·)\nto the problem of corollary 4.5. The fact that Uℓis inC∞(O ×[0,2π]) follows\nfrom the explicit formula (47) and assumption 3) by using the classica l results\nof differentiation in the integral. The estimates follow from the ones p roven in\nproposition 4.4. /squareTEMPERED-IN-TIME RESPONSE OF A PLASMA 27\nWe can now give the general result for equation (36) with right-han d side inC∞\nand then in the Schwartz space, which will immediately imply uniqueness inS′.\nUniqueness of the C∞solution, in particular, is a special case of lemma 4.1, but\nhere we give a different more explicit argument.\nProposition 4.6. Leta∈C∞(Rl×R3)satisfy condition (37a). Then for any\nψ∈C∞(Rl×R3), equation (36) has a unique solution ϕ∈C∞(Rl×R3).\nProof.Uniqueness of a C1solution. If ϕ∈C1(Rl×R3) is a solution of (36),\nevaluating the equations at ( u1,u2) = (0,0) yields\nϕ(θ,0,0,u3) =iψ(θ,0,0,u3)/a(θ,0,0,u3),\nwhile for (u1,u2)/\\e}atio\\slash= (0,0),\nϕ(θ,u) =U(r,φ),\nwhereUis the unique solution of (42) constructed in using corollary 4.5. Thes e\nconditions completely define the value of a C1solution everywhere in Rl×R3.\nExistence of a C∞solution. First we address the special case\n(50) ψ(θ,u) =/summationdisplay\nm+n=kum\n1un\n2˜ψm,n(θ,u),\nwhere˜ψm,n∈C∞(Rl×R3) andk≥2 is a given integer. Let us consider the\nordinary differential equation (42) with source term Vdetermined by the ψgiven\nin (50). This equation is a special cases of the problem addressed in c orollary 4.5\nwithℓ= 0; particularly, because of assumption (37a), M0(r) = ˜a(r) satisfies the\nhypotheses of the corollary. Therefore, equation (42) has a uniq ue solution U∈\nC∞(O ×[0,2π]). Due to the special choice of ψand the estimate in corollary 4.5,\none deduces that for any given point ( θ,u3) andδ>0, there are constants cU,θ,u 3,δ\nandcV,θ,u 3,δ>0 for which\n/vextendsingle/vextendsingleU(r,φ)/vextendsingle/vextendsingle≤uk\n⊥cU,θ,u 3,δ,/vextendsingle/vextendsingleV(r,φ)/vextendsingle/vextendsingle≤uk\n⊥cV,θ,u 3,δ,\nuniformly in u⊥∈(0,δ] andφ∈[0,2π]. Let us construct the function\nϕ(θ,u):=/braceleftigg\nU(r,φ),for (u1,u2)/\\e}atio\\slash= (0,0),\n0, for (u1,u2) = (0,0).\nSince polar coordinates in the region ( u1,u2)/\\e}atio\\slash= (0,0) define a diffeomorphism\nwhich maps the partial differential equation (36) into the ordinaryd ifferential equa-\ntion (42), the function ϕis of classC∞and solves equation (36) in the open set\n{(θ,u)∈Rl×R3: (u1,u2)/\\e}atio\\slash= (0,0)}. We also have |ϕ(θ,u)| ≤cU,θ,u 3,δ|(u1,u2)|k,\nand thusϕis continuous on whole domain Rl×R3.\nWe now show that ϕ∈Ck−1(Rl×R3). We need to check the existence\nof derivatives at ( u1,u2) = (0,0) and their continuity. With this aim we col-\nlect the derivatives of ϕwith the same order ℓof differentiation in ( u1,u2) into\nthe vector-valued functions ϕα,β,ℓ, as defined in (43); analogously let ψα,β,ℓbe\ngiven by (45). For ( u1,u2)/\\e}atio\\slash= (0,0),ϕis of classC∞and solves equation (36),\nso thatϕα,β,ℓsatisfies equation (44) with source ψα,β,ℓ. In polar coordinates\nthose equations amount to ordinarydifferential equations (46) fo r the vector-valued\nfunctions given by Uα,β,ℓ(r,φ) =ϕα,β,ℓ(θ,u⊥cosφ,−u⊥sinφ,u3) and with source\nVα,β,ℓ(r,φ) =ψα,β,ℓ(θ,u⊥cosφ,−u⊥sinφ,u3). From lemma E.1 we know that the\nmatricesAℓ, and thustAℓ, are diagonalizable with integer eigenvalues. It follows\nthat the matrices Mℓ(r) = ˜a(r) +tAℓin equation (46) are diagonalizable and the28 O. LAFITTE AND O. MAJ\nimaginary part of the eigenvalues coincides with Im˜ a. In view of assumption (37a),\nwe have |Im˜a| ≥η>0, and the hypotheses of corollary 4.5 are therefore satisfied.\nWe can conclude that\n(51)/vextendsingle/vextendsingleUα,β,ℓ(r,φ)/vextendsingle/vextendsingle\n∞≤κℓ\nηmax\nφ′/vextendsingle/vextendsingleVα,β,ℓ(r,φ′)/vextendsingle/vextendsingle\n∞.\nWe can use this estimate to show that, for ( u1,u2)/\\e}atio\\slash= (0,0) we have\n(52)|∂α\nθ∂β\nuϕ(θ,u)| ≤Kα,β(θ,u3)uk−ℓ\n⊥withℓ=β1+β2,and 0<|(u1,u2)| ≤δ.\nWe prove this by induction over α,β3andℓ=β1+β2. Forα= 0,β3= 0, and\nℓ= 0 the claim follows directly from the estimate in corollary 4.5 and V=uk\n⊥VR.\nFor the induction step, let us assume that the claim holds for all α′<α,β′\n3<β3,\nandℓ′=β′\n1+β′\n2<ℓ=β1+β2. Then from (45) we deduce\n/vextendsingle/vextendsingleVα,β,ℓ(r,φ)/vextendsingle/vextendsingle\n∞=/vextendsingle/vextendsingleψα,βℓ(θ,u)/vextendsingle/vextendsingle\n∞≤cψ\nα,β(θ,u3)uk−ℓ\n⊥,\nand from estimate (51) we deduce that, if β1+β2=ℓ,\n/vextendsingle/vextendsingle∂α\nθ∂β\nuϕ(θ,u)/vextendsingle/vextendsingle≤/vextendsingle/vextendsingleϕα,β,ℓ(θ,u)/vextendsingle/vextendsingle\n∞=/vextendsingle/vextendsingleUα,β,ℓ(r,φ)/vextendsingle/vextendsingle\n∞≤κℓ\nηcψ\nα,β(θ,u3)uk−ℓ\n⊥,\nwhich is (52) as claimed. Therefore,\n|∂α\nθ∂β\nuϕ(θ,u)|\n|(u1,u2)|≤Kα,β(θ,u3)|(u1,u2)|k−ℓ−1,forβ1+β2=ℓ≤k−2,\nwhich implies that the derivatives ∂α\nθ∂β\nuϕ(θ,0,0,u3) exist and are zero for β1+β2≤\nk−1. Continuity of ∂α\nθ∂β\nuϕfollows from inequality (52). Hence ϕ∈Ck−1(Rl×R3)\nas claimed. Since k≥2,ϕ∈C1(Rl×R3) and equation (36) is satisfied also at\n(u1,u2) = (0,0) since all terms vanish if u1=u2= 0.\nFor the general case ψ∈C∞(Rl×R3), for any integer k≥2 we write\nψ=ψk−1+ψr,k,\nwhereψk−1istheTaylorpolynomialofdegree k−1in (u1,u2) centeredat( u1,u2) =\n(0,0)andψr,kistheremainder,whichisoftheform(50). Hencetheaboveargume nt\napplies toψr,k. Letϕr,k∈Ck−1(Rl×R3) be the unique solution obtained with\nψr,kas a source term. On the other hand proposition 4.3 established the existence\nof a unique solution ϕk−1∈C∞(Rl×R3) for the case with source term ψk−1. The\nsumϕ=ϕk−1+ϕr,kis of classCk−1and it is the unique solution of (36). Since k\nis arbitrary we conclude that ϕ∈C∞(Rl×R3). /square\nCorollary 4.7. Leta∈C∞(Rl×R3)satisfy both conditions (37) and let m∈Rbe\nthe constant in (37b). Then the unique solution ϕ∈C∞(Rl×R3)of equation (36)\nobtained in proposition 4.6 is such that:\n(i)If there are m0∈Randn0∈N0such that\n/vextendsingle/vextendsingle∂α\nθ∂β\nuψ(θ,u)/vextendsingle/vextendsingle≤Cψ\nα,β(1+θ2+u2)m0,|α|+|β| ≤n0,\nthen\n/vextendsingle/vextendsingle∂α\nθ∂β\nuϕ(θ,u)/vextendsingle/vextendsingle≤Cϕ\nη,α,β(1+θ2+u2)mα,β,|α|+|β| ≤n0,\nwheremα,β=m0ifm≤0, andmα,β=m0+(|α|+|β|)mifm>0.\n(ii)Ifψ∈ S(Rl×R3), thenϕ∈ S(Rl×R3).TEMPERED-IN-TIME RESPONSE OF A PLASMA 29\nProof.(i) We proceedbyinduction asin the proofof(52). For α= 0andβ= 0, the\nclaim follows directly from the estimate in corollary 4.5 since for u2\n1+u2\n2≥ρ2>0,\n/vextendsingle/vextendsingleϕ(θ,u)/vextendsingle/vextendsingle=/vextendsingle/vextendsingleU(r,φ)/vextendsingle/vextendsingle≤1\nηmax\nφ/vextendsingle/vextendsingleV(r,φ)/vextendsingle/vextendsingle≤1\nηCψ\n0,0(1+θ2+u2)m0.\nThe constant is independent of the radius ρ, hence the claim. For the induction\nstep, if the claim is true for all multi-indices α′<αandβ′<β, whereαandβare\nany multi-indices satisfying |α|+|β| ≤n0, then, again for u2\n1+u2\n2≥ρ2>0, we\nestimate/vextendsingle/vextendsingleVα,β,ℓ(r,φ)/vextendsingle/vextendsingle\n∞=/vextendsingle/vextendsingleψα,β,ℓ(θ,u)/vextendsingle/vextendsingle\n∞by\n/vextendsingle/vextendsingleVα,β,ℓ/vextendsingle/vextendsingle\n∞≤max/braceleftig/vextendsingle/vextendsingle∂α\nθ∂β\nuψ/vextendsingle/vextendsingle+/summationdisplay\nα′<α/summationdisplay\nβ′<β/parenleftbiggα\nα′/parenrightbigg/parenleftbiggβ\nβ′/parenrightbigg/vextendsingle/vextendsingle∂α−α′\nθ∂β−β′\nua/vextendsingle/vextendsingle|∂α′\nθ∂β′\nuϕ/vextendsingle/vextendsingle/bracerightig\n≤max/braceleftig\nCψ\nα,β(1+θ2+u2)m0+/summationdisplay\nα′<α/summationdisplay\nβ′<βCα,β\nη,α′,β′(1+θ2+u2)m+mα′,β′/bracerightig\n,\nwhere the maximum is computed over all ( β1,β2) such that β1+β2=ℓ, holdingα\nandβ3fixed. We observe that, if m≤0, thenm+mα′,β′=m+m0≤m0=mα,β,\nwhile ifm>0,m+mα′,β′=m0+(|α′|+|β′|+1)m≤m0+(|α|+|β|)m=mα,β.\nIn both cases we have\nmax\nφ/vextendsingle/vextendsingleVα,β,ℓ(r,φ)/vextendsingle/vextendsingle\n∞≤˜Cψ\nη,α,β 3,ℓ(1+θ2+u2)mα,β.\nWe can now apply inequality (51) and deduce\n/vextendsingle/vextendsingle∂α\nθ∂β\nuϕ(θ,u)/vextendsingle/vextendsingle≤κℓ\nη˜Cψ\nη,α,β 3,ℓ(1+θ2+u2)mα,β,\nwhereℓ=β1+β2; this proves the claim for αandβsatisfying |α|+|β| ≤n0.\n(ii) Ifψ∈ S(Rl×R3), then it satisfies the assumption of item (i) for all m0∈R\nand for all α,β. For anyµ∈R,α∈Nl\n0, andβ∈N3\n0, let us apply the estimate\nprovenin item (i) with m0=µ−(|α|+|β|)m; we obtain/vextendsingle/vextendsingle∂α\nθ∂β\nuϕ(θ,u)/vextendsingle/vextendsingle≤Cϕ\nη,α,β(1+\nθ2+u2)µ. Henceϕ∈ S(Rl×R3) as claimed. /square\nThe existence of a solution ϕ∈ S(Rl×R3) of equation (36) implies (by duality)\nuniqueness in S′(Rl×R3) for the linear equation\n(53) −(u1∂u2−u2∂u1)h−iah=s,\nfors∈ S′(Rl×R3) andasatisfying conditions (37).\nProposition 4.8. Ifa∈C∞(Rl×R3)satisfies all conditions (37), equation (53)\nhas at most one solution in S′.\nProof.We show that the only solution of the associate homogeneous equat ion is\nthe trivial solution h= 0. Explicitly, this means that if\n/angbracketleftbig\nh,+(u1∂u2−u2∂u1)χ−iaχ/angbracketrightbig\n= 0,\nfor allχ∈ S(Rl×R3), thenh= 0.\nGiven an arbitrary test function ψ∈ S(Rl×R3) let us consider the equation\n(u1∂u2−u2∂u1)ϕ−iaϕ=ψ.\nSince−asatisfies conditions (37), we have established in corollary 4.7 that th is\nequation has a unique solution ϕ∈ S. Then for any ψ∈ S,\n/a\\}b∇acketle{th,ψ/a\\}b∇acket∇i}ht=/angbracketleftbig\nh,+(u1∂u2−u2∂u1)ϕ−iaϕ/angbracketrightbig\n= 0,30 O. LAFITTE AND O. MAJ\nand thush= 0 as a tempered distribution. /square\n5.Response of a uniform magnetized plasma\nIn this section we address the case ofa uniform magnetized plasmaa nd provethe\nresults stated in section 1.4. With this aim we shall rely heavily on the pr eparatory\nresults of section 4 and on a stationary-phase argument postpon ed to section 6.\n5.1.Notation. Weshallmakeuseofnormalizedmomentum(12)andfor( u1,u2)/\\e}atio\\slash=\n(0,0) we define the two additional systems of cylindrical coordinates\n(54) u1=u⊥cosφ, u 2=∓u⊥sinφ, u 3=u/bardbl,\nwithu⊥∈R+andφ∈[0,2π]. We re-write the linearized Vlasov equation in one\nof these two cylindrical coordinate systems depending on the elect ric charge of the\nconsidered particle species: we choose the sign −(resp., +) for a positively (resp.,\nnegatively) charged particle species.\nWith normalized Fourier variables\n(55)τ:=ω/ωc,s, ξ:=ck/ωc,s,and with κs(u):=γ(u)νs(u)/ωc,s,\nwe define the quantities\n(56)aε(τ,ξ,u):=ω+iεν−k3v3\nωc,s/γ=γ(u)τ−ξ3u3+iεκs(u),\nbi(ξ,u):=kiv⊥\nωc,s/γ=u⊥ξi, i= 1,2.\nWritten in terms of normalized variables, the functions a0:=aε|ε=0,bi, andγare\nindependent of the particle species.\nWithGsdefined in (14), the functions of u∈R3defined by\n(57) Fs(u):=1\nu⊥∂Gs\n∂u⊥(u/bardbl,u⊥),Gs(u):=∂Gs\n∂u/bardbl(u/bardbl,u⊥)−u/bardbl\nu⊥∂Gs\n∂u⊥(u/bardbl,u⊥),\nbelongS(R3) because of the assumptions on Gs.\nNext we define the first-order partial differential operators\n(58) Qs,j(τ,ξ,u,∂ξ):=Fs(u)Φj(τ,ξ,u,∂ξ)+Gs(u)Ψj(τ,ξ,u,∂ξ),\nwhere Φj,Ψj, forj= 1,2,3, are given in terms of the coefficients\nΓ0(τ,ξ,u):=τγ(u)e±i(ξ1u2−ξ2u1), (59a)\nΓj(τ,ξ,u):=ξje±i(ξ1u2−ξ2u1), j = 1,2,3, (59b)\nby\nΦ1:=±Γ0∂ξ2,\nΨ1:=±Γ3∂ξ2,Φ2:=∓Γ0∂ξ1,\nΨ2:=∓Γ3∂ξ1,Φ3:=iu/bardblΓ0,\nΨ3:=iΓ0∓/parenleftbig\nΓ1∂ξ2−Γ2∂ξ1/parenrightbig\n.\nThe coefficients Γ jall satisfy\n|∂α\nτ,ξ,uΓj(τ,ξ,u)| ≤Cα(1+τ2+ξ2+u2)1+|α|,\nfor allα∈N7\n0, and thus multiplication by Γ jis closed in S.\nWe also introduce the functions of b= (b1,b1) given by\n(60) A±\nn(b):=/summationdisplay\nk,ℓ∈Z:k±ℓ=n(±i)ℓJk(b1)Jℓ(b2),TEMPERED-IN-TIME RESPONSE OF A PLASMA 31\nwhereJkare the Bessel’s functions of the first kind. If b= (b1,b2) = (0,0), the\nonly non-zero term in the series is the one for k= 0 andℓ= 0, hence A±\nn(0) = 1\nforn= 0, andA±\nn(0) = 0 for n/\\e}atio\\slash= 0; if, on the other hand, b= (b1,b2)/\\e}atio\\slash= (0,0) we\ncan writeb1=|b|cosθ,b2=∓|b|sinθand\n(61) A±\nn(b) =Jn(|b|)e−inθ, b/\\e}atio\\slash= 0,\nThis identity follows from Jacobi-Anger expansions [1, p.361]\neizcosφ=/summationdisplay\nn∈ZinJn(z)einφ, eizsinφ=/summationdisplay\nn∈ZJn(z)einφ,\nforz∈Candφ∈R, that imply\n(62) e±i(b2cosφ±b1sinφ)=/summationdisplay\nn∈ZA±\nn(b)einφ.\nWithb1=|b|cosθandb2=∓|b|sinθ, one computes\n/summationdisplay\nn∈ZA±\nn(b)einφ=ei|b|sin(φ−θ)=/summationdisplay\nn∈ZJn(|b|)ein(φ−θ),\nwhich yields identity (61). At last let\n(63)r±\nε(τ,ξ,u):=/integraldisplay+∞\n0eiaε(τ,ξ,u)λ−i(ξ1u1+ξ2u2)sinλ±i(ξ2u1−ξ1u2)cosλdλ\n=i\n2e−iπaε(τ,ξ,u)\nsin/parenleftbig\nπaε(τ,ξ,u)/parenrightbigP±\nε(τ,ξ,u),\nwith\n(64)P±\nε(τ,ξ,u):=/integraldisplay2π\n0eiaε(τ,ξ,u)λ−i(ξ1u1+ξ2u2)sinλ±i(ξ2u1−ξ1u2)cosλdλ.\nThequantitiesdefinedinequations(59)-(64)areindependentont heparticlespecies.\nLemma 5.1. Forε>0and for each choice of the sign, the equation\n∓(u1∂u2−u2∂u1)r±\nε−iaεr±\nε=e±i(ξ2u1−ξ1u2),\nhas a unique solution in C∞(R7)given by (63) and for every α∈N4\n0,β∈N3\n0there\nare constants Cα,β,ε,ν>0andmα,β∈Rsuch that\n|∂α\nτ,ξ∂β\nur±\nε(τ,ξ,u)| ≤Cα,β,ε,ν(1+τ2+ξ2+u2)mα,β,\nuniformly in (τ,ξ,u).\nProof.For both choices of the sign, the equation for r±\nεis of the form (36) with\nright-hand side in C∞and withθ= (τ,ξ). (Forr−\nεin particular, one can multiply\nthe equation by −1 and seta=−aεandψ=−e±i(ξ2u1−ξ1u2).) Forε >0, the\nfunctionaεdefined in (56) is such that all conditions (37) are true. Therefore\nproposition 4.6 ensures the existence a unique solution r±\nε∈C∞(R7) for each\nchoice of the sign. For the claimed estimates it is enough to show that for any\nn0∈N0the right-hand side of the equation satisfies the hypothesis of cor ollary 4.7\n(i), and this is straightforward.\nThe integral expressions (63) can be checked by direct substitut ion into the\nequation. In fact if I±\nε(τ,ξ,u;λ) denotes the integrand in (63), we have the identity\n/bracketleftbig\n∓(u1∂u2−u2∂u1)−iaε(τ,ξ,u)/bracketrightbig\nI±\nε(τ,ξ,u;λ) =−∂\n∂λI±\nε(τ,ξ,u;λ),32 O. LAFITTE AND O. MAJ\nandI±\nε(τ,ξ,u;0) =e±i(ξ2u1−ξ1u2). For the second form of r±\nε, we notice that,\nfollowing Qin et al. [49],\nr±\nε(τ,ξ,u) =/integraldisplay+∞\n0eiaε(τ,ξ,u)λ−i(ξ1u1+ξ2u2)sin(λ+2π)±i(ξ2u1−ξ1u2)cos(λ+2π)dλ,\nand changing integration variable we have\nr±\nε(τ,ξ,u) =e−2πiaε(τ,ξ,u)/integraldisplay+∞\n2πeiaε(τ,ξ,u)λ−i(ξ1u1+ξ2u2)sinλ±i(ξ2u1−ξ1u2)cosλdλ\n=e−2πiaε(τ,ξ,u)/bracketleftbigg\nr±\nε(τ,ξ,u)\n−/integraldisplay2π\n0eiaε(τ,ξ,u)λ−i(ξ1u1+ξ2u2)sinλ±i(ξ2u1−ξ1u2)cosλdλ/bracketrightbigg\n,\nhence,\nr±\nε(τ,ξ,u) =−e−2πiaε(τ,ξ,u)\n1−e−2πiaε(τ,ξ,u)/integraldisplay2π\n0eiaε(τ,ξ,u)λ−i(ξ1u1+ξ2u2)sinλ±i(ξ2u1−ξ1u2)cosλdλ,\nfrom which the second expression for r±\nεfollows. /square\n5.2.The roots of a0−nand the distribution limε→0+(1/sinπaε).From ex-\npression (63), one can see that the main issue in computing the limit fo rε→0+of\nr±\nεconsists in the sets of points ( τ,ξ,u)∈R×R3×R3for which\na0(τ,ξ,u) =γ(u)τ−ξ3u3∈Z.\nWe note a few preliminary facts about such points.\nRemark 9.The condition a0(τ,ξ,u) =n∈Zis equivalent to\nω−k3v3=nΩs(u), n∈Z,\nwhich defines the cyclotron resonances: particles of the species sthat satisfy this\ncondition along their orbit, for some integer n, resonate to a plane wave with\nfrequency and wave vector ( ω,k). For resonant particles, the Doppler-shifted wave\nfrequencyω−k3v3matches a multiple (also referred to as a harmonic) of the\ngyration frequency Ω sof the particle’s orbit around the magnetic field.\nFor any given n∈Zand (τ,ξ)∈R×R3,τ/\\e}atio\\slash= 0, let\n(65) Rn(τ,ξ) ={u∈R3:a0(τ,ξ,u) =n}.\nPhysically Rn(τ,ξ) is the set of normalized particle momenta uthat resonate with\nthen-th harmonic of the cyclotron frequency when the wave field is a plan e wave\nwith frequency and wave vector are given by ( τ,ξ). Sincea0is constant in ( ξ1,ξ2),\nthe sets Rn(τ,ξ) for a fixed harmonic number ndepend only on τ, andξ3. In\naddition it is enough to study them for τ >0 andξ3≥0, in view of the symmetries\nof the function a0. (The hyperplane τ= 0 will be excluded in our main results.) A\nnecessary condition for u∈ Rn(τ,ξ) is\n(τ2−ξ2\n3)u2\n3+τ2(1+u2\n1+u2\n2)−2nξ3u3−n2= 0.\nwhich defines a family of surfaces of revolution obtained by the rota tion of conics\naround the u3-axis. Specifically we find ellipsoids for τ2> ξ2\n3, a paraboloid for\nτ2=ξ2\n3, and one branch of a hyperboloid for τ2< ξ2\n3. We shall speak of elliptic,TEMPERED-IN-TIME RESPONSE OF A PLASMA 33\nparabolic, and hyperbolic resonances, with reference these thre e conditions, respec-\ntively. In the elliptic case, Rn(τ,ξ) is non-empty if and only if nτ >0, that is\nwhen,τandnare both non-zero and have the same sign; ellipsoids degenerate to\na point when τ2−ξ2\n3=n2>0. In the hyperbolic case one can check that Rn(τ,ξ)\nis non-empty for all integers n. The special case n= 0 is a particular hyperbolic\nresonance, to be referred to as Landau resonance, for which R0(τ,ξ) is non empty\nonly ifτ2<ξ2\n3.\nWe shall study the limit in S′(R3) of functions of the form\n(66) ˜ rε(τ,ξ,u) =˜Pε(τ,ξ,u)\nsin/parenleftbig\nπaε(τ,ξ,u)/parenrightbig,\nwhere˜Pεis a family of functions parameterized by ε∈[0,ε0] for a fixed ε0>0.\nThe choices of ˜Pεrelevant to our analysis are ˜Pε=ie−iπaεΦjP±\nε/2 and˜Pε=\nie−iπaεΨjP±\nε/2, with notation of section 5.1. With this aim, it will be sufficient to\nconsider a family of functions ˜Pεthat satisfy the following conditions:\n(67)\n\n˜Pε∈C∞(R7), ∀ε∈[0,ε0],\n/vextendsingle/vextendsingle∂µ˜Pε(τ,ξ,u)/vextendsingle/vextendsingle≤ˆcµ(1+τ2+ξ2)ˆmµ(1+u2)ˆnµ,∀ε∈[0,ε0],∀µ∈N7\n0,\n˜Pε(τ,ξ,u)→˜P0(τ,ξ,u) forε→0+, ∀(τ,ξ,u)∈R7,\nwhere the constants ˆ cµ,ˆmµ,ˆnµ∈Rare independent of ε∈[0,ε0] and (τ,ξ,u)∈R7.\nWe state the main result for the limit of (66) as ε→0+. We shall see that it is\nsufficient to consider the case κs= 1, orνs=ωc,s/γ; thenaεis independent of the\nparticle species. This is a valid choice of the damping coefficient νs, since it satisfies\nconditions (17).\nProposition 5.2. Letε0,τ0>0, and let ˜rεbe the family of functions defined for\nε∈(0,ε0]in equation (66) with ˜Pεsatisfying condition (67), and with κs= 1.\nThen, for every (τ,ξ),τ/\\e}atio\\slash= 0, there is ˜r0(τ,ξ,·)∈ S′(R3), such that\n(i)in the limit ε→0+,˜rε(τ,ξ,·)→˜r0(τ,ξ,·)inS′(R3);\n(ii)for everyφ∈ S(R3)the function (τ,ξ)/ma√sto→/angbracketleftbig\n˜r0(τ,ξ,·),φ/angbracketrightbig\nis continuous on\n(R\\ {0})×R3, andC∞near points (τ,ξ)such thatτ2/\\e}atio\\slash=ξ2\n3+n2for all\nintegersn≥0;\n(iii)for anyφ∈ S(R3)there are reals K0,M >0, such that\n/vextendsingle/vextendsingle/angbracketleftbig\n˜rε(τ,ξ,·),φ/angbracketrightbig/vextendsingle/vextendsingle≤K0(1+τ2+ξ2)M,\nfor all(τ,ξ)with|τ| ≥τ0andε∈[0,ε0].\nRemark 10 (Degenerate resonances) .The varieties τ2=ξ2+n2forn≥1 in the\nFourier space correspond to plane waves for which the elliptic reson anceRn(τ,ξ)\ndegenerates to a point. The special case n= 0, that is τ2−ξ2= 0, corresponds\nto the parabolic resonance which separates elliptic and hyperbolic re sonances. The\nfunction (τ,ξ)/ma√sto→/angbracketleftbig\n˜r0(τ,ξ,·),φ/angbracketrightbig\nis smooth away from such topological transitions,\nwhere we can show continuity only.\nFor the proofofproposition5.2, weneed afew preparatoryresult s. Let us choose\na cut-off function χ∈C∞\n0(R) such that\n0≤χ(z)≤1, χ(z) = 1,for|z|<1\n4, χ(z) = 0,for|z| ≥1\n3,34 O. LAFITTE AND O. MAJ\nand for any n∈Z,δ∈(0,1) let\nχδ,n(τ,ξ,u) =χ/parenleftbig\n(a0(τ,ξ,u)−n)/δ/parenrightbig\n.\nOne can see that χδ,n∈C∞(R7) but it is not necessarily compactly supported; in\nfact,χδ,n(τ,ξ,·) is localized around Rn(τ,ξ) which is unbounded when τ2≤ξ2\n3.\nLemma 5.3. If(τ,ξ,u)∈supp(χδ,n)for a certain n∈Zthenχδ,m(τ,ξ,u) = 0for\nallm/\\e}atio\\slash=n.\nProof.The set supp χδ,nis given by the condition |a0−n| ≤δ/3, so that\n|a0(τ,ξ,u)−m| ≥ |m−n|−|a0(τ,ξ,u)−n|>1−δ/3>δ/3,\nand thusχδ,m(τ,ξ,u) = 0. /square\nIt follows from lemma 5.3 that the sum\n(68) χδ=/summationdisplay\nn∈Zχδ,n,\nis locally finite and thus it defines a family of smooth functions χδforδ∈(0,1).\nLemma 5.4. For everyδ∈(0,1), the cut-off function χδin (68) is such that\n/vextendsingle/vextendsinglecos/parenleftbig\nπa0(τ,ξ,u)/parenrightbig/vextendsingle/vextendsingle≥cos(πδ/3)>0, for(τ,ξ,u)∈supp(χδ),\n/vextendsingle/vextendsinglesin/parenleftbig\nπa0(τ,ξ,u)/parenrightbig/vextendsingle/vextendsingle≥sin(πδ/4)>0, for(τ,ξ,u)∈supp(1−χδ).\nProof.For every point ( τ,ξ,u) in the support of χδ, there is an integer nsuch that\n|a0−n| ≤δ/3,|cos(πa0)|=/vextendsingle/vextendsinglecos/parenleftbig\nπ(a0−n)/parenrightbig/vextendsingle/vextendsingle≥cos(πδ/3)>0.\nAnalogously, on the support of 1 −χδ,\n|a0−n| ≥δ/4,|sin(πa0)|=/vextendsingle/vextendsinglesin/parenleftbig\nπ(a0−n)/parenrightbig/vextendsingle/vextendsingle≥sin(πδ/4)>0,\nfor all integers n. /square\nThe cut-off function χδallowsus to isolatethe singularities of (66) when ε→0+.\nIfνsin equation (55) is chosen so that κs= 1, for (τ,ξ,u)∈supp(χδ) we can write\nsin(πaε) = cos(πa0)/bracketleftbig\ntan(πa0)cosh(πε)+isinh(πε)/bracketrightbig\n,\nand\n(69)1\nsin(πaε)=−i\ncos(πa0)/integraldisplay+∞\n0eiλ[tan(πa0)cosh(πε)+isinh(πε)]dλ.\nTherefore, for any ψ∈C∞\n0(R3),\n(70) /a\\}b∇acketle{t˜rε(τ,ξ,·),ψ/a\\}b∇acket∇i}ht=−i/integraldisplay+∞\n0e−λsinh(πε)Ic\nδ,ε(ψ)(τ,ξ,λ)dλ+Is\nδ,ε(ψ)(τ,ξ),\nwhere we have defined the functions\nϑε(τ,ξ,u):= tan/parenleftbig\nπa0(τ,ξ,u)/parenrightbig\ncosh(πε), (71a)\nIc\nδ,ε(ψ)(τ,ξ,λ):=/integraldisplay\nR3eiλϑε(τ,ξ,u)˜Pε(τ,ξ,u)ψ(u)\ncos/parenleftbig\nπa0(τ,ξ,u)/parenrightbigχδ(τ,ξ,u)du, (71b)\nIs\nδ,ε(ψ)(τ,ξ):=/integraldisplay\nR3˜Pε(τ,ξ,u)ψ(u)\nsin/parenleftbig\nπaε(τ,ξ,u)/parenrightbig/parenleftbig\n1−χδ(τ,ξ,u)/parenrightbig\ndu. (71c)TEMPERED-IN-TIME RESPONSE OF A PLASMA 35\nWe also define\nIc\nδ,0(ψ)(τ,ξ,λ):=/integraldisplay\nR3eiλtan(πa0(τ,ξ,u))˜P0(τ,ξ,u)ψ(u)\ncos/parenleftbig\nπa0(τ,ξ,u)/parenrightbigχδ(τ,ξ,u)du, (71d)\nIs\nδ,0(ψ)(τ,ξ):=/integraldisplay\nR3˜P0(τ,ξ,u)ψ(u)\nsin/parenleftbig\nπa0(τ,ξ,u)/parenrightbig/parenleftbig\n1−χδ(τ,ξ,u)/parenrightbig\ndu. (71e)\nThe main step in the proof of proposition 5.2 consists of an application of the\nstationary phase formula [29, Chapter 7] in order to prove that Ic\nδ,ε(ψ)(τ,ξ,·) is\nbounded in L1uniformly in ε. The real-valued phase is given by ϑεand the pa-\nrameter isλ≥1. All technical results needed in the proof of proposition 5.2 are\ncollected in section 6, below.\nProof of proposition 5.2. We start from identity (70),\n/a\\}b∇acketle{t˜rε(τ,ξ,·),ψ/a\\}b∇acket∇i}ht=−i/integraldisplay+∞\n0e−λsinh(πε)Ic\nδ,ε(ψ)(τ,ξ,λ)dλ+Is\nδ,ε(ψ)(τ,ξ),\nforψ∈C∞\n0(R3) and lemma 6.4 (i) shows that this is extended to φ∈ S(R3).\nFor everyδ∈(0,1), lemma 6.4 (iii) and (v) allows us to define the functional\n(72)/angbracketleftbig\n˜rδ,0(τ,ξ,·),φ/angbracketrightbig\n=−i/integraldisplay+∞\n0Ic\nδ,0(φ)(τ,ξ,λ)dλ+Is\nδ,0(φ)(τ,ξ),\noverS(R3) for every ( τ,ξ),τ/\\e}atio\\slash= 0.\nIn view of lemma 6.4 (ii), as ε→0+, we have Ic\nδ,ε(φ)(τ,ξ)→ Ic\nδ,0(φ)(τ,ξ) and\nIs\nδ,ε(φ)(τ,ξ)→ Is\nδ,0(φ)(τ,ξ) for allφ∈ S(R3). By the dominated convergence\ntheorem and lemma 6.4 (iii) we deduce\n/integraldisplay+∞\n0e−λsinh(πε)Ic\nδ,ε(φ)(τ,ξ,λ)dλ→/integraldisplay+∞\n0Ic\nδ,0(φ)(τ,ξ,λ)dλ,\nfor allφ∈ S(R3), hence ˜rε(τ,ξ,·)→˜rδ,0(τ,ξ,·) inS′(R3). By uniqueness of the\nlimit, we have that ˜ rδ,0(τ,ξ,·) is the same tempered distribution for all δ∈(0,1)\nwhich we denote by ˜ r0(τ,ξ,·). This proves (i).\nAs for the regularity of /a\\}b∇acketle{t˜r0(τ,ξ,·),φ/a\\}b∇acket∇i}htwith respect to ( τ,ξ), lemma 6.4 (i) show\nin particular that Is\nδ,0(φ)∈C∞, hence it is enough to address\n/a\\}b∇acketle{t˜rc\nδ,0(τ,ξ,·),φ/a\\}b∇acket∇i}ht:=/a\\}b∇acketle{t˜rδ,0(τ,ξ,·),φ/a\\}b∇acket∇i}ht−Is\nδ,0(φ)(τ,ξ) =−i/integraldisplay+∞\n0Ic\nδ,0(φ)(τ,ξ,λ)dλ.\nIn lemma 6.4 (i) we have established that Ic\nδ,0(φ) isC∞, and the inequality in item\n(iii) of the same lemma 6.4 gives a function ˜Bε0,δ∈L1(R+) such that\n|Ic\nδ,0(φ)(τ,ξ,λ)| ≤˜Bε0,δ(λ),\nuniformlyin ( τ,ξ)∈KwhereKis anycompact where |τ|>0. Thenthe dominated\nconvergence theorem can be applied to show that /a\\}b∇acketle{t˜rc\nδ,0(τ,ξ,·),φ/a\\}b∇acket∇i}htis continuous at\nany point (τ,ξ), where |τ|>0.\nConcerning the derivatives of Ic\nδ,0(φ) with respect to ( τ,ξ), let as fix a point\n(¯τ,¯ξ) such that |¯τ| /\\e}atio\\slash= 0 and ¯τ2/\\e}atio\\slash=¯ξ2\n3+n2for alln∈N0. For a sufficiently small\nradiusρ >0 the closed ball K={(τ,ξ) : (τ−¯τ)2+(ξ−¯ξ)2≤ρ2}satisfies the\nassumptions of lemma 6.4 item (iv). Therefore, there is a value of δdepending only\nonKand an upper bound |∂α\nτ,ξIc\nδ,ε(φ)(τ,ξ,λ)| ≤ B(α)\nε0,δ(λ,τ∗) withB(α)\nε0,δ(·,τ∗)∈L1,36 O. LAFITTE AND O. MAJ\nwhereτ∗= min|τ|inK. Hence, /a\\}b∇acketle{t˜rc\nδ,0(τ,ξ,·),φ/a\\}b∇acket∇i}htis of classC∞near any point where\n|τ|>0 andτ2/\\e}atio\\slash=ξ2\n3+n2, as claimed in (ii).\nAgain lemma 6.4, items (iii) and (v), imply\n|/a\\}b∇acketle{t˜rε(τ,ξ,·),φ/a\\}b∇acket∇i}ht| ≤ /ba∇dblBε0,δ(·,τ0)/ba∇dblL1(R+)(1+τ2+ξ2)m4+5\n2/ba∇dblφ/ba∇dbl2˜m4+10\n+Ks\n0(1+τ2+ξ2)ℓ0,0/ba∇dblφ/ba∇dbl2˜ℓ0,0+4,\nuniformly in ( τ,ξ) where|τ| ≥τ0>0, wheremj, ˜mj,ℓ0,0and˜ℓ0,0have been defined\nin lemma 6.3 and 6.4, respectively. This estimate is uniform in ε∈[0,ε0]. Since\nm4≥ℓ0,0, we obtain claim (iii) with constant K0depending in particular on τ0,ε0\nandδand withM=m4+5/2. /square\nRemark 11.The estimate in proposition 5.2 item (iii) could be replaced by\n/vextendsingle/vextendsingle/angbracketleftbig\n˜rε(τ,ξ,·),φ/angbracketrightbig/vextendsingle/vextendsingle≤K0(τ)(1+τ2+ξ2)M, τ/\\e}atio\\slash= 0,\nbutK0(τ) is not bounded near τ= 0. Hence our argument does not allow any\nconclusion for τ= 0 and we have excluded all frequencies in |τ|< τ0withτ0\narbitrarily small and fixed.\n5.3.Solution of the linear Vlasov equation for the magnetized ca se.We\naddress equation (15) for a given species s, and thus drop the subscript sfor sim-\nplicity. Particularly, we shall denote by Qjthe differential operator Qs,jdefined in\nequation (58), which depends on the equilibrium distribution function and the sign\nof the electric charge of the consider particle species.\nWe shall first addressthe existence of solutionsof the linear Vlasov equation (18)\nfor the time-derivative of the distribution function including a dampin g term and\naddress its dissipation-less limit. The result will then be used to compu te the\ntime-derivative of the induced current (19).\nTheorem 5.5. Letε >0and letνbe any function satisfying conditions (17).\nThen equation (18) has a solution gε∈ S(R7)which is unique as an element of\nS′(R7)and:\n(i)The Fourier transform of the unique tempered solution gεis\nˆgε(ω,k,u) =qn0\n(mc)43/summationdisplay\nj=1ˆEj(ω,k)Qj(τ,ξ,u,∂ξ)r±\nε(τ,ξ,u),\nwhere the sign +(resp.−) is chosen for q>0(resp.q<0).\n(ii)For(u1,u2)/\\e}atio\\slash= (0,0),\nˆgε(ω,k,u) =iqn0\n(mc)43/summationdisplay\nj=1ˆEj(ω,k)/summationdisplay\nn∈ZQj(τ,ξ,u,∂ξ)A±\nn(b)\naε−ne+inφ,\nwhereφis defined in (54) and b= (b1,b2)in (56).\nRemark 12.The expression of the solution in item (i) in terms of the integral\nr±\nεdefined in (63) was first proposed by Qin et al. [49], cf. also the subse quent\ndiscussion in the literature [40, 50].\nRemark 13.The solution ˆ gεgiven in item (ii) is in agreement with the standard\nexpression obtained in the physics literature [9, 14, 54]. For ε→0+it exposes a\ncountable number of poles for a0∈Zwhere sin(πa0) = 0. These poles correspond\nto cyclotron resonances briefly discussed in section 5.2.TEMPERED-IN-TIME RESPONSE OF A PLASMA 37\nProof of theorem 5.5. We look for tempered solutions of equation (18) and thus we\ncan equivalently consider the Fourier transform ˆ gεof the distribution function gε.\nIf we define\n(73) ˆhε=e±i(ξ2u1−ξ1u2)ˆgε,\nwhereτ=ω/ωc,s,ξ=ck/ωc,s, and with sign + (resp. −) forq >0 (resp.q <0),\nthe Fourier transform of equation (18) is equivalent to equation (5 3) with source\nˆs±=|q|n0\n(mc)4e±i(ξ2u1−ξ1u2)3/summationdisplay\nj=1ˆEjQj(τ,ξ,u,∂ξ)e±i(ξ2u1−ξ1u2),\nand with the sign chosen according to the electric charge q. (As in the proof of\nproposition5.1, in the case q<0 we canmultiply by −1the equationfor ˆhεin order\nrecast it into the form of (53); hence a=−aε.) Under the hypotheses, s±belongs\ntoS(R7). Proposition 4.6 gives a solution ˆhε∈ S(R7), and thus ˆ gε∈ S(R7). This\nis the unique solution in C∞, as proven in proposition 4.6, as well as in S′(R7), as\nproven in proposition 4.8.\n(i) Substitution of the claimed expression ˆ gεinto (73) gives\nˆhε(ω,k,u) =qn0\n(mc)43/summationdisplay\nj=1ˆEj(ω,k)e±i(ξ2u1−ξ1u2)Qj(τ,ξ,u,∂ξ)r±\nε(τ,ξ,u),\nwhereτ=ω/ωc,s,ξ=ck/ωc,s, and we observe that the operator e±i(ξ2u1−ξ1u2)Qj\ncommutes with u2∂u1−u1∂u2. Then upon using lemma 5.1, we have that ˆhε\nsolves (53) and thus ˆ gεis the solution of equation (15).\n(ii) Upon using coordinates (54), for either choice of the sign of the particle\ncharge the equation for ˆhεreduces to (42) with ˜ a=aεand with source\nV(φ) =qn0\n(mc)4e±i(ξ2u1−ξ1u2)3/summationdisplay\nj=1ˆEjQje±i(ξ2u1−ξ1u2).\nWe evaluate the Fourier coefficients of the source, that is,\nˆVn=qn0\n(mc)41\n2π/integraldisplay2π\n0e−inφe±i(ξ2u1−ξ1u2)3/summationdisplay\nj=1ˆEjQje±i(ξ2u1−ξ1u2)dφ\n=qn0\n(mc)4e±i(ξ2u1−ξ1u2)3/summationdisplay\nj=1ˆEjQj1\n2π/integraldisplay2π\n0e−inφe±i(ξ2u1−ξ1u2)dφ,\nand,inthesecondidentity, wehaveusedthefactthatthecoefficie ntsoftheoperator\ne±i(ξ2u1−ξ1u2)Qjare independent of φ, i.e., the exponential factor is canceled by the\ncorresponding factor in the definition of the coefficients in equation s (59). It is now\nsufficient to compute the Fourier coefficients of\ne±i(ξ2u1−ξ1u2)=e±i(b2cosφ±b1sinφ),\nand those are equal to A±\nn(b) as we have shown in equation (62). Then\nˆVn=qn0\n(mc)4e±i(ξ2u1−ξ1u2)3/summationdisplay\nj=1ˆEjQjA±\nn.\nThe Fourier expansion in lemma 4.4 then yields the claimed identity. /square38 O. LAFITTE AND O. MAJ\nWe shallnowshowthat the solution gεforε→0+approachesthe causalsolution\nof (15) as computed by integration along the characteristics curv es. This result in\naddition clarifies the relation between the analytical expressions fo r the solution in\nFourierandphysicalspaces. In the position-momentumvariables, thecharacteristic\ncurvet′/ma√sto→/parenleftbig\nX(t′;t,x,p),P(t′;t,x,p)/parenrightbig\nwith terminal condition ( x,p) att′=tis [9]\n(74)\n\nX1(t′;t,x,p) =x1−p1\nmγΩsin/parenleftbig\nΩ·(t−t′)/parenrightbig\n−p2\nmγΩ/bracketleftbig\ncos/parenleftbig\nΩ·(t−t′)/parenrightbig\n−1/bracketrightbig\n,\nX2(t′;t,x,p) =x2+p1\nmγΩ/bracketleftbig\ncos/parenleftbig\nΩ·(t−t′)/parenrightbig\n−1/bracketrightbig\n−p2\nmγΩsin/parenleftbig\nΩ·(t−t′)/parenrightbig\n,\nX3(t′;t,x,p) =x3−v3(t−t′),\nP1(t′;t,x,p) =p1cos/parenleftbig\nΩ·(t−t′)/parenrightbig\n−p2sin/parenleftbig\nΩ·(t−t′)/parenrightbig\n,\nP2(t′;t,x,p) =p1sin/parenleftbig\nΩ·(t−t′)/parenrightbig\n+p2cos/parenleftbig\nΩ·(t−t′)/parenrightbig\n,\nP3(t′;t,x,p) =p3,\nwhereγand thus Ω are constants of motion. Under assumption (17) νis con-\nstant along the characteristics as well. The characteristic flow has the semi-group\nproperty: for every t1,t2,t3witht1≤t2≤t3and (x,p),\n(75)X(t1;t2,X(t2;t3,x,p),P(t2;t3,x,p)) =X(t1;t3,x,p),\nP(t1;t2,X(t2;t3,x,p),P(t2;t3,x,p)) =P(t1;t3,x,p),\nwhich can be verified directly from (74). For EandF0at least in C1, lets=\n−q(∂tE−v× ∇ ×E)· ∇pF0for brevity, and F0=n0Gas in (14). We shall\nalways imply the relation u=p/(mc) between normalized and physical momentum\nvariables.\nProposition 5.6. For everyε >0,E∈[S(R4)]3,F0=n0G∈ S(R3), and\ng0,ε∈ S(R6), the Cauchy problem for equation (15) with initial conditio ng0,εat\ntimet=t0has a unique classical solution gε∈C∞(R7)and it holds that:\n(i)There is a unique Cauchy datum for which the solution gε∈ S′and that is\ngiven by\ng∗,ε(x,u):=/integraldisplayt0\n−∞e−εν(t0−t′)s/parenleftbig\nt′,X(t′;t0,x,p),P(t′;t0,x,p)/parenrightbig\ndt′,\nwhich belongs to S(R6).\n(ii)Forg0,ε=g∗,εthe solution is\ngε(t,x,u) =/integraldisplayt\n−∞e−εν(t−t′)s/parenleftbig\nt′,X(t′;t,x,p),P(t′;t,x,p)/parenrightbig\ndt′,\nandˆgεis the same function defined in theorem 5.5 (i).\n(iii)Forε→0+, the function gεhas a pointwise limit\ng(t,x,u) =/integraldisplayt\n−∞s/parenleftbig\nt′,X(t′;t,x,p),P(t′;t,x,p)/parenrightbig\ndt′,\nfor every (t,x,u)∈R7. The limit gbelongs toC∞\nb(R7), it is a classical\nsolution of equation (15), and it is independent of the choic e ofν.\nRemark 14.The first part of this statement is essentially Wollman’s result on the\nlinear Vlasov equation [57, theorem 3.1], but for the case of a uniform plasmaTEMPERED-IN-TIME RESPONSE OF A PLASMA 39\nequilibrium and with somewhat relaxed hypotheses on the support of the initial\ndatum.\nProof.By hypothesis the damping function νis constant along the characteristics.\nThe application of the standard method of characteristics [34] give s the unique\nclassical solution of the Cauchy problem,\ngε(t,x,u) =g0,ε/parenleftbig\nX(t0;t,x,p),P(t0;t,x,p)/parenrightbig\ne−εν(t−t0)\n+/integraldisplayt\nt0e−εν(t−t′)s/parenleftbig\nt′,X(t′;t,x,p),P(t′;t,x,p)/parenrightbig\ndt′,\nwhereXandPare given in equation (74), and gε∈C∞follows by Leibniz rule for\ndifferentiation of integrals. This completes the first part of the pro position.\nSincee−εν(t−t′)is integrable for t′∈(−∞,t], the integral\nHε(t,x,p) =/integraldisplayt\n−∞e−εν(t−t′)s/parenleftbig\nt′,X(t′;t,x,p),P(t′;t,x,p)/parenrightbig\ndt′\nis finite. If the electric field is given in Fourier transform, using\nE(t,x) =1\n(2π)4/integraldisplay\nR4e−iωt+ik·xˆE(ω,k)dωdk,\nand applying Fubini’s theorem, we get\nHε(t,x,p) =1\n(2π)4/integraldisplay\nR4e−iωt+ik·x˜Hν(t,ω,k,p)dωdk,\nwith, using F0=n0G,\n˜Hε(t,ω,k,p) =qn0\n(mc)43/summationdisplay\nj=1ˆEj(ω,k)Qj(τ,ξ,u,∂ξ)R±\nε(t,ω,k,p),\nwhereQjare the operators defined in (58) for the considered particle spec ies, and\nR±\nε(t,ω,k,p) =ωc\nγ/integraldisplayt\n−∞e−εν(t−t′)+iω(t−t′)−ik/bardblv/bardbl(t−t′)\ne±i(ξ2u1−ξ1u2)cos(ωc\nγ(t−t′))−i(ξ1u1+ξ2u2)sin(ωc\nγ(t−t′))dt′,\nwith sign chosen according to the particle charge q. The change of variable λ=\nωc\nγ(t−t′) shows that R±\nεis actually independent of time tandR±\nε(t,ω,k,p) =\nr±\nε(τ,ξ,u) wherer±\nεhas been defined in (63). Then, ˜Hεis also independent of time\nand˜Hε(t,ω,k,p) = ˆgε(ω,k,p) with ˆgεthe unique tempered solution established in\ntheorem 5.5 (i). We have ˆ gε∈ Sand thusHεis the inverse Fourier transform of\na Schwartz function, so that Hε∈ S(R7). By definition g∗,ε(x,p) =Hε(t0,x,p) =\ngε(t0,x,p), henceg∗,ε∈ Sand it is the Cauchy datum of the unique tempered\nsolutiongε=Hε. This proves (i) and item (ii) follows from the expression for Hε.\nAs for the pointwise limit of the solution, item (iii), we observe that the char-\nacteristic flow and the source term ssatisfy the hypothesis of proposition C.2; in\nparticular, the flow (74) satisfies assumptions (i)-(iv) of appendix C, while sis\ndefined as the sum of the products of a function in S(R4) in frequency and wave-\nvector and a function in S(R3) in momentum, hence s∈ S(R7). The function g40 O. LAFITTE AND O. MAJ\nis then the causal solution of (16) in the sense of appendix C and pro position C.2\ngivesg∈C∞\nb(R7). For every m∈N0one has the estimate\n(1+s2)m/vextendsingle/vextendsingles/parenleftbig\ns,X(s;t,x,p),P(s;t,x,p)/parenrightbig/vextendsingle/vextendsingle\n≤(1+s2+X(s;t,x,p)2+P(s;t,x,p)2)m/vextendsingle/vextendsingles/parenleftbig\ns,X(s;t,x,p),P(s;t,x,p)/parenrightbig/vextendsingle/vextendsingle,\nwhich gives\n/vextendsingle/vextendsingles/parenleftbig\ns,X(s;t,x,p),P(s;t,x,p)/parenrightbig/vextendsingle/vextendsingle≤/ba∇dbls/ba∇dbl2m\n(1+s2)m,\nwhere/ba∇dbl·/ba∇dblmare the semi-norms defined in appendix A. Therefore, for every ( t,x,p)\nand formlarge enough, dominated convergence allows us to pass to the limit\nε→0+in the integrand. /square\n5.4.Current density and conductivity operator. Firstweshowthat thefunc-\ntionsr±\nεin lemma 5.1 have a limit in S′. For simplicity let\n(76)ζ±(ξ1,ξ2,u1,u2,λ) = (ξ1u1+ξ2u2)sinλ∓(ξ2u1−ξ1u2)cosλ,\nand we recall the definition of a0in equation (56) and normalized Fourier vari-\nables (55).\nProposition 5.7. For every integer m≥2,\n(i)the linear map\nψ/ma√sto→/integraldisplay /integraldisplay1\n0eia0(τ,ξ,u)λ−iζ±(ξ1,ξ2,u1,u2,λ)dλψ(τ,ξ,u)dτdξdu\n+im/integraldisplay /integraldisplay+∞\n1λ−meia0(τ,ξ,u)λ−iζ±(ξ1,ξ2,u1,u2,λ)dλ∂m\nτψ(τ,ξ,u)\nγ(u)mdτdξdu,\nis continuous on S(R7)and thus defines a tempered distribution r±,m\n0;\n(ii)forε→0+,r±\nεhas a limit r±\n0inS′, andr±\n0=r±,m\n0for allm≥2. The\nlimit is independent of the choice of the function νs.\nProof.The two terms defining the linear map on ψare bounded by the norms /ba∇dblψ/ba∇dbl8\nand/ba∇dblψ/ba∇dblm+8, respectively, andthis showscontinuityinthe topologyof S(R7). Then\nthe map is a tempered distribution which is denoted by r±,m\n0. Since\n(−i)m\nγmλm∂m\nτeiaελ−iζ±=eiaελ−iζ±,\nafter integration by parts in τ,\n/a\\}b∇acketle{tr±\nε,ψ/a\\}b∇acket∇i}ht=/integraldisplay /integraldisplay1\n0eiaελ−iζ±dλψdτdξdu\n+im/integraldisplay/bracketleftig/integraldisplay+∞\n1eiaελ−iζ±\nλmdλ/bracketrightig∂m\nτψ\nγmdτdξdu,\nand ifm≥2 the integrand is uniformly bounded by an integrable function. We\ncan then pass to the limit in the integral and obtain\n/a\\}b∇acketle{tr±\nε,ψ/a\\}b∇acket∇i}htε→0+\n− −−− → /a\\}b∇acketle{tr±,m\n0,ψ/a\\}b∇acket∇i}ht,\nfor everyψ∈ S(R7) and every m≥2. Uniqueness of the limit implies that all\ndistributions r±,m\n0form≥2 are equal to the limit r±\n0= limε→0+r±\nε. /squareTEMPERED-IN-TIME RESPONSE OF A PLASMA 41\nThis result is already sufficient to compute the limit of the time derivativ e of the\ninducedcurrentdensityasatempereddistributionunderfairlygen eralassumptions.\nIn the following we define, for any slabeling a particle species,\nrs,ε(τ,ξ,u):=r±\nε(τ,ξ,ν), rs:=r±\n0,\nwith sign + (resp. −) forqs>0 (resp.qs<0), withr±\nεdefined in equation (63),\nand withr±\n0being the distributional limit in lemma 5.7.\nBy definition, cf. equation (19), the current density associated t o{gs,ε}is\n∂tjε=K({gs,ε}).\nWe recall that ωp,s=/radicalbig\n4πq2sns,0/msis the plasma frequency of the s-th species.\nWe also need the operator Qs,jdefined in equation (58). Relations (55) are also\nimplied.\nProposition 5.8. The function ∂tjεbelongs to [S(R4)]3and\n(77a) /hatwidest∂tjε(ω,k) = ˆςε(ω,k)ˆE(ω,k),\nwhere the tensor ˆςεis given component-wise by\n(77b) ˆςε,ij(ω,k) =/summationdisplay\nsω2\np,s\n4π/integraldisplay\nR31\nγ(u)uiQs,j(τ,ξ,u,∂ξ)rs,ε(τ,ξ,u)du.\nProof.Intheorem5.5wehaveshownthat gs,ε∈ S(R7), henceforevery l,m,n∈N0\nandα∈N4\n0,\n(1+t2+x2)m(1+u2)n/vextendsingle/vextendsingle∂l\nt∂α\nxgs,ε(t,x,u)/vextendsingle/vextendsingle≤ /ba∇dblgs,ε/ba∇dbl|α|+l+2m+2n.\nSince forn >3/2, (1 +u2)−nis integrable, from equation (19) we deduce that\n∂tjε=K({gs,ε}) isC∞and the same inequality implies that all derivatives are\nrapidly decaying at infinity, that is, ∂tjε∈[S(R4)]3.\nThen the Fourier transform of ∂tjεexists in the classical sense and, by Fubini’s\ntheorem, we have\n/hatwidest∂tjε(ω,k) =/summationdisplay\nsqs(msc)3/integraldisplay\nvs(u)ˆgs,ε(ω,k,u)du.\nUpon using the expressionfor ˆ gs,εin theorem 5.5 (i), we arriveat equation (77). /square\nRemark 15.Convergence of the integral in (77b) is ensured by the rapid decay in\nuat infinity of the coefficients (57) in the operators Qs,j.\nWe recall that Qs,jare first-order partial differential operators and we can define\nthe formal adjoint Q′\ns,jby\n/integraldisplay\nR7χ1(τ,ξ,u)Qs,jχ2(τ,ξ,u)dτdξdu=/integraldisplay\nR7χ2(τ,ξ,u)Q′\ns,jχ1(τ,ξ,u)dτdξdu,\nfor everyχ1,χ2∈ S(R7). The adjoint Q′\ns,jis again a first-order partial differential\noperator and it is continuous from S → S. Therefore we can define the distribution\n∂tj∈[S′(R4)]3by\n/a\\}b∇acketle{t/hatwider∂tj,ˆϕ/a\\}b∇acket∇i}ht:=3/summationdisplay\nj=1/summationdisplay\nsω4\nc,sω2\np,s\n4πc3/a\\}b∇acketle{trs,Q′\ns,j/parenleftbigˆϕ·u\nγˆEj/parenrightbig\n/a\\}b∇acket∇i}ht,\nfor every vector test-function ˆ ϕ∈[S(R4)]3.42 O. LAFITTE AND O. MAJ\nProposition 5.9. Asε→0+,∂tjε→∂tjin[S′(R4)]3and the limit satisfies\n∂tj=K({gs})∈[C∞\nb(R4)]3.\nProof.Step 1:∂tjε→∂tj. Forε>0, one has\n/a\\}b∇acketle{t/hatwidest∂tjε,ˆϕ/a\\}b∇acket∇i}ht=/summationdisplay\nsω4\nc,sω2\np,s\n4πc33/summationdisplay\nj=1/a\\}b∇acketle{trs,ε,Q′\ns,j/parenleftbigˆϕ·u\nγˆEj/parenrightbig\n/a\\}b∇acket∇i}ht,\nwhere the factor ω4\nc,sc−3comes for the chance of variable dωdk= (ω4\nc,s/c3)dτdξ.\nThe result then follows from the convergence of rs,εinS′proven in proposition 5.7.\nStep 2:gsbelongs to the domain of the operator K. Wee observe that the flow\n(74) is polynomially bounded and thus satisfies the hypothesis of lemm a C.3. Then,\nsince|v(u)|=c|u|/γ(u)≤c, for everyn,l∈N0andα∈Nd\n0,\n(78) (1+ u2)n|v(u)∂l\nt∂α\nxgs(t,x,u)| ≤c(1+u2)n|∂l\nt∂α\nxgs(t,x,u)| ≤C,\nand upon choosing n>3/2 we deduce that u/ma√sto→v(u)∂l\nt∂α\nxgs(t,x,u) is bounded by\nanL1-function. For l= 0 andα= 0, this shows that gshas finite first velocity\nmoment. Iterated application of the dominated convergence theo rem also shows\nthatK({gs})∈C∞\nb(R4).\nStep 3:∂tj=K({gs}). Let/tildewider∂tj=K({gs}). For every ϕ∈[S(R4)]3,\n/a\\}b∇acketle{t∂tj−/tildewider∂tj,ϕ/a\\}b∇acket∇i}ht=−/a\\}b∇acketle{t∂tjε−∂tj,ϕ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tK({gs,ε−gs}),ϕ/a\\}b∇acket∇i}ht.\nThe first terms on the right-hand side converges to zero for ε→0+. The second\nterm is estimated by\n/a\\}b∇acketle{tK({gs,ε−gs}),ϕ/a\\}b∇acket∇i}ht=/summationdisplay\nsqs(msc)3/integraldisplay\nR4/integraldisplay\nR3(gs,ε−gs)(t,x,u)ϕ(t,x)·v(u)dudtdx.\nWe know that gs,ε∈ Sand upon using estimate (78) with l= 0 andα= 0, we can\nbound the integrand by an L1-function independent on εso thus pass to the limit\nin the integrand. At last from proposition 5.6, we have gs,ε−gs→0. /square\nProposition 5.9 is sufficient to prove the continuity of ς: [S(R4)]3→[S′(R4)]3,\ncf. equation(20). However,wewishtounderstandunderwhichco nditionsςreduces\nto a Fourier multiplier and, when this is the case, we wish to address th e regularity\nof its symbol.\nSince the limit ∂tjis independent of the choice of the damping function, we\nchooseνssuch thatκs= 1; one can check that this satisfies condition (17).\nWe start from equation (77) which can be rewritten as\n(79) ˆςε,ij(ω,k) =/summationdisplay\nsω2\np,s\n4π/bracketleftig/integraldisplay\nR3ui\nγΦjrs,ε(τ,ξ,u)Fs(u)du\n+/integraldisplay\nR3ui\nγΨjrs,ε(τ,ξ,u)Gs(u)du/bracketrightig\n.\nSinceaεdepends on ξthroughξ3only, we have\n(80) Φ jrs,ε=ie−iπaεΦjP±\nε\n2sin(πaε),Ψjrs,ε=ie−iπaεΨjP±\nε\n2sin(πaε),\nwith sign + (resp., −) forqs>0 (resp.,qs<0).\nWe shall now apply proposition 5.2, which holds for generic functions o f the\nform (66) with ˜Pεsubject to condition (67), to the case of interest, which is givenTEMPERED-IN-TIME RESPONSE OF A PLASMA 43\nin equations (79) and (80). We choose a normalization time-scale com mon to all\nparticle species, that is, ˆ ω:= max sωc,s.\nProposition 5.10. Under the same hypotheses as in theorem 5.5, for ω/\\e}atio\\slash= 0, the\ntensorˆςε(ω,k)definedin (77), has a pointwise limit ˆς0(ω,k), which can be computed,\ncf. equation (81), and such that:\n(i)The limit ˆς0is continuous on (R\\{0})×R3and it isC∞whereω/\\e}atio\\slash= 0and\nω2/\\e}atio\\slash= (ck3)2+n2ω2\nc,sfor alln∈Zand species index s.\n(ii)For anyω0>0, there is a constant K0(ω0)for which\n|ˆς0(ω,k)| ≤K0(ω0)(1+(ω\nˆω)2+(ck\nˆω)2)M,\nuniformly for (ω,k)∈R4,|ω| ≥ω0, with exponent Mindependent of ω0.\n(iii)For allE∈[S(R4)]3such thatω/\\e}atio\\slash= 0insuppˆEwe have∂tj=F−1/parenleftbig\nˆς0ˆE/parenrightbig\n.\nProof.We start from equation (79) which reads\nˆςε,ij(ω,k) =/summationdisplay\nsω2\np,s\n4π/bracketleftig/angbracketleftig\nΦjrs,ε(τ,ξ,·),1\nγuiFs/angbracketrightig\n+/angbracketleftig\nΨjrs,ε(τ,ξ,·),1\nγuiGs/angbracketrightig/bracketrightig\n,\nwhereτ=ω/ωc,sandξ=ck/ωc,s, cf. equation (55).\nAs observed in equations (80), both Φ jrs,εand Ψjrs,εare special cases of (66)\ncorresponding to ˜Pε= (i/2)e−iπaεΦjP±\nεand˜Pε= (i/2)e−iπaεΨjP±\nε, respectively.\nLemma 6.5 shows that in both cases ˜Pεsatisfies condition (67). Proposition 5.2\ngives, for any ( τ,ξ) withτ/\\e}atio\\slash= 0, distributions rΦ\ns,j(τ,ξ,·),rΨ\ns,j(τ,ξ,·)∈ S′(R3) such\nthat\nΦjrs,ε(τ,ξ,·)→rΦ\ns,j(τ,ξ,·),Ψjrs,ε(τ,ξ,·)→rΨ\ns,j(τ,ξ,·),\nin the topology of S′(R3). In this case, let\n(81) ˆς0,ij(ω,k) =/summationdisplay\nsω2\np,s\n4π/bracketleftig/angbracketleftig\nrΦ\ns,j(τ,ξ,·),1\nγuiFs/angbracketrightig\n+/angbracketleftig\nrΨ\ns,j(τ,ξ,·),1\nγuiGs/angbracketrightig/bracketrightig\n,\nand we have ˆ ςε,ij(ω,k)→ˆς0,ij(ω,k) pointwise in Fourier space where ω/\\e}atio\\slash= 0.\n(i) Proposition 5.2 implies that ˆ ς0,ijis continuous in ( R\\ {0})×R3andC∞\nwhereτ/\\e}atio\\slash= 0 andτ2/\\e}atio\\slash=ξ2\n3+n2for all integers n, and for all particle species.\n(ii) We observe that for any s, ˆω/ωc,s≥1, hence,\n1+τ2+ξ2≤ˆω2\nω2c,s/parenleftig\n1+(ω\nˆω)2+(ck\nˆω)2/parenrightig\n.\nThen proposition 5.2 item (iii) implies that there are constants K0,M >0 inde-\npendent of εsuch that\n/vextendsingle/vextendsingleˆςε(ω,k)/vextendsingle/vextendsingle≤K0(ω0)/parenleftbig\n1+(ω\nˆω)2+(ck\nˆω)2/parenrightbigM,\nuniformly in ε∈[0,ε0] and|ω| ≥ω0.\n(iii) We use the estimate proven in (ii), which implies\n/vextendsingle/vextendsingleˆςε(ω,k)−ˆς0(ω,k)/vextendsingle/vextendsingle≤2K0(ω0)/parenleftbig\n1+(ω\nˆω)2+(ck\nˆω)2/parenrightbigM,\nuniformly in ε∈[0,ε0] and for |ω| ≥ω0. Since by hypothesis ω/\\e}atio\\slash= 0 and thus τ/\\e}atio\\slash= 0\nin suppˆE, we can choose ω0small enough that, for any ˆ ϕ∈[S(R4)]3,\n/vextendsingle/vextendsingle/bracketleftbig/parenleftbig\nˆςε(ω,τ)−ˆς0(ω,k)/parenrightbigˆE(ω,k)/bracketrightbig\n·ˆϕ(ω,k)/vextendsingle/vextendsingle\n≤2K0(ω0)/parenleftbig\n1+(ω\nˆω)2+(ck\nˆω)2/parenrightbigM/vextendsingle/vextendsingleˆE(ω,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆϕ(ω,k)/vextendsingle/vextendsingle,44 O. LAFITTE AND O. MAJ\nuniformly in ε∈[0,ε0] and (ω,k)∈R4,|ω| ≥ω0. SinceE,ϕ∈[S(R4)]3, the\nright-hand side of the above inequality belongs to L1(R4) and we have already\nestablished that ˆ ςε(ω,τ)−ˆς0(ω,k)→0 pointwise. Therefore the hypothesis of the\ndominated convergence theorem are satisfied and we can conclude\n/integraldisplay\nR4/bracketleftbig/parenleftbig\nˆςε(ω,τ)−ˆς0(ω,k)/parenrightbigˆE(ω,k)/bracketrightbig\n·ϕ(ω,k)dωdk→0,\nfor all ˆϕ∈[S(R4)]3which means that ˆ ςεˆE→ˆς0ˆEinS′(R4). At last proposition 5.9\nestablishes that the S′-limit of ˆςεˆEis exactly /hatwider∂tj. /square\n5.5.Proof of the results on the relativistic, three-dimensiona l case with\nuniform magnetic field (section 1.4). We can now collect the partial results\nobtained in this section and give a proof of the main theorems stated in section 1.4.\nProof of theorem 1.3. (i) For each sandε, existence of a solution gs,ε∈ S(R7) of\nthe Vlasov equation with damping and its uniqueness is established in th eorem 5.5.\nWe have shown that /hatwidest∂tjε∈[S(R4)]3in proposition 5.9.\n(ii) Proposition5.6 proves that gs,εconvergespointwise to gs∈C∞\nband the limit\nis independent of the damping function ν. Explicit expressions for gs,εandgsare\ngiven. Using the inequality, cf. the proof of proposition 5.6,\n(1+s2)m/vextendsingle/vextendsingles/parenleftbig\ns,X(s;t,x,p),P(s;t,x,p)/parenrightbig/vextendsingle/vextendsingle≤ /ba∇dbls/ba∇dbl2m,\nyields the bound/vextendsingle/vextendsinglegs,ε(t,x,u)/vextendsingle/vextendsingle≤C,\nuniformly in εand thus for any ϕ∈ S(R7),\n/vextendsingle/vextendsingle/parenleftbig\ngs,ε(t,x,u)−gs(t,x,u)/parenrightbig\nϕ(t,x,u)/vextendsingle/vextendsingle≤2C/vextendsingle/vextendsingleϕ(t,x,u)/vextendsingle/vextendsingle,\nand|ϕ|is inL1. Hence the dominated convergence theorem yields\n/a\\}b∇acketle{tgs,ε−gs,ϕ/a\\}b∇acket∇i}ht=/integraldisplay/parenleftbig\ngs,ε(t,x,u)−gs(t,x,u)/parenrightbig\nϕ(t,x,u)dtdxdu→0,\nfor allϕ∈ S(R7). This is equivalent to gs,ε→gsinS′. Existence of the S′-limit\nof∂tjεis proven in proposition 5.9, in which we also show that the limit ∂tjis\ndetermined by gsonly, hence it is independent of ν.\n(iii) The fact that the limit gsis a classical solution of the linearized Vlasov\nequation in C∞\nbis proven in proposition 5.6. At last, the identity ∂tj=K({gs}) is\nshown in proposition 5.9. /square\nProof of theorem 1.4. After Fouriertransform, the action of ς(E) on a test function\nisgivenin proposition5.9. Since rsisatempereddistribution and Q′\ns,jiscontinuous\nfromS → S, we have/vextendsingle/vextendsingle/a\\}b∇acketle{t/hatwider∂tj,ˆϕ/a\\}b∇acket∇i}ht/vextendsingle/vextendsingle≤C/ba∇dblu·ˆϕˆE/γ/ba∇dblℓ,\nand Leibniz rule for the derivative yields\n/vextendsingle/vextendsingle/a\\}b∇acketle{t/hatwider∂tj,ˆϕ/a\\}b∇acket∇i}ht/vextendsingle/vextendsingle≤˜C/ba∇dblˆϕ/ba∇dblℓ/ba∇dblˆE/ba∇dblℓ.\nIfˆE∈[S(R4)]3, then (1 −χ(ω))ˆE(ω,k) satisfies the hypothesis of proposition 5.10\nwhich gives ς(E) =F−1/parenleftbig\n(1−χ)ˆς0ˆE/parenrightbig\n. /squareTEMPERED-IN-TIME RESPONSE OF A PLASMA 45\n6.Stationary and non-stationary phase results for some integ rals\nThis section is devoted to the proof of four lemmas, in which we study the phase\nϑε= tan(πa0)cosh(πε) with the aim of evaluating integrals in uvia stationary\nphase results. It replaces the study of1\nsinπaǫby the study of −i\ncosπa0/integraltext+∞\n0eiλθǫdλ\nand exchanging integration in λanduin integrals defining the current, cf. equa-\ntions (70) and (71) in section 5. In addition, in this section, as the dis tribution\n1\nsinπaǫacts on the functions ˜Pε, we give results on the functions ˜Pεrelevant to our\napplications (lemma 6.5 on functions satisfying (67) below). In this se ction, we let\nκs= 1, cf. comments before proposition 5.2.\nWe note basic facts about the stationary phase points, that are t he same as the\ncriticalpoints of a0, since tan(πa0) behaves as π(a0−n). Becauseofthe symmetries\nof the function a0, it is sufficient to consider the case τ >0 andξ3≥0.\nLemma 6.1. Forε>0, letϑεbe defined in equation (71a) and let τ >0,ξ∈R3\nwithξ3≥0.\n(i)Forτ2−ξ2\n3≤0,ϑε(τ,ξ,·)has no critical points and\n|∇uϑε(τ,ξ,u)|>(πτ/2)(1+u2)−1,for allu∈R3.\n(ii)Forτ2−ξ2\n3>0,ϑε(τ,ξ,·)has an isolated non-degenerate critical point at\nu=uc(τ,ξ) =/parenleftbig\n0,0,ξ3/(τ2−ξ2\n3)1/2/parenrightbig\n,\nand at the critical point u=ucone has\n/vextendsingle/vextendsingledetϑ′′\nε(τ,ξ,uc)/vextendsingle/vextendsingle≥(πτ)3(1+u2\nc)−5/2,\n|/a\\}b∇acketle{tϑ′′\nε(τ,ξ,uc)−1∇u,∇u/a\\}b∇acket∇i}htΥ| ≤3\nπτ(1+u2\nc)3/2max\n|β|=2|∂β\nuΥ|,\nfor allΥ(τ,ξ,u)of classC2, whereϑ′′\nεdenotes the Hessian matrix of ϑε\nwith respect to u, and/a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htis the Euclidean product in R3.\nProof.A direct computation gives\n∇uϑε=π(1+tan2(πa0)/parenrightbig\ncosh(πε)∇ua0,\nand thus ∇uϑε= 0 if and only if ∇ua0= 0. On the other hand the critical points\nofa0for a given ( τ,ξ) are solution to\nu1=u2= 0,τu3/radicalbig\n1+u2\n3=ξ3.\nForτ/\\e}atio\\slash= 0, this equation is satisfied only if ξ2\n3/τ2≤1. In that case the solution is\ngiven byucas claimed. The Hessian matrix\nϑ′′\nε= 2π2tan(πa0)/parenleftbig\n1+tan2(πa0)/parenrightbig\ncosh(πε)∇a0⊗∇a0\n+π(1+tan2(πa0)/parenrightbig\ncosh(πε)a′′\n0,\nat the critical point reduces to\nϑ′′\nε(τ,ξ,uc) =π/parenleftbig\n1+tan2/parenleftbig\nπa0(τ,ξ,uc)/parenrightbig/parenrightbig\ncosh(πε)a′′\n0(τ,ξ,uc),\nwith\na′′\n0(τ,ξ,uc) =a0(τ,ξ,uc)\n1 0 0\n0 1 0\n0 0 1−ξ2\n3/τ2\n46 O. LAFITTE AND O. MAJ\nbeing the Hessian matrix of a0(τ,ξ,·) evaluated at u=ucand we have accounted\nfor the identity a0(τ,ξ,uc) = (τ2−ξ2\n3)1/2.\nProof of the inequality in (i). We write\n|∇ua0(τ,ξ,u)|2=τ2\n1+u2/bracketleftig\nu2\n1+u2\n2+/parenleftig\nu3−ξ3\nτ/radicalbig\n1+u2/parenrightig2/bracketrightig\n,\nand observe that for τ >0,ξ3≥0,τ2−ξ2\n3≤0 and allu∈R3,\n/vextendsingle/vextendsingle/vextendsingleu3−ξ3\nτ/radicalbig\n1+u2/vextendsingle/vextendsingle/vextendsingle≥/radicalig\n1+u2\n3−|u3|,\ntherefore (1+ u2)|∇ua0(τ,ξ,u)|2≥τ2ψ(u3), withψ(u3) =/parenleftbig/radicalbig\n1+u2\n3−|u3|/parenrightbig2. The\nfunctionu3/ma√sto→(1 +u2\n3)ψ(u3) is even and for u3≥0, it decreases monotonically\nstarting from the value ψ(0) = 1, and approaching 1 /4 asu3→+∞. Hence\n(1+u2\n3)ψ(u3)>1/4 for allu3∈R. This yields the claimed inequality.\nEstimates in (ii). We compute\ndeta′′\n0(τ,ξ,uc) = (τ2−ξ2\n3)5/2/τ2,\nand, for every function Υ ∈C2,\n/vextendsingle/vextendsingle/a\\}b∇acketle{t/parenleftbig\nϑ′′\nε(τ,ξ,uc)/parenrightbig−1∇u,∇u/a\\}b∇acket∇i}htΥ/vextendsingle/vextendsingle=|∂2\nu1Υ+∂2\nu2Υ+τ2\nτ2−ξ2\n3∂2\nu3Υ|\nπ(1+tan2(πa0))cosh(πε))(τ2−ξ2\n3)1/2\n≤3τ2\nπ(τ2−ξ2\n3)3/2max\n|α|=2|∂α\nuΥ|,\nsinceτ2/(τ2−ξ2\n3)≥1. At last it is enough noting that/radicalbig\nτ2−ξ2\n3=τ//radicalbig\n1+u2c./square\nRemark 16 (Non-relativistic limits) .In the non-relativistic and weakly relativistic\nlimits, one has\na0(τ,ξ,u)≈anr,0(τ,ξ,u):=τ−ξ3u3,\nand\na0(τ,ξ,u)≈awr,0(τ,ξ,u):=τ(1+u2/2)−ξ3u3,\nrespectively. In the first case (the non-relativistic limit), there is n o stationary\nphase point if ξ3/\\e}atio\\slash= 0, but the phase reduces to a constant (in u) whenξ3= 0;\nparticularly, when ξ3= 0, all values of u∈R3are either in resonance if τ∈Zor\nnot in resonanceif τ/\\e}atio\\slash∈Z, that is, the set R(τ,ξ)|ξ3=0is not a closedsurface. On the\nother hand, in the weakly non-relativistic limit there is a non-degener ate stationary\nphase point uc= (0,0,ξ3/τ) forτ/\\e}atio\\slash= 0. Then, uc∈suppχδ(τ,ξ,·), whereχδis the\ncut-off function introduced in equation (68) of section 5, only if the re is an integer\nnsuch thatτ−ξ2\n3/(2τ)∈[n−δ/3,n+δ/3]. The Hessian matrix of the phase at\nthe critical point in the weakly non-relativistic case amounts to\nϑ′′\nwr,ε(τ,ξ,uc) =πτ/parenleftbig\n1+tan2(πawr,0)/parenrightbig\ncosh(πε)I\nwhereIis the identity matrix. In a sense, the relativistic Lorentz factor re moves\nthe degeneracy of the case ξ3= 0. The stationary phase argument developed here\napplies to both the relativistic and weakly relativistic cases.TEMPERED-IN-TIME RESPONSE OF A PLASMA 47\nLemma 6.2. For anyα∈N4\n0,α/\\e}atio\\slash= 0, and any integer n≥0, there are polynomials\nπα,πα,j,j= 1,...,|α|, andπ(n)\nαof one real variable such that\n∂α\nτ,ξϑε=πα/parenleftbig\ntan(πa0)/parenrightbig\ncosh(πε)(∇τ,ξa0)α, (82)\ne−iλϑε∂α\nτ,ξeiλϑε=|α|/summationdisplay\nj=1/parenleftbig\niλcosh(πε)/parenrightbigjπα,j/parenleftbig\ntan(πa0)/parenrightbig\n(∇τ,ξa0)α, λ∈R, (83)\n/vextendsingle/vextendsingle∂α\nτ,ξ/bracketleftbig\nϑn\nεeiλϑε/bracketrightbig/vextendsingle/vextendsingle≤π(n)\nα(λ)(1+u2)|α|\n2, u∈suppχδ(τ,ξ,·), λ≥0, (84)\nwhereχδis the cut-off function defined in equation (68).\nProof.The key observation is that a0is a linear function of ( τ,ξ), hence∂β\nτ,ξa0= 0\nfor|β| ≥2. Identities (82) and (83) are true for |α|= 1 and can be extended to all\nαwith|α| ≥1 by induction. As for inequality (84), Leibniz formula applied to the\nfunctionGn(λ,z) =zneiλzwithλ∈Randz∈C, gives for any positive integer ℓ,\n∂ℓ\nzGn(λ,z) =eiλzℓ/summationdisplay\nm=0/parenleftbiggℓ\nm/parenrightbigg\n(iλ)ℓ−m∂m\nz(zn).\nBecause of lemma 5.4, for u∈suppχδ(τ,ξ,·) andε≤ε0,\n|ϑε| ≤ |tan(πa0)cosh(πε)|+|sinh(πε)| ≤cosh(πε0)/cos(πδ/3)+sinh(πε0),\ntherefore, for λ≥0,\n|∂ℓ\nzGn(λ,ϑε)| ≤˜π(n)\nℓ(λ),\nwhere ˜π(n)\nℓ(λ) is a polynomial of degree ℓinλand with positive real coefficients.\nSince/vextendsingle/vextendsingle∇τ,ξa0(τ,ξ,u)/vextendsingle/vextendsingle≤(1+u2)1/2, lemma 5.4 and identity (82) imply\n|∂α\nτ,ξϑε(τ,ξ,u)| ≤cα(1+u2)|α|/2,forε∈[0,ε0],|α| ≥1 andu∈suppχδ(τ,ξ,·).\nThe Fa` a di Bruno’s formula for ∂α\nτ,ξ[ϑn\nεeiλϑε] =∂α\nτ,ξ[Gn(λ,ϑε)] amounts to the sum\nof terms of the form\n∂ℓ\nzGn(λ,ϑε)∂α1\nτ,ξϑε···∂αℓ\nτ,ξϑε,\nwhere the multi-indices α1,...,αℓin theℓfactors are such that ∂α\nτ,ξ=∂α1\nτ,ξ···∂αℓ\nτ,ξ,\nand in articular, |α1|+···+|αℓ|=|α|. It follows that\n/vextendsingle/vextendsingle∂α\nτ,ξ[ϑn\nεeiλϑε]/vextendsingle/vextendsingle≤π(n)\nα(λ)(1+u2)|α|/2,\nwhereπ(n)\nαis a polynomials of degree |α|inλ, with positive real coefficients. /square\nNext, we establish estimates for the integrands in equations (71). (Let us recall\nthat/ba∇dblφ/ba∇dbljdenotes the Schwartz semi-norms of φ∈ Sas defined in appendix A.)\nLemma 6.3. Letε0>0andδ∈(0,1)be fixed and let aεandχδbe defined in\nequation (56) and (68). For every φ∈ S(R3),˜Pεsatisfying condition (67), α∈N4\n0,\nρ∈N3\n0, and integer m≥0, there are constants Cc\nα,ρandCs\nα,ρ, dependent on ε0,and\nδbut independent of m, such that\n(1+u2)m/vextendsingle/vextendsingle/vextendsingle∂α\nτ,ξ∂ρ\nu/bracketleftigχδ˜Pεφ\ncos(πa0)/bracketrightig/vextendsingle/vextendsingle/vextendsingle≤Cc\nα,ρ(1+τ2+ξ2)ℓα,ρ+|ρ|\n2/ba∇dblφ/ba∇dbl2(m+˜ℓα,ρ)+|α|+|ρ|\n(1+u2)m/vextendsingle/vextendsingle/vextendsingle∂α\nτ,ξ∂ρ\nu/bracketleftig(1−χδ)˜Pεφ\nsin(πaε)/bracketrightig/vextendsingle/vextendsingle/vextendsingle≤Cs\nα,ρ(1+τ2+ξ2)ℓα,ρ+|ρ|\n2/ba∇dblφ/ba∇dbl2(m+˜ℓα,ρ)+|α|+|ρ|,48 O. LAFITTE AND O. MAJ\nuniformly in (τ,ξ,u)andε∈[0,ε0], withℓα,ρ= min{ℓ∈N0:ℓ≥ˆmµ,∀µ:|µ| ≤\n|α|+|ρ|}and˜ℓα,ρ= min{ℓ∈N0:ℓ≥ˆnµ,∀µ:|µ| ≤ |α|+|ρ|}.\nProof.Forα= 0 andρ= 0, both inequalities follows immediately from (67) and\nlemma 5.4. For the first inequality with |α|+|ρ| ≥1, the Leibniz rule yields\n/vextendsingle/vextendsingle/vextendsingle∂α\nτ,ξ∂ρ\nu/bracketleftig˜Pεφ\ncos(πa0)χδ/bracketrightig/vextendsingle/vextendsingle/vextendsingle≤ˆCα,ρmax\nβ,ρ′/vextendsingle/vextendsingle/vextendsingle∂β\nτ,ξ∂ρ′\nu(χδ\ncos(πa0))/vextendsingle/vextendsingle/vextendsinglemax\nβ,ρ′/vextendsingle/vextendsingle∂β\nτ,ξ∂ρ′\nu˜Pε/vextendsingle/vextendsinglemax\nρ′/vextendsingle/vextendsingle∂ρ′\nuφ/vextendsingle/vextendsingle,\nwhere the max are on multi-indices β∈N4\n0andρ′∈N3\n0such thatβi≤αifor all\ni= 0,1,2,3 andρ′\ni≤ρifor alli= 1,2,3, and with constant depending only of α,ρ.\nThe function a0is linear in ( τ,ξ), hence∂β\nτ,ξa0= 0 for all multi-indices βwith\n|β| ≥2, while|∂τ,ξa0| ≤(1+u2)1/2, and\n|∂ρ\nua0| ≤(1+τ2+ξ2)1/2,|∂τ,ξ∂ρ\nua0| ≤˜cρ,for allρ∈N3\n0:|ρ| ≥1,\nwhere ˜cρ= max{1,supu|∂ρ\nu√\n1+u2|}.\nBecause of lemma 5.3, near any point ( τ,ξ,u)∈suppχδ, there is a unique\nintegernsuch thatχδ=χ/parenleftbig\n(a0−n)/δ/parenrightbig\n. One can estimate the derivatives of\nχ/parenleftbig\n(a0−n)/δ/parenrightbig\n/cos(πa0) by means of a multivariate version of the standard Fa` a di\nBruno’s formula [28]. However, we observethat each factor ∂β\nτ,ξ∂ρ′\nua0growsat most\nlike either (1+ τ2+ξ2)1/2or (1+u2)1/2regardless of the order of the derivative.\nTherefore it is sufficient to estimate the term in the Fa` a di Bruno’s f ormula with\nthe highest number of factors, that is,\n∂β\nτ,ξ∂ρ′\nu/bracketleftigχδ\ncos(πa0)/bracketrightig\n= (∇τ,ξ,ua0)µdN\ndzN/bracketleftigχ/parenleftbig\n(z−n)/δ/parenrightbig\ncos(πz)/bracketrightig\nz=a0+···\nwhereµ= (β,ρ′)∈N7\n0, andN=|µ|=|β|+|ρ′|. All the derivative with respect\ntozare bounded because of the support of χand its derivatives as in lemma 5.4.\nHence, for all β≤αandρ′≤ρ,/vextendsingle/vextendsingle/vextendsingle∂β\nτ,ξ∂ρ′\nu/bracketleftigχδ\ncos(πa0)/bracketrightig/vextendsingle/vextendsingle/vextendsingle≤cα,ρ(δ)(1+u2)|α|/2(1+τ2+ξ2)|ρ|/2.\nAssumption (67) then gives\n/vextendsingle/vextendsingle/vextendsingle∂α\nτ,ξ∂ρ\nu/bracketleftig˜Pεφ\ncos(πa0)χδ/bracketrightig/vextendsingle/vextendsingle/vextendsingle≤Cc\nα,ρ(1+τ2+ξ2)ℓα,ρ+|ρ|\n2(1+u2)˜ℓα,ρ+|α|\n2max\n|ρ′|≤|ρ|/vextendsingle/vextendsingle∂ρ′\nuφ/vextendsingle/vextendsingle,\nwhereℓα,ρand˜ℓα,ρare smallest integers larger than all the exponents ˆ mµand ˆnµ\nfor|µ| ≤ |α|+ρ|, respectively. Upon multiplying by (1 + u2)m, one obtains the\nclaimed inequality. The second inequality follows analogously. /square\nThe functions defined in (71) have the following properties.\nLemma 6.4. Letε0>0be fixed,aεandχδbe defined in equation (56) and (68),\nℓα,ρand˜ℓα,ρbe the integers given in lemma 6.3, and mj= min{ℓ∈N0:ℓ≥\nℓ0,ρ,|ρ| ≤j},˜mj= min{ℓ∈N0:ℓ≥˜ℓ0,ρ,|ρ| ≤j}. If˜Pε,ε∈[0,ε0], satisfies\ncondition (67),\n(i)for anyδ∈(0,1),Ic\nδ,ε(ψ),Is\nδ,ε(ψ),Ic\nδ,0(ψ), andIs\nδ,0(ψ)defined in (71) for\nψ∈C∞\n0are defined and of class C∞for anyψ∈ S(R3);\n(ii)for anyδ∈(0,1)andφ∈ S(R3),Ic\nδ,ε(φ)→ Ic\nδ,0(φ)andIs\nδ,ε(φ)→ Is\nδ,0(φ),\npointwise as ε→0+;TEMPERED-IN-TIME RESPONSE OF A PLASMA 49\n(iii)for anyδ∈(0,1), there exists a function Bε0,δ:R+×(R\\{0})→R+such\nthatBε0,δ(·,τ)∈L1(R+),Bε0,δ(λ,τ1)≤ Bε0,δ(λ,τ2)for|τ1| ≥ |τ2|, and\n|Ic\nδ,ε(φ)(τ,ξ,λ)| ≤ Bε0,δ(λ,τ)(1+τ2+ξ2)m4+5\n2/ba∇dblφ/ba∇dbl2˜m4+10,\nfor allφ∈ S(R3),(τ,ξ,λ)∈(R\\{0})×R3×R+, andε∈[0,ε0];\n(iv)forα∈N4\n0,|α| ≥1,φ∈ S(R3), and for any connected, compact set K⊂\nR4with non-empty interior, such that τ/\\e}atio\\slash= 0and|τ2−ξ2\n3−n2| ≥δK>0\nfor alln∈N0and(τ,ξ)∈K, there isδ0∈(0,1)depending on Ksuch that\n|∂α\nτ,ξIc\nδ,ε(φ)(τ,ξ,λ)| ≤ B(α)\nε0,δ(λ,τ),(τ,ξ)∈K, δ∈(0,δ0),\nwhereB(α)\nε0,δ:R+×(R\\ {0})→R+satisfies B(α)\nε0,δ(·,τ)∈L1(R+), and\ndecreases in |τ|, that is, B(α)\nε0,δ(λ,τ1)≤ B(α)\nε0,δ(λ,τ2)for|τ1| ≥ |τ2|;\n(v)for any multi-index α∈N4\n0, there isKs\nα>0dependent on ε0andδsuch\nthat\n|∂α\nτ,ξIs\nδ,ε(φ)(τ,ξ)| ≤Ks\nα(1+τ2+ξ2)ℓα,0/ba∇dblφ/ba∇dbl2˜ℓα,0+|α|+4,\nuniformly for ε∈[0,ε0], for allφ∈ S(R3).\nProof.(i) We shall showthat, forany φ∈ S(R3), the integrandsin equations(71b)-\n(71e) and their derivatives with respect to ( τ,ξ,λ) are uniformly bounded by an\nintegrable function of u∈R3for (τ,ξ) andλin a bounded set and for ε∈[0,ε0].\nThen it follows that all integrals in equations (71) are finite also when ψ∈C∞\n0(R3)\nis replaced by φ∈ S(R3), and the dominated convergence theorem allows us to\ndifferentiate in the integral.\nFor the case of Is\nδ,εandIs\nδ,0, the needed uniform upper bound follows directly\nfrom the second inequality of lemma 6.3 with m>3/2 andρ= 0.\nAs forIc\nδ,ε, ifφ∈ S(R3),\n∂α\nτ,ξ∂n\nλ/bracketleftig\neiλϑεχδ˜Pεφ\ncos(πa0)/bracketrightig\n=∂α\nτ,ξ/bracketleftig/parenleftbig\niϑε/parenrightbigneiλϑεχδ˜Pεφ\ncos(πa0)/bracketrightig\n=in/summationdisplay\nβ≤α/parenleftbiggα\nβ/parenrightbigg\n∂α−β\nτ,ξ/bracketleftbig\nϑn\nεeiλϑε/bracketrightbig\n∂β\nτ,ξ/bracketleftigχδ˜Pεφ\ncos(πa0)/bracketrightig\n.\nFor anyR>0, inequality (84) in lemma 6.2 gives\n/vextendsingle/vextendsingle∂α−β\nτ,ξ/bracketleftbig\nϑn\nεeiλϑε/bracketrightbig/vextendsingle/vextendsingle≤MR,α,n(1+u2)|α−β|\n2,\nforu∈suppχδ/parenleftbig\n(τ,ξ,·)/parenrightbig\nand|λ| ≤R. Then the first inequality provenin lemma 6.3\nwithρ= 0 gives\n(85)/vextendsingle/vextendsingle/vextendsingle∂α\nτ,ξ∂n\nλ/bracketleftig\neiλϑεχδ˜Pεφ\ncos(πa0)/bracketrightig/vextendsingle/vextendsingle/vextendsingle≤/tildewiderMR,α,n\n(1+u2)m/ba∇dblφ/ba∇dbl2(m+˜ℓα)+|α|\nuniformly for ε∈[0,ε0], (τ,ξ,λ) in the ball τ2+ξ2+λ2≤R2for allR, and with\n˜ℓα= maxβ≤α˜ℓβ,0. Form>3/2 the right-hand side is integrable over R3.\n(ii) The integrands in equations (71b)- (71c) are continuous for ε→0+and\nthe inequalities proven in lemma 6.3 with m >3/2,α= 0, andρ= 0 imply\nupper bounds by L1functions uniformly in ε∈[0,ε0]. Then, the hypothesis of the\ndominated convergence theorem are satisfied and one can pass to the limit in the\nintegral.50 O. LAFITTE AND O. MAJ\n(iii) Let us first address the case Ic\nδ,ε(ψ) forψ∈C∞\n0(R3). Since Ic\nδ,ε(ψ)(τ,ξ,·)\nis continuous, it is measurable and integrable on compact intervals. W e write\n/integraldisplay+∞\n0Ic\nδ,ε(ψ)(τ,ξ,λ)dλ=/integraldisplay1\n0Ic\nδ,ε(ψ)(τ,ξ,λ)dλ+/integraldisplay+∞\n1Ic\nδ,ε(ψ)(τ,ξ,λ)dλ.\nThe first integral on the right-hand side is bounded by\n/integraldisplay\nR3/vextendsingle/vextendsingle/vextendsingle˜Pεψ\ncos(πa0)χδ/vextendsingle/vextendsingle/vextendsingledu≤Cc\n0,0(1+τ2+ξ2)ℓ0,0/ba∇dblψ/ba∇dbl2(m+˜ℓ0,0)/integraldisplay\nR3du\n(1+u2)m,\nin view of the first inequality of lemma 6.3 with α= 0,ρ= 0, andm>3/2.\nFor the second integral, we estimate the decay in λofIc\nδ,ε(ψ)(τ,ξ,·) by means\nof the stationary phase formula.\nForτ/\\e}atio\\slash= 0 andτ2−ξ2\n3≤0, lemma 6.1 shows that there are no stationary phase\npoints, and we have a lower bound for the gradient of the phase. Th e standard\nstationary phase lemma [29, Theorem 7.7.1] gives\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3eiλϑεχδ˜Pεψ\ncos(πa0)du/vextendsingle/vextendsingle/vextendsingle≤csp,ℓ\nλℓ/summationdisplay\n|ρ|≤ℓsup/parenleftig1\n|∇uϑε|2ℓ−|ρ|/vextendsingle/vextendsingle/vextendsingle∂ρ\nu/bracketleftigχδ˜Pεψ\ncos(πa0)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle/parenrightig\n,\nfor any integer ℓ≥0. The right-hand side is integrable in λ∈[1,+∞) ifℓ≥2. We\nchooseℓ= 2 and lemma 6.1 (i) gives, for |ρ| ≤ℓ= 2,\n1\n|∇uϑε|2ℓ−|ρ|≤(1+u2)4−|ρ|\n(πτ/2)4−|ρ|≤16\nπ2τ4−|ρ|(1+u2)4,\nand lemma 6.3 with m= 4 andα= 0 implies that, for |ρ| ≤ℓ= 2,\n1\n|∇uϑε|2ℓ−|ρ|/vextendsingle/vextendsingle/vextendsingle∂ρ\nu/bracketleftigχδ˜Pεψ\ncos(πa0)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle≤16Cc\n0,ρ\nπ2τ4−|ρ|(1+τ2+ξ2)ℓ0,ρ+|ρ|/2/ba∇dblψ/ba∇dbl2˜ℓ0,ρ+|ρ|+8,\nwhich in turns yields\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3eiλϑεχδ˜Pεψ\ncos(πa0)du/vextendsingle/vextendsingle/vextendsingle≤ B′\nε0,δ(λ,τ)(1+τ2+ξ2)m2+1/ba∇dblψ/ba∇dbl2˜m2+10.\nwhere\nB′\nε0,δ(λ,τ) =16csp,2\nπ2λ2/summationdisplay\n|ρ|≤2Cc\n0,ρ\nτ4−|ρ|, λ≥1.\nSince by definition m2≤m4this proves the claim for τ2−ξ2\n3≤0.\nForτ2−ξ2\n3>0, there is an isolated non-degenerate stationary phase point\nu=uc(τ,ξ) as shown in lemma 6.1. In this case, the stationary phase lemma [29,\nTheorem 7.7.5] gives\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3eiλϑεχδ˜Pεψ\ncos(πa0)du/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingledet(λϑ′′\nε/2π)/vextendsingle/vextendsingle−1\n2ℓ−1/summationdisplay\nj=0λ−j|Lj,τ,ξ(ψ)|\n+c′\nsp,ℓ\nλℓ/summationdisplay\n|ρ|≤2ℓsup/vextendsingle/vextendsingle/vextendsingle∂ρ\nu/bracketleftigχδ˜Pεψ\ncos(πa0)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle,\nwhere the Hessian matrix ϑ′′\nεandLj,τ,ξ(ψ) are differential operators acting on ψ\nand evaluated at the stationary phase point u=uc(τ,ξ). We seek an upper boundTEMPERED-IN-TIME RESPONSE OF A PLASMA 51\ninL1, hence it is sufficient to choose ℓ= 2, and we have\nL0,τ,ξ(ψ) =χδ˜Pεψ\ncos(πa0),\nL1,τ,ξ(ψ) =−i\n2/a\\}b∇acketle{t(ϑ′′\nε)−1∇u,∇u/a\\}b∇acket∇i}ht/bracketleftigχδ˜Pεψ\ncos(πa0)/bracketrightig\n,\nevaluated at ( τ,ξ,u) = (τ,ξ,uc). Using the results of lemma 6.1 (ii) together with\nthe first inequality of lemma 6.3 yields\n|L0,τ,ξ(ψ)|\n|det(ϑ′′ε)|1/2≤1\n(πτ)3/2(1+u2\nc)5/4/vextendsingle/vextendsingle/vextendsingleχδ˜Pεψ\ncos(πa0)/vextendsingle/vextendsingle/vextendsingle\n≤1\n(πτ)3/2C0(1+τ2+ξ2)m0/ba∇dblψ/ba∇dbl2˜m0+4,\n|L1,τ,ξ(ψ)|\n|det(ϑ′′ε)|1/2≤3/2\n(πτ)5/2max\n|β|=2/vextendsingle/vextendsingle/vextendsingle(1+u2\nc)11/4∂β\nu/bracketleftigχδ˜Pεψ\ncos(πa0)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle\n≤3/2\n(πτ)5/2C2(1+τ2+ξ2)m2+1/ba∇dblψ/ba∇dbl2˜m2+8,\nand\n/vextendsingle/vextendsingle/vextendsingle∂ρ\nu/bracketleftigχδ˜Pεψ\ncos(πa0)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle≤C4(1+τ2+ξ2)m4+2/ba∇dblψ/ba∇dbl2˜m4+4,\nwhereCj= max |ρ|≤jCc\n0,ρ,j∈N0. Therefore,\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3eiλϑεχδ˜Pεψ\ncos(πa0)du/vextendsingle/vextendsingle/vextendsingle≤ B′′\nε0,δ(λ,τ)(1+τ2+ξ2)m4+5\n2/ba∇dblψ/ba∇dbl2˜m4+8,\nwith\nB′′\nε0,δ(λ,τ) =/parenleftig2π\nλ/parenrightig3/2/bracketleftigC0\n(πτ)3/2+3\n2C2\n(πτ)5/21\nλ/bracketrightig\n+C′\n4\nλ2, λ≥1,\nwith the constant C′\n4depending on δandε0. At last we can combine the estimates\nobtained in the three cases by defining Bε0,δ=B′\nε0,δ+B′′\nε0,δforλ≥1 and extending\nit to a constant for λ∈[0,1]. This gives the claimed estimate for ψ∈C∞\n0(R3).\nThen we observe that for any ( τ,ξ,λ) withτ/\\e}atio\\slash= 0, the map ψ/ma√sto→ Ic\nδ,ε(ψ)(τ,ξ,λ)\ndefines a linear functional on C∞\n0(R3), bounded by a Schwartz semi-norm. Since\nC∞\n0(R3) is dense in S(R3) the inequality remains true for ψ∈ S(R3): given a\nsequenceψi∈C∞\n0,i∈N, converging to φ∈ S(R3) in the topology of S, inequal-\nity (85) with α= 0 andn= 0 shows that |Ic\nδ,ε(ψi)| → |Ic\nδ,ε(φ)|, while by definition\nof convergence we have /ba∇dblψi/ba∇dblj→ /ba∇dblφ/ba∇dblj; then we can pass to the limit i→+∞on\nboth sides of the inequality.\n(iv) First, let ψ∈C∞\n0(R3). From (i), we have that\n∂α\nτ,ξIc\nδ,ε(ψ)(τ,ξ,λ) =in/summationdisplay\nβ≤α/parenleftbiggα\nβ/parenrightbigg/integraldisplay\nR3∂α−β\nτ,ξ/bracketleftbig\neiλϑε/bracketrightbig\n∂β\nτ,ξ/bracketleftigχδ˜Pεψ\ncos(πa0)/bracketrightig\ndu.\nWe exploit the second identity in lemma 6.2 for the factor ∂α−β\nτ,ξeiλϑεwith the result\nthat\n∂α\nτ,ξIc\nδ,ε(ψ)(τ,ξ,λ) =|α|/summationdisplay\nj=0λjcosh(πε)j/integraldisplay\nR3eiλϑεψδ,ε,j(τ,ξ,u)du,52 O. LAFITTE AND O. MAJ\nwhereψδ,ε,jare combinations of polynomials in tan( πa0) and (τ,ξ)-derivatives of\nthe function χδ˜Pεψ/cos(πa0). The assumption states that ( τ,ξ) varies in a con-\nnected, compact set Kwith non-empty interior, therefore the continuous function\n(τ,ξ)/ma√sto→τ2−ξ2\n3mapsKinto a closed interval [ c1,c2]⊂R. The assumption with\nn= 0 also implies that |τ2−ξ2\n3| ≥δK, hence zero is not in the interval, that is,\neitherc1< c2<0, or 0< c1< c2. In the first case, τ2−ξ3\n3<0 and lemma 6.1\nestablished that the phase ϑεhas no critical points. In the second case, let n0∈N0\nbe the unique non-negative integer for which\nn2\n0+δK≤τ2−ξ2\n3≤(n0+1)2−δK,(τ,ξ)∈K.\nWe have shown in the proof of lemma 6.1 that a0(τ,ξ,uc) =/radicalbig\nτ2−ξ2\n3is the value\nofa0at the stationaryphase point uc(τ,ξ). Ifnc∈ {n0,n0+1}is the closest integer\ntoa0(τ,ξ,uc), we find/vextendsingle/vextendsinglea0(τ,ξ,uc)−nc/vextendsingle/vextendsingle≥c(δK), and for any n∈Z,\n/vextendsingle/vextendsinglea0(τ,ξ,uc)−n/vextendsingle/vextendsingle≥/vextendsingle/vextendsinglea0(τ,ξ,uc)−nc/vextendsingle/vextendsingle≥c(δK).\nWe can now choose δsufficiently small that δ/3< c(δK), which depends only on\nthe setK, and we obtain χδ(τ,ξ,uc) = 0: there areno critical points of the phase in\nthe support of χδfor (τ,ξ)∈K. We can then apply the same argument as in (iii) in\nordertoshowthateachoftheintegralsontheright-handsidedec reaseslike1 /λℓfor\nallpositiveintegers ℓandsince(τ,ξ)variesinacompactset Kweobtaintheclaimed\ninequality with functions B(α)\nε0,δdepending on sup {(1+τ2+ξ2) : (τ,ξ)∈K}and\nthe Schwartz semi-norms of ψ. Then the inequality can be extended to φ∈ S(R3)\nby choosing a sequence ψn∈C∞\n0(R3) converging to φinS(R3).\n(v). The derivatives of Is\nδ,ε(φ) forφ∈ S(R3) are estimated by the second\ninequality in lemma 6.3 with ρ= 0, andm= 2. /square\nWe conclude this section with the proof that the functions defined in equa-\ntion (80) are of the form (66). The function in the following lemma are defined in\nsection 5.1.\nLemma 6.5. The functions e−iπaεΦjP±\nεande−iπaεΨjP±\nεsatisfy condition (67).\nProof.The considered functions are all defined as integrals in λover a compact\ninterval and with smooth integrands, hence they are of class C∞. As for the\npolynomial-growth estimate, the derivatives of any\nFε∈ {e−iπaεΦjP±\nε:j= 1,2,3}∪{e−iπaεΨjP±\nε:j= 1,2,3}\nare given by\n∂µFε(τ,ξ,u) =/integraldisplay2π\n0Πµ(τ,ξ,u,λ)ei(λ−π)aε−i˜ζdλ,\nwhere˜ζ(τ,ξ,u,λ) = (ξ1u1+ξ2u2)sinλ∓(ξ2u1−ξ1u2)(cosλ−1) and Πµis a linear\ncombination of monomials of the form\nγmτℓ0ξαuβλℓ1(cosλ)ℓ2(sinλ)ℓ3,\nform∈Z,α,β∈N3\n0,ℓj∈N0. This can be checked directly for µ= 0 and |µ|= 1,\nthen extended to all µ∈N7\n0by induction. We also have that Π µis independent of\nε. Forλ∈[0,2π] we readily have\n|γmτℓ0ξαuβλℓ1(cosλ)ℓ2(sinλ)ℓ3| ≤(2π)ℓ1(1+τ2+ξ2)(ℓ0+|α|)/2(1+u2)(m+|β|)/2,TEMPERED-IN-TIME RESPONSE OF A PLASMA 53\nand (since we have chosen κ= 1)\n|ei(λ−π)aε−i˜ζ| ≤eπε,forλ∈[0,2π],\nhence\n|∂µFε(τ,ξ,u)| ≤CF,µeπε(1+τ2+ξ2)ˆmF,µ(1+u2)ˆnF,µ,\nwith constant CF,µand exponents ˆ mF,µ,ˆnF,µdepending on the specific function F.\nThis is the claimed inequality if ε∈[0,ε0]. By means of the same identity one can\nalso check continuity of all derivatives for ε→0, since\n/vextendsingle/vextendsingle∂µFε(τ,ξ,u)−∂µF0(τ,ξ,u)/vextendsingle/vextendsingle≤/integraldisplay2π\n0/vextendsingle/vextendsingleΠµ(τ,ξ,u,λ)/vextendsingle/vextendsingle·/vextendsingle/vextendsinglee−(λ−π)ε−1/vextendsingle/vextendsingledλ\n≤˜CF,µ(τ,ξ,u)|eπε−1|,\nand thus, as ε→0+,∂µFε(τ,ξ,u)→∂µF0(τ,ξ,u) pointwise. /square\nRemark 17.This induction argument has the advantage of avoiding lengthy calcu -\nlations, but does not allows us to compute the exponents ˆ mF,µand ˆnF,µexplicitly.\nAppendix A.Notation and basic definitions\nPositive integers are denoted by N={1,2,...}and natural numbers by N0=\nN∪ {0}including zero. Integers including zero, reals and complex numbers a re\ndenoted by Z,R, andC, respectively. Strictly positive real numbers are denoted\nbyR+={x∈R:x>0}. Weusethestandardmulti-indexnotation,amulti-index\nofdimension n∈Nbeing anelement α= (α1,...,αn)∈Nn\n0. Thelengthofamulti-\nindex is|α|=/summationtext\niαi, and forx= (x1,...,xn)∈Rn, we writexα=xα1\n1···xαnnand\n∂α\nx=∂α1x1···∂αnxn.\nAs usualCk(Rn) denotes the space of k-times continuously differentiable func-\ntions ofnvariables,C∞(Rn) =/intersectiontext\nkCk(Rn).\nDefinition 1. We denote by C∞\nb(Rn) the space of functions u∈C∞(Rn) that\nare bounded with all derivatives bounded, that is, for every α∈Nn\n0there are real\nconstantsCα>0 such that |∂α\nxu(x)| ≤Cαuniformly.\nWe work with tempered distributions on Rn. As usually we denote by S(Rn)\nthe Schwartz space of rapidly decreasing functions on Rn. Those are functions\nϕ∈C∞(Rn) for which\nsup\nx∈Rn/vextendsingle/vextendsinglexα∂β\nxϕ(x)/vextendsingle/vextendsingle<∞.\nSemi-norms in Sare defined by\n/ba∇dblϕ/ba∇dblj= max\n|α|+|β|≤jsup\nx∈Rn/vextendsingle/vextendsinglexα∂β\nxϕ(x)/vextendsingle/vextendsingle, j∈N0.\n(With the condition |α|+|β| ≤j, instead of equality, these are actually norms.)\nThe countable set of semi-norms gives the Schwartz space the top ology of a Fr´ echet\nspaceanditstopologicaldual S′(Rn)isthespaceoftempereddistributions, namely,\nthe space of continuous linear functionals u:S(Rn)→C; the action of u∈ S′on a\ntest-function ϕ∈ Sis equivalently denoted by /a\\}b∇acketle{tu,ϕ/a\\}b∇acket∇i}ht=u(ϕ). In this casecontinuity\nof a linear functional umeans that there exists an integer j≥0 and a constant\nCj>0 such that |/a\\}b∇acketle{tu,ϕ/a\\}b∇acket∇i}ht|=|u(ϕ)| ≤Cj/ba∇dblϕ/ba∇dblj.54 O. LAFITTE AND O. MAJ\nWe also note that for any ϕ∈ S, for every m∈R, andβ∈Nd\n0, there is a\nconstantCm,β>0 such that\n(86) (1+ x2)m/vextendsingle/vextendsingle∂β\nxϕ(x)/vextendsingle/vextendsingle≤Cm,β,\nuniformly in x. Form≤0, this inequality follows from (1 + x2)m≤1, while for\nm>0, one can observe that (1+ x2)m≤(1+x2)lfor any integer l≥m, and use\nthe multi-nomial formula.\nThe Fourier transform of a function ϕ∈ S(Rn) is defined by\nˆϕ(ξ) =/integraldisplay\nRnϕ(x)e−iξ·xdx.\nThe Fourier transform F:ϕ/ma√sto→ˆϕis continuous from Sinto itself and extends to\nS′by duality. Specifically, this means\nˆu(ϕ) =u(ˆϕ).\nThat ˆuis a continuous linear functional follows from the continuity of uand of the\nFourier transform ϕ→ˆϕ. For a function ϕ(t,x) inS(R1+d) of physical time and\nspace (d= 3), we write\nˆϕ(ω,k) =/integraldisplay\nR1+dϕ(t,x)eiωt−ik·xdtdx,\nwhere,ω∈Ris the angular frequency and k∈Rdis the wave vector.\nDefinition 2 (Fourier multipliers) .Leta∈C(Rn) andm,C0∈R, be such that\n|a(ξ)| ≤C0(1+|ξ|)m. The Fourier multiplier with symbol ais the continuous linear\noperator on S(Rn) defined by\nAϕ(x) =F−1/parenleftbig\naF(ϕ)/parenrightbig\n(x) =1\n(2π)n/integraldisplay\nRneix·ξa(ξ)ˆϕ(ξ)dξ,\nwhere the inverse Fourier transform is an absolutely convergent in tegral, and the\nintegrand has partial derivatives in xof orderkbounded by the L1-functionξ/ma√sto→\n|ξ|k|a(ξ)||ˆϕ(ξ)|; hence,A:S(Rn)→C∞\nb(Rn)⊂ S′(Rn).\nIf in addition, a∈C∞(Rn) with all derivatives bounded by\n|∂α\nξa(ξ)| ≤Cα(1+|ξ|)mα,\nfor allξ∈Rnandα∈Nn\n0, with constants Cα,mα∈Rdepending only on α,\nthen for any ϕ∈ S(Rn),aˆϕ∈ S(Rn) and the inverse Fourier transform gives\nF−1(aˆϕ)∈ S(Rn). Such operations are continuous on S. Hence, the corresponding\nFouriermultiplierdefinedby aisacontinuouslinearoperatorform S(Rn)→ S(Rn).\nFourier multipliers are relevant to the case of a uniform plasma equilibr ium.\nSpecifically, the high-frequency component of the current densit y induced in a uni-\nform plasma by an electromagnetic disturbance is related to the elec tric field of the\ndisturbance by a Fourier multiplier.\nAppendix B.Proofs for the case study of section 2\nProof of proposition 2.1. With an initial condition u0inC∞\nb, the solution of this\nproblem is\nu(t,x) =u0(x)+/integraldisplayt\n0v(s,x)ds,TEMPERED-IN-TIME RESPONSE OF A PLASMA 55\nand we have u∈C∞(R1+d). In addition, u∈C∞\nb(R1+d), thanks to v∈ S.\nThis follows upon considering the derivatives ∂ℓ\nt∂α\nxu(t,x). Forℓ >1 one has\n∂ℓ\nt∂α\nxu(t,x) =∂ℓ−1\nt∂α\nxv(t,x) withv∈ S(R1+d), while for ℓ= 0 and for every\nm>1/2 we have\n|∂α\nxu(t,x)| ≤ |∂α\nxu0(x)|+sup\ns∈R/vextendsingle/vextendsingle(1+s2)m∂α\nxv(s,x)/vextendsingle/vextendsingle/integraldisplay+∞\n−∞ds\n(1+s2)m,\nwith∂α\nxu0bounded by hypothesis.\nThe initial condition\nu0(x) =/integraldisplay0\n−∞v(s,x)ds,\nbelongs to S(Rd) and thus to C∞\nb(Rd). The corresponding unique solution in\nC∞\nb(R1+d) is\nu(t,x) = (/integraldisplay0\n−∞+/integraldisplayt\n0)v(s,x)ds=/integraldisplayt\n−∞v(s,x)ds.\nFor every integers ℓ≥0 andt∈R,∂ℓ\ntu(t,·)∈ S(Rd) andusatisfies the condition\nlimt→−∞u(t,x) = 0. As for the uniqueness, if u∗∈C∞\nb(Rd) is another initial\ncondition such that the limit for t→ −∞of the corresponding solution vanishes,\nthen\n0 =u∗(x)+/integraldisplay−∞\n0v(s,x)ds=u∗(x)−u0(x),\nwhich shows that u∗=u0. Since, in particular, u∈L∞(R1+d), it defines a tem-\npered distribution by integration,\n/a\\}b∇acketle{tu,ϕ/a\\}b∇acket∇i}ht=/integraldisplay\nR1+du(t,x)ϕ(t,x)dtdx,∀ϕ∈ S(R1+d).\nThe continuity of the map ϕ/ma√sto→ /a\\}b∇acketle{tu,ϕ/a\\}b∇acket∇i}htfollows form>(1+d)/2 from\n|/a\\}b∇acketle{tu,ϕ/a\\}b∇acket∇i}ht| ≤ /ba∇dblu/ba∇dblL∞(R1+d)sup\ny∈R1+d/vextendsingle/vextendsingle(1+y2)mϕ(y)/vextendsingle/vextendsingle/integraldisplay\nR1+ddy\n(1+y2)m,\nand, ifm>1/2 is an integer,\nsup\ny∈R1+d/vextendsingle/vextendsingle(1+y2)mϕ(y)/vextendsingle/vextendsingle≤Cm/ba∇dblϕ/ba∇dbl2m,\nOn the other hand,\n|u(t,x)| ≤sup\nt∈R/vextendsingle/vextendsingle(1+t2)µv(t,x)/vextendsingle/vextendsingle/integraldisplay+∞\n−∞ds\n(1+s2)µ, µ>1/2,\nand thus\n/ba∇dblu/ba∇dblL∞(R1+d)≤(/integraldisplay+∞\n−∞ds\n(1+s2)µ) sup\n(t,x)∈R1+d/vextendsingle/vextendsingle(1+t2)µv(t,x)/vextendsingle/vextendsingle.\nSince, if we choose µ>1/2 inN,\nsup\n(t,x)∈R1+d/vextendsingle/vextendsingle(1+t2)µv(t,x)/vextendsingle/vextendsingle≤sup\n(t,x)∈R1+d/vextendsingle/vextendsingle(1+t2+x2)µv(t,x)/vextendsingle/vextendsingle≤Cµ/ba∇dblv/ba∇dbl2µ,\nwe have\n/ba∇dblu/ba∇dblL∞(R1+d)≤˜Cµ/ba∇dblv/ba∇dbl2µ,and|/a\\}b∇acketle{tu,ϕ/a\\}b∇acket∇i}ht| ≤Km,µ/ba∇dblv/ba∇dbl2µ/ba∇dblϕ/ba∇dbl2m,\nform>(1+d)/2 andµ>1/2, both integers. /square56 O. LAFITTE AND O. MAJ\nProof of proposition 2.2. Ifuis a solution in S′of the damped equation, its Fourier\ntransform satisfies\n−i(ω+iν)ˆuν= ˆv.\nForν >0, this has one and only one solution\nˆuν(ω,k) =iˆv(ω,k)\nω+iν,\nand we have ˆ uν∈ S(R1+d) since (ω+iν)−nis smooth and polynomially bounded\nforω∈Rand for all integers n>0. Hence, its inverse Fourier transform belongs\ntoS(R1+d). We recall that\nuν(−t,−x) = (2π)−(1+d)ˆˆuν(t,x),\nso that, if ˇϕ(t,x) =ϕ(−t,−x),\n/a\\}b∇acketle{tuν,ϕ/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t(2π)−(1+d)ˆˆuν,ˇϕ/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tˆuν,(2π)−(1+d)ˆˇϕ/a\\}b∇acket∇i}ht.\nLet us introduce, for any ϕ∈ S(R1+d), the function ˆψ∈ S(R) given by\nˆψ(ω) = (2π)−(1+d)/integraldisplay\nRdˆv(ω,k)ˆϕ(−ω,−k)dk.\nWe deduce\n/a\\}b∇acketle{tuν,ϕ/a\\}b∇acket∇i}ht=/integraldisplay\nRiˆψ(ω)\nω+iνdω.\nHowever, the sequence {uν}ν∈R+is not bounded in S. In order to take the limit,\nwe use the identity\ni\nω+iν=/integraldisplay+∞\n0ei(ω+iν)tdt,\nand note that the function ( t,ω)/ma√sto→ei(ω+iν)tˆψ(ω) belongs to L1(R+×R) so that,\nby Fubini’s theorem,\n/a\\}b∇acketle{tuν,ϕ/a\\}b∇acket∇i}ht=/integraldisplay+∞\n0e−νt(/integraldisplay+∞\n−∞eiωtˆψ(ω)dω)dt\n= 2π/integraldisplay+∞\n0e−νtψ(−t)dt.\nAlso the Fourier inversion theorem gives\nψ(t) =1\n2π/integraldisplay+∞\n−∞e−iωtˆψ(ω)dω\n=1\n2π/integraldisplay+∞\n−∞e−iωt/integraldisplay\nRdˆv(ω,k)(2π)−(1+d)ˆϕ(−ω,−k)dkdω\n=1\n(2π)d+2/integraldisplay+∞\n−∞e−iωt/integraldisplay\nRd/integraldisplay\nR1+de−i(k·x1−ωt1)v(t1,x1)dt1dx1\n×/integraldisplay\nR1+de+i(k·x2−ωt2)ϕ(t2,x2)dt2dx2dkdω\n=1\n(2π)2/integraldisplay+∞\n−∞e−iωt/integraldisplay\nR1+de+iωt1v(t1,x1)dt1/integraldisplay\nRe−iωt2ϕ(t2,x1)dt2dx1dω\n=1\n2π/integraldisplay\nR1+dv(t′,x1)ϕ(t′−t,x1)dt′dx1,TEMPERED-IN-TIME RESPONSE OF A PLASMA 57\nand this yields\n/a\\}b∇acketle{tuν,ϕ/a\\}b∇acket∇i}ht=/integraldisplay+∞\n0/integraldisplay\nR1+de−νt′′v(t′,x)ϕ(t′+t′′,x)dt′dxdt′′.\nBy the change of variables t′′=t−s,t′=s, one has\n/a\\}b∇acketle{tuν,ϕ/a\\}b∇acket∇i}ht=/integraldisplay\nR1+d/integraldisplayt\n−∞e−ν(t−s)v(s,x)ϕ(t,x)dsdtdx,\nwhich shows that the distribution uνis regular and equal to the C∞function\nuν(t,x) =/integraldisplayt\n−∞e−ν(t−s)v(s,x)ds,\nas claimed. In addition, e−ν(t−s)≤1 fors∈(−∞,t], and with udefined in\nproposition 2.1,\nlim\nν→0+/a\\}b∇acketle{tuν,ϕ/a\\}b∇acket∇i}ht=/integraldisplay\nR1+d/integraldisplayt\n−∞v(s,x)ϕ(t,x)dsdtdx=/integraldisplay\nu(t,x)ϕ(t,x)dtdx,\nfor everyϕ∈ S(R1+d), that is,\nuν→u,inS′(R1+d),\nand the limit is the causal solution of proposition 2.1. /square\nAppendix C.Causal solutions of linear kinetic equations\nLet Ω be a domain in Rdandϕ∈C∞(R×Ω). We denote ϕt=ϕ(t,·) and\nassume that\n(i)ϕt: Ω→Ω for every t∈R,\n(ii)ϕ0= Id is the identity map on Ω,\n(iii)ϕt+s=ϕt◦ϕsfor everyt,s∈R.\n(iv) For any multi-index α∈Ndthere are constants C,m∈Rsuch that\n/vextendsingle/vextendsingle∂α\nxϕt(x)/vextendsingle/vextendsingle≤C/parenleftbig\n1+t2+|ϕt(x)|2/parenrightbigm.\nRemark 18.In condition (iv) the case α= 0 is excluded because, for α= 0,\n|ϕt(x)| ≤/parenleftbig\n1+t2+|ϕt(x)|2/parenrightbig1/2for any map ϕ, and the inequality in (iv) is trivially\nverified.\nAs a consequence of properties (i)-(iii), {ϕt:t∈R}is a one-parameter Abelian\ngroup of diffeomorphisms of Ω. Particularly, ϕ−1\nt=ϕ−t. We can associate to ϕt\nthe autonomous vector field X∈C∞(Ω,Rd) defined by\n(87) X(x) =dϕt(x)\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0,\nand, withx=ϕt(x0) for every x0∈Ω, we have\nX/parenleftbig\nϕt(x0)/parenrightbig\n=dϕs\nds/parenleftbig\nϕt(x0)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ns=0=d\ndsϕt+s(x0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ns=0=dϕt(x0)\ndt,\nwhich shows that the orbit x(t) =ϕt(x0) of the group solves the Cauchy problem\ndx\ndt=X(x), x(0) =x0,58 O. LAFITTE AND O. MAJ\nglobally in time and for every initial point x0∈Ω. While conditions (i)-(iii) are\nstandard properties of flows of autonomous vector fields, condit ion (iv) is specific\nto the application considered here. An example of flow satisfying con dition (iv) is\n(88) ϕt(x) =A(t)x,\nwhereA(t) is ad×dmatrix satisfying suitable conditions.\nProposition C.1. If the matrix-valued function A∈C∞(R,Rd×d)is such that,\nA(0) =I,A(t+s) =A(t)A(s)and, in the norm induced by the standard Euclidean\nnorm in Rd,\n/ba∇dblA(t)/ba∇dbl ≤C(1+t2)m,\nfor given constants C,m>0, then the map (88) satisfies conditions (i)-(iv) above\nwithΩ =Rd.\nProof.Condition (i) is true because of definition (88), while (ii) and (iii) follow\ndirectly from the assumptions. As for condition (iv), since the flow is linear inx\nthe spatial derivative vanish for |α| ≥2. For|α|= 1, condition (iv) is implied by\nthe polynomial growth of A. /square\nRemark 19.The characteristics flow of the kinetic equation considered in sectio n 3\nand the non-relativistic version of the one in section 5 (i.e., equation ( 74) with\nγ= 1) are both special cases of (88). The relativistic flow (74) is not o f the same\nform since Ω cdepends on momentum. However, it satisfies assumptions (i)-(iv).\nIn this section we consider the linear advection equation\n(89) ∂tf+X·∇f=g,inR×Ω,\nwith given source g∈ S(R1+d). By construction the orbits of the group are the\ncharacteristics curves of (89) and exist globally in time.\nWe define the function\n(90) f(t,x):=/integraldisplayt\n−∞g/parenleftbig\ns,ϕ−1\nt−s(x)/parenrightbig\nds,(t,x)∈R×Ω,\nfor which we prove the following.\nProposition C.2. Letϕbe a map satisfying properties (i)-(iv) above. Then, for\nanyg∈ S(R1+d)the function fdefined in (90) belongs to C∞\nb(R×Ω)and is a\nclassical solution of the linear advection equation (89).\nProof.First we observe that, for every t∈Rthe function s/ma√sto→(1+s2)/(1+(t−s)2)\nis inC∞(R), strictly positive, tends to 1 for s→ ±∞, and fort/\\e}atio\\slash= 0 has two critical\npoints ats=s±= (t±(t2+4)1/2)/2 that correspond to a local minimum and a\nlocal maximum depending of the sign of t. The local maximum, in particular, is\nalso the global maximum and thus\n(1+s2)≤Ct(1+(t−s)2),\nuniformly in s. The constant Ctis the value of the function at the maximum\nwhich isCt=/bracketleftbig\n4 +t2+|t|(4 +t2)1/2/bracketrightbig\n//bracketleftbig\n4 +t2− |t|(4 +t2)1/2/bracketrightbig\n. The trivial case\nt= 0 is included with Ct= 1. The substitutions s→s/aandt→t/ayield\n(a2+s2)≤Ct/a(a2+(t−s)2), for alla/\\e}atio\\slash= 0. Explicitly, Ct/a= (ξ+|t|)/(ξ−|t|)\nwhereξ=√\n4a2+t2; this is a monotonicallydecreasingfunction of ξin the intervalTEMPERED-IN-TIME RESPONSE OF A PLASMA 59\n[√\n4+t2,+∞) corresponding to a≥1, and the maximum is exactly Ct. Therefore,\nfor everya≥1,\n(91) ( a2+s2)≤Ct(a2+(t−s)2),\nuniformly in s∈Rand the constant is independent of a.\nWe now apply this inequality to the function (90). After the change o f variable\ns′=t−s, we have\nf(t,x) =/integraldisplay+∞\n0g/parenleftbig\nt−s′,ϕ−1\ns′(x)/parenrightbig\nds′.\nWe shall show that all derivatives ∂n\nt∂α\nx/bracketleftbig\ng/parenleftbig\nt−s′,ϕ−1\ns′(x)/parenrightbig/bracketrightbig\nare uniformly bounded\nbyL1-functions of s. If this is the case, repeated use of the dominated convergence\ntheorem gives f∈C∞(R×Ω) with all derivative bounded as claimed.\nIn the case α= 0, for any real k≥0 and for any integer n≥0, we have\n/vextendsingle/vextendsingle∂n\ntg/parenleftbig\nt−s,ϕ−1\ns(x)/parenrightbig/vextendsingle/vextendsingle=(1+(t−s)2+|ϕ−1\ns(x)|2)k/vextendsingle/vextendsingle∂n\ntg/parenleftbig\nt−s,ϕ−1\ns(x)/parenrightbig/vextendsingle/vextendsingle\n(1+(t−s)2+|ϕ−1s(x)|2)k\n≤(1+(t−s)2+|ϕ−1\ns(x)|2)k/vextendsingle/vextendsingle∂n\ntg/parenleftbig\nt−s,ϕ−1\ns(x)/parenrightbig/vextendsingle/vextendsingle\n(1+(t−s)2)k\nand sinceg∈ Sinequality (86) gives\n/vextendsingle/vextendsingle∂n\ntg/parenleftbig\nt−s,ϕ−1\ns(x)/parenrightbig/vextendsingle/vextendsingle≤C\n(1+(t−s)2)k,\nwith constant Cindependent on sands/ma√sto→(1+(t−s)2)−kis inL1fork >1/2.\nWithn= 0, this also shows that f(t,x) is bounded.\nIncluding spatial derivative requires condition (iv). In the case |α|= 1,\n∂n\nt∂α\nx/bracketleftbig\ng/parenleftbig\nt−s,ϕ−1\ns(x)/parenrightbig/bracketrightbig\n=∂α\nxϕ−1\ns(x)·(∂n\nt∇g)/parenleftbig\nt−s,ϕ−1\ns(x)/parenrightbig\n≤d·|∂α\nxϕ−s(x)|·max\nj/vextendsingle/vextendsingle(∂n\nt∂xjg)/parenleftbig\nt−s,ϕ−1\ns(x)/parenrightbig/vextendsingle/vextendsingle.\nHypothesis (iv) and the estimate (91) give, for every k>0,\n|∂α\nxϕ−s(x)| ≤C(1+s2+|ϕ−1\ns(x)|2)m\n≤˜Ct(1+(t−s)2+|ϕ−1\ns(x)|2)m\n≤˜Ct(1+(t−s)2+|ϕ−1\ns(x)|2)m+k\n(1+(t−s)2)k.\nProceeding as before, we choose k>1/2 and obtain that the first-order derivatives\nare uniformly bounded by an L1-function.\nThe case of general αis complicated by the form of the chain rule. However, the\nsame argument can be applied to the explicit formula for the multi-var iate chain\nrule (Fa` a di Bruno formula) which is linear in the derivatives of gand polynomial\nin the derivatives of ϕ.\nThe fact that fis a classical solution of (89) can be checked by substitution. /square\nIf the flow is polynomially bounded, we can deduce that f, viewed as a distribu-\ntion on Ω, has finite moments at all orders. This is a consequence of t he following\nresult.60 O. LAFITTE AND O. MAJ\nLemma C.3. Letϕsatisfy assumptions (i)-(iv) and g∈ S(R1+d). In addition\nlet us assume the there are n∈N0andC >0such that |ϕt(x)| ≤C(1 +x2)n,\nuniformly in R×Ω. Then,p·∂l\nt∂α\nxf(t,·)∈L∞(Ω)for every polynomial p=p(x),\nl∈N0andα∈Nd\n0.\nProof.Itisenoughtoshowthat |x|k∂l\nt∂α\nxf(t,x)isinL∞(Ω). Sincex=ϕs/parenleftbig\nϕ−1\ns(x)/parenrightbig\n,\n|x|k≤Ck(1+|ϕ−1\ns(x)|2)kn≤Ck(1+(t−s)2+|ϕ−1\ns(x)|2)kn,\nand\n|x|k|f(t,x)| ≤Ck/integraldisplay+∞\n0(1+(t−s)2+|ϕ−1\ns(x)|2)kn∂l\nt∂α\nx/bracketleftbig\ng/parenleftbig\nt−s,ϕ−1\ns(x)/parenrightbig/bracketrightbig\nds\n≤Ck,l,m,α/integraldisplay+∞\n0ds\n(1+(t−s)2)m,\nwhere the derivatives of gare estimates as in proposition C.2. /square\nThe function (90) is referred to as the causal solution of equation (89).\nAppendix D.The Hilbert transform and its action on symbols\nThe Hilbert transform is defined by\nH(φ)(x) =1\nπp.v./integraldisplayφ(y)\nx−ydy=1\nπ(p.v.1\nx)∗φ(x),\nfor a function φ∈C∞\n0. It has the following properties, which we state without\nproof.\nProposition D.1. The Hilbert transform Hdefined above is extended to functions\ninS(R)through the equality\nH(φ)(x) =1\nπ/integraldisplay\nR1\n2u/bracketleftbig\nφ(x−u)−φ(x+u)/bracketrightbig\ndu.\nMoreover, it extends to an isometry of Sobolev spaces Hk(R)through the equality\n/hatwideH(u) =−isgn(ξ)ˆu,∀u∈Hk(R),\nfor every non-negative integer k. Particularly if u∈ S(R), thenH(u)∈H∞(R).\nWithν >0,G∈ S(R), andφ∈ S(R2) let us consider the integrals\nAν(ω,k):=/integraldisplay\nRG(v)\nω−kv+iνdv, Bν(v,k):=/integraldisplay\nRφ(ω,k)\nω−kv+iνdω,\nthat involve the same denominator as in (28).\nLemma D.2. LetG∈ S(R),φ∈ S(R2), andν >0.\n(i)There are constants CA,j,j= 0,1, depending on G, such that for k/\\e}atio\\slash= 0,\n/vextendsingle/vextendsingleAν(ω,k)/vextendsingle/vextendsingle≤CA,0+CA,1/|k|,\nAν(ω,k)ν→0+\n− −−− →1\nk/bracketleftig\nπH(G)(ω/k)−iπG(ω/k)/bracketrightig\n.\n(ii)For any integer m≥0there exists a constant CB, depending on φandm,\nsuch that/vextendsingle/vextendsingleBν(v,k)/vextendsingle/vextendsingle≤CB(1+k2)−m,\nBν(v,k)ν→0+\n− −−− → −πH/parenleftbig\nφ(·,k)/parenrightbig\n(kv)−iπφ(kv,k).TEMPERED-IN-TIME RESPONSE OF A PLASMA 61\nProof.LetG∈ S(R) andφ∈ S(R2). Fork/\\e}atio\\slash= 0, define the two functions G0and\nG1by\nG0/parenleftbigω\nk,u/parenrightbig:=1\n2/bracketleftbig\nG/parenleftbigω\nk−u/parenrightbig\n+G/parenleftbigω\nk+u/parenrightbig/bracketrightbig\n,\nuG1/parenleftbigω\nk,u/parenrightbig:=1\n2/bracketleftbig\nG/parenleftbigω\nk−u/parenrightbig\n−G/parenleftbigω\nk+u/parenrightbig/bracketrightbig\n.\nSimilarly, for all ( ω,k,v)∈R3we define\nφ0(kv,̟,k):=1\n2(φ(kv+̟,k)+φ(kv−̟,k)),\n̟φ1(kv,̟,k):=1\n2/bracketleftig\nφ(kv+̟,k)−φ(kv−̟,k)/bracketrightig\n.\nWe observe that,\n(92)/vextendsingle/vextendsingleG0(ω\nk,u)/vextendsingle/vextendsingle≤ /ba∇dblG/ba∇dbl0,(1+k2)m/vextendsingle/vextendsingleφ0(kv,̟,k)/vextendsingle/vextendsingle≤ /ba∇dblφ/ba∇dbl2m.\nThe first inequality follows directly from the definition of G0, and we have\n(1+k2)m/vextendsingle/vextendsingleφ0(kv,̟,k)/vextendsingle/vextendsingle≤1\n2/bracketleftbig\n(1+(kv+̟)2+k2)m/vextendsingle/vextendsingleφ(kv+̟,k)/vextendsingle/vextendsingle\n+(1+(kv−̟)2+k2)m/vextendsingle/vextendsingleφ(kv−̟,k)/vextendsingle/vextendsingle/bracketrightbig\n≤ /ba∇dblφ/ba∇dbl2m.\nMoreover,\n/vextendsingle/vextendsingleG1(ω\nk,u)/vextendsingle/vextendsingle≤ /ba∇dblG/ba∇dbl1,(1+k2)m/vextendsingle/vextendsingleφ1(kv,̟,k)/vextendsingle/vextendsingle≤ /ba∇dblφ/ba∇dbl1+2m,\nand this can be proven by Taylor’s formula and\n/vextendsingle/vextendsingleG1(ω\nk,u)/vextendsingle/vextendsingle≤1\n2/integraldisplay+1\n−1/vextendsingle/vextendsingleG′(ω\nk+λu)/vextendsingle/vextendsingledλ≤sup|G′(v)|,\n(1+k2)m/vextendsingle/vextendsingleφ1(kv,̟,k)/vextendsingle/vextendsingle≤1\n2/integraldisplay+1\n−1(1+k2)m/vextendsingle/vextendsingle∂ωφ(kv+λ̟,k)/vextendsingle/vextendsingledλ\n≤sup/bracketleftbig\n(1+(kv+λ̟)2+k2)m/vextendsingle/vextendsingle∂ωφ(kv+λ̟,k)/vextendsingle/vextendsingle/bracketrightbig\n.\nWhen|u|>1 we also have, for any M∈N0,\n/vextendsingle/vextendsingleG1(ω\nk,u)/vextendsingle/vextendsingle≤1\n2/bracketleftig/vextendsingle/vextendsingleG(ω\nk−u)/vextendsingle/vextendsingle+/vextendsingle/vextendsingleG(ω\nk+u)/vextendsingle/vextendsingle/bracketrightig\n≤1\n2/bracketleftbig1\n(1+(ω\nk−u)2)M+1\n(1+(ω\nk+u)2)M/bracketrightbig\nsup/bracketleftbig\n(1+v2)M|G(v)|/bracketrightbig\n,\nand, analogously, when |̟|>1,\n(1+k2)m/vextendsingle/vextendsingleφ1(kv,̟,k)/vextendsingle/vextendsingle≤1\n2/bracketleftbig1\n(1+(kv+̟)2)M+1\n(1+(kv−̟)2)M/bracketrightbig\n×sup/bracketleftbig\n(1+ω2+k2)M+m/vextendsingle/vextendsingleφ(ω,k)/vextendsingle/vextendsingle/bracketrightbig\n.\nTherefore,\n(93)/vextendsingle/vextendsingleG1(ω\nk,u)/vextendsingle/vextendsingle≤G∗\n1(ω\nk,u),(1+k2)m/vextendsingle/vextendsingleφ1(kv,̟,k)/vextendsingle/vextendsingle≤φ∗\n1(kv,̟),\nwhere\nG∗\n1(ω\nk,u):=/braceleftigg\n/ba∇dblG/ba∇dbl1, |u| ≤1,\n1\n2/bracketleftbig1\n(1+(ω\nk−u)2)M+1\n(1+(ω\nk+u)2)M/bracketrightbig\n/ba∇dblG/ba∇dbl2M,|u|>1.62 O. LAFITTE AND O. MAJ\nand\nφ∗\n1(kv,̟):=/braceleftigg\n/ba∇dblφ/ba∇dbl1+2m, |̟| ≤1,\n1\n2/bracketleftbig1\n(1+(kv−̟)2)M+1\n(1+(kv+̟)2)M/bracketrightbig\n/ba∇dblφ/ba∇dbl2(M+m),|̟|>1.\nForM≥1, one hasG∗\n1(ω\nk,·),φ∗\n1(kv,·)∈L1(R) with\n(94)/ba∇dblG∗\n1(ω\nk,·)/ba∇dblL1≤2/ba∇dblG/ba∇dbl1+/ba∇dblG/ba∇dbl2M/integraldisplay\nRdt\n(1+t2)M=:CM,G,\n/ba∇dblφ∗\n1(kv,·)/ba∇dblL1≤2/ba∇dblφ/ba∇dbl1+2m+/ba∇dblφ/ba∇dbl2(M+m)/integraldisplay\nRdt\n(1+t2)M=:CM,m,φ,\nwhereCM,Gis a constant depending only on the Schwartz semi-norms of Gand\nthe integer MandCM,m,φdepends only on M,m, and the Schwartz semi-norms\nofφ.\nThen, one has the identities\nAν(ω,k) =/integraldisplay\nRG0(ω\nk,u)+uG1(ω\nk,u)\nku+iνdu,\nBν(v,k) =/integraldisplay\nRφ0(kv,̟,k)+̟φ1(kv,̟,k)\n̟+iνd̟.\nAsG0,G1are even functions in u, andφ0,φ1are even functions in ̟, one deduces\nkAν(ω,k) =/integraldisplay\nRk2u2G1(ω\nk,u)−ikνG0(ω\nk,u)\nk2u2+ν2du\n=/integraldisplay\nRk2u2G1(ω\nk,u)\nk2u2+ν2du−ik/integraldisplay\nRG0(ω\nk,νt)\nk2t2+1dt,\nBν(v,k) =/integraldisplay\nR̟2φ1(kv,̟,k)−iνφ0(kv,̟,k)\n̟2+ν2d̟\n=/integraldisplay\nR̟2φ1(kv,̟,k)\n̟2+ν2d̟−i/integraldisplay\nRφ0(kv,νt,k)\nt2+1dt.\nWe observe thatk2u2\nk2u2+ν2and̟2\n̟2+ν2are uniformly bounded by 1. Moreover, t/ma√sto→\n1\nt2+1is inL1(R), and, for k/\\e}atio\\slash= 0,t/ma√sto→1\nk2t2+1belongs to L1(R), with the values/integraltext\nRdt\nt2+1=πand/integraltext\nRdt\nk2t2+1=π\nk.\nThen estimates (92), (93), and (94) give\n/vextendsingle/vextendsinglekAν(ω,k)/vextendsingle/vextendsingle≤ /ba∇dblG∗\n1(ω\nk,·)/ba∇dblL1+π|k|/ba∇dblG/ba∇dbl0≤CM,G+π|k|/ba∇dblG/ba∇dbl0,\n(1+k2)m/vextendsingle/vextendsingleBν(v,k)/vextendsingle/vextendsingle≤ /ba∇dblφ∗\n1(kv,·)/ba∇dblL1+π/ba∇dblφ/ba∇dbl2m≤CM,m,φ+π/ba∇dblφ/ba∇dbl2m.\nAs for the limits of AνandBνasν→0+, estimate (92) and (93) with M≥1\nimply that the integrands are bounded by an L1function uniformly in ν. Then the\ndominated convergence theorem allows us to conclude that, as ν→0+,\nkAν(ω,k)→/integraldisplay\nRG1(ω\nk,u)du−iπG0(ω\nk,0),\nBν(v,k)→/integraldisplay\nRφ1(kv,̟,k)d̟−iπφ0(kv,0,k).TEMPERED-IN-TIME RESPONSE OF A PLASMA 63\nUpon accounting for the definitions of G0,G1andφ0,φ1that reads\nkAν(ω,k)→πH/parenleftbig\nG)(ω/k)−iπG0(ω/k,0),\nBν(v,k)→ −πH/parenleftbig\nφ(·,k)/parenrightbig\n(kv)−iπφ(kv,k),\nas claimed. /square\nAppendix E.A useful linear algebra result\nIn this appendix we establish a linear algebra result which constitutes a key step\nin the proof of the results in section 4.\nLemma E.1. For any integer ℓ≥0, the matrices\nAℓ=\n0i 0 0 ...\n−iℓ0 2 i0...\n0−i(ℓ−1) 0 3 i ...\n0 0 −i(ℓ−2) 0...\n...............\n∈C(ℓ+1)×(ℓ+1)\nare diagonalizable with integer eigenvalues {(2s−ℓ) :s= 0,1,...,ℓ}. Forz∈C\\Z,\nAℓ−zis invertible and\n|(Aℓ−z)−1|1≤(ℓ+1)22ℓ/δz,\nwhere the norm | · |1is induced by the L1-norm on Cℓ+1andδz= min{|z−m|:\nm∈Z}.\nProof.The mapi1:Cℓ+1→C∞(T),T=R/(2πN), defined by\nx/ma√sto→U(φ) =ℓ/summationdisplay\nr=0xr/parenleftbig\ncosφ/parenrightbigℓ−r/parenleftbig\nsinφ)r,\nis an linear embedding of Cℓ+1and becomes an isomorphism when restricted to its\nrange. Injectivity in particular holds since, if x∈Cℓ+1is such that U=i1(x) = 0,\nthen for any R≥0\nRℓU(φ) =ℓ/summationdisplay\nr=0xruℓ−r\n1ur\n2= 0,\nuniformly for ( u1,u2) = (Rcosφ,Rsinφ)∈R2. Since monomials uℓ−r\n1ur\n2are lin-\nearly independent, we deduce x= 0 and thus i1is injective.\nA second embedding is i2:Cℓ+1→C∞(T) with\nv/ma√sto→V(φ) =ℓ/summationdisplay\ns=0vsei(ℓ−2s)φ,\nwhich is injective since the exponential are linearly independent.\nWe claim that i1,i2have the same range i.e., Vℓ:=i1(Cℓ+1) =i2(Cℓ+1). In\nfact,i1(Cℓ+1) is spanned by functions fr(φ) =/parenleftbig\ncosφ/parenrightbigℓ−r/parenleftbig\nsinφ)rwhilei2(Cℓ+1) is64 O. LAFITTE AND O. MAJ\nspanned by gs(φ) =ei(ℓ−2s)φ. On the one hand, the binomial formula gives\nfr(φ) =(−i)r\n2ℓ/parenleftbig\neiφ+e−iφ/parenrightbigℓ−r/parenleftbig\neiφ−e−iφ/parenrightbigr\n=(−i)r\n2ℓℓ−r/summationdisplay\nm=0r/summationdisplay\nn=0(−1)n/parenleftbiggℓ−r\nm/parenrightbigg/parenleftbiggr\nn/parenrightbigg\nei(ℓ−2(m+n))φ,\nand thus\nfr(φ) =ℓ/summationdisplay\ns=0Tsrgs(φ), Tsr=(−i)r\n2ℓ/summationdisplay\n(m,n)∈Σ(r,s)(−1)n/parenleftbiggℓ−r\nm/parenrightbigg/parenleftbiggr\nn/parenrightbigg\n,\nwhere the last sum is over the set of indices\nΣ(r,s) ={(m,n) :m= 0,...,ℓ−r, n= 0,...,r, m +n=s}.\nOn the other hand,\ngs(φ) =ei(ℓ−2s)φ=/parenleftbig\ncosφ+isinφ/parenrightbigℓ−s/parenleftbig\ncosφ−isinφ/parenrightbigs\n=ℓ−s/summationdisplay\nm=0s/summationdisplay\nn=0im−n/parenleftbiggℓ−s\nm/parenrightbigg/parenleftbiggs\nn/parenrightbigg/parenleftbig\ncosφ)ℓ−(m+n)/parenleftbig\nsinφ/parenrightbigm+n,\nor\ngs(φ) =ℓ/summationdisplay\nr=0Srsfr(φ), Srs=/summationdisplay\n(m,n)∈Σ(s,r)im−n/parenleftbiggℓ−r\nm/parenrightbigg/parenleftbiggr\nn/parenrightbigg\n.\nIn summary, we have obtained that\nfr=ℓ/summationdisplay\ns=0Tsrgs, gs=ℓ/summationdisplay\nr=0Srsfr,\nhence for any ℓ,{fr}ℓ\nr=0and{gs}ℓ\ns=0span the same linear space Vℓ. The matrices\n(Tsr) and (Ssr) correspond to the linear operators\nT=i−1\n2◦i1:Cℓ+1→Cℓ+1, S=i−1\n1◦i2:Cℓ+1→Cℓ+1.\nAt last, let us introduce the differential operator\nB=id\ndφ:C∞(T)→C∞(T),\nfor which a direct calculation shows that\nB◦i1(x) =ℓ/summationdisplay\nr=0(Aℓx)rfr(φ).\nThis means that Bcan be restricted to operators from Vℓ→Vℓand\n(95) Aℓ=i−1\n1◦B◦i1.\nOn the other hand\n(96) B◦i2(v) =ℓ/summationdisplay\ns=0(2s−ℓ)vsgs(φ),\nthat is, the operator i−1\n2◦B◦i2is diagonal with eigenvalues (2 s−ℓ).\nFor equation (95) it follows that\nAℓ=i−1\n1◦B◦i1=S◦(i−1\n2◦B◦i2)◦T,TEMPERED-IN-TIME RESPONSE OF A PLASMA 65\nwhich shows that Aℓis diagonalizable with eigenvalues a= (2s−ℓ) fors= 0,...ℓ.\nTherefore if z∈C\\Z, thenAℓ−zis invertible and\n(Aℓ−z)−1=S◦(i−1\n2◦(B−z)−1◦i2)◦T,\nhence\n(97) |(Aℓ−z)−1|1≤ |S|1·|i−1\n2◦(B−z)−1◦i2|1·|T|1.\nWe claim that\n(98)|i−1\n2◦(B−z)−1◦i2|1≤δ−1\nz,|T|1≤(ℓ+1),|S|1≤(ℓ+1)2ℓ.\nIf this is true, then inequality (97) becomes |(Aℓ−z)−1|1≤2ℓ(ℓ+1)2/δz, which is\nthe claimed estimate. Therefore, in order to complete the proof it is sufficient to\nshow that (98) holds.\nFrom (96), one deduces the expression for the inverse,\n(B−z)−1◦i2(v) =ℓ/summationdisplay\ns=0vs\n2s−ℓ−zei(ℓ−2s)φ,\nand thus, in the L1-norm,\n|i−1\n2◦(B−z)−1◦i2(v)|1=ℓ/summationdisplay\ns=0|vs|\n|2s−ℓ−z|.\nThe definition of δzimplies\n|2s−ℓ−z| ≥δz= min\nm∈Z|z−m|,\nso that\n(99) |i−1\n2◦(B−z)−1◦i2(v)|1≤1\nδzℓ/summationdisplay\ns=0|vs|=|v|1\nδz,\nor|i−1\n2◦(B−z)−1◦i2|1≤δ−1\na. As for the linear operator T,\n|Tsr| ≤1\n2ℓℓ−r/summationdisplay\nm=0r/summationdisplay\nn=0/parenleftbiggℓ−r\nm/parenrightbigg/parenleftbiggr\nn/parenrightbigg\n,\nand for the last sum we can use the identity,\n(a+b)ℓ= (a+b)ℓ−r(a+b)r=ℓ−r/summationdisplay\nm=0r/summationdisplay\nn=0/parenleftbiggℓ−r\nm/parenrightbigg/parenleftbiggr\nn/parenrightbigg\naℓ−m−nbm+n,\nwhich fora=b= 1 and together with the previous inequality implies |Tsr| ≤1;\nthen\n|Tx|1=ℓ/summationdisplay\ns=0/vextendsingle/vextendsingle/vextendsingleℓ/summationdisplay\nr=0Tsrxs/vextendsingle/vextendsingle/vextendsingle≤ℓ/summationdisplay\ns=0ℓ/summationdisplay\nr=0|Tsr||xs| ≤(ℓ+1)|x|1,\nor|T|1≤(ℓ+1). Analogously for S,\n|Srs| ≤ℓ−s/summationdisplay\nm=0s/summationdisplay\nn=0/parenleftbiggℓ−s\nm/parenrightbigg/parenleftbiggs\nn/parenrightbigg\n= 2ℓ,\nand|S|1≤(ℓ+1)2ℓ. /square66 O. LAFITTE AND O. MAJ\nAcknowledgments. 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Tan et al., \narXiv:0705.3502, (2007). ] and the damping -like terms [REF: H. Kurebayas hi et al., Nature \nNanotechnology 9, 211 (2014). ] that have been widely studied for applications in magnetic \nmemory. We focus , in this article, not on the spin orbit effect producing the above spin \ntorques, but on its magnifying the damping constant of all field like spin torques. As first \norder precession leads to second order damping, the Rashba constant is naturally co -opted, \nproducing a magnified field -like dam ping effect. The Landau -Liftshitz -Gilbert equations are \nwritten separately for the local magnet ization and the itinerant spin, allowing the \nprogression of magnetization to be self -consistently locked to the spin. \n \n \n \n \n \nPACS: 03.65.Vf, 73.63. -b, 73.43. -f \n† Correspondence author: \nSeng Ghee Tan \nEmail: Tan_Seng_Ghee@dsi.a -star.edu.sg \n Magnified Damping under Rashba Spin Orbit Coupling \nNovember 1, 2015 \n \n \n2 \n 1. Introduction \n In spintronic and magnetic physics, magnetization switching and spin torque [1] have \nbeen well -studied. The advent of the Rashba spin-orbit coupling ( RSOC) [2,3] due to \ninversion asymmetry at the i nterface of the ferromagnetic/heavy atom (FM/HA) \nheterostructure introduces new spin torque to the FM magnetization. The field -like [4-6] \nand the damping -like [7] SOC spin torque had been theoretically derived based on the gauge \nphysics and the Pancharatna m-Berry’s phase , as well as experimentally verified and resolved . \nThe numerous observation s of spin -orbit generation of spin torque [8-10], are all related to \nthe experimental resolutions [6,7] of their field -like and damping -like nature, thus ushering \nin the possibility of spin -orbit based magnetic memory. While the damping -like spin torque \ndue to Kurebayashi et al. [7] is dissipative in nature, the field -like due to Tan et al. [4,11], is \nnon-dissipative , and precession causing . Recent studies have even mo re clearly \ndemonstrated the physics and application promises of both the field -like and the damping -\nlike SOC spin torque [12-14]. Besides , similar SOC spin torque have also been studied \ntheoretically in FM/3D -Rashba [15] and FM -topological -insulator materi al [16,17] , and \nexperimentally shown [18, 19 ] in topological insulator materials. \n The dissipative physics of all field -like magnetic torque terms have been derived in \nsecond -order manifestation in a manner introduced by Gilbert in the 1950 ’s. Conven tional \nstud y of magnetization dynamics is based on a Gilbert damping constant which is \nincorporated manually into the Landau -Lifshit z-Gilbert (LLG) equation. In this paper, we will \nfocus our attention not so much on the spin-orbit effect producing the SOC spin torque, as \non the spin -orbit effect magnifying the damping constant of all field -like spin torques. As \nfield -like spin torques, regardless of origin s, generate first-order precession , the Rashba \nconstant will be co -opted in to the second -order damping effect, producing a mag nified \ndamping constant . On the other hand, c onventional incorporation of the dissipative \ndamping physics into the LLG would fail t o account for the spin-orbit magnification of the \ndamping strength . It would therefore be necessary to deriv e the LLG equations from a \nHamiltonian which describes electron due to the local FM magnetization (𝒎), and those \nitinerant (𝒔) and injected from external parts . We present a set of modified LLG equation s \nfor the 𝒎 and the 𝒔. This will be necessary for a more precise modeling of the 𝒎 trajectory \nthat simultaneously tracks the 𝒔 trajectory. In summary, the two central themes of this Magnified Damping under Rashba Spin Orbit Coupling \nNovember 1, 2015 \n \n \n3 \n paper is our presentation of a self-consistent set of LLG equations under the Rashba SOC \nand the derivation of the Ra shba -magnified d amping constant in the second -order damping -\nlike spin torque . \n \n2. Theory of Magnified Damping \n The system under consideration is a FM/HA hetero -structure with inversion asymmetry \nprovided by the interface. F ree electron denoted by 𝒔, is injected in an in -plane manner into \nthe device . The FM equilibrium electron is denoted by 𝒎. One considers the external \nsource -drain bias to inject electron of free-electron nature 𝒔 into the FM with kinetic, \nscattering , magnetic, and spin -orbit energ ies. The Hamiltonian is \n𝐻𝑓=𝑝2\n2𝑚+𝑉𝑖𝑚𝑝𝑠+𝐽𝑠𝑑𝑺.𝑴 +𝜇0𝑴.𝑯𝒂𝒏𝒊+(2(𝜆+𝜆′)\nℏ)(𝒔+𝒎).(𝒑×𝑬𝒕)\n−𝑖(𝜆+𝜆′)(𝒔+𝒎).(∇×𝑬𝒕) \n(1) \nwhere 𝒔,𝒎 have the unit s of angular momentum i.e. 𝑛ℏ\n2 , while 𝑴=(𝑔𝑠𝜇𝐵\nℏ)𝒎 has the unit \nof magnetic moment , and 𝜇𝐵=𝑒ℏ\n2𝑚 is the Bohr magneton . Note that (2𝜆\nℏ) is the vacuum SOC \nconstant, while (2𝜆′\nℏ=2𝜂𝑅\nℏ2𝐸𝑖𝑛𝑣) is the Rashba SOC constant. The SOC part of the Hamiltonian \nillustrates th e simultaneous presence of vacuum and Rashba SOC. The proportion of the \nnumber of electron subject to each coupling would depend on the degree of hybridization. \nBut s ince 𝜆′≫𝜆, the above can be written with just the Rashba SOC effect. Care is taken t o \nensure 𝜆,𝜆′ share the same dimension of 𝑇𝑒𝑠𝑙 𝑎−1, and 𝑬𝒕 is the total electric field , 𝐽𝑠𝑑 is \nthe s -d coupling constant, 𝑉𝑖𝑚𝑝𝑠 denotes the spin flip scattering potential, 𝑯𝒂𝒏𝒊 denotes the \naniso tropy field of the FM material. On the other hand, one needs to be aware that the \nabove is an e xpanded SOC expression that c omprises a momentum part as well as a \ncurvature part [20]. One can then consider the physics of the electric curvature as related to \nthe time dynamic of the spin moment , which bears a similar origin to the Faraday effect. In \nthe modern context of Rashba physics [21], one considers electron spin to lock to the orbital \nangular momentum 𝑳 due to intrinsic spin orbit coupling at the atomic level. Due to broken Magnified Damping under Rashba Spin Orbit Coupling \nNovember 1, 2015 \n \n \n4 \n inversion symmetry , electric field (𝑬𝒊𝒏𝒗) points perpendicular to the plane of the FM/HA \nhost . Because of hybridization, the 𝒔,𝑳,𝒑 of an electron is coupled in a complic ated way by \nthe electric field. In a simple way, one first considers 𝑳 to be coupled as 𝐻=(2𝜆\nℏ)𝑳.(𝒑×\n𝑬𝒊𝒏𝒗). As spin 𝒔 is coupled via atomic spin orbit locking to 𝑳, an effective coupling of 𝒔 to \n𝑬𝒊𝒏𝒗 can be expected to occur with strength as determined by the atomic electric field. We \nwill now take things a step further to make an assumption that 𝒔 is also coupled via 𝑳 to \nother sources of electric f ields e.g. those arising from spin dynamic (𝒅 𝑴\n𝒅𝒕,𝒅 𝑺\n𝒅𝒕), in the same \nway that it is coupled to 𝑬𝒊𝒏𝒗 . The actual extent of coupling will , however, be an \nexperimental parameter that measures the efficiency of Rashba coupling to 𝑬𝒊𝒏𝒗 as opposed \nto electric fields (𝑬𝒎 ,𝑬𝒔) arising due to spin dynamic . The total electric field in the system \nis now 𝑬𝒕=𝑬𝒊𝒏𝒗 +𝑬𝒎+𝑬𝒔 , where 𝑬𝒎 ,𝑬𝒔 arise due to 𝒅 𝑴\n𝒅𝒕,𝒅 𝑺\n𝒅𝒕, respectively. On the \nmomentum part of the Hamiltonian 2 𝜆′𝒔.(𝒌×𝑬𝒕), we only need to consider that \n𝑬𝒕=𝑬𝒊𝒏𝒗 as one can, for simplicity, consider 𝑬𝒎 and 𝑬𝒔 to simply vanish on average. Thus \nin this renewed treatment, the momentum part is : \n2𝜆′\nℏ𝒔.(𝒑×𝑬𝒊𝒏𝒗)=𝜂𝑅𝝈.(𝒌×𝒆𝒊𝒏𝒗) \n(2) \nwhere 𝜂𝑅=𝜆′ℏ𝐸𝑖𝑛𝑣 is the Rashba constant that has been vastly measured in many material \nsystems with experimental values ranging from 0.1 to 2 𝑒𝑉𝐴̇. On the curvature part, one \nconsiders 𝑬𝒕=𝑬𝒎+𝑬𝒔 without the 𝑬𝒊𝒏𝒗 as 𝑬𝒊𝒏𝒗 is spatially uniform and thus would have \nzero curvature. In summary, the theory of this paper has it that the time -dynamic of the \nspin in a Rashba system produces a curvature part o f 𝑖𝜆′(𝒔+𝒎).(∇×𝑬𝒕). Without the \nRashba effect, this energy term would just take on the vacuum constant of (2𝜆\nℏ) instead of \nthe magnified (2𝜆′\nℏ). The key physics is that in a Rashba FM/HA system , curvature \n𝑖𝜆′(𝒔+𝒎).(∇×𝑬𝒕) is satisfied by the first-order precession due to 𝒅𝑴\n𝒅𝒕,𝒅𝑺\n𝒅𝒕 which provide \nthe electric field curvature in the form of −𝜇0(1+𝜒𝑚)𝑑 𝑴\n𝑑𝑡=∇×𝑬𝒎 , and −𝜇0(1+\n𝜒𝑠)𝑑𝑺\n𝑑𝑡=∇×𝑬𝒔 , where we remind reader again that 𝑴,𝑺 have the unit of magnetic \nmoment. This results in spin becoming couple d to its own time dynamic, producing a spin -Magnified Damping under Rashba Spin Orbit Coupling \nNovember 1, 2015 \n \n \n5 \n orbit second -order damping -like spin torque. The electric field effect is illustrated in Fig. 1 \nbelow: \n \n \n \n \n \n \n \n \n \nFig.1 . Magnetic precession under the effect of electric fields due to inv ersion asymmetry, self -dynamic of 𝑑𝑴\n𝑑𝑡 \nand the spin dynamic of 𝑑𝑺\n𝑑𝑡 . Projecting 𝑑𝑀 to the heterostructure surface, one could visualize the emergence \nof an induced electric field in the form of 𝛻𝑋𝐸 in such orientation as to satisfy the law of electromagnetism. \n \n \n One notes that the LLG equation is normally derived by letting 𝑺 satisfy the physical \nrequirements of spin transport . One example of these requirements is assumed and \ndiscussed in REF 1 , with definitions contained therein : \n𝑺(𝒓,𝑡)=𝑆0𝒏+𝜹𝑺 \n𝑱(𝒓,𝑡)=−𝜇𝐵𝑃\n𝑒 𝑱𝒆⊗𝒏−𝐷0∇𝜹𝑺 \n(3) \nwhere 𝒏 is the unit vector of 𝑴, and 𝐷0 is the spin diffusion constant. Thus 𝑺=𝑺𝟎+𝜹𝑺 \nwould be the total spin density that contains , respectively, the equilibrium, the non-\nequilibrium adiabatic, non-adiabatic , and Rashba field -like terms , i.e. 𝜹𝑺=𝜹𝑺𝒂+𝜹𝑺𝒏𝒂+\n𝜹𝑺𝑹. One notes that 𝑺𝟎 is the equilibrium part of 𝒔 that is aligned to 𝒎, meaning 𝒔𝟎 could \nexist in the absence of external field and current in the system. The conditions to satisfy are \nrepresented explicitly by the equations of: 𝑑𝑀 \n𝐸 𝑓𝑖𝑒𝑙𝑑 𝑑𝑢𝑒 𝑡𝑜 \n𝑖𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦 \n𝛻𝑋𝐸 \n𝑑𝑀 Magnified Damping under Rashba Spin Orbit Coupling \nNovember 1, 2015 \n \n \n6 \n 𝜕𝜹𝑺 \n𝜕𝑡=0, 𝐷0∇2𝜹𝑺=0,−𝜇𝐵𝑃\n𝑒 𝛁.𝑱𝒆𝑴\n𝑀𝑠=0, 𝑠0𝑴(𝒓,𝑡)\n𝑡𝑓𝑀𝑠=0 \n (4) \nIn the steady state treatment where 𝜕 𝜹𝑺\n𝜕𝑡=0, one recover s the adiabatic component of \n𝜹𝑺𝒂=𝒏×𝒋𝒆.𝛁𝒏 , and the non -adiabatic component of 𝜹𝑺𝒏𝒂=𝒋𝒆.𝛁𝒏. We also take the \nopportunity here to reconcile this with the gauge physics of spin torque, in which case , the \nspin potential 𝐴𝜇𝑠𝑚=𝑒 [𝛼 𝑈𝐸𝑖𝜎𝑗𝜀𝑖𝑗𝜇𝑈†+𝑖ℏ\n𝑒𝑈𝜕𝜇𝑈†] would correspond , respectively, to \n𝜹𝑺𝑹+ 𝜹𝑺𝒂. In fact, t he emergent spin p otential [22, 23] can be considered to encapsulate \nthe physics of electron interaction with the local magnetization under the effect of SOC [4, 5 , \n24-26]. Here we caution that 𝜹𝒔𝑹 is restricted to the field -like spin -orbit effect only . \n However, in this paper , 𝑺 is defined to satisfy the transport equations in Eq.(4) except for \n𝜕𝜹𝑺 \n𝜕𝑡=0. Keeping the dynamic property of 𝑺 here allows a self -consistent equation set \n𝑑𝑺\n𝑑𝑡,𝑑𝑴\n𝑑𝑡 to be introduced . The energy as experienced by the 𝑺,𝑴 electron are, respectively, \n𝐻𝑓𝑠=𝑺.𝛿𝐻𝑓\n𝛿𝑺 , 𝐻𝑓𝑚=𝑴.𝛿𝐻𝑓\n𝛿𝑴 \n(5) \nwith caution that 𝐻𝑓𝑠≠𝐻𝑓𝑚 . Upon rearrangement, the 𝒔,𝒎 centric energ ies are, \nrespectively, \n𝐻𝑓𝑠=(𝑝2\n2𝑚+𝑉𝑖𝑚𝑝𝑠+𝐽𝑠𝑑𝑺.𝑴+𝑺.𝑩𝑹−𝒊𝜆′𝒔.(∇×𝑬𝒕)) \n𝐻𝑓𝑚=(𝐽𝑠𝑑𝑴.𝑺+𝜇0𝑴.𝑯𝒂−𝒊𝜆′𝒎.(∇×𝑬𝒕) ) \n(6) \nwhere 2𝜆′\nℏ𝒔.(𝒑×𝑬𝒕)=𝑺.𝑩𝑹, while 2𝜆′\nℏ𝒎.(𝒑×𝑬𝒕) vanishes . We particularly note that \nthere have been recent discussions on the field -like [4,6,11 ] spin orbit torque as well as the \ndamping [7] version. With 𝑑𝒔\n𝑑𝑡=−𝟏\n𝒊ℏ[𝒔,𝐻𝑓𝑠] ,𝑑𝒎\n𝑑𝑡=−𝟏\n𝒊ℏ[𝒎,𝐻𝑓𝑚], one would now have four \ndissipative torque t erms experienced by electron 𝒔,𝒎 as shown below : Magnified Damping under Rashba Spin Orbit Coupling \nNovember 1, 2015 \n \n \n7 \n ( 𝝉𝑺𝑺 𝝉𝑺𝑴\n𝝉𝑴𝑺 𝝉𝑴𝑴)=𝑖𝜆′𝜇0(𝒔×(1+𝜒𝑠−1)𝑑𝑺\n𝑑𝑡𝒔×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡\n𝒎×(1+𝜒𝑠−1)𝑑𝑺\n𝑑𝑡𝒎×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡) \n(7) \nTo be consistent with conventional necessity to preserve magnetization norm in the physics \nof the LLG equation, we will drop the off-diagonal terms which are norm -breaking (non -\nconservation) . This is in order to keep the LLG equation in its conventional norm -conserving \nform, simplifying physics and calculation therefrom. Nonetheless, the non -conserving parts \nrepresent new dynamic physics that can be analysed in the future with techniques other \nthan the familiar LLG equations. The self -consistent pair of spin torque equations in their \nopen forms are: \n𝜕𝑺\n𝜕𝑡=−(𝑺× 𝑩𝑹+𝑺\n𝑡𝑓)−1\n𝑒𝛻𝑎(𝑗𝑎𝒔 𝑺)−(𝑺×𝑴\n𝑚𝑡𝑒𝑥)−𝝉𝑺𝑺 \n𝜕𝑴\n𝜕𝑡=−𝛾𝑴×𝜇0𝑯𝒂−𝑴×𝑺\n𝑚 𝑡𝑒𝑥−𝝉𝑴𝑴 \n(8) \nwhere 𝐽𝑠𝑑=1\n𝑚𝑡𝑒𝑥 has been applied, 𝛾 is the gyromagnetic ratio, 𝜒𝑚 is the susceptibility. For \nthe stud y of Rashba -magnified damping i n this paper, we only need to keep the most \nrelevant term which is 𝝉𝑴𝑴=𝑖 𝜂𝑅\nℏ𝐸𝑖𝑛𝑣𝜇0(1+𝜒𝑚−1) 𝒎×𝑑 𝑴\n𝑑𝑡. In the phenomenological physics \nof Gilbert, the first-order precession leads inevitably to the second -order dissipative terms \nvia 𝒔.𝒅𝑺\n𝒅𝒕 ,𝒎.𝒅𝑴\n𝒅𝒕. But in this paper, the general SOC physics had been expanded as shown in \nearlier sections, so that the dissipative terms are to naturally arise fr om such expansion. The \nadvantage of the non -phenomenological approach is that, as said earlier, the Rashba \nconstant will be co -opted into the second -order damping effect, resulting in the \nmagnification of the damping constant associated with all field -like spin torque. \n \n Magnified Damping under Rashba Spin Orbit Coupling \nNovember 1, 2015 \n \n \n8 \n 3. Conclusion \n The im portant result in this paper is that the damping constants have been magnified by \nthe Rashba effect. This would not be possible if the damping constant was incorporated \nmanually by standard means of Gilbert. As the Rashba constant is larger than the vacuum \nSOC constant as can be deduced from Table 1 and shown below \n𝛼𝑅=𝛼𝜆′\n𝜆 , \n(9) \nmagnetization dynamics in FM/HA hetero -structure with inversion asymmetry (interface, or \nbulk) might have to be modelled with the new equations. It is important to remind that all \npreviously measured 𝜂𝑅 has had 𝐸𝑖𝑛𝑣 captured in the measured value. But w hat is needed in \nour study is the coupling of 𝑺 to a dynamic electric field, and that requires the value of just \nthe coupling strength (𝜆′). As most measurement is carr ied out for 𝜂𝑅, the exact knowledge \nof 𝐸𝑖𝑛𝑣 corresponding to a specific 𝜂𝑅 will have a direct impact on the actual value of 𝜆′. We \nwill, nonetheless, provide a quick, possibly exaggerated estimate. Noting that 𝜆=𝑒ℏ\n4𝑚2𝑐2 \nand 𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣, and taking one measured value of 𝜂𝑅=1×10−10𝑒𝑉𝑚 , corresponding to a \n𝐸𝑖𝑛𝑣=1010𝑉/𝑚, the magnification of 𝛼 works out to 104 times in magnitude , which may \nseem unrealistically strong . The caveat lies in the exact correspondence of 𝜂𝑅 to 𝐸𝑖𝑛𝑣, which \nremains to be determined experimentally. For example, if an experimentally determined \n𝜂𝑅 actual ly corresponds to a much larger 𝐸𝑖𝑛𝑣, that would mean that 𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣 which \nmagnifies the damping constant through 𝛼𝑅=𝛼𝜆′\n𝜆 might actually be much lower than \npres ent estimate. Therefore, it is worth remembering, for simplicity sake that 𝛼𝑅 actually \ndepends on the ratio of 𝜂𝑅\n𝐸𝑖𝑛𝑣 but not 𝜂𝑅. It has also been assumed that 𝑳 couples to 𝑬𝒔,𝑬𝒎 \nwith the same efficiency that it couples to 𝑬𝒊𝒏𝒗. This is still uncertain as th e Rashba \nconstant with respect to 𝑬𝒔,𝑬𝒎 might actually be lower than those 𝜂𝑅 values that have \nbeen experimentally measured mostly with respect to 𝑬𝒊𝒏𝒗. Last, we note that as damping \nconstant has been magnified here, and as increasingly high -precision, live monitoring of \nsimult aneous 𝒔,𝒎 evolution is no longer redundant in smaller devices, care has been taken Magnified Damping under Rashba Spin Orbit Coupling \nNovember 1, 2015 \n \n \n9 \n to present the LLG equations in the form of a self-consistent pair of dynamic equations \ninvolving 𝑴 and 𝑺. This will be necessary for the accurate modeling of the simultaneous \ntrajectory of both 𝑴 and 𝑺. \n \nTable 1. Summary of damping torque and damping con stant with and without Rashba effects. \n Hamiltonian Torque Damping constant \n1. 𝐻=(2𝜆\nℏ)𝒔.(𝒑×𝑬𝒕) \n𝜆=𝑒ℏ\n4𝑚2𝑐2 \n𝜕𝒎\n𝑑𝑡=𝑖𝜆𝜇0𝒎×(1+𝜒𝑚−1)𝜕𝑴\n𝑑𝑡 \n𝛼=𝑖𝜆\n2𝜇0𝑀𝑠(1+𝜒𝑚−1) \n2. 𝐻𝑅=(2𝜆′\nℏ)𝒔.(𝒑×𝑬𝒕) \n𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣 \n𝜕𝒎\n𝑑𝑡=𝑖𝜆′𝜇0𝒎×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡 \n𝛼𝑅=𝑖𝜆′\n2𝜇0𝑀𝑠(1+𝜒𝑚−1) \n \n \n \n \n \nREFERENCES \n \n \n[1] S. Zhang & Z. Li, “Roles of Non -equilibrium conduction electrons on the magnetization dynamics \nof ferromagnets”, Phys. Rev. Letts 93, 127204 (2004). \n[2] F.T. Vasko, “Spin splitting in the spectrum of two -dimensional electrons due to the surface \npotential”, Pis’ma Zh. Eksp. Teor. Fiz. 30, 574 (1979) [ JETP Lett. , 30, 541]. \n[3] Y.A. Bychkov & E.I. Rashba, “Properties of a 2D electron gas with lifted spectral degeneracy”, \nPis’ma Zh. Eksp. Teor. Fiz. , 39, 66 (1984) [ JETP Lett. , 39, 78]. \n[4] S. G. Tan, M. B. A. 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Chapter -5 (2012). \n \n " }, { "title": "2002.02865v1.Model_of_damping_and_anisotropy_at_elevated_temperatures__application_to_granular_FePt_films.pdf", "content": "Model of damping and anisotropy at elevated temperatures: application to granular FePt films\nMara Strungaru,1Sergiu Ruta,1Richard F. L. Evans,1and Roy W. Chantrell1\n1Department of Physics, University of York, York, YO10 5DD, UK\nUnderstanding the damping mechanism in finite size systems and its dependence on temperature is a critical\nstep in the development of magnetic nanotechnologies. In this work, nano-sized materials are modeled via atom-\nistic spin dynamics, the damping parameter being extracted from Ferromagnetic Resonance (FMR) simulations\napplied for FePt systems, generally used for heat-assisted magnetic recording media (HAMR). We find that the\ndamping increases rapidly close to TCand the effect is enhanced with decreasing system size, which is ascribed\nto scattering at the grain boundaries. Additionally, FMR methods provide the temperature dependence of both\ndamping and the anisotropy, important for the development of HAMR. Semi-analytical calculations show that,\nin the presence of a grain size distribution, the FMR linewidth can decrease close to the Curie temperature due\nto a loss of inhomogeneous line broadening. Although FePt has been used in this study, the results presented in\nthe current work are general and valid for any ferromagnetic material.\nI. INTRODUCTION\nThe magnetic damping parameter is important from both a\nfundamental and applications point of view as it controls the\ndynamic properties of the system such as magnetic relaxation,\nspin waves, domain wall propagation and magnetic reversal\nprocesses. Magnetic materials have a broad range of inter-\nest for nano-devices/nano-elements and exhibit a fast response\nto external excitations. In information technologies, damping\nplays a crucial role, especially for spin-transfer torque mag-\nnetic random access memories (STT-MRAM) where it con-\ntrols the switching current [1]. With the emerging field of\nmagnetisation switching via ultrafast laser pulses, the damp-\ning parameter can influence the fluence of the laser pulse nec-\nessary for demagnetising and switching of the sample [2].\nSpintronic devices such as race-track memories which are\nbased on domain wall propagation in magnetic nanowires are\nalso influenced by damping [3]. As current magnetic tech-\nnologies are based on nanostructures of smaller and smaller\nsizes, the finite size effects become more important and can\nsignificantly influence the magnetic properties including the\ndamping. Therefore understanding the dependence of damp-\ning on temperature in finite size systems is a critical step in\nthe development of magnetic nano-technologies.\nOne of the technologies that is strongly influenced by\ndamping is magnetic recording, where the damping constant\nof the storage medium controls the writing speeds and bit er-\nror rates [4, 5]. The next generation of ultra-high density stor-\nage technology is likely to be based on heat-assisted magnetic\nrecording (HAMR) [6–9]. The main candidate for HAMR\nmedia is L10ordered FePt [8, 10] due to its large perpendicu-\nlar anisotropy and low Curie temperature ( TC). For HAMR ap-\nplications, information is stored at room temperature (300K)\nbut the writing is done at elevated temperatures close to TC,\ntherefore providing a large range over which the temperature\ndependence of the damping needs to be understood. As the\nareal density increases, the grain size decreases and finite-size\neffects are becoming crucial. For this reason, FePt is an ideal\ncandidate for studying temperature and finite size effects on\nthe damping. Although FePt has been used in this study, the\nresults presented in the current work are general and valid for\nany ferromagnetic material.First investigations of the Gilbert damping for FePt in-\nvolved experimental measurements via optical pump-probe\ntechniques. The damping measured at room temperature\nvaries widely from one study to another. Becker et al. [11]\nreported an effective damping of 0.1, and an even larger value\n(0.21) was found by Lee et al. [12], while the measurments\nof Mizukami et al [13] gave a value of 0.055. It is important\nto note that these values include both intrinsic and extrinsic\ncontributions, the purely intrinsic damping being even smaller\nthan the reported values [13, 14]. Recently, Richardson et al\n[15] reported experimental measurements of damping at ele-\nvated temperatures showing an unexpected decrease of damp-\ning with temperature. A decrease in the effective damping can\nbe crucial in HAMR, as this can increase the switching time,\naffect the signal to noise ratio and negatively impact the per-\nformance of HAMR. Theoretical studies on how the damping\nvaries at elevated temperatures and in finite sized systems is\ntherefore a critically important problem.\nOstler et al [16] have successfully calculated the tem-\nperature dependence of damping in FePt bulk and thin-film\nsystems based on the Landau-Lifshitz-Bloch (LLB) equa-\ntion [17], showing an increased damping for thin-film sys-\ntems, in comparison with the bulk case. The LLB equation\nis derived for a bulk material. It is important to note that a\nmajor contribution to damping, especially at elevated temper-\natures, arises from magnon scattering. On the bulk scale these\nprocesses are reproduced by the LLB equation, but with de-\ncreasing linear dimension, finite size and surface effects be-\ncome important. Since these are not accounted for by the\nLLB equation it is necessary to use atomistic spin dynamics\n(ASD) simulations [18] for nanoscale grains, as ASD calcula-\ntions include magnon processes. Using atomistic spin dynam-\nics, ASD, we are able to calculate the FMR spectra for small\nsystem sizes at elevated temperatures. Ferromagnetic reso-\nnance simulations are computationally very expensive, hence\nwe have developed a more efficient method of calculating the\ndamping of these systems via a grid search method. We show\nthat both of the methods agree, and furthermore, that they\nare able to calculate both the dependence of damping and\nanisotropy as function of temperature. The dependence of\nanisotropy as function of temperature is crucial for HAMR as\nit defines the temperature at which the writing process occurs.arXiv:2002.02865v1 [cond-mat.mtrl-sci] 7 Feb 20202\nII. FERROMAGNETIC RESONANCE USING ATOMISTIC\nSPIN DYNAMICS\nTo calculate damping as function of temperature we per-\nform atomistic spin dynamics (ASD) simulations using the\nsoftware package VAMPIRE [18]. ASD simulations assume\na fixed lattice of atoms to which is associated a magnetic\nmoment or spin Si=mi=msthat can precess in an effective\nfield Hiaccording to the Landau-Lifshitz Gilbert (LLG) equa-\ntion. In our model, the Hamiltonian of the system contains a\nHeisenberg exchange term of strength Ji j, uniaxial energy of\nstrength kuand a Zeeman term as shown in Eq. 1:\nH=\u00001\n2å\ni;jJi j(Si\u0001Sj)\u0000kuå\ni(Si\u0001e)2\u0000å\nimi(Si\u0001B)(1)\nThe effective field can be calculated from the Hamiltonian of\nthe model to which we add a thermal noise xi, that acts as a\nLangevin thermostat:\nHi=\u00001\nmim0¶H\n¶Si+xi (2)\nThe thermal field is assumed to be a white noise, with the\nfollowing mean, variance and strength, as calculated from the\nFokker-Planck equation:\nhxia(t)i=0;hxia(t)xjb(s)i=2Dda;bdi jd(t\u0000s)(3)\nD=lkBT\ngmim0(4)\nwhere lrepresents the coupling to the heat bath, Tthe ther-\nmostat temperature, gthe gyromagnetic ratio. We note that\nthe heat bath coupling constant lis different from the effec-\ntive Gilbert damping a, as the latter includes contributions\nfrom magnon scattering and other extrinsic processes such as\ninhomogeneous line broadening.\nAfter calculating the effective field that acts on each atom,\nthe magnetisation dynamics is given by solving the LLG equa-\ntion (Eq. 5) applied at the atomistic level [19] using a numer-\nical integration based on the Heun scheme.\n¶Si\n¶t=\u0000g\n(1+l2)Si\u0002(Hi+lSi\u0002Hi) (5)\nTo calculate the damping using atomistic spin dynamics we\napply an out-of-plane magnetic field ( B) to the sample with\nan additional in-plane oscillating field ( Brf=B0sin(2pnt)),\nwhich is the default setup for ferromagnetic resonance experi-\nments. The oscillating field will induce a coherent precession\nof the spins of the system which will result in an oscillatory\nbehaviour of the in-plane magnetisation. By sweeping the fre-\nquency of the in-plane field, the amplitude of the oscillations\nof magnetisation will change, with a maximum corresponding\nto the resonance frequency (as shown in Fig. 1). By Fourier\nFIG. 1. Illustration of the setup used for ferromagnetic resonance ex-\nperiments. An out of plane magnetic field ( B) and an in-plane oscil-\nlating field ( Br f=B0sin(2pnt)) is applied to the sample, as shown\nin the right inset. By Fourier Transformation of the in-plane mag-\nnetisation the power spectrum as function of frequency is obtained.\nThe simulation is performed for a single FePt spin at T=0K, having\na damping of 0.01. This is equivalent with simulating a macrcospin\nat T=0K with equivalent properties. By fitting the power spectrum,\nthe input resonance frequency and damping can be reproduced.\ntransformation of the in-plane magnetisation, the power spec-\ntrum as a function of frequency is obtained. Fig. 1 shows the\nFMR spectra for a single spin of FePt at 0K. The spectra can\nbe fitted by a Lorentzian curve (Eq. 6) where wrepresents the\nwidth of the curve and A its amplitude. By fitting with Eq. 6,\nthe effective Gilbert damping aand resonance frequency f0\ncan be extracted.\nL(x) =A\np0:5w\n(x\u0000f0)2+ (0:5w)2;a=0:5w\nf0(6)\nThe model parameters for FePt are listed in Tab. 1. L10\nFePt has a face-centred tetragonal structure formed of alter-\nnating layers of Fe and Pt, which can be approximated to a\nbody-centered tetragonal structure with the central site occu-\npied by Pt. The ab-initio calculations by Mryasov et al [20]\nshowed that the Pt spin moment is found to be linearly de-\npendent on the exchange field from the neighbouring Fe mo-\nments. This dependence allows the Hamiltonian to be written\nonly considering the Fe degrees of freedom. Under these as-\nsumptions, by neglecting the explicit Pt atoms, the system can\nbe modelled as a simple cubic tetragonal structure with each\natomic site corresponding to an effective Fe+Pt moment. The\nmodel used for the FePt system is restricted only to nearest\nneighbour interaction to minimise the computational cost of\nFMR calculations, in contrast with the full Hamiltonian given\nby Mryasov et al [20]. The nearest-neighbour exchange value\nis chosen to give a Curie temperature of FePt of 720K, to be in\nagreement with reported values for nearest and long-range ex-\nchange magnetic Hamiltonian [21]. The damping parameter\nhas been chosen to approximate the experimentally measured\nvalue in recording media provided by Advanced Storage Re-\nsearch Consortium (ASRC). The L1 0phase of FePt has a very\nlarge uniaxial anisotropy, hence the increased thermal stability\nof the grains. The uniaxial anisotropy used in the simulation\ngives a anisotropy field of Hk=2ku=ms=17:55 T, slightly\nlarger than the value used by Ostler et al (15.69 T). The FMR3\nfields (0 :05 T) used in our simulations are generally larger\nthan experimental FMR fields to allow more accurate simu-\nlations with enhanced temperature. Our tests confirm that no\nnon-linear modes are excited during the FMR simulations.\nQuantity Symbol Value Units\nNearest-neighbours exchange Ji j 6:71\u000210\u000021J\nAnisotropy energy ku 2:63\u000210\u000022J\nMagnetic moment mS 3:23 mB\nThermal bath coupling l 0:05\nDC perpendicular field B 1 T\nRF in-plane field Br f 0:05 T\nTABLE I. Parameters used for the initial calculations of the damping\nconstant of FePt.\nAtT=0K, the damping we extract from the FMR spec-\ntrum should correspond to the input coupling las no ther-\nmal scattering effects are present, hence the effective damp-\ning of the system is given by the Gilbert damping which is\nthe coupling to the heat bath. For this simulation, we have\nused an input heat bath constant of l=0:01, which we then\nrecover by performing FMR calculations at T=0K, method\nthat serves as verification of our model.The damping obtained\nagrees within 0.1% fitting error. The resonance peak should\nappear exactly at the resonance frequency given by Kittel for-\nmula fKittel =g\n2p\u0001(B+2ku\nmS), depending on the applied field\nstrength (B)and on the perpendicular anisotropy of our sys-\ntem (Hk=2ku\nms). For an FePt system the resonance frequency\nwe obtain is 520 GHz within 1% fitting error, due to the ex-\nceptionally large magnetic anisotropy of the system.\nIII. GRID-SEARCH METHOD\nThe Gilbert damping can be also calculated by fitting the\ntime-traces of the magnetisation relaxation. The time-traces\ncan be obtained via pump-probe experiments [11], however\nthe dynamics of the magnetisation will include the effect of\nthe laser pulse, such as heating and induced local magnetisa-\ntions due to the inverse Faraday effect. To avoid the contribu-\ntions to the damping from the laser pulse, damping can be cal-\nculated by taking the system out of equilibrium, letting it relax\nand subsequently recording the time-trace of the magnetisa-\ntion. Ellis et al [22] have numerically studied the damping\nof rare-earth doped permalloy using the transverse relaxation\ncurves, by fitting them with the analytical solutions of the\nLLG equations. In the case of large anisotropy, exchange in-\nteraction and applied field, there is no simple general solution\nto the LLG equation. Pai et al [23] used an applied field much\nlarger than the anisotropy field so that the dynamics closely\napproximate that of the LLG equation with no anisotropy.\nHowever, this approach is unsuitable for FePt due to the very\nlarge fields required and also the influence of strong magnetic\nfields near the Curie temperature. Hence, we adopt a com-\nputational grid search method where we pre-calculate single\nFIG. 2. c2map calculated using the grid-search method based on\nsingle-spin simulations at T=0.1K. (inset) The input and fitted mag-\nnetisation relaxation curves showing the validation of the method.\nspin solutions for the LLG equation using ASD, build a data\nbase using these solutions and then build an algorithm that\ncan identify the damping and anisotropy parameters from any\ntransverse relaxation curve. The method we chose simply in-\nvolves sweeping through the parameter space, the solution be-\ning given by minimising the sum of the squared residuals, a\nmethod known as grid-search.\nThe grid search method can be used to fit time-dependent\nm(t)curves in the case where analytical solutions do not exist.\nThe numerical curves that need to be fitted are compared with\neach of the pre-calculated numerical curves with the single\nspin system. The best match will be given by the curve with\nlowest sum of squared residuals, the c2parameter, where c2\nis defined as:\nc2=N\nå\ni=1h\nm(ti)\u0000f(ti;p)i2\n(7)\nwhere mi(ti)is the value of the magnetisation at each moment\nin time ti,f(ti;p)are the pre-calculated single-spin depen-\ndences of the magnetisation at each moment ti,pis the list\nof parameters that have been varied (in our case p= (K,a)).\nThe minimum value of c2from all pparameters is the best-\nagreement numerical solution.\nFigure 2 shows the calculated c2as function of the main pa-\nrameters, specifically the anisotropy and damping, at T=0.1K.\nIn order to construct the single spin simulation data-base, we\nchose a resolution of Dku=0:015\u000210\u000022Jfor the anisotropy\nandDastep=0:001 for the damping. It can be seen that the\nanisotropy is very well resolved: there is a sharp minimum\natku=2:625\u000210\u000022, which is the closest value to the input\nanisotropy, ku=2:63\u000210\u000022taking into account the resolu-\ntion we use for the data base. In the case of damping, the\nminimum is wider, leading to an error of approximately 0 :017\nin determination of damping, which is slightly larger than the\nresolution used in the construction of the database.4\nFIG. 3. Comparison of FMR and grid-search fitting; a) Damping; b)\nAnisotropy;\nIV . HIGH-TEMPERATURE FMR: DAMPING AND\nANISOTROPY CALCULATIONS\nIn this section the damping and anisotropy are computed,\nfrom frequency dependent FMR spectra and via the grid\nsearch method. The aim is to investigate the damping close\ntoTcand in particular the effect of finite grain size. First,\nwe test the effectiveness of the grid search method which was\npresented in Section III. Fig. 3 shows the comparison between\nthe two methods of calculation of the damping as a function\nof temperature for a granular system of 15 non-interacting\ngrains of 5nm diameter and 10nm height. The variations of\nanisotropy with temperature agree very well between the two\nmethods, however the grid search method is far more com-\nputationally efficient. The enhanced computational efficiency\ncomes from the fact that instead of simulating multiple fre-\nquency points to obtain the FMR, a single transverse relax-\nation simulation is needed to calculate the same parameters.\nThe time-scales for the two simulations are also different: the\nfrequency dependent FMR requires around 3 ns for each data\npoint in the FMR spectra to perform the FFT analysis, while\nthe transverse relaxation method requires, depending on the\nmaterial, less than 1 ns. Extracted damping values agree rea-\nsonably well between the two methods, within the error bars.\nFor the grid search method, there will be a damping interval\nthat gives the same value of c2. For the FMR experiment,\nthe error bar is computed as the standard error of groups of 5\nnon-interacting grains.\nBecause of the large error bars, especially close to TCfor\nthe grid search method, we use the direct FMR simulations\nfor the remainder of the paper and later consider possible\nmeans of improvement of the reliability of the grid search\nmethod. For initial calculations we model a granular FePt\nsystem as a cylinder of 10nm height and 5nm diameter. For\ncomparison, the bulk FePt system is modelled via a system of\n32\u000232\u000232 atoms with periodic boundary conditions. Close\ntoTC, the thermal fluctuations become increasingly large for\nnon-periodic systems and can lead to large errors in the de-\ntermination of damping and anisotropy. For this reason, to re-\nduce the statistical fluctuations, a system of 15 non-interacting\ngrains is modelled. This significantly reduces the fluctua-\ntions in the magnetisation components and leads to statisti-\ncally improved results. The in-plane magnetisation time se-\nries is Fourier transformed, and the damping is extracted as\nFIG. 4. Damping as function of normalised temperature for bulk\nand granular FePt system. The granular system shows overall larger\ndamping than the bulk system, due to additional magnon scattering\nprocesses at the interface. (inset) Magnetisation curves for granular\nand bulk FePt. The Curie temperatures for the two systems are :\ngrain- TC=690K, bulk - TC=720K.\npresented in Section II.\nFig. 4 shows the damping as a function of temperature for\nbulk and granular systems. For comparison the temperature\nis normalised to the Curie Temperature of the systems, which\ndiffer due to finite size effects [21, 24]. The granular system\nwill have a reduced Curie Temperature due to the cutoff in the\nexchange interactions at the surface. This is shown as an inset\nin Fig. 4, where the magnetisation as a function of temper-\nature is computed for the two systems. The Curie tempera-\ntures for the two systems, determined from the susceptibility\npeak, are: TCfor the grain =690K, and for the bulk TC=720K.\nThe input Gilbert damping parameter is 0.05, this value be-\ning reproduced at T=0K as expected due to the quenching of\nmagnon excitations.\nFIG. 5. Damping as function of temperature for granular (5nm \u0002\n5nm\u000210nm) (a) and bulk (b) systems. The damping calculated\nvia FMR method is compared against the effective damping from\nparameterised the LLB formalism - Eq. 9, where m(T;D)andTc(D)\nare computed numerically from the atomistic model.\nWith increasing temperature, for both bulk and granular\nsystems, the effective damping increases. This can be under-\nstood as, with enhanced temperature, there is increasing exci-\ntation of magnons which can suffer more complex non-linear5\nFIG. 6. Temperature dependence of the damping constant for diameters of 4nm, 5nm and 6nm. Solid lines are calculations using the LLB\ndamping expression. Divergence from the LLB expression for small particle diameter is indicative of surface effects.\nscattering processes.\nMoving on to granular systems, the inclusion of the surface\nwill add extra magnon modes into the system, leading to more\nscattering effects that will increase the effective damping. In\norder to effect a qualitative illustration of surface effects we\nuse the damping calculated from the Landau-Lifshitz-Bloch\nequation [17]. An analytical solution to the variation of\ndamping with temperature exists in the LLB description, as\ngiven by Garanin [17] and Ostler et al [16]. The effective\ndamping as derived within the LLB description is given by:\na(T) =l\nm(T)\u0012\n1\u0000T\n3Tc\u0013\n; (8)\nwhere lis the input coupling to the thermal bath used in atom-\nistic spin dynamics simulations, TCthe Curie Temperature of\nthe system, m(T) =M(T)=MsVthe normalised magnetisa-\ntion. In principle Eq. 8 is strictly valid only for an infinite sys-\ntem. However, as a first approximation, finite size effects can\nbe introduced empirically using diameter dependent functions\nm(T;D)andTc(D)calculated using an atomistic model. In the\ndamping calculations considered here, the grain surfaces have\ntwo effects. Firstly, the loss of coordination at the surfaces\ndrive a reduction in Tcand loss of criticality of the phase tran-\nsition. This effect can be accounted for by using numerically\ncalculated m(T;D)andTc(D)for a given diameter D. The sec-\nond effect is the increased magnon scattering at the surfaces\nwhich is a dynamic effect and not included in the parameter-\nization of the static properties. Thus it seems reasonable to\nassociate deviations from the parameterized version of Eq. 8\nwith scattering at the grain surfaces.\nConsequently, we compare our numerical results for\na(T;D)with the parameterised version of Eq. 8 - Fig. 5 ,\nwhere m(T;D)is calculated numerically with the ASD model\n(shown in Fig. 4, inset) and the Curie Temperature ( Tc(D))\nis calculated from the peak of the susceptibility. For the\nbulk system, the numerical damping calculated from the FMR\ncurves with the atomistic model agree well with the damp-\ning calculated with the analytical formula given by Eq. 8.\nThis is consistent with the first comparison of atomistic and\nLLB models [25] which showed that the mean-field treatment\nof [17] agreed quantitatively well with atomistic model calcu-\nlations for the transverse and longitudinal damping. However,the granular system gives a consistently increased damping\ncompared to the analytical formula. Following the earlier rea-\nsoning, this enhancement can be attributed to the scattering\neffects at the grain surface.\nTo systematically study the effect of the scattering at the\nsurface, we have calculated the damping as a function of the\nsystem size. For simplicity, we consider cubic grains with\na volume varying from 4nm \u00024nm\u00024nm, 5nm\u00025nm\u0002\n5nm and 6nm\u00026nm\u00026nm. Fig. 6 shows the damping as\na function of the temperature for the different system sizes.\nWith decreasing grain size, the damping is enhanced, and sys-\ntematically diverges further from the LLB analytical damping.\nThe separation of the effects of the finite size on the static\nand dynamic properties through comparison with the param-\neterised version of Eq. 8 strongly suggests that this is due to\nsurface scattering of magnons. Clearly, the magnon contribu-\ntions to the damping give rise to an increase of the damping\nwith increasing temperature, which is inconsistent with the\nresults of Richardson et al. [15]. However, the experiments\ndescribed in ref [15] give the temperature dependence of the\nlinewidth which likely has contributions from inhomogeneous\nline broadening arising from dispersion of magnetic proper-\nties. In the following we develop a model accounting for the\ninhomogenous line broadening which gives good qualitative\nagreement with the experiments.\nV . MODEL INCLUDING INHOMOGENEOUS LINE\nBROADENING\nRealistic granular systems will present a distribution of\nproperties. In the simplest case, the distribution of mag-\nnetic properties can arise from a distribution of the size of the\ngrains, which can induce a distribution of TC,mandHk. Since\nit is computationally expensive to study a system of grains nu-\nmerically within the ASD model we can, in the first instance,\nmodel the effect of the distributions analytically. In the case\nof a distribution of grains of diameter D, the power spectrum\nof the system is expressed by:\nPsys(f;T) =Z+¥\n0P(f;D;T)F(D)dD (9)6\nFIG. 7. Field swept FMR for a lognormal distribution of grains of D=4 nm and sD=0:17. Input damping l=0:01, input H0\nk=0:66T,\nf=13:7 GHz. The anisotropy is lower than for bulk FePt to (a) allow resonance at a frequency of 13.7Ghz corresponding to experiment.\nThe figure shows (a) the variation of the FMR field, (b) FMR linewidth ( DH) and (c) system magnetisation and anisotropy field as function of\ntemperature. Close to TC, the linewidth shows a decrease which translates to a decrease in the damping of the system. No magnetostatic or\nexchange interaction between grains is considered.\nThe distribution of size, F(D), is considered lognormal.\nThe power spectrum of a grain of diameter Dcan be expressed\nby [16]:\nP(f;B0;D;T) =C˜mD2 f2gB2\n0˜a\n(˜a˜f0)2+ (f\u0000˜f0)2(10)\nwhere ˜ m=m(T;D);˜a=a(T;D);˜f0=f0(T;D) =g(B0+\nHk(T;D));C=ph\n16. This allows to model both fre-\nquency swept FMR ( B0=constant) and field swept FMR\n(f=constant).\nWe note that a Distribution of grain size leads to distri-\nbutions of further properties, starting, due to finite size ef-\nfects, with the Curie temperature. Each of these is introduced\ninto the analytical model as follows. Hovorka et al [21] have\nshown via finite size scaling analysis that the relation between\nthe size of a grain and its Curie temperature is given by:\nTC(D) =T¥\nC(1\u0000d0=D)1=n; (11)\nwhere d0=0:71 and n=0:79 [21] parametrised for nearest\nneighbours exchange systems and T¥\nC=720K. The variation\ninTCwill introduce a variation in the magnetisation curves\ngiven by:\nm(T;D) =\u0012\n1\u0000T\nTC(D)\u0013b\n;b=0:33: (12)\nAs a further consequence, the anisotropy will be dependent\non the diameter. The uniaxial anisotropy energy Khas a tem-\nperature dependence in the form of K(T)\u0018m(T)g. For FePt\nit was found that the exponent is equal to 2.1 by experimen-\ntal measurements [26][27] in agreement with later ab-initio\ncalculations [20] . Hence the anisotropy field will depend on\nm(T;D)with an exponent of 1.1;\nHK(T;D) =H0\nKm(T;D)1:1; (13)\nleading to a dispersion of HK(D;T).\nFinally the size distribution will produce different varia-\ntions of damping as a function of grain size, since\na(T;D) =l\nm(T;D)\u0012\n1\u0000T\n3Tc(D)\u0013\n: (14)In the presence of distributions of properties, the variation\nof damping with temperature can have a complex behaviour,\nespecially close to Tcwhere there is a strong variation of\nmagnetic properties with temperature and size. On reverting\nto a monodispersed system by setting the distributions to d-\nfunctions the damping is given by Eq. 14 resulting in an in-\ncrease of linewidth consistent with temperature Fig. 6.\nRichardson et al [15] have shown that, in the case of a\ngranular system of FePt, close to TCa decrease in damping/-\nlinewidth is observed. This effect was attributed to the com-\npetition between two-magnon scattering and spin-flip magnon\nelectron scattering. We have shown via atomistic spin dynam-\nics simulations that surface effects alone cannot be responsible\nfor a decrease in damping, scattering at the surface leading to\nincreased damping at high temperatures.\nIt is well known that, in the presence of a distribution of\nproperties in the system, the linewidth broadens. In our case\nthe distribution of size will lead to a distribution of anisotropy\nwhich increases the linewidth. Close to the Curie Temperature\nof the system, some grains will become superparamagnetic\nand will not contribute further to the FMR spectrum, hence it\nis possible that close to Tc, the linewidth can decrease. Fig. 7\npresents a case where a decrease in linewidth appears within\n30-40K of the Curie Temperature of the system, a similar\ntemperature interval as spanned by the experimental measure-\nments [15]. Fig. 7 a) and b) show the variation of the FMR\nfield and linewidth as functions of temperature. The FMR\nspectra are calculated at constant frequency of f=13:7GHz ,\nconsistent with the experimental value used in [15]. The aver-\nage magnetisation of distributed grains is calculated as\nM(T) =R¥\n0m(T;D)F(D)D2dDR¥\n0F(D)D2dD(15)\nand the anisotropy field is calculated as\nHK(T) =Z¥\n0HK(T;D)F(D)dD: (16)\nThe decrease in linewidth is associated with the fact that, close\nto the Curie Temperature of the system, the small grains be-\ncome superparamagnetic and do not contribute to the power7\nspectrum. The loss of the signal from the small grains is es-\npecially pronounced due to the enhanced damping of smaller\ngrains.\nVI. CONCLUSIONS AND OUTLOOK\nWe have calculated the temperature dependence of damping\nand anisotropy for small FePt grain sizes. These parameters\nwere calculated within the ASD framework, via simulation of\nswept frequency FMR processes and a fitting procedure based\non the grid search method. The grid search method offers a\nmuch faster determination of damping and anisotropy, param-\neters crucial for the development of future generation HAMR\ndrives. The method can be applied both for numerical data,\nas well for experimental relaxation curves obtained via pump-\nprobe experiments. The damping calculations at large tem-\nperatures showed an increased damping for uncoupled gran-\nular systems as expected due to increased magnon excitation\nat high temperature. Deviations from the parameterised ex-\npression for the temperature dependence of damping from the\nLLB equation with decreasing grain size suggest that scatter-\ning events at grain boundaries enhance the damping mecha-\nnism.\nThis increase in damping, however, is not consistent with\nthe experimental data of Richardson et al [15] which show\na decrease in linewidth at elevated temperatures. We have\ndeveloped a model taking into account inhomogeneous linebroadening arising from the size distribution of the grains\nwhich gives rise to concomitant dispersions of TC,mandK.\nThe model has been used to simulate swept field FMR as used\nin the experiments. Calculations have shown that, under the\neffect of distribution of properties, the linewidth can exhibit a\ndecrease towards large temperatures, in accordance with the\nexperiments of Ref. [15]. The decrease is predominantly due\nto a transition to superparamagnetic behaviour of small grains\nwith increasing temperature. This suggests inhomogeneous\nline broadening (likely a significant factor in granular films)\nas an explanation for the unusual decrease in linewidth mea-\nsured by Richardson et al [15]. As large damping is necessary\nfor good performance of HAMR and MRAM devices with\nthis work we further stress the importance of experimentally\ncontrolling the size distributions of the media.\nVII. ACKNOWLEDGEMENTS\nWe are grateful to Prof. M Wu and Stuart Cavill for helpful\ndiscussions. Financial support of the Advanced Storage Re-\nsearch Consortium is gratefully acknowledged. The atomistic\nsimulations were undertaken on the VIKING cluster, which\nis a high performance compute facility provided by the Uni-\nversity of York. We are grateful for computational support\nfrom the University of York High Performance Computing\nservice,VIKING and the Research Computing team.\n[1] J. C. Slonczewski et al. , “Current-driven excitation of magnetic\nmultilayers,” Journal of Magnetism and Magnetic Materials ,\nvol. 159, no. 1, p. L1, 1996.\n[2] U. Atxitia, T. A. Ostler, R. W. Chantrell, and O. Chubykalo-\nFesenko, “Optimal electron, phonon, and magnetic character-\nistics for low energy thermally induced magnetization switch-\ning,” Applied Physics Letters , vol. 107, no. 19, p. 192402, 2015.\n[3] H. Yuan, Z. Yuan, K. Xia, and X. 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Kel-\nlock, “Temperature dependent magnetic properties of highly\nchemically ordered fe 55- x ni x pt 45 l1 0 films,” Journal of\napplied physics , vol. 91, no. 10, pp. 6595–6600, 2002.\n[27] S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, Y . Shi-\nmada, and K. Fukamichi, “Chemical-order-dependent magnetic\nanisotropy and exchange stiffness constant of fept (001) epitax-\nial films,” Physical Review B , vol. 66, no. 2, p. 024413, 2002." }, { "title": "2005.06190v1.Magnetic_circular_dichroism_spectra_from_resonant_and_damped_coupled_cluster_response_theory.pdf", "content": "MCD-CC\nMagnetic circular dichroism spectra from resonant and damped coupled cluster\nresponse theory\nR. Faber,1S. Ghidinelli,2C. Hättig,3and S. Coriani1,a)\n1)DTU Chemistry, Technical University of Denmark, Kemitorvet Bldg 207,\nDK-2800 Kongens Lyngby, Denmark\n2)Department of Molecular and Translational Medicine, Università degli Studi di Brescia,\nViale Europa 11, I-25123 Brescia, Italy.\n3)Arbeitsgruppe Quantenchemie, Ruhr-Universität Bochum, D-44780,\nGermany\n(Dated: 14 May 2020)\nA computational expression for the Faraday Aterm of magnetic circular dichroism (MCD)\nis derived within coupled cluster response theory and alternative computational expres-\nsions for the Bterm are discussed. Moreover, an approach to compute the (temperature-\nindependent) MCD ellipticity in the context of coupled cluster damped response is pre-\nsented, and its equivalence with the stick-spectrum approach in the limit of infinite life-\ntimes is demonstrated. The damped response approach has advantages for molecular sys-\ntems or spectral ranges with a high density of states. Illustrative results are reported at the\ncoupled cluster singles and doubles level and compared to time-dependent density func-\ntional theory results.\na)Author to whom correspondance should be addressed: soco@kemi.dtu.dk\n1arXiv:2005.06190v1 [physics.comp-ph] 13 May 2020MCD-CC\nI. INTRODUCTION\nIn magnetic circular dichroism (MCD) spectroscopy, the sample is probed with circularly po-\nlarized light in presence of a relatively strong magnetic field oriented parallel to the direction of\npropagation of the light beam. The external magnetic field induces a differential absorption of the\nright- and left- circularly polarized light.1MCD can provide insight to the geometric, electronic,\nand magnetic properties of chemical systems. The applied magnetic field couples to the (spin\nand/or orbital) angular momentum, lifting the degeneracies among ground and excited states (by\nZeeman splitting), and giving rise to additional spectroscopic features compared to the zero-field\ncase. Since the MCD spectral features are signed and depend upon molecular magnetic moments\nin electronic states and the direction of the field, MCD yields additional information when com-\nbined with conventional absorption spectroscopy. MCD spectra can be obtained from gases, solu-\ntions, or isotropic solids. Also, MCD can be observed for any sample of molecules independent\nof whether they are chiral or not. One can use MCD to study molecules of high symmetry, and to\nprobe degenerate electronic ground and excited states.\nAbout fifty years ago, Buckingham and Stephens2described in an elegant and incisive way the\ntheoretical foundation of MCD. For an electronic transition, the intensity of the signal is given\nby the contribution of three effects called A,BandCterms. The Aterm originates from the\nZeeman splitting of degenerate excited states. The Bterm arises from the mixing of the zero-\nfield wavefunctions between nondegenerate states in the presence of a magnetic field. The C\nterm is a temperature-dependent effect and originates from the Zeeman splitting of a degenerate\nground state. Each term is associated with a characteristic band shape. After the seminal work of\nBuckingham and Stephens, the MCD spectra of several molecules were rationalized and under-\nstood qualitatively based on Hückel molecular orbital, the Pariser- Parr-Pople (PPP) model, and\nthe Complete Neglect of Differential Overlap/Spectroscopic (CNDO/S) method.3–5\nThe challenging aspect of the ab initio computation of MCD spectra derives from the need\nto consider both the perturbation of a static magnetic field and the perturbation of an oscillating\nelectric field. In the last twenty years, several approaches have been proposed for the simulation of\nMCD, see e.g. Ref. 6 for a review up to 2012. Among them, response theory7–9has been employed\nto formulate MCD in different forms, for instance as single residue of dipole-dipole-magnetic\nquadratic response functions,10as a complex polarization propagator,11,12or a damped response\nfunction,13to avoid divergences, and by magnetically-perturbed time-dependent density functional\n2MCD-CC\ntheory (MP-TDDFT) evaluating the perturbations induced into TDDFT excitation energies and\ntransition densities by a static magnetic field.12,14In the complex polarization propagator/damped\nresponse framework, the MCD signal is computed directly, without separation into MCD terms.15\nMCD spectra have also been calculated with sum-over-states (SOS) methods for the individual\nterms at the Hartree-Fock and DFT levels16of theory and within full configuration interaction\n(CI).17For the treatment of MCD arising from transition metals, DFT and HF may be inadequate,\nthus multi-configurational self-consistent-field with the treatment of spin-orbit coupling (SOC)\nand spin-spin coupling (SSC) using complete active space self-consistent field (CASSCF)18and\nrestricted active space (RAS)19wavefunctions have been implemented. Gauge-origin independent\nformulations of MCD using the perturbative approach with London orbitals have been developed\nwithin DFT,20,21Hartree-Fock,21and coupled cluster (CC) frameworks.22,23Calculations of MCD\nwithin a variational treatment of the magnetic field have also been proposed.24,25\nIn this work we re-analyze the derivation of the MCD Bterm within resonant CC response\nfunction theory and extend the theory to the computation of the Aterm. Then, we derive the CC\ndamped response expression for the MCD ellipticity. Compared to the computation of induced\ntransition strengths for stick spectra, the calculation of the damped response function is computa-\ntionally more efficient for large chromophores or spectral regions with a high density of states.9\nIn these cases the computation of the stick spectra requires the convergence of eigenvectors, and\nthe calculation of (derivatives of) transition moments for many states. The costs for the calcula-\ntions of the damped response function depends mainly on the size of the frequency range and the\nfrequency resolution, but is almost insensitive to the density of states. To show the equivalence\nof the two approaches, illustrative numerical results are reported at the coupled cluster singles and\ndoubles (CCSD) level for the molecular systems cyclopropane and urea. These are compared with\nTDDFT (CAM-B3LYP) results.\nII. THEORY\nA.AandBterms from resonant CC response theory\nFollowing Ref. 13, we write the ellipticity qof plane-polarized light traveling in the Zdirection\nof a space-fixed frame through a sample of randomly moving molecules in the presence of a\n3MCD-CC\nmagnetic field directed along Zas\nq=1\n6m0clNB zqMCD (1)\nwhere, in atomic units,\nqMCD=\u0000wå\nf\u001a¶g(w;wf)\n¶wA(0!f)+g(w;wf)B(0!f)\u001b\n(2)\nIn the equations above, Nis the number density, cis the velocity of light in vacuo, m0is the\npermeability in vacuo, lis the length of the sample, wis the circular frequency, Bzis the strength\nof the external magnetic field, and g(w;wj)is a lineshape function. We adopt the sign convention\nused by Michl.26Thus, the contribution to qMCD of a transition 0!fcan consists of a positive\n(when B<0) or negative (when B>0) band of absorption-like shape centered at the position of\nthe absorption band. If the transition is degenerate, the absorption-like band is superimposed to a\ns-like (dispersive) shape, centered at the position of the absorption band, with a positive wing at\nlower energies and a negative one at higher energies (when A<0) or with a negative wing at lower\nenergies and a positive one at higher energies (when A>0). Note that, in the expression of the\nMCD ellipticity qMCD in Eq. (2), we have omitted the temperature-dependent term, proportional\ntoC(0!f)\nkT, as it only contributes for systems with a degenerate ground state.\nThe spectral representation of the Aterm for a non-degenerate ground state 0 is2,27\nA(0!f) =1\n2eabgå\nf02DfIm\u0002\nh0jmajfihfjmgjf0ihf0jmbj0i\u0003\n(3)\nwhere maandmbare components of the electric dipole operator, mgis a component of the mag-\nnetic dipole operator, and eabgis the Levi-Civita tensor. Implicit summation over repeated Greek\nindices is assumed. Dfis the set of degenerate states of which fis a part.2TheAterm vanishes\nfor a non-degenerate excited state (as the magnetic moment is quenched).28\nThe spectral representation of the Bterm is given by2,27\nB(0!f) =eabgIm\"\nå\nk6=0hkjmgj0i\nwkh0jmajfihfjmbjki+å\nk=2Dfhfjmgjki\nwk\u0000wfh0jmajfihkjmbj0i#\n(4)\nA connection has previously been made between the Bterm of a non degenerate state and the\nderivative of the transition strength matrix.22We will here extend this definition in order to include\ntheA-term. For exact states, the magnetic-field derivative of the electric-dipole transition strength,\n4MCD-CC\nSmamb\no f=h0jmajfihfjmbj0i, is\n1\n2Im \ndSmamb\no f\ndBg!\n=Im\"\nå\nk6=0hkjmgj0i\nwkh0jmajfihfjmbjki+å\nk6=fhfjmgjki\nwk\u0000wfh0jmajfihkjmbj0i#\n(5)\nwhich is exactly the expression for the contributions to the Bterm if the state fis non-degenerate.\nIffis degenerate, however, the second sum contains additional terms, explicitly excluded from\nEq. (4), involving the states degenerate with the final state. If we assume that the degeneracy can\nbe broken by an infinitesimal amount, h=wf0\u0000wf, theAterm can be defined as the residue\nA(0!f) =1\n4eabglim\nh!0hIm \ndSmamb\n0f\ndBg!\n(6)\nSimilarly, the expression for the Bterm in Eq. (4) is obtained by defining the Bterm as what re-\nmains of the transition-moment derivative once the singularities are removed, i.e. any degeneracy\nis projected out of the excited state wavefunction response.\nIn CC response theory, the transition strength is given as the product of distinct left and right\ntransition moments8,10,29,30\nSmamb\n0f=1\n2Tma\n0fTmb\nf0+1\n2(Tmb\n0fTma\nf0)\u0003; (7)\nTma\n0f=hmaRf+Mfxma; (8)\nTmb\nf0=Lfxmb: (9)\nThe eigenvectors are obtained by solving the right and left eigenvalue equations\n(A\u0000wf1)Rf=0 (10)\nLf(A\u0000wf1) =0 (11)\nunder the biorthogonality condition LkRl=dlk, and the transition multipliers Mf(wf)are the so-\nlution of the linear equation\nMf(wf)(A+wf1) =\u0000FRf: (12)\nFor ease of notation we have omitted the overbar on the transition multiplier. The definitions of\nthe Jacobian matrix A, the matrix Fand of the property gradients, xXandhX, for any generic\noperator X, can be found, e.g., in Ref. 29.\n5MCD-CC\nLet us start by considering the case where the final state fis not degenerate. Straightforward\ndifferentiation of the CC left and right ground-to-excited-state transition moments yields\ndTma\n0f\ndBg=\u0000Tmamg\n0f=hmaRmg\nf+(Fmatmg+¯tmgAma)Rf+Mmg\nfxma+MfAmatmg(13)\ndTmb\nf0\ndBg=\u0000Tmbmg\nf0=Lmg\nfxmb+LfAmbtmg(14)\nwhere tmgand¯tmgare the zero-frequency derivatives, with respect to the magnetic field, of the CC\namplitudes and Lagrangian multipliers, respectively, obtained solving usual right and left response\nequations:\n(A\u0000wA1)tX(wX) =\u0000xX(15)\n¯tX(wX)(A+wX1) =\u0000(hX+FtX(wX))\n=\u0000¯xX(wX) (16)\nfor operator Xequal to mgandwX=0.\nThe equations determining the magnetic-field derivatives, Lmg\nfandRmg\nf, of the left and right\neigenvectors, as well as the magnetic-field derivative Mmg\nfof the transition multipliers, are\n(A\u0000wf1)Rmg\nf=\u0000\u0010\nAmg+Btmg\u0000wmg\nf1\u0011\nRf; (17)\nLmg\nf(A\u0000wf1) =\u0000Lf\u0010\nAmg+Btmg\u0000wmg\nf1\u0011\n; (18)\nMmg\nf(A+wf1) =\u0000FRmg\nf\u0000(Fmg+Gtmg+¯tmgB)Rf\u0000Mf(Amg+wmg\nf1+Btmg) (19)\nwhere\nwmg\nf=Lf(Amg+Btmg)Rf: (20)\nSee again Ref. 29 for the definition of the remaining CC matrices.\nWhile (A\u0000wf1)in Eqs. (17) and (18) is singular, it is easy to show that the right hand sides are\northogonal to RfandLf, respectively. It is sufficient to insert wmg\nfin their definition and to project\nthem against LfandRf, respectively. Thus, for non-degenerate final states f, Eqs. (17) and (18)\ncan be solved in the orthogonal complement to the singularity without loss of generality.22,30,31In\npractice, this is achieved by introducing the projector\nPf=1\u0000RfLf (21)\n6MCD-CC\nand the projected derivative eigenvectors\n?Rmg\nf=PfRmg\nf(22)\n?Lmg\nf=Lmg\nfPf (23)\nwhich are obtained solving\nPf(A\u0000wf)?Rmg\nf=\u0000Pf(Amg+Btmg)Rf; (24)\n?Lmg\nf(A\u0000wf)Pf=\u0000Lf(Amg+Btmg)Pf: (25)\nIn addition, we use the notation?Mmg\nfto emphasize that the Lagrange multiplier responses are\ncalculated using the non-singular derivative of the eigenvector, i.e.\n?Mmg\nf(A+wf) =\u0000F?Rmg\nf\u0000(Fmg+Gtmg+¯tmgB)Rf\u0000Mf(Amg+wmg\nf+Btmg)(26)\nIf the final state fis degenerate (i.e., it belongs to the set Df), the projector is generalized as\nPf=1\u0000å\nf02DfRf0Lf0: (27)\nThen, we introduce a distinction between the two kinds of contributions, i.e., the Aand the B\nterm: In accordance with exact theory, we define the Bterm as the term obtained by projecting\nout the singularity and otherwise continuing as in the non-degenerate case. The Aterm, on the\nother hand, will be defined as the residue of the term involving the singularity.\nThus, the CC Bterm will be obtained as\nBCC(0!f) =\u00001\n2eabg\u0010\n?Tmamg\n0fTmb\nf0+Tma\n0f?Tmbmg\nf0\u0011\n(28)\nwhere the perpendicular perturbed transition moments (?Tmamg\n0fand?Tmbmg\nf0) are defined by intro-\nducing?Rmg\nf,?Lmg\nfand?Mmg\nfin place of their non- ?equivalents into Eqs. (13) and (14). The\nformulation of the derivative transition moments as in Eqs. (13) and (14) is attractive as all depen-\ndencies on the electric dipole components maandmbare explicit, allowing for the identification\nof derivative left and right transition densities.\nAn alternative expression of the (orthogonal) left moment is obtained by eliminating Mmg\nf(or\n7MCD-CC\n?Mmg\nf) from Eq. (13) using Eq. (19) (or Eq (26))\nMmg\nfxma=\u0000h\nFRmg\nf+(Fmg+Gtmg+¯tmgB)Rf\n+Mf(Amg+wmg\nf1+Btmg)i\u0010\nA+wf1\u0011\u00001\nxma\n=h\nFRmg\nf+(Fmg+Gtmg+¯tmgB)Rf\n+Mf(Amg+wmg\nf1+Btmg)i\ntma(\u0000wf); (29)\nTmamg\n0f=\u0002\nGtmgtma(\u0000wf)+Fmgtma(\u0000wf)+Fmatmg\u0003\nRf\n+Mf\u0002\nAmatmg+Amgtma(\u0000wf)+Btmgtma(\u0000wf)\u0003\n+\u0002\nhma+Ftma(\u0000wf)\u0003\nRmg\nf\n+wmg\nf\u0001Mftma(\u0000wf)\n+¯tmg\u0002\nAma+Btma(\u0000wf)\u0003\nRf (30)\nThe last term in Eq. (30) can be further replaced by\n\u0000\nhmg+Ftmg\u0001\nRma\nf(\u0000wf) =¯xmg(0)Rma\nf(\u0000wf) (31)\nwhich now involves Rma\nf(\u0000wf), the first-order response to the electric field of the right eigenvector\nin a non-phase-isolated (i.e. unprojected) form.31Similarly, the third term can be recast as\n\u0002\nhma+Ftma(\u0000wf)\u0003?Rmg\nf=\u0000?¯t(\u0000wf)\u0010\nAmg+Btmg\u0000wmg\nf1\u0011\nRf (32)\nEq. (30) is formally the approach taken in the implementation in Dalton22,32and Turbomole,33,34\nthe latter also employing Eq. (32).33,34If the final states fare non-degenerate, both approaches\n(Eq. (13) and (30)) require the solution of the same amount of linear equations. In the case of\ndegenerate states, however, the latter is advantageous as the dipole response amplitudes tma(\u0000wf)\nneed to be calculated only once for each degenerate set.\nTo obtain the CC expression for the Aterm, we perform a residue analysis according to Eq. (6),\ni.e.\nACC(0!f) =\u00001\n4eabglim\nh!0hIm\u0010\nTmamg\n0fTmb\nf0+Tma\n0fTmbmg\nf0\u0011\n(33)\nwhich requires the residues\nkTmamg\n0f=lim\nh!0hTmamg\n0f=\u0000hmakRmg\nf\u0000kMmg\nfxma; (34)\nkTmbmg\nf0=lim\nh!0hTmbmg\nf0=\u0000kLmg\nfxmb: (35)\n8MCD-CC\nThe response of the eigenvectors parallel to the degenerate set Dfare defined as residues of non-\nphase isolated derivatives of the eigenvectors31\nkRmg\nf=lim\nh!0hRmg\nf=\u0000å\nf02Df;\nf06=fRf0Lf0(Amg+Btmg)Rf=\u0000å\nf02Df;\nf06=fRf0Tmg\nf0f(36)\nkLmg\nf=lim\nh!0hLmg\nf=\u0000å\nf02Df;\nf06=fLf(Amg+Btmg)Rf0Lf0=\u0000å\nf02Df;\nf06=fTmg\nf f0Lf0 (37)\nand\nkMmg\nf=\u0000FkRmg\nf(A+wf1)\u00001=\u0000å\nf02Df;\nf06=fMf0Tmg\nf0f; (38)\nIn the equations above, simplifications have been made by identifying the conventional CC expres-\nsion for transition moments between excited states, e.g. Tmg\nf f0=Lf(Amg+Btmg)Rf0. This allows us\nto write the ACCterm as\nACC(0!f) =\u00001\n4eabgImå\nf02Df;\nf06=f\u0010\nTma\n0f0Tmg\nf0fTmb\nf0+Tma\n0fTmg\nf f0Tmb\nf00\u0011\n(39)\nor, when summed over the whole degenerate set,\nACC(0!Df) =\u00001\n2eabgImå\nf0;f002Df(1\u0000df0f00)\u0010\nTma\n0f0Tmg\nf0f00Tmb\nf000\u0011\n: (40)\nNote that the Aterm has previously been formulated as the derivative of the excitation fre-\nquency13,20,27\nA(0!f) =\u00001\n2eabgå\nf2Df\u0012¶wf\n¶Bg\u0013\nImn\nm0˜f\nam˜f0\nbo\n(41)\nwhere the real degenerate states fare (typically) expanded in complex states ˜f, which diagonalize\nthe imaginary operator mg.13,20This is consistent with our derivation, as we can identify\n¶wf\n¶Bg=Lf(Amg+Btmg)Rf0=Tf f0 (42)\nOur derivation highlights how the transformation to the diagonal basis for mgcan be avoided.\n9MCD-CC\nB. MCD spectra from CC damped response theory\nWithin damped response theory, the MCD ellipticity can be obtained directly from the magnetic\nfield derivative of the damped polarizability:\nqMCD=\u0000weabgRe\u0012dhhma;mbiiw+iv\ndBg\u0013\nB=0: (43)\nIn coupled cluster theory, the damped polarizability can be written as given in Refs. 35–37:\nhhma;mbiiw+iv=1\n2C\u0006w\b\nhmatmb(w+iv)+ (44)\nhmbtma(\u0000w\u0000iv)+\nFtmb(w+iv)tma(\u0000w\u0000iv)\t\n:\nThe complex amplitudes are found solving the complex linear equations:\n[A\u0000(w+iv)1]tma(w+iv) =\u0000xma: (45)\nWe refer to our previous work35–37for details on how to solve the complex equations in Eq. (45).\nTypically, the CC response functions need to be explicitly symmetrized,29as indicated in\nEq. (44) by the1\n2C\u0006woperator. However, the Levi-Civita symbol in Eq. (43) makes this sym-\nmetrization redundant. Taking the first derivative of the non-symmetric CC linear response func-\ntion, i.e. the term in brackets in Eq. (44), we obtain:\ndhhma;mbiiw+iv\ndBg=Fmgtmb(w+iv)tma(\u0000w\u0000iv)\n+h\nFmatmb(w+iv)+Fmbtma(\u0000w\u0000iv)\n+Gtmb(w+iv)tma(\u0000w\u0000iv)i\ntmg\n+¯tmgh\nAmatmb(w+iv)+Ambtma(\u0000w\u0000iv)\n+Btmb(w+iv)tma(\u0000w\u0000iv)i\n+[Ftma(\u0000w\u0000iv)+hma]tmbmg(w+iv)\n+[Ftmb(w+iv)+hmb]tmamg(\u0000w\u0000iv)(46)\nThe above expression contains the doubly perturbed amplitudes, which are defined by the second-\norder response equations\n[A+(w+iv)]tmamg(\u0000w\u0000iv) =\u0000Amatmg\u0000Amgtma(\u0000w\u0000iv)\u0000Btmgtma(\u0000w\u0000iv):(47)\n10MCD-CC\nHowever, the expression by which tmamg(\u0000w\u0000iv)is multiplied is exactly the right hand side\nof the equations that determine ¯tmb(w+iv), so that this term can be eliminated according to:\n[Ftmb(w+ig)+hmb]tmamg(\u0000w\u0000iv) =\n¯tmb(w+ig)[Amatmg+Amgtma(\u0000w\u0000iv)+Btmgtma(\u0000w\u0000iv)]:(48)\nThis leads to a more convenient computational expression, which shows the symmetry between\nthe perturbations:\ndhhma;mbiiw+iv\ndBg=Fmgtmb(w+iv)tma(\u0000w\u0000iv)\n+h\nFmatmb(w+iv)+Fmbtma(\u0000w\u0000iv)+Gtmb(w+iv)tma(\u0000w\u0000iv)i\ntmg\n+¯tmgh\nAmatmb(w+iv)+Ambtma(\u0000w\u0000iv)+Btmb(w+iv)tma(\u0000w\u0000iv)i\n+¯tma(\u0000w\u0000iv)\u0002\nAmbtmg+Amgtmb(w+iv)+Btmgtmb(w+iv)\u0003\n+¯tmb(w+iv)\u0002\nAmatmg+Amgtma(\u0000w\u0000iv)+Btmgtma(\u0000w\u0000iv)\u0003\n(49)\nThe connection to the quadratic response function expression hhma;mb;mgiiw;0(in the limit of\nv=0) is apparent.22,29\nIII. RESULTS AND DISCUSSION\nThe calculation of the ACC(0!f)andBCC(0!f)terms in gas phase according to the\nexpressions in Eqs. (28) and (40) as well as that of the MCD ellipticity according to the CPP\nalgorithm discussed in Section II B have been implemented at CCSD level in the our stand-alone\npython CC response platform.36,38Two illustrative cases were considered, cyclopropane, C 3H6,\nand urea, H 2N(CO)NH 2. Cyclopropane has D 3hsymmetry and thus possesses degenerate excited\nstates, yielding spectral features that arise from the A-term. Urea belongs to the C 2v(or lower)\npoint group and does not support degenerate excited states per symmetry. Experimental results in\ngas phase as well as computational (TDDFT and SOS-HF) results for cyclopropane are available\nin the literature.11,39To the best of our knowledge, the MCD spectrum of urea has neither been\nmeasured nor simulated before.\nThe geometry of urea was optimized at the CCSD/aug-cc-pVDZ level, whereas the geometry\nof cyclopropane was optimized at the CCSD(T)/aug-cc-pVTZ level.\n11MCD-CC\nThe MCD spectra resulting from the calculated individual ACC(0!f)andBCC(0!f)terms\nwere generated according to Eq. (2) with a Lorentzian lineshape function\ng(w;wf) =v\np1\n(w\u0000wf)2+v2(50)\n¶g(w;wf)\n¶w=\u00002v\npw\u0000wf\u0002\n(w\u0000wf)2+v2\u00032(51)\nand the same v=0:0045563 a :u:\u00191000 cm\u00001was used for the broadening adopted in the\ndamped response calculations. The CCSD results are compared with CAM-B3LYP results ob-\ntained using LSDalton32. The values of the excitation energies and MCD terms for cyclopropane,\nobtained from resonant response theory, are collected in Table I.\nTABLE I. Computed spectral parameters for cyclopropane: excitation energies ( wf), dipole oscillator\nstrengths (f), and MCD AandBterms.\nSymm wf/eV (f) A/a.u. B/a.u.\nCCSD/aug-cc-pVDZ\nE07.686 (0.0001) \u00000.00018785 0.24123551\nE08.305 (0.16) 0.05915707 2.57266636\nE09.361 (0.009) \u00000.01141625 3.48280434\nA00\n2 9.557 (0.0098) 0.00000000 \u00004.51643887\nCAM-B3LYP/aug-cc-pVDZ\nE07.476 (0.0001) \u00000.00003014 0.18798269\nE08.105 (0.156) 0.05278849 1.97266712\nE09.168 (0.0088) \u00000.01079584 2.96404589\nA00\n2 9.286 (0.0096) 0.00000000 \u00003.61714300\nBased on the values in Table I, Figure 1 illustrates the relative importance of the AandB\nterms. Clearly, the bisignate spectral feature centered at 8.30 eV is dominated by the positive\nAterm contribution of the second E0excited state, where the Bterm is causing the slightly\nasymmetry of the dispersion band. The second bisignate feature at around 9.5 eV is the result of\nthe fine balance of the negative Aterm for the third E0state and the oppositely signed pseudo A\ndue to the Bterms of the close-lying third E0state and the non-degenerate A00\n2state.\n12MCD-CC\nMCD (A)\n MCD (B)\n7.0 7.5 8.0 8.5 9.0 9.5 10.0\neVMCD\nResonant\nExp\nFIG. 1. Cyclopropane. CCSD relative contributions of ACC(upper panel) and BCC(mid panel) terms of\nresonant response theory to the total (lower panel) broadened MCD spectrum. The experimental spectrum\nwas taken from Ref. 40.\nThe total MCD spectrum generated by Lorentzian broadening of the AandBterms is com-\npared with the spectrum obtained directly from damped response theory in Figure 2. The broad-\nened spectrum is basically identical to the one of damped response theory. The CAM-B3LYP\nspectrum is red-shifted compared to the CCSD one, and with slightly weaker intensities, but oth-\nerwise the spectral profiles are similar.\n13MCD-CC\n7.0 7.5 8.0 8.5 9.0 9.5 10.0\neVMCD\nDamped\nResonant\nDFT\nFIG. 2. Cyclopropane. CCSD/aug-cc-pVDZ MCD spectra from damped and resonant response theory,\nand comparison with the CAM-B3LYP spectrum from resonant response theory.\nTable II collects the values of the excitation energies and MCD spectral parameters for urea,\nas obtained from resonant response theory. Only Bterms are possible by symmetry. The corre-\nsponding MCD spectra, including the ones obtained from damped response, are shown in Figure 3.\nAlso in this case, damped and resonant theories yield almost identical spectra up to the number of\nfrequencies that have been considered. The CAM-B3LYP spectrum is qualitatively very similar,\nyet with larger intensities from the two excited states at around 8 eV .\nIV . CONCLUSIONS\nWe have presented a computational approach to obtain the Aterm of MCD within CC (reso-\nnant) response theory, together with alternative computational recipes for the Bterm, Moreover,\nwe have derived the computational expression of the MCD ellipticity (temperature-independent\npart) within CC damped response theory. The latter can prove particularly convenient when the\nsystem under investigation is characterized by a large density of excited states. Illustrative results\nhave been reported for cyclopropane and urea, and compared with results from a previous B3LYP\nimplementation of the AandBterms. The spectral profiles were found qualitatively similar,\nthough with noticeable differences on the intensity scale and the usual shifts in the position of the\nexcited states.\n14MCD-CC\nTABLE II. Urea. Computed spectral parameters: excitation energies ( wf), oscillator strengths (f) and and\nMCDBterms. Basis set aug-cc-pVDZ.\nCCSD CAM-B3LYP\nwf/eV (f) B/a.u.wf/eV (f) B/a.u.\n6.420 (0.031)\u00004.78769 6.354 (0.025)\u00004.91873\n6.754 (0.033) 5.23536 6.612 (0.033) 5.66315\n7.524 (0.009)\u00000.67620 7.371 (0.012)\u00000.62543\n7.623 (0.036)\u00007.46470 7.463 (0.021)\u00007.63975\n7.731 (0.014) 1.53604 7.587 (0.020) 1.83269\n7.832 (0.002) 5.05075 7.674 (0.009) 4.95339\n8.019 (0.14) 84.0208 7.928 (0.18) 80.3061\n8.054 (0.20)\u000084.0705 8.010 (0.13)\u000080.2552\n8.634 (0.059)\u00004.23649 8.518 (0.07)\u00004.09754\n8.671 (0.003) 3.05857 8.557 (0.004) 3.01059\n5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0\neVMCD\nDamped\nResonant\nDFT\nFIG. 3. Urea. CCSD/aug-cc-pVDZ MCD spectrum from damped and resonant response theory, and\ncomparison with the CAM-B3LYP one from resonant response theory.\n15MCD-CC\nACKNOWLEDGMENTS\nS.C. thanks Antonio Rizzo for useful discussion. S.G. thanks the University of Brescia and\nMIUR for a visiting grant under the auspices of the Doctorate program. R.F. and S.C. acknowledge\nfinancial support from the Independent Research Fund Denmark – Natural Sciences, Research\nProject 2, grant no. 7014-00258B. C.H. acknowledges financial support by the DFG through grant\nno. HA 2588/8.\nDATA A VAILABILITY\nThe data that support the findings of this study are available from the corresponding author\nupon reasonable request.\nV . APPENDIX: THE RESIDUES OF THE DERIVATIVE OF THE DAMPED CC\nLINEAR RESPONSE FUNCTION\nFor the analysis of the residues of the derivative of hhma;mbiiw+ivwith respect to Bgwe define\nthe non-phase-isolated derivatives of the eigenvectors with respect to the electric fields:\nRma\nf(\u0000w) =\u0000\u0000\nA\u0000(wf\u0000w)1\u0001\u00001\u0000\nAma+Btma(w)\u0001\nRf (52)\nOf the amplitude and Lagrange multiplier vectors in Eq. (46) only the vectors tmb(w),¯tmb(w), and\n¯tma(\u0000w)have nonvanishing residues in the limit w!wf:\nlimw!wf(w\u0000wf)tmb(w) =å\nf02DfRf0Tmb\nf00(53)\nlimw!wf(w\u0000wf)¯tmb(w) =å\nf02DfTmb\nf00\u0001Mf0 (54)\nlimw!wf(w\u0000wf)¯tma(\u0000w) =å\nf02DfTma\n0f0\u0001Lf0 (55)\n16MCD-CC\nWe get for the non-singular part of residue (simple pole) for the limit w!wf:\nlim\nw+iv!wf(w+iv\u0000wf)dhhma;mbiiw+iv\ndeg\f\f\f\f\nnon-res\n=1\n2Pabå\nf02Df\u001a\u0000\nFmgtma(\u0000wf)Rf0\u0001\n\u0001Tmb\nf00(56)\n+\u0010h\nFma+Gtma(\u0000wf)i\nRf0tmg\u0011\n\u0001Tmb\nf00\n+\u0010\n¯tmgh\nAma+Btma(\u0000wf)i\nRf0\u0011\n\u0001Tmb\nf00\n+\u0010\n?¯tma(\u0000wf)h\nAmg+Btmgi\nRf0\u0011\n\u0001Tmb\nf00\n+Lf0h\nAmbtmg+Amg?tmb(wf)+Btmg?tmb(wf)i\n\u0001Tma\n0f0\n+Mf0h\nAmatmg+Amgtma(\u0000wf)+Btmgtma(\u0000wf)i\n\u0001Tmb\nf00\u001b\n=1\n2Pabå\nf02Df\u001a\u0010¯xmgRma\nf0(\u0000wf)\u0011\n\u0001Tmb\nf00(57)\n+\u0010¯xma(\u0000wf)?Rmg\nf0\u0011\n\u0001Tmb\nf00\n+\u0010h\nFmgtma(\u0000wf)+Fmatmg+Gtma(\u0000wf)tmgi\nRf0\u0011\n\u0001Tmb\nf00\n+Mf0h\nAmatmg+Amgtma(\u0000wf)+Btmgtma(\u0000wf)i\n\u0001Tmb\nf00\n+Tma\n0f0\u0001\u0010\nLf0Ambtmg+?Lmg\nf0xmb\u0011\u001b\n=1\n2Pabå\nf02Df\u001adTma\n0f0\ndegTmb\nf00+Tma\n0f0\u0001dTmb\nf00\ndeg\u001b\n(58)\nOnly one contribution to the derivative of the damped linear response function, ¯tma(\u0000w\u0000\niv)\u0002\nAmg+Btmg]tmb(w+iv), contains two vectors that become singular for w!wfand con-\ntributes to the second-order residue:\nlim\nw+iv!wf(w+iv\u0000wf)2dhhma;mbiiw+iv\ndeg(59)\n=1\n2Pabå\nf;f02DfTma\n0f0\u0001\u0010\nLf0h\nAmg+Btmgi\nRf\u0011\nTmb\nf0\n=Pabå\nf0. The second term in Eq. (35) is nonzero only for \u000b=\f=y. By contrast, for a\ndiagonal spiral with Q= (\u0019\u00002\u0019\u0011;\u0019\u00002\u0019\u0011), we have J(2)\nxy=J(2)\nyx6= 0, andJ(2)\nxx=J(2)\nyy. In\nthis case the second term in Eq. (35) does not depend on \u000band\f.\nWe now determine the momentum and frequency dependence of the leading imaginary\nterm describing the damping of the in-plane Goldstone mode for small q. Imaginary con-\ntributions to the diagonal susceptibilities arise from the \u000e-function contributions ~ \u001fab\n0ito\n~\u001fab\n0. For small frequencies (and small q), only intraband terms ( `=`0) contribute since\nE+\nk\u0000E\u0000\nk>2\u0001. We expand the imaginary part of 1 =~\u001f22(q;!) for small q, keeping the ratio\n^!=!=jqj\fxed. The coupling to the 3-component can be neglected, since the intraband\ncoherence factor A23\n``(k\u0000q=2;q) is of orderjqj2for small q. Hence, for the imaginary part,\nthe expansion Eq. (33) can be generalized to\nIm1\n~\u001f22(q;!)=\u00004U2\u0014\n~\u001f22\n0i(q;!) + ImX\na;b=0;1~\u001f2a\n0(q;!)\u0016\u0000ab(0;0) ~\u001fb2\n0(q;!)\u0015\n+O(jqj3) (36)\nfor small qand \fxed \fnite ^ !. We will now show that both terms in Eq. (36) are of order\njqj2at \fxed ^!.\nShifting the integration variable kin Eq. (22) by\u0000q=2, the imaginary part of ~ \u001f22\n0(q;!)\ncan be written as\n~\u001f22\n0i(q;!) =i\u0019\n8Z\nkX\n`;`0A22\n``0(k\u0000q=2;q)\u0002\nf(E`\nk\u0000q=2)\u0000f(E`0\nk+q=2)\u0003\n\u000e(!+E`\nk\u0000q=2\u0000E`0\nk+q=2):(37)\nFor small frequencies, only intraband terms contribute. The intraband coherence factor\nA22\n``(k\u0000q=2;q) = 1\u0000hk\u0000q=2hk+q=2+ \u00012\nek\u0000q=2ek+q=2(38)\nis of orderjqj2for small q. Expanding E`\nk+q=2\u0000E`\nk\u0000q=2=q\u0001rkE`\nk+O(jqj3), and using\n\u000e(jqjx) =jqj\u00001\u000e(x), we \fnd that ~ \u001f22\n0i(q;!) is of orderjqj2.12\nSince \u0016\u0000ab(0;0) is real, the second term in Eq. (36) receives contributions from the cross\nterms ~\u001f2a\n0r(q;!)\u0016\u0000ab(0;0) ~\u001fb2\n0i(q;!) and ~\u001f2a\n0i(q;!)\u0016\u0000ab(0;0) ~\u001fb2\n0r(q;!). For small !, only intra-\nband terms contribute to ~ \u001f2a\n0i(q;!) and ~\u001fb2\n0i(q;!). Both are of order qfor small qat \fxed ^!,\nbecause the intraband coherence factors A02\n``(k;q) =\u0000A20\n``(k;q) andA12\n``(k;q) =\u0000A21\n``(k;q)\nare of order q. Moreover, ~ \u001f2a\n0r(q;!) and ~\u001fb2\n0r(q;!) are antisymmetric in qand thus of order\nq, too. Hence, the second term in Eq. (36) is of order jqj2.\nIn summary, we have shown that the damping term of the in-plane Goldstone mode has\nthe scaling form\nImm2\n~\u001f22(q;!)=\u0000jqj2\r(^q;^!) +O(jqj3); (39)\nwhere\r(^q;^!) is a function of ^q=q=jqjand ^!=!=jqj. The scaling function \r(^q;^!) has the\nsame sign as ^ !, and it vanishes for ^ != 0. The damping of the in-plane mode thus has the\nsame form as the Landau damping of the two Goldstone modes in a N\u0013 eel antiferromagnet\n[7]. It is of the same order as the leading real terms near the Goldstone pole. Hence, the\ndamping of the in-plane Goldstone mode is of the same order as its excitation energy, that\nis, of orderjqj. Asymptotically stable low-energy quasi-particles require damping rates that\nvanish faster than their excitation energy in the low-energy limit. The in-plane Goldstone\nmode in a metallic spiral state and the Goldstone mode in a metallic N\u0013 eel state violate this\ncriterion, albeit only marginally.\n2. Out-of-plane mode\nWe derive the RPA expression for the out-of-plane spectral weight factor Z(3)in a rotated\nspin basis spanned by S\u0006\nj=1\n2(S1\nj\u0006iS2\nj) instead of S1\njandS2\nj. The coherence factors in this\nbasis,Aab\n``0witha;b2f0;+;\u0000;3g, are all real. The matrix elements of the bare interaction\nmatrix \u0000 0with indices + and \u0000are \u0000+\u0000\n0= \u0000\u0000+\n0= 4Uand \u0000++\n0= \u0000\u0000\u0000\n0= 0. The components\n\u001f+\u0000and\u001f\u0000+of the physical susceptibility are diagonal in momentum space, while \u001f++and\n\u001f\u0000\u0000are o\u000b-diagonal, with a momentum shift Q.\nForq=Qand \fnite!the 3-component of the spin couples to all the other spin compo-\nnents and the charge channel. The function ~ \u001f33(Q;!) can again be extracted from the RPA\nexpression (20) by using a suitable Schur complement. Expanding the matrix elements to13\nsecond order in !, one obtains\n1\n~\u001f33(Q;!)= 2U\u0014\n1\u00002U~\u001f33\n0(Q;!)\u00002UX\na;b=0;+;\u0000~\u001f3a\n0(Q;!)\u0016\u0000\u0016ab(Q;0) ~\u001f\u0016b3\n0(Q;!)\u0015\n+O(!3);(40)\nwhere the bar over the indices aandbleaves the index 0 unchanged, while it exchanges\nthe indices + and \u0000. Here, the matrix \u0016\u0000(q) represents the RPA e\u000bective interaction in the\nsubspace spanned by the charge channel and the in-plane spin channel in the basis spanned\nbyS+andS\u0000,\n\u0016\u0000(q) =2\n666413\u00000\nBBB@\u00002U0 0\n0 4U0\n0 0 4U1\nCCCA0\nBBB@~\u001f00\n0(q) ~\u001f0\u0000\n0(q) ~\u001f0+\n0(q)\n~\u001f+0\n0(q) ~\u001f+\u0000\n0(q) ~\u001f++\n0(q)\n~\u001f\u00000\n0(q) ~\u001f\u0000\u0000\n0(q) ~\u001f\u0000+\n0(q)1\nCCCA3\n7775\u000010\nBBB@\u00002U0 0\n0 4U0\n0 0 4U1\nCCCA:(41)\nInserting Eq. (40) into Eq. (30), the out-of-plane spectral weight factor can be expressed in\nthe form\nZ(3)= 2\u00012h\n@2\n!~\u001f33\n0(Q;!)\f\f\n!=0+ 2X\na;b=0;+;\u0000\u0000\n@!~\u001f3a\n0(Q;!)\f\f\n!=0\u0001\u0016\u0000\u0016ab(Q;0)\u0010\n@!~\u001f\u0016b3\n0(Q;!)\f\f\n!=0\u0011i\n:\n(42)\nThis expression is real, because ~ \u001f33\n0i(q;!) is antisymmetric in !, while ~\u001f3a\n0i(q;!) witha6= 3\nis symmetric.\nThe expression for the out-of-plane spin sti\u000bness is comparatively simple, since all o\u000b-\ndiagonal susceptibilities involving the 3-component of the spin vanish at != 0. Expanding\naround q=\u0006Qone obtains\nJ(3)\n\u000b\f=\u00002\u00012@2\nq\u000b\f~\u001f33\n0(q;0)\f\f\nq=\u0006Q: (43)\nFor the most common spiral states in two dimensions with wave vectors of the form Q=\n(\u0019;\u0019\u00002\u0019\u0011) and Q= (\u0019\u00002\u0019\u0011;\u0019\u00002\u0019\u0011), the spatial structure of J(3)\n\u000b\fis the same as for the\nin-plane sti\u000bness J(2)\n\u000b\fdiscussed above.\nWe \fnally determine the asymptotic momentum and frequency dependence of the imag-\ninary part of 1 =~\u001f33(q;!) forqnear\u0006Q, which determines the damping of the out-of-plane\nGoldstone modes. We discuss the case q\u0018Q. The behavior for q\u0018\u0000Qis equivalent.\nWe \frst analyze the low frequency asymptotics for q=Qand show that all contributions\nto the imaginary part of 1 =~\u001f33(Q;!) in Eq. (40) are of order !3. The \frst contribution is\ndetermined by the imaginary part of the bare out-of-plane spin susceptibility,\n~\u001f33\n0i(Q;!) =i\u0019\n8Z\nkX\n`;`0A33\n``0(k;Q)\u0002\nf(E`\nk)\u0000f(E`0\nk+Q)\u0003\n\u000e(!+E`\nk\u0000E`0\nk+Q): (44)14\nFor small frequencies !, only momenta corresponding to small energies E`\nkandE`0\nk+Qof\norder!contribute to the k-integral. These momenta are restricted to a small neighborhood\nofhot spots kHde\fned by the equations\nE`\nkH=E`0\nkH+Q= 0: (45)\nGeometrically, the hot spots are the intersection points of the Fermi surface of E`\nkand the\nQ-shifted Fermi surface of E`0\nk. In our two-dimensional case studies (see below) we have only\nfound intraband ( `=`0) hot spots. While we cannot exclude the existence of interband hot\nspots in general, we restrict the subsequent analysis to intraband contributions.\nFor`=`0, the equations (45) are equivalent to\nE`\nkH= 0 and \u0018kH=\u0018kH+2Q: (46)\nWe note that for a N\u0013 eel state, where 2 Qis a reciprocal lattice vector, the second equation\nis always satis\fed, so that all momenta on the Fermi surface of E`\nkare hot spots. The\ncondition\u0018kH=\u0018kH+2Qimplies that hkH+Q=\u0000hkH. As a direct consequence, we \fnd\nthatA33\n``(kH;Q) = 0 and alsorkA33\n``(k;Q)\f\f\nk=kH= 0. Hence, the coherence factor leads to a\nstrong suppression of ~ \u001f33\n0i(Q;!) at low frequencies. For small !, the momenta kcontributing\nto the integral in Eq. (44) are situated at a distance of order !away from the hot spots.\nFor such momenta the coherence factor A33\n``(k;Q) is of order !2, sinceA33\n``(k;Q) and also\nits gradient vanish at k=kH. Multiplying this with the usual factor !coming from the\ndi\u000berence of Fermi functions, we obtain\n~\u001f33\n0i(Q;!)/!3(47)\nfor small!.\nWe now turn to the second contribution to the imaginary part of 1 =~\u001f33(Q;!) in Eq. (40),\nwhich involves the o\u000b-diagonal bare susceptibilities ~ \u001f3a\n0and ~\u001fa3\n0witha2f0;+;\u0000g. Since\nthe static RPA e\u000bective interaction in Eq. (40) is real, contributions to the imaginary part\nof 1=~\u001f33(Q;!) are due to products of real and imaginary parts of the o\u000b-diagonal bare\nsusceptibilities. The real parts ~ \u001f3a\n0r(Q;!) are antisymmetric in the frequency argument and\nthus of order !for small!. The coherence factors A3a\n``(k;Q) vanish at the hot spots, but\ntheir gradientsrkA3a\n``(k;Q) are \fnite at k=kH. Hence, following the above arguments\nused to determine the low frequency dependence of ~ \u001f33\n0i(Q;!), we obtain\n~\u001f3a\n0i(Q;!)/!2(48)15\nfora2f0;+;\u0000gand small!. The product of imaginary and real parts of o\u000b-diagonal\nbare susceptibilities is thus of order !3. Combining all terms, we have thus shown that the\nout-of-plane damping term at q=Qobeys\nImm2\n~\u001f33(Q;!)/!3(49)\nat low frequencies.\nForq6=Q, the coherence factors remain \fnite at the hot spots (now determined by the\nequationsE`\nkH=E`0\nkH+q= 0) so that\n~\u001f3a\n0i(q;!) =\u0000p3a(q)! (50)\nfor small!anda2f0;+;\u0000;3g. However, the prefactor of this linear frequency dependence\nvanishes as qapproaches the ordering wave vector Q. For the diagonal intraband coherence\nfactorA33\n``(k;q), also the gradient with respect to qvanishes at k=kHandq=Q. Hence\np33(q) is of order ( q\u0000Q)2forq!Q. Fora6= 3, the gradientrqA3a\n``(k;q) is \fnite at k=kH\nandq=Q, so thatp3a(q) is of orderjq\u0000Qjforq!Q. Eq. (40) can be generalized in the\nsame form for q6=Q. For!!0 the contribution from ~ \u001f33\n0i(q;!) is leading and yields\nImm2\n~\u001f33(q;!)=\u0000\r(q)!+O(!2); (51)\nwhere\r(q)/(q\u0000Q)2forq!Q. The o\u000b-diagonal contributions to the damping term are\nof order!2forq6=Q, with a prefactor that is linear in jq\u0000Qj. Taking the limit !!0,\nq!Qat a \fxed ratio ^ !=!=jq\u0000Qj, diagonal and o\u000b-diagonal contributions are both of\norderjq\u0000Qj3. The Landau damping of out-of-plane Goldstone modes thus scales to zero\nmore rapidly than their excitation energy, so that these modes remain asymptotically stable\nquasi-particles.\nThe above results for the Landau damping hinge on the existence of hot spots. If Eq. (45)\nhas no solution, the imaginary parts of the RPA susceptibilities are strictly zero below a\ncertain threshold frequency. Higher order terms beyond RPA, such as fermionic self-energy\ncontributions, will however yield a small low-frequency damping in any case.\nAlthough electron and hole pockets coexist in the Brillouin zone for certain model pa-\nrameters, we have not found any interband hot spots in spiral states for the two-dimensional\nHubbard model. If interband hot spots existed in a suitable system, an exceptionally large\nLandau damping would follow. Since the interband coherence factor A33\n`;\u0000`(k;Q) remains16\n\fnite at the interband hot spots, the Landau damping term would be linear in !even at\nq=Q, leading to a strong overdamping of the out-of-plane Goldstone mode.\nD. Special case: N\u0013 eel state\nThe N\u0013 eel state can be viewed as a special case of the spiral state where the ordering wave\nvector Qassumes the special value Q= (\u0019;\u0019) in two dimensions and Q= (\u0019;\u0019;\u0019 ) in three\ndimensions. In this section we analyze how the properties of the Goldstone modes derived\nabove change in this case. In particular, we will see that the number of Goldstone modes is\nreduced to two, and their properties are equivalent.\nThe special properties of the N\u0013 eel state are due to the fact that Qand\u0000Qare identical\nwave vectors in the Brillouin zone if all components of Qare equal to \u0019. In other words 2 Qis\nidentical to 0. As a \frst consequence, in the relation between the physical susceptibilities \u001f11\nand\u001f22and the susceptibilities ~ \u001fabin the rotated spin basis, see Eq. (15), terms which in the\nspiral state contribute only to o\u000b-diagonal (in momentum) susceptibilities \u001faa(q\u00062Q;q;!)\ncontribute to the momentum diagonal susceptibilities \u001faa(q;q;!) in the N\u0013 eel state. Hence,\ninstead of Eq. (15) one obtains\n\u001f11(q;!) = ~\u001f11(q\u0006Q;!); (52)\n\u001f22(q;!) = ~\u001f22(q\u0006Q;!): (53)\nIn the spiral state we found three distinct Goldstone modes, an in-plane mode associated\nwith a divergence of ~ \u001f22(q;!) forq!0and!!0, and two out-of-plane modes leading\nto divergencies of ~ \u001f33(q;!) forq!\u0006Qand!!0. In the N\u0013 eel state the two singularities\nof ~\u001f33(q;!) collapse to one, since Qand\u0000Qare now identical. Hence, only two Goldstone\nmodes survive. This is in agreement with the fact that in the N\u0013 eel state the continuous\nSU(2) spin rotation invariance is not completely broken: an U(1) symmetry associated with\nrotations around the spin orientation axis remains. Moreover, in the N\u0013 eel state the notion\nof \\in-plane\" and \\out-of-plane\" modes is meaningless since the N\u0013 eel order singles out a\nparticular axis, not a plane. The two Goldstone modes correspond to \ructuations of that\naxis in two orthogonal directions. By symmetry they must have the same sti\u000bness, spectral\nweight and damping. We will now show that the properties of the in-plane and out-of-plane\nmodes derived in the preceding section are indeed degenerate in the N\u0013 eel limit.17\nIn Appendix E we show that the o\u000b-diagonal bare susceptibilities ~ \u001f02\n0, ~\u001f03\n0, ~\u001f12\n0, and ~\u001f13\n0\nvanish identically in the N\u0013 eel state. Hence, the sectors 0 and 1 are completely decoupled\nfrom the sectors 2 and 3 for all momenta and frequencies. The expression (35) for the\nin-plane sti\u000bness thus simpli\fes to\nJ(2)\n\u000b\f=\u00002\u00012@2\nq\u000bq\f~\u001f22\n0(q;0)\f\f\nq=0: (54)\nComparing with Eq. (43) for the out-of-plane sti\u000bness, and using the relation ~ \u001f22\n0(q;!) =\n~\u001f33\n0(q+Q) derived in Appendix E, one \fnds J(2)\n\u000b\f=J(3)\n\u000b\fas expected.\nDue to the decoupling of the sectors 0 and 1 from the sectors 2 and 3, one can write Z(3)\nin a form analogous to the expression (32), that is,\nZ(3)= 2\u00012\u0014\n@2\n!~\u001f33\n0(Q;!)\f\f\n!=0+4U\n1\u00002U~\u001f22\n0(Q;0)\f\f@!~\u001f23\n0(Q;!)\f\f\n!=0\f\f2\u0015\n: (55)\nUsing once again ~ \u001f22\n0(q;!) = ~\u001f33\n0(q+Q), and ~\u001f23\n0(q;!) = ~\u001f23\n0(q+Q) derived in Appendix E,\none obtains Z(2)=Z(3).\nWe \fnally turn to the damping terms. In the N\u0013 eel state, the intraband coherence factor\nA23\n``(k\u0000q=2;q) is not only suppressed (of order jqj2) for small q, but also for q!Q, where\nit is of orderjq\u0000Qj2. Combining this with the decoupling of the sectors 0 and 1 from the\nsectors 2 and 3, one obtains\nIm1\n~\u001f22(q;!)=\u00004U2~\u001f22\n0i(q;!) +O(jqj3) (56)\nfor small q, and\nIm1\n~\u001f33(q;!)=\u00004U2~\u001f33\n0i(q;!) +O(jq\u0000Qj3) (57)\nfor small q\u0000Q. The relation ~ \u001f22\n0(q;!) = ~\u001f33\n0(q+Q) then implies that the damping of\nthe 2-mode and the 3-mode is identical. Returning to the susceptibilities in the physical\n(unrotated) spin basis one obtains\nImm2\n\u001f22(q;!)= Imm2\n\u001f33(q;!)=\u0000jq0j2\r(^q0;^!) +O(jq0j3) (58)\nfor small q0=q\u0000Qand \fxed ^!=!=jq0j. This form of the Landau damping in a N\u0013 eel state\nhas already been derived by Sachdev et al. [7].18\nFIG. 1. Magnetization m(left axis, solid line) and incommensurability \u0011(right axis, dashed line) as\na function of the electron density nin the mean-\feld ground state of the two-dimensional Hubbard\nmodel with parameters t0=t=\u00000:16 andU=t= 2:5.\nE. Numerical results in two dimensions\nTo complement our general results, and to get an idea about the typical size of the spin\nsti\u000bnesses and the damping terms, we now present some numerical results as obtained by\nevaluating the analytic expressions derived above for a speci\fc model in two dimensions: the\nrepulsive Hubbard model on the square lattice with nearest and next-to-nearest neighbor\nhopping amplitudes ( tandt0, respectively). We choose tas our unit of energy, that is, all\nresults with an energy dimension are presented for t= 1.\nWe compute only ground state properties. We choose t0=\u00000:16tand a relatively weak\nHubbard interaction U= 2:5t. For this choice of parameters mean-\feld theory yields a\nhomogeneous spiral magnetic state over an extended density range between n\u00190:61 and\nn= 1 (half-\flling). At half-\flling and for electron doping up to n\u00191:15 the simple N\u0013 eel\nstate minimizes the mean-\feld energy. In the spiral state for n<1 the ordering wave vector\nhas the form Q= (\u0019\u00002\u0019\u0011;\u0019 ). The incommensurability \u0011increases monotonically upon\nreducing the density, and vanishes continuously for n!1. The onset of the spiral order at\nn\u00190:61 is continuous, while the transition between the N\u0013 eel state and the paramagnetic\nstate atn\u00191:15 is of \frst order, albeit with a relatively small jump of the order parameter.\nThe magnetization mand the incommensurability \u0011are plotted as functions of the electron\ndensitynin Fig. 1.19\nFIG. 2. Quasiparticle Fermi surfaces in the magnetic ground state at various electron densities.\nBlue lines correspond to momenta satisfying E+\nk= 0, red lines to momenta satisfying E\u0000\nk= 0. The\ndashed vertical lines at kx= 2\u0019\u0011andkx= 2\u0019\u0011\u0000\u0019are solutions of the equation \u0018k+2Q=\u0018k. For\nn= 0:84 andn= 0:63 there are hot spots on the Fermi surfaces (black dots) which are connected\nto other points on the Fermi surfaces (grey dots) by a momentum shift Q. The numbers indicate\nthe pairwise connection. In the N\u0013 eel state at n= 1:1 all points on the Fermi surface are connected\nto each other by Q= (\u0019;\u0019).\nIn Fig. 2 we show the quasiparticle Fermi surfaces in the magnetic ground state at various\nelectron densities from n= 0:63 ton= 1:1. Forn<1 these are given by momenta satisfying\nthe equation E\u0000\nk= 0, forn > 1 by solutions of E+\nk= 0. In the spiral state for n < 1 hot\nspots corresponding to solutions of Eqs. (45) or (46) exist only for su\u000eciently large hole\ndoping atn= 0:84 andn= 0:63. Hence, for low hole doping, such as n= 0:95, there is no\nLandau damping of the out-of-plane magnons.\nIn Fig. 3 we show the in-plane and out-of-plane spin sti\u000bnesses J(a)\n\u000b\fas a function of the\nelectron density. Both in the spiral state for n < 1 and in the N\u0013 eel state for n\u00151 only\ndiagonal components J(a)\n\u000b\u000bwith\u000b=x;yare non-zero. In the N\u0013 eel state the sti\u000bnesses are\nisotropic (independent of \u000b) and degenerate ( J(2)\n\u000b\u000b=J(3)\n\u000b\u000b), as dictated by symmetry. The\nspin sti\u000bnesses are positive for all densities where a magnetic solution exists, showing that20\nFIG. 3. In-plane and out-of-plane spin sti\u000bnesses as a function of the electron density. In the N\u0013 eel\nstate forn\u00151 all sti\u000bnesses assume the same value.\nthe spiral state for n<1 and the N\u0013 eel state for n\u00151 are at least meta stable. In the spiral\nstate the in-plane and out-of-plane sti\u000bnesses di\u000ber signi\fcantly among each other, except\nfor the lowest densities (where m!0) and near half-\flling. Both exhibit a slight nematicity\n(dependence on \u000b) which comes from the di\u000berence between QxandQy. All spin sti\u000bnesses\nJ(a)\n\u000b\u000bexhibits a pronounced jump at half-\flling. More precisely, upon approaching half-\flling\nfrom below ( n <1) the sti\u000bnesses converge to a value that di\u000bers from J(a)\n\u000b\u000bat half-\flling.\nThis discontinuity is caused by the sudden appearance of hole-pockets upon hole-doping,\nwhich allow for intraband processes with small excitation energies. A discontinuity due to\nelectron pockets upon approaching half-\flling from above ( n>1) is prevented by vanishing\nprefactors at the momenta ( \u0019;0) and (0;\u0019) where the electron pockets pop up.\nThe density dependence of the spectral weights of the magnon modes m2=Z(a)is shown\nin Fig. 4. In the spiral state for n<1 there is a pronounced di\u000berence between the in-plane\nand the out-of-plane modes. The discontinuity of m2=Z(3)at half-\flling is again due to\nintraband contributions within the hole pockets emerging for n <1. By contrast, m2=Z(2)\nis continuous, since only interband terms contribute to the in-plane Z-factor. Both spectral\nweights are positive in the entire ordered phase. The spectral weights decrease near the\nedges of the magnetic regime, since m2vanishes more rapidly than Z(a)upon approaching\nthe edges. The weight of the out-of-plane mode m2=Z(3)exhibits a dip at the density\nn\u00190:84 where two hole pockets merge.21\nFIG. 4. In-plane and out-of-plane spectral weights as a function of the electron density. In the\nN\u0013 eel state for n\u00151 both weights assume the same value.\nFIG. 5. In-plane and out-of-plane magnon velocities c(a)\n\u000b\u000b=\u0002\nJ(a)\n\u000b\u000b=Z(a)\u00031=2as a function of the\nelectron density.\nIn Fig. 5 we show the magnon velocities c(a)\n\u000b\u000b=\u0002\nJ(a)\n\u000b\u000b=Z(a)\u00031=2. The velocities exhibit only\na moderate density dependence. Their size is over order one (in units of t) in the entire\nmagnetic regime.\nIn Fig. 6 we plot the in-plane damping term Im[ m2=~\u001f22(q;!)] as a function of jqjat\ntwo \fxed values of ^ !=!=jqjand three \fxed directions ^q=q=jqj. The density is \fxed at\nn= 0:84. One can see the quadratic dependence on jqjin agreement with Eq. (39). The22\nFIG. 6. Damping term of the in-plane Goldstone mode as a function of jqjfor two \fxed values of\n^!and a density n= 0:84. Various directions of qare parametrized by the angle \u0012between qand\ntheqx-axis. The prefactor \rof the leading quadratic dependence on jqjis shown in the inset.\nprefactors\r(^!;^q) are shown in the inset.\nThe frequency dependence of the out-of-plane damping Im[ m2=~\u001f33(q;!)] is shown in\nFig. 7 for various \fxed momenta qat and near Q. For q=Qthe damping is proportional\nto!3for low frequencies, in agreement with Eq. (49). For q6=Qone can see the linear\nfrequency dependence in agreement with Eq. (51). The prefactors of the leading cubic and\nlinear terms are listed in the inset.\nIV. CONCLUSION\nIn summary, we have investigated the properties of the Goldstone modes (that is,\nmagnons) in metallic electron systems with spiral magnetic order. Our analysis is based\non the RPA susceptibilities of tight binding electrons with an arbitrary dispersion and\na local Hubbard interaction. In agreement with general arguments and previous studies\n[26{29] we have identi\fed three Goldstone poles in the susceptibilities, one associated with\nin-plane, and two associated with out-of-plane \ructuations of the order parameter. The\nenergy-momentum relations of all the modes are linear.\nWe have derived expressions for the spin sti\u000bnesses and the spectral weights of the\nmagnons, from which the magnon velocities can be obtained, too. The expressions for23\nFIG. 7. Damping term of the out-of-plane Goldstone mode as a function of !for various \fxed\nwave vectors qnearQ= (0:82\u0019;\u0019) and \fxed density n= 0:84. The prefactors \r1of the linear\nfrequency dependence for q6=Qand the prefactor \r3of the cubic frequency dependence for q=Q\nare shown in the inset.\nthe spin sti\u000bnesses are also useful for checking the stability of the spiral state, for example,\nagainst an out-of-plane canting of the spins. Moreover, we have determined the size of the\ndecay rates of the magnons due to Landau damping. The Landau damping of the in-plane\nmode has the same form as for the Goldstone modes in a N\u0013 eel antiferromagnet [7] and is of\nthe same order as the energy !of the mode. By contrast, the Landau damping of the out-of-\nplane modes is smaller, of the order !3=2. Hence, the out-of-plane modes are asymptotically\nstable excitations in the low energy limit.\nWe have complemented our general analysis with a numerical evaluation of the spin\nsti\u000bnesses, spectral weights, and decay rates for a speci\fc two-dimensional model system.\nSome of the quantities exhibit peaks and discontinuities as a function of the electron density\nwhich are related to changes of the Fermi surface topology and special contributions in the\nN\u0013 eel state.\nMagnons and their decay rates can in principle be detected by inelastic neutron scattering.\nOur analysis indicates that out-of-plane magnon branches in a metallic spiral magnet should\nbe sharper than the in-plane branch at low excitation energies.24\nACKNOWLEDGMENTS\nWe are grateful to A. Chubukov, L. Classen, L. Debbeler, B. Keimer, E. K onig,\nJ. Mitscherling, O. Sushkov, J. Sykora, and D. Vilardi for valuable discussions.\nAppendix A: Coherence factors\nThe coherence factors entering the bare susceptibilities ~ \u001fab\n0in Eq. (22) are de\fned as\nAab\n``0(k;q) =1\n2trh\n\u001bau`\nk\u001bbu`0\nk+qi\n; (A1)\nwhere`;`0are the quasi-particle band indices, a2f0;1;2;3glabels the charge and spin\ncomponents, and the functions u`\nkare the linear combinations of Pauli matrices de\fned\nin Eq. (9). Performing the trace we obtain explicit expressions. For the charge-charge\ncoherence factor we get\nA00\n``0(k;q) = 1 +``0hkhk+q+ \u00012\nekek+q; (A2)\nwhile for the charge-spin ones we \fnd\nA01\n``0(k;q) =`\u0001\nek+`0\u0001\nek+q; (A3)\nA02\n``0(k;q) =i``0\u0001hk\u0000hk+q\nekek+q; (A4)\nA03\n``0(k;q) =`hk\nek+`0hk+q\nek+q: (A5)\nThe diagonal coherence factors in the spin subsector are given by\nA11\n``0(k;q) = 1\u0000``0hkhk+q\u0000\u00012\nekek+q; (A6)\nA22\n``0(k;q) = 1\u0000``0hkhk+q+ \u00012\nekek+q; (A7)\nA33\n``0(k;q) = 1 +``0hkhk+q\u0000\u00012\nekek+q; (A8)\nand the o\u000b-diagonal ones by\nA12\n``0(k;q) =i`hk\nek\u0000i`0hk+q\nek+q; (A9)\nA13\n``0(k;q) =``0\u0001hk+hk+q\nekek+q; (A10)\nA23\n``0(k;q) =i`\u0001\nek\u0000i`0\u0001\nek+q: (A11)25\nThe coherence factors for a > b are obtained from the general relation Aab\n``0(k;q) =\n\u0002\nAba\n``0(k;q)\u0003\u0003. The coherence factors are purely imaginary if (and only if) exactly one\nof the indices a;bis equal to two, and they are real otherwise. Hence, the exchange of the\nindicesaandbyields\nAba\n``0(k;q) =papbAab\n``0(k;q); (A12)\nwherepa= +1 fora= 0;1;3, andpa=\u00001 fora= 2.\nFrom\u0018k=\u0018\u0000kone obtains the relations h\u0000k\u0000Q=\u0000hk,g\u0000k\u0000Q=gk,e\u0000k\u0000Q=ek, and\nu`\n\u0000k\u0000Q=\u001b1u`\nk\u001b1. From Eq. (24) we then see that\nAab\n`0`(\u0000k\u0000Q\u0000q;q) =1\n2trh\n~\u001bbu`\nk+q~\u001bau`0\nki\n; (A13)\nwith ~\u001ba=\u001b1\u001ba\u001b1=sa\u001ba, wheresa= +1 fora= 0;1, andsa=\u00001 fora= 2;3. Using\nEq. (A12), we then obtain\nAab\n`0`(\u0000k\u0000Q\u0000q;q) =sasbAba\n``0(k;q) =sabAab\n``0(k;q); (A14)\nwhere\nsab=sasbpapb= (1\u00002\u000ea3)(1\u00002\u000eb3): (A15)\nThe relation (A14) will be useful in the following section.\nAppendix B: Symmetries of the bare susceptibilities\nIn this appendix we derive the behavior of the bare susceptibilities under sign changes of\nthe frequency and the momentum arguments.\n1. Parity under frequency sign change\nWe decompose the expression (22) for the susceptibility components in intraband and\ninterband contributions\n~\u001fab\n0(q;!) =\u00001\n8X\n`Z\nkAab\n``(k;q)f(E`\nk)\u0000f(E`\nk+q)\nE`\nk\u0000E`\nk+q+z\u00001\n8X\n`Z\nkAab\n`;\u0000`(k;q)f(E`\nk)\u0000f(E\u0000`\nk+q)\nE`\nk\u0000E\u0000`\nk+q+z;\n(B1)26\nwherez=!+i0+. Substituting k!\u0000k\u0000Q\u0000q, the intraband term can be rewritten as\n[~\u001fab\n0(q;!)]intra=\u00001\n8X\n`Z\nkAab\n``(k;q)f(E`\nk)\nE`\nk\u0000E`\nk+q+z\n\u00001\n8X\n`Z\nkAab\n``(\u0000k\u0000Q\u0000q;q)\u0000f(E`\n\u0000k\u0000Q)\n\u0000(E`\n\u0000k\u0000Q\u0000E`\n\u0000k\u0000Q\u0000q\u0000z):(B2)\nUsing Eq. (A14) and E`\n\u0000k\u0000Q=E`\nk, we obtain\n[~\u001fab\n0(q;!)]intra=\u00001\n8X\n`Z\nkAab\n``(k;q)f(E`\nk) \n1\nE`\nk\u0000E`\nk+q+z+sab\nE`\nk\u0000E`\nk+q\u0000z!\n:(B3)\nSimilarly, the interband term can be rewritten as\n[~\u001fab\n0(q;!)]inter=\u00001\n8X\n`Z\nkAab\n`;\u0000`(k;q)f(E`\nk)\nE`\nk\u0000E\u0000`\nk+q+z\n\u00001\n8X\n`Z\nkAab\n\u0000`;`(\u0000k\u0000q\u0000Q;q)\u0000f(E`\n\u0000k\u0000Q)\n\u0000(E`\n\u0000k\u0000Q\u0000E\u0000`\n\u0000k\u0000Q\u0000q\u0000z):(B4)\nIn the second term we have also made the substitution `!\u0000`. Using Eq. (A14) for `0=\u0000`,\nwe get\n[~\u001fab\n0(q;!)]inter=\u00001\n8X\n`Z\nkAab\n`;\u0000`(k;q)f(E`\nk) \n1\nE`\nk\u0000E\u0000`\nk+q+z+sab\nE`\nk\u0000E\u0000`\nk+q\u0000z!\n;(B5)\nwithsabas de\fned in Eq. (A15). Summing the intraband and the interband terms we obtain\n~\u001fab\n0r(q;\u0000!) =sab~\u001fab\n0r(q;!) (B6)\n~\u001fab\n0i(q;\u0000!) =\u0000sab~\u001fab\n0i(q;!): (B7)\n2. Parity under momentum sign change\nSubstituting k!k\u0000q=2, we rewrite the bare susceptibility as\n~\u001fab\n0(q;!) =\u00001\n8X\n``0Z\nkAab\n``0\u0010\nk\u0000q\n2;q\u0011f(E`\nk\u0000q\n2)\u0000f(E`0\nk+q\n2)\nE`\nk\u0000q\n2\u0000E`0\nk+q\n2+!+i0+: (B8)\nUsing\nAab\n`0`\u0010\nk+q\n2;\u0000q\u0011\n=Aba\n``0\u0010\nk\u0000q\n2;q\u0011\n=papbAab\n``0\u0010\nk\u0000q\n2;q\u0011\n; (B9)\nwithpaas de\fned in Appendix A, we immediately see that\n~\u001fab\n0(\u0000q;\u0000!) =papb~\u001fab\n0(q;!): (B10)27\nCombining this with Eqs. (B6) and (B7), we obtain\n~\u001fab\n0r(\u0000q;!) =pab~\u001fab\n0r(q;!) (B11)\n~\u001fab\n0i(\u0000q;!) =\u0000pab~\u001fab\n0i(q;!); (B12)\npab=papbsab=sasb= (1\u00002\u000ea2)(1\u00002\u000eb2)(1\u00002\u000ea3)(1\u00002\u000eb3): (B13)\nAppendix C: Calculation of ~\u001f33\n0(\u0006Q;0)\nIn this Appendix we prove the relation (28) for ~ \u001f33\n0(\u0000Q;0). The corresponding relation\nfor ~\u001f33\n0(Q;0) follows from the parity of ~ \u001f33\n0(q;!) under q!\u0000q. Using the general expres-\nsion (22) for the bare susceptibility, and Eq. (A8) for the coherence factor A33\n``0(k;q), one\nobtains\n~\u001f33\n0(\u0000Q;0) =\u00001\n8Z\nk\u0014\n1 +hkhk\u0000Q\u0000\u00012\nekek\u0000Q\u0015 \nf(E+\nk)\u0000f(E+\nk\u0000Q)\nE+\nk\u0000E+\nk\u0000Q+f(E\u0000\nk)\u0000f(E\u0000\nk\u0000Q)\nE\u0000\nk\u0000E\u0000\nk\u0000Q!\n\u00001\n8Z\nk\u0014\n1\u0000hkhk\u0000Q\u0000\u00012\nekek\u0000Q\u0015 \nf(E+\nk)\u0000f(E\u0000\nk\u0000Q)\nE+\nk\u0000E\u0000\nk\u0000Q+f(E\u0000\nk)\u0000f(E+\nk\u0000Q)\nE\u0000\nk\u0000E+\nk\u0000Q!\n=\u00001\n4X\n`=\u0006Z\nk(\u0014\n1\u0000hkh\u0000k+ \u00012\neke\u0000k\u0015f(E`\nk)\nE`\nk\u0000E`\n\u0000k+\u0014\n1 +hkh\u0000k+ \u00012\neke\u0000k\u0015f(E`\nk)\nE`\nk\u0000E\u0000`\n\u0000k)\n=X\n`=\u0006Z\nk(\u0000`)f(E`\nk)\n4ek(\n2`ek(gk\u0000g\u0000k) + 2hk(hk\u0000h\u0000k)\n(E`\nk\u0000E\u0000`\n\u0000k)(E`\nk\u0000E`\n\u0000k))\n: (C1)\nIn the second equation we have used hk\u0000Q=\u0000h\u0000k,ek\u0000Q=e\u0000k, andE\u0006\nk\u0000Q=E\u0006\n\u0000k. It is\neasy to see that the linear combinations g\u0000\nk=gk\u0000g\u0000k,h\u0006\nk=hk\u0006h\u0000k, ande\u0006\nk=ek\u0006e\u0000k\nobey the relations h\u0000\nkh+\nk=h2\nk\u0000h2\n\u0000k=e2\nk\u0000e2\n\u0000k=e\u0000\nke+\nk, andh\u0000\nk=\u0000g\u0000\nk. Using these\nrelations, we \fnally get\n~\u001f33\n0(\u0000Q;0) =X\n`=\u0006Z\nk(\u0000`)f(E`\nk)\n4ek\u001a2`ekg\u0000\nk+ 2hkh\u0000\nk\n(g\u0000\nk+`e+\nk)(g\u0000\nk+`e\u0000\nk)\u001b\n=X\n`=\u0006Z\nk(\u0000`)f(E`\nk)\n4ek\u001a2`ekg\u0000\nk+e\u0000\nke+\nk+ (g\u0000\nk)2\n(g\u0000\nk+`e+\nk)(g\u0000\nk+`e\u0000\nk)\u001b\n=X\n`=\u0006Z\nk(\u0000`)f(E`\nk)\n4ek=Z\nkf(E\u0000\nk)\u0000f(E+\nk)\nek: (C2)28\nAppendix D: Expansion of 1=~\u001f22(q;0)for small q\nHere we derive the expansion of 1 =~\u001f22(q;0) for small qto quadratic order by using a block\nform of the susceptibility matrix and Schur's complement. The expansion of 1 =~\u001f33(Q;!)\nfor small!proceeds in close analogy.\nSince ~\u001fa3\n0(q;0) = ~\u001f3a\n0(q;0) = 0 fora6= 3, the 3-component is decoupled from all the other\ncomponents for != 0, so that we need to consider only matrix elements with indices 0 ;1;2.\nHence, in this appendix, ~ \u001f, ~\u001f0and \u0000 0denote 3\u00023 matrices formed only by these matrix\nelements. We write ~ \u001f0and \u0000 0in block form\n~\u001f0=0\n@\u0016~\u001f0v\nvy~\u001f22\n01\nA;\u00000=0\n@\u0016\u000000\n0 2U1\nA; (D1)\nwhere\n\u0016~\u001f0=0\n@~\u001f00\n0~\u001f01\n0\n~\u001f10\n0~\u001f11\n01\nA;\u0016\u00000=0\n@\u00002U0\n0 2U1\nA; (D2)\nand\nv=0\n@~\u001f02\n0\n~\u001f12\n01\nA; vy=\u0000\n~\u001f20\n0;~\u001f21\n0\u0001\n: (D3)\nTo compute the RPA susceptibility ~ \u001f= ~\u001f0[1\u0000\u00000~\u001f0]\u00001we need to invert\n13\u0000\u00000~\u001f0=0\n@12\u0000\u0016\u00000\u0016~\u001f0\u0000\u0016\u00000v\n\u00002Uvy1\u00002U~\u001f22\n01\nA: (D4)\nThe inverse of a block matrix\nM=0\n@A B\nC D1\nA (D5)\nwith matrices A;B;C;D can be written as [32]\nM\u00001=0\n@A\u00001+A\u00001BS\u00001CA\u00001\u0000A\u00001BS\u00001\n\u0000S\u00001CA\u00001S\u000011\nA; (D6)\nwhereS=D\u0000CA\u00001Bis the so-called Schur complement. The inverse of 13\u0000\u00000~\u001f0is\nthus given by Eq. (D6) with A=12\u0000\u0016\u00000\u0016~\u001f0,B=\u0000\u0016\u00000v,C=\u00002Uvy, andD= 1\u00002U~\u001f22\n0.\nMultiplying by ~ \u001f0on the left, one obtains\n~\u001f22(q;0) =vy(q;0)\u0001w(q;0) + ~\u001f22\n0(q;0)=S(q;0); (D7)29\nwherew=\u0000A\u00001BS\u00001.\nAconverges to a \fnite 2 \u00022 matrix for q!0,BandCare linear in qfor small q, and\nDis of order q2. Hence, the second term in Eq. (D7) diverges as 1 =q2forq!0, while\nthe \frst term tends to a constant and thus becomes irrelevant. Using ~ \u001f22\n0(0;0) = (2U)\u00001we\nthus obtain\n1\n~\u001f22(q;0)= 2Uh\n1\u00002U~\u001f22\n0(q;0)\u00002Uvy(q;0)\u0002\n12\u0000\u0016\u00000\u0016~\u001f0(0;0)\u0003\u00001\u0016\u00000v(q;0)i\n+O(jqj3):\n(D8)\nDe\fning \u0016\u0000 =\u0002\n12\u0000\u0016\u00000\u0016~\u001f0\u0003\u00001\u0016\u00000, one obtains Eq. (33).\nAppendix E: Bare susceptibilities in the N\u0013 eel state\nSince Qand\u0000Qare equivalent wave vectors in the N\u0013 eel state, the functions gkandhk\nobey the relations gk+Q=gkandhk+Q=\u0000hk, respectively, and ek+Q=ek. Hence, the\nquasi-particle energies E`\nkand the functions F``0(k;q;!) de\fned in Eq. (23) are invariant\nunder a momentum shift by Q, that is,E`\nk+Q=E`\nkandF``0(k+Q;q;!) =F``0(k;q;!).\nThe coherence factors A02\n``0(k;q),A03\n``0(k;q),A12\n``0(k;q), andA13\n``0(k;q) change sign under\na momentum shift k!k+Q. Hence, in the momentum integral in Eq. (22) for the\ncorresponding bare susceptibilities, contributions from kandk+Qcancel, such that\n~\u001f02\n0(q;!) = ~\u001f03\n0(q;!) = ~\u001f12\n0(q;!) = ~\u001f13\n0(q;!) = 0: (E1)\nFrom the obvious relation A22\n``0(k;q) =A33\n``0(k;q+Q) one obtains\n~\u001f22\n0(q;!) = ~\u001f33\n0(q+Q;!): (E2)\nSimilarly,A23\n``0(k;q) =A23\n``0(k;q+Q) yields\n~\u001f23\n0(q;!) = ~\u001f23\n0(q+Q;!): (E3)\n[1] J. 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Dwarkadas3, and M. Pohl1;2\n1DESY , 15738 Zeuthen, Germany\n2Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany\n3University of Chicago, Department of Astronomy& Astrophysics, 5640 S Ellis Ave, ERC 569, Chicago, IL 60637, U.S.A.\nReceived ; accepted\nABSTRACT\nContext. Tycho’s supernova remnant (SNR) is associated with the historical supernova (SN) event SN 1572 of Type Ia. The explosion\noccurred in a relatively clean environment, and was visually observed, providing an age estimate. Therefore it represents an ideal\nastrophysical test-bed for the study of cosmic-ray acceleration and related phenomena. A number of studies suggest that shock accel-\neration with particle feedback and very e \u000ecient magnetic-field amplification combined with Alfv ´enic drift are needed to explain the\nrather soft radio spectrum and the narrow rims observed in X-rays.\nAims. We show that the broadband spectrum of Tycho’s SNR can alternatively be well explained when accounting for stochastic\nacceleration as a secondary process. The re-acceleration of particles in the turbulent region immediately downstream of the shock\nshould be e \u000ecient enough to impact particle spectra over several decades in energy. The so-called Alfv ´enic drift and particle feedback\non the shock structure are not required in this scenario. Additionally, we investigate whether synchrotron losses or magnetic-field\ndamping play a more profound role in the formation of the non-thermal filaments.\nMethods. We solve the full particle transport equation in test-particle mode using hydrodynamic simulations of the SNR plasma flow.\nThe background magnetic field is either computed from the induction equation or follows analytic profiles, depending on the model\nconsidered. Fast-mode waves in the downstream region provide the di \u000busion of particles in momentum space.\nResults. We show that the broadband spectrum of Tycho can be well explained if magnetic-field damping and stochastic re-\nacceleration of particles are taken into account. Although not as e \u000ecient as standard di \u000busive shock acceleration (DSA), stochastic\nacceleration leaves its imprint on the particle spectra, which is especially notable in the emission at radio wavelengths. We find a\nlower limit for the post-shock magnetic-field strength \u0018330\u0016G, implying e \u000ecient amplification even for the magnetic-field damping\nscenario. For the formation of the filaments in the radio range magnetic-field damping is necessary, while the X-ray filaments are\nshaped by both the synchrotron losses and magnetic-field damping.\nKey words. Acceleration of particles – cosmic rays : supernova remnants – ISM\n1. Introduction\nSupernova remnants (SNRs) are among the most exciting astro-\nphysical objects, because they may shed light on a major ques-\ntion of physics: the origin of cosmic rays. An interesting ob-\nject in this respect is the remnant of the historical Type Ia event\nSN 1572, first described by Tycho Brahe. Due to his observation,\nthe age of Tycho’s SNR (SNR G120.1 +1.4, hereafter Tycho) is\naccurately determined to be \u0018440 years. Tycho originated from\na Type Ia supernova (SN) (Krause et al. 2008), and is assumed to\nhave a canonical explosion energy of \u00181051erg. These details,\ntogether with the available broadband spectrum, make Tycho one\nof the best astrophysical laboratories to study particle accelera-\ntion. Nevertheless, several questions still remain unsolved.\nIndeed, despite the success of di \u000busive shock acceleration\n(DSA) theory, it fails to explain the observed radio spectrum S\u0017.\nThe measured radio spectral index \u001b\u00190.65, with S\u0017/\u0017\u0000\u001b\n(Kothes et al. 2006), significantly deviates from the standard\nDSA prediction \u001b\u00190.5. This discrepancy is generally ac-\ncounted for using the concept of Alfv ´enic drift (Bell 1978).\nHence, various authors postulate Alfv ´enic drift only in the up-\nstream region (V ¨olk et al. 2008; Morlino & Caprioli 2012), or\nin the upstream and downstream (Slane et al. 2014) regions of\n?Corresponding author, e-mail: alina.wilhelm@desy.deTycho’s forward shock. The proper motion of cosmic-ray scat-\ntering centers that proceed with Alfv ´en speed is assumed to de-\ncrease the compression ratio felt by the particles and therefore to\ncause a softening of their spectra. To be more precise, the e \u000bec-\ntive compression ratio seen by particles is\nre\u000b=u1+Hc1vA1\nu2+Hc2vA2; (1)\nwhere u1,vA1are the plasma velocity and the Alfv ´en velocity\nin the upstream and u2,vA2in the downstream regions, respec-\ntively. The relative direction of the propagation of the Alfv ´en\nwaves in the upstream (downstream) is reflected by the cross\nhelicity, Hc1(2). In the above global models its value is chosen\nrather freely1, in order to reduce the e \u000bective compression ratio\nand thereby to account for the soft particle spectrum. On closer\ninspection, however, the concept of Alfv ´enic drift as an explana-\ntion for the spectral softening emerges as misleading.\nFirst of all, Vainio & Schlickeiser (1999) performed a de-\ntailed calculation on self-consistent transmission of the Alfv ´en\nwaves through the shock and found that the presence of waves\n1Usually it is taken Hc1=\u00001 and Hc2=0 (V¨olk et al. 2008; Morlino\n& Caprioli 2012) or Hc1=\u00001 and Hc2=1 (Slane et al. 2014).\n1arXiv:2006.04832v1 [astro-ph.HE] 8 Jun 2020A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\nresults in a harder particle spectra than predicted by the stan-\ndard theory (meaning that Alfv ´en waves infer exactly the oppo-\nsite e \u000bect as claimed in V ¨olk et al. (2008), Morlino & Caprioli\n(2012) and Slane et al. (2014)). The reason is that Alfv ´en waves\nthat move in the upstream region in the opposite direction as the\nbackground plasma ( Hc1=\u00001), propagate also predominately\nin the opposite direction in the downstream ( Hc2\u0019\u00001)(Vainio\n& Schlickeiser 1999). Therefore, even despite the modifica-\ntions induced by the strong magnetic-field pressure, the e \u000bec-\ntive compression ratio seen by particles (Eq. 1) exceeds the\nstandard strong-shock value ( re\u000b>4). Investigating the im-\npact of the Alfv ´enic drift within the framework of Vainio &\nSchlickeiser (1999) we find that for the Alfv ´enic Mach num-\nbers, MA\u0011u1=v1;A, in the range 10-13 (as presented in V ¨olk\net al. (2008) and Morlino & Caprioli (2012)) the particle spec-\ntral index for a strong shock results in s\u00191:9 instead of the\ns\u00192:3, required by radio observations of Tycho. Furthermore,\nthe negative downstream helicity, Hc2\u0019\u00001, predicted by Vainio\n& Schlickeiser (1999) is exactly reversed scenario as assumed\nby Slane et al. (2014), who used Hc2=1.\nSecondly, the Alfv ´enic-drift phenomenon in the global mod-\nels of Tycho is often referred to result from the non-resonant\nstreaming instability of cosmic rays (Bell 2004). According to\nBell (2004), the phase speed of the non-resonant modes is neg-\nligible compared to the shock velocity. But in case of Alfv ´enic\ndrift, the Alfv ´en velocity required to account for the Tycho’s ra-\ndio spectra has to be enormous.\nThe following estimation should demonstrate the corre-\nsponding discrepancy. Let us assume that Alfv ´enic drift occurs\nonly in the upstream region ( Hc2=0 and Hc1=\u00001), as in V ¨olk\net al. (2008) and Morlino & Caprioli (2012), although it contra-\ndicts the findings of Vainio & Schlickeiser (1999). In this case,\nwe can rearrange Eq. 1 to\nMA=(1\u0000re\u000b=rsh)\u00001; (2)\nwhere MA\u0011u1=v1;Ais the Alfv ´enic Mach number and rsh\u0011\nu1=u2is the gas compression ratio of the shock. The sub-shock\ncompression ratio in V ¨olk et al. (2008) and Morlino & Caprioli\n(2012) is in the range rsh=3:7\u00003:9, even if the non-linear\ne\u000bects of the DSA are included. The e \u000bective compression\nratio, seen by particles, required by the radio observations is\nre\u000b\u00193:3. Inserting these values into Eq. 2 provides a relatively\nlow Alfv ´enic Mach number, MA=6\u00009, making clear that the\nAlfv ´en speed exhibits a significant fraction of the shock veloc-\nity. Thus, to explain Tycho’s radio data with Alfv ´enic drift, the\nAlfv ´en phase speed has to attain 11% \u000016% of the shock ve-\nlocity. Obviously this value is in conflict with the phase speed\nof the non-resonant mode, v\u001e\u00190, as described by Bell (2004).\nTherefore, Alfv ´enic drift in the models for Tycho cannot be as-\nsociated with the non-resonant streaming instability.\nIt is important to note here that V ¨olk et al. (2008) and\nMorlino & Caprioli (2012) applied in their modeling somewhat\nsmaller magnetic-field values than required to best match the\nradio data with Alfv ´enic drift. The post-shock magnetic fields\nof\u0018300\u0016G (Morlino & Caprioli 2012) and \u0018400\u0016G (V ¨olk\net al. 2008) provide su \u000ecient flux in the radio range, but fit the\nspectral shape of the observed data only moderately well. The\nAlfv ´en velocity in these models (with corresponding Alfv ´enic\nMach numbers of MA\u001913 and MA\u001910) is still\u001810% of\nthe shock velocity. The phase speed of the non-resonant mode is\nmuch less than that and can not support Alfv ´enic drift, of course.\nA post-shock magnetic field above \u0018300\u0016G, when com-\nbined with the relatively low ambient density of 0 :3\u00000:4 cm\u00003(V¨olk et al. 2008; Morlino & Caprioli 2012), becomes dynami-\ncally important. The corresponding magnetic-field pressure will\na\u000bect the shock compression ratio, which results in rsh<3:9.\nAside from the work of Morlino & Caprioli (2012), this e \u000bect\nhas been neglected in global models for Tycho.\nThe above arguments illustrate that the Alfv ´enic-drift con-\ncept is problematic to explain Tycho’s soft radio spectrum con-\nsistently. An alternative way to explain the softening of the par-\nticle spectra in collisionless shocks is accounting for neutral hy-\ndrogen in the surrounding medium, first proposed by Blasi et al.\n(2012). Analytic calculations from Ohira (2012) plus later simu-\nlations (Ohira 2016) show that neutrals can leak from the down-\nstream into the upstream region and modify the shock struc-\nture. This results in a steeper particle spectrum as produced by\nstandard DSA. Morlino & Blasi (2016) build on that idea to\nmodel the rather soft \r-ray spectrum of Tycho. However, the\nleakage of neutral particles is significant for shock velocities\nVsh<3000 km s\u00001(Ohira 2012), which is considerably below\nthe value ascertained for Tycho. Morlino & Blasi (2016) ar-\ngued that certain regions of Tycho can feature slower shocks that\npropagate into dense, partially neutral material. The regions with\nslower shock velocities would have to provide nearly all of the\nobserved overall emission, while contributions from the regions\nwith the fast shock velocities would have to be weak, otherwise\nthe integrated emission would reflect the hard spectra expected\nfor fast shocks in an ionized medium.\nIn this work we suggest a new approach: besides standard\nDSA, we consider an additional acceleration process, namely\nthe stochastic re-acceleration of particles in the immediate post-\nshock region of the SNR. It has been shown that fast-mode\nwaves that survive the transit-time damping by the background\nplasma are e \u000ecient modes to accelerate charged particles via\ncyclotron resonance (Yan & Lazarian 2002; Liu et al. 2008).\nPohl et al. (2015) demonstrated that particles may be stochas-\ntically re-accelerated by fast-mode turbulence, which occurs in\nthe downstream region, after escaping from the forward shock.\nIn this work we build on that idea and include it into a detailed\nmodeling of Tycho.\nFast-mode turbulence may arise behind the shock from ve-\nlocity fluctuations of the plasma flow, via e.g shock rippling\n(Giacalone & Jokipii 2007), and build a thin turbulent region\nbehind the blast wave. The width of this zone is regulated by the\nenergy transfer from the background plasma into turbulence as\nwell as damping induced by the re-acceleration of cosmic rays.\nWe show that fast-mode turbulence that carries a few percent\nof the energy density of the background plasma in the down-\nstream region is strong enough to modify the spectrum of parti-\ncles that have already been accelerated by the shock. Thus, in our\ntreatment stochastic acceleration and DSA operate together and\nproduce a particle spectrum consistent with the observed radio\nspectral index. An additional advantage of our approach is that it\nis fully time-dependent.2We solve the time-dependent transport\nequation for cosmic rays that contains a DSA term and di \u000busion\nin momentum space, and is coupled to hydrodynamic simula-\ntions.\nAnother interesting question regarding Tycho is to what ex-\ntent the magnetic field is amplified inside the remnant. A rela-\ntively high post-shock magnetic field, 300 \u0000400\u0016G, is postulated\nby several global models (V ¨olk et al. 2008; Morlino & Caprioli\n2Time-dependent hydrodynamics for Tycho was already used by\nSlane et al. (2014). In their approach, however, a steady-state solution\nfor DSA was used to numerically inject a specific cosmic-ray spectrum\nat the location of the forward shock.\n2A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\n2012). One of the reasons is the afore-mentioned Alfv ´enic drift,\nwhich demands rather large magnetic-field values to account\nfor the radio spectrum. Since we do not postulate any Alfv ´enic\ndrift in our model, our approach of inferring the magnetic-field\nstrength is an alternative to that of previous works on Tycho.\nA major argument for a high magnetic field in the down-\nstream of Tycho are the observed narrow non-thermal X-ray fil-\naments (Hwang et al. 2002; Parizot et al. 2006; Helder et al.\n2012). Since electrons can only propagate for a finite distance\nbefore they lose their energy due to synchrotron radiation, the\nrim widths may reflect the magnetic-field strength in the imme-\ndiate downstream of the SNR. An alternative scenario is pro-\nvided by damping of the turbulent magnetic field in the inte-\nrior of the remnant (Pohl et al. 2005; Ressler et al. 2014; Tran\net al. 2015), in which the narrowness of the non-thermal rims\ncan be explained by the damping of the turbulently amplified\nmagnetic field. For Tycho, a distinction between these scenar-\nios by means of the energy-dependence of X-ray filaments is\ndi\u000ecult (Rettig & Pohl 2012; Tran et al. 2015). Magnetic-field\ndamping is widely considered as the scenario that allows for a\nweak magnetic-field strength inside SNRs. Nevertheless, Tran\net al. (2015) find that in either case the minimum downstream\nmagnetic-field value inferred from the Tycho’s non-thermal fil-\naments is at least \u001820\u0016G. Assuming that the electron accel-\neration is limited by the age of the remnant, the work from\nNuS tar collaboration (Lopez et al. 2015) estimates \u001830\u0016G for\nthe downstream magnetic field. However, the majority of studies\ncited above favor the loss-limited interpretation for particle ac-\nceleration in Tycho. Furthermore, the most realistic limit is ob-\ntained from an analysis of the entire spectral energy distribution\n(SED), which provides the minimal post-shock magnetic field\nvalue of\u001880\u0016G (Acciari et al. 2011). Attempts to simultane-\nously fit the radio and the \r-ray data infer that, any weaker mag-\nnetic field would cause an overproduction of \r-ray photons gen-\nerated via inverse-Compton scattering. Therefore, the question\nabout the magnetic field value is automatically tied to the ques-\ntion of whether Tycho’s \r-ray emission has a predominately lep-\ntonic or hadronic origin. The hadronic scenario has been strongly\nfavored in the literature (Morlino & Caprioli 2012; Zhang et al.\n2013; Berezhko et al. 2013; Caragiulo & Di Venere 2014; Slane\net al. 2014), as opposed to a leptonic model (Atoyan & Dermer\n2012).\nFor our modeling, we start from the minimal magnetic field\ncompatible with the entire SED. The evolution of the SNR,\nwhich is computed using hydrodynamical simulations, occurs in\na medium with a constant density. We explicitly model the accel-\neration of each particle species in the test-particle limit, taking\nshock acceleration as well as stochastic acceleration in the down-\nstream region into account, with both acceleration processes be-\ning time-dependent. We explicitly model advection and di \u000busion\nof cosmic rays and take synchrotron losses for electrons into ac-\ncount. Furthermore, we consider the non-thermal radio and X-\nray filaments and investigate whether they arise from extensive\nsynchrotron losses or magnetic-field damping. For the study of\nthe X-ray filaments it is especially important to include di \u000busion\nof the particles, otherwise the distance that they propagate and\nthus the rim width would be underestimated.\nThis paper is organized as follows: Section 2 describes\nour method: hydrodynamical picture, magnetic profiles as well\nas particle acceleration via DSA and Fermi II processes. In\nSection 3.1 we examine the model with the minimal ampli-\nfied magnetic field compatible with the \r-ray observations. We\njustify the necessity for magnetic field damping that we intro-\nduce in Section 3.2. Therein we discuss our favorite model forTycho, and deduce a new theoretical minimum for the magnetic-\nfield strength based on investigation of the multi-frequency spec-\ntrum along with non-thermal filaments. The \r-ray spectrum of\nthe resulting model comprises both hadronic and leptonic com-\nponents. In Section 3.4 we discuss the potential for a purely\nhadronic scenario with a strongly amplified magnetic field.\n2. Modeling\n2.1. Hydrodynamics\nTo accurately calculate the acceleration of particles to relativis-\ntic energies and the high-energy emission from the remnant, we\nstart with an evolutionary model of the remnant. This model pro-\nvides the dynamical and kinematic properties of the remnant as\na function of time. We model the hydrodynamic evolution of the\nSN shock wave in Tycho using the VH-1 code, a 1, 2 and 3 di-\nmensional finite-di \u000berence code that solves the hydrodynamic\nequations using the Piecewise Parabolic Method of Colella &\nWoodward (1984). Dwarkadas & Chevalier (1998) showed that\nthe ejecta structure of Type-Ia SNRs such as Tycho could be\nbest approximated by an exponential density profile. Using this\nprofile, the parameters needed to model the SNR are the ex-\nplosion energy of the SN, the ejected mass, and the density of\nthe medium into which the supernova is expanding. We use the\ncanonical value of 1051ergs for the energy of the explosion. Type\nIa’s are presumed to arise from the thermonuclear deflagration\nand detonation of a white dwarf, so we assume an ejecta mass\nof 1.4 M\f, appropriate for a Chandrashekhar-mass white dwarf.\nThe remaining parameter necessary to model the evolution is the\ndensity of the ambient medium.\nAlthough the density around Tycho varies (Williams et al.\n2013), as expected for such an extended structure, the remnant\nappears to expand in a clean environment without any large in-\nhomogeneities, such as molecular clouds (Tian & Leahy 2011).\nTherefore, an explosion in a medium with a constant density pro-\nvides a reasonable description for the dynamics of the remnant.\nThe average ambient density varies in the literature, depending\non how it was measured. X-ray measurements of the expansion\nrate suggest an ambient density of 0.2-0.6 cm\u00003(Hughes 2000).\nLater observations infer an upper limit of 0.2 cm\u00003(Katsuda\net al. 2010). X-ray observations of Cassam-Chena ¨ı et al. (2007)\nreveal a lack of thermal emission in the post-shock region of\nTycho, inferring an ambient density below 0.6 cm\u00003. A low den-\nsity of 0.2 cm\u00003around Tycho is obtained by Williams et al.\n(2013), who determined the post-shock temperature from mid-\ninfrared emission of the remnant. On another hand, e \u000ecient par-\nticle acceleration in SNRs can reduce the downstream temper-\nature of the plasma (O’C. Drury et al. 2009), leading to a sup-\npression of thermal emission and hence an underestimation of\nthe ambient density. Higher values for the ambient density are\nadditionally supported by Dwarkadas & Chevalier (1998), who\nfound that densities in the range of 0.6-1.1 cm\u00003better matched\nthe observations of Tycho. Furthermore, Kozlova & Blinnikov\n(2018) and Badenes et al. (2006) favor a delayed detonation\nmodel with\u00181:0 cm\u00003for the X-ray morphology of Tycho. The\ndensity uncertainty of Tycho (0.2-1.0 cm\u00003) is likewise reflected\nin the uncertainty in the distance to the remnant (see Hayato et al.\n(2010) for a review and Tian & Leahy (2011)), since both quan-\ntities are interdependent. For our modeling we choose a value\nfor the hydrogen number density of nH=0.6 cm\u00003, which gives\na good fit to the observed shock radii and velocities.\nExplosion energy, ejecta mass and ambient density, together\nwith the exponential density profile, form the suite of param-\n3A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\n50 100 150 200 250 300 350 400\nAge[yr]1.02.03.04.05.06.07.08.010.020.0\nShock speed \n Vsh[103kms1]\nShock radius \n Rsh[pc]Shock temperature \n T[109K]\nFig. 1. The shock speed (blue solid line), radius (green dashed\nline) and temperature (red dotted line) as functions of time.\neters necessary to model the complete hydrodynamical evolu-\ntion of Tycho. Our simulations are spherically symmetric, and\nare similar to those described in Dwarkadas & Chevalier (1998)\nand Telezhinsky et al. (2012). The evolution of the remnant into\nthe surrounding medium gives rise to a double-shocked struc-\nture consisting of a forward shock expanding into the surround-\ning medium, and a reverse shock that propagates back into the\nejecta. The two are separated by a contact discontinuity that di-\nvides the shocked ejecta from the shocked surrounding medium.\nThe time-evolution of the shock speed, radius and temperature\nderived from our simulations is depicted in Fig. 1. At 440 years\nour simulations provide a forward shock radius Rsh\u00193:5 pc,\nwhich implies a distance to the remnant, d\u00192:9 kpc. The ve-\nlocity of the forward shock yields Vsh\u00194100 km s\u00001. The posi-\ntion of the reverse shock, \u00180:69Rsh, is in good agreement with\nmeasurements of Warren et al. (2005). By contrast, the position\nof the contact discontinuity (CD), RCD\u00190:78Rsh, falls below\nthe value identified by Warren et al. (2005), who interpreted the\ncloseness of the CD to the blast wave as evidence for the e \u000ecient\nback-reaction of cosmic rays. However, global models that in-\ncorporate non-linear DSA (NLDSA) e \u000bects (Morlino & Caprioli\n2012; Slane et al. 2014) fail to reproduce the CD position for\nTycho. The discrepancy for the CD position can be attributed\nto the decelerating CD being unstable to the Rayleigh-Taylor in-\nstability. Two-dimensional hydrodynamical simulations with the\nexponential profile show that Rayleigh-Taylor structures can ex-\ntend almost halfway from the CD to the outer shock (Dwarkadas\n2000; Wang & Chevalier 2001). Furthermore, Orlando et al.\n(2012) have shown that Rayleigh-Taylor instabilities and ejecta\nfingers that extend far beyond the CD can misleadingly suggest\nthat the CD is further out than its actual location. Therefore, we\nconclude that the position of the CD obtained in our model is\nquite reasonable.\nAccording to our simulations, at the age of 440 years the rem-\nnant accumulated\u00183:8M\fof ambient gas, indicating that in our\nmodeling Tycho is in the transition between ejecta-dominated\nand Sedov-Taylor stages. The total thermal energy in the rem-\nnant at 440 years is Eth\u00195:6\u00021050erg.\nThe shock profiles and velocity distribution from the sim-\nulation are used in the calculation for the particle acceleration,\ndescribed in Section 2.3.2.2. Magnetic field\nIn this work, the canonical value for the magnetic-field strength\nof the interstellar medium (ISM), BISM=5\u0016G, in the far up-\nstream is used. Its exact value is, however, insignificant for our\nmodeling, since the magnetic field is amplified in the vicinity of\nthe shock, e.g. due to streaming instabilities (Bell 2004; Lucek\n& Bell 2000) or turbulent dynamos (Giacalone & Jokipii 2007).\nThus, the amplified magnetic field represents a physically rele-\nvant parameter. The exact treatment of the magnetic-field ampli-\nfication is beyond the scope of this work. We adopt instead for\nthe magnitude of the magnetic-field strength in the immediate\nupstream region B1=\u000bBISM, where\u000b, a generic amplification\nfactor, is a free parameter in our model. Following the ansatz\nused by Zirakashvili & Ptuskin (2008) and Brose et al. (2016)\nthe magnetic-field profile in the precursor is assumed to drop ex-\nponentially to the interstellar field, BISM, at the distance of 5%\nof the shock:\nB(r)=8>><>>:\u000bBISM\u0001exp\u0010\u0000(r=Rsh\u00001)\n0:05\u0001ln(\u000b)\u0011\nforRsh\u0014r\u00141:05\u0001Rsh\nBISM forr\u00151:05\u0001Rsh:\n(3)\nWe stress that the magnetic-precursor length scale is not related\nto a free-escape boundary, which is usually introduced in the\nglobal modeling of SNRs (V ¨olk et al. 2008; Morlino & Caprioli\n2012; Slane et al. 2014). The precursor merely reflects typical\ncharacteristics of the spatial profile of the amplified magnetic\nfield in the upstream region. In contrast to the previous mod-\nels, our approach does not need an escape boundary, since our\nsystem is large enough (ca. 65 shock radii) to retain all injected\nparticles in the simulation. The pre-shock magnetic field is as-\nsumed to be isotropic, i.e its individual components are equal in\ntheir magnitudes (for more details see Telezhinsky et al. (2013)).\nShock compression for the strong unmodified shock yields the\nimmediate downstream value, B2=p\n11B1.\nWe consider two scenarios for the magnetic-field distribution in-\nside the SNR. In the first case, the immediate downstream value\nof the magnetic field is transported inside the SNR with the\nplasma flow and evolves according to the induction equation for\nideal magnetohydrodynamics (MHD) (Telezhinsky et al. 2013):\n@B\n@t=r\u0002(u\u0002B): (4)\nHere Bis the magnetic-field vector and uthe plasma velocity,\nobtained from hydrodynamic simulations. Eq. 4 accounts for ad-\nvection, stretching and compression of the field, implying that\nmagnetic flux is conserved during the entire evolution of the\nSNR.\nFor the second case, we assume that after being amplified in\nthe upstream region and compressed at the shock, the magnetic\nfield decays due to the damping of magnetic turbulence in the\ndownstream region. Magnetic-field damping is one of the key\nprocesses in astrophysical plasmas (Kulsrud & Cesarsky 1971;\nThrelfall et al. 2011), and is expected to operate in SNRs, where\nit dissipates turbulently amplified magnetic field (Pohl et al.\n2005). We do not know the turbulence mode that is most relevant\nfor magnetic-field amplification in SNRs, and hence the particu-\nlars of damping are beyond the scope of this paper. Instead, we\nadopt a simple parametrization for the magnetic profile in the\ndownstream region:\nB(r)=B0+(B2\u0000B0)\u0001exp \u0000(Rsh\u0000r)\nld!\n; (5)\n4A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\nwhere ldis the characteristic damping length of magnetic turbu-\nlence. The residual field level, B0, is the limit value, to which the\nmagnetic-field tends in the far downstream, behind the CD and\nthe reverse shock of the SNR. In contrast to the previous case, the\nmagnetic flux is not conserved in the scenario described by Eq. 5.\nFar inside the remnant the advection of magnetic field has to be-\ncome important as the magnetic field approaches the constant\nvalue B0. This inner field is, however, physically non-relevant.\nFirst of all, we do not include the acceleration of particles at the\nreverse shock, since we find the radiative contribution of those\nparticles to be negligible. Furthermore, the bulk of the acceler-\nated particles resides in the vicinity of the forward shock and\nhence do not experience the constant magnetic field, B0. Most\nof the particles cannot cross the CD, which separates the swept-\nup material from the ejecta, since their dynamics is governed by\nadvection. Only a marginal fraction of the very-high energetic\nparticles overcome the CD via di \u000busive transport. Their radia-\ntive contribution to the synchrotron spectrum does not exceed\n1% of the overall emission, and thus is negligible for the global\nSED modeling. Therefore, we can omit the advection of mag-\nnetic field for the second scenario that includes magnetic-field\ndamping.\nThe separate treatment of hydrodynamics and magnetic field\ndoes not allow us to include dynamic feedback from magnetic\npressure. Therefore, we investigate the range where the impact\nof magnetic field on the shock structure remains negligible.\nWe analytically solve the classical MHD-equations for a plane-\nperpendicular shock in steady-state, assume frozen-in plasma,\nand derive the dependency of the shock compression ratio on\nthe magnetic-field strength. Apart from the magnetic field the\nresulting gas compression ratio depends mainly on shock speed\nand upstream density, which in our model are Vsh\u00194100 km s\u00001\n(at 440 years) and nH=0:6 cm\u00001(see Sec. 2.1). As the exten-\nsive magnetic-field pressure reduces the compression ratio of the\nshock, it consequentially softens the particle spectra. Hence, the\ncorresponding particle spectral index deviates from the canoni-\ncal DSA solution ( s=2:0) by \u0001s, resulting in s=2+ \u0001s. The\nresulting radio spectral index, \u001b=0:5+\u0001\u001b, is accordingly mod-\nified by the value \u0001\u001b= \u0001s=2. We choose \u0001\u001b=0:01 (and thus\n\u0001s=0:02) as the limit within which the magnetic-field pressure\ncan be neglected, since the corresponding changes are hardly\nvisible in the radio spectrum when compared to the observed\ndata. We find for the sonic upstream Mach number of M\u00193500,\nwhich corresponds to the hydrodynamic quantities we use in this\nwork for Tycho, that the magnetic field is dynamically unimpor-\ntant for Alfv ´enic Mach numbers above MA;c\u001917. For lower\nAlfv ´enic Mach numbers the deviation from the classical DSA-\nsolution exceeds the values of s=2:02 and\u001b=0:51 and hence\ncannot be neglected. Taking into account that the magnetic-field\npressure is exerted only by the tangential components, we find\nthe magnetic-field limits B1;max\u0019120\u0016G for the upstream and\nB2;max\u0019400\u0016G for the downstream regions, respectively. Any\nhigher magnetic field would e \u000eciently lower the shock compres-\nsion ratio and produce particle spectra with s>2:02 and radio\nspectra with \u001b>0:51, and thus demand an MHD treatment.\n2.3. Particle acceleration\nTo determine the evolution of the particle number density,\nN(r;p;t), we numerically solve the time-dependent transport\nequation for cosmic raysr(DrrN\u0000uN)\u0000@\n@p \nN˙p\u0000ru\n3N p!\n+@\n@p \np2Dp@\n@pN\np2!\n+Q=@N\n@t: (6)\nHere Dr=\u0018c rg=3 is the spatial di \u000busion coe \u000ecient, with the\nspeed of light cand the Larmor radius rg. The parameter \u0018\nis the ratio between the spatial di \u000busion coe \u000ecient and that\nfor Bohm di \u000busion, sometimes referred as the gyrofactor. The\nLarmor radius and hence the spatial di \u000busion coe \u000ecient are cal-\nculated using the local magnetic-field strength. Since in our ap-\nproach the magnetic field is spatially variable (see Sec. 2.2),\nthe spatial di \u000busion coe \u000ecient also varies with position. The\nsynchrotron losses for electrons are included in Eq. 6 via ˙ p=\n(4e4B2)=(9c6m4\ne)p2, with the elementary charge eand the elec-\ntron rest mass me. The plasma velocity, u, is obtained from the\nhydrodynamic simulations.\nThe injection of particles is a complex issue, which is not fully\nunderstood. For simplicity, we use the thermal leakage injection\nmodel as presented in Blasi et al. (2005). The source term for\nelectrons and protons is given by\nQi=\u0011in1;iVsh\u000e(r\u0000Rsh)\u000e(p\u0000pinj;i); (7)\nwhere index idenotes the particle species (electrons and pro-\ntons),\u0011ithe corresponding injection e \u000eciency and n1;ithe parti-\ncle number density in the upstream region. The particles in our\napproach are mono-energetically injected. The associated injec-\ntion momentum is multiple of their thermal momentum:\npinj;i=4:45pth;i\u00114:45p\n2mikBT; (8)\nwhere kBis the Boltzmann constant and Tis the temperature of\nthe plasma.\nElectrons and ions are not in equilibrium at collisionless\nshocks. In fact, the current Rankine-Hugoniot ion temperature\nis about 30 keV , whereas the observed electron temperature in\nthe post-shock medium is on the order of a keV after years of\nresidence in the downstream region. The thermal leakage model\nis based on the requirement that particles see the shock as a sharp\ndiscontinuity and hence are injected only if their Larmor radius\nexceeds the width of the shock wave. This condition requires\nthat electrons need to be pre-accelerated to around 100 MeV to\nparticipate in DSA. Particle-in-cell simulations provide evidence\nthat thermal electrons can indeed be pre-accelerated by shock-\nsurface and shock-drift acceleration (Matsumoto et al. 2017;\nBohdan et al. 2017). Tests demonstrate (Katou & Amano 2019)\nthat this process yields an electron spectra shaped as a power low\nwith a spectral index in the range \u0018(1:5\u00005:5). The exact value of\nthe spectral index depends on the internal structure of the shock\ntransition region. We approximate the two-step energization of\nelectrons by DSA at all momenta and electron injection at very\nlow momentum, which intends to replace an advanced, compu-\ntationally far more expensive treatment of the pre-acceleration.\nThus, the same particle temperature set for Eq. 8 is supposed to\nmimic the pre-acceleration process of electrons. Although, we\ndo not know the exact value of the spectral index provided by\nthe shock-drift acceleration, it only marginally a \u000bects our final\nspectra, as it was shown in Pohl et al. (2015)(Fig. 4).\nSince the main goal of this work is to explain the broad SED\nof Tycho, we can neglect the thermal and supra-thermal protons.\nIndeed, as the hadronic \r-rays production starts from the thresh-\nold energy\u00180.3 GeV , the protons with momenta below pinj;pare\n5A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\ninsignificant for the final photon spectrum. In contrast, the elec-\ntrons at lower energies play an important role, as the correspond-\ning synchrotron emission is visible in the radio range.\nBesides the full time dependence of the particle accelera-\ntion and transport, the novel aspect in our work is the stochastic\nacceleration of particles presented in Eq. 6 by the momentum-\ndi\u000busion coe \u000ecient Dp. Pohl et al. (2015) explicitly derived\nthe momentum-di \u000busion coe \u000ecient for the fast-mode waves and\ndemonstrated that for low-energy particles the acceleration time\nis independent of momentum and can be of the order of a few\nyears. At higher energy the process is expected to become less\ne\u000ecient. Stochastic acceleration is an important damping pro-\ncess of fast-mode waves (cf. Thornbury & Drury 2014), and so\nit is e \u000ecient only in a thin region behind the forward shock with\nthe thickness, Lfm. A useful parametrization of the momentum\ndi\u000busion coe \u000ecient, Dp, is then (Pohl et al. 2015)\nDp(r;p)=8>><>>:0 for r<(Rsh\u0000Lfm)\np2\n\u001caccf(p) for ( Rsh\u0000Lfm)\u0014r\u0014Rsh:(9)\nHere,\u001caccdenotes the acceleration time at small momenta, and\nf(p) approximates the loss in acceleration e \u000eciency at higher\nenergies as a power law,\nf(p)=8>><>>:1 for p\u0014p0\u0010p\np0\u0011\u0000mforp\u0015p0:(10)\nThe critical momentum p0and the power-low index mare free\nparameters of our model for now. It is important to note here that\nthe energy in fast-mode waves has mostly a kinetic character. As\nthe magnetic component of the fast-mode turbulence in the typ-\nical post-shock plasma is weak, it cannot amplify the magnetic\nfield su \u000eciently. Therefore, in our approach other type of waves,\nsuch as streaming instabilities (Bell 2004; Lucek & Bell 2000)\nor turbulent dynamos (Giacalone & Jokipii 2007) are required to\nprovide the magnetic-field amplification at the shock, which is\nan important scale factor for scattering (see Sec. 2.2). There is\nnot necessarily a simple scaling between momentum and spatial\ndi\u000busion (e.g. Yan & Lazarian 2004, Eq.14). In addition, dif-\nferent types of turbulence may be responsible for the di \u000busive\ntransport and stochastic re-acceleration (e.g. Shalchi 2009, page\n23). Therefore, the spatial and the momentum di \u000busion coe \u000e-\ncients are independent in our model.\nThe energy in the fast-mode turbulence that occurs in the\npost-shock region can be primarily provided by the background\nplasma. To simplify matters, in our approach the energy density\nin the fast-mode waves scales with the thermal energy density of\nthe post-shock background plasma\nUfm=\u000fUth: (11)\nHere\u000fis the energy-conversion factor, which is assumed to be of\nthe order of a few percent. The minor value of \u000fprovides that the\nenergy transfer from the main plasma flow can be neglected in\nour hydrodynamic simulations. For a strong shock expanding in\na cold plasma the downstream thermal energy is approximately\nUth\u00199\n8\u001aISMV2\nsh; (12)\nwhere\u001aISMdenotes the ambient gas density.\nThe acceleration time for the fast-mode waves, derived\nin Pohl et al. (2015), is given by\n\u001cacc\u0019(0:63 yr) Uth\n10Ufm!\n: (13)If energy density of the fast-mode waves scales with the thermal\npost-shock energy density of the plasma, as reflected by Eq. 11,\nEq. 13 simplifies to\n\u001cacc\u0019(6:3 yr)\u0012\u000f\n0:01\u0013\u00001\n; (14)\nproviding that the acceleration time scale depends only on the\nenergy fraction that is transferred to the fast-mode waves.\nThe re-acceleration of cosmic rays will inevitably damp the\nfast-mode waves and a \u000bect the width of the turbulent region,\nLfm. The latter quantity can be estimated by equating the en-\nergy density of the fast-mode waves, Ufm, and the total energy\nper volume that went into cosmic rays. Its value can be obtained\nfrom the energy transfer rate from waves to particles, ˙Etr, and\nthe time period that particles spend in the turbulent region inter-\nacting with fast-mode waves, \u0001t, providing\nUfm=˙Etr\u0001\u0001t=Ucr\n\u001cacc\u0001Lfm\nu2: (15)\nHere Ucris the cosmic-ray energy density in the immedi-\nate post-shock region induced by the stochastic re-acceleration.\nCombining Eqs. 13 and 15 we finally obtain\nLfm\u0019(5\u00021013cm) Vsh\n1000 km=s! Uth\nUcr!\n: (16)\nSummarizing, the thickness of the re-acceleration region is lim-\nited by damping of the turbulence caused by the re-acceleration\nof particles. Therefore, in our method Ucrfrom Eq. 16 is cou-\npled to the intermediate results from Eq. 6 for the immediate\npost-shock area. Calculating the energy density of cosmic rays,\nUcr, we do not take the contribution from electrons into account,\nsince in our model their energy is negligible compared to that of\nprotons.\nThe physically essential quantity for stochastic re-\nacceleration is the amount of energy available for it, here\nin the form of fast-mode waves and described by the parameter\n\u000f. The size of the turbulence region, Lfm, follows from the\nacceleration time scale, \u001cacc, and only their ratio is relevant for\nthe resulting particle spectra. The reason is that re-acceleration,\nonce e \u000ecient, becomes the main damping mechanism for the\nwaves and hence ceases when the energy supply is exhausted.\nMore details on general discussion on stochastic re-acceleration\nin SNRs can be found in Pohl et al. (2015).\n2.4. Radiative processes\nTo account for the entire SED from Tycho, we calculate syn-\nchrotron radiation of electrons and \r-ray radiation resulting from\nleptonic and hadronic interactions. The synchrotron emission is\ncalculated following Blumenthal & Gould (1970). Care must be\nexercised to properly account for magnetic field fluctuations in\ncalculating the synchrotron emissivity function. The standard\ncalculation, although not applicable to the turbulent magnetic\nfield, assumes by default a delta function for the probability dis-\ntribution of local magnetic-field amplitudes. For the turbulent\nmagnetic-field the amplitudes distribution is dispersed compared\nto the standard case. Its exact form is unknown. In the litera-\nture, however, one finds modified emissivities for exponential\n(Zirakashvili & Aharonian 2010) and power-law (Kelner et al.\n2013) distributions. In this work we use the Gaussian distribution\nof magnetic-field amplitudes and the corresponding synchrotron\nemissivity function (Pohl et al. 2015). Given the low gas density,\n6A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\nnon-thermal bremsstrahlung (Blumenthal & Gould 1970) yields\ntoo low a flux for it to be relevant, and so leptonic gamma-ray\nemission is solely provided by inverse Compton scattering, for\nwhich we consider the microwave background as the target pho-\nton field (Blumenthal & Gould 1970; Sturner et al. 1997). Tycho\nshows evidence for infrared emission (Douvion et al. 2001). Its\ncontribution for the overall inverse-Compton spectrum, investi-\ngated by Acciari et al. (2011) was, however, found to be negli-\ngible. Hadronic \r-rays are the result of decays of neutral pions\nand other secondaries produced in interactions of cosmic rays\nwith nuclei of the ISM. To calculate its spectrum we follow the\nmethod from Huang et al. (2007).\n3. Results\nThe method to solve the transport equation (Eq. 6) separately for\nelectrons and protons is described in Telezhinsky et al. (2012).\nWe emphasize that, in contrast to previous attempts to model\nTycho, we follow the full temporal evolution of the remnant\nstarting at an age of 25 years. For conciseness, however, we show\nand discuss only results for the current age of Tycho. We find that\nthe reverse-shock contribution to the particle spectrum in Tycho\nis negligible in the framework of our modeling at an age of 440\nyears (Telezhinsky et al. 2012), in full agreement with Warren\net al. (2005). Therefore we do not discuss it further.\nWe present two models for the multiwavelength emission of\nTycho, which both adequately fit the SED. Model I allows for the\nweakest magnetic field in the immediate downstream region of\nthe shock, B2=150\u0016G, that is compatible with the entire \r-ray\nflux observed with Fermi -LAT in GeV-range and VERITAS in\nTeV-band (Archambault et al. 2017). In Model I, the resulting \r-\nray flux consists of both leptonic and hadronic components. The\nmagnetic profile deeper inside the remnant is determined by ad-\nvection of frozen-in magnetic field and corresponds to the MHD\nsolution for negligible magnetic pressure and energy density. As\nwe shall demonstrate, this model fails to explain radio and X-ray\nintensity profiles.\nTherefore we present a second model, Model II, involving\nmagnetic field damping and derive constraints on the magnetic\nfield in Tycho. Generic technical details can be found in subsec-\ntion 3.1, where we introduce Model I. They also apply to the\nfollowing subsections listing the results for Model II.\n3.1.Model I: Moderate advected magnetic field\nThe e \u000ecacy of stochastic acceleration is fully determined by\nthree parameters: mandp0, and the energy fraction of the plasma\ntransferred to the turbulence, \u000f. The acceleration time scale,\n\u001cacc, and the width of the turbulent region, Lfm, are then self-\nconsistently calculated according to Eqs. 14 and 16. The param-\neter p0provides the critical limit up to which the momentum\ndi\u000busion coe \u000ecient Dpis energy-independent (cf. Eq. 10). It\nshould be derived from exact scattering theory, which is, how-\never, beyond the scope of this work. We tested several values\nofp0and found that for p0&10\u00003mpcthe resulting curvature\nof the radio synchrotron spectrum becomes too strong to remain\nin agreement with radio data. Therefore, in this work we adopt\np0=9:3\u000210\u00004mpc, which corresponds to an electron energy\nof 500 keV . The power-law index, m, may vary up to 25% (to\nstay in agreement with observations) and can be compensated\nby suitable choice of the energy-conversion factor, \u000f, implying\nthat these quantities exhibit degeneracy to a certain degree. In\nthe following calculations we will fix m=0:25 and\u000f=0:027\n(corresponding to 2.7% of the thermal energy density), whichadequately reproduce the observed radio spectrum. The acceler-\nation time scale is determined by the energy-conversion factor,\nwhich for\u000f=0:027 provides \u001cacc\u00192:3 yr.\nThe thickness of the region, in which stochastic re-\nacceleration is operating, is fairly small. For 2.7% of the ther-\nmal energy density of the post-shock plasma transferred to the\nturbulence, it results into Lfm\u00181015\u00001016cm, as illustrated\nin Fig. 2. The width of the re-acceleration region decays with\ntime as the SNR shock slows down and thus the energy density\nin the turbulence decreases. At the age of 440 years the width\nof the turbulent region comprises only \u00183\u000210\u00004Rsh. Still, the\ncontribution to the particle spectra is substantial.\n50 100 150 200 250 300 350 400\nAge[yr]1.0\n0.40.50.60.70.80.92.03.04.05.06.07.08.09.0\nUfm[108ergcm3]\n Lfm[1016cm] \nFig. 2. The energy density in fast-mode waves (red solid line)\nand width of the turbulent re-acceleration region (blue dashed\nline) as functions of time for Model I.\nFig. 3 shows the di \u000berential number density for electrons and\nprotons as obtained from solving Eq. 6. As a result of stochastic\nre-acceleration, the spectra strongly deviate from the canonical\nsolution, N/p\u00002, expected for DSA in the test-particle limit.\nFor comparison, a standard case with stochastic re-acceleration\n-10.0-9.5-9.0-8.5-8.0-7.5-7.0\n-3-2-1012345logp2N[mpc/cm3]\nlogp[m pc]Protonsrescaledby1/8 5\nElectrons\nFig. 3. The number density of protons (red solid) and electrons\n(green dashed) for Model I involving weakest advected magnetic\nfield with B2=150\u0016G. The proton number density is multiplied\nwith the factor Ke=p=1=85.\n7A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\nTable 1. Summary of the model parameters.\nVarying parameters Fixed for both models\nModel B2 B0 ld\u0018 \u0011 e \u0011p nH p0 m\u000f\n(\u0016G) (\u0016G) ( Rsh) (cm\u00003) (10\u00004mpc)\nI 150 - - 10 9 :4\u000210\u000062:4\u000210\u000050.6 9.3 0.25 0.027\nII 330 70 0.01 16 10 :2\u000210\u000062:4\u000210\u000050.6 9.3 0.25 0.027\nB2 magnetic field in the immediate post-shock region \u0011p injection e \u000eciency of protons\nB0 residual level of the magnetic field nH ambient hydrogen number density\nld damping scale of the downstream magnetic field p0 critical momentum of the momentum-di \u000busion coe \u000ecient\n\u0018 spatial di \u000busion coe \u000ecient parameter (gyrofactor) m power-low index of the momentum-di \u000busion coe \u000ecient\n\u0011e injection e \u000eciency of electrons \u000f energy-conversion factor for turbulence\n-10.0-9.5-9.0-8.5-8.0-7.5-7.0\n-3-2-1012345logp2N[mpc/cm3]\nlogp[m pc]Protonsrescaledby1/8 5\nElectrons\nFig. 4. The number density of protons (red solid) and elec-\ntrons (green dashed) for Model I with turned o \u000bstochastic re-\nacceleration. A comparison with Fig. 3 illustrates the quantita-\ntive e \u000bect of the second order acceleration in cosmic-ray spectra.\nturned o \u000bbut the same remaining parameters is depicted in\nFig. 4. To be noted in Fig. 3 are distinct bumps for both parti-\ncle species at low energies with concave tails that extend up to\nthe maximum momenta of the spectra, \u0018103mpc. Deviations be-\ntween electron and proton spectra at lower momenta results from\nthe di \u000berent injection criteria for the particles. Since we use the\nthermal leakage model, electrons and protons feature injection\nmomenta di \u000bering by a factor\u001840 due to pinj;i/pmi, as seen\nfrom Eq. 8. In other words, we do not explicitly treat electron\nacceleration by, e.g., stochastic shock drift acceleration below\n100 MeV (Katou & Amano 2019) and subsume the entire accel-\neration from suprathermal to very high energies under DSA, as\nmentioned above. Electrons and protons at energies between the\nthermal peak and the injection threshold a factor 4.45 higher in\nmomentum are considered part of the thermal bulk and ignored.\nThe electrons between their injection threshold and the proton\ninjection momentum are very numerous and with stochastic re-\nacceleration can form a larger bump in the particle spectrum than\nwould be observed for protons.\nThe total energy that went into electrons, Etot;e\u00196:8\u0002\n1047erg, is marginal compared to that of protons, Etot;p\u0019\n2:7\u00021049erg. Reasons for this are the small rest mass of elec-\ntrons and di \u000berent injection e \u000eciencies\u0011e=9:4\u000210\u00006and\n\u0011p=2:4\u000210\u00005, which are chosen to fit the entire SED. The elec-\ntron injection e \u000eciency simultaneously accounts for the maxi-\nmum IC peak consistent with the \r-ray data and su \u000ecient ra-\ndio emission for Bd=150\u0016G (amplification factor \u000b\u00199),\nwhile injection of protons provides an adequate hadronic con-\ntribution in the GeV range. The injection e \u000eciencies determine\n-10.0-9.5-9.0-8.5-8.0\n3.43.63.8 44.24.44.6logp2N[mpc/cm3]\nlogp[m pc]Protonsrescaledby1/8 5\nElectronsFig. 5. The number density of protons (red solid) and electrons\n(green dashed) for Model I as in Fig. 3 zoomed to the cuto \u000b\nregion.\nthe electron-to-proton flux ratio, Ke=p\u0011Ne=Np\u00191=85 in the\nrange (10\u0000103)mpc.\nThe proton spectrum cuts o \u000bat the maximum momentum,\npmax\u0019104mpc'10 TeV=c, which is limited by the age\nof the remnant and particle di \u000busion in the upstream region.\nThe latter process is determined by two factors: the magnetic\nprecursor which flattens the cosmic-ray precursor, discussed in\nSec. 2.2, and the gyrofactor, \u0018, which beside the magnetic-\nfield strength co-determines the di \u000busive transport of particles.\nSince the electrons additionally experience e \u000bective synchrotron\nlosses, their maximum momentum, \u00183 TeV=c, is lower than the\nlimit (10 TeV =c) imposed by the spatial di \u000busion and the age of\nthe remnant. The exact shape of electron and proton cuto \u000bs is\nshown in Fig. 5, which allows a direct comparison of the cuto \u000b\nforms. As one can see, the electron density is steeper than that of\nprotons due to e \u000bective synchrotron losses. Note that the shape\nof the spectral cuto \u000bin our approach deviates from the simple\nexponential and super-exponential forms (e.g. Blasi (2010)) due\nto the full time-dependency of our method. In contrast to the pre-\nvious works on Tycho, our approach includes time-dependent\ntransport equation and hydrodynamics. In fact, the maximum\ncosmic-ray energy in the age-limited case scales linearly with\ntime, and as the square of the shock velocity, V2\nsh. As the shock\nof the remnant slows down in our simulations, the maximum en-\nergy of cosmic rays grows more slowly than one would expect\nfor an approach with a constant shock velocity. As a result, the\nfinal particle spectrum in our work, which is qualitatively an av-\nerage of the instantaneous spectra, gives a sharper cuto \u000bthan\npredicted within the hydrodynamics in steady-state.\n8A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\n2345678910\n-7 -6 -5 -4 -37 8 9 10 11E1.6F10-6[erg3/5cm-2s-1]\nlogE[eV]logν[Hz]\nModelI ModelII\nFig. 6. Synchrotron plus gray-body emission in the radio and mi-\ncrowave band for Models I and II. The radio (black error bars)\nand microwave (black triangles) data are taken from Reynolds &\nEllison (1992) and Planck Collaboration et al. (2016).\nInstead of showing the entire SED at one singe plot, we\npresent figures for particular energy bands (radio, X-ray and \r-\nray) to provide a detailed comparison of data and models. The\nsynchrotron emission from electrons in the radio and microwave\nband for Model I is presented as red solid line in Fig. 6, where\nthe data are taken from Reynolds & Ellison (1992) and Planck\nCollaboration et al. (2016). Tycho’s radio spectrum is frequently\nconsidered to be distorted in response to shock modification\nby cosmic rays (Reynolds & Ellison 1992; V ¨olk et al. 2008).\nFig. 6 clearly demonstrates that this is not the only viable in-\nterpretation: stochastic re-acceleration in the downstream region\nof a test-particle shock can reproduce the observed radio spec-\ntrum without invoking non-linear e \u000bects.3It was noted earlier\n(Planck Collaboration et al. 2016) that the radiation from Tycho\nin the microwave band consists of at least two components: syn-\nchrotron emission from the non-thermal electrons and thermal\ndust emission. The latter process is responsible for the sharp rise\nin flux above\u001830 GHz. We account for the thermal dust emis-\nsion by calculating the gray-body radiation with a temperature of\n25 K and a normalization chosen to fit the flux density measured\nwith Herschel (Gomez et al. 2012). Thus, the red line in Fig. 6\nrepresents the sum of the synchrotron and thermal spectra. While\nthe slightly concave part of the radio spectrum below \u001830 GHz\nis dominated by the synchrotron emission from electron pop-\nulation shaped by the stochastic re-acceleration, the steep flux\nincrease above\u001830 GHz results from the thermal dust emission.\nThe X-ray emission is generated by electrons beyond the cut-\no\u000benergy, Emax;e\u00193 TeV , via synchrotron radiation. It is pre-\nsented for Model I as red solid line in Fig. 7, where the exper-\nimental data at 90% confidence level are taken from Tamagawa\net al. (2009). To achieve a good fit for the fixed magnetic field\n(B2=150\u0016G) and electron injection e \u000eciency (\u0011e=9:4\u000210\u00006)\nwe adapt the spatial di \u000busion coe \u000ecient parameter and set \u0018=\n10. As already discussed in Section 2.4, for our calculation of\nthe synchrotron emission we use a Gaussian distribution func-\ntion for the amplitudes of the turbulent magnetic field. The dif-\nferences to the standard formula are mostly seen at the cuto \u000bof\nthe synchrotron emission. Indeed, the standard emissivity func-\n3Shock modification by cosmic-ray feedback is neglected in our ap-\nproach since the cosmic-ray pressure remains well below 10% of the\nshock ram pressure (Kang & Ryu 2010). In fact it is at most 2.1% of the\nshock ram pressure (cf. Sec. 3.3).\n-11.5-11.0-10.5-10.0\n10 15 20 25logE2F[ergcm-2s-1]\nE[keV]Model I Model IIFig. 7. Synchrotron emission in the X-ray band for Models\nI and II with Gaussian distribution of magnetic-field ampli-\ntudes. Experimental data at 90% confidence level are taken from\nTamagawa et al. (2009).\ntion produces the steepest slope in the spectral tail while any ex-\ntended distribution smears the spectral cuto \u000band causes spectral\nhardening in the X-ray band. Nevertheless, the predicted X-ray\nspectrum above 10 keV is softer than that observed. This can\nbe attributed to the spherical symmetry of our model geometry.\nLopez et al. (2015) analyzed sixty-six regions across Tycho and\nfind that their X-ray emission exhibits a varying roll-o \u000benergy,\nwhich is defined as\nErollo\u000b'7 eV B2\n100\u0016G! Emax;e\nTeV!2\n: (17)\nAs the magnetic field and the maximum energy of electrons may\nvary across Tycho’s perimeter, so do the corresponding syn-\nchrotron spectra. Integration over the individual regions, as in\nLopez et al. (2015), results in a harder total spectrum, because\nthe variations in the roll-o \u000benergy will invariably smear out the\ncuto\u000b. This hardening can not be accounted for in our model due\nto the 1-D geometry, and hence our X-ray spectrum is softer than\nthat observed by Suzaku . Note, since we use a Gaussian distri-\nbution function to calculate the synchrotron emission, its cuto \u000b\ndeviates from the usually assumed exponential profile and thus\nEq. 17 is no longer valid in our case.\nThe minimum downstream magnetic field that provides the\nmaximum inverse-Compton contribution compatible with the \r-\nray data and su \u000ecient flux contribution in the radio range on\naccount of stochastic re-acceleration is \u0018150\u0016G. Any weaker\nmagnetic field would imply an overshooting of \r-ray emis-\nsion at\u0018100 GeV induced by the inverse Compton process.\nNevertheless, it is not able to account for the GeV-scale emission\nand hence the hadronic channel is required. The resulting \r-ray\nspectra of Tycho and the corresponding \r-ray data are given at\nthe top panel of Fig. 8. The pion bump is represented by the\nblue dotted and the inverse-Compton peak by the green dashed\nlines, respectively. The total \r-ray spectral distribution (red solid\nline) is rather flat with spectral index \u0000\u00192. The impact from\nthe stochastic re-acceleration of protons is hardly visible in the\npion bump. The reason is a relatively small energy fraction in the\nfast-mode waves. In our model for Tycho parameters relevant for\nthe stochastic re-acceleration are dictated by the radio data. For\nother SNRs their value can di \u000ber, potentially resulting in more\ne\u000ecient re-acceleration of protons and consequentially into a\nsofter hadronic spectrum. Thus, stochastic acceleration of pro-\ntons may provide an alternative explanation for the softening of\n9A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\n-13.5-13.0-12.5-12.0-11.5-11.0\nModel IIC PD PD+IC\n-13.5-13.0-12.5-12.0-11.5-11.0\n-1 0 1 2 3 4logE2F[ergcm-2s-1]\nlogE[GeV]Model II\nFig. 8. The calculated \r-ray emission from Tycho in comparison\nwith measurements of Fermi -LAT (black solid) and VERITAS\n(black circles) from Archambault et al. (2017). The top panel is\nfor Model I involving a moderate transported magnetic field and\nthe bottom panel presents results for Model II, which includes\ndamping of magnetic field.\nhadronic emission, as opposed to high-density structures in the\nambient medium (Berezhko et al. 2013; Morlino & Blasi 2016),\nif enough energy is available for re-acceleration. Nevertheless,\nthe impact from stochastic re-acceleration is more prominent\nfor electron than proton spectra. The cuto \u000bin the leptonic and\nhadronic\r-ray contributions is linked to that in the synchrotron\nspectrum via the gyrofactor, \u0018, and hence constrained by X-\nray data. Thus, in our approach, we do not specifically adjust\nthe model parameters to fit the hadronic cuto \u000b. Nevertheless, it\nshows a remarkable match with the observed VERITAS data.\nThe corresponding parameters for Model I are summarized in\nthe first row of Table 1. The amplification e \u000eciency (i.e. the\nratio between cosmic-ray pressure and the amplified magnetic-\nfield pressure in the immediate upstream region) is roughly \u001874\nfor 440 years.\nAs already mentioned in Sec 2.4, the contribution from non-\nthermal bremsstrahlung is negligible for the density of gas at\nTycho. Therefore it is not shown in Fig. 8 and we do not discuss\nit further.\nTo further test Model I we consider the spatial profiles of\nradio and X-ray synchrotron radiation at 1.4 GHz and 1 keV , re-\nspectively. The results are depicted in top panel of Fig. 9. The\nradio (Slane et al. 2014) and X-ray (Cassam-Chena ¨ı et al. 2007)\nbrightness profiles are extracted from the western rim of the rem-\nnant and normalized to their peak values. The measured X-ray\n0.20.40.60.81.01.2\nModel IRadio Xray\n0.20.40.60.81.01.2\n0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01F/Fmax\nr/RshModel IIFig. 9. Radial intensity profiles for Model I (top panel) and\nModel II (bottom panel). X-ray emission at 1 keV (blue dashed\nline) was observed with Chandra (Cassam-Chena ¨ı et al. 2007),\nand radio data at 1.4 GHz (red solid line) were taken from Slane\net al. (2014). Following Slane et al. (2014) the radio data were\nslightly shifted to account for the expansion of the remnant. The\ngrey dotted line in the bottom panel represents a radio profile\nproduced without magnetic-field damping.\nbrightness is strongly peaked at the shock and rapidly decreases\ntowards the contact discontinuity. The radio profiles are slightly\nwider but still exhibit a narrow structure close to the shock. The\nenhancement of the emission towards the interior that is evident\nin the radio profiles might be attributed to the afore-mentioned\nRayleigh-Taylor distortions operating in the vicinity of CD. As\nseen in the figure, the predictions of Model I involving the small-\nest possible advected magnetic field are in disagreement with the\ndata. The model can explain neither the narrow X-ray rims nor\nthe structure of the radio emission.\nThere are at least two potential solutions to this problem that\nwe shall explore in the following section. Turbulent magnetic-\nfield damping would a \u000bect the synchrotron emissivity and thus\ncreate narrow structures close to the shock. Alternatively, a very\nstrong magnetic field at the shock would impose strong syn-\nchrotron losses and thus limit the width of the rims.\n3.2.Model II: High damped magnetic field\nAs Model I involving a weak advected magnetic field fails to\nreproduce the observed intensity profiles of the synchrotron\nemission in both the radio and the X-ray band, we introduce\nnow damping of the turbulent magnetic field. Therefore we set\n10A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\nmagnetic-field profile described by Eq. 5. Obviously, in this\nscenario the magnetic field strength in the immediate down-\nstream of the forward shock must be larger than for Model I,\nbecause damping suppresses synchrotron emission from the far-\ndownstream region. Since inverse Compton radiation is pro-\nduced wherever high-energy electrons reside, an overproduction\nof photons in the \r-ray band would arise unless the magnetic\nfield is scaled up. Especially when combined with stochastic re-\nacceleration as an explanation for the radio data, the magnetic-\nfield damping requires an extensive increase of the overall mag-\nnetic field. As reflected by the decreasing width of the turbulent\nregion, the stochastic re-acceleration of particles is more e \u000ecient\nat the early stages of SNR. As the re-accelerated electrons are ad-\nvected away from the shock, they mostly experience the damped\nmagnetic field. While the total synchrotron emission depends on\nthe absolute values of the immediate post-shock field strength,\nB2, and the residual level B0, the shape of the radio filaments\nis determined (apart from the damping scale ld) by the ratio of\nB2andB0. We find that the immediate post-shock field strength\nand the residual level must be at least B2=330\u0016G (amplifica-\ntion factor\u000b=20) and B0=70\u0016G, in order to maximize the\nfilling factor, and thus to produce su \u000ecient overall synchrotron\nemission, and to fit the radio filaments simultaneously. Model II\nis based on these minimum values, and results are displayed as\ngreen dashed lines in Figs. 6 and 7, as well as in the bottom\npanels of Figs. 8 and 9.\nParticle acceleration and propagation is modeled following\nthe same procedure as in Model I, but some of the parameters\nare slightly adjusted. The second row of Table 1 lists the rele-\nvant model parameters, which now include the damping length,\nld, and the residual field level, B0. Note, that the damping length,\nld, provides spatial characteristics for the turbulence responsible\nfor the magnetic-field amplification. Likewise, the width of the\nre-acceleration region, Lfm, represents the damping scale for the\nfast-mode waves. As both quantities are associated with di \u000ber-\nent types of turbulence, they are independent of each other. At\n440 years the amplification e \u000eciency for the second model is\n\u001814.\nAlong the shock surface certain variations in the parameter\nvalues are to be expected, as not all filaments are the same (cf.\nRessler et al. 2014). Also, Tycho is not a perfectly spherically-\nsymmetric SNR and the projection e \u000bects may impose a bias.\nTherefore our choice for the residual field level and the damp-\ning scale rather provide a reasonable order of their magnitudes\nas in this work we consider only one particular rim of Tycho.\nFitting the Suzaku data with an enhanced post-shock magnetic-\nfield B2=330\u0016G requires a moderate increase of the gyrofactor\ncompared to Model I ( \u0018=16).\nDamping quenches the synchrotron emissivity in the deep\ndownstream region. While the suppression is achromatic for\npower-law electron spectra, i.e. at frequencies below the cuto \u000bin\nthe synchrotron spectrum, it becomes progressively stronger be-\nyond the roll-o \u000benergy. A competing process for quenching the\nemissivity at the roll-o \u000bfrequency is energy losses preventing\nthe propagation of electron from the shock to the deep down-\nstream region. The synchrotron loss time can be expressed in\nterms of the magnetic-field strength and the energy of photons,\nEsy, that the electrons would typically emit\n\u001closs'50 yr B2\n100\u0016G!\u00003=2 Esy\nkeV!\u00001=2\n: (18)\nThe distance electrons travel during the loss time roughly equals\nthe thickness of the filaments. The particle transport is governedby di\u000busion and advection. The latter process dominates at lower\nenergies and its length scale is given by\nladv=u2\u001closs=Vsh\nrsh\u001closs\n'2\u00021017cm Vsh\n5000 km=s! B2\n100\u0016G!\u00003=2 Esy\nkeV!\u00001=2\n:\n(19)\nAt high energies di \u000busion with Bohmian energy scaling be-\ncomes the dominant propagation process. The corresponding\ndistance is energy-independent and given by\nldi\u000b=p\nDr\u001closs\n'1017cmp\n\u0018 B2\n100\u0016G!\u00003=2\n: (20)\nEquating Eqs. 19 and 20 one can find the critical photon en-\nergy, where the transition from the advection-dominated into\ndi\u000busion-dominated transport occurs\nEsy;c'4 keV\n\u0018 Vsh\n5000 km=s!2\n: (21)\nWith\u0018=16 and Vsh=4100 km=s we obtain for Model II Esy;c\u0019\n0:2 keV , implying that both, advection and di \u000busion impact the\ndistance covered by electrons that account for the 1 keV-rims.\nEstimating the propagation length for the electrons that radiate\n1 keV-photons, we find ladv'3\u00021016cm and ldi\u000b'7\u00021016cm,\nand hence di \u000busive transport is more important, but advective\ntransport is not negligible. Accounting for both advective and\ndi\u000busive terms, the total transport length that the electrons travel\nbefore expending their energy is then given by (Parizot et al.\n2006)\nlloss=0BBBBBBB@s\nu2\n2\n4D2r+1\nDr\u001closs\u0000u2\n2Dr1CCCCCCCA\u00001\n'1017cm; (22)\nwhere the last expression applies to electron emitting 1-keV pho-\ntons and Model II. The underlying assumption of a constant\nmagnetic-field strength is violated though, as the synchrotron\nenergy losses decrease over a length scale 5 \u00011016cm. Therefore\nthe e\u000bective loss-length of electrons is much larger than 1017cm,\nand the synchrotron filaments are largely shaped by magnetic-\nfield damping.\nTo account for the radio profiles magnetic-field damping is\nclearly needed. Radio-emitting electrons have energy-loss times\nfar in excess of the age of SNRs, and the radio rims cannot\narise from synchrotron losses. The model with magnetic-field\ndamping (Model II) fits the spectral data reasonably well. To be\nnoted from Fig. 9 is that an e \u000bective damping length of ld=\n0:01Rsh'1017cm can indeed reproduce the sharply peaked ra-\ndio profiles in the shock vicinity, but somewhat underpredicts the\nradio intensity in the deep downstream region where Rayleigh-\nTaylors fingers from the contact discontinuity may provide mag-\nnetic field amplification (Jun & Norman 1995; Bj ¨ornsson &\nKeshavarzi 2017) that is not included in our model. In contrast, a\nradio emission profile calculated for non-damped magnetic field\n(grey dotted line in bottom panel of Fig. 9) obviously contra-\ndicts the observed data. As a relatively high magnetic-field value\nis required to maintain the total radio and \r-ray data, the width\nof the X-ray rims is inevitably a \u000bected by synchrotron losses.\n11A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\nStill, magnetic-field damping is more important for the forma-\ntion of the X-ray filaments since lloss>ld. Without magnetic-\nfield damping, the synchrotron losses for the post-shock mag-\nnetic field B2=330\u0016G are able to produce the thin X-ray rims,\nbut fail to form the radio profiles. When introduced as a nat-\nural explanation for the radio profiles, magnetic-field damping\nmoreover becomes the dominant process for the production of\nthe X-ray filaments.\nWe conclude that magnetic-field damping is essential for\nthe radio filaments. Furthermore, for the minimum downstream\nmagnetic field that can explain the complete observed data, B2=\n330\u0016G, magnetic-field damping is more important for the X-ray\nprofiles than radiative losses. Nevertheless, for 1-keV emission\nsynchrotron losses provides a subordinate additional process for\nX-ray filaments formation. Simultaneously the width of the ra-\ndio rim is solely determined by the magnetic-field damping. In\nagreement with our results, the joint X-ray and radio analysis of\nTran et al. (2015) finds magnetic-field damping as the preferable\nscenario. Both studies find similar damping lengths: 1%-2% of\nthe SNR radius.\n3.3. Cosmic-ray pressure\nNext, we verify that the test-particle approximation is valid for\nModel II. Therefore we calculate the cosmic-ray pressure, given\nby\nPcr(r;t)=c\n3Z\nN(r;p;t)p2dpp\np2+(mpc)2; (23)\nwhere mpis the rest mass of proton and N(r;p;t) the di \u000berential\nproton number density. As mentioned before, the pressure ex-\nerted by the non-thermal electrons is negligible in our model.\nThe relative velocity change of the plasma in the shock rest-\nframe in a particular region between r1andr2caused by the\nparticle pressure is\n\u000eu\nu=\u00001\nu2\u001aZr2\nr1dx@Pcr(x)\n@x\n=Pcr(r2)\u0000Pcr(r1)\nPflow\u0011\u000ePcr\nPflow: (24)\nHere\u001adenotes the density, uthe velocity and Pflow=\u001au2the\ndynamical pressure of the plasma. Eq. 24 is universally valid\nand holds in any region of the plasma flow.\nAccording to Kang & Ryu (2010), the test-particle approx-\nimation is justified as long as the cosmic-ray pressure in the\nprecursor does not exceed 10% of the shock ram pressure.\nConsequently, according to Eq. 24 the test-particle regime re-\nquires\u000eu=u=\u000ePcr=Pflow\u00140:1 in the upstream region. Fig. 10\nshows the cosmic-ray pressure normalized by the shock ram\npressure, Psh=\u001a1u2\n1, as a function of the position, r=Rsh, for\ndi\u000berent times. To be noted from the figure is that the rel-\native velocity change in the upstream region at 440 years is\n\u000eu=u=\u000ePcr=Psh\u00190:021, and hence clearly below the test-\nparticle threshold. Similarly at the earlier stages, the cosmic-ray\npressure at the shock does not exceed \u000eu=u<0:1, verifying that\nthe dynamic cosmic-ray feedback is indeed negligible in our ap-\nproach.\nBehind the shock wave the pressure of particles is signifi-\ncantly enhanced due to stochastic re-acceleration. In contrast to\nstandard DSA, where the maximal cosmic-ray pressure occurs\nat the shock, in our model it reaches its maximum in the im-\nmediate downstream region. The corresponding position is de-\ntermined by the current width of the turbulent zone where the\n0.010.020.030.04\n0.998 0.9985 0.999 0.9995 1 1.0005Pcr/Psh\nr/Rsh440yr\n350yr\n250yr\n150yrFig. 10. Cosmic-ray pressure, Pcr, normalized by the shock ram\npressure, Psh, for Model II (damped magnetic field with B2=\n330\u0016G) for di \u000berent times.\nfast-mode waves operate. Taking into account that the dynamic\npressure in the post-shock region is a factor 4 lower than the\nshock ram pressure in the shock rest-frame, we estimate for the\ntotal velocity change at 440 years: \u000eu=u\u00190:01. We neglect\nthe dynamical feedback from the re-accelerated particles in this\nwork. Nevertheless, this e \u000bect may be of interest for future in-\nvestigations. Indeed, when strong enough, the stochastic accel-\neration can have significant e \u000bect on the post-shock structure.\nThe cosmic-ray pressure in the immediate downstream region\nenhanced by the stochastic acceleration has to press the plasma\naway from the re-acceleration region and force it to accumu-\nlate directly behind the shock. This process increases the plasma\ndensity and the associated magnetic-field strength at the shock\nof the SNR. It is clear that this phenomena fundamentally di \u000bers\nfrom the standard dynamical reaction of particles described in\nthe NLDSA theory.\nSummarizing, we conclude that we find a viable self-\nconsistent scenario for Tycho, represented by Model II.\n3.4. Hadronic model\nIn the previous subsection we presented a lepto-hadronic sce-\nnario that requires the lowest possible magnetic field in the rem-\nnant compatible with radio filaments and the SED. More e \u000ecient\nmagnetic-field amplification suppresses the inverse-Compton\ncomponent until the \r-ray spectrum becomes purely hadronic.\nIn our approach, where we use magnetic-field damping to fit\nthe radio filaments, the magnetic-field values have to be at least\nB0\u0019120\u0016G and B2\u0019600\u0016G. The total magnetic energy in\nthe remnant is still moderate, \u00181:1\u00021049erg, on account of the\nmagnetic-field damping.\nAs discussed in Sec. 2.2, the separate treatment of mag-\nnetic field and hydrodynamics for the ambient density of nH=\n0:6 cm\u00003is justified for post-shock magnetic field below 400 \u0016G.\nThus, for the hadron-dominated case the dynamical feedback\nfrom the magnetic field becomes significant. The correspond-\ning pressure, with the Alfv ´enic Mach number MA\u00199:6, would\nlower the shock compression ratio to rsh\u00193:87 and thus soften\n12A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\nthe particle and radio spectra indices to s\u00192:05 and\u001b\u00190:52,\nrespectively. As the electron spectrum becomes slightly softer\non account of magnetic-field impact, the contribution from the\nstochastic re-acceleration becomes less necessary. Hence, in\ngeneral, as competing explanation for the softening of the spec-\ntra, very high magnetic field inevitably decreases the energy\nfraction converted to turbulence, \u000f.\nDue to the very high magnetic field, X-ray filaments at 1 keV\nwould be primarily governed by the radiative losses, with cor-\nresponding propagation scale lloss'4\u00021016cm. Thus, the\nbrightness profile would lack the X-ray flux due to extreme syn-\nchrotron losses. Nevertheless, as pointed out above, the radio as\nwell the X-ray rims can strongly vary with position along the\nperimeter due to the natural asymmetry of the remnant. As the\nelectrons emitting the synchrotron radiation in the radio range\ndo not experience significant losses, the radio profiles do require\nmagnetic-field damping.\nSummarizing, we conclude that a purely hadronic model is\npossible for Tycho, but requires an elaborate and cautious treat-\nment that among other e \u000bects includes the dynamical feedback\nfrom magnetic-field pressure. Nevertheless, the lepto-hadronic\nscenario, referred to as Model II, is able to explain the broad-\nband observations of Tycho in a satisfactory manner.\n4. Summary and conclusion\nIn this work we have conducted extensive multiwavelength mod-\neling of Tycho. For the very first time we accounted for stochas-\ntic re-acceleration in the downstream region of the forward\nshock, which provides a consistent explanation to the soft parti-\ncle spectra without resorting to Alfv ´enic drift (V ¨olk et al. 2008;\nMorlino & Caprioli 2012; Slane et al. 2014) and its inherent\nproblems, or deducing the particle spectral index from the ra-\ndio observations (Atoyan & Dermer 2012; Zhang et al. 2013;\nCaragiulo & Di Venere 2014). As discussed in the introduction,\nwe find the concept of Alfv ´enic drift contradictory, although it\nhas been widely used in the global models for various SNRs. We\npresented instead a new approach that adds di \u000busion in momen-\ntum space to the standard DSA approach. Furthermore, in our\nmethod both hydrodynamics and particle acceleration are fully\ntime-dependent .\nThe stochastic acceleration of particles in the immediate\ndownstream region is assumed to arise from the fast-mode tur-\nbulence, which is supplied by the energy of the background\nplasma. We found that 2.7% of the thermal energy density of the\ndownstream background plasma is su \u000ecient to explain Tycho’s\nsoft radio spectra. Simultaneously, the magnetic field is assumed\nto be amplified by streaming instabilities (Bell 2004; Lucek &\nBell 2000) or turbulent dynamos (Giacalone & Jokipii 2007).\nBased on this, we have presented a self-consistent global model\n(Model II) that is able to accurately reproduce the observed ra-\ndio, X-ray, and \r-ray emission, and simultaneously account for\nthe non-thermal filaments in radio and X-ray range. The radio\nfilaments are generated due to magnetic-field damping, which\nis widely considered to allow a relatively low magnetic-field\nvalue inside a remnant. Combining this scenario with stochas-\ntic re-acceleration, we found that the minimum magnetic-field\nrequired to explain the entire observed dataset of Tycho is B2\u0019\n330\u0016G. This value is similar to the results of Morlino & Caprioli\n(2012), V ¨olk et al. (2008) and Zhang et al. (2013) although the\nunderlying physical assumptions are quite di \u000berent. We find that\nfor this minimum magnetic-field strength of B2\u0019330\u0016G, the X-\nray filaments at 1 keV are primarily produced by magnetic-field\ndamping, while the synchrotron losses play a secondary role.This finding is inconsistent with the work of Morlino & Caprioli\n(2012), who concluded that X-ray filaments shaped by magnetic-\nfield damping are not possible for Tycho. An important criterion\nhere is that the propagation length and hence synchrotron loss\nscale for electrons radiating at 1 keV is dominated by di \u000busion .\nTherefore, the di \u000busive transport of particles, which has been\npreviously neglected in all global models of Tycho, has to be\ntaken into account to adequately model the X-ray filaments. We\nstress that an accurate modeling of the filaments is seamlessly\ntied to determination of the post-shock magnetic field. Magnetic-\nfield damping is additionally needed for its unique capability to\nexplain the radio rims. In our model for Tycho the magnetic-\ndecay length is of the order of ld\u00180:01Rsh, which is consistent\nwith the estimate of Tran et al. (2015). The total magnetic en-\nergy in the remnant for Model II is 3 :4\u00021048erg. This value is\nmoderate on account of magnetic-field damping, in spite of the\ne\u000ecient magnetic-field amplification at the shock.\nIn the framework of our model we predict relatively inef-\nficient Bohm di \u000busion, reflected in the value of the gyrofactor\n\u0018\u001916. This value is required to consistently fit the synchrotron\ncuto\u000bobserved in the X-ray range. In line with most previous\nworks on Tycho, we also conclude that acceleration of protons\nis required to explain the \r-ray flux observed by VERITAS and\nFermi -LAT. Our research cannot account for a purely leptonic\norigin for the \r-ray emission as in Atoyan & Dermer (2012).\nWe favor instead a mixed model with a composite flat spectrum\n(\u0000\u00192). The electron-to-proton ratio for Model II is Ke=p\u00191=80\nand the maximum energy for protons is Emax;p\u001910 TeV, since\nit is linked to the roll-o \u000benergy of the synchrotron emission\nvia gyrofactor, \u0018. This result falls below previously presented\n\u0018500 TeV (Morlino & Caprioli 2012) and \u001850 TeV (Slane et al.\n2014). The maximum energy of electrons is \u00183 TeV , which is\nsimilar to the value 5 - 6 TeV suggested by Zhang et al. (2013)\nand Caragiulo & Di Venere (2014). The total energy in protons\nin our model is Etot;p\u00192:7\u00021049erg, implying that a few per-\ncent of the explosion energy went into cosmic rays. This value\nis significantly below the 16% claimed by Slane et al. (2014).\nAs a relatively marginal energy fraction is transferred to parti-\ncles and the cosmic-ray pressure at the shock does not exceed\n2.1% of the shock ram pressure during the entire evolution of\nthe remnant, we can use the test-particle approximation.\nFurthermore, we explicitly show that NLDSA e \u000bects are\nnot required, neither to explain the hydrodynamic structure of\nTycho nor to produce its slightly concave radio spectrum. First,\nthe hydrodynamic simulations with an ambient gas density of\nnH=0:6 cm\u00003and canonical explosion energy of 1051erg pro-\nvide a decent fit to the observed radii and a reasonable rem-\nnant distance\u00182:9 kpc. Second, the radio spectrum is pro-\nduced by synchrotron emission generated by electrons that are\nre-accelerated in the immediate downstream region. In general,\nthe imprint left by stochastic re-acceleration is more prominent\nin electrons than in protons. Thus, future \r-ray observations\nthat can successfully discriminate between leptonic and hadronic\nmodels for various SNRs may also be able to distinguish be-\ntween NLDSA and stochastic re-acceleration scenarios.\nWe find that a purely hadronic model may also be possi-\nble for Tycho for the immediate post-shock magnetic-field B2\u0019\n600\u0016G and far-downstream field B0\u0019120\u0016G. However, we do\nnot explicitly model the hadronic scenario in this work, because\nthe dynamical reaction of the magnetic field has to be taken\ninto account, an e \u000bect that is not yet included in our method.\nNevertheless, we favor a lepto-hadronic scenario (Model II) as\nit might better produce the X-ray filaments. Additionally, ex-\n13A. Wilhelm et al.: Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant\ntremely e \u000ecient magnetic-field amplification is not required, as\nin the case of a purely hadronic model.\nIn our model for Tycho, the contribution of the stochastic\nre-acceleration in the proton spectrum is marginal due to the\nsmall energy fraction converted into downstream turbulence. For\na larger amount of energy in fast-mode waves, the correspond-\ning\r-ray spectrum has to be naturally softer than predicted by\nthe standard DSA, which can provide a viable explanation for\nthe observed soft \r-ray spectra of some SNRs. This subject is\nbeyond the scope of this work, but could be of interest for future\ninvestigations.\nFinally, we emphasize that the dynamical feedback on the\nbackground plasma from the cosmic rays re-accelerated in the\nimmediate downstream region of the SNR is of great interest\nfor future studies, since it clearly di \u000bers from the classical non-\nlinear e \u000bects discussed in the literature.\nSummarizing, we find that stochastic re-acceleration is in-\ndeed a promising and natural alternative to explain the soft spec-\ntra of Tycho, and potentially other SNRs.\nAcknowledgements. We are very grateful to Asami Hayato for kindly providing\nus the Suzaku data. Furthermore, we acknowledge NASA and Suzaku grants, and\nsupport by the Helmholtz Alliance for Astroparticle Physics HAP funded by the\nInitiative and Networking Fund of the Helmholtz Association. VVD is supported\nby NSF grant 1911061.\nReferences\nAcciari, V . 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S. 2008, ApJ, 678, 939\n14" }, { "title": "1911.00728v1.Tuning_Non_Gilbert_type_damping_in_FeGa_films_on_MgO_001__via_oblique_deposition.pdf", "content": "Tuning Non -Gilbert -type damping in FeGa film s on MgO(001) via oblique \ndeposition \nYang Li1,2, Yan Li1,2, Qian Liu3, Zhe Yuan3, Qing -Feng Zhan4, Wei He1, Hao-Liang \nLiu1, Ke Xia3, We i Yu1, Xiang-Qun Zhang1, Zhao -Hua Cheng1,2,5 a) \n1State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed \nMatter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, \nChina \n2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing \n100049, China \n3The Center for Advanced Quantum Studies and Department of Physics, Beijing \nNormal University, 100875 China \n4State Key Laboratory of Precision Spectroscopy, School of Physics and Materials \nScience, East Ch ina Normal University, Shanghai 200241, China \n5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China \na) Corresponding author , e-mail: zhcheng@iphy.ac.cn \nAbstract \nThe ability to tailor the damping factor is essential for spintronic and spin- torque \napplication s. Here, we report an approach to manipulate the damping factor of \nFeGa/MgO(001) films by oblique deposition. Owing to the defects at the surface or \ninterface in thin films , two -magnon scatterin g (TMS) acts as a non -Gilbert damping \nmechanism in magnetization relaxation. In this work, the contribution of TMS was \ncharacterized by in-plane angul ar dependent ferromagnetic resonance (FMR) . It is \ndemonstrated that the intrinsic Gilbert damping is isotropic and invariant , while the \nextrinsic mechanism related to TMS is anisotropic and can be tuned by oblique \ndeposition. Furthermore, the two and fourfold TMS related to the uniaxial magnetic \nanisotropy (UMA) and m agnetocrystalline anisotropy were discussed. Our result s open \nan avenue to manipulate magnetization relaxation in spintronic devices. \n1 \n Keywords : Gilbert damping , two -magnon scattering, FMR, oblique deposition, \nmagnetic anisotropy \n2 \n 1. Introduction \nIn the past decades, controlling magnetization dynamics in magnetic \nnanostructures has been extensively studied due to its great importance for spintronic \nand spin- torque applications [1,2] . The magnetic relaxation is described within the \nframework of the Landa u-Lifshitz Gilbert (LLG) phenomenology using the Gilbert \ndamping factor α [3]. The intrinsic Gilbert damping depends primarily on the spin- orbit \ncoupling (SOC) [4,5] . It has been demonstrated that alloying or doping with non-\nmagnetic transition metals provides an opportunity to tune the intrinsic damping [6,7] . \nUnfortunately, in this way the soft magnetic properties will reduce . In addition to the \nintrinsic damping, the two-magnon scatterin g (TMS) process se rves as a n important \nextrinsic mechanism i n magnetization relaxation in ultrathin films due to the defects at \nsurface or interface [8,9] . This process describes the scattering between the uniform \nmagnons and degener ate final -state spin wave modes [10]. The existence of TMS has \nbeen demonstrated in many systems of ferrites [11-13]. Since the anisotropic scattering \ncenters , the angular dependence of the extrinsic TMS process exhibi ts a strong in -plane \nanisotropy [14], which allows us to adjust the overall magnetic relaxation , including \nboth the int ensity of relaxation rate and the anisotropic behavior. \nHere, we report an approach to engineer the damping factor of Fe81Ga19 (FeGa ) \nfilms by oblique deposition. The FeGa alloy exhibits large magnetostriction and narrow \nmicrowave resonance linewidth [15] , which could assure it as a promising material for \nspintronic devices. For the geometry of off -normal deposition, it has been demonstrated \nto provoke shadow effects and create a periodic stripe defect matrix. This can introduce \na strong uniaxial magnetization anisotropy (UMA) pe rpendicular to the projection of \nthe atom flux [16-19]. Even though some reports have shown oblique deposition \nprovokes a twofold TMS channel [20-22], the oblique angle dependence of the intrinsic \n3 \n Gilbert damping and the TMS still remain in doubt. For our case, on the basis of the \nfirst-principles calculation and the in -plane angular -dependent FMR measurements, we \nfound that the intrinsic Gilbert damping is isotropic and invariant with varying oblique \ndeposition angles, while the extrinsic mechanism related to the two -magnon -scattering \n(TMS ) is anisotropic and can be tuned by oblique deposition. In addition, importantly \nwe firstly observe a phenomenon that the cubic magnetocrystalline anisotropy \ndetermines the area including degenerate magnon modes, as well as the intensity of fourfold TMS. In general , the strong connection between the extrinsic TMS and the \nmagnetic anisotropy , as well their direct impact on the damping constants , are \nsystem ically investigated, which offer us a useful approach to tailor the damping factor.\n \n2. Experimental details \nFeGa thin films with a thickness of 20 nm were grown on MgO(001) substrates in \na magnetron sputtering system with a base pressure below 3 × 10−7 Torr. Prior to \ndeposition, t he substrates were annealed at 700 °C for 1 h in a vacuum chamber to \nremove surface contaminations and then held at 250 °C during deposition. The incident \nFeGa beam was at different obl ique angles of ψ =0°, 15°, 30°, and 45°, with respect to \nthe surface normal , and named S1, S2, S3, and S4 in this paper , respectively. The \nprojection of FeGa beam on the plane of the substrates was set perpendicular to the \nMgO[110] direction, which induces a UMA perpendicular to the projection of FeGa \nbeam , i.e., parallel to the MgO[110] direction, due to the we ll-known self-shadowing \neffect. Finally , all the samples were covered with a 5 nm Ta capping layer to avoid \nsurface oxidation [see figure 1(a)]. The epitaxial relation of \nFeGa(001)[ 110]||MgO (001)[ 100] was characterized by using t he X -ray in -plane Φ-\nscans , as described elsewhere [23]. Magnetic hy steresis loops were measured at various \nin-plane magnetic field orientations φ H with respect to the FeGa [100] axis using \n4 \n magneto -optical Kerr effect (MOKE) technique at room temperature . The d ynamic \nmagnetic properties were investigated by broadband FMR measurements based on a \nbroadband vector network analyzer (VNA) with a transmission geometry coplanar \nwaveguide (VNA- FMR ) [24]. This setup allows both frequency and field- sweeps \nmeasurements with external field applied parallel to the sample plane. During \nmeasurements, the sampl es were placed face down on the coplanar wavegu ide and the \ntransmission coefficient S 21 was recorded. \n3. Results and discussion \nFigure 1(b) displays the Kerr hysteresis loops of sample S1 and S4 recorded along \nwith the main crystallographic directions of FeGa [100], [110], and [010] . The sample \nS1 exhibit s rectangular hysteresis curves with sm all coercivities for the magnetic field \nalong [100] and [010] easy axes. In contrast, the S4 displays a hysteresis curve with \ntwo step s for the magnetic field along the [010] axis, which indicates a UMA along the \nFeGa[100 ] axis superimposed on the four fold magnetocrystalline anisotropy . As a \nresult, with increasing the oblique angle, the angular dependence of normalized remnant \nmagnetization ( Mr/Ms) gradually reveals a four fold symmetry combined with a uniaxial \nsymmetry, as shown i n the inset of f igure 1(b). \nSubsequently, the magnetic anisotropic properties can be further precisely \ncharacterized by the in -plane angular -dependent FMR measurements. Figure 1(c) and \n1(d) show typical FMR spectra for the real and imaginary part s of coefficient S 21 for \nthe sample S2 . Recorded FMR spectra contain a symmetric and an antisymmetric \nLorentzian peak , from which the resonant field H r with linewidth ∆𝐻𝐻 can be obtained \n[24,25] . \nFigure 2(a) shows the in -plane angular dependence of H r measured at 13 .0 GHz \nand can be fitted by the following expression [26,27] : \n5 \n 𝑓𝑓=𝛾𝛾𝜇𝜇0\n2𝜋𝜋�𝐻𝐻𝑎𝑎𝐻𝐻𝑏𝑏 (1 ) \nHere,𝐻𝐻𝑎𝑎=𝐻𝐻4(3+𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M)/4+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H)+𝑀𝑀eff and 𝐻𝐻𝑏𝑏=\n𝐻𝐻4𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H), H4 and Hu represent the fourfold \nanisotropy field and the UMA field caused by the self -shadowing effect , respectively. \n𝜑𝜑H(𝜑𝜑M) is the azimuthal angles of the applied field ( the tipped magnetization ) with \nrespect to the [100] direction , as depicted in figure 1(a). 𝜇𝜇0𝑀𝑀eff=𝜇𝜇0𝑀𝑀𝑠𝑠−2𝐾𝐾out\n𝑀𝑀𝑠𝑠, Ms is \nthe saturation magnetization and Kout is the out -of-plane uniaxial anisotropy constant . \n𝑓𝑓 is the resonance frequency , 𝛾𝛾 is the gyromagnetic ratio and here used as the accepted \nvalue for Fe films, 𝛾𝛾=185 rad GHz/T [28] . \nThe angular dependent Hr reveals only a fourfold symmetry for the none -\nobliquely deposited sample , which indicates the cubic lattice texture of FeGa on MgO . \nWith increasing the oblique angle , a uniaxial symmetry is found to be superimposed \non the four fold symmetry, clearly confirming a UMA is produced by the oblique \ngrowth , which agrees with the MOKE’ results. The fitted parameter 𝜇𝜇0𝑀𝑀eff=1.90±\n 0.05T is found to be independent on the oblique deposition and close to 𝜇𝜇0𝑀𝑀𝑠𝑠=\n1.89 ± 0.02 T estimated using VSM, which is almost same as the value of the \nliterature [29] . This indicates negligible out- of-plane ma gnetic anisotropy in the thick \nFeGa films . As shown in figure 2(b), it is observed that the UMA (Ku=HuMs/2) exhibits \na general increasing trend with oblique angle , which coincides with the fact the \nshadowing effect is stronger at larger angles of incidence [16-19]. Interestingly the \noblique deposition also affects the cubic anisotropy K4 (K4=H4Ms/2). Different from \nthe K4 increases slightly with deposition angle in Co/Cu system [16], here the value of \nK4 is the lowest at a n oblique angle of 15°. It is well known th at film stress significantly \ninfluences the crystallization tendency [30,31] . FeGa alloy is highly stress ed sensitive \n6 \n due to its larger magnetostriction . Thus, t he change in K4 of FeGa films may be \nattributed to the anisotropy dispersion created due to the stress variations during grain \ngrowth. It should be mentioned that the best way to determine magnetic parameters is \nto measure the out -of-plane FMR . But the effective saturation magnetization 𝜇𝜇0𝑀𝑀eff=\n1.90T of FeGa alloy leads to the perpendicular applied field beyond our instrument \nlimit. Meanwhile, t he results obtained above are also in accord with those extracted \nby fitting field dependence of the resonance frequency with H//FeGa[100] shown in \nfigure 2(c). \nThe effective Gilbert damping 𝛼𝛼eff is extracted by linearly fitting the dependence \nof linewidth on frequency : 𝜇𝜇0∆H=𝜇𝜇0∆H0+2𝜋𝜋𝜋𝜋𝛼𝛼𝑒𝑒ff\n𝛾𝛾, where ∆𝐻𝐻0 is the inhomogeneous \nbroadening. For the sake of clarity , figure 3(a) only shows the frequency dependence \nof linewidth for the samples S1 and S2 along [110] and [100] axes. It is evident that, \nfor the sample S1, both linear slopes of two direction s are almost same. While w ith \nregard to the sample S2 , the slope of the ∆H-f curve along the easy axis is approximately \na factor of 2 greater than that of the hard axis. The obtained values of 𝛼𝛼eff are shown in \nfigure 3(b). Firstly, the results clearly indicate that the effective damping exhibits \nanisotropy , with higher value along the easy axis . Secondly, f or the easy axis, the \noblique angle dependence on the damping parameter indicates an extraordinary trend \nand has a peak at deposition angle 15°. However, the damping shows an increasing \ntrend with the oblique angle for the field along the hard axis. In the following part, we \nwill explore the effect of oblique deposition on the mechanism of the anisotropic \ndamping and the magnetic relaxation pr ocess. \nSo far, convincing experimental evidence is still lacking to prove the existence of \nanisotropic damping in bulk magnets. Chen et al. have shown the emergence of \nanisotropic Gilbert damping in ultrathin Fe (1.3nm)/GaAs and its anisotropy disappears \n7 \n rapidly when the Fe thickness increases [32]. We perform the first-principles \ncalculation of the Gilber t damping of Fe Ga alloy considering the effect induced by the \nlattice distortion. W e artificially make a tetragonal lattice with varying the lattice \nconstant of the c -axis. The electronic structure of Fe -Ga alloy is calculated self -\nconsistently using the coherent potential approx imation implemented with the tight-\nbinding linear muffin- tin orbitals. Then the atomic potentials of Fe and Ga are randomly \ndistributed in a 5× 5 lateral supercell, which is connected to two semi -infinite Pd leads . \nA thermal lattice disorder is included via displacing atoms randomly from the perfect \nlattice sites following a Gaussian type of distribution [ 33]. The root -mean -square \ndisplacement at room temperature is determined by the Debye model with the Debye \ntemperature 470 K. The length of the supercell is variable and the calculated total \ndamping is scaled linearly with this length. Thus, a linear least- squares fitting can be \nperformed to extract the bulk damping of the Fe -Ga alloy [34]. The calculated Gilber t \ndamping is plotted in f igure 3(c) as a funct ion of the lattice distortion (𝑐𝑐−𝑎𝑎)𝑎𝑎⁄. The \nGilbert damping is nearly independent of the lattice distortion and there is no evidence of anisotropy in t he intrinsic bulk damping of Fe Ga alloy. \nSo the extrinsic contributions are responsible for the anisotropic behavior of \ndamping , which can be separated from the in -plane angular dependent linewidth. The \nrecorded FMR linewidth have the following different cont ributions [11] : \n 𝜇𝜇0∆𝐻𝐻=𝜇𝜇0∆𝐻𝐻inh+2𝜋𝜋𝛼𝛼𝐺𝐺𝑓𝑓\n𝛾𝛾𝛾𝛾+�𝜕𝜕𝐻𝐻r\n𝜕𝜕𝜑𝜑H∆𝜑𝜑H�+�Γ<𝑥𝑥𝑖𝑖>𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>�\n<𝑥𝑥𝑖𝑖>𝑎𝑎𝑎𝑎𝑐𝑐𝑐𝑐𝑎𝑎𝑎𝑎 \n �(�𝜔𝜔2+(𝜔𝜔0\n2)2−𝜔𝜔0\n2)/(�𝜔𝜔2+(𝜔𝜔0\n2)2+𝜔𝜔0\n2)+Γtwofoldmaxcos4(φM- φtwofold) (2) \n∆Hinh is both frequency and angle independent term due to the sample \ninhomogeneity . The second term is the intrinsic Gilbert damping (𝛼𝛼𝐺𝐺) contribution. 𝛾𝛾 \n8 \n is a correction factor owing to the field dragging effect caused by magnetic anisotropy \n[12], 𝛾𝛾 =cos (φM-φH). The 𝜑𝜑M as a function of φH for the sample S2 at fixed 13 GHz \nis calculated and show n in figure 4(a). Note that the draggi ng effect vanishes (𝜑𝜑M=\nφH) when the field is along the hard or easy axes . The third term describes the mosaicity \ncontribution originating from the angular dispersion of the crystallographic cubic axes \nand yield s a broader linewidth [35]. The four th term is the TMS contribution. The \nΓ<𝑥𝑥𝑖𝑖> signifies the intensity of the TMS along the principal in -plane crystallographic \ndirection <𝑥𝑥𝑖𝑖>. The 𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>� term indica tes the TMS contribution depending \non the in- plane direction of the field rel ative to <𝑥𝑥𝑖𝑖> and commonly expressed as \ncos2[2(φM-φ)] [14]. In addition, 𝜔𝜔 is the angular resonant frequency and 𝜔𝜔0=\n𝛾𝛾𝜇𝜇0𝑀𝑀eff. In our case, besides the fourfold TMS caused by expected lattice geometric \ndefects, the other twofold TMS channel is induced by the dipolar fields emerging from \nperiodic stripelike defects [20,21] . This term is parameterized by its strength Γtwofoldmax \nand the axis of maximal scattering rate φtwofold. \nAs an example, t he angle- dependent linewidth measured at 13 .0 GHz for the \nsample S2 is shown in f igure 4(b). It clearly exhibits a strong in -plane anisotropy, and \nthe linewidth along the [100] direction is significantly larger than that along the [110] \ndirection . Taking only isotropic Gilbert damping into account , the dragging effect \nvanishes with field applied along the hard and easy axes . Meanwhile, the mosaicity \nterm gives an angular variation of the linewidth proportional to |𝜕𝜕𝐻𝐻𝑟𝑟𝜕𝜕𝜑𝜑𝐻𝐻|⁄ , which is \nalso zero along with the principal <100> and < 110> directions. This gives direct \nevidence that the rel axation is not exclusively governed only considering the intrinsic \nGilbert mechanism and mosaicity term. Because the probability of defect formation \nalong with <100> directions is higher than that along the <110> directions [12], the \n9 \n TMS contribution is stronger along the easy axes , which is in accordance with t he fact \nthat the linewidth s along the [100] and [110] direction s are non -equivalent. Moreover , \nthe linewidth of [010] direction is slightly larger than that along the [100] dire ction, \nsuggesting that another twofold TMS channel is induce d by oblique deposition. As \nindicated by the red solid line in figure 4(b), the linewidth can be well fitted. D ifferent \nparts making sense to the linewidth can therefore be sepa rated and summarized in Tab le \nI. As we know, the TMS predicts the curved non- linear frequency dependence of \nlinewidth, which not appear in a small frequency range for our case (as shown in f igure \n3(a)). The linewidth as function of frequency was also well fitted including the TMS -\ndamping using the parameters in Table I (not shown here) . \nThe larger strength of TMS along the easy axis can clearly explain the anisotropic \nbehavior of da mping , with higher value along the easy axis shown in f igure 3(b). The \nobtained Gilbert damping factor of ~ 7×10-3 is isotropic and invariant with different \noblique angle s. The value of damping is slightly larger than the bulk value of 5.5×10-3 \n[29], which may be attributed to spin pumping of the Ta capping layer. \n The obtained maxi ma of twofold TMS exhibits an increasing trend with the \noblique angle [shown in f igure 4(c)]. According to previous works on the shadowing \neffect [16-19], the larger deposition angle makes the shadow ing effect stronger , and the \ndipolar fields within stripe like defects increase just like the UMA. This can clearly \nexplain that the intensity of two fold TMS follows exactly the same trend with the \ndeposition angle as the UMA . The axis of the maximal intensity of two fold TMS is \nparal lel to the projection of the FeGa atom flux from the fitting data. As shown in Table \nI, amazingly the modified growth conditions also influence the fourfold TMS, \nespecially the strength of TMS along the <100> axis. Figure 4(c) also presents the \nchanges of the fourfold TMS intensity as the deposition angle and shows a peak at 15° , \n10 \n which follows a similar trend as that of 𝛼𝛼eff along [100] axis as shown in f igure 3(b). \nThis indeed confirm s TMS -damping plays an important role in FeGa thin films. \nFor the dispersion relation ω(k∥) in thin magnetic films , the propagation angle \n𝜑𝜑𝑘𝑘∥����⃗ defined as the angle between k∥���⃗ and the projection of the saturation magnetization \nMs into the sample plane is less than the critical value : 𝜑𝜑max =\n𝑐𝑐𝑎𝑎𝑎𝑎−1�𝜇𝜇0𝐻𝐻r(𝜇𝜇0𝐻𝐻r+𝜇𝜇0𝑀𝑀eff) ⁄ [9,36,37] . This implies no degenerate modes are \navailable for the angle 𝜑𝜑𝑘𝑘∥����⃗ larger than φmax. Based on this theory, we propose a \nhypothesis that the crystallographic anisotropy determine s the area including \ndegenerate magnon modes , as well as the intensity of the fourfold TMS. The resonance \nfield along <100> axis change s due to the various crystallographic anisotropy , which \nhas a great effect on the φmax. The values of φmax of samples are shown in f igure 4(d). \nThe data follow the same trend with the oblique angle as Γ<100>. During the grain \ngrowth, the cubic anisotropy is influenced possibly since the anisotropy dispersion due \nto the stress. For the lower anisotropy of sample S2 , a relatively larger amount of stress \nand defects present in the sample and lead to a larger four fold TMS. \n4. Conclusions \nIn conclusion, the effects of oblique deposition on the dynamic properties of FeGa \nthin films have been investigated systematically . The pronounced TMS as non-Gilbert \ndamping results in an anisotropic magnetic relaxation . As the oblique angle increases, \nthe magnitude of the twofold TMS increases due to the larger shadowing effect . \nFurthermore, the cubic anisotropy dominates the area including degenerate magnon \nmodes, as well as the intensity of fourfold TMS. The reported results confirm that the \nmodified anisotropy can influence the extrinsic relaxation pr ocess and open a n avenue \nto tailor magnetic relaxation in spintronic devices. \n11 \n Acknowledgments \nThis work is supported by the National Key Research Program of China (Grant Nos. \n2015CB921403, 2016YFA0300701, and 2017YFB0702702), the National Natural \nSciences Foundation of China (Grant Nos. 91622126, 51427801, and 51671212) and \nthe Key Research Program of Frontier Sciences, CAS (Grant Nos. QYZDJ -SSW -\nJSC023, KJZD -SW-M01 and ZDYZ2012 -2). The work at Beijing Norma l University \nis partly supported by the National Natural Sciences Foundation of China (Grant Nos. 61774017, 61704018, and 11734004), the Recruitment Program of Global Youth \nExperts and the Fundamental Research Funds for the Central Universities (Grant No. \n2018EYT03). \n12 \n References \n[1] Slonczewski J C 1996 J. Magn. Magn. Mater. 159 L1 \n[2] Žutić I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. 76 323 \n[3] Gilbert T L2004 IEEE Trans. Magn. 40 3443 \n[4] He P, Ma X, Zhang J W, Zhao H B, Lüpke G, Shi Z and Zhou S M 2013 Phys. \nRev. Lett. 110 077203 \n[5] Heinrich B, Meredith D J and Cochran J F 1979 J. Appl. Phys. 50 7726 \n[6] Lee A J, Brangham J T, Cheng Y, White S P, Ruane W T, Esser B D, \nMcComb D W, Hammel P C and Yang F Y 2017 Nat. Commun. 8 234 \n[7] Scheck C, Cheng L, Barsukov I, Frait Z and Bailey W E 2007 Phys. Rev. Lett. \n98 117601 \n[8] Azzawi S, Hindmarch A and Atkinson D 2017 J. Phys. D: Appl. Phys. 50 \n473001 \n[9] Arias R and Mills D L 1999 Phys. Rev. B 60 7395 \n[10] Lenz K, Wende H, Kuch W, Baberschke K, Nagy K and Jánossy A 2006 Phys. \nRev. B 73 144424 \n[11] Kurebayashi H et al 2013 Appl. Phys. Lett. 102 062415 \n[12] Zakeri K et al 2007 Phys. Rev. B 76 104416 \n[13] Lindner J, Lenz K, Kosubek E, Baberschke K, Spoddig D, Meckenstock R, \nPelzl J, Frait Z and Mills D L 2003 Phys. Rev. B 68 060102(R) \n[14] Woltersdorf G and Heinrich B 2004 Phys. Rev. B 69 184417 \n[15] Parkes D E et al 2013 Sci. Rep. 3 2220 \n[16] Dijken S, Santo G D and Poelsema B 2001 Phys. Rev. B 63 104431 \n[17] Shim Y and Amar J G 2007 Phys. Rev. Lett. 98 046103 \n13 \n [18] Zhan Q F, Van Haesendonck C, Vandezande S and Temst K 2009 Appl. Phys. \nLett. 94 042504 \n[19] Fang Y P, He W, Liu H L, Zhan Q F, Du H F, Wu Q, Yang H T, Zhang X Q \nand Cheng Z H 2010 Appl. Phys. Lett. 97 022507 \n[20] Barsukov I, Meckenstock R, Lindner J, Möller M, Hassel C, Posth O, Farle M \nand Wende H 2010 IEEE Trans. Magn. 46 2252 \n[21] Barsukov I , Landeros P, Meckenstock R, Lindner J, Spoddig D, Li Z A, \nKrumme B, Wende H, Mills D L and Farle M 2012 Phys. Rev. B 85 014420 \n[22] Mendes J B S, Vilela -Leão L H, Rezende S M and Azevedo A 2010 IEEE Trans. \nMagn. 46(6) 2293 \n[23] Zhang Y, Zhan Q F, Zuo Z H, Yang H L, Zhang X S, Yu Y, Liu Y W, Wang J , \nWang B M and Li R W 2015 IEEE Trans. Magn. 51 1 \n[24] Kalarickal S S, Krivosik P, Wu M Z, Patton C E, Schneider M L, Kabos P, Silva \nT J and Nibarger J P 2006 J. Appl. Phys. 99 093909 \n[25] Bai L H, Gui Y S, Wirthmann A, Recksiedler E, Mecking N, Hu C-M, \nChen Z H and Shen S C 2008 Appl. Phys. Lett. 92 032504 \n[26] Suhl H 1955 Phys. Rev. 97 555 \n[27] Farle M 1998 Rep. Prog. Phys. 61 755 \n[28] Butera A, Gómez J, Weston J L and Barnard J A 2005 J. Appl. Phys. 98 033901 \n[29] Kuanr B K, Camley R E, Celinski Z, McClure A and Idzerda Y 2014 J. Appl. \nPhys. 115 17C112 \n[30] Jhajhria D, Pandya D K and Chaudhary S 2018 J. Alloy Compd. 763 728 \n[31] Jhajhria D, Pandya D K and Chaudhary S 2016 RSC Adv. 6 94717 \n[32] Chen L et al 2018 Nat. Phys . 14 490 \n[33] Liu Y, Starikov A A, Yuan Z and Kelly P J 2011 Phys. Rev. B 84 014412 \n14 \n [34] Starikov A A, Liu Y, Yuan Z and Kelly P J 2018 Phys. Rev. B 97 214415 \n[35] McMichael R D, Twisselmann D J and Kunz A 2003 Phys. Rev. Lett. 90 227601 \n[36] Arias R and Mills D L 2000 J. Appl. Phys. 87 5455 \n[37] Lindner J, Barsukov I, Raeder C, Hassel C, Posth O, Meckenstock R, Landeros \nP and Mills D L 2009 Phys. Rev. B 80 224421 \n \n \n \n \n \n15 \n Figure Captions \nFigure 1 (color online) (a) Schematic illustration of the film deposition geometry and \ncoordinate system (b) I n-plane hysteresis loops of samples S1 and S4 with the field \nalong [100], [110], and [010]. The inset shows the polar plot of the normalized \nremanence (M r/Ms) as a functi on of the in- plane angle. FMR spectrum for the sample \nS2 with H along [100] and [110] axes showing the real (c ) and imaginary (d ) part s of \nthe S 21. \nFigure 2 (color online) (a) H r vs. φH for FeGa films. (b) The anisotropy constants K4 \nand Ku vs. deposition angle. (c) f vs. Hr plots measured at H //[100], Symbols are \nexperimental data and the solid lines are the fitted results. \nFigure 3 (color online) (a) ∆H as a function of f for samples S1 and S2 with field along \neasy and hard axis. (b) The dependence of the damping parameter on the oblique angle \nwith field along [100] and [110] directions. (c) The calculated damping of FeGa alloy \nas a function of lattice distortion. Figure 4 (color online) (a) φ\nM and (b) ∆H as a function of φH for the sample S2 \nmeasured at 13.0 GHz. (c) Oblique angle dependences of Γ<100> and Γtwofoldmax. (d) The \nlargest angle including degenerate magnon modes as a function of the oblique angle \nwith the applied field along <100> direction. \nTable Caption \nTable I. The magnetic relaxation parameters of the FeGa films prepared via oblique \ndeposition (with experimental errors in parentheses). \n \n \n16 \n Figure 1 \n \n \n \n \n \n \n \n \n \n \n17 \n Figure 2 \n \n \n \n \n \n \n \n \n \n \n \n \n \n18 \n Figur e 3 \n \n \n \nFigure 4 \n \n \n \n \n \n \n \n19 \n TableⅠ \nSample 𝜇𝜇0ΔHinh\n(mT) 𝛼𝛼G Δ𝜑𝜑H \n(deg.) Γ<100> \n(107Hz) Γ<110> \n(107Hz) Γtwofoldmax\n(107Hz) 𝜑𝜑twofold \n(deg. ) \nS1 0 0.007 0.62 17(3) 5.8(1.8) 0(2) 90 \nS2 0.7 0.007 1.2 81.4(3.7) 9.3(1.9) 7.4(3) 90 \nS3 0 0.007 1.0 59.2(4.5) 11.1(2) 13(3.7) 90 \nS4 0 0.007 1.1 33.3(6) 14.8(3.7) 26(4) 90 \n \n20 \n " }, { "title": "2008.08196v1.Survey_of_360____circ___domain_walls_in_magnetic_heterostructures__topology__chirality_and_current_driven_dynamics.pdf", "content": "arXiv:2008.08196v1 [cond-mat.mes-hall] 18 Aug 2020Survey of 360◦domain walls in magnetic heterostructures: topology, chir ality and current-driven\ndynamics\nMei Li1and Jie Lu2,∗\n1Physics Department, Shijiazhuang University, Shijiazhua ng, Hebei 050035, People’s Republic of China\n2College of Physics and Hebei Advanced Thin Films Laboratory ,\nHebei Normal University, Shijiazhuang 050024, People’s Re public of China\n(Dated: August 20, 2020)\nChirality and current-driven dynamics of topologically no ntrivial 360◦domain walls (360DWs) in magnetic\nheterostructures (MHs) are systematically investigated. For MHs with normal substrates, the static 360DWs are\nN´ eel-type with no chirality. While for those with heavy-me tal substrates, the interfacial Dzyaloshinskii-Moriya\ninteraction (iDMI) therein makes 360DWs prefer specific chi rality. Under in-plane driving charge currents,\nas the direct result of “full-circle” topology a certain 360 DW does not undergo the “Walker breakdown”-type\nprocess like a well-studied 180◦domain wall as the current density increases. Alternativel y, it keeps a fixed\npropagating mode (either steady-flow or precessional-flow, depending on the effective damping constant of the\nMH) until it collapses or changes to other types of solition w hen the current density becomes too high. Similarly,\nthe field-like spin-orbit torque (SOT) has no effects on the d ynamics of 360DWs, while the anti-damping SOT\nhas. For both modes, modifications to the mobility of 360DWs b y iDMI and anti-damping SOT are provided.\nI. INTRODUCTION\nThe invention and great development of non-volatile mag-\nnetic nanodevices have led to a profound revolution in the in -\nformation industry[ 1–3]. In these nanodevices, various mag-\nnetic solitons or magnetic domains they separate play the ro les\nof 0 and 1 in binary world. In wide magnetic nanostrips,\nskyrmions/antiskyrmions[ 4–8], bimerons[ 9–13] and so on are\ntwo-dimensional (2D) isolated topologically nontrivial m ag-\nnetic solitions surrounded by connected domains with uni-\nform orientation. Under the standard definition of 2D topo-\nlogical charge\nW2D(m)=1\n4π/integraldisplay\nR2m·/parenleftbigg∂m\n∂x×∂m\n∂y/parenrightbigg\nd(x,y), (1)\nin which mis aR3unit magnetization field locating on the\n(x,y)plane, these solitions have an integer W2Dand they\nthemselves are the information carriers. While in narrow\nenough nanostrips which are quasi-one dimensional (Q1D)\nsystems, the most studied magnetic solitons are the 1D (N´ ee l\nor Bloch) 180◦domain walls (180DWs) bearing 1/2 1D topo-\nlogical charge, which is defined as\nW1D(m)=1\n2π/integraldisplay\nR/parenleftbigg\nm1∂m2\n∂ρ−m2∂m1\n∂ρ/parenrightbigg\ndρ, (2)\nwhere m1,2are magnetization components in the wall plane\nand the Q1D systems are supposed to extend in ρ−direction.\n180DWs separate two opposite oriented domains whose ori-\nentations can be defined as 0 and 1, meantimes the wall mo-\ntion leads to the transformation of information. Since the\nfamous Walker analysis[ 14], tremendous progress has been\nmade on statics and dynamics of 180DWs driven by various\nexternal stimuli[ 15–29]. The corresponding results have laid\nthe foundation for many mature commercial and developing\nmagnetic nanodevices.\nInterestingly, even in Q1D systems we also have some\nkinds of isolated magnetic solitions which have integer W1D.Among them, the simplest ones are the so-called 360◦do-\nmain walls (360DWs) in which the magnetization rotates over\none full circle across the intermediate region thus bearing\nW1D=±1. In the beginning of 1960s, 360DWs were first\nfound to appear in the magnetization reversal process of thi n\nfilms and their existence seem to be a nuisance since they may\ncomplicate the reversal process[ 30–32]. However, studies in\nthe past three decades revealed that 360DWs themselves in 2D\nmagnetic films have more interesting physics[ 33–38]. Now\nwe know that 360DWs in lower dimensional systems, such\nas nanorings[ 39–41,43,44,57] and nanostrips[ 45–50], can\nbe qualified candidates to store and process information in\nmagnetic nanodevices due to its “full-circle” topology. Fr om\nthe viewpoint of application, the energy barrier of nucleat ing\na 360DW in single-domain nanorings or nanostrips is much\nlower than a 180DW since in the latter case one should reverse\nthe magnetic moments in entire half. Also in many cases,\na 360DW emerges from the combination of two neighboring\n180DWs with opposite polarity due to the long-range magne-\ntostatic interaction or external magnetic fields.\nFrom the beginning of this century, a series of analytical\nworks focus on the question whether 360DWs are genuine\nstable magnetization textures or just long-lived metastab le\nstates[ 35,51,52]. For 2D ferromagnetic (FM) films, the main\nresults are as follows: (i) the magnetostatics is crucial fo r\nthe existence of 360DWs; (ii) if the long-range component of\nmagnetostatics is neglected, an in-plane external field mus t be\napplied to stabilize a 1D front of 360DW whose energy is in-\ndependent of wall orientation[ 35]. As the films fade into nar-\nrow enough nanostrips, changes in boundary conditions fur-\nther require that the external field should align with the eas y\naxis to guarantee the existence of 360DWs. In recent device\napplications, narrow FM metallic nanostrips often serve as the\ncentral components of magnetic heterostructures (MHs) wit h\nheavy-metal (HM) substrates. Then the effects of interfaci al\nDzyaloshinskii-Moriya interaction (iDMI)[ 53,54] therein to\nthe chirality preference of 360DWs need to be clarified.2\nOnce nucleated, 360DWs in MHs can be driven by certain\nexternal stimuli. First, external magnetic fields along eas y\naxis can not finish this job. This can be understood by our\nroadmap of field-driven domain wall motion since the Zee-\nman energy densities in the two domains on both sides of\n360DWs are the same[ 16]. Then, current-induced motion of\n360DWs becomes the next choice. Indeed, it is the most com-\nmon way to implement and manipulate in real MHs. Numer-\nical investigations on this issue have been widely preforme d\nin the past decade[ 55–61]. Alternatively, there are few ana-\nlytical studies due to the complexity from the coexistence o f\niDMI, spin-transfer torque (STT) and spin-orbit torque (SO T)\ntherein. In this paper, by adopting the Lagrangian-based\ncollective coordinate models (LB-CCMs) and adequate wall\nansatz, the current-driven dynamics of 360DWs is systemati -\ncally explored which constitutes the second part of this wor k.\nThe paper is organized as follows. First, the magnetic La-\ngrangian and dissipation functional of MHs are introduced\nin Sec. II. Both perpendicular magnetic anisotropy (PMA)\nand in-plane magnetic anisotropy (IPMA) for the central FM\nmetallic layers are considered. Then in Sec. III.A we define\nthe proper ansatz for 1D topologically nontrivial 360DWs an d\nthen introduce three typical candidates. After integratin g over\nthe long axis of MHs, a set of unified dynamical equations\nis obtained in Sec. III.B and serves as the startpoint of our\nwork. In Sec. III.C, chirality preference of 360DWs selecte d\nby iDMI is investigated. After that, the propagation mode of\n360DWs under in-plane currents are systematically explore d\nin Sec. III.D. Also, for both modes the effects of iDMI and\nSOT to the dynamics of 360DWs are analytically calculated.\nFinally discussions and concluding remarks are provided in\nSec. IV and V , respectively.\nII. FORMULISM\nA MH under consideration is shown in Fig. 1, which is\ncomposed of three layers: a HM substrate, a central FM metal-\nlic layer and a normal caplayer. We suppose that the MH is\nlong and narrow enough so that it can be viewed as a Q1D\nsystem extended in the long axis. For central FM layers with\nPMA (which will be referred to as “PMA sytems”), the easy\naxis lies in z−axis (out-of-plane normal), the long axis of MH\nis along x−axis and ey=ez×exbeing the hard axis. While for\ncentral FM layers with IPMA (named as “IPMA sytems”), the\neasy axis coincides with long axis and is defined as z−axis, the\nhard axis is along out-of-plane normal and denote as y−axis,\nat last ex=ey×ez. In these two coordinate systems, the crys-\ntalline aisotropy energy density for PMA and IPMA systems\nshares the same form. However, the iDMI should be treated\ncarefully since it is determined by the out-of-plane normal\ncomponent of the magnetization vector. By setting “ n” as the\nout-of-plane normal, the iDMI energy density can be written\nas[62]\nEiDMI=Di{[m(r)·n]∇·m(r)−[m(r)·∇][m(r)·n]},(3)\naJPMA: \n360DW: (a)\nHeavy Metal FM Caplayer \nM\nIT\nxz(easy) \ny(hard) \nI\nT\nz(easy) xIPMA: \n360DW: (b)\ny(hard) \nTTTTy)\nTTTTTTTTTTTTTTTTTTTTTTTTx\nzH\nzHM\nFIG. 1. (Color online) Sketch of a MH in which a 360DW is formed\nin its central FM metallic layer with (a) PMA and (b) IPMA. A ty p-\nical MH is composed of a three-layer structure: a HM substrat e, a\ncentral FM metallic layer and a normal caplayer. The corresp onding\ncoordinate system is depicted at the up-right and bottom-le ft corners\nin the respective subfigure. In each case, the easy (hard) axi s lies in\nthez(y)−direction. An external magnetic field Hz=Hzezis applied\nto guarantee the existence of 360DWs. When in-plane charge c urrent\nJa=jaeρis applied, magnetization vectors will be driven to tilt fro m\ntheir static locations meantime the 360DW will be driven to p ropa-\ngate along the long axis. Gray (orange) planes describe the p lanar\nϕ−distribution of static (dynamical) magnetization texture .\nwhere Diis the iDMI strength and m(r)is the unit magnetiza-\ntion vector at position r. Accordingly, the total magnetic en-\nergy density E0includes the exchange, crystalline anisotropy,\nmagnetostatic, Zeeman and iDMI energies. In narrow enough\nstrips, most of the magnetostatic energy can be described by\nlocal quadratic terms of Mx,y,zby means of three average de-\nmagnetization factors Dx,y,z[25]. In addition, for Q1D systems\n∇≡∂\n∂ρeρin which ρ=x(z)for PMA (IPMA) systems. Thus\none has\nE0[m]=A/parenleftbigg∂m\n∂ρ/parenrightbigg2\n+µ0M2\ns/parenleftbigg\n−1\n2kEm2\nz+1\n2kHm2\ny/parenrightbigg\n−µ0Msm·Hz+EiDMI,(4)\nin which Ais the exchange stiffness, µ0is the permeability of\nvacuum, Msis the saturation magnetization and Hz=Hzez\nis the external magnetic field along the easy axis with the\nstrength Hz. At last, kE(kH)denotes the total anisotropy co-\nefficient along the easy (hard) axis of the central FM layer,\nnamely kE=k1+(Dx−Dz)andkH=k2+(Dy−Dx)with\nk1(2)being the crystalline anisotropy coefficient in easy (hard)\naxis.\nThe in-plane charge current flows along “ eρ” with density\nja. As passing through the MH, the charge current splits\ninto two parts. Suppose jF(jH) to be the component in FM\n(HM) layer. A simple circuit model delivers that jF=ja(tF+\ntH)σF/(tFσF+tHσH)and jH=ja(tF+tH)σH/(tFσF+tHσH),\nwhere tF(tH) and σF(σH) are the thickness and conductivity3\nof the FM (HM) layer, respectively. For the most common FM\nmetal (Co, Ni, Fe) and HM (Pt, Ta, Ir) materials, the conduc-\ntivity varies from 10 to 20 (µΩm)−1. For simplicity, we set\nσF≈σHthus jF=jH=ja. The charge current component\n(jH) in HM substrate will induce a spin current into the FM\nlayer which is polarized in the direction of “ mp≡n×eρ”,\nhence generate the SOT. On the other hand, in the global\nCartesian coordinate system, the unit vector of magnetizat ion\nin the FM layer can be fully described by its polar angle θ\nand azimuthal angle φ, as shown in Fig. 1. The resulting local\nspherical coordinate system is denoted as ( em,eθ,eφ). Then\nmpcan be decomposed as\nmp=pmem+pθeθ+pφeφ. (5)\nBase on all these preparations, the Lagrangian density Land\ndissipation functional density Fof this magnetic system can\nbe expressed as\nL\nµ0M2s=−cosθ\nγMs∂φ\n∂t−BJφ\nγMs∂(cosθ)\n∂ρ+HFL\nMspm−E0\nµ0M2s,\n(6)\nand\nF\nµ0M2s=α\n2γMs/braceleftbigg/bracketleftbigg∂\n∂t−βBJ\nα∂\n∂ρ/bracketrightbigg\nm/bracerightbigg2\n−HADL\nMs/parenleftbig\nm×mp/parenrightbig\n·∂m\n∂t,\n(7)\nin which γ=µ0γewith γebeing the electron gyromagnetic\nratio, BJ=µBP ja/(eMs)with e,µBbeing respectively the ab-\nsolute value of electron charge and Bohr magneton, Pis the\nspin polarization of jF,αis the Gilbert damping constant, βis\nthe dimensionless coefficient describing the relative stre ngth\nof the nonadiabatic STT over the adiabatic one, at last HFL\nandHADL denotes the strength of ��eld-like (FL) and anti-\ndamping-like (ADL) SOT, respectively.\nThe dynamics of magnetization in the central FM layers of\nMHs is then described by the Lagrangian-Rayleigh equation\nd\ndt/parenleftbiggδL\nδ˙X/parenrightbigg\n−δL\nδX+δF\nδX=0, (8)\nin which Xis any related local or collective coordinate. In\nparticular, when X=θandφ(the most common local coordi-\nnates), the resulting two equations can be combined to recov er\nthe familiar Landau-Lifshitz-Gilbert equation\n∂m\n∂t=−γm×Heff+αm×∂m\n∂t+TSTT+TSOT,(9)\nwhere Heff=−(µ0Ms)−1δE0/δm, and\nTSTT=BJ∂m\n∂ρ−βBJm×∂m\n∂ρ, (10)\nas well as\nTSOT=−γHFLm×mp−γHADLm×/parenleftbig\nm×mp/parenrightbig\n. (11)\nHowever, θ(ρ,t)andφ(ρ,t)vary from point to point, hence\ngenerating a huge number of degrees of freedom. To ob-\ntain collective behaviors of magnetization system, LB-CCM sare adopted which need preset ansatz. In the beginning of\nnext section, we will define aqequate ansatz for topological ly\nnontrivial 360DWs and introduce several typical trial profi les\nwhich contain reasonable collective coordinates. Based on\nthem, a set of dynamical equations can be obtained, which\nlays the foundation of our work in this paper.\nIII. RESULTS\nIII.A Adequate ansatz for topologically nontrivial 360DWs\nAs we mentioned above, earlier studies confirm that in Q1D\nMHs if the long-range component of magnetostatics is ne-\nglected, then an external field along the easy axis is crucial\nfor forming a 360DW. Accordingly, an analytical profile of\nstatic 360DWs has been provided based on the requirement\nthat at equilibrium the eθcomponent of Heffdisappears[ 35].\nIn this solution the azimuthal angle takes a fixed value while\nthe polar angle changes monotonously from 0 to πasρruns\nfrom one end of MH to the wall center and then decreases\nback to 0 as ρgoes further to the other end. This nonmono-\ntonic behavior comes from the consideration that polar angl es\nare defined in spherical coordinate system thus can not excee d\nπ. However, we would like to point out that: a 360DW defined\nlike this must be a topologically trivial one . In PMA (IPMA)\nsystems, this corresponds to a “ ↑→↓→↑ ” (“→↑←↑→ ”) type\nwall which first rotates half a circle and then returns back, t hus\nleading to W1D=0.\nOne possible remedy is to add a fixed value πto the az-\nimuthal angle when the polar angle crosses the South Pole.\nIn principle this new set of polar and azimuthal is indeed the\nreal spherical coordinates that realizes a topologically n ontriv-\nial “↑→↓←↑ ” (“→↑←↓→ ”) type wall for PMA (IPMA) sys-\ntems, however it will artificially bring a discontinuity poi nt in\nexchange energy. For the convenience of comparisons below,\nwe denote them as ϑrealandϕreal. To remove the artificial dis-\ncontinuity, we propose a monotonically increasing “0 to 2 π”\npolar angle profile meanwhile keep the azimuthal angle a con-\nstant value which are defined as ϑansatz andϕansatz . We focus\non the “ πto 2π” part since this is the main region where dif-\nferences occur. Obviously, we have\nϑreal=2π−ϑansatz,ϕreal=ϕansatz+π, (12)\nand they lead to the same magnetization component as follows\nreal spherical ansatz\nmx: sin ϑrealcosϕreal≡sinϑansatz cosϕansatz\nmy: sin ϑrealsinϕreal≡sinϑansatz sinϕansatz\nmz: cos ϑreal≡cosϑansatz.(13)\nIn addition, the polar and azimuthal profiles in our proposal\nare not bothered by discontinuities in continuous Heisenbe rg\nexchange interaction, meanwhile provide W1D= +1. Based4\non these facts, we conclude that a 360DW profile with a mono-\ntonically increasing “0 to 2 π” polar angle and a constant az-\nimuthal angle should be an adequate ansatz for 360DWs.\nIn this work, we use three trial profiles of 360DWs to ex-\nplore their chirality preference and current-driven dynam ics.\nThe first one is inspired by the work of Muratov in 2008[ 35],\nbut has been generalized to [0,2π)as we proposed above. By\nintroducing the “traveling coordinate” ξ≡ρ−q(t)\n∆(t)where q(t)\nand∆(t)are respectively the center position and width of the\n360DW, it can be written as\nϑ=2cot−1/bracketleftBigg/radicalbigg\nh\n1+hsinh/parenleftBig\n−√\n1+hξ/parenrightBig/bracketrightBigg\n,φ(r,t)=ϕ(t),\n(14)\nin which h≡Hz\nkEMsand the “cot−1” function takes the range of\n0 toπ. Note that Eq. ( 14) is accurate in the absence of driv-\ning current. When in-plane currents are applied, this solut ion\nbecomes an approximation since it may not hold everywhere\nbut it does grasp the main features of dynamical 360DWs. In\nparticular, Eq. ( 14) clearly ascertains the conclusion that in\nthe absence of external magnetic fields along the easy axis\n(i.e.h=0),ϑkeeps a constant value thus 360DWs disappear.\nAlso, we have two other options. The second trial profile is\ndirectly generalized from the Walker ansatz, which reads\nϑ=4tan−1eξ,φ(r,t)=ϕ(t), (15)\nand the third one is\nϑ=2π\n1+e−ξ,φ(r,t)=ϕ(t). (16)\nObviously, the latter two do not depend on h, thus can not\nbe rigorous even in the absence of driving currents. However ,\ndue to their mathematical simplicity, they can be used as ref er-\nences. In particular when h=1 (h=π2/16), d ϑ/dξatξ=0\nin Eq. ( 14) coincides with that of Eq. ( 15) [Eq. ( 16)]. The\ncorresponding ϑprofiles are plotted in Fig. 2. Also, curves\nwith h=0.1 and h=5 have been appended to illustrate the\ndependence of polar angle profile in Eq. ( 14) on h: as hin-\ncreases the effective width [not the parameter ∆(t)] of 360DW\nis compressed.\nIII.B Dynamical equations\nIn all three trial profiles, the wall center position q(t), tilt-\ning angle ϕ(t)and wall width ∆(t)are the three collective\ncoordinates. In Eq. ( 8), by letting Xtake q(t),ϕ(t),∆(t)\nsuccessively, and integrating over the long axis of MHs (i.e ./integraltext+∞\n−∞dρ), a set of dynamic equations can be obtained and ex-\npressed in a unified form for both PMA and IPMA systems:/s45/s52 /s45/s50 /s48 /s50 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s32/s32/s32/s40 /s41/s32/s69/s113/s46/s32/s40/s49/s52/s41/s32/s119/s105/s116/s104/s32/s104/s61/s48/s46/s49\n/s32/s69/s113/s46/s32/s40/s49/s52/s41/s32/s119/s105/s116/s104/s32/s104/s61/s50\n/s47/s49/s54\n/s32/s69/s113/s46/s32/s40/s49/s52/s41/s32/s119/s105/s116/s104/s32/s104/s61/s49/s46/s48\n/s32/s69/s113/s46/s32/s40/s49/s52/s41/s32/s119/s105/s116/s104/s32/s104/s61/s53/s46/s48\n/s32/s69/s113/s46/s32/s40/s49/s53/s41\n/s32/s69/s113/s46/s32/s40/s49/s54/s41\nFIG. 2. (Color online) Trial polar angle profiles in Eq. ( 14) - Eq.\n(16). Four solid curves are those from Eq. ( 14) with different h,\nwhile the magenta (blue) dashed curve shows Eq. ( 15) [Eq. ( 16)].\n0=(α˙q+βBJ)−2π\nI1γHADL∆·f(ϕ), (17a)\n0=α\nγMs˙ϕ+kHsinϕcosϕ\n+I2\nI31\nγMs˙∆\n∆+2π\nI3Di\nµ0M2s∆df(ϕ)\ndϕ,(17b)\nαI4\nγMs˙∆\n∆=I2\nγMs˙ϕ−/parenleftbig\nkE+kHsin2ϕ/parenrightbig\nI5−kEI2h+λl2\n0\n∆2,(17c)\nwhere an overdot means ∂/∂tandl0=/radicalbig\n2A/(µ0M2s). For\nPMA systems f(ϕ)=cosϕwhile for IPMA systems f(ϕ)=\n−sinϕ. The five integrals ( I1toI5) can be defined in a gen-\neral way without depending on the specific form of trial pro-\nfiles (see the first column of Table I). Their values and the pa-\nrameter λunder each profile have been listed in the last three\ncolumns of Table I. We also plot them in Fig. 3 as functions\nofhto show their evolution as hincreases.\nEq. ( 17) is the starting point for our investigations on chiral-\nity and current-driven dynamics of 360DWs in Q1D MHs. Be-\nfore explicitly solving it, we would like to discuss its qual ita-\ntive properties first. In the dynamical equations for 180DWs ,\nthe iDMI, FL-SOT and ADL-SOT are all present. However\nin Eq. ( 17), the FL-SOT disappears. This can be under-\nstood based on the mathematical form of SOTs in Eq. ( 11).\nThe main difference lies in the fact that the FL-term is lin-\near to the magnetization mwhile the ADL-term is quadratic\n(thus is nonlinear). When integrating over the whole strip, the\nconstant “−γHFLmp” factor can be brought up, leaving “ m”\nto be integrated over a full circle thus canceled out. How-\never, this procedure fails for the ADL-term since the consta nt\n“−γHADLmp” factor can not be brought up there. This ex-\nplains the presence (absence) of HADL(HFL) in Eq. ( 17). Sim-\nilar analysis can be made to explain the presence of both HADL\nandHFLin 180DW case. Furthermore, a general rule can be\nsummarized as follows: When dealing with current-driven dy-5\nTABLE I. Summary of parameters in Eq. ( 17): definitions and values based on the three trial profiles in E qs. (14) to ( 16). First to fifth rows:\nIntegrals I1toI5. Last row: Parameter λ.\nParameter: Definition Value on Eq. ( 14) Value on Eq. ( 15) Value on Eq. ( 16)\nI1: ∆/integraltext2π\n0∂ ϑ\n∂ ρdϑ 4√\n1+h+2hln√\n1+h+1√\n1+h−182\n3π2\nI2:/integraltext2π\n0sinϑ(−ξ)dϑ 2ln√\n1+h+1√\n1+h−14 2/integraltext2π\n01−cost\ntdt≈4.8753\nI3:1\n∆/integraltext2π\n0sin2ϑ\n∂ ϑ/∂ ρdϑ 4√\n1+h−2hln√\n1+h+1√\n1+h−18\n3/integraltext4π\n01−cost\ntdt≈3.1144\nI4: ∆/integraltext2π\n0∂ ϑ\n∂ ρξ2dϑ8h\n(1+h)3/2/integraltext+∞\n0√\n1+x2(sinh−1x)2\n(1+h\n1+hx2)2dx2\n3π2(2π)2/integraltext+∞\n0x(lnx)2\n(1+x)4dx≈8.4870\nI5:/integraltext2π\n0sinϑcosϑ(−ξ)dϑI3\n2\nλ: I1−hI2−I5 I1−I2I1\n2\n/s48/s49/s48/s50/s48\n/s48/s53/s49/s48\n/s48/s50/s52\n/s48/s49/s48/s50/s48/s51/s48\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s52/s56/s32\n/s32/s32\n/s73\n/s49/s40/s97/s41\n/s32/s58/s32/s69/s113/s46/s32/s40/s49/s52/s41\n/s32/s58/s32/s69/s113/s46/s32/s40/s49/s53/s41\n/s32/s58/s32/s69/s113/s46/s32/s40/s49/s54/s41/s40/s99/s41/s32/s40/s98/s41\n/s32/s73\n/s50\n/s32\n/s32/s73\n/s51\n/s40/s100/s41\n/s32\n/s32/s73\n/s52\n/s40/s101/s41\n/s32\n/s104\nFIG. 3. (Color online) Evolution of I1,2,3,4andλashincreases for\nEq. ( 14) - Eq. ( 16). Note that I5is omitted since it is always half of\nI3.\nnamical equations of magnetic domain walls, only for “ 2nπ”\nwalls H FLdisappears; otherwise H ADLand H FLcoexist. Par-\nallel discussions can be performed to the anisotropic field p ro-\nportional to kH, which profoundly affect the dynamic behav-\niors of 360DWs. We will revisit this issue in Section 3.4.1.III.C iDMI-induced chirality for static 360DWs\nBy first choosing the easy-axis-oriented single-domain sta te\nas reference, and then integrating over the Q1D MH, the\n“renormalized magnetic energy” Ere\n0of the central FM layer\nreads\nEre\n0\nµ0M2sS=I1\n2l2\n0\n∆+/bracketleftbiggI3\n2/parenleftbig\nkE+kHsin2ϕ/parenrightbig\n+kEI2h/bracketrightbigg\n∆\n+2πDif(ϕ)\nµ0M2s,(18)\nwhere Sis the cross section of the central FM layer. Combing\nwith Eq. ( 17) for ja=0 (thus HADL=0 and BJ=0), the\nchirality preference of static 360DWs can be analyzed.\niDMI is absent\nFirst we review the simplest case where the iDMI is ab-\nsent ( Di=0). Physically this corresponds to MHs with nor-\nmal substrates. Then the dynamical equations, as well as the\nrenormalized magnetic energy for PMA and IPMA systems\nare the same. Since HADL=0 and BJ=0, Eq. ( 17a) provides\n˙q=0 meaning that the 360DW keeps static. A static wall also\nrequires that ˙ϕ=0 and ˙∆=0. Putting them into Eq. ( 17b),\none has sin2 ϕ=0 which means ϕ=nπ\n2. However, Eq. ( 18)\nclearly tells us that only ϕ=nπ(i.e. sin ϕ=0) minimizes\nEre\n0. Therefore, in the absence of iDMI, 360DWs should be\nN´ eel type, but have no chirality preference. At last, Eq. ( 17c)\nprovides the static wall width ∆0as\n∆0=l0√kE/radicalBigg\nλ\nI2h+I5. (19)\nNote that ∆0should not be obtained from the direct minimiza-\ntion of the first two terms in Eq. ( 18) since the result may\nnot satisfy the dynamical equations. This argument also hol ds\nwhen iDMI appears.6\nPMA systems with iDMI\nFor PMA systems, f(ϕ) =cosϕ. The combination of Eq.\n(17b) and the static requirement ( ˙ϕ=0 and ˙∆=0) leads to\n\n\ncase(a): sin ϕ=0 or\ncase(b): cos ϕ=2πDi\nkHI3µ0M2s∆(20a)\n(20b)\nTo determine which solution provides the real tilting angle ,\nwe must compare the corresponding “renormalized magnetic\nenergy” in Eq. ( 18). For case (a), sin ϕ=0⇔ϕ=nπ. How-\never, the existence of iDMI [the last term in Eq. ( 18)] breaks\nthe two-fold degeneracy of Ere\n0upon azimuthal angle. To min-\nimize Ere\n0, one must have\ncosϕ=−sgn(Di), (21)\nwhere “sgn” denotes the sign function. Correspondingly in\nthis case the renormalized magnetic energy becomes\n/parenleftbig\nEre\n0/parenrightbig\na\nµ0M2sS=I1\n2l2\n0\n∆+kE/parenleftbiggI3\n2+I2h/parenrightbigg\n∆−2π|Di|\nµ0M2s. (22)\nFor case (b), direct calculation yields\n/parenleftbig\nEre\n0/parenrightbig\nb\nµ0M2sS=I1\n2l2\n0\n∆+kE/parenleftbiggI3\n2+I2h/parenrightbigg\n∆\n+/bracketleftBigg\n(2πDi)2\n2kHI3(µ0M2s)2∆+I3\n2kH∆/bracketrightBigg\n.(23)\nObviously for any positive ∆, we always have/parenleftbig\nEre\n0/parenrightbig\na0.(28)\nThis is the fundamental equation when dealing with current-\ndriven 360DW dynamics.7\niDMI is absent\nFirst we consider the simplest case where the iDMI is ab-\nsent ( Di=0), which corresponds to a 360DW residing in a\nMH with a normal substrate. Note that at this moment the\nSOT is also absent. Now Eq. ( 27) can be directly integrated\nout and the result depends on the value of Γ.\nWhen|Γ|<1,\ntan/parenleftBig\nϕ+κ\n2/parenrightBig\n=1\nΓ−√\n1−Γ2\nΓC1e√\n1−Γ2χt+1\nC1e√\n1−Γ2χt−1, (29)\nwith\nC1=Γtan/parenleftBig\nϕ0+κ\n2/parenrightBig\n−1−√\n1−Γ2\nΓtan/parenleftBig\nϕ0+κ\n2/parenrightBig\n−1+√\n1−Γ2,\nandϕ=ϕ0att=0. Obviously when t→+∞the azimuthal\nangle approaches the following value\nϕ∞=arctan/parenleftBigg\n1\nΓ−√\n1−Γ2\nΓ/parenrightBigg\n−κ\n2. (30)\nThis means that in this case the 360DW will eventually fall\ninto the “steady-flow” mode. By letting ˙ϕ=0 and ˙∆=0,\nwe know that the wall propagates with a constant velocity\n−βBJ/αand a fixed width\n∆(ϕ∞)=l0√kE/radicalBigg\nλ\nI2h+I5+(kH/kE)I5sin2ϕ∞. (31)\nWhen|Γ|>1,\nϕ=arctan/bracketleftBigg√\nΓ2−1\nΓtan/parenleftBigg√\nΓ2−1\n2χt+C2/parenrightBigg\n+1\nΓ/bracketrightBigg\n−κ\n2,\n(32)\nwith\nC2=arctanΓtan/parenleftBig\nϕ0+κ\n2/parenrightBig\n−1\n√\nΓ2−1.\nNow the azimutial angle rotates periodically with the perio d\nT0=4π\nχ√\nΓ2−1, (33)\nwhich means that the 360DW takes a “precessional-flow”\nmode with the same constant velocity −βBJ/αand a peri-\nodically changing width.\nIt is worth noting that for a certain 360DW as the current\ndensity increases the wall always takes a specific mode (eith er\nsteady-flow or precessional-flow) rather than going through\na process of mode change, which is quite different from the\ncommonly studies 180DWs. This is the direct consequence\nof the “full-circle” topology that 360DWs hold. Similar wit hthe discussions in Sec. 3.2, for 180DWs [or other “ (2n+1)π”\nwalls] the incomplete cancellation over a half-circle rota tion\nofmleads to the appearance of “ kHsin2ϕ” term, and then\nresults in the famous “Walker breakdown” process. However\nfor 360DWs (or other “2 nπ” walls), the full cancellation of\nanisotropic field leads to the absence of “ kHsin 2ϕ” term in\nEq. ( 17), thus results in the “fixed mode” behavior. Interest-\ningly, both modes share the same wall mobility which is equal\nto that in steady-flow mode of 180DWs. This explains nearly\nall existing numerical observations before 360DWs change t o\nother magnetic solitons (for example vortices) under too hi gh\ncurrents [ 55,60,61].\nFurthermore, we provide the sufficient but non-necessary\ncondition for the steady flow of 360DWs. Under the presup-\nposition|Γ|<1, by putting the wall width [see Eq. ( 31)] back\ninto the definition of Γ[see Eq. ( 28)], we obtain\nΓ=I2I5\nαI3I4cos2 ϕ∞. (34)\nThus the sufficient but non-necessary condition for |Γ|<1\nshould be I2I5/(αI3I4)<1, which corresponds to α>αc≡\nI2I5/(I3I4). For the second and third trial profiles, one has\nαc=3/π2≈0.304 and αc=0.287, respectively. While for\nthe first profile, αcis the function of h. We have plotted their\ndependence on hin Fig. 4. One can clear see that for all\nthree cases αchas a upper limit 3 /π2even when hincreases\nto 5 which is a quite high value in real experiments. In many\nMHs, existing measurements show that the effective damping\nin FM strips is enhanced from 0.001-0.01 to 0.3-0.9[ 63,64].\nThis guarantees that experimentally 360DWs should take the\nsteady-flow mode. As for precessional flow, since the explici t\nform of wall width is hard to obtain, thus it is difficult to obt ain\nthe definite range of its existence. However, from the above\ndiscussion we can reasonably infer that for sufficient small\nα, 360DWs precess. This prediction needs to be verified by\nfuture experiments and numerical simulations.\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s32/s32\n/s99\n/s104/s32/s69/s113/s46/s32/s40/s49/s52/s41\n/s32/s69/s113/s46/s32/s40/s49/s53/s41\n/s32/s69/s113/s46/s32/s40/s49/s54/s41\nFIG. 4. (Color online) Dependence of αconhbased on Eq. ( 14) -\nEq. ( 16).8\niDMI and ADL-SOT are present\nNext we study the effects of iDMI and ADL-SOT on the\ncurrent-driven dynamics of 360DWs in MHs with HM sub-\nstrates. In the presence of iDMI, in principle the azimuthal\nangle ϕcan not be integrated out explicitly from Eq. ( 27). Re-\ncently a phase diagram has been drawn to show how the types\nof solutions are determined by the DMI and the anisotropic\nparameters[ 65]. However in real MHs, generally the iDMI is\nweaker than other magnetic interactions, thus can be reason -\nably viewed as a small quantity. Depending on the value of Γ,\ndifferent approximate treatments will be used.\nWhen|Γ|<1 or under the stronger condition α>αc, at the\nlowest level of approximation the 360DW should eventually\npropagate like a rigid body with the finial azimtuhal angle ϕ∞,\nwidth ∆(ϕ∞)and velocity\n˙q=−β\nαBJ+2π\nαI1γHADL∆(ϕ∞)·f(ϕ∞). (35)\nObviously, the wall mobility is modified by the second term.\nHowever the effect of iDMI is totally submerged since it has\nbeen dropped when obtaining ϕ∞. Note that the form of ∆(ϕ∞)\nin Eq. ( 31) is not effected by this dropping.\nWhen|Γ|>1, the 360DW precesses. In this case for a\nphysical quantity O, its time average\n/an}bracketle{tO/an}bracketri}ht≡1\nT/integraldisplayT\n0X(t)dt=1\nT/integraldisplay2π\n0X\n˙ϕdϕ (36)\ncorresponds to experimental observables, where Tis the pre-\ncession period. Under the assumption of small iDMI, we cal-\nculate the time-averaged wall velocity /an}bracketle{t˙q/an}bracketri}ht. First the period\nTis replaced by T0in Eq. ( 33). Then the approximation\n(1−x)−1≈1+x+x2for|x|<1 is used to simplify 1 /˙ϕin\nEq. ( 27) hence the integral in Eq. ( 36) can be calculated.\nAfter standard algebra, we have\n/an}bracketle{t˙q/an}bracketri}ht=−β\nαBJ−η√\nΓ2−1\nΓ3γHADLDi\nµ0M2s4π2cosκ\nαkHI1I3,(37)\nwhere η=+1 (−1) for PMA (IPMA) systems. Clearly, Eq.\n(37) provides the effects of both ADL-SOT and iDMI to the\nwall velocity in precessional flows.\nIV . DISCUSSIONS\nFirst, one should note that the premise of all our analyti-\ncal results is the existence of 360DWs. The constant mobil-\nity (whether adjusted by iDMI and SOT or not) upon current\nincrease is the direct manifestation of the wall’s “full-ci rcle”\ntopology. Accordingly, strong enough external stimuli wou ld\ndestroy the configuration of 360DWs, thereby greatly change\nthe mobility of domain walls (not 360DWs any more). This\nexplains the huge reduction of 360DW mobility under high\ncurrents in existing numerics[ 55,60,61].Second, our analytics presented here is based on “0 to 2 π”\nmonotonic profiles of polar angle. If ϑis no longer monotonic\nbut its overall change across the wall region keeps 2 π(W1D=\n+1 still holds), then the results will be unchanged. In additi on,\nfor a 360DW with W1D=−1 mathematically its profile can be\ntransfer to that with W1D=+1, except for an increase by πin\nthe azimuthal angle. The following procedure is similar to\nwhat we have presented in the main text and will not provide\nnew physics, so we won’t repeat it.\nAt last, topologically the 1D 360DWs in narrow MHs un-\nder investigation here are analogous to the 1D domain wall\nskyrmions (DWSs) evolved from vertical Bloch lines in wide\nMHs with PMA[ 66]. Both magnetic solitons carry integer 1D\ntopological charges ( W1D=±1), hence should belong to the\nsame topology class. The effective field of iDMI in that work\nplays the role of external fields along easy axis here, theref ore\nis crucial to the formation of 1D DWSs. The current-driven\nresults here may provide insights for exploring dynamical b e-\nhaviors of 1D DWSs under external stimuli.\nV . CONCLUSION\nIn this work, the topology, chirality and current-driven dy -\nnamics of 360DWs in Q1D MHs are systematically investi-\ngated. On one hand, the iDMI uniquely select the chirality of\nstatic 360DWs. On the other hand, the “full-circle” topolog y\nof 360DWs makes them completely different from the tradi-\ntional 180DWs. For 360DWs, effective fields which are linear\nto the magnetization have been fully canceled out and disap-\npear in the dynamical equations. In particular, the full can cel-\nlation of magnetic anisotropic fields directly results in th e ab-\nsence of “Walker breakdown”-type process under increasing\ncurrents. In a certain MH, 360DWs will take either steady-\nflow or precessional-flow mode, depending on the strength\nof effective Gilbert damping constant therein. In MHs with\nnormal substrates, the wall mobility of both modes are the\nsame as that in the steady-flow mode of STT-driven propa-\ngation of 180DWs. While in MHs with HM substrates, the\nmobility will be modified by the ADL-SOT and iDMI. These\nresults should deepen our understanding of topological sol i-\ntons in low-dimensional magnetic systems, meanwhile pro-\nvide necessary theoretical basis for expanding the applica tion\nof 360DWs in the field of magnetic nanodevices.\nACKNOWLEDGEMENT\nM. L. is supported by the National Natural Science Founda-\ntion of China (Grants No. 11947023) and the Project of Hebei\nProvince Higher Educational Science and Technology Pro-\ngram (QN2019309). J. 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He, T.-F. Xue, B. Wu, J. Magn. Magn.\nMater. 512(2020) 166981.\n[66] R. Cheng, M. Li, A. Sapkota, A. Rai, A. Pokhrel, T. Mewes,\nC. Mewes, D. Xiao, M. De Graef, V . Sokalski, Phys. Rev. B 99\n(2019) 184412." }, { "title": "1107.2208v1.Mode_conversion_of_radiatively_damped_magnetogravity_waves_in_the_solar_chromosphere.pdf", "content": "Mon. Not. R. Astron. Soc. 000, 000{000 (0000) Printed 1 February 2018 (MN L ATEX style \fle v2.2)\nMode conversion of radiatively damped magnetogravity\nwaves in the solar chromosphere\nMarie E. Newington1?and Paul S. Cally1;2y\n1Monash Centre for Astrophysics, School of Mathematical Sciences, Monash University, Victoria, Australia 3800\n2High Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307, USA\n1 February 2018\nABSTRACT\nModelling of adiabatic gravity wave propagation in the solar atmosphere showed that\nmode conversion to \feld guided acoustic waves or Alfv\u0013 en waves was possible in the\npresence of highly inclined magnetic \felds. This work aims to extend the previous\nadiabatic study, exploring the consequences of radiative damping on the propagation\nand mode conversion of gravity waves in the solar atmosphere. We model gravity waves\nin a VAL-C atmosphere, subject to a uniform, and arbitrarily orientated magnetic\n\feld, using the Newton cooling approximation for radiatively damped propagation.\nThe results indicate that the mode conversion pathways identi\fed in the adiabatic\nstudy are maintained in the presence of damping. The wave energy \ruxes are highly\nsensitive to the form of the height dependence of the radiative damping time. While\nsimulations starting from 0.2 Mm result in modest \rux attenuation compared to the\nadiabatic results, short damping times expected in the low photosphere e\u000bectively\nsuppress gravity waves in simulations starting at the base of the photosphere. It is\ndi\u000ecult to reconcile our results and observations of propagating gravity waves with\nsigni\fcant energy \rux at photospheric heights unless they are generated in situ , and\neven then, why they are observed to be propagating as low as 70 km where gravity\nwaves should be radiatively overdamped.\nKey words: Sun: oscillations { Sun: chromosphere { waves { magnetic \felds.\n1 INTRODUCTION\nRecent multi-height observations of low frequency oscilla-\ntions in the low solar chromosphere suggest that the energy\n\rux carried by upward propagating gravity waves, with fre-\nquencies between 0.7 and 2.1 mHz, comfortably exceeds the\nco-spatial acoustic wave \rux (Straus et al 2008). In our pre-\nvious paper (Newington & Cally 2010), (referred to hence-\nforth as paper I), we explored the propagation, re\rection and\nmode conversion of gravity waves in a VAL C related atmo-\nsphere, permeated by uniform, inclined magnetic \feld. The\nsigni\fcant \fnding of that study was that in regions of highly\ninclined magnetic \feld, gravity waves experience mode con-\nversion to up-going (\feld-guided) acoustic or Alfv\u0013 en waves.\nWhile acoustic waves are likely to shock before reaching the\nupper chromosphere, Alfv\u0013 en waves can propagate to greater\natmospheric heights, perhaps contributing to the observed\ncoronal Alfv\u0013 enic oscillations (De Pontieu et al. 2007; Tom-\nczyk et al. 2007).\nA limitation of the work presented in paper I was the\n?E-mail: Marie.Newington@monash.edu\nyE-mail: Paul.Cally@monash.eduassumption of adiabatic wave propagation, which is known\nto be invalid in the photosphere and low chromosphere. A\nmore realistic investigation of the propagation and mode\nconversion of gravity waves in the solar atmosphere would\ninclude the e\u000bects of radiative damping.\nRadiatively damped atmospheric gravity waves have\nbeen considered by a small number of authors previously,\n(see Sou\u000brin 1966, Stix 1970, and the exhaustive study by\nMihalas & Toomre 1982), but all of those studies have been\npurely hydrodynamic in nature, with no applied magnetic\n\feld, hence there is no possibility of mode conversion. Our\nprincipal objective in this work is to explore the conse-\nquences of radiative damping on the mode conversion of\ngravity waves, and this requires a magnetohydrodynamic\n(MHD) treatment.\nIn this paper we extend our earlier MHD study, relaxing\nthe assumption of adiabatic wave propagation by incorpo-\nrating radiative damping using the Newton cooling approx-\nimation. We recognise that this approximation is strictly\nvalid only for optically thin perturbations of a homogeneous,\nin\fnite, and isothermal gas, and that none of these condi-\ntions are realised in the photosphere and chromosphere. The\nadvantage of using this simple treatment is that the same\nc\r0000 RASarXiv:1107.2208v1 [astro-ph.SR] 12 Jul 20112M. E. Newington and P. S. Cally\nmathematical tools used in paper I can be employed with\nminor modi\fcations. The insight gained from this simple\nmodel may direct our attention to cases that are interesting\nenough to warrant a more thorough treatment of the e\u000bects\nof radiation.\nThe questions central to our investigation are: How are\nthe mode conversion pathways a\u000bected by radiative damp-\ning? Do we still get appreciable mode conversion to Alfv\u0013 en\nwaves when damping is present?\nThis paper is organised as follows: Section 2 presents the\natmospheric model and the mathematical tools used in this\ninvestigation, namely the damped dispersion relation and\nnumerical solution of the damped linear MHD wave equa-\ntions. Section 3 presents the results. Dispersion diagrams\nreveal insights into the mode conversion pathways in z-kz\nphase space, and numerical integration of the wave equations\nquantify the acoustic and magnetic \ruxes. The conclusions\nare stated in Section 4.\n2 MODEL AND EQUATIONS\nThis section describes the atmospheric model, the coordi-\nnate system and the equations used to generate the results\nof this paper. As this work is an extension of the adiabatic\ninvestigation of paper I, the reader will be referred to that\npaper to avoid repetition of material, where appropriate.\n2.1 Atmospheric model and coordinate system\nWe use the same atmospheric model as employed in paper I\n{ the Schmitz & Fleck (2003) adaptation of the horizontally\ninvariant VAL C model (Vernazza et al. 1981) up to a height\n1.6 Mm above the base of the photosphere. An isothermal\ntop is appended above 1.6 Mm. No transition region is in-\ncluded. A uniform inclined magnetic \feld is imposed upon\nthis atmosphere.\nThe origin of the coordinate system is located at the\nbase of the photosphere. The coordinate system is orientated\nsuch that the wavevector ( k) lies in the x-zplane, and the\norientation of the magnetic \feld ( B) is described in terms of\nthe inclination from the vertical ( \u0012) and the azimuthal angle\n(\u001e) (see Fig. 1). We distinguish between two dimensional\n(2D) and three dimensional (3D) con\fgurations by the angle\nbetween the magnetic \feld and the vertical plane of wave\npropagation ( \u001e): 2D:\u001e= 0; 3D:\u001e6= 0.\nFollowing paper I, our tools for investigation of radia-\ntively damped gravity wave propagation are the dispersion\nrelation and numerical integration of the linearised MHD\nwave equations. Ray theory is not directly employed as it\nis di\u000ecult to institute and interpret in the presence of dis-\nsipation. The derivation of the appropriate forms of these\nequations including Newton cooling are described below.\n2.2 Mathematical tools incorporating Newton\ncooling\n2.2.1 Energy equation and expression for the pressure\nperturbation\nIn the Newton cooling approximation, the radiative damp-\ning is parametrized by the radiative relaxation time ( \u001c), as-\nkB\nθ\nφz\nxyFigure 1. The coordinate system and geometry adopted for this\nanalysis. Bis the magnetic \feld vector and kis the wavevector.\nsumed a function of height zonly. As\u001c!1 , the motions\nbecome adiabatic. The continuity equation and the momen-\ntum equation are unchanged in the Newton cooling approxi-\nmation, but the energy equation is modi\fed by the inclusion\nof a term proportional to the temperature perturbation, as\nfollows (see, for example, Cally 1984):\nDp\nDt=c2D\u001a\nDt\u0000p0\n\u001cT1\nT0: (1)\nHere\u001ais the density, pis the pressure, Tis the temperature\nandcis the sound speed. The subscripts 0 and 1 denote the\nbackground values and Eulerian perturbations respectively.\nUpon linearising equation (1), applying the WKBJ ap-\nproximation, and making use of the perfect gas law, the def-\ninition of the adiabatic sound speed c2=\rp0=\u001a0(where\r\nis the ratio of speci\fc heats), and the magentohydyrostatic\nbalance, the following equation for the pressure perturbation\nis obtained.\np1=\u00161\u001a0g\u0010\u0000\u00162\u001a0c2r\u0001\u0018; (2)\nwhere\n\u00161= (1 +i\n\r!\u001cc2\ngH)(1 +i\n!\u001c)\u00001(3)\n\u00162= (1 +i\n\r!\u001c)(1 +i\n!\u001c)\u00001(4)\nandgis the acceleration due to gravity. Hthe density scale\nheight and\u0018= (\u0018;\u0011;\u0010 ) is the displacement vector. In the\nadiabatic limit \u00161and\u00162reduce to 1.\n2.2.2 Height dependence of the radiative relaxation time \u001c\nAlthough expressions for the radiative damping time have\nbeen provided by Spiegel (1957) and Stix (1970), (for con-\ntinuum and line emission, respectively), the actual values for\nthe radiative relaxation time in the photosphere and chro-\nmosphere are only crudely known.\nIn this paper it will su\u000ece to adopt the simple linear\nc\r0000 RAS, MNRAS 000, 000{000Radiatively damped gravity waves 3\n0.000.050.100.150.200.2501234\nzHMmLDamping ratio-1H2gNtRL\nFigure 2. The behaviour of the damping ratio as a function of\nheight for the VAL-C atmosphere and the Mihalas & Toomre\n(1982) prescription for the radiative damping times.\nradiative damping time introduced by Mihalas & Toomre\n(1982) to generate the results in that paper (curve 2):\n\u001c(z) = 50 + (2200 =3)z; (5)\nwhere\u001cis expressed in seconds and zin megameters.\nVery short damping times are known to suppress the\npropagation of gravity waves. When Newton cooling is in-\ncluded in the energy equation, the velocity of a vertically dis-\nplaced \ruid element may be described in terms of a damped\nharmonic oscillator, (see, Sou\u000brin 1966 and Bray & Lough-\nhead 1974). This allows identi\fcation of the damping ratio\nas 1=(2\rN\u001c), whereNis the Brunt-V ais al a or buoyancy\nfrequency. If the damping ratio is greater than one, the mo-\ntion is overdamped and the \ruid parcel will return to the\nequilibrium position without oscillating. Applying this con-\ndition to their atmosphere, Mihalas & Toomre (1982) noted\nthat the wave was overdamped below 0.1 Mm and so they\nstarted their simulations from this height. The atmosphere\nin this paper has a di\u000berent height dependence of the Brunt-\nV ais al a frequency, and the location where the damping ratio\nis 1 is higher at about 0.2 Mm (see \fg. 2). Hence, we expect\ngravity waves to be underdamped and propagating above\n0.2 Mm, and overdamped below this height.\nWe also ran simulations using constant relaxation times\nfrom 200s to 1 ks to gauge the sensitivity of the wave prop-\nagation to the form of \u001c, but (5) was used to generate all\nthe \fgures in this paper.\n2.2.3 Dispersion relation\nThe adiabatic 3D MHD dispersion relation described in\nNewington & Cally (2010) was derived using a Lagrangian\nformulation, which is not easily extended to nonadiabatic\nwave propagation. However, a dispersion relation for the\nnonadiabatic case can be readily constructed from the adia-\nbatic relation, if the simplifying assumption of an isothermal\natmosphere is adopted. This is not a bad approximation for\nthe region in question (see \fgure 3).\nIn an isothermal atmosphere, the density scale height\nisH=c2=\rg, and so\u00161=1. The equation for the Eulerian\npressure perturbation, equation (2), then becomes,\np1=\u001a0g\u0010\u0000\u001a0^c2r\u0001\u0018; (6)\n0.00.51.01.52.00100020003000400050006000\nzHMmLTemperature HKLFigure 3. Height dependence of temperature in the model atmo-\nsphere used here. The VAL-C model is employed below 1.6 Mm ;\nan isothermal layer is appended above 1.6 Mm.\nwhere, following Cally (1984), we de\fne a new (complex)\nquantity1\n^c2=\u00162c2: (7)\nBecause the other perturbation equations are un-\nchanged in the Newton cooling approximation, this suggests\nthat for an isothermal atmosphere, the equations describ-\ning the damped system will have the identical form to the\nadiabatic equations, with the modi\fcation that the sound\nspeed squared c2is replaced by ^ c2. The damped form of the\ndispersion relation for an isothermal atmosphere is therefore\ntaken as\n!2^!2\nca2\nyk2\nh+ (!2\u0000a2k2\nk)\u0002\n\u0002\n!4\u0000(a2+ ^c2)!2k2+a2^c2k2k2\nk\n+ ^c2^N2k2\nh\u0000(!2\u0000a2\nzk2)^!2\nci\n= 0;(8)\nwhereazis the vertical component of the Alfv\u0013 en velocity and\nayis the component perpendicular to the plane containing\nkandg.^Nis de\fned by ^N2=g=H\u0000g2=^c2, ^!c= ^c=2H,\nandkhis the horizontal component of the wavevector.\nAlthough ad hoc , the above dispersion relation will\nprove useful in Section 3.1 in understanding the connectiv-\nity of gravity waves to \feld guided acoustic (slow) waves or\nAlfv\u0013 en waves in the upper atmosphere where a\u001dc. (See\nNewington & Cally (2010) for the low- \fasymptotic solu-\ntions of the 3D MHD dispersion relation.) As in paper I,\nwe corroborate the dispersion diagrams with numerical so-\nlutions of the linearized wave equations, the derivation of\nwhich does not assume an isothermal atmosphere. This is\ndescribed in the following section.\n2.2.4 Numerical solution of linearised, damped MHD\nequations\nThe linearised, MHD equations incorporating Newton cool-\ning are obtained from using the (non-isothermal) expression\nfor the pressure perturbation (2), in the momentum equa-\ntion, and eliminating the density by means of the continuity\n1Note that this is equivalent to rede\fning the ratio of speci\fc\nheats, as in Bunte & Bogdan (1994).\nc\r0000 RAS, MNRAS 000, 000{0004M. E. Newington and P. S. Cally\nequation. Expressed in terms of the cartesian components of\nthe Lagrangian displacement vector, \u0018= (\u0018;\u0011;\u0010 ) the equa-\ntions are as follows:\n!2\u0018\u0000kx\u0000\nig\u0010\u00161+c2\u00162\u0000\nkx\u0018\u0000i\u00100\u0001\u0001\n+a2\u0002\nk2\nxcos\u0012cos\u001esin\u0012\u0010+k2\nxcos\u001esin2\u0012sin\u001e\u0011\n\u0000k2\nxcos2\u0012\u0018\u0000k2\nxsin2\u0012sin2\u001e\u0018+ikxsin2\u0012sin2\u001e\u00100\n\u0000ikxcos\u0012sin\u0012sin\u001e\u00110\u0000cos\u0012cos\u001esin\u0012\u001000+ cos2\u0012\u001800\u0003\n= 0;\n(9)\n\u0000c2\nH\u00162\u0000\nikx\u0018+\u00100\u0001\n+g\u0000\n(1\u0000\u00161)\u00100+ ikx\u0018\u0001\n\u0000g\nH\u0010\u0000\n1\u0000\u00161+H\u00160\n1\u0001\n+\u0000\n!2\u0000k2\nxa2cos2\u001esin2\u0012\u0000\u00161g0\u0001\n\u0010+ ikx\u0018\u00162c20+\u00162c20\u00100\n+ ikxc2\u00162\u00180+ ikxc2\u0018\u00160\n2+c2\u00100\u00160\n2+c2\u00162\u001000\n+a2sin\u0012\u0002\nk2\nxcos\u001ecos\u0012\u0018\u0000cos\u0012\u0000\nsin\u001e\u001100+ cos\u001e\u001800\u0001\n+ sin\u0012\u0000\nikxsin\u001e\u0000\nsin\u001e\u00180\u0000cos\u001e\u00110\u0001\n+\u001000\u0001\u0003\n= 0;(10)\n\u0000\n!2\u0000k2\nxa2cos2\u001esin2\u0012\u0001\n\u0011\n+a2\u0002\nsin\u0012\u0000\nkxcos\u001esin\u0012sin\u001e\u0000\nkx\u0018\u0000i\u00100\u0001\n+ cos\u0012\u0000\n2ikxcos\u001e\u00110\u0000sin\u001e\u0000\nikx\u00180+\u001000\u0001\u0001\u0001\n+ cos2\u0012\u001100\u0003\n= 0\n(11)\nThe Alfv\u0013 en speed is denoted by a.\nThe purely horizontal case \u0012= 90\u000eis not examined\nhere as it is singular in nature, with the governing equations\nreduced in order resulting in `critical levels' producing res-\nonant absorption at the Alfv\u0013 en and cusp resonances (Cally\n1984). This `absorption' may in fact be interpreted as a mode\nconversion, where the absorption coe\u000ecient is continuous\nwith\u0012as it reaches 90\u000e(as found by Cally & Hansen (2011)\nin the case of the Alfv\u0013 en resonance), so physically there is\nnothing particularly special about horizontal \feld despite its\nmathematical peculiarity. The singularities in the solutions\nare regularised in practice due to solar atmospheric waves\nnot being strictly monochromatic.\nAs in paper I, a driven wave scenario is envisaged, where\nmonochromatic upward propagating (in terms of group ve-\nlocity) gravity or acoustic waves are assumed to be excited\nat the bottom of our region of interest. In the interest of\nisolating the e\u000bects of mode conversion of acoustic and grav-\nity waves during their propagation through the atmosphere,\nthe condition that no slow magneotacoustic waves or Alfv\u0013 en\nwaves entered the photosphere from below is imposed. Con-\nversely, no waves of any sort are allowed to enter from the\ntop.\nThe choice of a boundary value problem means that we\nconsidered the spatial damping of these waves (see Sou\u000brin\n(1972)). Temporal invariance of the atmospheric model con-\nsidered means that the wave frequency !is constant; this is\ndetermined by the driving force and is real. Horizontal in-\nvariance ensures constancy of the horizontal component of\nthe wave number kxduring the wave's propagation. Damp-\ning results in the zcomponent of the wavenumber kzbeing\ncomplex. The magnitude of the imaginary component of kz\nincreases with the strength of the damping.\nSolution of the equations requires an arbitrary normal-\nisation condition be applied. In paper I this was chosen tobe normalisation of the vertical velocity, wto 1 km\u00001at\nthe base of the photosphere ( z= 0). For most of the re-\nsults presented in section 3, we chose to normalise wto 1\nkm\u00001at the base of the underdamped region (0.2 Mm), but\nwe also ran simulations where the wave was normalised at\nthe base of the photosphere (0 Mm) to observe the e\u000bect of\noverdamping on the transmitted \ruxes.\nThe motion is assumed adiabatic ( \u001c!1 ) at heights\nabove 1.6 Mm, where the radiative decay time becomes very\nlong, and below the height of application of the normalisa-\ntion condition. This simpli\fcation allows us to apply the\nsame boundary conditions as in paper I, without modi\fca-\ntion. The reader is referred to that paper for details.\nThe acoustic and magnetic \ruxes were calculated in the\nsame manner as in paper I.\n3 RESULTS AND DISCUSSION\n3.1 Dispersion diagrams\nWe present diagnostic dispersion diagrams for four di\u000berent\norientations of the magnetic \feld. Figure 4 presents the adi-\nabatic results. This is the same diagram as \fgure 4 in paper\nI, except that in this paper the z axis has been truncated at\nthe lower boundary of 0.2 Mm, and an essentially evanes-\ncent branch is included (yellow). Figure 5 displays results\nobtained for damped wave propagation, using the isother-\nmal dispersion relation, (8). Recall that the lower branch\nof the dispersion curve corresponds to the up-going gravity\nwave. The asymptotic solutions for the \feld guided acoustic\nwave and the Alfv\u0013 en wave are indicated in the region above\nthe equipartition level, by the dashed blue and dotted red\ncurves, respectively.\nIn both \fgures, the curves are colour coded according\nto the damping per wavelength (Im fkg=Refkg), where the\nmore yellow the curve, the larger the value, and the heavier\nthe damping. In the adiabatic results, (\fg. 4), the yellow\ncurves represent a predominantly evanescent mode that is\nabsent from the \fgure in paper I. We draw attention to it so\nthat its counterpart can be identi\fed in the damped \fgure,\nbut we are not concerned with this mode in this paper. The\ncolour coding in \fgure 5 shows that the up-going gravity\nwave is most heavily damped lower in the atmosphere, which\nis to be expected from the linear form of \u001c.\nThe dispersion diagrams imply that the behaviours\nfound in the adiabatic case, concerning the wave propaga-\ntion behaviour with the \feld orientation and the character\nof the dominant mode conversion, are largely una\u000bected by\nthe damping. Comparison of \fgures 4 and 5 show that the\nwave paths in z\u0000kzspace are preserved when damping is\nincluded. This implies that the conclusions drawn in paper\nI about gravity wave behaviour with various \feld orienta-\ntions are also valid when the gravity wave experiences radia-\ntive damping. The connectivities to the asymptotic solutions\n(which are indicative of the dominant mode conversion) are\npreserved despite the introduction of damping. Numerical\nintegration is used to con\frm the dominant mode conver-\nsions in the following subsection.\nThe paths in z{kzspace are largely unchanged when\nconstant values of \u001c(ranging from 200s to 1ks) are used\ninstead of the linear form (5). The strength of the damping\nc\r0000 RAS, MNRAS 000, 000{000Radiatively damped gravity waves 5\n0.20.40.60.81.01.21.41.6-50050\nzHMmLkzHMm-1L1mHz,kx=2Mm-1,q=0°,f=0°,10.G\n0.20.40.60.81.01.21.41.6-50050\nzHMmLkzHMm-1L1mHz,kx=2Mm-1,q=80°,f=0°,10.G\n0.20.40.60.81.01.21.41.6-50050\nzHMmLkzHMm-1L1mHz,kx=2Mm-1,q=-80°,f=0°,10.G\n0.20.40.60.81.01.21.41.6-50050\nzHMmLkzHMm-1L1mHz,kx=2Mm-1,q=80°,f=50°,10.GÈIm8k200 km; the region below 200{300 km\nis where the waves are driven.\nACKNOWLEDGMENT\nThe authors would like to gratefully acknowledge Stuart Jef-\nferies and Thomas Straus for helpful discussion and clari\f-\ncation of details of their simulations.\nThe National Center for Atmospheric Research is spon-\nsored by the National Science Foundation.\nREFERENCES\nBray R. J., Loughhead R. E., 1974, The Solar Chromo-\nsphere. Chapman and Hall, London\nprocesses such as wave breaking and the production of turbulent\ncascades (Lane & Sharman 2006), and non-linear wave-wave in-\nteractions (Dong & Yeh 1988; Huang et al. 2007). Such processes\ndo not seem to be available to us in the low solar atmosphere.\nc\r0000 RAS, MNRAS 000, 000{0008M. E. Newington and P. S. Cally\n70727476788082840.00.20.40.60.81.01.2\nqH°LFac,FmagHkWm-2L1mHz,kx=2Mm-1,f=0°,B=10G\n70727476788082840.00.20.40.60.81.01.2\nqH°LFac,FmagHkWm-2L1mHz,kx=2Mm-1,f=30°,B=10G\n70727476788082840.00.20.40.60.81.01.2\nqH°LFac,FmagHkWm-2L1mHz,kx=2Mm-1,f=60°,B=10G\nFigure 6. Acoustic (full) and magnetic (dashed) wave-energy \rux\n(kW m\u00002) as functions of magnetic \feld inclination \u0012and three\ndi\u000berent orientations \u001e(0\u000e, 30\u000eand 60\u000e, respectively), for B=10\nG and 1 mHz waves with kx= 2 Mm\u00001. The black curves indicate\nthe results obtained for wave propagation subjected to damping,\nwhile the grey curves are the results of adiabatic simulations for\ncomparison. We apply the normalisation condition at 0.2 Mm.\nThere the vertical velocity wis normalised to 1 km s\u00001.\nBunte, M., Bogdan, T .J., 1994, A&A, 283, 642\nCally P. S., 1984, A&A, 136, 121\nCally P. S, Hansen S. C., 2011, ApJ, in press\nDe Pontieu B., McIntosh S. W., Carlsson M., Hansteen V.\nH., Tarbell T. D., Schrijver C. J., Title A. M., Shine R.\nA., Tsuneta S., Katsukawa Y., Ichimoto K., Suematsu Y.,\nShimizu T., Nagata S., 2007, Science, 318, 1574\nDintrans, B., Brandenburg, A., Nordlund, \u0017A., Stein, R.F.,\n2003, Astrophys. Space Sci., 284, 237\nDintrans, B., Brandenburg, A., Nordlund, \u0017A., Stein, R.F.,\n2005, A&A, 438, 365\nDong B., Yeh K. C., 1988, J. Geophys. Res., 93, 3729\n70727476788082840.00.20.40.60.81.01.21.4\nqH°LFac,FmagHkWm-2L1mHz,kx=2Mm-1,f=0°,B=100G\n70727476788082840.00.20.40.60.81.01.21.4\nqH°LFac,FmagHkWm-2L1mHz,kx=2Mm-1,f=30°,B=100G\n70727476788082840.00.20.40.60.81.01.21.4\nqH°LFac,FmagHkWm-2L1mHz,kx=2Mm-1,f=60°,B=100GFigure 8. As for Figure 6, but with B=100G\nHuang K. M., Zhang S. D., Yi F., J. Geophys. Res., 112,\nD11115\nKomm R., Mattig W., Nesis, A., 1991, AnAp, 252, 827\nLane T. P., Sharman R. D., 2006 Geophys. Res. Lett., 33,\nL23813\nMihalas B. W., Toomre J., 1982, ApJ, 263, 386\nNewington M. E., Cally P. S., 2010, MNRAS, 402, 386\nSchmitz F., Fleck B., 2003, A&A,399, 623\nSou\u000brin P., 1966, AnAp, 29, 55\nSou\u000brin P., 1972, AnAp, 17, 458\nSpiegel E. A., 1957, ApJ,126, 202\nStix M., 1970, A&A, 4, 189\nStodilka M. I., 2008, MNRAS, 390, L83\nStraus T., Fleck B., Je\u000beries S. M., Cauzzi G., McIntosh\nS. W., Reardon K., Severino G., Ste\u000ben M., 2008, ApJL,\n681, L125\nTomczyk S., McIntosh S. W., Keil S. L., Judge P. G., Schad\nT., Seeley D. H., Edmondson J., 2007, Science, 317, 1192\nVernazza J. E., Avrett E. H., Loeser R., 1981, ApJS, 45,\nc\r0000 RAS, MNRAS 000, 000{000Radiatively damped gravity waves 9\n70727476788082840.00.20.40.60.81.01.2\nqH°LFac,FmagHkWm-2L0.7mHz,kx=2Mm-1,f=30°,B=10G\n30405060708001234567\nqH°LFac,FmagHkWm-2L2.1mHz,kx=2Mm-1,f=30°,B=10G\nFigure 9. Acoustic (full) and magnetic (dashed) damped (black)\nand adiabatic (grey) wave-energy \ruxes (kW m\u00002) as functions\nof magnetic \feld inclination \u0012with\u001e= 30\u000efor B=10 G, kx= 2\nMm\u00001for waves of two di\u000berent frequencies. Top panel: 0.7 mHz;\nLower panel: 2.1 mHz. Note the di\u000berent \u0012scales on the two\ngraphs.\n635\n70727476788082840.00.51.01.52.02.53.03.5\nqH°LFac,FmagHkWm-2L1mHz,kx=2Mm-1,f=0°,B=10G\n70727476788082840.00.51.01.52.02.53.0\nqH°LFac,FmagHkWm-2L1mHz,kx=2Mm-1,f=30°,B=10G\n70727476788082840.00.51.01.52.02.53.0\nqH°LFac,FmagHkWm-2L1mHz,kx=2Mm-1,f=60°,B=10G\nFigure 10. As for 6, but with the normalisation condition now\napplied deeper at z=0. Very little \rux is measured at the top\nwhen the gravity wave is damped. With the chosen form of \u001c(z),\nthe short damping times in the low photosphere very e\u000bectively\nmitigate gravity wave progression.\nc\r0000 RAS, MNRAS 000, 000{000" }, { "title": "1411.4100v4.Spin_transfer_torque_through_antiferromagnetic_IrMn.pdf", "content": "1 \n \nSpin -transfer -torque through antiferromagnet ic IrMn \n \n \nT. Moriyama*1, M. Nagata1, K. Tanaka1, K-J Kim1, H. Almasi2, W. G. Wang2, and T. \nOno†1 \n1 Institute for Chemical Research, Kyoto University, Japan. \n2 Department of Physics, The University of Arizona, USA \n \nAbstract \nSpin-transfer -torque, a transfer of angular momentum between the electron spin and the \nlocal magnetic moments, is a promising and key mechanism to control ferromagnetic \nmaterials in modern spintronic devices . However, much less attentio n has been paid to \nthe same effect in antiferromagnets . For the sake of investigating how the spin current \ninteracts with the magnetic moment s in antiferromagnets, w e perform spin -torque \nferromagne tic resonance measurements on Co 20Fe60B20/Ir25Mn 75/Pt multilayers under a \nspin Hall effect of Pt. The effective magnetic damping in CoFeB is modified by the spin \ncurrent injected from the Pt layer via the IrMn layer . The results indicate that the spin \ncurrent interacts with IrMn magnetic moments and exerts the anti-damping torque on \nthe magnetic moments of CoFeB through the IrMn . It is also found that the reduction of \nthe exchange bias in the IrMn/Pt interface degrades the anti-damping torque exerted on \nthe Co FeB layer , suggesting the transmission of the spin torque becomes less efficient \nas the interface exchange coupling degrade s. Our work infers that the magnetic \nmoments i n IrMn can be manipulated by spin torque similarly to the one in a \nferromagnetic layer. \n*mtaka@scl.kyoto -u.ac.jp †ono@scl.kyoto -u.ac.jp 2 \n Antiferromagnet (AFM) is a magnetic material which has local anisotropic \nmagnetic moments but has no net magnetic moment as a whole. Since AFMs have no \nspontaneous magnetization unlik e ferromagnetic materials (FMs) and the magnetic \nsusceptibility is very small, it is generally not easy to manipulate the magnetic moments \nin AFMs by an external magnetic field. However, recent theoretical studies 1,2,3,4,5 \nsuggest a possibility of manipulating the magnetic moments in AFMs by \nspin-transfer -torque (STT) in a similar man ner to FMs6,7. \nWhile a number of theoretical studies have been reported on i nteraction between \nspin current and order parameters of AFM , only few experimental work have been \nreported so far. Wei et al. 8 reported that the electric current flowing in the \nFeMn/CoFe /Cu/CoFe spin valve can alter the exchange -bias, implying the magnetic \nmoments in AFM is influenced by STT. Urazhdin et al.9 also reported that under strong \ncurrent injection in the nano -patterned spin valves the exchange bias at AFM /FM can be \naltered due to both the spin transfer torque and electron -magnon scattering. These \nprevious reports8,9 focused on the current -perpendicular -to-the-plane (CPP) \nmeasurements in the FM/AFM bilayer where the current flowing through FM is \nspin-polarized and interacts with the AFM magnetic moments. However, it is not clear \nwhether or not the alternation of the exchange bias is due to the STT acting on the bulk \nAFM magnetization . It remains possible that the effect is due to the uncompensated \nmoment at the FM/AFM interface which interacts with the STT . It is still of great \ninterest to further experimentally investigate the interaction of spin current and AFM \nmagnetic moments and to pursue potential applications in emerging antiferromagnetic \nspintronics10,11. \nThe spin -torque ferromagnetic resonance (ST -FMR) measurement techn ique 3 \n developed for spin valves12 and magnetic tunnel junctions13,14 and later applied to the \nsystem involved with spin Hall effect (SHE)15 is a useful method for quantifying the \nspin torque exerting in the system. As above mentioned previous report s8,9 focus only \non the response of the magnetization direction under the application of the spin torque , \nthe ST -FMR can quantify another important parameter, magnetic damping , which is \ndirectly connected to the interaction between the magnetic moments and the spin \nangular momentum carried by the spin current . In this study, we perform ST-FMR \nmeasurements on AFM/FM bilayers to experimentally investigate the transfer of the \nspin angular momentum into the magnetic moments in AFM. Our ST -FMR \nmeasurement enables to inject spin current directly to AFM without a help of \nspin-polarizing FM layer, which makes this work essentially clearer and different than \nthe previous reports. \nWe prepare Co20Fe60B20 5 nm/Ir25Mn 75 tIrMn nm/Pt 4 nm multilayers on a thermally \noxidized Si substrate by magnetron sputtering. The film is patterned into 4 ~ 10 μm \nwide strips attached to a coplanar waveguide facilitating both the r .f. and d .c. current \ninjection into the strip. The d .c. electric current flowing in Pt layer invokes a spin Hall \neffect and injects a pure spin current into the neighboring IrMn layer as shown in Fig. \n1(a). ST-FMR is performed by sweeping the external magnetic field at a fixed frequency \nof the r.f. current . Figure 1(b) shows the measurement configuration together with our \ncoordinate system. The p ositive electric current is defined when it flows along the \npositive y direction. The e xternal positive magnetic field is applied in the sample plane \nand at 45º away from x axis. We apply nominal r .f. power up to 14 dBm and d .c. current \nup to 2 mA to the strip. All the measurements are performed at room temperature. \nThe expected rectified dc voltage Vdc is written as15, 4 \n 𝑉𝑑𝑐=1\n4(𝑑𝐻\n𝑑𝑓)|\n𝐻𝑒𝑥=𝐻0𝑑𝑅\n𝑑𝜃𝛾(𝐼𝑅𝐹)2sin𝜃\n2𝜋𝜎 (𝑃𝑠𝑆(𝜔)+𝑃𝐴 𝐴(𝜔) ), (1) \nwhere γ = 1.76 x 1011 T-1s-1 is the gyromagnetic ratio, Hex is the external field, and IRF is \nr.f. current flowing in the strip . 𝑆(𝜔)=1((𝐻𝑒𝑥𝑡−𝐻0)2𝜎2⁄ +1) ⁄ and 𝐴(𝜔)=\n((𝐻𝑒𝑥𝑡−𝐻0)𝜎⁄)((𝐻𝑒𝑥𝑡−𝐻0)2𝜎2⁄ +1) ⁄ are the symmetric and the antisymmetric \nLorentzian, respectively . The prefactors are 𝑃𝑠=(𝜕𝜏𝑆𝐻𝐸 𝜕𝐼⁄)(1𝑀𝑠𝑉𝑜𝑙 ⁄ ) and \n 𝑃𝐴=(𝜕ℎ𝜕𝐼⁄)√1+4𝜋𝑀𝑒𝑓𝑓 𝐻𝑒𝑥⁄ , where 𝜏𝑆𝐻𝐸 is the ant i-damping torque due to the \nSHE , ℎ is sum of the field torque and field -like torque17,18, Ms is saturation \nmagnetization of the CoFeB , and 4𝜋𝑀𝑒𝑓𝑓 is the effective demagnetizing field . We \nexpect 𝑑𝑅 𝑑𝜃⁄ to be a finite value due to anisotropic magnetoresistance (AMR) of the \nCoFeB. The resonant field 𝐻0 and resonant frequency 𝜔0 follow the Kittel equation \nas, \n𝜔0 =𝛾√(𝐻0+𝐻𝑒𝑏)(𝐻0+𝐻𝑒𝑏+ 4𝜋𝑀𝑒𝑓𝑓 ) , (2) \nwhere 𝐻𝑒𝑏 is the unidirectional exchange bias field . All parameters except 𝜕𝜏𝑆𝐻𝐸 𝜕𝐼⁄ \nand 𝜕ℎ𝜕𝐼⁄ in Eq. 1 are experimentally obtainable. The Lorentzian linewidth 𝜎 is \nproportional to the intrinsic dam ping α0 as, \n𝜎 =𝜔0\n𝛾𝛼0. (3) \nThe linewidth is modified by the spin current density (ℏ2𝑒⁄)𝐽𝑠 under the spin Hall \nangle 𝜃𝑆𝐻=𝐽𝑠𝐽𝑐⁄ where Jc is the electric current density flowing in the spin Hall \nmaterial ; \n𝜎 =𝜔0\n𝛾(𝛼0+cos𝜃\n(𝐻𝑒𝑥+2𝜋𝑀𝑒𝑓𝑓)𝑀𝑠𝑡𝐹𝑀ℏ\n2𝑒𝜃𝑆𝐻𝐽𝑐) , (4) \nwher e ℏ is the reduced Planck constant, e is the elementary charge , and tFM is the \nthickness of the magnetic layer . It should be noted that the linewidth is sensitive to the \nanti-damping torque not to the field -like torque. 5 \n Figure 2(a) show s typical spectr a of Vdc as a function of the external field at the \ndifferent r.f frequencies in CoFeB 4nm/IrMn 23 nm/Pt 4 nm . As Eq. 1 indicates , the \ncombination o f symmetric and anti -symmetric L orentizan can be seen around the \nresonant field at each r.f. frequency . We confirm that there are no hysteretic behaviors in \nthe resonant spectra above |500| Oe (see the inset of Fig. 2(a)) . It is found that the \nresonant fields differ in positive and negative external field as shown in Fig. 2(b), \nimplying that there is a unidirectional exchange bias in the CoFeB layer . By fitting with \nEq. 2 , we extract the unidirectional anisotropy to be 47 Oe and 4𝜋𝑀𝑒𝑓𝑓 to be 1 .4 Tesla \nwhich is consistent with the saturation magnetization of 1.4 Tesla measured by SQUID \nmagnetomet ry. More precise measurements determine the unidirectional field direction \nto be along the positive y direction and the magnitude to be about 100 Oe (see \nsupplementary information ). In the case of the CoFeB 4nm/Pt 4 nm bilayer, w e found \n4𝜋𝑀𝑒𝑓𝑓 to be 1.2 Tesla due to a sizable interfacial perpendicular anisotropy at the \nCoFeB/Pt interface16. \nFigure 3(a) shows the ratio of PS to PA for CoFeB 4nm/IrMn tIrMn nm /Pt 4 nm with \ntIrMn = 0, 11, and 23 nm. Since PS and PA are proportional to 𝜕𝜏𝑆𝐻𝐸 𝜕𝐼⁄ and 𝜕ℎ𝜕𝐼⁄, \nrespectively, 𝑃𝑆𝑃𝐴⁄ represents the ratio of the anti -damping torque over the sum of the \nfield torque and field -like torque . Finite value of PS /PA indicates that we have an \nanti-damping torque in the system. The monotonous increase in PS /PA with the external \nfield originates from the field dependence of PA. It is found that PS /PA is largest with \ntIrMn = 0 nm and become s very small with tIrMn = 11 nm. It is remarkable that the PS /PA \nagain takes the appreciable value with thicker IrMn ( tIrMn = 23nm). We also confirm that \nthe sign of the 𝜕𝜏𝑆𝐻𝐸 𝜕𝐼⁄ derived from PS for the positive current is consistent with the \nspin torque direction created by the spin Hall effect in Pt layer. It would be possible to 6 \n extract the magnitude of the spin torque from 𝑃𝑆𝑃𝐴⁄ itself in a self -consistent way15. \nHowever, as it might mislead the final results without accurately evaluating the \nfield-like torque arising from Rashba effect and spin-orbit torque17,18 which may \npossibly be expected in our system, in the following discussion we take advantage of \nthe linewidth analysis based on Eq. 4 rather than using PS /PA . \nFigure s 3(b), (c) and (d) shows the dc current Icd dependence of the linewidth at f = \n9 GHz for CoFeB 4nm/IrMn tIrMn nm /Pt 4 nm with tIrMn =0, 1 1, and 2 3 nm. As one \nexpect s from the finite value of PS /PA discussed above , the effective damping is indeed \nmodified by the spin current even with tIrMn ≠ 0 nm. Assuming that IrMn is not \ntransparent for the spin polarized current (the spin diffusion length is reported to be \nabout < 1nm19), the results suggest that the angular momentum of the spin current is \ntransferred into the collection of AFM magnetic moments and it modifies the effective \ndamping of the CoFeB layer via the exchange coupling at IrMn/Co FeB. The slope of \nthe linear fitting Δσ/Idc as shown in Figs. 3 (a), (b), and (c) is plotted against the IrMn \nthickness in Fig. 4(a). Δσ/Idc is largest with tIrMn = 0 nm and the insertion of IrMn as thin \nas 2 nm suddenly drops Δσ/Idc. It is then subsided until 1 6 nm of IrMn. This dependence \nis not typically observed by the insertion of spin-diffusive paramagnetic materials like \nCu20 where injected spin current decays with a length scale of the spin relaxation of the \nparamagnetic materials . In this case, one would expect a monotonous decrease of Δσ/Idc \nwith the thickness of the paramagnetic material insertion . \nΔσ/Idc represents how efficient the spin current is transferred into the CoFeB layer \nvia the IrMn layer with the form transformed from Eq. 4 ; \n∆𝜎𝐼𝑑𝑐⁄ =𝜔0\n𝛾cos𝜃\n(𝐻𝑒𝑥+2𝜋𝑀𝑒𝑓𝑓)𝑀𝑠𝑡𝐹𝑀 𝑤ℏ\n2𝑒(𝛽𝜃𝑆𝐻,𝑃𝑡𝑟𝑃𝑡\n𝑡𝑃𝑡), (5) \nwhere w is the width of the strip , 𝑟𝑃𝑡is the shunt current ratio in the Pt layer , and 𝑡𝑃𝑡 is 7 \n the thickness , and (0 ≤ ≤ 1) is the spin -transfer efficiency in the IrMn layer . = \n1 when the all injected spin current exerts a spin torque on the CoFeB layer through the \nIrMn layer and = 0 when it is dissipated within the IrMn layer before reaching to the \nCoFeB layer . To accurately evaluate Δσ/Idc, we separately measure the resistivity of \neach layer to be 1.4 x 10-6, 1.9 x 10-6, and 3.0 x 10-7 Ω m for CoFeB , IrMn, and Pt , \nrespectively , and then calculate the shu nt current ratio of the Pt layer . We estimated the \nspin Hall angle for the Pt layer to be 0.09 0.01 by a CoFeB 4nm/Pt 4nm bilayer . We \nalso characterized the spin Hall angle for the IrMn layer and concluded that the effect is \nnegligible (See the supplementary information) . For the sample with tIrMn ≠ 0 nm, the \nshunt current ratio in Pt decreases with increasing the IrMn thickness. \nThe dotted curve in Fig. 4 is the calculated Δσ/Idc based on Eq. (5) by assuming = \n1. The calculated Δσ/Idc decreases with increasing Ir Mn thickness because of the \ndecrease in the shunt current ratio in Pt by the current shunting in the IrMn layer . In \nother word s, this theoretical curve represents the spin torque which the CoFeB layer \nreceives based on the current flowing in the Pt layer as if there exist no IrMn insertion \nlayer which may dissipate the spin angular momentum . The = 1 curve does not \nreproduce t he measured Δσ/Idc until 16 nm. Our results surprisingly indicate ~ 1 for \nIrMn thicker than 16 nm, implying that the spin torque is interacting with the Co FeB \nmagnetization through the IrMn layer . We speculate that this peculiar spin transfer \ntorque through the antiferromagne t may be attributed to the angular momentum transfer \nmediated by antiferromagnetic spin fluctuations21. First, the injected spin current \ninteract s with the magnetic moment at the Pt/IrMn interface and induce s magnetic \nexcitations. The magnetic excitation then propagates and transfers the spin angular \nmomentum into the CoFeB layer. The discrepancy between the measured Δσ/Idc and the 8 \n calculation in the thinner Ir Mn regime then indicate s the breakdown of this hypothesis \nand can be explained by the following mechanisms . \nThe important factors are the exchange coupling at the IrMn/CoFeB interface and \nthe IrMn thickness dependence of the blocking temperature22,23,24. Even if the magnetic \nexcitations are transmitted from the Pt/IrMn interfa ce, absence of the magnetic coupling \nat the IrMn/CoFeB interface cannot effectively transfer the angular momentum to the \nCoFeB layer or influence the linewidth . Our results clearly show that the strength of the \nexchange bias as a function of the IrMn thickness coincides with the trend of Δσ/Idc (see \nFig. 4(a ) and (b)). Namely, small exchange bias yields small β. It is also possible that \nthe small β in thinner IrMn layer is due to the Néel temperature degradation25 at which \nthe angular momentum t ransfer by the antiferromagnetic excitation becomes inefficient. \nIn addition, we estimate d the IrMn thickness dependence of the intrinsic damping \nconstant in absence of the spin current as shown in Fig.4 (c) . The intrinsic damping also \nhas a similar trend t o the exchange bias as well as Δσ/Idc. Here, the intrinsic damping is \nestimated from the slope of the linear fitting to the frequency dependence of the \nlinewidth (examples are shown in the inset of Fig. 4 ) so that the extrinsic damping \ncontributions such as the two -magnon scattering26 due to magnetic inhomogeneity \ninduced at the AFM/FM interface27 is successfully ruled out . The increase of the \nintrinsic damping coinciding with the emergence of the exchange bias strongly suggest s \nthat the dynamics of the FM momen t is nonlocally influenced by the AFM28. Namel y, \nthe angular momentum of the precessing FM is dissipated into the AFM through the \nexchange coupl ed AFM/FM interface . This is another evidence that there is a channel \nfor the angular momentum flow through the magnetic coupling at AFM/FM . Thus, all \nthe observation s support our scenario that the spin torque is mediated by the 9 \n antiferromagnetic excitation through the IrMn layer . \nIn summary, we investigate the interaction between spin current and magnetic \nmoment s in the antiferromagnetic Ir Mn in the structure of CeFeB 4 nm/IrMn tIrMn nm/Pt \n4 nm by using ST -FMR technique. We f ind that the linewidth, that is proportional to the \neffective damping , changes as a function of the spin current even with tIrMn ≠ 0 nm. The \nresults indicate that the spin current is transferred and exerts a spin torque on the CoFeB \nlayer through the IrMn layer. We speculate the spin current interacts with IrMn \nmagnetic moments and exerts the anti -damping torque on the magnetic moments of \nCo20Fe60B20 through the IrMn. The decrease of the spin -transfer efficiency in the Ir Mn \nthickness < 16 nm is explained by the lack of the interfacial exchange coupling . Our \nresults manifest that a remarkable interaction between spin current and the \nmagnetizat ion in the antiferromagnet as theories have predicted1,2,3,4,5. Our work infers \nthat the magnetic moments of the antiferromagnetic Ir Mn can be manipulated by the \nspin transfer torque similarly to the ferromagnetic materials, raising those relatively \naban doned materials to a n emerging antiferromagnetic spintronics11. \n \n \nAcknowledgements \nWe would like to thank Dr. So Takei and Prof. Yaroslav Tserkovnyak for fruitful \ndiscussions. This work was partly supported by Grants -in-Aid for Scientific Research \n(S), Grant -in-Aid for Young Scientists (B) from Japan Society for the Promotion of \nScience and by US NSF (ECCS -1310338) . \n 10 \n \nFigure 1 Moriyama et al. \n \n11 \n \n \nFigure 2 Moriyama et al. \n \n \n12 \n \nFigure 3 Moriyama et al. \n \n \n13 \n \nFigure 4 Moriyama et al. \n \n \n14 \n FIGURE CAPTIONS : \n \nFIG. 1 Schematic illustrations of the sample and measurement setup. (a) T he sample \ncross section of CoFeB/IrMn/Pt deposited on a Si/SiO x substrate. The electric current \nflowing laterally in the Pt layer invokes the spin Hall effect and injects a spin current \ninto the neighboring Ir Mn layer. (b) The sample is patterned into 4 ~ 10μm-wide strip \nand is connected to coplanar waveguide so that ST-FMR measurement can be \nperformed. \n \nFIG. 2 (a) Vdc spectra from the ST -FMR measurements on CoFeB 4nm/IrMn 23 nm/Pt 4 \nnm with Idc = 0 mA. The r.f. frequency is varied up to 15 GHz . The black curves are \nfitting with Eq. 1 . The inset shows the full Vdc spectrum at 9 GHz with back and forth \nscanning between -2400 Oe and 2400 Oe. (b) The resonant frequency as a function of \nthe positive ( red) and negative field ( blue). \n \nFIG. 3 (a) PS/PA as a function of Hex for tIrMn = 0 nm (black), 11 nm (blue), 23 nm (red). \nChange in linewidth Δσ as a function of Idc for (b) tIrMn = 0 nm, (c) tIrMn = 11 nm, (d) \ntIrMn = 23 nm at the resonant frequency of 9 GHz. The open and solid circles are \nrespectively for the positive field and negative field. Fitted slope is Δσ/Idc = 1.19 0.04 \n(Oe/mA) in the positive field and Δσ/Idc = 1.11 0.03 (Oe/mA) in the negative field for \ntIrMn = 0 nm. Δσ/Idc = 0.23 0.05 (Oe/mA) in the positive field and Δσ/Idc = 0.15 0.06 \n(Oe/mA) in the negative field for tIrMn = 11 nm. Δσ/Idc = 0.44 0.08 (Oe/mA) in the \npositive field and Δσ/Idc = 0.64 0.1 (Oe/mA) in the negative field for tIrMn = 23 nm. \n \nFIG. 4 (a) Δσ/Idc as a function of Ir Mn thickness. The dotted blue curve is the calculated 15 \n Δσ/Idc based on Eq. 4 with = 1. (b) Exchange bias field as a function of IrMn thickness. \n(c) The intrinsic damping at Idc = 0 mA as a function of IrMn thickness. The inset shows \nexamples of the linear fitting on the frequency dependence of the linewidth for \nestimating the intrinsic damping26. \n \n 16 \n REFERENCES \n \n1 A. H. MacDonald and M. Tsoi, Phil. Trans. R. Soc. A 369, 3098 (2011). \n2 A. S. Núñez, R. A. Duine, P. Haney, and A. H. MacDonald, Phys. Rev. B 73, 214426 (2006). \n3 P. M. Haney and A. H. MacDonald, Phys. Rev. Lett. 100, 196801 (2008). \n4 H. V. Gomonay and V. M. Loktev, Phys. Rev. B 81, 144427 (2010). \n5 A. C. Swaving and R. A. Duine, Phys. Rev. B 83, 054428 (2011). \n6 J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996) \n7 D. C. 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Determination of the exchange bias in Pt/IrMn/CoFeB layers \nTo determine the exchange bias field Heb, we performed anisotropic \nmagnetoresistance (AMR) measurements in a rotating magnetic field1. Figure S1 \nshows the AMR as a function of the field angle θ with the field strength of Hex = 400 \nOe (see the definition for Fig. 1 in the main text. ). When the crystalline anisotropy \nand the in -plane shape anisotropy are negligible comparing with the external field, \nthe AMR curve is given as1 \n𝜌=𝜌0+∆𝜌cos(tan−1(cos𝜃+(𝐻𝑒𝑏𝐻𝑒𝑥⁄ )cos𝜃𝑒𝑏\nsin𝜃+(𝐻𝑒𝑏𝐻𝑒𝑥⁄ )sin𝜃𝑒𝑏)), (S1) \nwhere 𝜌0 is the resistivity at 𝜃=90° and 𝜃𝑒𝑏 is the direction of the exchange \nbias. The fitting with Eq. S1 yields 𝜃𝑒𝑏=90° and 𝐻𝑒𝑏= 100 Oe for CoFeB 5 \nnm/IrMn 2 3 nm/Pt 4 nm and 𝜃𝑒𝑏=90° and 𝐻𝑒𝑏= 17 Oe for CoFeB 5 nm/IrMn \n11 nm/Pt 4 nm . We used this technique to quantify the exchange bias field for all the \nsamples. \nAlthough we conduct ed no field cool process for any of the samples , for thicker t IrMn \nwe clearly observe the unidirectional anisotropy . We suspect that the sample stage \nmotion during the sputtering deposition may have induced the exchange bias field. We additionally checked the existence of the exchange bias by conducting a field \ncool. The field cool did not alt er the thickness dependence of the exchange bias \nshown in Fig. 4(b). \n \nFigure S1 AMR as a function of the field angle θ for (a) CoFeB 5 nm/IrMn 2 3 nm/Pt \n4 nm and (b) CoFeB 5 nm/IrMn 11 nm/Pt 4 nm . The blue curves are the fitting with \nEq. S1. \n \n2. Estimation of the resistivity in each layer \nThe conductivity of each layer is determined by measuring a set of samples varying \nthe layer thickness of each material . The measurements are performed by a \nconventional two probe method. Assuming the parallel conductance model as \nΣ𝑡𝑜𝑡=(𝑊/𝐿)∑𝑡𝑋𝜎𝑋 𝑋 , (S2) \nwhere X = CoFeB, IrMn, Pt, and σ is the conductivity of each material. Note that we \nonly assume the bulk conductance and did not take int o account any interfacial \nconductance. We obtain the resistivity 1/σCoFeB = 1.4 x 10-6Ω m, 1/σIrMn = 1.9 x 10-6 \nΩ m, and σPt = 3.0 x 10-7 Ω m. \n3. Determination of the spin Hall angle in the Pt layer \nDetermination of the spin Hall angle has been a controversial issue2,3,4,5,6,7. Differ ent \nmeasurement techniques give different spin Hall angle ranging from 0.01 to 0.12 for \nPt. The large spread in the reported values may be due to a variation of other \nimportant parameters used to derive the spin Hall angle such as the interfacial spin \nmixing conductan ce at FM/Pt and t he spin diffusion length of Pt . These parameters \ncan be influence d significantly by the quality of the sample. In other words, how the \nsamples are prepared can strongly affect the spin Hall angle estimation. In our \nexperiment, we refer to the d.c. current dependence of the linewidth to estimate the \nspin Hall angle of the Pt layer. Fitting the data in Fig. 3 (b) with \n∆𝜎𝐼𝑑𝑐⁄ =𝜔0\n𝛾cos𝜃\n(𝐻𝑒𝑥+2𝜋𝑀𝑒𝑓𝑓)𝑀𝑠𝑡𝐹𝑀 𝑤ℏ\n2𝑒(𝜃𝑆𝐻,𝑃𝑡𝑟𝑃𝑡\n𝑡𝑃𝑡) , (𝑆3) \nyields 𝜃𝑆𝐻,𝑃𝑡 = 0.09 0.01, which is reasonably within the range of the reported \nvalues. Note that we did not measure the spin diffusion length for the Pt or the spin \nmixing conductance in the sample , and that we presumed the spin diffusion length to \nbe much smaller than the thickness and we ignore d the spin mixing conductance for \nour spin Hall angle estimation . Therefore, w e do not intend to claim a definite \naccuracy of the intrinsic spin Hall angle of Pt in this work . \n \n4. The spin Hall angle in the IrMn layer \nIt would be possible that IrMn have a spin Hall effect as Ir can give rise to the spin \nHall effect8. We preformed ST -FMR measurements on CoFeB/IrMn /SiO 2 \nmultilayers to check the possibility of IrMn spin Hall effect. If the IrMn layer gives \nrise to the spin Hall effect, the injected spin current into the CoFeB layer decreases \nor increases the linewidth of CoFeB . We tested two different IrMn thicknesses: 10 nm and 20 nm. As plotted in Fig. S2, we did not observe a clear d.c. current \ndependence on the linewidth. Using 1/σCoFeB = 1.4 x 10-6Ω m, 1/σIrMn = 1.9 x 10-6 Ω \nm, and presumably 𝜃𝑆𝐻,𝐼𝑟𝑀𝑛 ~ 0.05, we estimated the expected slope Δσ/Idc as \nshown in Fig. S2. Our measurements are indeed not sensitive enough to determine \nspin Hall angle in the IrMn layer with two decimal places because of its high \nresistivity. In the discussion in the main text, we disregard the Ir Mn spin Hall effect \nfor calculating Δσ/Idc show n in Fig. 4(a) considering the uncertain ty of the value . \nThe calculated line in Fig. 4 deviate s by at most 9% even if the s pin Hall angle as \nmuch as ~0.05 for IrMn is taken into account because of the small shunt current \nratio in the IrMn layer . \n \nFigure S2 Δσ as a function of Idc for (a) CoFeB 5 nm/IrMn 10 nm/SiO 2 5 nm and (b) \nCoFeB 5 nm/IrMn 20 nm/SiO 2 5 nm. The red and blue circles are respectively for \nthe po sitive field and negative field. The dotted lines are estimated slope Δσ/Idc with \n1/σCoFeB = 1.4 x 10-6 Ω m, 1/σIrMn = 1.9 x 10-6 Ω m, and presumably 𝜃𝑆𝐻,𝐼𝑟𝑀𝑛 ~ 0.05 . \n \n \n1 B. H. Miller and E. Dan Dahlberg, Appl. Phys. Lett. 69, 3932 (1996) \n2 L. Q. Liu, T. Moriyama, D. C. Ralph, R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). \n3 M. Morota, Y . Niimi, K. Ohnishi, D. Wei, T. Tanaka, H. Kontani, T. Kimura, and Y . Otani, Phys. Rev. B \n83, 174405 (2011).) \n4 L. Bai, P. Hyde, Y . S. Gui, and C. -M. Hu, V . Vlaminck,J. E. Pearson, S. D. Bader, and A. Hoffmann, \n \nPhys. Rev. Lett. 111, 217602 (2013). \n5 O. Mosendz, G. Woltersdorf, B. Kardasz, B. Heinrich, and C. H. Back, Phys. Rev. B 79, 224412 (2009). \n6 M. Obstbaum, M. H¨artinger, H. G. Bauer, T. Meier, F. Swientek, C. H. Back, and G. Woltersdorf, Phys. \nRev. B 89, 060407 (2014). \n7 M. Kawaguchi, K. Shimamura, S. Fukami, F. Matsukura, H. Ohno, T. Moriyama, D. Chiba, and T. Ono, \nAppl. Phys. Exp. 6, 113002 (2013) \n8 Y . Niimi, M. Morota, D. H. Wei, C. Deranlot, M. Basletic, A. Hamzic, A. Fert, and Y . Otani, Phys. Rev. \nLett. 106, 126601 ( 2011) " }, { "title": "0803.2175v1.Current_induced_noise_and_damping_in_non_uniform_ferromagnets.pdf", "content": "arXiv:0803.2175v1 [cond-mat.mes-hall] 14 Mar 2008Current-induced noise and damping in non-uniform ferromag nets\nJørn Foros,1Arne Brataas,1Yaroslav Tserkovnyak,2and Gerrit E. W. Bauer3\n1Department of Physics, Norwegian University of Science and Technology, 7491 Trondheim, Norway\n2Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n3Kavli Institute of NanoScience, Delft University of Techno logy, 2628 CJ Delft, The Netherlands\n(Dated: November 1, 2018)\nIn the presence of spatial variation of the magnetization di rection, electric current noise causes a\nfluctuatingspin-transfer torque that increases the fluctua tions of the ferromagnetic order parameter.\nBy the fluctuation-dissipation theorem, the equilibrium flu ctuations are related to the magnetiza-\ntion damping, which in non-uniform ferromagnets acquires a nonlocal tensor structure. In biased\nferromagnets, shot noise can become the dominant contribut ion to the magnetization noise at low\ntemperatures. Considering spin spirals as a simple example , we show that the current-induced noise\nand damping is significant.\nPACS numbers: 72.70.+m, 72.25.Mk, 75.75.+a\nElectric currents induce magnetization dynamics in\nferromagnets. Three decades ago, Berger1,2showed that\nan electric current passing through a ferromagnetic do-\nmain wall exerts a torque on the wall. The cause of this\nspin-transfer torque is the reorientation of spin angular\nmomentum experienced by the electrons as they adapt to\nthe continually changingmagnetization. Subsequently, it\nwas realized that the same effect may also be present in\nmagnetic multilayers3. In the latter case, the torque may\ncause reversal of one of the layers, while in the former,\nit may cause domain wall motion. The early ideas have\nbeen confirmed both theoretically and experimentally4.\nRecently, the importance of noise in current-induced\nmagnetization dynamics has drawn attention. Although\noften noise is undesired, it may in some cases be quite\nuseful. Wetzels et al.5showed that current-induced mag-\nnetization reversal of spin valves is substantially sped up\nby an increased level of current noise. The noisy cur-\nrent exerts a fluctuating torque on the magnetization6.\nRavelosona et al.7reported observation of thermally as-\nsisted depinning of a narrow domain wall under a cur-\nrent. Thermally-assistedcurrent-driven domain wall mo-\ntion has also been studied theoretically8,9.\nThe present paper addresses current-induced magneti-\nzation noise in non-uniformly magnetized ferromagnets.\nThe spatial variationof the magnetization direction gives\nrise to increased magnetization noise; by a fluctuating\nspin-transfer torque, electric current noise causes fluc-\ntuations of the magnetic order parameter. We take\ninto account both thermal current noise and shot noise,\nand show that the resulting magnetization noise is well\nrepresented by introducing fictitious stochastic magnetic\nfields. By the fluctuation-dissipation theorem (FDT),\nthe thermal stochastic field is related to the dissipation\nof energy, or damping, of the magnetization. The FDT\nhence constitutes a simple and efficient way to evaluate\nthe damping, providing also a physical explanation in\ntermsofcurrentnoiseand spin-transfertorque. Since the\ncorrelator of the stochastic field in general is inhomoge-\nnous and anisotropic, the damping is a nonlocal tensor.\nAs a simple and illuminating example we consider ferro-magnetic spin spirals, for which the field correlator and\ndamping become spatially independent. It is shown that\nfor spirals with relatively short wavelength ( ∼20nm),\nthe current-induced noise and damping is substantial.\nSince half a wavelength of a spin spiral can be consid-\nered as a simple model for a domain wall, this suggests\nthat current-induced magnetization noise and damping\nshould be an issue for narrow domain walls.\nIt is instructive to start with an introduction to the\nFDT for uniform (single-domain) ferromagneticsystems,\ncharacterized by a time-dependent unit magnetization\nvectorm(t) and magnetization magnitude Ms(the sat-\nuration magnetization). The spontaneous equilibrium\nnoise of such macrospins is convenientlydescribed by the\ncorrelator Sij(t−t′) =/angbracketleftδmi(t)δmj(t′)/angbracketright, whereδmi(t) =\nmi(t)−/angbracketleftmi(t)/angbracketrightis the random deviation of the magne-\ntization from the mean value at time t. The brackets\ndenote statistical averaging at equilibrium, while iandj\ndenote Cartesian components. The magnetization fluc-\ntuations are assumed weak, so that they to first order are\npurely transverse to the equilibrium (average) direction\nof magnetization. Applying a weak external magnetic\nfieldh(ext)(t), the magnetization can be excited from the\nequilibrium state. Assuming linear response, the result-\ning transverse change in magnetization is\n∆mi(t) =/summationdisplay\nj/integraldisplay\ndt′χij(t−t′)h(ext)\nj(t′),(1)\ndefining the transverse magnetic susceptibility χij(t−t′)\nas the causal response function. The FDT relates this\nsusceptibility to the equilibrium noise correlator10:\nSij(t−t′) =kBT\nMsV/integraldisplay\ndωe−iω(t−t′)χij(ω)−χ∗\nji(ω)\ni2πω,(2)\nwhereTis the temperature and Vis the volume of the\nferromagnet.\nThe spontaneous equilibrium fluctuations δm(t) may\nbe regardedto be caused by a fictitious random magnetic\nfieldh(t) with zero mean. We can derive an alternative\nform of the FDT in terms of the correlator /angbracketlefthi(t)hj(t′)/angbracketright.2\nTo do so, simply note that Eq. (1) implies that δmi(ω) =/summationtext\njχij(ω)hj(ω) in Fourier space. Inverting this relation,\nit follows from Eq. (2) that\n/angbracketlefthi(t)hj(t′)/angbracketright=kBT\nMsV/integraldisplay\ndωe−iω(t−t′)[χ−1\nji(ω)]∗−χ−1\nij(ω)\ni2πω,\n(3)\nwhereχ−1\nij(ω) is theij-component of the Fourier trans-\nformed inverse susceptibility tensor.\nThe magnetic susceptibility can be found from the\nLandau-Lifshitz-Gilbert (LLG) equation of motion. The\nstochasticLLGequationdescribesmagnetizationdynam-\nics and noise in both uniform as well as non-uniform fer-\nromagnets, and reads\ndm\ndt=−γm×[Heff+h+h(ext)]+α0m×dm\ndt.(4)\nHereγisthegyromagneticratio, Heffisaneffectivestatic\nmagnetic field determining the equilibrium state, h(t) is\nthe aboverandomnoise-field, h(ext)(t) is the weakexcita-\ntionintroducedin Eq. (1), and α0isthe Gilbert damping\nconstant. Linearizing this equation in the magnetic re-\nsponse to h(ext)(t), we find the inverse susceptibility\nχ−1=1\nγ/bracketleftbigg\nγ|Heff|−iωα0iω\n−iω γ|Heff|−iωα0/bracketrightbigg\n(5)\nwritteninmatrix(tensor)formintheplanenormaltothe\nequilibriummagnetizationdirection. Notethat the static\nfield has here been assumed local and magnetization in-\ndependent. While not valid in most realistic situations,\nthis simple form for the effective field captures the key\nphysics of interest here, since only the dissipative part\nof the susceptibility (the Gilbert damping term) affects\nthe noise. Inserting Eq. (5) into Eq. (3), we get the\nwell-known result11\n/angbracketlefthi(t)hj(t′)/angbracketright=2kBTα0\nγMsVδijδ(t−t′),(6)\nwhereiandjdenote components orthogonalto the equi-\nlibrium magnetization direction. This expression relates\nthe equilibrium noise, in terms of h, to the damping or\ndissipationofenergyin the ferromagnet. It maybe noted\nthat in thin ferromagneticfilms in good electrical contact\nwith a metal, the equilibrium noise and corresponding\nGilbert damping has been shown to be substantially en-\nhanced. This is due to the transfer of transverse spin\ncurrent fluctuations in the neighbouring metal to the\nmagnetization6,12.\nWe now turn our attention to a more complex sys-\ntem, i.e., a metallic ferromagnet whose direction of mag-\nnetization mis varying along some direction in space,\nsay, the y-axis. It is assumed that the spatial variation\nis adiabatic, i.e., slow on the scale of the ferromagnetic\ncoherence length. The ferromagnet is furthermore as-\nsumed to be translationally invariant in the x- andz-\ndirections, and its magnetization magnitude is taken to\nbe constant and equal to the saturation magnetizationMs. In general, the dynamics and fluctuations of such\na magnetization texture depend on position. Due to the\nspatial variation of the magnetization, longitudinal (i.e.,\npolarized parallel with the magnetization) spin current\nfluctuations transfer spin angular momentum to the fer-\nromagnet. The resulting enhancement of the magnetiza-\ntion noise is described by introducing a random magnetic\nfield, whose correlator is inhomogenous and anisotropic,\nunlike Eq. (6). By the FDT, the correlator is related\nto the magnetization damping, that acquires a nonlo-\ncal tensor structure. In the following we make use of\nthe fact that the time scale of electronic motion is much\nshorter than the typical precession period of magneti-\nzation dynamics, as implicitly done already in Eq. (6).\nWe shall disregard spin-flip processes and the associated\nnoise. Spin-flipcorrectionsinFe, Ni, andCoareexpected\nto be small because the spin-flip lengths are long com-\npared to the length scale of spatial variation (domain\nwall width) we consider. Spin-flip is important in Py.\nHowever, domain walls in Py are so wide that the effects\ndiscussed here are not important anyway. We therefore\ndo not discuss spin-flip scattering.\nIt is convenient to transform the magnetization tex-\nture to a rotated reference frame, defined in terms of the\nequilibrium (average) magnetization direction m0(y) =\n/angbracketleftm(y,t)/angbracketrightof the texture. The three orthonormal unit\nvectors spanning this position-dependent frame is ˆ v1=\nˆ v2׈ v3,ˆ v2= (dm0/dy)/|dm0/dy|andˆ v3=m0. The\nlocal gauge\nU(y) =/bracketleftbigˆ v1(y)ˆ v2(y)ˆ v3(y)/bracketrightbigT, (7)\ntransforms the magnetization, and hence the relevant\nequations involving the magnetization, to this frame.\nThat is, Um0(y)≡˜m0=ˆ z, where the tilde indicates\na vector in the transformed frame. We note also that\nUˆ v1=ˆ xandUˆ v2=ˆ y, and that Uis orthogonal, i.e.,\nU−1=UT= [ˆ v1ˆ v2ˆ v3].\nWe consider a charge current Iflowing through the\nferromagnet along the y-axis. Assuming that the equi-\nlibrium magnetization direction m0(y) changes adiabat-\nically, the electrons spins align with the changing mag-\nnetization direction when propagating through the tex-\nture. Thespincurrentisthenanywherelongitudinal,and\nhence given by Is(y) =Ism0(y). The alignment of the\nelectrons spins causes a torque τ(y) =dIs(y)/dyon the\nferromagnet. Since dIs(y)/dyis perpendicular to m0(y),\nthe torque can be written τ(y) =−m0(y)×[m0(y)×\ndIs(y)/dy], or˜τ(y) =Uτ(y) =−˜m0×[˜m0×UdIs(y)/dy]\nin the transformed representation. When I= 0, which\nwe will take in the following, Is= 0 and ˜τ= 0, on aver-\nage. However, at T/negationslash= 0 thermal fluctuations of the spin\ncurrent result in a fluctuating spin-transfer torque\n∆˜τ(y,t) =−∆Is(t)˜m0×[˜m0×Udm0(y)\ndy],(8)\nwhere ∆Is(t) are the time-dependent spin current fluctu-\nationswith zeromean, propagatingalongthe y-direction.3\nThe action of the fluctuating torque on the magnetiza-\ntion is described by the LLG equation if we, by conserva-\ntion of angular momentum, add the term γ∆τ/(MsA)\non the right hand side. Here Ais the cross section\n(in thexz-plane) of the ferromagnet. Linearizing and\ntransforming the LLG equation to the rotated reference\nframe, it is seen that the fluctuating torque (8) can\nbe represented by a random magnetic field ˜h′(y,t) =\n∆Is(t)/MsA)[˜m0×Udm0(y)/dy], analogous to h(t) dis-\ncussed above. Using Eq. (7)\n˜h′(y,t) =−∆Is(t)\nMsA/vextendsingle/vextendsingle/vextendsingle/vextendsingledm0(y)\ndy/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆx, (9)\ni.e., the (transformed) current-induced random field\npoints in the x-direction.\nThe longitudinal spin current fluctuations ∆ Is(t) can\nbe found by Landauer-B¨ uttiker scattering theory6,13.\nDisregarding spin-flip processes, the spin-up and spin-\ndown electrons flow in different and independent chan-\nnels. In the low-frequency regime, in which charge is in-\nstantly conserved, longitudinal spin current fluctuations\nare perfectly correlated throughout the entire ferromag-\nnet. Hence, the thermal spin current fluctuations are\ngiven by6,13\n/angbracketleft∆Is(t)∆Is(t′)/angbracketright=¯h2\n(2e)22kBT(G↑+G↓)δ(t−t′),(10)\nwhereG↑(↓)istheconductanceforelectronswiththespin\naligned (anti)parallel with the magnetization. This ex-\npression is simply the Johnson-Nyquist noise generalized\nto spin currents6. We find from Eqs. (9) and (10)\n/angbracketleft˜h′\nx(y,t)˜h′\nx(y′,t′)/angbracketright=2kBTξxx(y,y′)\nγMsVδ(t−t′) (11)\nfor the correlator of the current induced random field.\nHere we have defined\nξxx(y,y′) =γ¯h2σ\n4e2Ms/vextendsingle/vextendsingle/vextendsingle/vextendsingledm0(y)\ndy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledm0(y′)\ndy/vextendsingle/vextendsingle/vextendsingle/vextendsingle(12)\nwithσ= (G↑+G↓)L/Athe total conductivity. Recall\nthat˜h′\ny(t) =˜h′\nz(t) = 0. Eq. (11) describes the nonlo-\ncal anisotropic magnetization noise due to thermal cur-\nrent fluctuations in adiabatic non-uniform ferromagnets.\nThis excess noise vanishes with the spatial variation of\nthe magnetization. As a consequence of Eq. (10), the\nrandom field correlator depends nonlocally on the mag-\nnetization gradient.\nAccording to the FDT, the thermal noise is related to\nthe magnetization damping. Since the noise correlator\n(11) is inhomogeneous and anisotropic, the correspond-\ning damping must in general be a nonlocal tensor. To\nevaluate the damping, we hence need the spatially re-xyz\n/c108/c47/c50/c113\nFIG. 1: An example of a non-uniform ferromagnet. The mag-\nnetization rotates with wavelength λin theyz-plane, forming\na spin spiral.\nsolved version of the FDT, which reads\n/angbracketleftδ˜mi(y,t)δ˜mj(y′,t′)/angbracketright=kBT\nMsA/integraldisplay\ndωe−iω(t−t′)\n×χij(y,y′,ω)−χ∗\nji(y′,y,ω)\ni2πω,\n(13)\nin the transformed representation. Here δ˜m(y,t) =\nUδm(y,t) =δmx(y,t)ˆ x+δmy(y,t)ˆ yare the spatially\ndependent transformed magnetization fluctuations. The\nsusceptibility is defined by\n∆˜mi(y,t) =/summationdisplay\nj/integraldisplay /integraldisplay\ndy′dt′χij(y,y′,t−t′)˜h(ext)\nj(y′,t′),\n(14)\nanalogous to Eq. (1), but with the external field\nand magnetic excitations transformed: ˜h(ext)\nj(y,t) =\nUh(ext)\nj(y,t) and ∆ ˜m(y,t) =U∆m(y,t). The suscep-\ntibility in the local gauge frame differs from Eq. (5) and\nhas to be determined. It is straightforward to gener-\nalize Eqs. (13) and (14) to the case of general three-\ndimensional dynamics.\nWe may substitute ˜h(ext)\nj(y′,t′) by˜h′\nj(y′,t′) in Eq. (14)\ntofindthe fluctuations δ˜m(y,t)ofthemagnetizationvec-\ntor caused by the spin-transfer torque. Combining this\nexpression with Eqs. (13) and (11), we arrive at an inte-\ngral equation for the unknown susceptibility, from which\nthe nonlocal tensor damping follows. Instead of finding\na numerical solution for an arbitrary texture, we con-\nsider here a ferromagnetic spin spiral as shown in Fig. 1,\nfor which the description of magnetization noise can be\nmappedontothemacrospinproblem. Asimpleanalytical\nresult can then be found, allowing for a comparison with\nEq. (6), and hence an estimate of the relative strength\nand importance of the current-induced noise and damp-\ning.\nSpin spirals can be found in some rare earth metals14\nand in the γ-phase of iron15, and are described by\nm0(y) = [0,sinθ(y),cosθ(y)], where θ(y) = 2πy/λ=qy,\nwithλthe wavelength of the spiral. Then dm0(y)/dy=\nq[0,cosθ(y),−sinθ(y)] so that|dm0(y)/dy|=q. As em-\nphasized earlier, our theory is applicable when the wave-\nlengthismuchlargerthanthemagneticcoherencelength.4\nFor transition metal ferromagnets, the coherence length\nis of the order of a few ˚ angstr¨ om. From Eq. (12) we\nfindξxx=γ¯h2σq2/(4e2Ms). The current-induced noise\ncorrelator (11) for spin spirals is hence homogeneous,\n/angbracketleft˜h′\nx(t)˜h′\nx(t′)/angbracketright=2kBTξxx\nγMsVδ(t−t′),(15)\nsimilar to Eq. (6), but anisotropic. The problem of relat-\ning noise to damping in terms of the FDT can therefore\nbe mapped exactly onto the macrospin problem: The\ntransformation (7) can be used to show that equations\nanalogous to Eqs. (1)-(6) are valid for the spin spiral,\nwhen analyzed in the local gauge frame. It is then seen\nthat the damping term corresponding to Eq. (15) is\n˜m×← →ξd˜m\ndt(16)\nin the transformed representation. Here\n← →ξ=/parenleftbigg\nξxx0\n0 0/parenrightbigg\n(17)\nis the 2×2 tensor Gilbert damping in the xy-plane.\nHence,ξxxis the enhancement of the Gilbert damping\ncaused by the spatial variation of the magnetization and\nthe spin-transfer torque. Due to its anisotropic nature,← →ξis inside the cross product in Eq. (16), ensuring that\nthe LLG equation preserves the length of the unit mag-\nnetization vector ˜m.\nIn order to get a feeling for the significance of the\ncurrent-induced noise and damping, we evaluate← →ξnu-\nmerically for a spin spiral with wavelength 20 nm, and\ncompare with α0. Taking parameter values for α0,Ms,\nandσfrom Refs.16,17,18,19, we find ξxx≈5α0for Fe (with\nα0= 0.002), and ξxx≈4α0for Co (with α0= 0.005).\nHence, current-induced noise and damping in spin spi-\nrals can be substantial. Considering half a wavelength\nof the spin spiral as a simple domain wall profile, these\nresults furthermore suggest that a significant current-\ninduced magnetization noise and damping should be ex-\npected in narrow (width ∼10 nm) domain walls in typ-\nical transition metal ferromagnets. The increased noise\nlevel should assist both field- and current-induced do-\nmain walldepinning7,9,20. The increaseddamping shouldbe important for the velocity of current-driven walls,\nwhich recent theoretical and experimental advances sug-\ngest is inversely proportional to the damping4. The in-\ncreased noise and the tensor nature of the Gilbert damp-\ning should be taken into account in micromagnetic sim-\nulations.\nSo far we have only considered thermal current noise;\nlet us finally turn to shot noise. With the voltage U\nacross the ferromagnet turned on, a nonzero current I\nflows in the y-direction. Disregarding spin-flip processes,\nthe resulting spin current shot noise is6,13\n/angbracketleft∆I(sh)\ns(t)∆I(sh)\ns(t′)/angbracketright=¯h2\n(2e)2eUFGδ(t−t′) (18)\nat zero temperature. Here the superscript (sh) empha-\nsizesthatwearenowlookingatshotnoise. TheFanofac-\ntorFis between 0 and 1 for non-interacting electrons21.\nWhen the length of the metal exceeds the electron-\nphonon scattering length λep, shot noise vanishes13,21.\nλepis strongly temperature dependent, and can at low\ntemperatures exceed one micron in metals. To find the\ncontribution from shot noise to the magnetization noise,\nsimply replace Eq. (10) with Eq. (18) in the above cal-\nculation of the random-field correlator. In experiments\non current-induced domain wall motion, typical applied\ncurrent densities are about j= 108A/cm24. At low\ntemperatures, the ratio of shot noise to thermal current\nnoise,eUF/2kBT, can then exceed unity for long (but\nnot longer than λep) ferromagnetic (e.g. Fe) wires. Shot\nnoise can hence be expected to be the dominant contri-\nbution to the magnetization noise at low temperatures.\nIn summary, we have calculated current-induced mag-\nnetization noise and damping in non-uniform ferromag-\nnets. Taking into account both thermal and shot noise,\nwe evaluated the fluctuating spin-transfer torque on the\nmagnetization. The resulting magnetization noise was\ncalculated in terms of a random magnetic field. Em-\nploying the FDT, the corresponding enhancement of the\nGilbert damping was identified for spin spirals.\nThis work was supported in part by the Research\nCouncil of Norway, NANOMAT Grants No. 158518/143\nand 158547/431, and EC Contract IST-033749 “Dyna-\nMax”.\n1L. Berger, J. Appl. Phys. 49, 2156 (1978).\n2L. Berger, J. Appl. Phys. 55, 1954 (1984).\n3J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n4C. H. Marrows, Adv. Phys. 54, 585 (2005).\n5W. Wetzels, G. E. W. Bauer, and O. N. Jouravlev, Phys.\nRev. Lett. 96, 127203 (2006).\n6J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 95, 016601 (2005).\n7D. Ravelosona, D. Lacour, J. A. Katine, B. D. Terris, and\nC. Chappert, Phys. Rev. Lett. 95, 117203 (2005).8G. Tatara, N. Vernier, and J. Ferr´ e, Appl. Phys. Lett. 86,\n252509 (2005).\n9R. A. Duine, A. S. N´ u˜ nez, and A. H. MacDonald, Phys.\nRev. Lett. 98, 056605 (2007).\n10L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Statis-\ntical Physics, Part 1 (Pergamon Press, 1980), 3rd ed.\n11W. F. Brown, Phys. Rev. 130, 1677 (1963).\n12Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n13Y. M. Blanter and M. B¨ uttiker, Phys. Rep. 336, 1 (2000).5\n14J. Jensen and A. K. Mackintosh, Rare Earth Magnetism\n(Oxford University Press, 1991).\n15M. Marsman and J. Hafner, Phys. Rev. B 66, 224409\n(2002).\n16S. M. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974).\n17K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n18CRC Handbook of Chemistry and Physics (CRC Press,1985), 66th ed.\n19American Institute of Physics Handbook (McGraw-Hill,\n1963), 2nd ed.\n20J. P. Attan´ e, D. Ravelosona, A. Marty, Y. Samson, and\nC. Chappert, Phys. Rev. Lett. 96, 147204 (2006).\n21A. H. Steinbach, J. M. Martinis, and M. H. Devoret, Phys.\nRev. Lett. 76, 3806 (1996)." }, { "title": "2112.08163v1.An_Innovative_Transverse_Emittance_Cooling_Technique_using_a_Laser_Plasma_Wiggler.pdf", "content": "arXiv:2112.08163v1 [physics.acc-ph] 15 Dec 2021An Innovative Transverse Emittance Cooling Technique usin g a Laser-Plasma Wiggler\n1,†O. Apsimon,∗2D. Seipt,1,5,†M. Yadav,1,†A. Perera,3Y. Ma,\n4,†D.A. Jaroszynski,3A. G. R. Thomas,6,†G. Xia, and1†C. P. Welsch\n1The University of Liverpool, Liverpool L69 3BX, United King dom\n2Helmholtz-Institut Jena, Fr¨ obelstieg 3, 07743 Jena, Germ any\n3G´ erard Mourou Center for Ultrafast Optical Sciences and De partment of Nuclear Engineering and Radiological Sciences ,\nUniversity of Michigan, Ann Arbor, MI 48109, USA\n4University of Strathclyde, Glasgow G11XQ, Scotland\n5University of California, Los Angeles, CA 90095, USA\n6University of Manchester, M13 9PL, Manchester, United King dom and\n†The Cockcroft Institute of Accelerator Science and Technol ogy, Warrington, WA4 4AD, United Kingdom\nWe propose an innovative beam cooling scheme based on laser d riven plasma wakefields to address\nthe challenge of high luminosity generation for a future lin ear collider. For linear colliders, beam\ncooling is realised by means of damping rings equipped with w iggler magnets and accelerating\ncavities. This scheme ensures systematic reduction of phas e space volume through synchrotron\nradiation emission whilst compensating for longitudinal m omentum loss via an accelerating cavity.\nInthis paper, the concept ofa plasmawiggler and its effectiv e model analogous toa magnetic wiggler\nare introduced; relation of plasma wiggler characteristic s with dampingproperties are demonstrated;\nunderpinning particle-in-cell simulations for laser prop agation optimisation are presented. The\noscillation of transverse wakefields and resulting sinusoi dal probe beam trajectory are numerically\ndemonstrated. The formation of an order of magnitude larger effective wiggler field compared to\nconventional wigglers is successfully illustrated. Poten tial damping ring designs on the basis of\nthis novel plasma-based technology are presented and perfo rmance in terms of damping times and\nfootprint was compared to an existing conventional damping ring design.\nI. INTRODUCTION\nAfter the discovery of the Higgs boson at the Large\nHadron Collider (LHC) at CERN [1, 2], the global vi-\nsion is a successor collider in order to explore the prop-\nerties of this newly discovered boson in great detail. The\nphysics of the Higgs field still holds secrets of funda-\nmental interactions and can only be addressed in the\ncontrolled environment of an electron-positron collider\nthat allows elementary particle collisions with minimum\nsynchrotron radiation losses and well defined initial mo-\nmenta. Over the last few decades, two different de-\nsign approaches, using normal conducting (CLIC, [3])\nand superconducting (ILC, [4]) radio frequency struc-\ntures, revealed the size of such a machine to be about\n30−50km. The electron-electron option of the Future\nCircular Collider (FCC) project envisages a 100km cir-\ncumference machine for collisions at centre-of-mass en-\nergy of 90GeV −350TeV [5]. The limits of conventional\ntechnologies for high energy demand of particle physics\nexplorationpersuade us to new frontiers ofparticle accel-\nerator science. Paradigm shifting technologies are being\ndeveloped such as plasma acceleration [6–8]. However,\nthere are still many challenges to be addressed before\nthe maturation of plasma technology for large-scale ac-\ncelerator applications. Advanced and novel accelerators\ncommunity (ICFA- ANAR2017) underlined and priori-\ntised the following technological challenges; the repeti-\ntion rate, efficiency and beam quality of the lasers; scala-\n∗email: oznur.apsimon@liverpool.ac.ukbility of the system; delivery of high collision luminosity;\nresolution of diagnostics; comprehensive simulations and\nthe availability of dedicated test facilities [9]. Driven by\nhigh-power lasers, plasma can generate orders of mag-\nnitudes larger electric fields compared to metallic radio-\nfrequency cavities[6, 10, 11]. In this paper, an innovative\ncooling method is presented which shows great promise\nofreplacingthe magneticwigglersin adampingringwith\nplasma wigglers and providing superior beam quality for\nthe future linear collider based on plasma technology.\nIn order to achieve high luminosity, a the ILC design\nrelies on emittance cooling via damping rings followed\nby a long transfer line from ring to main linear accel-\nerator followed by beam delivery system to the final in-\nteraction point. Emittance cooling is achieved through\nradiation damping due to synchrotron radiation emitted\nfrom beam particles moving along curved trajectories of\na circular accelerator (damping ring) that has a circum-\nference of a few hundred meters to kilometers. Particles\nemit synchrotron radiation depending on the local cur-\nvature of the orbit within a cone of angle γ, whereγis\nthe relativistic Lorentz factor. Longitudinal momentum\nis restored by radio-frequency cavities in the ring, while\ntransverse momentum is damped on every turn through\nradiationdamping and quantum excitation areequal and\nan equilibrium value is reached [12], typically in millisec-\nonds [13, 14]. A faster damping is achieved by increasing\nthe energy loss per turn by adding high field periodic\nmagnetic structures (wigglers or wigglers, depending on\nthe magnetic strength) [15].\nAsan example, the InternationalLinearCollider(ILC)\nproject proposes a machine with 250-500GeV centre-of-2\nmass energy over about a 31km footprint [16]. Electrons\nandpositronsemergingfromdifferentsourcesundergoan\ninitial acceleration up to 5GeV before they are injected\ninto a their respective damping rings with a circumfer-\nence of 3.2km, housed in the same tunnel. ILC damping\nrings are designed in a race track shape to accommodate\ntwo straight sections. A radiative section comprising 54\nsuper-ferric wigglers is located in one of these straight\nsections. Each wiggler is 2.1m long and generates a\n2.16T peak magnetic field when operating at 4.5K and\nradiates 17 kW radiation power [16]. This straight sec-\ntion also houses a superconducting radio-frequency sys-\ntemtoreplenishthelongitudinalmomentumofthebeam.\nA compact synchrotronradiation insertion device with\nsignificantly larger fields than the current wigglers would\nlimit the footprint and the cost of the ring by reducing\nthe required number and size of these insertion devices.\nThere are several methods for plasma assisted radiation\ngeneration from a relativistic particle beam, such as, be-\ntatron oscillations, Compton scattering, bremsstrahlung\nand transition radiation [17]. We propose to incorporate\nthe concept of plasma wiggler as radiators in a damping\nring to benefit from their large effective magnetic fields\nand compactness. There are various concepts to conceive\na plasma wiggler [18–21]. Following the one proposed in\n[18] , a plasma wiggler is formed when a short laser pulse\nis injected into plasma off-axis or at an angle that causes\nthe centroid of the laser pulse to oscillate. Given that\nthe product of the plasma wave number and the charac-\nteristic Rayleigh length of the laser is much larger than\none, the ponderomotively driven plasma wake will follow\nthis centroid. This oscillating transverse wakefield works\nas an wiggler forcing particles to follow sinusoidal trajec-\ntories and emit synchrotron radiation. In addition, the\ndamping time is inversely proportional to the square of\nthemagneticfieldofthedampingdevice. Theoretically,a\nplasmawigglercangenerateorderofmagnitudelargeref-\nfective magnetic fields than conventional wigglers, hence\ncan reduce the length of the damping units by a factor\nof hundred while providing the same damping times.\nIn this paper, we layout the fundamental design cri-\nteria including discussion on laser-plasma channel cou-\npling and average dephasing length in parabolic channel\nto achieve a plasma wiggler in Section II. This is followed\nby numerical demonstration of oscillation laser centroid\nin the channel and sinusoidal probe beam trajectory in\nSectionIII. In this section, we present the baseline de-\nsign that achieves 20T effective magnetic field compared\nto a typical 2T in a superconducting magnetic wiggler,\nas well as 4MW average radiated power by two orders of\nmagnitudeimprovementfromtheconventionalapproach.\nThis also includes studies discussing correlation of laser\ninjection offset with laser radius and plasma skin depth\nlaying out interesting phenomena of radial phase shift in\nfocusing force. In Section IV, we presented three differ-\nent dampingring designsthat offercompromisesbetween\nlow damping time or small footprint.II. PLASMA WIGGLER\nThe optimisation criteria for a plasma wiggler include\nlaser and plasma parametrisation ensuring the matching\nbetween the laser spot size and channel radius; probe\nbeam emittance to the channel width as well as limi-\ntations such as dephasing length in a channel. In this\nsection, characteristics of a plasma wiggler following the\noptimisation criteria is numerically demonstrated.\nLaser propagation through preformed plasma channels\nwere extensively studied, previously [22–31]. A plasma\nwiggler is formed when a short laser pulse is injected into\na parabolic plasma channel off-axis or at an angle that\ncauses the centroid of this laser pulse to oscillate [18]\nwhenthenormalisedvectorpotential a0<1. AGaussian\nlaser pulse will be guided in a parabolic channel in Eq. 1,\nn(r) =n0(1+(∆n/n0)r2/w2\n0), (1)\nat the matched spot size when the channel depth is equal\nto the critical channel depth as in ∆ n= (πrew2\n0)−1\n[10, 32]. In this expression re=e2/mc2is the classi-\ncal electron radius, n0is the on-axis plasma density and\nw0is the laser radius.\nFor a laser with sufficiently low power to satisfy P <\nPc, the self-focusing is avoided and the laser spot size\nis preserved during the propagation for an initial spot\nsize matched to the channel radius. If the laser power\nis higher than the critical power, Pc[GW]≈17(kL/kp)2,\nthen relativistic self-focusing will occur [10], where kL\nandkparethe wavenumbersfor the laserand the plasma\nwave, respectively.\nThe period of the laser centroid oscillation is deter-\nmined by the Rayleighlength ofthe laser, ZR=πw2\n0/λL,\nand given by 2 πZR. According to this, the centroid os-\ncillation follows,\nx=xcicos(z/ZR+φ), (2)\nwhereφis the initial arbitraryphase and xciis the initial\ntransverse offset. The wiggler period, λu, is equal to the\nwavelength of the laser centroid oscillation,\nλu= 2π2w2\n0/λL, (3)\nwhereλLis the wavelength of the driver laser.\nFurthermore, when kpZR≫1, the ponderomotively\ndriven wakefields follow the oscillating laser centroid. In\nthe case of an electron bunch injected into these oscil-\nlating wakefields at an appropriate phase such that the\nlongitudinal field Ex= 0 and the transverse field Eyis\nnonzero, then these probe electronswill follow oscillatory\ntrajectories correlated with the laser centroid.\nThe strength parameter corresponding to a plasma\nwiggler is derived in [18] and given in Eq. 4.\nau≈4πa2\n0Cxci/λL (4)\nwhereC=/radicalbig\nπ/8e≈0.38 for an optimised laser pulse\nduration and C→2Cfor a circularly polarised laser.3\nInamagneticwiggler,assumingthatthemagneticfield\nis sinusoidal, integrating the equation of motion of the\nelectrons in transverse plane, ¨ z=eBy/p, yields the max-\nimum horizontal deflection or the wiggler strength pa-\nrameter,\nK=B0\nmeceλu\n2π, (5)\nwhereByis the transverse field with a peak value of B0,\npis the momentum and meis the mass of an electron.\nTherefore, an effective magnetic field for a plasma wig-\ngler can be defined in practical units as in\nBeff[T] =au/(0.0934λu[mm]) (6)\nwhereKis replaced by the strength of plasma wiggler,\nauandλunow representing the period length of the\nplasma wiggler. Eq. 6can be translated into Beff[T] =\na2\n0Cxci[m]/(46.7π(w0[m])2) by substituting Eq. 3and\nEq.4in Eq. 6. Therefore, theoretically, for a nor-\nmalised vector potential, a0= 0.36, and the laser spot\nsize,w0= 10µm, a plasma wiggler can generate an order\nof magnitude larger wiggler field strength than supercon-\nducting wigglers designed for ILC. For a0values close to\nunity, strong transverse wakefields might cause ejection\nof probe electrons outside the wiggler and act as a de-\nflectingmagnet asshownin Section III.Therefore, higher\neffective wiggler fields may be achieved by reducing the\nlaser spot size. However, one should note that for small\nvalues of w0, fields created behind the laser pulse occupy\na narrow transverse region. Therefore capturing probe\nelectrons can be challenging. The general rule of thumb\nis that the transverse extend of both probe electrons and\nlaser centroid offset should be smaller than the laser spot\nsize.\nA. Coupling into the Channel\na. Laser Matching For the laser to propagate\nthrough the plasma with a constant spot size, the ini-\ntial spot size and the channel radius should match. A\ndetailed discussion is given in [33] by introducing dimen-\nsionlessvariablesin units ofplasmafrequency in SI units,\nωp=/radicalbig\nn0e2/meǫ0and the wavenumber, kp=ωp/c. Ac-\ncording to this, the evolution of the laser spot size, w, is\ngiven by,\nw2=w2\n0\n2+2R2\nw2\n0+/parenleftbiggw2\n0\n2−2R2\nw2\n0/parenrightbigg\ncos2Ωτ,(7)\nwherew0is the laser spot size, Ris the dimensionless\nchannel radius, Ω is the characteristic frequency of the\nlasercentroidoscillationand τisdimensionlesstime. The\nmatched case, where the spot size, w, is constant, re-\nquires the second term to be zero. Hence, for a matched\ncaseR=r2\nm/2 where rmis the dimensionless matchedlaser radius. Furthermore, Ω m= 2/(Mpr2\nm) is the di-\nmensionless frequency of the centroid oscillation, where\nMp=kL/kp. The coefficient Mp/2πtransforms the\nspace and time into units measured by plasma wave-\nlength and period, respectively.\nFIG. 1. Evolution of the dimensionless laser spot size for\nthe cases where the plasma channel radius is matched ( w0=\nrm) and unmatched ( w0= 1.2rm) to the laser spot size (red\nand blue curves, respectively). Laser centroid oscillatio ns in\nthe units of laser wavelength for the matched case is also\npresented with the green curve. The plot is generated for\na0= 0.36,w0= 10µm andn0= 7×1023m−3.\nFigure1presentsthe evolutionofthe laserspotsizeac-\ncording to Eq. 7, for matched ( w0=rm) and unmatched\n(w0= 1.2rm) cases in units of laser wavelength. For the\nunmatched case (blue curve), laser spot size oscillates\nwith an amplitude of ±5λL. Whereas it is constant at\n12.5λLduring its propagation for a matched condition.\nFigure1also shows the evolution of the centroid for the\nmatched case which oscillates at Ω m.\nb. Beam Envelope Matching The beam envelope\nevolves as a function of the beam emittance and energy\nas well as any external focusing or defocusing (such as\nspace charge) and dispersive effects. Ignoring the adi-\nabatic damping, any dispersive contribution and space\ncharge, the beam size, rb, is given as below.\nd2rb\nd(ct)2=ε2\nn\nγ2r3\nb−k2\nβrb (8)\nwhereεn=γrbθbisthenormalisedtransversebeamemit-\ntance with θbis the rms beam angle and γis the Lorentz\nfactor. To maintain a constant beam size through the\nchannel, i.e., d2rb/dct2= 0, the matched beam size\nshould be rbm= (εn/γkβ)1/2[34]. This matching condi-\ntion is also employed for the simulations presented here.4\nB. Dephasing Length in a Parabolic Channel\nEffective length of propagation for laser-plasma inter-\nactions is generally limited by laser diffraction. Guid-\ning high power laser pulses through a parabolic plasma\nchannel can circumvent this. Here, the channel acts as\neffective radio-frequency cavities that confine and shape\nthe electromagnetic fields in conventional accelerators.\nWhen diffraction is eliminated, the limiting factor for the\neffective interaction length is the dephasing length,\nλd=λ3\np/λ2\nL, (9)\nwhereλp= 2π/kpistheon-axisplasmawavelength. This\nis the distance over which a trailing electron bunch gains\nsufficient energyto outrun the driving laserpulse to cross\ninto decelerating fields. The dephasing length scales with\nn−3/2\n0wheren0is the plasma densityon the channelaxis.\nTo prevent witness electrons overtaking the driver laser\npulse, the dephasinglengthshould belargerthan, orsim-\nilar to the length of the plasma wiggler. Therefore, one\nmust optimise the wiggler target by establishing a com-\npromise between the interaction length and plasma den-\nsity [35, 36].\nHowever,inthecaseofasinusoidaltrajectoryofalaser\npulse in a channel, plasma density hence the dephasing\nlength for the laser has a radial dependence. Therefore,\nsuch a laser pulse would experience an averagedephasing\nlength that is shorter than a laser travelling on channel\naxis through the channel. The axial dephasing length\ncan be expanded as in Eq. 10,\nλd(r) =1\nλ2\nL(2πc)3/parenleftbiggǫ0me\ne2/parenrightbigg3/2\nn(r)−3/2,(10)\nby substituting the plasma wavelength and isolating the\nplasma density as a function of radial coordinate. The\naverage dephasing length is then calculated as,\n¯λ=1\nyi−yf/integraldisplayyf\nyiλd(y)dy, (11)\nresulting in,\n¯λd=An−3/2\n0/parenleftbigg\n1+∆n\nn0y2\nf\nw2\n0/parenrightbigg−1/2\n, (12)\nwhere the initial injection location yi= 0,A=\n(1/λ2\nL)(2πc)3(ǫ0me/e2)3/2andw2\n0/(∆n/n0) is defined as\nthe channel width, rc, for practicality. Using a Taylor\nexpansion on the expression in parenthesis in Eq. 12, it\nis simplified to Eq. 13,\n¯λd=An−3/2\n0/parenleftbigg\n1−y2\nf\n2r2c/parenrightbigg\n. (13)\nwhere the average dephasing length equals its on-axis\nvalue when the radial extend of the beam, yf, is zero.\nGiven that λuis the wiggler period and Nis the num-\nber ofperiodsofthe wiggler, to preventdephasingduringthe length of the wiggler, one can introduce the relation\n¯λd≥λuNand substitute ¯λd=λuNin Eq.13to de-\nduce an average plasma density, np,d, for a given channel\nwidth,rc, as given in Eq. 14,\nnp,d=/parenleftbigg4πc3\nw2\n0λLN/parenrightbigg2/3/parenleftbiggǫ0me\ne2/parenrightbigg/parenleftbigg\n1−y2\nf\n3r2c/parenrightbigg\n(14)\nThe expression will help determine the plasma density\nto achieve a certain dephasing length in a channel and\nentails two terms. The first term is an on-axis plasma\ndensity which can be deduced from Eq. 10for ar= 0\nand the second term is that scales with the radial extend\nof the beam and channel width.\nThe other factor effecting the dephasing length is the\nlongitudinal velocity reduction of the witness particles\ndue to their sinusoidal trajectories which is approxi-\nmatelyequalto ∝angbracketleftvz∝angbracketright/c≃1−(∝angbracketleftp2\n⊥∝angbracketright+m2c2)/2p2\nz. However,\nthis effect is not considered for the results presented in\nnext section.\nIII. SIMULATIONS AND DISCUSSIONS\nAplasmawiggleranditsinteractionwithaprobebeam\nare simulated using the particle-in-cell code, EPOCH\n[37]. A 200 µm long simulation window is defined with\n±70µm transverse acceptance that moves at the 99.9%\nof the speed of light.\nThe domain is set up with 10 cells per laserwavelength\nin the longitudinal, and per plasma skin depth for trans-\nverse directions, allowing the smallest features in those\ndirections to be resolved. Electrons in the plasma are\nrepresented with 5 macroparticles per cell assuming an\nimmobile neutralizing background. All boundaries are\nabsorbing for particles and fields. Fields are calculated\nwith a 2nd-order Yee solver while currents are smoothed\nusing a 3-point nearest-neighbour low-pass filter in each\ndirection [38]. QED photon emission by particles un-\ndergoing sinusoidal trajectories is calculated using an\noptical-depth-like model through local electromagnetic\nfield strengths at macroparticlepositions and incorporat-\ning radiation reaction and energy straggling effects [39].\nThe wavelengths of classical fields represented by such\nphotons are orders of magnitude too small to be resolved\nbythegridandarenotincorporatedintothemacroscopic\nfields of the simulation.\nThe physical characteristics of the laser, plasma and\nprobe beam are given in the following subsection.\nA. Initial simulation settings\na. Plasma A plasma profile comprising a vacuum\nsection followed by a 10 µm up-ramp section prior to a\nparabolic distribution was implemented. The zero den-\nsity section is provided for initialisation of a laser pulse\nby ensuring the consistency of the Maxwell’s equations.5\nThe plasma density profile for the first 200 µm of the\nsimulation is presented in Fig. 2.\nFIG. 2. The initial plasma density configuration including a\nzero density and up-ramp region prior to parabolic distribu -\ntion.\nb. Laser A custom defined laser pulse is used in\nEPOCH defining 2D Gaussian beam profiles. Leading\norder corrections on longitudinal electric field for the\ndiffraction angle expansion as well as for short pulse du-\nration (first derivative of pulse envelope in longitudinal\nfield) are implemented. The model featured control on\nparameters for focal spot position and temporal pulse\ncentre as well as longitudinal magnetic fields for out-of-\nplane laser polarisation.\nThe longitudinal pulse profiles are given as,\nEpulse=−E0e−(x−xt0)2/2(σtc)2, (15)\nBpulse=−E0e−(x−(xt0−δ))2/2(σtc)2/c,(16)\nwhereE0is the peak electric field of the laser pulse, xt0\nis the spatial position of the temporal pulse centre at\nt= 0,σtis the pulse duration and δis the offset for the\nmagnetic field. The initial distribution of the Gaussian\nlaser field can be seen in Fig. 3with a peak electric field\nof 3.2TV/m.\nc. Probe beam A hypothetical 10pC electron probe\nbeam with a γ=1000, a matched spot size of ∼1µm and\nrms length of 1 µm. The injection phase is determined so\nthat no accelerating field acts on the probe beam while\nthe focusing field is larger than zero and at its maxi-\nmum value. Thetransverse( ErandBθ) andlongitudinal\nwakefields ( Ex), that are related to each other through\nPanofsky-Wenzel theorem, are presented in Eq. 17[10].\nEr−Bθ∼/parenleftbigg4r\nkpw2\n0/parenrightbigg\nexp/parenleftbigg\n−2r2\nw2\n0/parenrightbigg\nsin(kpξ)\nEx∼exp/parenleftbigg\n−2r2\nw2\n0/parenrightbigg\ncos(kpξ),(17)\nFIG. 3. Initial laser pulse profile for a0= 0.8,λL= 800nm\nandw0= 30µm.\nwhereξ=x−ct. Theconditions, Ex= 0andEr−Bθ>0\ncan be satisfied simultaneously when kpξ= 3π/2. Ac-\ncording to this, the first possible location at λp/4 behind\nthe head of the laser generally is located inside the laser\ntherefore is an undesirable injection spot. Therefore, the\nnext possible location 5 λp/4 behind the head of the laser\nis used for the results reported in this paper.\nIt is non-trivial to exactly determine the start of the\nwakefields with respect to the laser pulse front. There-\nfore, the probe beam injection location is manually stud-\nied the field patterns and determined as approximately\n89.5µm behind with respect to the centre of the laser.\nThe field distribution experienced by the probe beam is\npresented in Fig. 4.\nFIG. 4. An example case depicting an optimum injection\npoint with Ex≈0 andEy>0. The colour map presents the\ntransverse (top) and longitudinal (bottom) field distribut ions.\nThe injection location is shown with black circles with resp ect\nto the centre of the laser (red circle). Blue line-outs show t he\non-axis field values.6\nB. Probe beam trajectory and radiated power\nThree merits for initial optimisation of the laser-\nplasma wiggler prior to injection of a probe beam are\neffective magnetic field, transverse offset of laser injec-\ntion point and dephasing length.\nFIG. 5. Induced effective magnetic field provided by the\nplasma wiggler (black curve) and average emitted radiation\nby the probe beam (blue curve) as a function of a0. The\ncurves are calculated for w0= 10µm andxci= 6µm.\nThe effective magnetic field has a strong dependence\non the normalisedvector field ofthe laser, injection offset\nand radius of the laser. The power radiated by the probe\nbeam performing lateral acceleration under this field is\ngiven in Eq. 18\nPavg[W] = 6.336E2[GeV]B2\neff[kG]IbLu,(18)\nwhereIbis the peak beam current and Luis the length of\nthe radiator(wiggler)which is taken equal to the dephas-\ning length for each given wiggler solution in this work.\nFigure5shows that an order of magnitude larger effec-\ntive magnetic field, compared to current ILC radiators is\ntheoretically possible at around a0≈0.36, with two or-\ndersofmagnitude largerradiatedaveragepower(4MW).\nInsummary,inaparabolicchannel, higheffectivemag-\nnetic fields canbe generatedby laserswith high a0. How-\never, there is an upper limit to maintain the probe beam\ntrajectory. The probe beam injected into such a chan-\nnel might be ejected in one transverse direction before it\nreaches to the next polarity of its sinusoidal trajectory.\nTherefore, for a plasma wiggler a0should be a fraction\nof unity. Example trajectories for a0=0.2, 0.5 and 0.8\nare presented in Fig. 6and the ejection of the probe\nbeam outside the field regions after a few millimeters of\npropagation for these three cases are shown in Fig. 7.\nNevertheless, in order to generate bending strengths sig-\nnificantly larger than the conventional technology, one\nmight consider smaller laser spot sizes and hence injec-\ntion offsets, to compensate against the limit on a0.(a)\nFIG.6. Beam trajectory following injection atoptimumphas e\nfor increasing a0.\nTABLE I. Analytical solutions for plasma wigglers yielding\ndifferent radiation power, Pav, and dephasing length, λd.\nSolution a0w0xcineauλuB0λdPav\n(µm) (µm) (m−3) (mm) (T) (cm) (kW)\n(I) 0.2 30 5 1 ×10241.2 22 0.6 3 2\n(II) 0.5 30 5 1 ×10247.5 22 3.6 3 75\n(III) 0.8 30 5 1 ×102419 22 9.2 3 500\n(IV) 0.3 10 3 0 .7×10241.6 2.5 7 5 500\n(V) 0.3 10 6 0 .7×10243.2 2.5 14 5 2000\n(VI) 0.36 10 6 0 .7×10244.6 2.5 20 5 4000\n(VII) 0.3 40 5.32 1 ×10242.8 39 0.9 3 3.5\nTableIsummarisesasetofnumericalsolutionsforphe-\nnomena observed during the numerical optimisation of a\nplasma wiggler to study witness ejection from the chan-\nnel and radial phase shift of the wakefields. Although,\nthe effective magnetic field scales with the square of a0\nandis proportionateto xci, weshowedthat a0∼1causes\nthe ejection of witness particles. Hence, angle integrated\naverage power yield through generation of a large Beff\nis achieved by optimising xci. We observed that a ra-\ndial phase shift is introduced in the transverse fields if\nthe conditions of xci/(c/ωp)≈1 andxci/w0≪1 are\nnot met simultaneously. This is demonstrated in Fig. 8\nusing a set of cases representing different values of the\nabove mentioned ratios. It is also observed that the ratio\nof the injection offset and the plasma skin depth has a\nmore dominant effect in this phenomena.\nCounter-intuitively, the radial phase shift seems to ac-\ncommodate witness propagation at least throughout the\ndephasing length as demonstrated in Fig. 9for the base-\nline case given in Table I-(VI). In the figure, the per-\npendicular propagation distance in the plasma is 10mm.\nHowever, the effective laser propagation is almost 10cm\ndue to the sinusoidal trajectory, achieveing the laser de-\nphasing length at about 5mm of perpendicular propa-\ngation. Figure also shows that the witness centroid fol-\nlows a sinusoidal trajectory as expected. For this base-7\n(a)\n(b)\n(c)\nFIG. 7. The ejection of probe beam out of the channel with\nincreasing a0due to strong transverse focusing field, eEy+\ncBz, driven in plasma for (a) a0= 0.2, (b)a0= 0.5 and\n(c)a0= 0.8 (Table I-(I), (II) and (III), respectively). All\nthree cases are for an xci= 5µm,w0= 30µm andne=\n1×1024m−3.\nline case, the characteristics of the laser, plasma and the\nprobe beam as well as the performance of the resulting\nplasmawiggleraresummarisedin Table II.We confirmed\nthe feasibility of generating this parameter space in a\nfacility such as Scottish Centre for the Application of\nPlasma-based Accelerators (SCAPA) [40] and tests willbe planned for future.\nTABLE II. Characteristics of the initial setup, resulting\nplasma wiggler and the emitted radiation.\nLaser\nWavelength (nm), λL 800\nNormalised vector potential, a0 0.36\nRadius at focus( µm),w0 10\nPulse length (fs), σL 25\nInjection offset ( µm) 6\nPlasma\nDensity ( m−3),n0 7×1023\nProbe beam\nNormalised beam energy γ, 1000\nDensity (m−3),ne n0/1000\nCharge (pC), Q 10\nMatched radius ( µm),σy 0.6\nBunch length ( µm),σx 1\nDephasing length (cm) 5\nPlasma wiggler\nPeriod (mm), λu 2.5\nStrength parameter, au 4.6\nEffective field (T), Beff 20\nRadiation\nWavelength (nm), λγ 14\nAngle integrated average power (MW) 4\nIn this study, we concentrated on numerically demon-\nstrating the feasibility of a plasma wiggler that is pre-\ndicted by the theoretical optimisation. We demonstrated\na baseline setting achieving an order of magnitude larger\neffective magnetic field (20T) and two orders of magni-\ntude larger average radiation power (4MW) in a single\npass compared to those aimed with a conventional wig-\ngler.\nThe phase space cooling occurs after tens of hundreds\nof passages through the radiator. Hence observation of\ncooling directly from particle-in-cell simulations is pro-\nhibitively computing-heavy with the current capabilities\nand will be feasible in parallel to studies in exascale com-\nputing such as the WARPX project [41]. Therefore, in\nthe next section, we will analytically demonstrate the\nimpact of the designed baseline plasma wiggler on the\ncooling performance of a linear collider damping ring.\nIV. TRANSVERSE EMITTANCE DAMPING\nAn electron beam circulating in a damping ring losses\nenergyat eachturndue to the synchrotronradiationgen-\neratedfrom its movement aroundthe circularorbit ofthe\nring as well as the radiation generated from lateral ac-\nceleration in the wiggler magnet. The beam emittance\ndecreases exponentially as given in Eq. 19,\nε(t) = (εinj−εequ)e−2t/τ+εequ, (19)\nfrom its value during injection into damping ring, εinj, to\nan equilibrium value εequwhereτis the damping time.8\n(a)\n (b)\n(c)\n (d)\nFIG. 8. The radial phase of transverse fields for four differen t wiggler solutions providing a) xci/w0= 0.2,xci/(c/ωp) = 0.94,\nb)xci/w0= 0.6,xci/(c/ωp) = 0.94, c)xci/w0= 0.3,xci/(c/ωp) = 0.5 and d) xci/w0= 0.13,xci/(c/ωp) = 1 (Table I-(I), (IV),\n(V), (VII), respectively).\nFIG. 9. The trajectories of the rms centroids of the laser\npulse (blue dots), its theoretical prediction (red dashed l ine)\nand the witness for baseline solution given in Table I-(VI).\nThe emission of synchrotron radiation is a statistical\nprocess involving quantum fluctuation of beam parame-ters. Anequilibriumemittanceisreachedwhenthequan-\ntum excitation is equal to damping [12].\nA. Contribution from wigglers\nThe energy loss in wigglers constitutes nearly 95% of\nthe total energy loss. This energy loss of a beam during\nits each passage through a wiggler is given in Eq. 20\nU0=Cγ\n2πE4I2w (20)\nwhich is proportional to the forth power of its energy\nEand second radiation integral for a wiggler I2w. This\nintegral can be written as in Eq. 21\nI2w=/integraldisplayLu\n01\nρ2ds=1\n(Bρ)2/integraldisplayLw\n0B2\nwds=1\n(Bρ)2B2\nwLw\n2.\n(21)\nfor a wiggler with an effective length of Lwand a radius\nof curvature ρfor the deflected trajectory. To evaluate\nI2w, beam rigidity ( Bρ=p/e) is used to simplify the\nintegral.9\nFrom Eq. 21, it is straightforward to see that a wiggler\ncontributes to the energy loss per turn in a damping ring\nproportionately to the square of its magnetic field and\nthe effective length, i.e.,\nU0∝B2\nwLw. (22)\nFurthermore, the damping time, which is the time\nneeded for the emittance to decrease by 1/e of its in-\njected value, is given as,\nτ= 2E0T0\nU0, (23)\nwhereE0is the injection energy of the electrons into the\ndamping ring and T0is the revolution period of the elec-\ntronsinthering. Thisimpliesthatanorderofmagnitude\nlarger field enables the same amount of damping with a\n100 times shorter wiggler. Alternatively, for the same\nradiative length, a 100 times faster damping might be\nachieved with a wiggler providing an order of magnitude\nlarger wiggler strength.\nB. Cooling scenarios for an ILC-like machine\nThe impact of a plasma wiggler on the damping pro-\ncess was studied using parameters similar to the baseline\nfor ILC damping ring to demonstrate its potential for fu-\nture colliders. To study this, an injection emittance of\n6mmmrad was considered to be damped down to 20nm\n(µmmrad). The ILC-like baseline case consists of a to-\ntal radiative length of 113m including 54 wigglers with\n2.16T (as defined for electrons at ILC in the technical\ndesign report). The evolution of emittance is studied as\ndescribed in Section IVand presented in Fig. 10for dif-\nferent scenarios. For ILC damping rings, the equilibrium\nemittance should be reached below 200ms, between ma-\nchine pulses, which is shown in solid black curve in Fig.\n10.\nThe first scenario proposes to replace the superferric\nwigglers by plasma wigglers with characteristics given in\nTableIIover the same 113m radiative section. This is\npresented with hollow dots and provides the largest re-\nductionindampingtimebelow1ms. Thesecondscenario\nemploys the advantage in significantly increased bending\nstrength and proposes to decrease the radiative section\nlength anorderofmagnitudedown to 10m, that achieves\ndampingwithinabout5ms(dashedlightcurve). Eventu-\nally, the third scenario reduces this down to a one meter-\nlong radiative section (dashed bold curve) for a larger\ndamping time than the conventional ILC option. Never-\ntheless, it still provides an 80ms damping time which is\nadequate for the ILC machine cycle.\nA damping ring design with a higher ring energy pro-\nvides shorter damping times. However the natural emit-\ntance scales with the cube of the energy deeming a lower\nenergy ring favourable. Keeping in mind that the collec-\ntive effects are more severe at low energies, the plasma10010210410-210-1100101100101102103\n10-210-1100101\nFIG. 10. Performance of different cooling scenarios using\nplasma wigglers for ILC damping ring in comparison to base-\nline with superferric wigglers.\nwiggleroffersa solutionwhere at least twoordersofmag-\nnitude shorter damping times are possible while keeping\nenergylowenoughtoensureaminimalnaturalemittance\nfor the ring. Therefore a next generation damping ring\nequipped with plasma wigglers rather than conventional\ncounterparts provides an exciting scenario for the future\nlinear collider design studies.\nC. Multi-stage single-pass cooling\nThe circumference, hence the cost of a damping ring\nincreases with the number of bunches per linac pulse.\nThe ILC design envisages bunches to be injected into a\nring in a compressed time structure to limit its circum-\nference. After cooling, these bunches should be decom-\npressedbyusingstate-of-the-artdipole magnets(kickers)\nduring extraction to provide the desired bunch spacing\nfor the experiments. Previous studies showed that, in a\ndamping ring, certain instabilities are introduced during\ninjection/extraction [42] and circulation until a design\norbit is established before cooling.\nFurthermore, due to the quantum emission, there is\nan equilibrium energy spread of the beam that is estab-\nlished in the ring. Because the ring is dispersive, this\nenergy spread results in an equilibrium bunch length.\nThis equilibrium bunch length produced in conventional\ndamping rings (9 mm for compressor entrance) is much\nlonger than the typical values for operating conditions of\na plasma-based accelerator. Reducing the bunch length\nis difficult, and results in intra-beam and Touschek scat-\ntering (due to high current), which are sources of un-\nacceptable emittance growth. Therefore one must use\nbunch compression after a damping ring. For example,\nILC compresses about a factor of 30 after the ring. For\na plasma accelerator requiring bunch lengths a fraction\nof the plasma wavelength, e.g., as small as ∼10 microns,10\nthis would require bunch compression factors of order\n∼1000. It is difficult to imagine any bunch compression\nsystem that could achieve this while still preserving the\nbeam emittance.\nAs such, a scheme eliminating the use of a ring alto-\ngether would revolutionise the cooling process for a col-\nlider based on plasma technology. This future work will\nrequireextensive computing-heavynumerical workto ex-\nplore the beam dynamics of a single-pass linear damping\nsection with adjacent plasma stages for radiation and ac-\nceleration. Staging is already a hot topic in the field,\nproviding the community with many challanges to tackle\n[43–47].\nV. CONCLUSIONS\nAn innovative concept of using a plasma wiggler for\nbeam damping was explored. Initial results demonstrate\nan order of magnitude larger effective magnetic field\nachievable with realistic laser and plasma parameters.\nThis has direct implication of several scenarios compro-\nmising between orders of magnitude faster damping or\nsmallerfootprint. Acomparisonto the ILCdampingring\ndesign, as a conventional example, was presented demon-\nstrating, for extreme scenarios, damping within <1ms or\nalternatively reduction of footprint of radiative section\ndown to 1m from 113m.\nUsing plasma wigglers for emittance damping brings\nan exciting prospect for future linear colliders especially\nbased on plasma technology. This may include muon col-\nliders as well, where cooling time might compete againstdecay time. This opens up new avenues for exploration\nof the implementation at large-scale, numerically, where\nreduced models or extensive high performance comput-\ning resources will be required as well as studies regarding\nthe multi-staging.\nThe reduction in damping time provided by a plasma\nwiggler can compensate against the need for large\nenergy in a conventional set-up, allowing to limit\nthe natural emittance in the ring. That being said,\nimplications on feasibility of the concept regarding insta-\nbilities in adampingringat lowenergieswill be explored.\nACKNOWLEDGMENTS\nThis work is supported by the European Union’s Hori-\nzon 2020 research and innovation programme and U.S.\nNSFgrant1804463andAFOSRgrantFA9550-19-1-0072.\nComputing resources provided by STFC Scientific Com-\nputingDepartment’s SCARF cluster. Authorswouldlike\ntothank toJonRoddom andDerekRossfortheir endless\nsupport on implementation of EPOCH on SCARF clus-\nters. 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Lett. 120, 154801 (2018) ." }, { "title": "1605.08195v2.Thickness_and_temperature_dependence_of_the_magnetodynamic_damping_of_pulsed_laser_deposited___text_La___0_7__text_Sr___0_3__text_MnO__3__on__111__oriented_SrTi__text_O__3_.pdf", "content": "Thickness and temperature dependence of the magnetodynamic damping of pulsed\nlaser deposited La 0.7Sr0.3MnO 3on (111)-oriented SrTiO 3\nVegard Flovik,1,∗Ferran Maci` a,2, 3Sergi Lend´ ınez,2Joan Manel\nHern` andez,2Ingrid Hallsteinsen,4Thomas Tybell,4and Erik Wahlstr¨ om1\n1Department of Physics, NTNU, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n2Grup de Magnetisme, Dept. de F´ ısica Fonamental, Universitat de Barcelona, Spain\n3Institut de Ci` encia de Materials de Barcelona (ICMAB-CSIC), Campus UAB, 08193 Bellaterra, Spain\n4Department of Electronics and Telecommunications, NTNU,\nNorwegian University of Science and Technology, N-7491 Trondheim, Norway\n(Dated: October 5, 2018)\nWe have investigated the magnetodynamic properties of La 0.7Sr0.3MnO 3(LSMO) films of thick-\nness 10, 15 and 30 nm grown on (111)-oriented SrTiO 3(STO) substrates by pulsed laser deposition.\nFerromagnetic resonance (FMR) experiments were performed in the temperature range 100–300 K,\nand the magnetodynamic damping parameter αwas extracted as a function of both film thickness\nand temperature. We found that the damping is lowest for the intermediate film thickness of 15 nm\nwithα≈2·10−3, where αis relatively constant as a function of temperature well below the Curie\ntemperature of the respective films.\nI. INTRODUCTION\nThe magnetodynamic properties of nanostructures\nhave received extensive attention, from both fundamental\nand applications viewpoints1–3. Nanometer sized mag-\nnetic elements play an important role in advanced mag-\nnetic storage schemes4,5, and their static and most im-\nportantly their dynamic magnetic properties are being\nintensely studied6–8.\nComplex magnetic oxides display intriguing proper-\nties that make these materials promising candidates for\nspintronics and other magnetic applications9. Mangan-\nites have received attention due to a large spin polariza-\ntion, the appearance of colossal magneto-resistance and a\nCurie temperature above room temperature10–13. Within\nmanganites, LSMO has been regarded as one of the pro-\ntotype model systems. Transport and static magnetic\nproperties of LSMO are well studied, but less attention\nhas been paid to the magnetodynamic properties. For\napplications in magnetodynamic devices a low magnetic\ndamping is desirable, and having well defined magnetic\nproperties when confined to nanoscale dimensions is cru-\ncial.\nThe dynamic properties can be investigated by ferro-\nmagnetic resonance spectroscopy (FMR), which can be\nused to extract information about e.g. the effective mag-\nnetization, anisotropies and the magnetodynamic damp-\ning. Earlier studies on LSMO have investigated the dy-\nnamic properties of the magnetic anisotropies14, also pro-\nviding evidence for well defined resonance lines, which is\na prerequisite for magnetodynamic devices.\nThe magnetodynamic damping is an important mate-\nrial parameter that can be obtained through FMR spec-\ntroscopy by measuring the resonance linewidth as a func-\ntion of frequency. The linewidth of the resonance peaks\nhas two contributions; an inhomogeneous contribution\nthat does not depend on the frequency, and the dynamic\ncontribution that is proportional to the precession fre-quency and to the damping parameter α.\nEarlier studies by Luo et. al. have investigated\nthe magnetic damping in LSMO films grown on (001)-\noriented STO capped by a normal metal layer15,16, and\nfound a damping parameter of α≈1.6·10−3for a 20 nm\nthick LSMO film at room temperature. Typical ferro-\nmagnetic metals have damping values of α≈10−2, and\nthe low damping indicate LSMO as a promising material\nfor applications in magnetodynamic devices. The stud-\nies by Luo et. al. were performed at room temperature,\nwhereas the Curie temperature of LSMO is around 350\nK. However, the Curie temperature depends strongly on\nfilm thickness and approaches room temperature as the\nthickness is decreased. Being able to control the tempera-\nture is thus important in order to accurately characterize\nthe damping in thin LSMO films.\nThe thickness and temperature dependence of static\nand dynamic magnetic properties of thin film LSMO\ngrown on (001)-oriented STO have been investigated in\na previous study by Monsen et. al17. The dynamic prop-\nerties were characterized from the FMR linewidth mea-\nsured in a cavity based FMR setup at a fixed frequency\nof 9.4 GHz, and provided evidence that the magnetic\ndamping is dominated by extrinsic effects for thin films.\nThe properties of complex magnetic oxides are very\nsensitive to the structural parameters, hence thin film\ngrowth can be used to engineer the magnetic properties.\nWe have previously shown that LSMO grown on (001)-\noriented STO results in a biaxial crystalline anisotropy,\ncompared with almost complete in-plane isotropy for\nLSMO grown on (111)-oriented STO18. The possibil-\nity to control the functional properties at the nm-scale\nmake these materials promising for spintronics and other\nmagnetic based applications. Hence, detailed studies on\nhow film thickness affect the magnetic properties is im-\nportant.\nHere, we investigate the magnetodynamic properties\nof pulsed laser deposited LSMO films of thickness 10, 15arXiv:1605.08195v2 [cond-mat.mes-hall] 16 Jul 2016ii\na)\nb) c)\n38 39 40 41 42 43\n2θ6[degrees] Temperature6(K)Magnetic6moment6(emu/cc)\n100 150 200 250 300 350 4000200400600Magneticpmomentp[emu/cm³]600\n400\n200\n030pnm\n15pnm\n10pnm\n38pppppppppppppp39ppppppppppppppp40pppppppppppppp41ppppppppppppppp42pppppppppppppp43pppp\n2ppp[degrees]\n100ppppppp150ppppppp200ppppppp250ppppppp300ppppppp350ppppppp400pppp\nTemperaturep[K]p30pnm15pnm10pnm10pnm 15pnm 30pnmIntensityp[A.U]\nFIG. 1. a) AFM topography for films of thickness 10 nm, 15 nm and 30 nm. b) XRD θ−2θscans of the Bragg reflection for the respective\nfilm thicknesses. c) Magnetic moment measurements (performed with a vibrating sample magnetometer, (VSM)), showing the difference\nin magnetic moment and Curie temperature for the various film thicknesses.\nand 30 nm grown on (111)-oriented STO (in contrast to\nthe previous study by Monsen el. al for LSMO grown on\n(001)-oriented STO17). By performing broadband FMR\nexperiments we separate the inhomogeneous linewidth\nbroadening and the dynamic contribution to the FMR\nlinewidth, and extract the magnetodynamic damping pa-\nrameterα. The main objective of our study is to charac-\nterizeαfor the various film thicknesses. However, as Tc\nchanges with film thickness, it is difficult to compare the\nabsolute values at a single temperature. We thus per-\nformed experiments in the temperature range T=100–\n300 K using the FMR setup described in section II B,\nallowing us to extract αas a function of both tempera-\nture and film thickness.\nII. SAMPLE GROWTH AND EXPERIMENTAL\nSETUP\nA. Sample growth and characterization\nLa0.7Sr0.3MnO 3thin films were deposited by pulsed\nlaser deposition on (111)-oriented SrTiO 3substrates. A\nKrF excimer laser (=248 nm) with a fluency of ≈2 Jcm−2\nand repetition rate 1 Hz was employed, impinging on\na stoichiometric La 0.7Sr0.3MnO 3target. The deposition\ntook place in a 0.35 mbar oxygen ambient, at 500 Celsius\nand the substrate-to-target separation was 45 mm, result-\ning in thermalized ad-atoms19,20. After the deposition,the films were cooled to room temperature in 100 mbar of\noxygen at a rate of 15 K/min. Atomic force microscopy\n(AFM) was used to study the surface topography. The\nAFM topography images shown in Fig. 1a confirm the\nstep and terrace morphology of the films after growth for\nthe 10 nm and 15 nm thick films. For the 30 nm film,\nwe observe a surface relaxation and transition to a more\n3 dimensional growth and rougher surface compared to\nthe 2 dimensional layer by layer growth for thinner films.\nThe crystalline structure of the films was investigated\nusing x-ray diffraction (XRD), and the XRD scans of the\nrespective films are shown in Fig. 1b.\nMagnetic moment measurements were performed with\na vibrating sample magnetometer (VSM). In Fig. 1c the\ntemperature dependence of the saturation magnetization\nis plotted against temperature, taken during warm-up\nafter field cooling in 2 T. Both saturated moment and\nTcincrease with film thickness as expected for thin films,\nand the values are comparable to similar thicknesses of\nLSMO films grown on (001)-oriented SrTiO 317.\nB. FMR experiments\nThe FMR experiments were performed using a vec-\ntor network analyzer (VNA) setup in combination with a\ncoplanar waveguide (CPW) that created microwave mag-\nnetic fields of different frequencies to the film’s structure.\nWe used a He cryostat with a superconducting magnetiii\ncapable of producing bipolar bias fields up to 5 T. The\nexperiments were performed in the temperature range\nT=100–300 K using a CPW designed specially for the\ncryostat and using semi-rigid coaxial cables capable to\ncarry up to 50 GHz electrical signals.\nThe static external field, H0, was applied in the sample\nplane, and perpendicular to the microwave fields from\nthe CPW. We measured the microwave transmission and\nreflection parameters as a function of field and frequency\nin order to obtain a complete map of the ferromagnetic\nresonances. Magnetic fields were varied from -500 to 500\nmT, and microwave frequencies from 1 to 20 GHz.\nBy measuring the microwave absorption as a function\nof both microwave frequency and the applied external\nfield, one can obtain the FMR dispersion. The field vs.\nfrequency dispersion when the field is applied in the film\nplane is described by the Kittel equation21given by the\nfollowing:\nfFMR =γ\n2π/radicalbig\nH0(H0+Heff). (1)\nHere,fFMR is the FMR frequency and γis the gyro-\nmagnetic ratio, where γ/2π≈28 GHz/T. H0is the ap-\nplied external field and Heff= 4πMs−Hkis the effective\nfield given by the saturation magnetization Msand the\nanisotropy field Hk.\nThe FMR absorption lineshape is often assumed to\nhave a symmetric Lorentzian lineshape. However, in con-\nductive samples, induced microwave eddy currents in the\nfilm can affect the linshape symmetry22,23. In an experi-\nmental setup containing waveguides, coaxial cables etc.,\nthe relative phase between the electric and magnetic field\ncomponents can also affect the lineshape24,25. We thus\nfit the FMR absorption, χ, to a linear combination of\nsymmetric and antisymmetric contributions, determined\nby theβparameter in Eq. (2).\nχ=A1 +β(HR−H0)/∆H\n(HR−H0)2+ (∆H/2)2. (2)\nHere A is an amplitude prefactor, HRandH0are the\nresonance field and external field respectively and ∆ H\nthe full linewidth at half maximum (FWHM). By mea-\nsuring the FMR linewidth as a function of the microwave\nfrequencyfmw, we extract the damping parameter αand\nthe inhomogeneous linewidth broadening ∆ H0from the\nfollowing relation26:\n∆H=4π\nγαfmw+ ∆H0. (3)\nIII. RESULTS AND DISCUSSION\nThe measured FMR absorption shows a good agree-\nment with the Kittel dispersion given by Eq. (1). InFig. 2a we show as an example the FMR dispersion for\nthe 15 nm film measured at room temperature, with the\nfit to Eq. (1) as dotted line. A typical absorption line-\nshape and the fit to Eq. (2) is shown as inset. Fitting\nthe FMR dispersion to Eq. (1) allows us to extract the\neffective field Heff, given by the saturation magnetiza-\ntionMsand the anisotropy field Hkthrough the relation\nHeff= 4πMs−Hk. The extracted values of Hefffor the\nvarious films are shown in Fig. 2b. From HeffandMs\n(shown in Fig. 1c) we calculate the anisotropy field Hk.\nThe obtained values for Hkare shown in Fig. 2c and\nindicate a strain induced negative perpendicular magne-\ntocrystalline anisotropy (PMA), as expected for epitaxial\nfilms. The negative PMA is in agreement with that ob-\nserved in LSMO grown on (001)-oriented STO27. As in-\ndicated in Fig. 2c, the PMA is strongest for the thinnest\nfilm and decreases as one approaches the Curie temper-\nature.\nThe main objective of our study is to characterize\nthe magnetodynamic damping, as measured through the\nFMR linewidth. As an example we show in Fig. 2d the\nlinewidth vs. frequency for the 15 nm sample at T=300\nK. The linear relation between linewidth and frequency\nfrom Eq. (3) allows us to extract the damping parameter\nαand the inhomogeneous linewidth broadening ∆ H0.\nTo ensure all samples are in a fully ferromagnetic (FM)\nstate, we first compare the damping well below Tcof the\nrespective films. We find that the damping is relatively\nconstant as a function of temperature well below Tc, as\nshown in Fig. 2e, and the lowest damping was found for\nthe intermediate film thickness of 15 nm with α≈2·10−3.\nThis value of αis comparable to that found in a previous\nstudy by Luo et. al.15for an LSMO film of thickness 20\nnm, but in their case grown on (001)-oriented STO.\nWe observe an increased damping as the temperature\napproaches the Curie temperature of the respective films,\nwhich we attribute to the coexistence of ferromagnetic\nand paramagnetic domains as T→Tc. The increase in\ndamping is largest for the 10 nm film, in agreement with\nthe reduced Tcfor this film compared to the thicker 15\nnm and 30 nm films, as shown in Fig. 1c. This behavior\nis consistent with data from LSMO single crystals28, and\nwith previous work by Monsen et. al.17for LSMO films\ngrown on (001)-oriented STO.\nThe difference in damping for the various film thick-\nnesses is observed well below Tc(see Fig. 2e), and is thus\nnot caused by the coexistence of ferromagnetic and para-\nmagnetic domains. The increased damping for the 10\nnm film compared to the 15 nm film is rather attributed\nto the increased importance of surface defects for the\nthinnest film. Surface/interface imperfections/scatterers\ninduce a direct thickness dependent broadening due to\nlocal variations in resonance field, or indirectly through\ntwo-magnon processes, as these contributions become\nmore dominating as the film thickness is reduced29. Pre-\nvious studies of the thickness dependence of the FMR\nlinewidth in LSMO films grown on (001)-oriented STO\nby Monsen et. al.17show similar behavior, with a mini-iv\ns 5 Us U5 2ss2468Us\nFrequencym[GHz]Linewidthm[mT]\nLSMOm3U5nmKkmT=3ssK\nFit\ns Uss 2ss 3ss 4ss 5sss5UsU52s\nFieldm[mT]Frequencym[GHz]LSMOm3U5nmKkmT=3ssK\nFit\naK cK\ndKeK\nUss U5s 2ss 25s 3ss−2sss−U5ss−Usss−5sss\nTemperaturem[K]Hk[Oe]\n3smnm\nU5mnm\nUsmnm\nUss U5s 2ss 25s 3sssUsss2sss3sss4sss5sss6sss7sss\nTemperaturem[K]Heff[Oe]\n3smnm\nU5mnm\nUsmnm\nbKcK\n6s 7s 8s−s.U5−s.U−s.s5s\nFieldm[mT]Abs.m[A.U]\nUss U5s 2ss 25s 3sssU23456\nTemperaturem[K]∆Hs[mT]3snm\nU5nm\nUsnm\nUss U5s 2ss 25s 3ssss.ss2s.ss4s.ss6s.ss8s.sU\nTemperaturem[K]α3snm\nU5nm\nUsnm\nfKeK\nFIG. 2. a) FMR frequency vs. external magnetic field. Experimental datapoints as red squares, and fit to Eq. (1) as dotted line. Inset:\nTypical absorption lineshape and the Fit to Eq. (2). b) Extracted values of Hefffrom the fit to Eq. (1). c) Calculated anisotropy fields\nHk. d) FMR absorption linewidth vs. frequency. Experimental datapoints shown as red squares, and the fit to Eq. (3) as dotted line. e)\nExtracted damping parameter αand f) inhomogeneous linewidth broadening ∆ H0as a function of film thickness and temperature.\nmum in the linewidth for an intermediate film thickness\nand increased linewidth for thicknesses below approxi-\nmately 10 nm.\nFor the 30 nm film we also observe an increased damp-\ning compared to the 15 nm film. Interface effects should\nbe less important for the 30 nm film, and the increased\ndamping is attributed to other effects as the thickness\nis increased. It is known that LSMO films can expe-\nrience strain relief relaxation as the film thickness is\nincreased30,31, and for the 30 nm film we observe a\nrougher surface compared to the thinner films. This\nchange in film structure can be observed in the AFM\ntopography images in Fig. 1a, and the difference in film\nstructure could thus cause an increased damping for the\nthickest film. Another consideration is the eddy-current\ncontribution to the damping in conducting films32. The\neddy-current contribution scales with the film thickness\nd, asd2, and separating the various contributions to the\ndamping would thus require a more detailed study of the\nscaling of damping vs. film thickness.\nIn a homogeneous strain-free ferromagnet one expects\nthat the inhomogeneous linewidth broadening, ∆ H0,\nshould be independent of temperature well below Tc33.\nThe relatively temperature independence of ∆ H0for the\n15 nm and 30 nm films shown in Fig. 2f indicate high\nquality samples, with ∆ H0<1 mT. For the 10 nm film\nthere is a slight increase in ∆ H0at low temperature, with\nan inhomogenous broadening of ∆ H0≈2 mT. The in-\ncrease in ∆ H0at 300 K is attributed to the reduced Tcof the 10 nm film compared to the 15 and 30 nm films,\nand the coexistence of ferromagnetic and paramagnetic\ndomains as T→Tc.\nIV. SUMMARY\nWe have characterized the magnetic damping param-\neterαin 10, 15 and 30 nm thick LSMO films grown\non (111)-oriented STO for temperatures T=100–300 K.\nWe found that αis relatively independent of tempera-\nture well below the Curie temperature of the respective\nfilms, with a significant increase as T→Tcdue to the\ncoexistence of ferromagnetic and paramagnetic domains.\nThe lowest damping was found for the intermediate film\nthickness of 15 nm, with α≈2·10−3. For the 10 nm film,\nwe attribute the increased damping to the increased im-\nportance of surface/interface imperfections/scatterers for\nthinner films. For the 30 nm film the increased damping\nis attributed to changes in the film structure with an in-\ncreased surface roughness compared to the thinner films,\nas well as additional eddy-current contributions to the\ndamping as the film thickness is increased.\nThe damping of α≈2·10−3for the 15 nm film is\nlower than that of e.g Permalloy (Ni 80Fe20) which is one\nof the prototype materials for magnetodynamic devices.\nThe low damping, in addition to other intriguing material\nproperties like large spin polarization, indicate LSMO as\na promising material for applications in magnetodynamicv\ndevices.\nACKNOWLEDGEMENTS\nThis work was supported by the Norwegian Research\nCouncil (NFR), project number 216700. V.F acknowl-edge partial funding obtained from the Norwegian PhD\nNetwork on Nanotechnology for Microsystems, which is\nsponsored by the Research Council of Norway, Division\nfor Science, under contract no. 221860/F40. F.M. ac-\nknowledges financial support from RYC-2014-16515 and\nfrom the MINECO through the Severo Ochoa Program\n(SEV- 2015-0496). J.M and F.M aslo acknowledge fund-\ning from MINECO through MAT2015-69144.\n∗vflovik@gmail.com\n1R. L. Stamps et al. J. Phys. D: Appl. Phys. 47 333001\n(2014)\n2J.˚Akerman, Science 308, 508 (2005)\n3S. D. Bader Rev. Mod. Phys. 78, 1 (2006)\n4A. Moser, K. Takano, D. T. Margulies, M. Albrecht, Y.\nSonobe, T. Ikeda, S. Sun, E. E. Fullerton, J. Phys. D.\nAppl. Phys. 35 R157, (2002)\n5S. S. P. Parkin, K. P. Roche, M. G. Samant, P. M. Rice,\nR. B. Beyers, R. E. Scheuerlein, E. J. O‘Sullivan, S. L.\nBrown, J. Bucchigano, D. W. Abraham, Y. Lu, M. Rooks,\nP. L. Trouilloud, R. A. Wanner, W. J. Gallagher, J. Appl.\nPhys. 85, 5828, (1999)\n6R. L. Stamps et al. J. Phys. D: Appl. Phys. 47 333001\n(2014)\n7V. V. Kruglak, S. O. Demokritov, D. Grundler J. Phys. D.\nAppl. Phys. 43 264001 (2010)\n8V. Flovik, F. Maci` a, J. M. Hern` andez, R. Bru˘ cas, M. Han-\nson, E. 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Appl.\nPhys. 81, 5737 (1997)" }, { "title": "1307.5155v1.Damping_and_non_linearity_of_a_levitating_magnet_in_rotation_above_a_superconductor.pdf", "content": "Damping and non-linearity of a levitating magnet in\nrotation above a superconductor\nJ. Druge, O. Laurent, M. A. Measson and I. Favero\nMat\u0013 eriaux et Ph\u0013 enomenes Quantiques, Universit\u0013 e Paris Diderot, CNRS UMR 7162,\nSorbonne Paris Cit\u0013 e, 10 rue Alice Domon et L\u0013 eonie Duquet, 75013 Paris, France\nE-mail: ivan.favero@univ-paris-diderot.fr,\nmarie-aude.measson@univ-paris-diderot.fr\nAbstract. We study the dissipation of moving magnets in levitation above a\nsuperconductor. The rotation motion is analyzed using optical tracking techniques. It\ndisplays a remarkable regularity together with long damping time up to several hours.\nThe magnetic contribution to the damping is investigated in detail by comparing 14\ndistinct magnetic con\fgurations, and points towards amplitude-dependent dissipation\nmechanisms. The non-linear dynamics of the mechanical rotation motion is also\nrevealed and described with an e\u000bective Du\u000eng model. The obtained picture of the\ncoupling of levitating magnets to their environment sheds light on their potential as\nultra-low dissipation mechanical oscillators for high precision physics.arXiv:1307.5155v1 [cond-mat.mes-hall] 19 Jul 20132\nMechanical oscillators with ultra-low dissipation \fnd applications as frequency\nstandards, probes of minute forces like in Atomic Force Microscopy, or signal processing\nwhere they can serve as \fne radio-frequency \flters. At a basic science level, they received\nattention in the past for their impact in high precision physics and metrology [1, 2]. More\nrecently, they have become a central subject for a whole community of physicists aiming\nat observing the quantum behavior of mesoscopic mechanical systems [3, 4]. This quest\nhas seen impressive advances notably thanks to optomechanical systems [5, 6, 7, 8] that\nuse the concepts of coupling light and mechanical motion, notably in the regime where\nthe quantumness of the mechanics starts being tangible [9, 10]. In all these situations,\nlow dissipation of the mechanical degree of freedom is required, to protect it against\nquantum decoherence or classical \ructuations of its environment.\nSources of dissipation are manyfold, but one ubiquitous amongst mechanical sys-\ntems is the anchoring (clamping) loss, which stems from the fact that the system is me-\nchanically attached to a support. To circumvent this source of loss, one obvious solution\nis to levitate the mechanical system. If we disregard optical levitation [11, 12, 13, 14],\nwhich restricts to nanoscopic mass objects, diamagnetic e\u000bects in superconductors are\nthe most established technique to levitate a macroscopic mass and hence isolate a me-\nchanical oscillator from its support. Superconducting magnetic levitation is been exten-\nsively studied in the context of bearings for transportations [15] but surprisingly, the\napplications of these systems in high precision physics remains relatively scarce [16].\nTo our knowledge, there is for example no complete picture of what ultimate level of\ndissipation such system could reach in a re\fned low-amplitude force sensing experiment\nor in the quantum regime. This paper is a \frst step to try to answer these questions.\nTo that aim, we carry out simple but systematic experiments on one of the most ubiqui-\ntous systems: a magnet levitating over a high critical temperature (Tc) superconductor.\nWe employ commercial NdFeB sphere magnets [17] of diameter varying between\n5 mm and 25 mm and position them one after another over a high-Tc Y1.65Ba2Cu3O7-\nx superconducting cylinder pad of diameter of about 40 mm [18]. As illustrated in Fig.1,\nthe superconducting pad is kept at low temperature below its critical temperature Tc\nby contacting it with apiezon grease to a large copper cylinder immersed into a liquid\nnitrogen bath. In order to increase thermal inertia and stability of the set-up during\nhour-long measurements, a polystyrene box of very large liquid capacity is used as the\nbath and placed on a mechanically isolating bench. Air turbulence and convection are\nkept to a low level during measurements.\nIn this work, we focus on the rotation motion of the sphere magnet around its\nmagnetic axis because this motion presents a very low dissipation. Indeed, in case\nof a perfectly homogeneous and spherical magnet, the magnetic con\fguration of the\nexperiment is invariant upon rotation of the sphere around this axis. As a consequence,\nstrictly no magnetic damping of the sphere rotation motion should occur. As we will\nsee, even in a real and non-ideal experimental situation, this argument still holds to\nsome extent and very long damping times of hours can be observed for this rotation3\nSC CameraSM\nScreen\nM\nM\nCu Nitrogen\nFigure 1. Schematics of the experiment. A spherical magnet (SM) is levitating and\nexperiences a rotational motion over a superconductor (SC) pad thermally connected\nto a copper (Cu) cylinder plunged into liquid Nitrogen. ~Mis the magnetic dipole\nmoment of the magnet. Inset shows a typical video image after treatment with the\nmark spot (in black) within the rotating magnet boundaries (dashed circle).\nmotion. Note that other types of motion involving millimeter scale displacements of the\nmagnet center of mass are systematically observed to damp rapidly in much less than a\nminute, and will hence not be considered here.\nA video camera is positioned near the magnet to register its motion. The camera\noptical axis is superposed to the sphere magnetic axis de\fned by the magnetic dipole\n~M. In order to allow systematic data analysis of the motion, a black mark spot is\ndrawn on the magnet and a treatment of each video image is performed to increase its\ncontrast. After treatment, a typical image displays the isolated black mark spot over a\nwhite background where the magnet spherical boundary is hardly visible (see inset of\nFig.1). This strong contrast allows for each image to run an automated search of the\nmark spot barycentre leading its radial coordinate (amplitude and phase) with respect\nto the center of the spherical boundary. In a rotation around ~Mthe amplitude remains\nconstant and the motion is analyzed by registering the phase evolution upon time. In the\nfollowing, we use this method to measure both rotation motion (with increasing phase\nupon time) and rotation-oscillation motion (with oscillating phase upon time). The\nobserved motion damping is highly dependent on the orientation of the magnet dipole\n~Mwith respect to the rest of the set-up with a minimum damping consistently observed\nwhen the axis is horizontal to the pad plane. We therefore adopt this orientation for all\nreported experiments (Fig.1). Second, the damping also depends on the height at which\nthe magnet is levitating above the superconducting pad. Hence, in our experiments we4\n01 02 03 04 05 06 0050010001500200025003000350040004500 \n Phase (rad)T\nime (min)405 06 0-0.80.00.8 \n Phase (rad)T\nime (min)4\n6.04 6.2-0.60.00.6 \n Phase (rad)T\nime(min)\nFigure 2. Typical rotation motion of a levitating sphere magnet. Time evolution\nof the phase for a sphere magnet of diameter 12.7 mm levitating over a soft pinning\nsuperconductor. Solid lines are employed for experimental data in order to illustrate\ntheir remarkable regularity, while cross symbols are employed for the exponential \ft.\nThe top-right inset is a \frst close-up on the rotation-oscillation motion, where the\nenvelope of the phase trace is apparent. The lower inset is a second close-up focusing\non the time oscillation, where a time period of a few seconds is visible.\nemploy a constant levitation height, de\fned as being the distance between the upper\npad plane and the bottom of the sphere. To that purpose every sphere magnet is \frst\ndeposited at room temperature on an interstitial te\ron element of \fxed height of about\n12 mm sitting on the pad. The system is then cooled below Tc and the interstitial\nelement removed before setting the levitating magnet in rotation.\nFig.2 shows a typical measurement of the phase upon time after the levitating\nsphere magnet has been put in rotation manually. The time zero in the measurement is\nset by the sampling rate of the video camera (24 images/s), which precludes acquisition\nof the motion before the rotation speed has decreased down to about 48 \u0019 rad=s . As\na consequence all rotation measurements shown hereafter e\u000bectively start with the\nsame rotation speed of about 48 \u0019 rad=s at time zero. In Fig.2, a rotation motion\n\frst lasts about 35 min before reaching a plateau where the phase oscillates upon time\ncorresponding to a rotation-oscillation motion, as seen clearly in the lower inset. During\nthe \frst rotation part, the rotation speed progressively decreases and the phase is very\naccurately described by a time exponential function typical of a linear damping model.5\n51 01 52 02 50102030405060 \n Damping time (min)S\nphere magnet diameter (mm)2550751000\n50100 \n \n BS (mT)Intrinsic damping time (min)\nFigure 3. Damping of the rotation motion for di\u000berent magnetic con\fgurations. The\nmain plot shows the total damping time as a function of the sphere diameter for a\nsoft pinning (squares) and hard pinning (circles) superconductor. The inset shows the\nintrinsic damping time as a function of the magnetic \feld on the top surface of the\nsuperconducting pad (see text for details), showing the role of magnetic e\u000bects in the\ndamping.\nIn the second part where rotation-oscillation takes place, the oscillation amplitude is\nitself damped in a time exponential manner as seen in the top right inset of Fig.2. This\nbehavior is again reminiscent of the harmonic oscillator model with linear damping. It\nis worth noting that the rotation motion of some magnets is observed to last for more\nthan 8 hours, reaching the limit of our ability to measure it reliably. In the following\nwe study in details the origin of the observed damping.\nFirst, as our experiments are run under ambient conditions, the surrounding air\ncan be a source a friction for the rotation motion. However we do not expect air\nto produce a restoring torque, while the existence of such torque is implied by the\nobservation of oscillatory rotation motion in the experiments. The forces responsible\nfor this torque act at distance on the spheres and may be magnetic in nature. If\ntrue, this would contradict the picture of a rotation invariance of the sphere around its\nmagnetic axis, which would imply no magnetic restoring torque on the rotational degree\nof freedom. The rotation invariance is also questioned by our independent observation\nof an inhomogeneous magnetic \feld as one rotates the magnetic sphere, which we could\nreveal in the orientation of iron \flings and by using a magnetic foil. We conclude that the6\nemployed magnetic spheres are not perfectly rotation-invariant around their magnetic\naxis and that a strictly null magnetic damping of the rotation can not be expected.\nTo study the role of magnetic e\u000bects in the damping of the rotation motion, we now\nvary both the magnetic con\fguration of the spheres and their magnetic environment. To\nthat aim we vary the diameter of the spheres and employ two types of superconductor\nfor levitation, one with strong vortices pinning and one with soft pinning. In each\ncon\fguration, we systematically measure the rotation damping during the \frst part of\nthe motion where the phase decays exponentially.\nFig.3 reports the measured damping time as a function of the sphere diameter,\nexploring 14 di\u000berent con\fgurations in total. The open square (circle) symbols\ncorrespond to the superconductor with soft (hard) pinning of the vortices. The dash-\ndotted line is a theoretical value for the air damping contribution, which is obtained\nusing the Stokes model for a rotating sphere in a non-turbulent incompressible \ruid\n[19]. In Fig.3 the damping measured for the spheres of small diameter 5 and 6 mm\nseem to be explained by air damping, but as the magnet diameter increases there is a\nstrong departure from this contribution. Air damping has a negligible contribution on\nthe spheres of large diameter like 12.7, 18 and 25 mm. On these spheres, the measured\ndamping time is systematically shorter (longer) when using the strong (soft) pinning\nsuperconductor, showing that the magnetic properties of the superconductor play a\nkey role in the rotation dynamics. To reveal even more clearly these magnetic e\u000bects,\nwe plot in the inset of Fig.3 the damping time as a function of the sphere magnetic\n\feld on the superconductor pad. To that aim we measure the magnetic \feld of the\nsphere alone (with no superconductor) with a Teslameter at a distance from the sphere\ncorresponding to the levitation height, on the magnetic equatorial line where the pad\nlies in the levitating con\fguration. The theoretical air damping contribution is removed\nin these data to make the \"intrinsic damping time\" directly appear. The plot clearly\nreveals that this \"intrinsic damping\" increases as the magnetic \feld on the pad increases,\nand as one passes from a soft pinning to a strong pinning superconductor.\nThese observations ascertain the importance of vortices in the damping of\nrotation motion. Indeed the larger the magnetic \feld applied by the magnet to the\nsuperconductor pad, the larger the amount of vortices accommodated in the pad. The\nvortices are known to be responsible for the rigidity of the superconducting levitation\ncon\fguration but this rigidity contribution is also accompanied by a dissipative\ncontribution. In early experiments a magnet was displaced above a superconductor and\nlossy hysteric behavior was observed that revealed energy dissipation as vortices move\nin the superconductor [20, 21, 22]. In our experiments, because the sphere magnet is\nnot perfectly symmetric, its rotation motion modulates the vortices con\fguration in the\nsuperconductor and produces dissipation. Our measurements show that this dissipation\nis stronger when there are more vortices and when the vortices are strongly pinned in\nthe superconductor.\nIf the observed magnetic damping \fnds its roots in the vortices dissipative\ndynamics, it should then depend on the amplitude of the magnet motion. Indeed7\n04 8 1 2567 \n ω (t) (rad/s)T\nime (min)(c)0\n24681012-0.8-0.6-0.4-0.20.00.20.40.6 \nPhase (rad)T\nime (min)05 1 01 52 0-101 \n Phase (rad)T\nime (min)(\nb)(a)\nFigure 4. Non linearities in the rotation of magnets levitating above a superconductor.\n(a) shows the time evolution of the phase of a rotation-oscillation motion, in a case\nwhere the envelope experiences several abrupt changes upon time. (b) is the \fnal\nevolution after the last abrupt change, displaying an asymmetry of the envelope. (c)\nis the corresponding time evolution of the instantaneous angular frequency. The open\ncircles are data and the dashed line is the \ft function predicted by the e\u000bective Du\u000eng\nmodel (see text for details).\nin case of a large motion amplitude, vortices are forced to hope between di\u000berent\npinning sites hence producing a strong dissipative creep. In case of a smaller amplitude,\neach vortex rests on its pinning site and only the dissipative part of the pinning force\nparticipates to the damping. An amplitude-dependent damping is then expected. There\nhas been a recent strong interest in such non-linear damping mechanisms in nanoscale\nmechanical systems, which would make them depart from the conventional damped\nharmonic oscillator behavior [23, 24]. Here we will not speci\fcally focus on these non-\nlinear aspects of the damping but show on a more general foot that strong non-linearities\nare indeed present in the dynamical behavior of rotating magnets in levitation above\nsuperconductors.\nFig.4a shows the phase time evolution of a 19 mm diameter sphere magnet\nin rotation-oscillation above a strong pinning superconductor. The evolution is\nqualitatively di\u000berent from the one shown in Fig.2, in that several abrupt changes are\nnow visible in the envelope evolution. These abrupt changes cannot be explained by an\nharmonic oscillator model and convey the picture of an oscillation motion within multiple8\nadjacent potential wells. As the mechanical energy dissipates upon time, the system\nprogressively restricts its motion to fewer wells until it resides within a single of these,\nwhere the mechanical energy \fnishes to be dissipated. Fig.4b shows such \fnal evolution\nin the last well for a 25 mm diameter sphere levitating over the same superconductor.\nEven in this case where a unique well is involved, the phase evolution reveals a non-\nlinearity. The envelope amplitude is strongly asymmetric with respect to the zero axis,\nimplying an anharmonicity in the related trapping potential. Indeed the harmonic\noscillator with linear damping predicts a symmetric envelope and a constant angular\nfrequency of the oscillation upon time. In our experiments, the potential anharmonicity\nis also witnessed by the time-evolution of the angular frequency !(t), which is reported in\nFig.4c for the damped motion of Fig.4b. Each value of !(t) is obtained by analyzing the\noscillation motion over 10 oscillations. The measured angular frequency is not constant\nbut follows an exponential time evolution. We analyze this behavior by adopting a\nsimple Du\u000eng model with damping:\n\u0002(t) +\u0015_\u0002(t) +!02\u00022(t) +B! 02\u00023(t) = 0 (1)\nwith B<0 the Du\u000eng coe\u000ecient. To deal with this non-linear equation, we propose\na mathematical Ansatz inspired by our experimental results. We inject the following\nexpression for the phase evolution\n\u0002(t) =Aexp(\u0000\u0015t\n2) cos(!(t)t) (2)\nin the equation and try to solve for !(t). To that aim, we make several\nsimpli\fcations which are again validated by our experimental results. These\nsimpli\fcations, valid for any time t of our experimental analysis, are the followings\n\u0015<0 is the positive elementary\ncharge.\nIn the case of the perturbation of the type Eq. (7) the\nsecond derivative∂2ˆM\n∂ri∂tis perpendicular to M. In this\ncase it is convenient to rewrite Eq. (6) as\nJICIT2a\ni=/summationdisplay\njkχICIDMI\nijkˆej·/bracketleftBigg\nˆM×∂2ˆM\n∂rk∂t/bracketrightBigg\n,(19)\nwhere the coefficients χICIDMI\nijkare given by\nχICIDMI\nijk=ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nviRvkRRTjR+viRRvkRTjR+\n−viRRTjRvkR−viRvkRTjAA\n+viRTjAvkAA+viRTjAAvkA\n−viRvkRRTjA−viRRvkRTjA\n+viRRTjAvkA+viAvkATjAA\n−viATjAvkAA−viATjAAvkA/bracketrightBig\n,(20)\nand\nT=ˆM×∂H\n∂ˆM(21)\nis the torque operator. In Sec. IIC we will explain that\nχICIDMI\nijkdescribes the inverse of current-induced DMI\n(ICIDMI).\nIn the case of the perturbation of the type of Eq. (10)\nthe second derivative∂2ˆMj\n∂rk∂tmay be rewritten as product\nof the first derivatives∂ˆMl\n∂tand∂ˆMl\n∂rk. This may be seen5\nas follows:\n∂H\n∂ˆM·∂2ˆM\n∂ri∂t=∂2H\n∂t∂ri=\n=∂\n∂t/bracketleftBigg/parenleftbigg\nˆM×∂H\n∂ˆM/parenrightbigg\n·/parenleftBigg\nˆM×∂ˆM\n∂ri/parenrightBigg/bracketrightBigg\n=\n=/bracketleftBigg/parenleftBigg\n∂ˆM\n∂t×∂H\n∂ˆM/parenrightBigg\n·/parenleftBigg\nˆM×∂ˆM\n∂ri/parenrightBigg/bracketrightBigg\n=\n=/bracketleftBigg/parenleftBigg/parenleftBigg\nˆM×∂ˆM\n∂t/parenrightBigg\n׈M/parenrightBigg\n×∂H\n∂ˆM/bracketrightBigg\n·/bracketleftBigg\nˆM×∂ˆM\n∂ri/bracketrightBigg\n=\n=−/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n·/bracketleftBigg\nˆM×∂ˆM\n∂ri/bracketrightBigg/bracketleftbigg\nˆM·∂H\n∂ˆM/bracketrightbigg\n=\n=−∂ˆM\n∂t·∂ˆM\n∂ri/bracketleftbigg\nˆM·∂H\n∂ˆM/bracketrightbigg\n.\n(22)\nThis expression is indeed satisfied by Eq. (11), Eq. (12)\nand Eq. (13):\n∂ˆM\n∂ri·∂ˆM\n∂t=−∂2ˆM\n∂ri∂t·ˆM (23)\natr= 0,t= 0. Consequently, Eq. (6) can be rewritten\nas\nJICIT2a\ni=/summationdisplay\njkχICIT2a\nijk∂2ˆMj\n∂rk∂t=\n=−/summationdisplay\njklχICIT2a\nijk∂ˆMl\n∂rk∂ˆMl\n∂t[1−δjl]\n=/summationdisplay\njklχICIT2a\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n,\n(24)\nwhere\nχICIT2a\nijkl=−/summationdisplay\nmχICIT2a\niml[1−δjm]δjk.(25)\nThus, Eq. (24) and Eq. (25) can be used to express\nJICIT2a\niin the form of Eq. (4).\nC. Direct and inverse CIDMI\nEq. (20) describes the response of the electric current\nto time-dependent magnetization gradients of the type\nEq. (15). The reciprocal process consists in the current-\ninduced modification of DMI. This can be shown by ex-\npressing the DMI coefficients as [10]\nDij=1\nV/summationdisplay\nnf(Ekn)/integraldisplay\nd3r(ψkn(r))∗Dijψkn(r)\n=1\nV/summationdisplay\nnf(Ekn)/integraldisplay\nd3r(ψkn(r))∗Ti(r)rjψkn(r),\n(26)where we defined the DMI-operator Dij=Tirj. Using\nthe Kubo formalism the current-induced modification of\nDMI may be written as\nDCIDMI\nij=/summationdisplay\nkχCIDMI\nkijEk (27)\nwith\nχCIDMI\nkij=1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(28)\nwhere\n∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) =−i∞/integraldisplay\n0dteiωt∝an}bracketle{t[Dij(t),vk(0)]−∝an}bracketri}ht(29)\nis the Fourier transform of a retarded function and Vis\nthe volume of the unit cell.\nSince the position operator rin the DMI operator\nDij=Tirjis not compatible with Bloch periodic bound-\nary conditions, we do not use Eq. (28) for numerical\ncalculations of CIDMI. However, it is convenient to use\nEq. (28) in order to demonstrate the reciprocity between\ndirect and inverse CIDMI.\nInverseCIDMI (ICIDMI) describes the electric current\nthat responds to the perturbation by a time-dependent\nmagnetization gradient according to\nJICIDMI\nk=/summationdisplay\nijχICIDMI\nkijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\n.(30)\nThe perturbation by a time-dependent magnetization\ngradient may be written as\nδH=−/summationdisplay\njm·∂2ˆM\n∂t∂rjrjΩxc(r)sin(ωt)\nω=\n=/summationdisplay\njT·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\nrjsin(ωt)\nω\n=/summationdisplay\nijDijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\nsin(ωt)\nω.(31)\nConsequently, the coefficient χICIDMI\nkijis given by\nχICIDMI\nkij=1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n.(32)\nUsing\n∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,ˆM) =−∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,−ˆM) (33)\nwe find that CIDMI and ICIDMI are related through the\nequations\nχCIDMI\nkij(ˆM) =−χICIDMI\nkij(−ˆM). (34)\nIn order to calculate CIDMI we use Eq. (20) for ICIDMI\nand then use Eq. (34) to obtain CIDMI.6\nThe perturbation Eq. (16) describes a different kind\nof time-dependent magnetization gradient, for which the\nreciprocaleffect consists in the modification of the expec-\ntation value ∝an}bracketle{tσ·ˆMrj∝an}bracketri}ht. However, while the modification\nof∝an}bracketle{tTirj∝an}bracketri}htby an applied current can be measured [8, 9]\nfrom the change of the DMI constant Dij, the quantity\n∝an}bracketle{tσ·ˆMrj∝an}bracketri}hthas not been considered so far in ferromagnets.\nIn noncollinear magnets the quantity ∝an}bracketle{tσrj∝an}bracketri}htcan be used\ntodefinespintoroidization[36]. Therefore,whiletheper-\nturbation of the type of Eq. (15) is related to CIDMI and\nICIDMI, which are both accessible experimentally [8, 9],\nin the case of the perturbation of the type of Eq. (16)\nwe expect that only the effect of driving current by the\ntime-dependent magnetization gradient is easily accessi-\nble experimentally, while its inverse effect is difficult to\nmeasure.\nD. Direct and inverse dynamical DMI\nNot only applied electric currents modify DMI, but\nalso magnetization dynamics, which we call dynamical\nDMI (DDMI). DDMI can be expressed as\nDDDMI\nij=/summationdisplay\nkχDDMI\nkijˆek·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n.(35)\nIn Sec. IIG we will show that the spatial gradient of\nDDMI contributes to damping and gyromagnetism in\nnoncollinear magnets. The perturbation used to describe\nmagnetization dynamics is given by [24]\nδH=sin(ωt)\nω/parenleftBigg\nˆM×∂ˆM\n∂t/parenrightBigg\n·T.(36)\nConsequently, the coefficients χDDMI\nkijmay be written as\nχDDMI\nkij=−1\nVlim\nω→0/bracketleftbigg1\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;Tk∝an}bracketri}ht��an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg\n.(37)\nSince the position operator in Dijis not compatible\nwith Bloch periodic boundary conditions, we do not use\nEq. (37) for numerical calculations of DDMI, but instead\nwe obtain it from its inverse effect, which consists in the\ngeneration of torques on the magnetization due to time-\ndependent magnetization gradients. These torques can\nbe written as\nTIDDMI\nk=/summationdisplay\nijχIDDMI\nkijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\n,(38)\nwhere the coefficients χIDDMI\nkijare\nχIDDMI\nkij=1\nVlim\nω→0/bracketleftbigg1\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg\n,(39)becausethe perturbationby the time-dependent gradient\ncan be expressed in terms of Dijaccording to Eq. (31)\nand because the torque on the magnetizationis described\nby−T[23]. Consequently,DDMIandIDDMIarerelated\nby\nχDDMI\nkij(ˆM) =−χIDDMI\nkij(−ˆM). (40)\nFor numerical calculations of IDDMI we use\nχIDDMI\nijk=i\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvkRRTjR+TiRRvkRTjR+\n−TiRRTjRvkR−TiRvkRTjAA\n+TiRTjAvkAA+TiRTjAAvkA\n−TiRvkRRTjA−TiRRvkRTjA\n+TiRRTjAvkA+TiAvkATjAA\n−TiATjAvkAA−TiATjAAvkA/bracketrightBig\n,(41)\nwhichisderivedinAppendix A. InordertoobtainDDMI\nwecalculateIDDMIfromEq.(41)andusethereciprocity\nrelation Eq. (40).\nEq.(38)is validfortime-dependent magnetizationgra-\ndients that lead to perturbations of the type of Eq. (15).\nPerturbations of the second type, Eq. (16), will induce\ntorques on the magnetization as well. However, the in-\nverse effect is difficult to measure in that case, because it\ncorresponds to the modification of the expectation value\n∝an}bracketle{tσ·ˆMrj∝an}bracketri}htby magnetization dynamics. Therefore, while\nin the case of Eq. (15) both direct and inverse response\nare expected to be measurable and correspond to ID-\nDMI and DDMI, respectively, we expect that in the case\nof Eq. (16) only the direct effect, i.e., the response of the\ntorque to the perturbation, is easy to observe.\nE. Dynamical orbital magnetism (DOM)\nMagnetization dynamics does not only induce DMI,\nbut also orbital magnetism, which we call dynamical or-\nbital magnetism (DOM). It can be written as\nMDOM\nij=/summationdisplay\nkχDOM\nkijˆek·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n,(42)\nwhere we introduced the notation\nMDOM\nij=e\nV∝an}bracketle{tvirj∝an}bracketri}htDOM, (43)\nwhich defines a generalized orbital magnetization, such\nthat\nMDOM\ni=1\n2/summationdisplay\njkǫijkMDOM\njk (44)7\ncorresponds to the usual definition of orbital magnetiza-\ntion. The coefficients χDOM\nkijare given by\nχDOM\nkij=−1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvirj;Tk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(45)\nbecause the perturbation by magnetization dynamics is\ndescribed by Eq. (36). We will discuss in Sec. IIF that\nthe spatial gradient of DOM contributes to the inverse\nCIT. Additionally, we will show below that DOM and\nCIDMI are related to each other.\nIn order to obtain an expression for DOM it is conve-\nnient to consider the inverse effect, i.e., the generation of\natorquebythe applicationofa time-dependent magnetic\nfieldB(t) that actsonly onthe orbitaldegreesoffreedom\nof the electrons and not on their spins. This torque can\nbe written as\nTIDOM\nk=1\n2/summationdisplay\nijlχIDOM\nkijǫijl∂Bl\n∂t, (46)\nwhere\nχIDOM\nkij=−1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;virj∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(47)\nbecause the perturbation by the time-dependent mag-\nnetic field is given by\nδH=−e\n2/summationdisplay\nijkǫijkvirj∂Bk\n∂tsin(ωt)\nω.(48)\nTherefore, thecoefficientsofDOMandIDOM arerelated\nby\nχDOM\nkij(ˆM) =−χIDOM\nkij(−ˆM). (49)\nIn Appendix A we show that the coefficient χIDOM\nijkcan\nbe expressed as\nχIDOM\nijk=−ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvkRRvjR+TiRRvkRvjR+\n−TiRRvjRvkR−TiRvkRvjAA\n+TiRvjAvkAA+TiRvjAAvkA\n−TiRvkRRvjA−TiRRvkRvjA\n+TiRRvjAvkA+TiAvkAvjAA\n−TiAvjAvkAA−TiAvjAAvkA/bracketrightBig\n.(50)\nEq. (50) and Eq. (20) differ only in the positions of\nthe two velocity operators and the torque operator be-\ntween the Green functions. As a consequence, IDOM\nare ICIDMI are related. In Table I and Table II we list\nthe relations between IDOM and ICIDMI for the Rashba\nmodel Eq. (83). We will explain in Sec. III that IDOM\nandICIDMI arezeroin the Rashbamodel when themag-\nnetization is along the zdirection. Therefore, we discussin Table I the case where the magnetization lies in the xz\nplane, and in Table II we discuss the case where the mag-\nnetization lies in the yzplane. According to Table I and\nTable II the relation between IDOM and ICIDMI is of\nthe formχIDOM\nijk=±χICIDMI\njik. This is expected, because\nthe indexiinχIDOM\nijkis connected to the torque operator,\nwhile the index jinχICIDMI\nijkis connected to the torque\noperator.\nTABLEI:Relations betweentheinverseofthemagnetization -\ndynamics induced orbital magnetism (IDOM) and inverse\ncurrent-inducedDMI (ICIDMI)in the 2d Rashbamodel when\nˆMlies in the zxplane. The components of χIDOM\nijk(Eq. (50))\nandχICIDMI\nijk(Eq. (20)) are denoted by the three indices ( ijk).\nICIDMI IDOM\n(211) (121)\n(121) (211)\n-(221) (221)\n(112) (112)\n-(212) (122)\n-(122) (212)\n(222) (222)\n(231) (321)\n(132) (312)\n-(232) (322)\nTABLE II: Relations between IDOM and ICIDMI in the 2d\nRashba model when ˆMlies in the yzplane.\nICIDMI IDOM\n(111) (111)\n-(211) (121)\n-(121) (211)\n(221) (221)\n-(112) (112)\n(212) (122)\n(122) (212)\n-(131) (311)\n(231) (321)\n(132) (312)\nF. Contributions from CIDMI and DOM to direct\nand inverse CIT\nIn electronic transport theory the continuity equation\ndetermines the current only up to a curl field [37]. The\ncurl of magnetization corresponds to a bound current\nthat cannot be measured in electron transport experi-\nments such that\nJ=JKubo−∇×M (51)\nhastobeusedtoextractthetransportcurrent Jfromthe\ncurrentJKuboobtained from the Kubo linear response.8\nThe subtraction of ∇×Mhas been shown to be impor-\ntant when calculating the thermoelectric response [37]\nand the anomalous Nernst effect [20]. Similarly, in the\ntheory of the thermal spin-orbit torque [10, 18] the gra-\ndients of the DMI spiralization have to be subtracted in\norder to obtain the measurable torque:\nTi=TKubo\ni−/summationdisplay\nj∂\n∂rjDij, (52)\nwhere the spatial derivative of the spiralization arises\nfrom its temperature dependence and the temperature\ngradient.\nSince CIDMI and DOM depend on the magnetization\ndirection, they vary spatially in noncollinear magnets.\nSimilar to Eq. (52) the spatial derivatives of the current-\ninduced spiralization need to be included into the theory\nof CIT. Additionally, the gradients of DOM correspond\ntocurrentsthatneedtobeconsideredinthetheoryofthe\ninverse CIT, similar to Eq. (51). In section IV we explic-\nitly show that Onsager reciprocity is violated if spatial\ngradients of DOM and CIDMI are not subtracted from\nthe Kubo response expressions. By trial-and-error we\nfind that the following subtractions are necessary to ob-\ntain response currents and torques that satisfy this fun-\ndamental symmetry:\nJICIT\ni=JKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂MDOM\nij\n∂ˆM(53)\nand\nTCIT\ni=TKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂DCIDMI\nij\n∂ˆM,(54)\nwhereJICIT\niis the current driven by magnetization dy-\nnamics, and TCIT\niis the current-induced torque.\nInterestingly, we find that also the diagonal elements\nMDOM\niiare nonzero. This shows that the generalized def-\ninition Eq. (43) is necessary, because the diagonal ele-\nmentsMDOM\niido not contribute in the usual definition\nofMiaccording to Eq. (44). These differences in the\nsymmetry properties between equilibrium and nonequi-\nlibrium orbital magnetism can be traced back to sym-\nmetry breaking by the perturbations. Also in the case\nof the spiralization tensor Dijthe nonequilibrium cor-\nrectionδDijhas different symmetry properties than the\nequilibrium part (see Sec. III).\nThe contribution of DOM to χICIT2\nijklcan be written as\nχICIT2c\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χDOM\njil\n∂ˆM/bracketrightBigg\n(55)\nand the contribution of CIDMI to χCIT2\nijklis given by\nχCIT2b\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χCIDMI\njil\n∂ˆM/bracketrightBigg\n.(56)G. Contributions from DDMI to gyromagnetism\nand damping\nThe response to magnetization dynamics that is de-\nscribed by the torque-torque correlation function con-\nsists of torques that are related to damping and gyro-\nmagnetism [24]. The chiral contribution to these torques\ncan be written as\nTTT2\ni=/summationdisplay\njklχTT2\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n,\n(57)\nwhere the coefficients χTT2\nijklsatisfy the Onsager relations\nχTT2\nijkl(ˆM) =χTT2\njikl(−ˆM). (58)\nSinceDDMIdependsonthemagnetizationdirection,it\nvaries spatially in noncollinear magnets and the resulting\ngradients of DDMI contribute to the damping and to the\ngyromagnetic ratio:\nTTT\ni=TKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂DDDMI\nij\n∂ˆM.(59)\nThe resulting contribution of the spatial derivatives of\nDDMI to the coefficient χTT2\nijklis\nχTT2c\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χDDMI\njil(ˆM)\n∂ˆM/bracketrightBigg\n.(60)\nH. Current-induced torque (CIT) in noncollinear\nmagnets\nThe chiral contribution to CIT consists of the spatial\ngradient of CIDMI, χCIT2b\nijklin Eq. (56), and the Kubo\nlinear response of the torque to the applied electric field\nin a noncollinear magnet, χCIT2a\nijkl:\nχCIT2\nijkl=χCIT2a\nijkl+χCIT2b\nijkl. (61)\nIn orderto determine χCIT2a\nijkl, we assume that the magne-\ntization direction ˆM(r) oscillates spatially as described\nby\nˆM(r) =\nηsin(q·r)\n0\n1\n1/radicalBig\n1+η2sin2(q·r),(62)\nwherewewilltakethelimit q→0attheendofthecalcu-\nlation. Since the spatial derivative of the magnetization\ndirection is\n∂ˆM(r)\n∂ri=\nηqicos(q·r)\n0\n0\n+O(η3),(63)9\nthe chiralcontributiontothe CIToscillatesspatiallypro-\nportional to cos( q·r). In order to extract this spatially\noscillating contribution we multiply with cos( q·r) and\nintegrate over the unit cell. The resulting expression for\nχCIT2a\nijklis\nχCIT2a\nijkl=−2e\nVηlim\nq→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n,\n(64)\nwhereVis the volume of the unit cell, and\nthe retarded torque-velocity correlation function\n∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) needs to be evaluated in the\npresence of the perturbation\nδH=Tkηsin(q·r) (65)\ndue to the noncollinearity (the index kin Eq. (65) needs\nto match the index kinχCIT2a\nijkl).\nIn Appendix B we show that χCIT2a\nijklcan be written as\nχCIT2a\nijkl=−2e\n/planckover2pi1Im/bracketleftBig\nW(surf)\nijkl+W(sea)\nijkl/bracketrightBig\n,(66)\nwhere\nW(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nTiGR\nk(E)vlGR\nk(E)vjGA\nk(E)TkGA\nk(E)\n+TiGR\nk(E)vjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−TiGR\nk(E)vjGA\nk(E)TkGA\nk(E)vlGA\nk(E)\n+/planckover2pi1\nmeδjlTiGR\nk(E)GA\nk(E)TkGA\nk(E)/bracketrightBigg(67)\nis a Fermi surface term ( f′(E) =df(E)/dE) and\nW(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)/bracketleftBigg\n−Tr[TiRvlRRvjRTkR]−Tr[TiRvlRTkRRvjR]\n−Tr[TiRRvlRvjRTkR]−Tr[TiRRvjRvlRTkR]\n+Tr[TiRRvjRTkRvlR]+Tr[TiRRTkRvjRvlR]\n+Tr[TiRRTkRvlRvjR]−Tr[TiRRvlRTkRvjR]\n−Tr[TiRvlRRTkRvjR]+Tr[TiRTkRRvjRvlR]\n+Tr[TiRTkRRvlRvjR]+Tr[TiRTkRvlRRvjR]\n−/planckover2pi1\nmeδjlTr[TiRRRTkR]−/planckover2pi1\nmeδjlTr[TiAAATkA]\n−/planckover2pi1\nmeδjlTr[TiAATkAA]/bracketrightBigg(68)\nis a Fermi sea term.I. Inverse CIT in noncollinear magnets\nThe chiral contribution JICIT2(see Eq. (4)) to the\ncharge pumping is described by the coefficients\nχICIT2\nijkl=χICIT2a\nijkl+χICIT2b\nijkl+χICIT2c\nijkl,(69)\nwhereχICIT2a\nijkldescribes the response to the time-\ndependentmagnetizationgradient(seeEq.(18),Eq.(25),\nand Eq. (24)) and χICIT2c\nijklresults from the spatial gra-\ndient of DOM (see Eq. (55)). χICIT2b\nijkldescribes the re-\nsponseto the perturbation bymagnetizationdynamics in\na noncollinear magnet. In order to derive an expression\nforχICIT2b\nijklwe assume that the magnetization oscillates\nspatially as described by Eq. (62). Since the correspond-\ning response oscillates spatially proportional to cos( q·r),\nwe multiply by cos( q·r) and integrate over the unit cell\nin order to extract χICIT2b\nijklfrom the retarded velocity-\ntorque correlation function ∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω), which\nis evaluated in the presence of the perturbation Eq. (65).\nWe obtain\nχICIT2b\nijkl=2e\nVηlim\nq→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n,\n(70)\nwhich can be written as (see Appendix B)\nχICIT2b\nijkl=2e\n/planckover2pi1Im/bracketleftBig\nV(surf)\nijkl+V(sea)\nijkl/bracketrightBig\n,(71)\nwhere\nV(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBig\nviGR\nk(E)vlGR\nk(E)TjGA\nk(E)TkGA\nk(E)\n+viGR\nk(E)TjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−viGR\nk(E)TjGA\nk(E)TkGA\nk(E)vlGA\nk(E)/bracketrightBig(72)\nis the Fermi surface term and\nV(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\n−Tr[viRvlRRTjRTkR]−Tr[viRvlRTkRRTjR]\n−Tr[viRRvlRTjRTkR]−Tr[viRRTjRvlRTkR]\n+Tr[viRRTjRTkRvlR]+Tr[viRRTkRTjRvlR]\n+Tr[viRRTkRvlRTjR]−Tr[viRRvlRTkRTjR]\n−Tr[viRvlRRTkRTjR]+Tr[viRTkRRTjRvlR]\n+Tr[viRTkRRvlRTjR]+Tr[viRTkRvlRRTjR]/bracketrightBig(73)\nis the Fermi sea term.\nIn Eq. (70) we use the Kubo formula to describe the\nresponse to magnetization dynamics combined with per-\nturbation theory to include the effect of noncollinearity.10\nThereby, the time-dependent perturbation and the per-\nturbation by the magnetization gradient are separated\nand perturbations of the form of Eq. (15) or Eq. (16)\nare not automatically included. For example the flat cy-\ncloidal spin spiral\nˆM(x,t) =\nsin(qx−ωt)\n0\ncos(qx−ωt)\n (74)\nmoving inxdirection with speed ω/qand the helical spin\nspiral\nˆM(y,t) =\nsin(qy−ωt)\n0\ncos(qy−ωt)\n (75)\nmovinginydirectionwith speed ω/qbehavelikeEq.(10)\nwhentandraresmall. Thus, these movingdomainwalls\ncorrespond to the perturbation of the type of Eq. (10)\nand the resulting contribution JICIT2afrom the time-\ndependent magnetization gradient is not described by\nEq. (70) and needs to be added, which we do by adding\nχICIT2a\nijklin Eq. (69).\nJ. Damping and gyromagnetism in noncollinear\nmagnets\nThe chiral contribution Eq. (57) to the torque-torque\ncorrelation function is expressed in terms of the coeffi-\ncient\nχTT\nijkl=χTT2a\nijkl+χTT2b\nijkl+χTT2c\nijkl, (76)\nwhereχTT2c\nijklresults from the spatial gradient of DDMI\n(see Eq. (60)), χTT2a\nijkldescribes the response to a time-\ndependent magnetization gradient in a collinear magnet,\nandχTT2b\nijkldescribes the response to magnetization dy-\nnamics in a noncollinear magnet.\nIn order to derive an expression for χTT2b\nijklwe as-\nsume that the magnetization oscillates spatially accord-\ning to Eq. (62). We multiply the retarded torque-torque\ncorrelation function ∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) with cos(qlrl)\nand integrate over the unit cell in order to extract the\npart of the response that varies spatially proportional to\ncos(qlrl). We obtain:\nχTT2b\nijkl=2\nVηlim\nql→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n.\n(77)\nIn Appendix B we discuss how to evaluate Eq. (77) in\nfirst order perturbation theory with respect to the per-\nturbation Eq.(65) and showthat χTT2b\nijklcan be expressedas\nχTT2b\nijkl=2\n/planckover2pi1Im/bracketleftBig\nX(surf)\nijkl+X(sea)\nijkl/bracketrightBig\n,(78)\nwhere\nX(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nTiGR\nk(E)vlGR\nk(E)TjGA\nk(E)TkGA\nk(E)\n+TiGR\nk(E)TjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−TiGR\nk(E)TjGA\nk(E)TkGA\nk(E)vlGA\nk(E)/bracketrightBigg(79)\nis a Fermi surface term and\nX(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBigg\n−(TiRvlRRTjRTkR)−(TiRvlRTkRRTjR)\n−(TiRRvlRTjRTkR)−(TiRRTjRvlRTkR)\n+(TiRRTjRTkRvlR)+(TiRRTkRTjRvlR)\n+(TiRRTkRvlRTjR)−(TiRRvlRTkRTjR)\n−(TiRvlRRTkRTjR)+(TiRTkRRTjRvlR)\n+(TiRTkRRvlRTjR)+(TiRTkRvlRRTjR)/bracketrightBigg\n(80)\nis a Fermi sea term.\nThe contribution χTT2a\nijklfrom the time-dependent gra-\ndients is given by\nχTT2a\nijkl=−/summationdisplay\nmχTT2a\niml[1−δjm]δjk,(81)\nwhere\nχTT2a\niml=i\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvlRROmR+TiRRvlROmR+\n−TiRROmRvlR−TiRvlROmAA\n+TiROmAvlAA+TiROmAAvlA\n−TiRvlRROmA−TiRRvlROmA\n+TiRROmAvlA+TiAvlAOmAA\n−TiAOmAvlAA−TiAOmAAvlA/bracketrightBig\n,(82)\nwithOm=∂H/∂ˆMm(see Appendix A).\nIII. SYMMETRY PROPERTIES\nIn this section we discuss the symmetry properties of\nCIDMI, DDMI and DOM in the case of the magnetic\nRashba model\nHk(r) =/planckover2pi12\n2mek2+α(k׈ez)·σ+∆V\n2σ·ˆM(r).(83)11\nAdditionally, we discuss the symmetry properties of the\ncurrents and torques induced by time-dependent magne-\ntization gradients of the form of Eq. (10).\nWe consider mirror reflection Mxzat thexzplane,\nmirror reflection Myzat theyzplane, and c2 rotation\naround the zaxis. When ∆ V= 0 these operations leave\nEq. (83) invariant, but when ∆ V∝ne}ationslash= 0 they modify the\nmagnetization direction ˆMin Eq. (83), as shown in Ta-\nble III. At the same time, these operations affect the\ntorqueTandthecurrent Jdrivenbythe time-dependent\nmagnetization gradients (see Table III). In Table IV and\nTable V we show how ˆM×∂ˆM/∂rkis affected by the\nsymmetry operations.\nAflat cycloidalspin spiralwith spinsrotatingin the xz\nplane is mapped by a c2 rotation around the zaxis onto\nthe same spin spiral. Similarly, a flat helical spin spiral\nwith spins rotating in the yzplane is mapped by a c2 ro-\ntationaroundthe zaxisontothesamespinspiral. There-\nfore, when ˆMpoints inzdirection, a c2 rotation around\nthezaxis does not change ˆM×∂ˆM/∂ri, but it flips the\nin-plane current Jand the in-plane components of the\ntorque,TxandTy. Consequently, ˆM×∂2ˆM/∂ri∂tdoes\nnot induce currents or torques, i.e., ICIDMI, CIDMI, ID-\nDMI and DDMI are zero, when ˆMpoints inzdirection.\nHowever, they become nonzero when the magnetization\nhas an in-plane component (see Fig. 1).\nSimilarly, IDOM vanishes when the magnetization\npoints inzdirection: In that case Eq. (83) is invariant\nunder the c2 rotation. A time-dependent magnetic field\nalongzdirection is invariant under the c2 rotation as\nwell. However, TxandTychange sign under the c2 rota-\ntion. Consequently, symmetryforbidsIDOM inthiscase.\nHowever, when the magnetization has an in-plane com-\nponent, IDOM and DOM become nonzero (see Fig. 2).\nThat time-dependent magnetization gradients of the\ntype of Eq. (7) do not induce in-plane currents and\ntorqueswhen ˆMpoints inzdirectioncan alsobe seendi-\nrectly from Eq. (7): The c2 rotation transforms q→ −q\nandMx�� −Mx. Since sin( q·r) is odd in r, Eq. (7) is in-\nvariantunder c2rotation, whilethe in-planecurrentsand\ntorques induced by time-dependent magnetization gradi-\nents change sign under c2 rotation. In contrast, Eq. (10)\nis not invariant under c2 rotation, because sin( q·r−ωt)\nis not odd in rfort>0. Consequently, time-dependent\nmagnetization gradients of the type of Eq. (10) induce\ncurrents and torques also when ˆMpoints locally into\nthezdirection. These currents and torques, which are\ndescribed by Eq. (24) and Eq. (82), respectively, need to\nbe added to the chiral ICIT and the chiral torque-torque\ncorrelation. While CIDMI, DDMI, and DOM are zero\nwhen the magnetization points in zdirection, their gra-\ndients are not (see Fig. 1 and Fig. 2). Therefore, the gra-\ndients of CIDMI, DOM, and DDMI contribute to CIT, to\nICIT and to the torque-torque correlation, respectively,\neven when ˆMpoints locally into the zdirection.TABLE III: Effect of mirror reflection Mxzat thexzplane,\nmirror reflection Myzat theyzplane, and c2 rotation around\nthezaxis. The magnetization Mand the torque Ttransform\nlike axial vectors, while the current Jtransforms like a polar\nvector.\nMxMyMzJxJyTxTyTz\nMxz−MxMy−MzJx−Jy−TxTy−Tz\nMyzMx−My−Mz−JxJyTx−Ty−Tz\nc2-Mx-MyMz-Jx−Jy−Tx−TyTz\nTABLE IV: Effect of symmetry operations on the magneti-\nzation gradients. Magnetization gradients are described b y\nthree indices ( ijk). The first index denotes the magnetiza-\ntion direction at r= 0. The third index denotes the di-\nrection along which the magnetization changes. The second\nindex denotes the direction of ∂ˆM/∂rkδrk. The direction of\nˆM×∂ˆM/∂rkis specified by the number below the indices\n(ijk).\n(1,2,1) (1,3,1) (2,1,1) (2,3,1) (3,1,1) (3,2,1)\n3-2 -3 1 2-1\nMxz(-1,2,1)(-1,-3,1) (2,-1,1) (2,-3,1) (-3,-1,1) (-3,2,1)\n-3 -2 3 -1 2 1\nMyz(1,2,1) (1,3,1)(-2,-1,1) (-2,3,1) (-3,-1,1) (-3,2,1)\n3-2 -3 -1 2 1\nc2(-1,2,1)(-1,-3,1) (-2,1,1) (-2,-3,1) (3,1,1) (3,2,1)\n-3 -2 3 1 2-1\n.\nTABLE V: Continuation of Table IV\n(1,2,2) (1,3,2) (2,1,2) (2,3,2) (3,1,2) (3,2,2)\n3 -2 -3 1 2-1\nMxz(-1,-2,2) (-1,3,2) (2,1,2) (2,3,2)(-3,1,2)(-3,-2,2)\n3 2-3 1-2 -1\nMyz(1,-2,2) (1,-3,2) (-2,1,2)(-2,-3,2) (-3,1,2)(-3,-2,2)\n-3 2 3 1-2 -1\nc2(-1,2,2) (-1,-3,2) (-2,1,2)(-2,-3,2) (3,1,2) (3,2,2)\n-3 -2 3 1 2-1\nA. Symmetry properties of ICIDMI and IDDMI\nInthefollowingwediscusshowTableIII,TableIV,and\nTable V can be used to analyze the symmetry of ICIDMI\nandIDDMI.AccordingtoEq.(19)thecoefficient χICIDMI\nijk\ndescribes the response of the current JICIT2a\nito the time-\ndependent magnetization gradient ˆej·[ˆM×∂2ˆM\n∂rk∂t]. Since\nˆM×∂2ˆM\n∂rk∂t=∂\n∂t[ˆM×∂ˆM\n∂rk] fortime-dependent magnetiza-\ntion gradients of the type Eq. (7) the symmetry proper-\nties ofχICIDMI\nijkfollow from the transformation behaviour\nofˆM×∂ˆM\n∂rkandJunder symmetry operations.\nWe consider the case with magnetization in xdirec-\ntion. The component χICIDMI\n132describes the current in x\ndirection induced by the time-dependence of a cycloidal\nmagnetizationgradientin ydirection(withspinsrotating12\nFIG. 1: ICIDMI in a noncollinear magnet. (a) Arrows illus-\ntrate the magnetization direction. (b) Arrows illustrate t he\ncurrentJyinduced by a time-dependent magnetization gra-\ndient, which is described by χICIDMI\n221. When ˆMpoints in z\ndirection, χICIDMI\n221andJyare zero. The sign of χICIDMI\n221and\nofJychanges with the sign of Mx.\nFIG. 2: DOM in a noncollinear magnet. (a) Arrows illustrate\nthe magnetization direction. (b) Arrows illustrate the orb ital\nmagnetization induced by magnetization dynamics (DOM).\nWhenˆMpoints in zdirection, DOM is zero. The sign of\nDOM changes with the sign of Mx.\nin thexyplane).Myzflips both ˆM×∂ˆM\n∂yandJx, but\nit preserves ˆM.Mzxpreserves ˆM×∂ˆM\n∂yandJx, but it\nflipsˆM. A c2 rotation around the zaxis flips ˆM×∂ˆM\n∂y,\nˆMandJx. Consequently, χICIDMI\n132(ˆM) is allowed by\nsymmetry and it is even in ˆM. The component χICIDMI\n122\ndescribes the current in xdirection induced by the time-\ndependence of a helical magnetization gradient in ydi-\nrection (with spins rotating in the xzplane).Myzflips\nˆM×∂ˆM\n∂yandJx, but it preserves ˆM.MzxflipsˆM×∂ˆM\n∂y\nandˆM, but it preserves Jx. A c2 rotation around the z\naxis flipsJxandˆM, but it preserves ˆM×∂ˆM\n∂y. Conse-\nquently,χICIDMI\n122is allowed by symmetry and it is odd in\nˆM. The component χICIDMI\n221describes the current in y\ndirection induced by the time-dependence of a cycloidal\nmagnetization gradient in xdirection (with spins rotat-\ning in thexzplane).Mzxpreserves ˆM×∂ˆM\n∂x, but it flipsJyandˆM.Myzpreserves ˆM,Jy, andˆM×∂ˆM\n∂x. The\nc2 rotation around the zaxis preserves ˆM×∂ˆM\n∂x, but\nit flipsˆMandJy. Consequently, χICIDMI\n221is allowed by\nsymmetry and it is odd in ˆM. The component χICIDMI\n231\ndescribes the current in ydirection induced by the time-\ndependence of a cycloidal magnetization gradient in xdi-\nrection (with spins rotating in the xyplane).Mzxflips\nˆM×∂ˆM\n∂x,ˆM, andJy.Myzpreserves ˆM×∂ˆM\n∂x,ˆMand\nJy. The c2 rotation around the zaxis flips ˆM×∂ˆM\n∂x,Jy,\nandˆM. Consequently, χICIDMI\n231is allowed by symmetry\nand it is even in ˆM.\nThese properties are summarized in Table VI. Due to\nthe relations between CIDMI and DOM (see Table I and\nTable II), they can be used for DOM as well. When the\nmagnetization lies at a general angle in the xzplane or in\ntheyzplaneseveraladditionalcomponentsofCIDMIand\nDOMarenonzero(seeTableIandTableII,respectively).\nTABLE VI: Allowed components of χICIDMI\nijkwhenˆMpoints\ninxdirection. + components are even in ˆM, while - compo-\nnents are odd in ˆM.\n132 122 221 231\n+ - - +\nSimilarly, one can analyze the symmetry of DDMI. Ta-\nble VII lists the components of DDMI, χDDMI\nijk, which are\nallowed by symmetry when ˆMpoints inxdirection.\nTABLEVII:Allowedcomponentsof χDDMI\nijkwhenˆMpointsin\nxdirection. +componentsareevenin ˆM, while -components\nare odd in ˆM.\n222 232 322 332\n- + + -\nB. Response to time-dependent magnetization\ngradients of the second type (Eq. (10))\nAccording to Eq. (13) the time-dependent magneti-\nzation gradient is along the magnetization. Therefore,\nin contrast to the discussion in section IIIA we can-\nnot use ˆM×∂2ˆM\n∂rk∂tin the symmetry analysis. Eq. (24)\nand Eq. (25) show that χICIT2a\nijjldescribes the response of\nJICIT2a\nitoˆej·/bracketleftBig\nˆM×∂ˆM\n∂t/bracketrightBig\nˆej·/bracketleftBig\nˆM×∂ˆM\n∂rl/bracketrightBig\nwhileχICIT2a\nijkl=\n0 forj∝ne}ationslash=k. According to Eq. (23) the symmetry prop-\nerties of/bracketleftBig\nˆM×∂ˆM\n∂t/bracketrightBig\n·/bracketleftBig\nˆM×∂ˆM\n∂rl/bracketrightBig\nagree to the symmetry\nproperties of ˆM·∂2ˆM\n∂rl∂t. Therefore, in order to under-\nstand the symmetry properties of χICIT2a\nijjlwe consider\nthe transformation of JandˆM·∂2ˆM\n∂rl∂tunder symmetry\noperations.\nWe consider the case where ˆMpoints inzdirection.\nχICIT2a\n1jj1describes the current driven in xdirection, when13\nthe magnetization varies in xdirection. MxzflipsˆM,\nbut preserves JxandˆM·∂2ˆM/(∂x∂t).MyzflipsˆM,Jx,\nandˆM·∂2ˆM/(∂x∂t). c2 rotation flips ˆM·∂2ˆM/(∂x∂t)\nandJx, but preserves ˆM. Consequently, χICIT2a\n1jj1is al-\nlowed by symmetry and it is even in ˆM.\nχICIT2a\n2jj1describes the current flowing in ydirection,\nwhen magnetization varies in xdirection. MxzflipsˆM\nandJy, but preserves ˆM·∂2ˆM/(∂x∂t).MyzflipsˆM,\nandˆM·∂2ˆM/(∂x∂t), but preserves Jy. c2 rotation\nflipsˆM·∂2ˆM/(∂x∂t) andJy, but preserves ˆM. Conse-\nquently,χICIT2a\n2jj1is allowed by symmetry and it is odd in\nˆM.\nSimilarly, one can show that χICIT2a\n1jj2is odd in ˆMand\nthatχICIT2a\n2jj2is even in ˆM.\nAnalogously, one can investigate the symmetry prop-\nerties ofχTT2a\nijjl. We find that χTT2a\n1jj1andχTT2a\n2jj2are odd\ninˆM, whileχTT2a\n2jj1andχTT2a\n1jj2are even in ˆM.\nIV. RESULTS\nIn the following sections we discuss the results for the\ndirect and inverse chiral CIT and for the chiral torque-\ntorque correlation in the two-dimensional (2d) Rashba\nmodel Eq. (83), and in the one-dimensional (1d) Rashba\nmodel [38]\nHkx(x) =/planckover2pi12\n2mek2\nx−αkxσy+∆V\n2σ·ˆM(x).(84)\nAdditionally, we discuss the contributions of the time-\ndependent magnetization gradients, and of DDMI, DOM\nand CIDMI to these effects.\nWhile vertex corrections to the chiral CIT and to\nthe chiral torque-torque correlation are important in the\nRashba model [38], the purpose of this work is to show\nthe importance ofthe contributionsfrom time-dependent\nmagnetization gradients, DDMI, DOM and CIDMI. We\ntherefore consider only the intrinsic contributions here,\ni.e., we set\nGR\nk(E) =/planckover2pi1[E −Hk+iΓ]−1, (85)\nwhere Γ is a constant broadening, and we leave the study\nof vertex corrections for future work.\nThe results shown in the following sections are ob-\ntained for the model parameters ∆ V= 1eV,α=2eV˚A,\nand Γ = 0 .1Ry = 1.361eV, when the magnetization\npoints inzdirection, i.e., ˆM=ˆez. The unit of χCIT2\nijkl\nis charge times length in the 1d case and charge in the\n2d case. Therefore, in the 1d case we discuss the chiral\ntorkance in units of ea0, wherea0is Bohr’s radius. In the\n2d case we discuss the chiral torkance in units of e. The\nunit ofχTT2\nijklis angular momentum in the 1d case and\nangular momentum per length in the 2d case. Therefore,\nwe discussχTT2\nijklin units of /planckover2pi1in the 1d case, and in units\nof/planckover2pi1/a0in the 2d case.-2 -1 0 1 2\nFermi energy [eV]-0.02-0.0100.010.020.030.040.05χijklCIT2 [ea0]2121\n1121\n2121 (gauge-field)\n1121 (gauge-field)\nFIG. 3: Chiral CIT in the 1d Rashba model for cycloidal gra-\ndients vs. Fermi energy. General perturbation theory (soli d\nlines) agrees to the gauge-field approach (dashed lines).\nA. Direct and inverse chiral CIT\nIn Fig. 3 we show the chiral CIT as a function of the\nFermi energyfor cycloidalmagnetization gradients in the\n1d Rashba model. The components χCIT2\n2121andχCIT2\n1121are\nlabelled by 2121 and 1121, respectively. The component\n2121ofCITdescribesthe non-adiabatictorque, while the\ncomponent 1121 describes the adiabatic STT (modified\nby SOI). In the one-dimensional Rashba model, the con-\ntributionsχCIT2b\n2121andχCIT2b\n1121(Eq. (56)) from the CIDMI\nare zero when ˆM=ˆez(not shown in the figure). For cy-\ncloidal spin spirals, it is possible to solve the 1d Rashba\nmodel by a gauge-field approach [38], which allows us to\ntest the perturbation theory, Eq. (66). For comparison\nwe show in Fig. 3 the results obtained from the gauge-\nfield approach, which agree to the perturbation theory,\nEq. (66). This demonstrates the validity of Eq. (66).\nIn Fig. 4 we show the chiral ICIT in the 1d Rashba\nmodel. The components χICIT2\n1221andχICIT2\n1121are labelled\nby 1221and 1121, respectively. The contribution χICIT2a\n1221\nfrom the time-dependent gradient is of the same order of\nmagnitude as the total χICIT2\n1221. Comparison of Fig. 3 and\nFig. 4 shows that CIT and ICIT satisfy the reciprocity\nrelationsEq. (5), that χCIT2\n1121is odd in ˆM, and thatχCIT2\n2121\nis even in ˆM, i.e.,χCIT2\n2121=χICIT2\n1221andχCIT2\n1121=−χICIT2\n1121.\nThe contribution χICIT2a\n1221from the time-dependent gradi-\nents is crucial to satisfy the reciprocity relations between\nχCIT2\n2121andχICIT2\n1221.\nIn Fig. 5 and Fig. 6 we show the CIT and the ICIT, re-\nspectively, for helical gradients in the 1d Rashba model.\nThe components χCIT2\n2111andχCIT2\n1111are labelled 2111 and\n1111, respectively, in Fig. 5, while χICIT2\n1211andχICIT2\n1111\nare labelled 1211 and 1111, respectively, in Fig. 6. The\ncontributions χCIT2b\n2111andχCIT2b\n1111from CIDMI are of the14\n-2 -1 0 1 2\nFermi energy [eV]-0.0200.020.04χijklICIT2 [ea0]1221 \n1121\nχ1221ICIT2a\nFIG. 4: Chiral ICIT in the 1d Rashba model for cycloidal\ngradients vs. Fermi energy. Dashed line: Contribution from\nthe time-dependent gradient.\nsame order of magnitude as the total χCIT2\n2111andχCIT2\n1111.\nSimilarly, the contributions χICIT2c\n1211andχICIT2c\n1111from\nDOM are of the same order of magnitude as the to-\ntalχICIT2\n1211andχICIT2\n1111. Additionally, the contribution\nχICIT2a\n1111from the time-dependent gradient is substantial.\nComparisonofFig.5andFig.6showsthatCITandICIT\nsatisfy the reciprocity relation Eq. (5), that χCIT2\n2111is odd\ninˆM, and thatχCIT2\n1111is even in ˆM, i.e.,χCIT2\n1111=χICIT2\n1111\nandχCIT2\n2111=−χICIT2\n1211. These reciprocity relations be-\ntween CIT and ICIT are only satisfied when CIDMI,\nDOM, and the response to time-dependent magnetiza-\ntion gradients are included. Additionally, the compar-\nison between Fig. 5 and Fig. 6 shows that the contri-\nbutions of CIDMI to CIT ( χCIT2b\n1111andχCIT2b\n2111) are re-\nlated to the contributions of DOM to ICIT ( χICIT2c\n1111and\nχICIT2c\n1211). These relations between DOM and ICIT are\nexpected from Table I.\nIn Fig. 7 and Fig. 8 we show the CIT and the ICIT,\nrespectively, for cycloidal gradients in the 2d Rashba\nmodel. In this case there are contributions from CIDMI\nand DOM in contrast to the 1d case with cycloidal gra-\ndients (Fig. 3). Comparison between Fig. 7 and Fig. 8\nshows that χCIT2\n1121andχCIT2\n2221are odd in ˆM, thatχCIT2\n1221\nandχCIT2\n2121are even in ˆM, and that CIT and ICIT sat-\nisfy the reciprocity relation Eq. (5) when the gradients\nof CIDMI and DOM are included, i.e., χCIT2\n1121=−χICIT2\n1121,\nχCIT2\n2221=−χICIT2\n2221,χCIT2\n1221=χICIT2\n2121, andχCIT2\n2121=χICIT2\n1221.\nχCIT2\n1121describesthe adiabatic STT with SOI, while χCIT2\n2121\ndescribes the non-adiabatic STT. Experimentally, it has\nbeen found that CITs occur also when the electric field\nis applied parallel to domain-walls (i.e., perpendicular to\ntheq-vector of spin spirals) [39]. In our calculations, the\ncomponents χCIT2\n2221andχCIT2\n1221describe such a case, where\nthe applied electric field points in ydirection, while the-2 -1 0 1 2\nFermi energy [eV]-0.04-0.0200.020.040.06χijklCIT2 [ea0]1111\n2111\nχ1111CIT2b\nχ2111CIT2b\nFIG. 5: Chiral CIT for helical gradients in the 1d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.0200.020.040.06χijklICIT2 [ea0]1111\n1211\nχ1111ICIT2a\nχ1111ICIT2c\nχ1211ICIT2c\nFIG. 6: Chiral ICIT for helical gradients in the 1d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted line: Contribution from the time-\ndependent magnetization gradient.\nmagnetization direction varies with the xcoordinate.\nIn Fig. 9 and Fig. 10 we show the chiral CIT and\nICIT, respectively, for helical gradients in the 2d Rashba\nmodel. The component χCIT2\n2111describes the adiabatic\nSTT with SOI and the component χCIT2\n1111describes the\nnon-adiabatic STT. The components χCIT2\n2211andχCIT2\n1211\ndescribe the case when the applied electric field points\ninydirection, i.e., perpendicular to the direction along\nwhich the magnetization direction varies. Comparison\nbetween Fig. 9 and Fig. 10 shows that χCIT2\n1111andχCIT2\n2211\nare even in ˆM, thatχCIT2\n1211andχCIT2\n2111are odd in ˆMand\nthat CIT andICIT satisfythe reciprocityrelationEq.(5)\nwhenthegradientsofCIDMIandDOMareincluded, i.e.,15\n-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]1121\n2221\n1221\n2121\nχ2221CIT2b\nχ1221CIT2b\nχ2121CIT2b\nFIG. 7: Chiral CIT for cycloidal gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklICIT2 [e]1121\n1221\n2121\n2221\nχ2221ICIT2a\nχ1221ICIT2a\nχ2121ICIT2c\nχ1221ICIT2c\nχ2221ICIT2c\nFIG. 8: Chiral ICIT for cycloidal gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted lines: Contributions from the time-\ndependent gradients.\nχCIT2\n1111=χICIT2\n1111,χCIT2\n2211=χICIT2\n2211,χCIT2\n1211=−χICIT2\n2111, and\nχCIT2\n2111=−χICIT2\n1211.\nB. Chiral torque-torque correlation\nIn Fig. 11 we show the chiral contribution to the\ntorque-torque correlation in the 1d Rashba model for\ncycloidal gradients. We compare the perturbation the-\nory Eq. (78) plus Eq. (82) to the gauge-field approach\nfrom Ref. [38]. This comparison shows that perturba-\ntion theory provides the correct answer only when the\ncontribution χTT2a\nijkl(Eq. (82)) from the time-dependent-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]2211\n1111\n1211\n2111\nχ2111CIT2b\nχ1211CIT2b\nχ2211CIT2b\nχ1111CIT2b\nFIG. 9: Chiral CIT for helical gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.004-0.00200.0020.0040.006χijklICIT2 [e]1111\n1211\n2111\n2211\nχ1111ICIT2a\nχ2221ICIT2a\nχ1111ICIT2c\nχ2111ICIT2c\nχ1211ICIT2c\nχ2211ICIT2c\nFIG. 10: Chiral ICIT for helical gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted lines: Contributions from the time-\ndependent gradient.\ngradients is taken into account. The contributions χTT2a\n1221\nandχTT2a\n2221fromthe time-dependent gradientsarecompa-\nrable in magnitude to the total values. In the 1d Rashba\nmodel the DDMI-contribution in Eq. (60) is zero for cy-\ncloidal gradients (not shown in the figure). The compo-\nnentsχTT2\n2121andχTT2\n1221describe the chiral gyromagnetism\nwhile the components χTT2\n1121andχTT2\n2221describe the chi-\nral damping [38, 40, 41]. The components χTT2\n2121and\nχTT2\n1221are odd in ˆMand they satisfy the Onsagerrelation\nEq. (58), i.e., χTT2\n2121=−χTT2\n1221.\nIn Fig. 12 we show the chiral contributions to the\ntorque-torque correlation in the 1d Rashba model for\nhelical gradients. In contrast to the cycloidal gradients16\n-2 -1 0 1 2\nFermi energy [eV]-0.00500.0050.01χijklTT2 [h_]2121\n1221\n2221\n1121\nχ1221TT2a\nχ2221TT2a\n2121 (gf)\n1221 (gf)\n1121 (gf)\n2221 (gf)\nFIG. 11: Chiral contribution to the torque-torque correla-\ntion for cycloidal gradients in the 1d Rashba model vs. Fermi\nenergy. Perturbation theory (solid lines) agrees to the gau ge-\nfield (gf) approach (dotted lines). Dashed lines: Contribut ion\nfrom the time-dependent gradient.\n(Fig. 11) there are contributions from the spatial gra-\ndients of DDMI (Eq. (60)) in this case. The Onsager\nrelation Eq. (58) for the components χTT2\n2111andχTT2\n1211is\nsatisfied only when these contributions from DDMI are\ntaken into account, which are of the same order of mag-\nnitude as the total values. The components χTT2\n2111and\nχTT2\n1211are even in ˆMand describe chiral damping, while\nthe components χTT2\n1111andχTT2\n2211are odd in ˆMand de-\nscribe chiral gyromagnetism. As a consequence of the\nOnsager relation Eq. (58) we obtain χTT2\n1111=χTT2\n2211= 0\nfor the total components: Eq. (58) shows that diagonal\ncomponents of the torque-torque correlation function are\nzero unless they are even in ˆM. However, χTT2a\n1111,χTT2c\n1111,\nandχTT2b\n1111=−χTT2a\n1111−χTT2c\n1111are individually nonzero.\nInterestingly, the off-diagonal components of the torque-\ntorquecorrelationdescribechiraldampingforhelicalgra-\ndients, while for cycloidal gradients the off-diagonal ele-\nments describe chiral gyromagnetism and the diagonal\nelements describe chiral damping.\nIn Fig. 13 we show the chiral contributions to the\ntorque-torque correlation in the 2d Rashba model for cy-\ncloidal gradients. In contrast to the 1d Rashba model\nwith cycloidal gradients (Fig. 11) the contributions from\nDDMIχTT2c\nijkl(Eq.(60))arenonzerointhiscase. Without\nthesecontributionsfromDDMI theOnsagerrelation(58)\nχTT2\n2121=−χTT2\n1221is violated. The DDMI contribution is\nof the same order of magnitude as the total values. The\ncomponents χTT2\n2121andχTT2\n1221are odd in ˆMand describe\nchiral gyromagnetism, while the components χTT2\n1121and\nχTT2\n2221are even in ˆMand describe chiral damping.\nIn Fig. 14 we show the chiral contributions to the\ntorque-torque correlation in the 2d Rashba model for he-\nlical gradients. The components χTT2\n1211andχTT2\n2111are even-2 -1 0 1 2\nFermi energy [eV]-0.00500.0050.01χijklTT2 [h_]1111\n2111\n1211\n2211\nχ1111TT2c\nχ2111TT2c\nχ1211TT2c\nχ2211TT2c\nχ1111TT2a\nχ2111TT2a\nFIG. 12: Chiral contribution to the torque-torque correla-\ntion for helical gradients in the 1d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\n-2 -1 0 1 2\nFermi energy [eV]-0.000500.00050.001χijklTT2 [h_/a0]1121\n2121\n1221\n2221\nχ1221TT2a\nχ2221TT2a\nχ2121TT2c\nχ1221TT2c\nFIG. 13: Chiral contribution to the torque-torque correla-\ntion for cycloidal gradients in the 2d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\ninˆMand describe chiral damping, while the compo-\nnentsχTT2\n1111andχTT2\n2211are odd in ˆMand describe chiral\ngyromagnetism. The Onsager relation Eq. (58) requires\nχTT2\n1111=χTT2\n2211= 0 andχTT2\n2111=χTT2\n1211. Without the\ncontributions from DDMI these Onsager relations are vi-\nolated.17\n-2 -1 0 1 2\nFermi energy [eV]-0.000500.00050.001χijklTT2 [h_ /a0]1111\n2111\n1211\n2211\nχ1111TT2a\nχ2111TT2a\nχ1111TT2c\nχ1211TT2c\nχ2211TT2c\nFIG. 14: Chiral contribution to the torque-torque correla-\ntion for helical gradients in the 2d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\nV. SUMMARY\nFinding ways to tune the Dzyaloshinskii-Moriya inter-\naction (DMI) by external means, such as an applied elec-\ntriccurrent,holdsmuchpromiseforapplicationsinwhich\nDMI determines the magnetic texture of domain walls or\nskyrmions. In order to derive an expression for current-\ninduced Dzyaloshinskii-Moriya interaction (CIDMI) we\nfirst identify its inverse effect: When magnetic textures\nvary as a function of time, electric currents are driven by\nvarious mechanisms, which can be distinguished accord-\ningtotheirdifferentdependenceonthetime-derivativeof\nmagnetization, ∂ˆM(r,t)/∂t, and on the spatial deriva-\ntive∂ˆM(r,t)/∂r: One group of effects is proportional\nto∂ˆM(r,t)/∂t, a second group of effects is propor-\ntional to the product ∂ˆM(r,t)/∂t ∂ˆM(r,t)/∂r, and\na third group is proportional to the second derivative\n∂2ˆM(r,t)/∂r∂t. We show that the response of the elec-\ntric current to the time-dependent magnetization gradi-\nent∂2ˆM(r,t)/∂r∂tcontais the inverse of CIDMI. We\nestablish the reciprocity relation between inverse and di-\nrectCIDMI and therebyobtainan expressionforCIDMI.\nWe find that CIDMI is related to the modification of\norbital magnetism induced by magnetization dynamics,\nwhich we call dynamical orbital magnetism (DOM). We\nshow that torques are generated by time-dependent gra-\ndients of magnetization as well. The inverse effect con-\nsists in the modification of DMI by magnetization dy-\nnamics, which we call dynamical DMI (DDMI).\nAdditionally, we develop a formalism to calculate the\nchiral contributions to the direct and inverse current-\ninduced torques (CITs) and to the torque-torque correla-tion in noncollinear magnets. We show that the response\nto time-dependent magnetization gradients contributes\nsubstantially to these effects and that the Onsager reci-\nprocityrelationsareviolated when it is not takeninto ac-\ncount. InnoncollinearmagnetsCIDMI,DDMIandDOM\ndepend on the local magnetization direction. We show\nthat the resulting spatial gradients of CIDMI, DDMI\nand DOM have to be subtracted from the CIT, from\nthe torque-torque correlation, and from the inverse CIT,\nrespectively.\nWe apply our formalism to study CITs and the torque-\ntorque correlation in textured Rashba ferromagnets. We\nfind that the contribution of CIDMI to the chiral CIT is\noftheorderofmagnitudeofthe totaleffect. Similarly, we\nfind that the contribution of DDMI to the chiral torque-\ntorque correlation is of the order of magnitude of the\ntotal effect.\nAcknowledgments\nWeacknowledgefinancialsupportfromLeibnizCollab-\norative Excellence project OptiSPIN −Optical Control\nofNanoscaleSpin Textures. Weacknowledgefundingun-\nder SPP 2137 “Skyrmionics” of the DFG. We gratefully\nacknowledge financial support from the European Re-\nsearch Council (ERC) under the European Union’s Hori-\nzon 2020 research and innovation program (Grant No.\n856538, project ”3D MAGiC”). The work was also sup-\nported by the Deutsche Forschungsgemeinschaft (DFG,\nGerman Research Foundation) −TRR 173 −268565370\n(project A11). We gratefully acknowledge the J¨ ulich\nSupercomputing Centre and RWTH Aachen University\nfor providing computational resources under project No.\njiff40.\nAppendix A: Response to time-dependent gradients\nIn this appendix we derive Eq. (18), Eq. (20), Eq. (41),\nand Eq. (82), which describe the response to time-\ndependent magnetization gradients, and Eq. (50), which\ndescribesthe responsetotime-dependentmagneticfields.\nWe consider perturbations of the form\nδH(r,t) =Bb1\nqωsin(q·r)sin(ωt).(A1)\nWhenweset B=∂H\n∂ˆMkandb=∂2ˆMk\n∂ri∂t, Eq.(A1)turnsinto\nEq. (17), while when we set B=−eviandb=1\n2ǫijk∂Bk\n∂t\nwe obtain Eq. (48). We need to derive an expression for\nthe response δA(r,t) of an observable Ato this pertur-\nbation, which varies in time like cos( ωt) and in space like\ncos(q·r), because∂2ˆM(r,t)\n∂ri∂t∝cos(q·r)cos(ωt). There-\nfore, weusethe Kubolinearresponseformalismtoobtain18\nthe coefficient χin\nδA(r,t) =χcos(q·r)cos(ωt), (A2)\nwhich is given by\nχ=i\n/planckover2pi1qωV/bracketleftBig\n∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n−∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(−/planckover2pi1ω)/bracketrightBig\n,(A3)\nwhere∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) is the retarded\nfunction at frequency ωandVis the volume of the unit\ncell.\nThe operator Bsin(q·r) can be written as\nBsin(q·r) =1\n2i/summationdisplay\nknm/bracketleftBig\nB(1)\nknmc†\nk+nck−m−B(2)\nknmc†\nk−nck+m/bracketrightBig\n,\n(A4)\nwherek+=k+q/2,k−=k−q/2,c†\nk+nis the cre-\nation operator of an electron in state |uk+n∝an}bracketri}ht,ck−mis the\nannihilation operator of an electron in state |uk−m∝an}bracketri}ht,\nB(1)\nknm=1\n2∝an}bracketle{tuk+n|[Bk++Bk−]|uk−m∝an}bracketri}ht(A5)\nand\nB(2)\nknm=1\n2∝an}bracketle{tuk−n|[Bk++Bk−]|uk+m∝an}bracketri}ht.(A6)\nSimilarly,\nAcos(q·r) =1\n2/summationdisplay\nknm/bracketleftBig\nA(1)\nknmc†\nk+nck−m+A(2)\nknmc†\nk−nck+m/bracketrightBig\n,\n(A7)\nwhere\nA(1)\nknm=1\n2∝an}bracketle{tuk+n|/bracketleftbig\nAk++Ak−/bracketrightbig\n|uk−m∝an}bracketri}ht(A8)\nand\nA(2)\nknm=1\n2∝an}bracketle{tuk−n|/bracketleftbig\nAk++Ak−/bracketrightbig\n|uk+m∝an}bracketri}ht.(A9)\nIt is convenient to obtain the retarded response func-\ntion in Eq. (A3) from the correspondingMatsubarafunc-\ntion in imaginary time τ\n1\nV∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(τ) =\n=1\n4i/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/bracketleftBig\nA(1)\nknmB(2)\nkn′m′Z(1)\nknmn′m′(τ)\n−A(2)\nknmB(1)\nkn′m′Z(2)\nknmn′m′(τ)/bracketrightBig\n,\n(A10)\nwhered= 1,2 or 3 is the dimension,\nZ(1)\nknmn′m′(τ) =∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(0)ck+m′(0)∝an}bracketri}ht\n=−GM\nm′n(k+,−τ)GM\nmn′(k−,τ),\n(A11)Z(2)\nknmn′m′(τ) =∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(0)ck−m′(0)∝an}bracketri}ht\n=−GM\nm′n(k−,−τ)GM\nmn′(k+,τ),\n(A12)\nand\nGM\nmn′(k+,τ) =−∝an}bracketle{tTτck+m(τ)c†\nk+n′(0)∝an}bracketri}ht(A13)\nis the single-particle Matsubara function. The Fourier\ntransform of Eq. (A10) is given by\n1\nV∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=i\n4/planckover2pi1β/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\np/bracketleftBig\nA(1)\nknmB(2)\nkn′m′GM\nm′n(k+,iEp)GM\nmn′(k−,iEp+iEN)\n−A(2)\nknmB(1)\nkn′m′GM\nm′n(k−,iEp)GM\nmn′(k+,iEp+iEN)/bracketrightBig\n,\n(A14)\nwhereEN= 2πN/βandEp= (2p+ 1)π/βare bosonic\nandfermionicMatsubaraenergypoints, respectively, and\nβ= 1/(kBT) is the inverse temperature.\nIn order to carry out the Matsubara summation over\nEpwe make use of\n1\nβ/summationdisplay\npGM\nmn′(iEp+iEN)GM\nm′n(iEp) =\n=i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iEN)GM\nm′n(E′+iδ)\n+i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iδ)GM\nm′n(E′−iEN)\n−i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iEN)GM\nm′n(E′−iδ)\n−i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′−iδ)GM\nm′n(E′−iEN),(A15)\nwhereδis a positive infinitesimal. The retarded function\n∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(ω) is obtained from the Mat-\nsubara function ∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) by the\nanalytic continuation iEN→/planckover2pi1ωto real frequencies. The\nright-hand side of Eq. (A15) has the following analytic\ncontinuation to real frequencies:\ni\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′+/planckover2pi1ω)GR\nm′n(E′)\n+i\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′)GA\nm′n(E′−/planckover2pi1ω)\n−i\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′+/planckover2pi1ω)GA\nm′n(E′)\n−i\n2π/integraldisplay\ndE′f(E′)GA\nmn′(E′)GA\nm′n(E′−/planckover2pi1ω).(A16)\nTherefore, we obtain\nχ=−i\n8π/planckover2pi12qω/integraldisplayddk\n(2π)d[Zk(q,ω)−Zk(−q,ω)\n−Zk(q,−ω)+Zk(−q,−ω)],(A17)19\nwhere\nZk(q,ω) =\n=/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′+/planckover2pi1ω)BkGR\nk+(E′)/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′)BkGA\nk+(E′−/planckover2pi1ω)/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′+/planckover2pi1ω)BkGA\nk+(E′)/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGA\nk−(E′)BkGA\nk+(E′−/planckover2pi1ω)/bracketrightBig\n.(A18)\nWe consider the limit lim q→0limω→0χ. In this limit\nEq. (A17) may be rewritten as\nχ=−i\n2π/planckover2pi12/integraldisplayddk\n(2π)d∂2Zk(q,ω)\n∂q∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=ω=0.(A19)\nThe frequency derivative of Zk(q,ω) is given by\n1\n/planckover2pi1∂Zk\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAk∂GR\nk−(E′)\n∂E′BkGR\nk+(E′)/bracketrightBigg\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAkGR\nk−(E′)Bk∂GA\nk+(E′)\n∂E′/bracketrightBigg\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAk∂GR\nk−(E′)\n∂E′BkGA\nk+(E′)/bracketrightBigg\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAkGA\nk−(E′)Bk∂GA\nk+(E′)\n∂E′/bracketrightBigg\n.\n(A20)\nUsing∂GR(E)/∂E=−GR(E)GR(E)//planckover2pi1we obtain\n∂Zk\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−GR\nk−BkGR\nk+/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−BkGA\nk+GA\nk+/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−GR\nk−BkGA\nk+/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGA\nk−BkGA\nk+GA\nk+/bracketrightBig\n.\n(A21)\nMaking use of\nlim\nq→0∂GR\nk+\n∂q=1\n2GR\nkv·q\nqGR\nk (A22)we finally obtain\nχ=−i\n2π/planckover2pi12/integraldisplayddk\n(2π)dlim\nq→0lim\nω→0∂2Z(q,ω)\n∂q∂ω=\n=−i\n4π/planckover2pi12q\nq·/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nAkRvRRBkR+AkRRvRBkR\n−AkRRBkRvR−AkRvRBkAA\n+AkRBkAvAA+AkRBkAAvA\n−AkRvRRBkA−AkRRvRBkA\n+AkRRBkAvA\n+AkAvABkAA−AkABkAvAA\n−AkABkAAvA/bracketrightBig\n,(A23)\nwhere we use the abbreviations R=GR\nk(E) andA=\nGA\nk(E). When we substitute B=∂H\n∂ˆMj,A=−evi, and\nq=qkˆek, we obtain Eq. (18). When we substitute B=\nTj,A=−evi, andq=qkˆek, we obtain Eq. (20). When\nwe substitute A=−Ti,B=Tj, andq=qkˆek, we obtain\nEq. (41). When we substitute B=−evj,A=−Ti,\nandq=qkˆek, we obtain Eq. (50). When we substitute\nB=∂H\n∂ˆMj,A=−Ti, andq=qkˆek, we obtain Eq. (82).\nAppendix B: Perturbation theory for the chiral\ncontributions to CIT and to the torque-torque\ncorrelation\nIn this appendix we derive expressionsfor the retarded\nfunction\n∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) (B1)\nwithin first-orderperturbation theory with respect to the\nperturbation\nδH=Bηsin(q·r), (B2)\nwhich may arise e.g. from the spatial oscillation of the\nmagnetization direction. As usual, it is convenient to ob-\ntain the retarded response function from the correspond-\ning Matsubara function\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ) =−∝an}bracketle{tTτcos(q·r)A(τ)C(0)∝an}bracketri}ht.\n(B3)\nThe starting point for the perturbative expansion is\nthe equation\n−∝an}bracketle{tTτcos(q·r)A(τ1)C(0)∝an}bracketri}ht=\n=−Tr/bracketleftbig\ne−βHTτcos(q·r)A(τ1)C(0)/bracketrightbig\nTr[e−βH]=\n=−Tr/braceleftbig\ne−βH0Tτ[Ucos(q·r)A(τ1)C(0)]/bracerightbig\nTr[e−βH0U],(B4)20\nwhereH0is the unperturbed Hamiltonian and we con-\nsider the first order in the perturbation δH:\nU(1)=−1\n/planckover2pi1/integraldisplay/planckover2pi1β\n0dτ1Tτ{eτ1H0//planckover2pi1δHe−τ1H0//planckover2pi1}.(B5)\nThe essentialdifference between Eq. (A3) and Eq. (B4) is\nthat in Eq. (A3) the operator Benters together with the\nfactor sin( q·r)sin(ωt) (see Eq. (A1)), while in Eq. (B4)\nonly the factor sin( q·r) is connected to Bin Eq. (B2),\nwhile the factor sin( ωt) is coupled to the additional op-\neratorC.\nWe use Eq. (A4) and Eq. (A7) in order to express\nAcos(q·r) andBsin(q·r) in terms of annihilation and\ncreation operators. In terms of the correlators\nZ(3)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(τ1)ck−m′(τ1)c†\nk−n′′ck−m′′∝an}bracketri}ht\n(B6)\nand\nZ(4)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(τ1)ck−m′(τ1)c†\nk+n′′ck+m′′∝an}bracketri}ht\n(B7)\nand\nZ(5)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(τ1)ck+m′(τ1)c†\nk+n′′ck+m′′∝an}bracketri}ht\n(B8)\nand\nZ(6)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(τ1)ck+m′(τ1)c†\nk−n′′ck−m′��∝an}bracketri}ht\n(B9)\nEq. (B4) can be written as\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1) =\n=ηV\n4i/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay/planckover2pi1β\n0dτ/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′/bracketleftBigg\n−B(2)\nknmA(1)\nkn′m′Ck−n′′m′′Z(3)\nknmn′m′n′′m′′(τ,τ1)\n−B(2)\nknmA(1)\nkn′m′Ck+n′′m′′Z(4)\nknmn′m′n′′m′′(τ,τ1)\n+B(1)\nknmA(2)\nkn′m′Ck+n′′m′′Z(5)\nknmn′m′n′′m′′(τ,τ1)\n+B(1)\nknmA(2)\nkn′m′Ck−n′′m′′Z(6)\nknmn′m′n′′m′′(τ,τ1)/bracketrightBigg(B10)\nwithin first-order perturbation theory, where we de-\nfinedCk−n′′m′′=∝an}bracketle{tuk−n′′|C|uk−m′′∝an}bracketri}htandCk+n′′m′′=\n∝an}bracketle{tuk+n′′|C|uk+m′′∝an}bracketri}ht.\nNote that Z(5)can be obtained from Z(3)by replac-\ningk−byk+andk+byk−. Similarly, Z(6)can be\nobtained from Z(4)by replacing k−byk+andk+by\nk−. Therefore, we write down only the equations forZ(3)andZ(4)in the following. Using Wick’s theorem\nwe find\nZ(3)\nknmn′m′n′′m′′(τ,τ1) =\n=−GM\nm′n(k−,τ1−τ)GM\nmn′(k+,τ−τ1)GM\nm′′n′′(k−,0)\n+GM\nmn′(k+,τ−τ1)GM\nm′′n(k−,−τ)GM\nm′n′′(k−,τ1)\n(B11)\nand\nZ(4)\nknmn′m′n′′m′′(τ,τ1) =\n=−GM\nmn′(k+,τ−τ1)GM\nm′n(k−,τ1−τ)GM\nm′′n′′(k+,0)\n+GM\nmn′′(k+,τ)GM\nm′n(k−,τ1−τ)GM\nm′′n′(k+,−τ1).\n(B12)\nThe Fourier transform\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1)(B13)\nof Eq. (B10) can be written as\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=ηV\n4i/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′/bracketleftBigg\n−B(2)\nknmA(1)\nkn′m′Ck−n′′m′′Z(3a)\nknmn′m′n′′m′′(iEN)\n−B(2)\nknmA(1)\nkn′m′Ck+n′′m′′Z(4a)\nknmn′m′n′′m′′(iEN)\n+B(1)\nknmA(2)\nkn′m′Ck+n′′m′′Z(5a)\nknmn′m′n′′m′′(iEN)\n+B(1)\nknmA(2)\nkn′m′Ck−n′′m′′Z(6a)\nknmn′m′n′′m′′(iEN)/bracketrightBigg(B14)\nin terms of the integrals\nZ(3a)\nknmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β\n0dτ/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1×\n×GM\nmn′(k+,τ−τ1)GM\nm′′n(k−,−τ)GM\nm′n′′(k−,τ1) =\n=1\n/planckover2pi1β/summationdisplay\npGM\nk+mn′(iEp)GM\nk−m′′n(iEp)GM\nk−m′n′′(iEp+iEN)\n(B15)\nand\nZ(4a)\nknmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β\n0dτ/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1×\n×GM\nmn′′(k+,τ)GM\nm′n(k−,τ1−τ)GM\nm′′n′(k+,−τ1) =\n=1\n/planckover2pi1β/summationdisplay\npGM\nk+mn′′(iEp)GM\nk−m′n(iEp)GM\nk+m′′n′(iEp−iEN),\n(B16)\nwhereEN= 2πN/βis a bosonic Matsubara energy point\nand we used\nGM(τ) =1\n/planckover2pi1β∞/summationdisplay\np=−∞e−iEpτ//planckover2pi1GM(iEp),(B17)21\nwhereEp= (2p+1)π/βis a fermionic Matsubara point.\nAgain Z(5a)is obtained from Z(3a)by replacing k−by\nk+andk+byk−andZ(6a)is obtained from Z(4a)in\nthe same way.\nSummation overMatsubarapoints Epin Eq.(B15) and\nin Eq. (B16) and analytic continuation iEN→/planckover2pi1ωyields\n2πi/planckover2pi1Z(3a)\nknmn′m′n′′m′′(/planckover2pi1ω) =\n−/integraldisplay\ndEf(E)GR\nk+mn′(E)GR\nk−m′′n(E)GR\nk−m′n′′(E+/planckover2pi1ω)\n+/integraldisplay\ndEf(E)GA\nk+mn′(E)GA\nk−m′′n(E)GR\nk−m′n′′(E+/planckover2pi1ω)\n−/integraldisplay\ndEf(E)GA\nk+mn′(E−/planckover2pi1ω)GA\nk−m′′n(E−/planckover2pi1ω)GR\nk−m′n′′(E)\n+/integraldisplay\ndEf(E)GA\nk+mn′(E−/planckover2pi1ω)GA\nk−m′′n(E−/planckover2pi1ω)GA\nk−m′n′′(E)\n(B18)\nand\n2πi/planckover2pi1Z(4a)\nknmn′m′n′′m′′(/planckover2pi1ω) =\n−/integraldisplay\ndEf(E)GR\nk+mn′′(E)GR\nk−m′n(E)GA\nk+m′′n′(E−/planckover2pi1ω)\n+/integraldisplay\ndEf(E)GA\nk+mn′′(E)GA\nk−m′n(E)GA\nk+m′′n′(E−/planckover2pi1ω)\n−/integraldisplay\ndEf(E)GR\nk+mn′′(E+/planckover2pi1ω)GR\nk−m′n(E+/planckover2pi1ω)GR\nk+m′′n′(E)\n+/integraldisplay\ndEf(E)GR\nk+mn′′(E+/planckover2pi1ω)GR\nk−m′n(E+/planckover2pi1ω)GA\nk+m′′n′(E).\n(B19)\nIn the next step we take the limit ω→0 (see Eq. (64),\nEq. (70), and Eq. (77)):\n−1\nVlim\nω→0Im∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ω=\n=η\n4/planckover2pi1Im/bracketleftBig\nY(3)+Y(4)−Y(5)−Y(6)/bracketrightBig\n,(B20)where we defined\nY(3)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(2)\nknmA(1)\nkn′m′Ck−n′′m′′×\n×∂Z(3a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(4)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(2)\nknmA(1)\nkn′m′Ck+n′′m′′×\n×∂Z(4a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(5)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(1)\nknmA(2)\nkn′m′Ck+n′′m′′×\n×∂Z(5a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(6)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(1)\nknmA(2)\nkn′m′Ck−n′′m′′×\n×∂Z(6a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\n(B21)\nwhich can be expressed as Y(3)=Y(3a)+Y(3b)and\nY(4)=Y(4a)+Y(4b), where\n2π/planckover2pi1Y(3a)=1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)\n+AkGR\nk−(E)Ck−GA\nk−(E)GA\nk−(E)BkGA\nk+(E)/bracketrightBigg\n=/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)×\n×Tr/bracketleftBig\nAkGR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)/bracketrightBig\n(B22)\nand\n2π/planckover2pi1Y(3b)=−1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGA\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)Ck−GR\nk−(E)BkGR\nk+(E)\n+AkGA\nk−(E)Ck−GA\nk−(E)GA\nk−(E)BkGA\nk+(E)/bracketrightBigg\n.(B23)22\nSimilarly,\n2π/planckover2pi1Y(4a)=1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)GA\nk+(E)\n−AkGR\nk−(E)GR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)\n−AkGR\nk−(E)BkGR\nk+(E)GR\nk+(E)Ck+GA\nk+(E)/bracketrightBigg\n=/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)×\n×Tr/bracketleftBig\nAkGR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)/bracketrightBig\n(B24)\nand\n2π/planckover2pi1Y(4b)=−1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGA\nk−(E)BkGA\nk+(E)Ck+GA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)BkGR\nk+(E)Ck+GR\nk+(E)\n+AkGR\nk−(E)BkGR\nk+(E)GR\nk+(E)Ck+GR\nk+(E)/bracketrightBigg\n.(B25)\nWe call Y(3a)andY(4a)Fermi surface terms and Y(3b)\nandY(4b)Fermi sea terms. Again Y(5)is obtained from\nY(3)by replacing k−byk+andk+byk−andY(6)is\nobtained from Y(4)in the same way.\nFinally, we take the limit q→0:\nΛ =−2\n/planckover2pi1VηIm lim\nq→0lim\nω→0∂\n∂ω∂\n∂qi∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n=1\n2/planckover2pi1lim\nq→0∂\n∂qiIm/bracketleftBig\nY(3)+Y(4)−Y(5)−Y(6)/bracketrightBig\n=1\n2/planckover2pi1Im/bracketleftBig\nX(3)+X(4)−X(5)−X(6)/bracketrightBig\n,\n(B26)\nwhere we defined\nX(j)=∂\n∂qi/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=0Y(j)(B27)\nforj= 3,4,5,6. Since Y(4)andY(6)are related by\nthe interchange of k−andk+it follows that X(6)=\n−X(4). Similarly, since Y(3)andY(5)arerelated by the\ninterchange of k−andk+it follows that X(5)=−X(3).\nConsequently, we need\nΛ =1\n/planckover2pi1Im/bracketleftBig\nX(3a)+X(3b)+X(4a)+X(4b)/bracketrightBig\n,(B28)\nwhere X(3a)andX(4a)are the Fermi surface terms and\nX(3b)andX(4b)are the Fermi sea terms. The Fermisurface terms are given by\nX(3a)=−1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nAkGR\nk(E)vkGR\nk(E)CkGA\nk(E)BkGA\nk(E)\n+AkGR\nk(E)CkGA\nk(E)vkGA\nk(E)BkGA\nk(E)\n−AkGR\nk(E)CkGA\nk(E)BkGA\nk(E)vkGA\nk(E)\n+AkGR\nk(E)∂Ck\n∂kGA\nk(E)BkGA\nk(E)/bracketrightBigg(B29)\nand\nX(4a)=−/bracketleftBig\nX(3a)/bracketrightBig∗\n. (B30)\nThe Fermi sea terms are given by\nX(3b)=−1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBigg\n−(ARvRRCRBR)+(AACAABAvA)\n−(ARRvRCRBR)−(ARRCRvRBR)\n+(ARRCRBRvR)−(AAvACABAA)\n−(AACAvABAA)+(AACABAvAA)\n+(AACABAAvA)−(AAvACAABA)\n−(AACAvAABA)−(AACAAvABA)\n−(ARR∂C\n∂kRBR)−(AA∂C\n∂kAABA)\n−(AA∂C\n∂kABAA)/bracketrightBigg(B31)\nand\nX(4b)=−/bracketleftBig\nX(3b)/bracketrightBig∗\n. (B32)\nIn Eq. (B31) we use the abbreviations R=GR\nk(E),A=\nGA\nk(E),A=Ak,B=Bk,C=Ck. It is important\nto note that Ck−andCk+depend on qthrough k−=\nk−q/2 andk+=k+q/2 . Theqderivative therefore\ngenerates the additional terms with ∂Ck/∂kin Eq. (B29)\nand Eq. (B31). In contrast, AkandBkdo not depend\nlinearly on q.\nEq. (B28) simplifies due to the relations Eq. (B30) and\nEq. (B32) as follows:\nΛ =2\n/planckover2pi1Im/bracketleftBig\nX(3a)+X(3b)/bracketrightBig\n. (B33)\nIn order to obtain the expression for the chiral con-\ntribution to the torque-torque correlation we choose the\noperators as follows:\nB→ Tk\nA→ −Ti\nC→ Tj\n∂C\n∂k= 0\nv→vl.(B34)23\nThis leads to Eq. (78), Eq. (79) and Eq. (80) of the main\ntext.\nIn order to obtain the expression for the chiral contri-\nbution to the CIT, we set\nB→ Tk\nA→ −Ti\nC→ −evj\n∂C\n∂k→ −e/planckover2pi1\nmδjl\nv→vl.(B35)\nThis leads to Eq. (66), Eq. (67) and Eq. (68).\nIn order to obtain the expression for the chiral contri-\nbution to the ICIT, we set\nB→ Tk\nA→ −evi\nC→ Tj\n∂C\n∂k→0\nv→vl.(B36)\nThis leads to Eq. (71), Eq. (72) and Eq. (73).\n∗Corresp. author: f.freimuth@fz-juelich.de\n[1] K. Nawaoka, S. 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The s pin absorption was characterized by the\nmagnetic damping of the heterostructure. We show that the ma gnetic damping of a Ni 81Fe19film\nis clearly enhanced by attaching Pt-oxide on the Ni 81Fe19film. The damping enhancement is\ndisappeared by inserting an ultrathin Cu layer between the N i81Fe19and Pt-oxide layers. These\nresults demonstrate an essential role of the direct contact between the Ni 81Fe19and Pt-oxide to\ninduce sizable interface spin-orbit coupling. Furthermor e, the spin-absorption parameter of the\nNi81Fe19/Pt-oxide interface is comparable to that of intensively st udied heterostructures with strong\nspin-orbit coupling, such as an oxide interface, topologic al insulators, metallic junctions with Rashba\nspin-orbit coupling. This result illustrates strong spin- orbit coupling at the ferromagnetic-metal/Pt-\noxide interface, providing an important piece of informati on for quantitative understanding the spin\nabsorption and spin-charge conversion at the ferromagneti c-metal/metallic-oxide interface.\nI. INTRODUCTION\nAn emerging direction in spintronics aims at discover-\ning novel phenomena and functionalities originatingfrom\nspin-orbit coupling (SOC)1. An important aspect of the\nSOC is the ability to convert between charge and spin\ncurrents. The charge-spin conversion results in the gen-\neration of spin-orbit torques in heterostructures with a\nferromagnetic layer, enabling manipulation of magneti-\nzation2–4. Recent studies have revealed that the oxida-\ntion of the heterostructure strongly influences the gen-\neration of the spin-orbit torques. The oxidation of the\nferromagnetic layer alters the spin-orbit torques, which\ncannot be attributed to the bulk spin Hall mechanism5–7.\nThe oxidation of a nonmagnetic layer in the heterostruc-\nture also offers a route to engineer the spin-orbit devices.\nDemasius et al. reported a significant enhancement of\nthe spin-torque generation by incorporating oxygen into\ntungsten, which is attributed to the interfacial effect8.\nThe spin-torque generation efficiency was found to be\nsignificantly enhanced by manipulating the oxidation of\nCu, enablingto turn the light metal into anefficient spin-\ntorque generator, comparable to Pt9. We also reported\nthat the oxidation of Pt turns the heavy metal into an\nelectrically insulating generatorof the spin-orbit torques,\nwhich enables the electrical switching of perpendicular\nmagnetization in a ferrimagnet sandwiched by insulating\noxides10. These studies have provided valuable insights\ninto the oxide spin-orbitronics and shown a promising\nway to develop energy-efficient spintronics devices based\non metal oxides.\nThe SOC in solids is responsible for the relaxation of\nspins, as well as the conversion between charge and spin\ncurrents. The spin relaxation due to the bulk SOC of\nmetals and semiconductors has been studied both ex-\nperimentally and theoretically11–14. The influence of the\nSOC at interfaces on spin-dependent transport has also\nbeen recognized in the study of giant magnetoresistance\n(GMR). The GMR in Cu/Pt multilayers in the current-perpendicular-to-plane geometry indicated that there\nmust be a significant spin-memory loss due to the SOC\nat the Cu/Pt interfaces15. The interface SOC also plays\na crucial role in recent experiments on spin pumping.\nThe spin pumping refers to the phenomenon in which\nprecessing magnetization emits a spin current to the sur-\nrounding nonmagnetic layers12. When the pumped spin\ncurrent is absorbed in the nonmagnetic layer due to the\nbulk SOC or the ferromagnetic/nonmagnetic interface\ndue to the interface SOC, the magnetization damping\nof the ferromagnetic layer is enhanced because the spin-\ncurrent absorption deprives the magnetization of the an-\ngularmomentum16. Althoughthedampingenhancement\ninduced by the spin pumping has been mainly associated\nwith the spin absorption in the bulk of the nonmagnetic\nlayer, recent experimental and theoretical studies have\ndemonstrated that the spin-current absorption at inter-\nfaces also provides a dominant contribution to the damp-\ning enhancement17. Since the absorption of a spin cur-\nrent at interfaces originates from the SOC, quantifying\nthe damping enhancement provides an important infor-\nmation for fundamental understanding of the spin-orbit\nphysics.\nIn this work, we investigate the absorption of a spin\ncurrent at a ferromagnetic-metal/Pt-oxide interface. We\nshow that the magnetic damping of a Ni 81Fe19(Py) film\nis clearly enhanced by attaching Pt-oxide, Pt(O), despite\nthe absence of the absorption of the spin current in the\nbulk of the Pt(O) layer. The damping enhancement dis-\nappears by inserting an ultrathin Cu layer between the\nPyand Pt(O)layers. This resultindicates that the direct\ncontact between the ferromagnetic metal and Pt oxide is\nessential to induce the sizable spin-current absorption, or\nthe interface SOC. We further show that the strength of\nthe damping enhancement observedfor the Py/Pt(O) bi-\nlayer is comparable with that reported for other systems\nwith strong SOC, such as two-dimensional electron gas\n(2DEG) at an oxide interface and topological insulators.2\nII. EXPERIMENTAL METHODS\nThree sets of samples, Au/SiO 2/Py,\nAu/SiO 2/Py/Pt(O) and Au/SiO 2/Py/Cu/Pt(O),\nwere deposited on thermally oxidized Si substrates\n(SiO2) by RF magnetron sputtering at room tempera-\nture. To avoid the oxidation of the Py or Cu layer, we\nfirst deposited the Pt(O) layer on the SiO 2substrate in\na mixed argon and oxygen atmosphere. After the Pt(O)\ndeposition, the chamber was evacuated to 1 ×10−6Pa,\nand then the Py or Cu layer was deposited on the top\nof the Pt(O) layer in a pure argon atmosphere. For\nthe Pt(O) deposition, the amount of oxygen gas in the\nmixture was fixed as 30%, in which the flow rates of\nargon and oxygen were set as 7.0 and 3.0 standard cubic\ncentimeters per minute (sccm), respectively. The SiO 2\nlayer was deposited from a SiO 2target in the pure argon\natmosphere. The film thickness was controlled by the\ndeposition time with a precalibrated deposition rate.\nWe measured the magnetic damping using current-\ninduced ferromagnetic resonance (FMR). For the fabri-\ncation of the devices used in the FMR experiment, the\nphotolithography and lift-off technique were used to pat-\nternthefilmsintoa10 µm×40µmrectangularshape. A\nblanket Pt(O) film on a 1 cm ×1 cm SiO 2substrate was\nfabricatedforthecompositionconfirmationbyx-raypho-\ntoelectron spectroscopy (XPS). We also fabricated Pt(O)\nsingle layer and SiO 2/Py/Pt(O) multilayer films with a\nHall bar shape to determine the resistivity of the Pt(O)\nand Py using the four-probe method. The resistivity of\nPt(O) (6.3 ×106µΩ cm) is much larger than that of Py\n(106µΩ cm). Because of the semi-insulating nature of\nthe Pt(O) layer, we neglect the injection of a spin cur-\nrent into the Pt(O) layer from the Py layer; only the\nPy/Pt(O) interface can absorb a spin current emitted\nfrom the Py layer. Transmission electron microscopy\n(TEM) was used to directly observe the interface and\nmultilayer structure of the SiO 2/Py/Pt(O) film. All the\nmeasurements were conducted at room temperature.\nIII. RESULTS AND DISCUSSION\nFigure 1(a) exhibits the XPS spectrum of the Pt(O)\nfilm. Previous XPS studies on Pt(O) show that bind-\ning energies of the Pt 4 f7/2peak for Pt, PtO and PtO 2\nare around 71.3, 72.3 and 74.0 eV, respectively18. Thus,\nthe Pt 4 f7/2peak at 72.3 eV in our Pt(O) film indi-\ncates the formation of PtO. By further fitting the XPS\nspectra, we confirm that the Pt(O) film is composed of\na dominant structure of PtO with a minor portion of\nPtO2. Figure 1(b) shows the cross-sectional TEM image\nof the SiO 2(4 nm)/Py(8 nm)/Pt(O)(10 nm) film. As can\nbe seen, continuous layer morphology with smooth and\ndistinct interfaces is formed in the multilayer film. Al-\nthough we deposited the Py layer on the Pt(O) layer to\navoid the oxidation of the Py, it might still be possible\nthat the Py layer is oxidized by the Pt(O) layer. There-\nFIG. 1. (a) The XPS spectrum of the Pt(O) film. The gray\ncurve is the experimental data, and the red fittingcurve is th e\nmerged PtO and PtO 2peaks. (b) The cross-sectional TEM\nimage of the SiO 2(4 nm)/Ni 81Fe19(8 nm)/Pt(O)(10 nm) film.\nfore, we measured the resistance of the Au/SiO 2/Py and\nAu/SiO 2/Py/Pt(O) samples used in the FMR experi-\nment. The resistance of both samples show the same\nvalue (60 Ω). Furthermore, as described in the follow-\ning section, the saturation magnetization for each device\nwas obtained by using Kittel formula (0.746 T and 0.753\nT for the Au/SiO 2/Py and Au/SiO 2/Py/Pt(O), respec-\ntively). The only 1% difference indicates that the minor\noxidationofthePylayerdue tothe presenceofthe Pt(O)\nlayer can be neglected.\nNext, we conduct the FMR experiment to investigate\nthe absorption and relaxation of spin currents induced\nby the spin pumping. Figure 2(a) shows a schematic\nof the experimental setup for the current-induced FMR.\nWe applied an RF current to the device, and an in-plane\nexternal magnetic field µ0Hwas swept with an angle\nof 45ofrom the longitudinal direction. The RF charge\ncurrentflowingintheAulayergeneratesanOerstedfield,\nwhich drives magnetization precession in the Py layer\nat the FMR condition. The magnetization precession\ninduces an oscillation of the resistance of the device due\nto the anisotropic magnetoresistance (AMR) of the Py\nlayer. We measured DC voltage generated by the mixing\nof the RF current and the oscillating resistance using a\nbias tee.\nFigures 2(b), 2(c) and 2(d) show the FMR spec-\ntra for the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and\nAu/SiO 2/Py/Cu/Pt(O) films, respectively. For the\nFMRmeasurement, asmallRFcurrentpower P= 5mW\nwas applied. Around P= 5 mW, the FMR linewidth is\nindependent of the RF power as shown in the inset to\nFig. 3(a). This confirms that the measured linewidth\nis unaffected by additional linewidth broadening due to\nnonlinear damping mechanisms and Joule heating. As\nshown in Fig. 2, clear FMR signals with low noise are\nobtained, allowing us to precisely fit the spectra and ex-\ntract the magnetization damping for the three samples.\nHere, the mixing voltage due to the FMR, Vmix, is ex-3\nFIG. 2. Schematic illustration of the experimental setup\nfor the current-induced FMR. Mis the magnetization in\nthe Py layer. The FMR spectra of the (b) Py(9 nm),\n(c) Py(9 nm)/Pt(O)(7.3 nm), and (d) Py(9 nm)/Cu(3.6\nnm)/Pt(O)(7.3 nm) films by changing the RF current fre-\nquency from 4 to 10 GHz. All the films are capped with 3\nnm-thick SiO 2and 10 nm-thick Au layers. The RF current\npower was set as 5 mW. The schematic illustrations of the\ncorresponding films are also shown.\npressed as\nVmix=Vsym(µ0∆H)2\n(µ0H−µ0HR)2+(µ0∆H)2\n+Vasyµ0∆H(µ0H−µ0HR)\n(µ0H−µ0HR)2+(µ0∆H)2,(1)\nwhereµ0∆Handµ0HRare the spectral width and res-\nonance field, respectively19.VsymandVasymare the\nmagnitudes of the symmetric and antisymmetric com-\nponents. The symmetric and antisymmetric components\narise from the spin-orbit torques and Oersted field. In\nthe devices used in the present study, the Oersted field\ncreated by the top Au layer dominates the RF effective\nfields acting on the magnetization in the Py layer [see\nalso Fig. 2(a)]. The large Oersted field enables the elec-\ntric measurement of the FMR even in the absence of the\nspin-orbit torques in the Au/SiO 2/Py film.\nThe damping constant αof the Py layer\nin the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and\nAu/SiO 2/Py/Cu/Pt(O) films can be quantified byfitting the RF current frequency fdependence of the\nFMR spectral width µ0∆Husing\nµ0∆H=µ0∆Hext+2πα\nγf, (2)\nwhere ∆ Hextandγare the inhomogeneous linewidth\nbroadening of the extrinsic contribution and gyromag-\nnetic ratio, respectively19,20. Figure 3(a) shows the f\ndependence of the FMR linewidth µ0∆H, determined by\nfitting the spectra shown in Fig. 2 using Eq. (1). As\nshown in Fig. 3(a), the frequency dependence of the\nlinewidth is well fitted by Eq. (2). Importantly, the slope\nof thefdependence of µ0∆Hfor the Py/Pt(O) film is\nclearly larger than that for the Py and Py/Cu/Pt(O)\nfilms. This indicates larger magnetic damping in the\nPy/Pt(O) film. By using Eq. (2), we determined the\ndamping constant αas 0.0126, 0.0169 and 0.0124 for the\nPy, Py/Pt(O) and Py/Cu/Pt(O)films, respectively. The\ndifference in αbetween the Py and Py/Cu/Pt(O)films is\nvanishingly small, which is within an experimental error.\nIn contrast, the damping of the Py/Pt(O) film is clearly\nlarger than that of the other films, indicating an essen-\ntial role of the Py/Pt(O) interface on the magnetization\ndamping.\nThe larger magnetic damping in the Py/Pt(O) film\ndemonstrates an important role of the direct contact be-\ntween the Py and Pt(O) layers in the spin-current ab-\nsorption. If the bottom layers influence the magnetic\nproperties of the Py layer, the difference in the mag-\nnetic properties can also result in the different magnetic\ndamping in the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and\nAu/SiO 2/Py/Cu/Pt(O) films. However, we have con-\nfirmed that the difference in the magnetic damping is not\ncaused by different magnetic properties of the Py layer.\nIn Fig. 3(b), we plot the RF current frequency fdepen-\ndence of the resonance field µ0HR. As can be seen, the\nfdependence of µ0HRis almost identical for the differ-\nent samples, indicating the minor change of the magnetic\npropertiesofthe Pylayerdueto the different bottomlay-\ners. In fact, by fitting the experimental data using Kittel\nformula21, 2πf/γ=/radicalbig\nµ0HR(µ0HR+µ0Ms), the satura-\ntion magnetizationis obtainedto be µ0Ms= 0.746, 0.753\nand 0.777 T for the Py, Py/Pt(O) and Py/Cu/Pt(O)\nfilms, respectively. The minor difference ( <5%) in the\nsaturation magnetization indicates that the larger damp-\ning of the Py/Pt(O) film cannot be attributed to possi-\nble different magnetic properties of the Py layer. Thus,\nthe larger magnetic damping of the Py/Pt(O) film can\nonly be attributed to the efficient absorption of the spin\ncurrent at the interface. Notable is that the additional\ndamping due to the spin-current absorption disappears\nby inserting the 3.6 nm-thick Cu layer between the Py\nand Pt(O) layers. Here, the thickness of the Cu layer is\nmuchthinnerthanitsspin-diffusionlength( ∼500nm)22,\nallowing us to neglect the relaxation of the spin current\nin the Cu layer. This indicates that the directcontactbe-\ntween the Py and Pt(O) layersis essential for the absorp-\ntion of the spin current at the interface, or the interface\nSOC.4 0∆H (mT) \nµ\n3 6 9 12 0246\nf (GHz) (a) (b)\n8\n610 \n0\n 0HR (mT) µf (GHz) \n4\n100150 50 Py/Cu/Pt(O) Py/Pt(O)Py \nPy/Cu/Pt(O) Py/Pt(O) Py \n036\n10 010 110 2\nP (mW) 0∆H (mT) \nµ\nFIG. 3. (a) The RF current frequency fdependence of\nthe half-width at half-maximum µ0∆Hfor the Py, Py/Pt(O)\nand Py/Cu/Pt(O) samples. The solid lines are the linear fit\nto the experimental data. The inset shows RF current power\nPdependence of µ0∆Hfor the Py film at f= 7 GHz. (b)\nThe RF currentfrequency fdependenceof the resonance field\nµ0HRfor the three samples. The solid curves are the fitting\nresult using the Kittel formula.\nTABLE I. The summarized spin-absorption parameter Γ 0η\nin different material systems. In order to directly compare\nthis work with previous works, we used International Sys-\ntem of Units. We used the magnetic permeability in vac-\nuumµ0= 4π×10−7H/m. ∆ αand Γ 0ηfor the Sn 0.02-\nBi1.08Sb0.9Te2S/Ni81Fe19is the values at T <100 K.\nHeterostructure ∆ αΓ0η[1/m2] Ref.\nBi/Ag/Ni 80Fe20 0.015 8.7 ×1018[25]\nBi2O3/Cu/Ni 80Fe20 0.0045 1.5 ×1018[26]\nSrTiO 3/LaAlO 3/Ni81Fe19 0.0013 2.3 ×1018[27]\nPt(O)/Ni 81Fe19 0.0044 2.3 ×1018This work\nα-Sn/Ag/Fe 0.022 1.2 ×1019[28]\nSn0.02-Bi1.08Sb0.9Te2S/Ni81Fe190.013 1.4 ×1019[29]\nBi2Se3/Ni81Fe19 0.0013 2.5 ×1018[30]\nTo quantitatively discuss the spin absorption at the\nPy/Pt(O)interfaceand comparewith othermaterialsys-\ntems, we calculate the spin absorption parameters. In a\nmodel of the spin pumping where the interface SOC is\ntaken into account, the additional damping constant is\nexpressed as23\n∆α=gµBΓ0\nµ0Msd/parenleftbigg1+6ηξ\n1+ξ+η\n2(1+ξ)2/parenrightbigg\n.(3)\nHere,g= 2.11 is the gfactor24,µB= 9.27×10−24Am2\nis the Bohr magneton, dis the thickness of the Py layer,\nandΓ0isthemixingconductanceattheinterface. ξisthe\nback flow factor; no backflow refers to ξ= 0 and ξ=∞indicates that the entire spin current pumped into the\nbulk flows back across the interface. ηis the parameter\nthat characterizes the interface SOC. For the Py/Pt(O)\nfilm,ξcan be approximated to be ∞because of the spin\npumping into the bulk of the semi-insulating Pt(O) layer\ncan be neglected. Thus, Eq. (3) can be simplified as\n∆α=6gµBΓ0η\nµ0Msd. (4)\nHere, 6Γ 0ηcorresponds to the effective spin mixing\nconductance g↑↓\neff. From the enhancement of magnetic\ndamping ∆ α, we obtain Γ 0η= 2.3×1018m−2for the\nPy/Pt(O) film. We further compared this value with\nΓ0ηfor other systems where efficient interface charge-\nspin conversion has been reported. As shown in Table\nI, the spin-absorption parameter Γ 0ηof the Py/Pt(O)\nfilm is comparable with that of the heterostructures with\nthe strong SOC, such as the 2DEG at an oxide interface,\ntopological insulators, as well as metal/oxide and metal-\nlic junctions with the Rashba SOC. This result therefore\ndemonstrates the strong SOC at the Py/Pt(O) interface.\nIV. CONCLUSIONS\nIn summary, we have investigated the spin-current ab-\nsorption and relaxation at the ferromagnetic-metal/Pt-\noxide interface. By measuring the magnetic damping for\nthe Py, Py/Pt(O)and Py/Cu/Pt(O)structures, we show\nthat the direct contact between Py and Pt(O) is essential\nfor the absorption of the spin current, or the sizable in-\nterface SOC. Furthermore, we found that the strength of\nthe spin-absorption parameter at the Py/Pt(O) interface\nis comparable to the value for intensively studied het-\nerostructureswithstrongSOC,suchas2DEGatanoxide\ninterface, topological insulators, metallic junction with\nRashba SOC. The comparable value with these material\nsystems illustrates the strong SOC at the ferromagnetic-\nmetal/Pt-oxide interface. This indicates that the oxida-\ntion of heavy metals provides a novel approach for the\ndevelopment of the energy-efficient spintronics devices\nbased the SOC.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI Grant\nNumbers 26220604, 26103004, the Asahi Glass Founda-\ntion, JGC-SScholarshipFoundation, andSpintronicsRe-\nsearch Network of Japan (Spin-RNJ). H.A. is JSPS In-\nternational Research Fellow (No. P17066) and acknowl-\nedges the support from the JSPS Fellowship (Grant No.\n17F17066).\n∗ando@appi.keio.ac.jp1A. Soumyanarayanan, N. Reyren, A. Fert, and\nC. 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Son, K. Banerjee, N. Koirala, M. Brahlek,\nS. Oh, and H. Yang, Phys. Rev. B 90, 094403 (2014)." }, { "title": "1802.06874v1.Landau_Damping_in_a_strong_magnetic_field__Dissociation_of_Quarkonia.pdf", "content": "arXiv:1802.06874v1 [hep-ph] 16 Feb 2018Landau Damping in a strong magnetic field: Dissociation of Qu arkonia\nMujeeb Hasan†1, Binoy Krishna Patra†2, Bhaswar Chatterjee†3, and Partha Bagchi∗4\n†Department of Physics, Indian Institute of Technology Roorke e, Roorkee 247 667, India\n∗Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata 700 064, India\nAbstract\nIn this article we have investigated the effects of strong magnetic fi eld on the properties of quarko-\nnia immersed in a thermal medium of quarks and gluons and studied its q uasi-free dissociation\ndue to the Landau-damping. Thermalizing the Schwinger propagato r in the lowest Landau levels\nfor quarks and the Feynman propagator for gluons in real-time for malism, we have calculated\nthe resummed retarded and symmetric propagators, which in turn give the real and imaginary\ncomponents of dielectric permittivity, respectively. Thus the effec t of a strongly magnetized hot\nQCD medium have been encrypted into the real and imaginary parts o f heavy quark interaction\nin medium, respectively. The magnetic field affects the large-distanc e interaction more than the\nshort-distance interaction, as a result, the real part of potent ial becomes more attractive and the\nmagnitude of imaginary part too becomes larger, compared to the t hermal medium in absence\nof strong magnetic field. As a consequence the average size of J/ψ’s andψ′’s are increased but\nχc’s get shrunk. Similarly the magnetic field affects the binding of J/ψ’s andχc’s discriminately,\ni.e.it decreases the binding of J/ψand increases for χc. However, the further increase in mag-\nnetic field results in the decrease of binding energies. On contrary t he magnetic field increases\nthe width of the resonances, unless the temperature is sufficiently high. We have finally studied\nhow the presence of magnetic field affects the dissolution of quarko nia in a thermal medium due\nto the Landau damping, where the dissociation temperatures are f ound to increase compared to\nthe thermal medium in absence of magnetic field. However, further increase of magnetic field\ndecreases the dissociation temperatures. For example ,J/ψ’s andχc’s are dissociated at higher\n1hasan.dph2014@iitr.ac.in\n2binoyfph@iitr.ac.in\n3bhaswar.mph2016@iitr.ac.in\n4p.bagchi@vecc.gov.in\n1temperatures at 2 Tcand 1.1Tcat a magnetic field eB≈6 and 4m2\nπ, respectively, compared to\nthe values 1.60 Tcand 0.8Tcin the absence of magnetic field, respectively.\nPACS: 12.39.-x,11.10.St,12.38.Mh,12.39.Pn 12.75.N, 12.38.G\nKeywords : Thermal QCD, Retarded Propagator, Symmetric Propagator, D ielectric Permittiv-\nity, heavy quark potential\n1 Introduction\nQuantumChromodynamics (QCD)predictsthatatsufficiently highte mperatures and/ordensities\nthequarksandgluonconfinedinside thehadronsareliberatedintoa mediumofquarksandgluons,\nknown as Quark-gluon Plasma (QGP). Over the decades a large numb er of activities have been\ndirected towards the production and identification of this new stat e of matter theoretically and\nexperimentally inultra relativistic heavy-ion collisions (URHIC) with the increasing center ofmass\nenergies (√s) at BNL AGS, CERN SPS, BNL RHIC, and CERN LHC experiments. Howe ver, for\nthe non-central events in the above URHICs, a very strong magn etic field is generated at the very\nearlystages ofthecollisions dueto very highrelative velocities ofthe spectator quarks with respect\nto the fireball [ 1,2]. Depending on the centralities of the collisions, the strength of the magnetic\nfields may vary from m2\nπ(∼1018Gauss) at RHIC to 10 m2\nπat LHC. However, at extreme cases\nthe magnetic field may even reach 50 m2\nπat LHC and even much larger values ∼105m2\nπin the\nearly universe during electroweak phase transition [ 3]. Naive classical estimates of the lifetime of\nthese magnetic fields show that it only exists for a small fraction of t he lifetime of QGP. However\ndepending on the transport properties of the plasma the magnetic field may remain strong during\nthe lifetime of QGP [ 4].\nOne particularly suited probe to infer the properties of nuclear mat ter under extreme condi-\ntions is the heavy-quarkonia. The heavy quark and antiquark ( Q¯Q) pairs are produced in URHICs\non a very short time-scale tprod∼1/2mQ. Subsequently they develop into a physical resonance\nover a formation time tform∼1/Ebind(Ebindis the binding energy of the state). They traverse\nthe plasma and later the hadronic matter before decaying into the d ilepton, which is eventually\ndetected. This long journey is fairly ‘hazardous’ for the quarkoniu m because even before the for-\nmation of resonances, the cold nuclear matter may dissociate the n ascentQ¯Qpairs. However, even\n2after the resonances are formed, they react to the presence o f a thermal medium with the smaller\nbinding energies. Since the mass of the charm or bottom quarks is lar ger than the temperature\nof QGP created in current heavy-ion collisions, viz.TLHC≤0.6 GeV heavy quarkonium bound\nstates may survive while traversing the collision center. In that pro cess they accumulate informa-\ntion about their environment which is imprinted on their depleted prod uction yields, which may\nopen up a direct window on the vital properties of the deconfined me dium,namelythe temper-\nature and the presence of strong magnetic fields. Therefore the goal of the present work is to\nunderstand theoretically the properties of heavy quarkonium und er realistic conditions existing in\nan environment at high temperatures in the presence of strong ma gnetic fields.\nOur understanding of heavy quarkonium has made a significant step forward with the compu-\ntations of effective field theories (EFT) from the underlying theory - QCD,such asnon-relativistic\nQCD (NRQCD) and potential NRQCD, which are synthesized by separ ating the intrinsic scales\nof heavy quark bound states ( e.g.mass, velocity, binding energy) as well as the additional scales\nof thermal medium ( e.g.T,gT,g2T) in weak-coupling regime, in overall comparison with Λ QCD.\nHowever, the separation of scales in EFT is not always evident in realis tic conditions achieved\nat URHICs, so one needs the first-principle lattice QCD simulations to study the quarkonia in\na medium even without the potential models rather by the spectral functions in terms of the\nEuclidean meson correlation functions [ 5]. However the reconstruction of the spectral functions\nturns out to be very difficult because the temporal extent decrea ses at large temperature. Thereby\nthe studies of quarkonia using the potential models at finite temper ature complement the lattice\nstudies.\nFor a long time phenomenological potential models had been deployed in the literature, which\nwere not based on the systematic derivations from QCD. The color s inglet free energies extracted\nfrom the correlation function of Polyakov loops, which is computed f rom the first-principle lattice\nQCD simulations, has been commonly advocated as an appropriate po tential to study the quarko-\nnia in vacuum as well as in medium. The perturbative computations of t he potential at high\ntemperatures show that the Q¯Qpotential becomes complex, where the real part gets screened d ue\nto the presence of deconfined color charges [ 6] and the imaginary-part [ 7,8,9,10] attributes the\nthermal width of the resonance. The physics of quarkonium dissoc iation in a medium has been\nrefined over the last decade, where the resonances were initially th ought to be dissociated when\nthe screening becomes sufficiently strong, the potential becomes too weak to hold Q¯Qtogether.\nNowadays the dissociation is thought to be mainly due to the broaden ing of the width of reso-\n3nances inamedium. The broadeningarises mainlyeither bytheinelastic partonscattering process\nmediated by the spacelike gluons, known as Landau damping [ 9] or due to the gluo-dissociation\nprocess in which the color singlet state undergoes into a color octet state by a hard thermal\ngluon [11]. The later processes becomes dominant when the temperature of medium is smaller\nthan the binding energy of the particular resonance. Recently one of us estimated the imaginary\ncomponent of the potential perturbatively in resummed thermal fi eld theory, where the inclusion\nof a confining string term makes the (magnitude) imaginary compone nt smaller [ 12] , compared\nto the medium modification of the perturbative term alone [ 13]. Even in strong coupling limit the\npotential extracted through AdS/CFT correspondence develop s an imaginary component beyond\na critical separation of Q¯Qpair [14,15]. In a similar calculation, generalized Gauss law relates the\nnumerically simulated values of the potential to the in-medium permitt ivity of the QCD medium\nconventionally parameterized by the so called Debye mass pair [ 16].\nThe discussions referred above were limited for the simplest possible setting in heavy-ion phe-\nnomenology for fully central collisions but most events occur with a fi nite impact parameter where\nan extremely large magnetic fields may be produced. Recently some o f us have explored the effects\nof strong magnetic field on the properties of heavy-quarkonium by computing the real part of Q¯Q\npotential [ 17] as well as on the QCD thermodynamics [ 18]. However, such purely real potential\nalone cannot capture the physics relevant for in-medium modificatio n of quarkonium states so we\naim to estimate the imaginary component of the potential perturba tively in the real-time formal-\nism and investigate how the properties of quarkonia in a thermal QCD medium get affected by\nthe presence of strong magnetic field. Recently there was an atte mpt to derive the complex heavy\nquark potential due to an external strong magnetic field in a gener alised Gauss law [ 19], where\nthe imaginary part of in-medium permittivity, ǫ(k) is heuristically obtained by simply replacing\nthe Debye mass in the absence of magnetic field by the same in the pre sence of strong magnetic\nfield. In our calculation, we aim to calculate meticulously the imaginary p art of retarded gluon\nself-energy due to quark loop and gluon loop separately, similar to th e calculation of the real part.\nIt is found that the imaginary part due to quark loop is proportional to the square of the quark\nmasses and does not depend on the temperature directly (apart f rom the Debye mass). As a\nresult, the momentum dependence will be completely different from t heir calculation [ 19], which\ncan be understood by the dimensional reduction caused by the effe ct of magnetic field to quark\ndynamics, notthe gluon dynamics.\nOur work proceeds as follows. First we calculate the resummed reta rded/advanced and sym-\n4metric gluon propagator by calculating the real and imaginary part o f retarded/advanced gluon\nself-energies for a thermal QCD medium in the presence of strong m agnetic field in subsections\n2.1 and 2.2, respectively. Next the real and imaginary component of dielectric permittivities are\nobtained by taking the static limit of the resummed retarded and sym metric propagators, whose\ninverse Fourier transform gives the real and imaginary parts of he avy quark potential in the coor-\ndinate space in subsection 3.1 and 3.2, respectively. The real part o f potential is thereafter solved\nnumerically by the Schr¨ odinger equation to obtain both the energy eigenvalues and eigenfunc-\ntions to calculate the size and binding energies of quarkonia in subsec tion 4.1. In Section 4.2 we\ndeals with the imaginary component in a time-independent perturbat ion theory to estimate the\nmedium-induced thermal width of the resonances, which facilitates to study the dissociation due\nto the Landau damping. Finally we will conclude in Section 5.\n2 The resummed gluon propagator in strong magnetic\nfield\nIn Keldysh representation of real-time formalism, the retarded (R ), advanced (A) and symmetric\n(S) propagators are written as the linear combination of the compo nents of matrix propagator:\nD0\nR=D0\n11−D0\n12, D0\nA=D0\n11−D0\n21, D0\nS=D0\n11+D0\n22. (1)\nSimilarrepresentationforself-energiescanalsobeworkedoutinte rmsofcomponentsofself-energy\nmatrix through the retarded (Π R), advanced (Π A) and symmetric (Π S) self energies.\nThe resummation for the above propagators is done by the Dyson- Schwinger equation. For\nthe static potential, we need only the temporal (longitudinal) compo nent of the propagator and\nits evaluation is easier in the Coulomb gauge so the temporal compone nt of retarded/advanced\npropagator is resummed as\nDL\nR,A=DL(0)\nR,A+DL(0)\nR,AΠL\nR,ADL\nR,A, (2)\nwhereas the resummation for symmetric propagator is done as\nDL\nS=DL(0)\nS+DL(0)\nRΠL\nRDL\nS(0)+D0\nSΠADA+D0\nRΠSDA. (3)\nThus the resummed retarded (advanced) and symmetric propaga tors can be expressed explicitly\n5by the self-energies as\nDL\nR,A(k) =1\nk2−ReΠL\nR(k)∓iImΠL\nR(k), (4)\nDL\nS(k) = (1+2 nB(k0)) sgn(k0)/parenleftbig\nDL\nR(k)−DL\nA(k)/parenrightbig\n, (5)\nwhere the factor, (1 + 2 nB(k0))sgn(k0) and the difference,/parenleftbig\nDL\nR(k)−DL\nA(k)/parenrightbig\ncan be obtained as\n[13,20]\n(1+2nB(k0))sgn(k0) =2T\nk0, (6)\n/parenleftbig\nDL\nR(k)−DL\nA(k)/parenrightbig\n=2iImΠL\nR(k)/bracketleftbig\nk2−ReΠL\nR(k)/bracketrightbig2+/bracketleftbig\nImΠL\nR(k)/bracketrightbig2, (7)\nwith the following identities\nReΠL\nR(k) = ReΠL\nA(k), (8)\nImΠL\nR(k) =−ImΠL\nA(k). (9)\nIt is thus learnt that only the real and imaginary parts of the longitu dinal component of retarded\nself-energy suffice to calculate the resummed retarded, advance d and symmetric propagator in a\nstrongly magnetized hot QCD medium.\nForcalculatingtheretardedself-energy, weneedtoevaluatethe matrixpropagatorinathermal\nmedium in the presence of strong magnetic field for quarks and gluon s. The magnetic field affects\nonly the quark propagator viathe projection operator and its dispersion relation. So we will now\nrevisit the vacuum quark propagator in a strong magnetic field and t hen thermalize it in a real-\ntime formalism, which in turn computes the gluon self-energy for the quark-loop diagram. We\nstart with the vacuum quark propagator in coordinate-space, us ing the Schwinger’s proper-time\nmethod [ 21]\nS(y,y′) =φ(y,y′)/integraldisplayd4p\n(2π)4e−ip(y−y′)S(p), (10)\nwhere the phase factor, φ(y,y′) defined by\nφ(y,y′) =ei|qf|/integraltexty\ny′Aµ(ζ)dζµ. (11)\nis a gauge-dependent quantity, which is responsible for breaking of translational invariance. For\na single fermion line, it is possible to gauge away the phase factor by an appropriate gauge\n6transformation for a symmetric gauge in a magnetic field directed alo ng thezaxis. Thus one can\nexpress the propagator in the momentum-space [ 22,23] as an integral over the proper-time ( s)\niS(p) =/integraldisplay∞\n01\neBds\ncos(s)e−is[m2\nf−p2\n/bardbl+tan(s)\nsp2\n⊥]/bracketleftbigg\n(cos(s)+γ1γ2sin(s))(mf+γ·p/bardbl)−γ·p⊥\ncos(s)/bracketrightbigg\n,(12)\nwhich can be expressed more conveniently in a discrete form by the a ssociated Laguerre polyno-\nmials\niSn(p) =/summationdisplay\nn−idn(α)D+d′\nn(α)¯D\np2\nL+2neB+iγ·p⊥\np2\n⊥, (13)\nwith the notations in Ref[ 23].\nIn a strong magnetic field (SMF) limit both parallel and perpendicular c omponents of quark\nmomenta are smaller than the magnetic field (i.e. p2\n/bardbl,p2\n⊥≪ |qfB| ≫T2) so the transitions to\nthe higher Landau levels ( n≥1) are suppressed. Therefore only the lowest Landau levels (LLL)\nare populated, hence the vacuum propagator for quarks in the mo mentum-space for LLL ( n= 0)\nbecomes\niS0(p) =(1+γ0γ3γ5)(γ0p0−γ3pz+mf)\np2\n/bardbl−m2\nf+iǫe−p2\n⊥\n|qfB|, (14)\nwheremfandqfarethemass andelectric chargeof fthflavour, respectively. However, inreal-time\nformalism, the propagator in a thermal medium acquires a (2 ×2) matrix structure [ 20]\nS(p) =/parenleftbiggS0(p)+nF(p0)(S∗\n0(p)−S0(p))/radicalbig\nnF(p0)(1−nF(p0))(S∗\n0(p)−S0(p))\n−/radicalbig\nnF(p0)(1−nF(p0)(S∗\n0(p)−S0(p))−S∗\n0(p)+nF(p0)(S∗\n0(p)−S0(p))/parenrightbigg\n,(15)\nwherenF(p0) is the quark distribution function. Thus, the 11- and 12-compone nts can be read off\niS11(p) =/bracketleftBigg\n1\np2\n/bardbl−m2\nf+iǫ+2πinF(p0)δ(p2\n/bardbl−m2\nf)/bracketrightBigg\n(1+γ0γ3γ5)(γ0p0−γ3pz+mf)e−p2\n⊥\n|qfB|,(16)\nS12(p) =−2π/radicalbig\nnF(p0)(1−nF(p0))δ(p2\n/bardbl−m2\nf)(1+γ0γ3γ5)(γ0p0−γ3pz+mf)e−p2\n⊥\n|qfB|.(17)\nHowever, for gluons, the form of the vacuum propagator remains unaffected by the magnetic\nfield,i.e.\nDµν\n0(p) =igµν\np2+iǫ. (18)\nSimilar to thermalization of quark propagator, the gluon propagato r at finite temperature also\ntakes the matrix structure in the real-time formalism [ 20] in terms of the gluon distribution\nfunction,nB(p0)\nDµν(p) =/parenleftbigg\nDµν\n0(p)+nB(p0)(D∗µν\n0(p)+Dµν\n0(p))/radicalbig\nnB(p0)(1+nB(p0))(D∗µν\n0(p)+Dµν\n0(p))/radicalbig\nnB(p0)(1+nB(p0))(D∗µν\n0(p)+Dµν\n0(p))D∗µν\n0(p)+nB(p0)(D∗µν\n0(p)+Dµν\n0(p))/parenrightbigg\n.\n(19)\n7The above matrices ( 15,19) will be used to calculate the retarded/advanced and symmetric se lf\nenergies due to quark loop and gluon loops, respectively in the next s ection.\n2.1 Real part of retarded gluon self energy in real-time form alism\nIn Keldysh representation of real-time formaism, the evaluation of the real part of retarded gluon\nself-energy requires only the real part of 11-component of self- energy matrix\nReΠR(k) = ReΠ 11(k). (20)\nThere are four Feynman diagrams, e.g.tadpole, gluon-loop, ghost-loop and quark-loop, which\ncontribute to the gluon self-energy. Since only the quark-loop diag ram is affected by the presence\nof the magnetic field in the thermal medium so we first calculate the qu ark-loop in SMF limit and\nthen obtain the thermal contributions due to the remaining gluon-lo op diagrams.\nUsing the matrix propagator ( 15) for quarks in real-time formalism, the 11-component of the\ngluon self-energy matrix for the quark-loop (omitting the prefix 11 ) can be written as\nΠµν(k) =ig2\n2/summationdisplay\nf/integraldisplayd2p⊥d2p/bardbl\n(2π)4Tr/bracketleftbig\nγµ(1+γ0γ3γ5)/parenleftbig\nγ0p0−γ3pz+mf/parenrightbig\nγν(1+γ0γ3γ5)/parenleftbig\nγ0q0−γ3qz+mf/parenrightbig/bracketrightbig\n×/bracketleftBigg\n1\np2\n/bardbl−m2\nf+iǫ+2πinF(p0)δ/parenleftBig\np2\n/bardbl−m2\nf/parenrightBig/bracketrightBigg\ne−p2\n⊥\n|qfB|\n×/bracketleftBigg\n1\nq2\n/bardbl−m2\nf+iǫ+2πinF(q0)δ/parenleftBig\nq2\n/bardbl−m2\nf/parenrightBig/bracketrightBigg\ne−q2\n⊥\n|qfB|, (21)\nwherethefactor1 /2arisesduetothetraceincolorspaceandthemomentum, ( p+k)isreplacedby\nq. Here we use the one-loop running QCD coupling ( g=/radicalbig\n4παs(eB)), which, in strong magnetic\nfield limit, runs exclusively with the magnetic field because the most dom inant scale for quarks\nis no more the temperature of the medium rather the scale associat ed with the strong magnetic\nfield. This is exactly Ferrar et. al has recently explored the depend ence of running coupling on\nthe magnetic field only by decomposing the momentum into parallel and perpendicular to the\nmagnetic field [ 26].\nSince the momentum integration is factorizable into parallel and perp endicular components\nwith respect to the direction of magnetic field therefore the compo nent, which depends only the\ntransverse momentum, is given by\nΠ⊥(k⊥) =π|qfB|\n2e−k2\n⊥\n2|qfB|, (22)\n8and the self energy, which depends only the parallel component of t he momentum, Πµν(k/bardbl) is\ndecomposed into vacuum and medium contributions\nΠµν\n/bardbl(k/bardbl)≡Πµν\nvacuum(k/bardbl)+Πµν\nn(k/bardbl)+Πµν\nn2(k/bardbl). (23)\nThe vacuum and medium contributions having the linear and quadratic dependence on the distri-\nbution function, respectively are given by\nΠµν\nvacuum(k/bardbl) =ig2\n2(2π)4/integraldisplay\ndp0dpzLµν/bracketleftBigg\n1\n(p2\n/bardbl−m2\nf+iǫ)(q2\n/bardbl−m2\nf+iǫ)/bracketrightBigg\n, (24)\nΠµν\nn(k/bardbl) =−g2\n2(2π)3/integraldisplay\ndp0dpzLµν/bracketleftBigg\nnF(p0)δ(p2\n/bardbl−m2\nf)\n(q2\n/bardbl−m2\nf+iǫ)+nF(q0)δ(q2\n/bardbl−m2\nf)\n(p2\n/bardbl−m2\nf+iǫ)/bracketrightBigg\n,(25)\nΠµν\nn2(k/bardbl) =−ig2\n2(2π)2/integraldisplay\ndp0dpzLµν/bracketleftbig\nnF(p0)nF(q0)δ(p2\n/bardbl−m2\nf)δ(q2\n/bardbl−m2\nf)/bracketrightbig\n, (26)\nwhere the trace over γ-matrices,Lµνis\nLµν= 8/bracketleftBig\npµ\n/bardbl·qν\n/bardbl+pν\n/bardbl·qµ\n/bardbl−gµν\n/bardbl/parenleftBig\npµ\n/bardbl·q/bardblµ−m2\nf/parenrightBig/bracketrightBig\n. (27)\nNow we calculate the real part of the vacuum contribution ( 24) as [17]\nRe Πµν\nvacuum(k/bardbl) =/parenleftBigg\ngµν\n/bardbl−kµ\n/bardblkν\n/bardbl\nk2\n/bardbl/parenrightBigg\nΠ(k2\n/bardbl), (28)\nwhere the form factor, Π( k2\n/bardbl) is given by\nΠ(k2\n/bardbl) =g2\n2π3/summationdisplay\nf\n2m2\nf\nk2\n/bardbl/parenleftBigg\n1−4m2\nf\nk2\n/bardbl/parenrightBigg−1/2\nln\n\n/parenleftBig\n1−4m2\nf\nk2\n/bardbl/parenrightBig1/2\n+1\n/parenleftBig\n1−4m2\nf\nk2\n/bardbl/parenrightBig1/2\n−1\n\n+1\n. (29)\nThus multiplying the transverse momentum dependent part ( 22) to the parallel momentum de-\npendent component ( 28) and taking the static limit ( k0= 0,kx,ky,kz→0), the longitudinal\ncomponent of the vacuum part in the limit of massless flavours becom es\nRe ΠL\nvacuum=−g2\n4π2/summationdisplay\nf|qfB|, (30)\nwhereas for the physical quark masses, it vanishes\nRe ΠL\nvacuum= 0. (31)\n9Next the real part of the thermal contribution having linear depen dence on the distribution\nfunction in static limit for the massless quarks can be obtained [ 17] as\nRe ΠL\nn=g2\n4π2/summationdisplay\nf|qfB|−g2\n8π2/summationdisplay\nf|qfB|, (32)\nand for the physical quark masses, it becomes\nRe ΠL\nn=−g2\n4π2T/summationdisplay\nf|qfB|/integraldisplay∞\n0dpzeβ√\np2z+m2\nf\n/parenleftBig\n1+eβ√\np2z+m2\nf/parenrightBig2. (33)\nThe medium contribution having quadratic dependence on the distrib ution function ( 26) does not\nyield any contribution to the real-part, i.e.\nRe Πµν\nn2(k/bardbl) = 0. (34)\nThus the vacuum ( 30) and medium contributions ( 32) are combined together to give the\nlongitudinal component due to the quark-loop in the limit of massless q uarks\nRe ΠL\nquark loop=−g2\n8π2/summationdisplay\nf|qfB|, (35)\nwhich depends on the magnetic field only in the SMF limit ( eB >> T2) and is independent of\ntemperature even in the thermal medium. The above form have also been calculated through the\ndifferent approaches [ 24,25,17].\nSimilarly for the physical quark masses, the vacuum ( 31) and medium contributions ( 33) due\nto the quark loop are added to give the longitudinal component in the static limit\nRe ΠL\nquark loop=−g2\n4π2T/summationdisplay\nf|qfB|/integraldisplay∞\n0dpzeβ√\np2z+m2\nf\n/parenleftBig\n1+eβ√\np2z+m2\nf/parenrightBig2, (36)\nwhich now depends on both magnetic field and temperature. Howeve r it becomes independent of\ntemperature beyond a certain temperature [ 17].\nWe will now calculate the retarded/advanced gluon self-energy ten sor due to gluon loops using\nthe 11-component of matrix propagator for gluons ( 19) in a thermal medium. The longitudinal\ncomponent of the same [ 12] is obtained by the HTL perturbation theory as\nΠL\ngluon loops(k) =g′2T2/parenleftbiggk0\n2klnk0+k±iǫ\nk0−k±iǫ−1/parenrightbigg\n, (37)\n10with the prescriptions + iǫ(−iǫ) for the retarded and advanced self-energies, respectively. He re\nwe takeg′=/radicalbig\n4πα′s(T) as the one-loop strong running coupling, where the dominant scale for\ngluonic degrees of freedom is the temperature so the renormalizat ion scale is taken as 2 πT.\nThus the real part of longitudinal component due to the gluon-loop s in the static limit reduces\nto [12]\nRe ΠL\ngluon loops =−g′2T2(38)\nThus the Debye mass is obtained from static limit of quark-loop ( 35) and gluon-loops ( 38)\ncontributions for the massless quarks\nm2\nD=g′2T2+g2\n8π2/summationdisplay\nf|qfB|, (39)\nTherefore the collective behaviour of the thermal medium in the pre sence of magnetic field is\naffected both by the temperature and strong magnetic field, mainly through the gluon loop and\nquark loop contributions, respectively. Similarly for the physical qu ark masses, the Debye mass is\nobtained\nm2\nD=g′2T2+g2\n4π2T/summationdisplay\nf|qfB|/integraldisplay∞\n0dpzeβ√\np2z+m2\nf\n/parenleftBig\n1+eβ√\np2z+m2\nf/parenrightBig2. (40)\n 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1\n 1 1.5 2 2.5 3mD (GeV)\nT/Tc Pure Thermal\n eB=5 mπ2\n eB=15 mπ2\n eB=40 mπ2\nFigure 1: Variation of Debye mass with temperature\nTo see the competition between the temperature and the magnetic field, we have plotted the\nDebye mass as a function of temperature at the different strengt h of magnetic fields in Figure\n111. At lower temperatures the magnetic field contributes more to th e screening mass than the\ntemperature whereas as the temperature increases within the SM F limit (eB≫T2) the thermal\npart plays more dominant role than the magnetic field unless the magn etic field is sufficiently\nstrong.\nTherefore the real part of retarded gluon self-energy ( 40) gives the real-part of the retarded\nresummed gluon propagator for realistic quark masses\nReDL\nR(k0→0) =1\nk2+m2\nD. (41)\n2.2 Imaginary part of retarded gluon self-energy\nSimilar to the calculation of real part, the imaginary part of retarde d self-energy is obtained from\nthe real-time formalism\nIm ΠR(k0,k) =Im¯Π(k0,k)\nε(k0), (42)\nwhere Im ¯Π(k0,k) is derived from the off-diagonal element of self-energy matrix as\nIm¯Π(k0,k) =−sinh(βk0/2)Im Π 12(k0,k), (43)\nandε(k0) is the theta function.\nLike the evaluation of the real-part we will first evaluate the contrib ution due to quark-loop\nand then calculate for the gluon loops. Therefore the off-diagonal element ( 17) of the propagator\nmatrix (15) for quarks gives the 12-component of self-energy matrix\niΠL\n12(k) =−g2\n2/summationdisplay\nf/integraldisplaydpxdpy\n(2π)2e−(p+k)2\n⊥\n|qfB|e−p2\n⊥\n|qfB|\n×/integraldisplay\ndp0dpzeβ|p0+k0|\n2eβ|p0|\n2nF(p0)nF(p0+k0)L00δ/parenleftbig\n(p/bardbl+k/bardbl)2−m2\nf/parenrightbig\nδ(p2\n/bardbl−m2\nf),(44)\nwherein we use the equality/radicalbig\nnF(p0)(1−nF(p0)) =eβp0\n2nF(p0) and the trace, L00is evaluated as\nL00= 8/bracketleftbig\np0(p+k)0+pz(p+k)z+m2\nf/bracketrightbig\n. (45)\nThe magnetic field again facilitates the calculation of the imaginary par t by separating the\nmomentum integration into components perpendicular and parallel t o the magnetic field,\nIm ΠL\n12(k) =g2\n2/summationdisplay\nfIm Π/bardbl(k/bardbl) Im Π ⊥(k⊥), (46)\n12where the transverse component, Π ⊥is integrated out as\nIm Π⊥(k⊥) =|qfB|\n8πe−k2\n⊥\n2|qfB|, (47)\nand after performing the p0integration, the parallel component is given by\nIm Π/bardbl(k/bardbl) =/integraldisplaydpz\n2ωpeβ|ωp|\n2eβ|k0+ωp|\n2n(ωp)n(k0+ωp)L00(p0=ωp)δ(k2\n0−k2\nz+2p0ωp−2pzkz)\n+/integraldisplaydpz\n2ωpeβ|ωp|\n2eβ|k0−ωp|\n2n(ωp)n(k0−ωp)L00(p0=−ωp)δ(k2\n0−k2\nz−2k0ωp−2pzkz).(48)\nThus, in the static limit ( k0→0), the longitudinal component of the imaginary part of retarded\nself-energy ( 42) assumes the form\nlim\nk0→0ImΠL\nR(k)\nk0=−g2/summationdisplay\nf2m2\nf\nT|kz|Ekz/2nF(Ekz/2)/parenleftbig\n1−nF(Ekz/2)/parenrightbig\nIm Π⊥(k⊥),(49)\nwithEkz\n2=/radicalBig\nm2\nf+k2z/4.\nInweakcouplinglimit, theleading-ordercontributioninSMFlimitcomesf romthemomentum-\ntransferred - |k|2∼αseB, thus the exponential factor in transverse component becomes unity -\nexp(−k2\n⊥\n2|qfB|)∼1. Thus the transverse component of the imaginary part of the se lf-energy is\napproximated into\nIm Π⊥≈|qfB|\n8��, (50)\nand the dispersion relation is simplified too:\nEkz\n2≈|kz|\n2. (51)\nFurthermore using the identity\nnF(Ekz\n2)/bracketleftBig\n1−nF(Ekz\n2)/bracketrightBig\n=1\n2/bracketleftbig\n1+cosh(βEkz\n2/bracketrightbig, (52)\nthe imaginary component is rewritten as\nlim\nk0→0/bracketleftbiggIm ΠL\nR(k)\nk0/bracketrightbigg\n=−g2\n4πT/summationdisplay\nfm2\nf|qfB|1\nk2z/bracketleftbig\n1+cosh(βEkz\n2/bracketrightbig, (53)\nMoreover in SMF limit the longitudinal component ( |kz|) of the momentum is of the or-\nder (αseB)1/2, which is much smaller than the temperature ( << T). Therefore, the imaginary\ncomponent of retarded self energy due to quark-loop takes furt her lucid form\nlim\nk0→0/bracketleftBigImΠL\nR(k)\nk0/bracketrightBig\nquark loop=−g2/summationtext\nfm2\nf|qfB|\n8πT1\nk2z, (54)\n13Similarly we will now calculate the imaginary part due to the gluon loops fr om the off-diagonal\nelement of the self-energy matrix by the off-diagonal element of glu on propagator matrix ( 19).\nHowever, it will be easier to calculate it directly from the imaginary par t of the retarded self-\nenergy from the gluon-loop contribution ( 37). Thus using the identity\n1\nx±y±iǫ= P/parenleftbigg1\nx±y/parenrightbigg\n∓iπδ(x±y), (55)\nthe imaginary part due to the gluon-loop is extracted from ( 37)\nlim\nk0→0/bracketleftBigImΠL\nR(k)\nk0/bracketrightBig\ngluon loops=−g′2πT2\n21\nk. (56)\nThus the longitudinal component of the imaginary part of gluon self- energy due to both quark\nand gluon-loop always factorizes into k0times ImΠL\nR(k) so it vanishes in the static limit ( k0→0).\nTherefore, using the factors in ( 6,7), the resummed symmetric propagator ( 5) in the static limit\nreduces to\nDL\nS(k) = [1+2 nB(k0)] sgn(k0)/parenleftbig\nDL\nR(k)−DL\nA(k)/parenrightbig\n=i4TImΠL\nR(k)/bracketleftbig\nk2−ReΠL\nR(k)/bracketrightbig2, (57)\nwhich is however decomposed into the contributions due to the quar k and gluon loop\nDL\nS(k) =DL\nS(k)quark loop+DL\nS(k)gluon loop (58)\nwith\nDL\nS(k)quark loop =−ig2\n2πk2\nz/summationtext\nf|qfB|m2\nf\n(k2+m2\nD)2(59)\nDL\nS(k)gluon loop =−2iπg′2T3\nk(k2+m2\nD)2. (60)\n3 Heavy quark potential\nThe derivation of potential between a heavy quark Qand its anti-quark ( ¯Q) from effective field\ntheory,namelypNRQCD may not be plausible because the hierarchy of non relativistic scales\nand thermal scales assumed in weak coupling EFT calculations may not be satisfied. Even in the\nfirst principle QCD study, the adequate quality of the data is not ava ilable in the present lattice\ncorrelator studies so one may use the potential model to circumve nt the problems. Since the mass\n14of the heavy quark ( mQ) is very large, so the requirement - mQ≫T≫ΛQCDis satisfied for the\ndescription of the interactions between a pair of heavy quark and a nti-quark at finite temperature\nin strong magnetic field limit in terms of quantum mechanical potential. Thus we can obtain the\nmedium-modification to the vacuum potential in the presence of mag netic field by correcting both\nits short and long-distance part with a dielectric function ǫ(k) as\nV(r;T,B) =/integraldisplayd3k\n(2π)3/2(eik.r−1)V(k)\nǫ(k), (61)\nwhere we have subtracted a r-independent term to renormalize the heavy quark free energy, w hich\nis the perturbative free energy of quarkonium at infinite separatio n. The Fourier transform, V(k)\nof the Cornell potential is given by\nV(k) =−4\n3/radicalbigg\n2\nπαs\nk2−4σ√\n2πk4, (62)\nand the dielectric permittivity, ǫ(k) encodes the effects of deconfined medium in the presence of\nmagnetic field, which is going to be calculated next.\n3.1 The complex permittivity for a hot QCD medium in a strong\nmagnetic field\nThe dielectric permittivity is defined by the static limit of 11-componen t of longitudinal resummed\ngluon propagator by the following equation\n1\nǫ(k)= lim\nk0→0k2DL\n11(k0,k), (63)\nwhere the real and imaginary parts of DL\n11(k) are obtained by the retarded (or advanced) and\nsymmetric propagator, respectively\nReDL\n11(k) = ReDL\nR(k)\nImDL\n11(k) = ImDL\nS\n2(k), (64)\nwhich will in turn gives the real and imaginary part of dielectric permitt ivity, respectively.\nThus the static limit of resummed retarded propagator ( 41) gives the real part of dielectric\npermittivity\n1\nReǫ(k)=k2\nk2+m2\nD. (65)\n15Similarly the static limit of resummed symmetric propagators ( 59,60) gives the imaginary part\nof dielectric permittivity, due to quark and gluon loop contributions\n1\nImǫ(k)quark loop=−g2\n4π/summationdisplay\nfm2\nf|qfB|k2\nk2z(k2+m2\nD)2, (66)\n1\nImǫ(k)gluon loop=−g′2πT3k2\nk(k2+m2\nD)2, (67)\nrespectively.\nTherefore the real and imaginary part of dielectric permittivities giv e the real and imaginary\npart of the complex potential, respectively in the next subsection.\n3.2 Real and Imaginary Part of the potential\nThe real-part of dielectric permittivity ( 65) is substituted into the definition ( 61) to give the real\npart ofQ¯Qpotential in the presence of strong magnetic field [ 17] (with ˆr=rmD)\nReV(r;T,B) =−4\n3αsmDe−ˆr\nˆr+2σ\nmD(e−ˆr−1)\nˆr\n−4\n3αsmD+2σ\nmD, (68)\nwhere the dependence of temperature and magnetic field enter th rough the Debye mass. The\nnonlocal terms insure the potential in medium V(r;T,B) to reduce to the potential in ( T,B)→0\nlimit, which are, however, required to compute the masses of quark onium states. The additional\neffect due to the strong magnetic field on the potential in a hot QCD m edium is displayed as\na function of interparticle distance ( r) for different strength of magnetic fields in Figure 2, after\nexcluding the constant terms from ( 68). The solid line represents the potential in a pure thermal\nmedium ( i.e.in the absence of magnetic field) whereas the dashed and dotted line s denote the\neffect of strong magnetic fields, 10 and 25 m2\nπto a thermal medium, respectively. We have found\nthat the magnetic field (eB=10 m2\nπ) affects the linear string term more than the Coulomb term, as\na result, the overall potential at small and intermediate rbecomes less screened than the potential\nin pure thermal medium. However, further increase of magnetic fie ld (i.e.eB= 25m2\nπ), the\npotential becomes less attractive than eB=10 m2\nπ. However for large rthe effect of magnetic field\ndiminishes gradually.\nSimilarly the imaginary part of the potential is obtained by plugging the imaginary part of\ndielectricpermittivitiesduetoquark-loop( 66)andgluon-loop( 67)contributionsintothedefinition\n16-0.5-0.4-0.3-0.2-0.1 0\n 0 0.5 1 1.5 2Real Potential (GeV) \nr (fm)T=2 Tc\n Pure Thermal\n eB=10 mπ2\n eB=25 mπ2\n-1.6-1.4-1.2-1-0.8-0.6-0.4-0.2 0\n 0 1 2 3 4 5 6Imaginary Potential (GeV) \nr (fm)T=2 Tc\n Pure Thermal\n eB=10 mπ2\n eB=25 mπ2\nFigure 2: Real (left) and imaginary (right) part of the potential\nof potential ( 61). The imaginary component of the potential consists of Coulomb an d string terms\nIm V(r;T,B) = Im V C(r;T,B)+Im V S(r;T,B), (69)\nwhere each term is again split into quark-loop ( q) and gluon-loop ( g) contributions. We will first\ncalculate due to the quark loop from ( 66)\nImVq\nC(r;T,B) =/integraldisplayd3k\n(2π)3/2/parenleftbig\neik.r−1/parenrightbig/parenleftBigg\n−4\n3/radicalbigg\n2\nπαs\nk2/parenrightBigg/parenleftBigg\n−g2k2\n4πk2\nz/summationtext\nf|qfB|m2\nf\n(k2+m2\nD)2/parenrightBigg\n=αsg2\n3π2/parenleftBigg/summationdisplay\nf|qfB|m2\nf/parenrightBigg\nIC, (70)\nwhere the momentum integral, ICis integrated as\nIC=/integraldisplay∞\n0dk\n(k2+m2\nD)2/integraldisplay1\n−1dx(eikrx−1)\nx2\n=/integraldisplay∞\n0dk\n(k2+m2\nD)2[2−2cos(kr)−2kr Si(kr)]\n≡IC1+IC2+IC3, (71)\n17where\nIC1= 2/integraldisplay∞\n0dk\n(k2+m2\nD)2=π\n2m3\nD(72)\nIC2=−2/integraldisplay∞\n0coskr dk\n(k2+m2\nD)2=−/bracketleftbiggπe−ˆr\n2m3\nD+ˆrπe−ˆr\n2m3\nD/bracketrightbigg\n(73)\nIC3=−2r/integraldisplay∞\n0dk k\n(k2+m2\nD)2Si(kr)\n=−2ˆr\nmD/integraldisplay∞\n0dk k\n(k2+m2\nD)2/integraldisplaykr\n0dxsinx\nx, (74)\nrespectively. Similarly the string part of the imaginary potential is\nImVq\nS(r;T,B) =/integraldisplayd3k\n(2π)3/2/parenleftbig\neik.r−1/parenrightbig/parenleftbigg\n−4σ√\n2πk4/parenrightbigg/parenleftBigg\n−g2k2\n4πk2\nz/summationtext\nf|qfB|m2\nf\n(k2+m2\nD)2/parenrightBigg\n=σg2\n2π2/parenleftBigg/summationdisplay\nf|qfB|m2\nf/parenrightBigg\nIS, (75)\nwhere the integral, ISis evaluated as\nIS=/integraldisplay∞\n0dk\nk2(k2+m2\nD)2/integraldisplay1\n−1dx(eikrx−1)\nx2\n=/integraldisplay∞\n0dk\nk2(k2+m2\nD)2[2−2cos(kr)−2krSi(kr)]\n≡IS1+IS2, (76)\nwhereIS1andIS2are given by\nIS1=/integraldisplay∞\n0dk\nk2(k2+m2\nD)2(2−2cos(kr))\n=π\n2m5\nD/bracketleftbig\nˆre−ˆr−3(1−e−ˆr)+2ˆr/bracketrightbig\n(77)\nIS2=−2ˆr\nmD/integraldisplay∞\n0dk\nk(k2+m2\nD)2/integraldisplaykr\n0sinx\nxdx, (78)\nNext we calculate the imaginary part due to the gluon loop contributio n (67) for the Coulomb\nand string terms, respectively [ 12] as\nImVg\nC(r;T,B) =−8αs′T\n3/integraldisplay∞\n0dz z\n(z2+1)2/parenleftbigg\n1−sinzˆr\nzˆr/parenrightbigg\n(79)\nImVg\nS(r;T,B) =−4σT\nm2\nD/integraldisplay∞\n0dz\nz(z2+1)2/parenleftbigg\n1−sinzˆr\nzˆr/parenrightbigg\n, (80)\n18where the Debye mass is given by Eq.( 40).\nThus the equations ( 70) and (79) give the Coulombic contribution whereas the equations ( 75),\n(80) give the string contribution\nImVC(r;T,B) = ImVq\nC(r;T,B)+ImVg\nC(r;T,B) (81)\nImVS(r;T,B) = ImVq\nS(r;T,B)+ImVg\nS(r;T,B) (82)\nto the imaginary component of the potential, respectively. Like the real-part of potential, how\ndoes the imaginary part get affected by the additional presence of magnetic field we have plotted\nit as a function of interquark distance in the right panel of Figure 2. In pure thermal medium\n(denoted by solid line), both Coulomb and string term are larger powe rs of ˆrand counter each\nother, resulting the overall magnitude very small. Now the strong m agnetic field not only reduces\nthe power of ˆ rin both terms compared to the pure thermal medium, it induces Coulo mb and\nstring terms to contribute additively, resulting the overall magnitu de of imaginary part larger.\nThe above observation ultimately translates into the enhancement of thermal width of resonance\nstates due to the ambient strong magnetic field.\n4 Properties of Quarkonia\n4.1 Wavefunction and Binding Energy\nTo investigate the properties of quarkonia in a strong magnetic field , we first solve the Schr¨ odinger\nequation numerically by employing the real part of the potential ( 68) to see how the eigenstates to\nJ/ψ,ψ′andχcstates in a thermal QCD medium get affected by the presence of str ong magnetic\nfieldinfigures3-5,respectively. Inthepresenceofmagneticfieldb othwavefunctionandprobability\ndistribution of quarkonia becomes sharply peaked compared to qua rkonia in absence of magnetic\nfield.\nThus the medium effects encoded into the wavefunctions (Φ( r)) and the corresponding proba-\nbilitydistributionsexplorehowtheaveragesizeofaparticularquark onia(√ri2=(/integraltext\ndτ r2|Φi(r)|2)1/2)\nget affected due to a thermal medium in absence (presence) of mag netic field in left (right) panel\nof Figure 6, respectively. The magnetic field in general causes swellin g of all resonances unless the\ntemperature is very large (Figure 7).\nFinally we have studied how the binding energies of quarkonia change w ith the temperature in\n19 0 1 2 3 4 5 6\n 0 0.5 1 1.5 2 2.5 3 3.5 4Radial Wave Function (J/ ψ) \nr (fm)T=2 Tc\n Pure Thermal\n eB=10 mπ2\n eB=25 mπ2\n 0 0.2 0.4 0.6 0.8 1 1.2\n 0 0.5 1 1.5 2 2.5 3 3.5 4Probability Density (J/ ψ) \nr (fm)T=2 Tc\n Pure Thermal\n eB=10 mπ2\n eB=25 mπ2\nFigure 3: The wavefunction and the radial probability density of J/ψstate\nabsence (presence) of magnetic field in left (right) panel of Figure 8, respectively. The immediate\nobservation is that the magnetic field causes the binding energy to d ecrease with the temperature\nslowly, compared to the medium in absence of magnetic field. Moreove r the competition between\nthe scales associated to the temperature and magnetic field affect s the binding of quarkonia dis-\ncriminately, viz.J/ψbecomes less bound and χcbecomes more bound due to the presence of\nmagnetic field. However, the binding energy decreases with the mag netic field too (Figure 9).\n20-0.5 0 0.5 1 1.5 2 2.5\n 0 1 2 3 4 5 6Radial Wave Function ( ψ(2S)) \nr (fm)T=2 Tc\n Pure Thermal\n eB=10 mπ2\n eB=25 mπ2\n 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45\n 0 1 2 3 4 5 6Probability Density ( ψ(2S)) \nr (fm)T=2 Tc\n Pure Thermal\n eB=10 mπ2\n eB=25 mπ2\nFigure 4: The wavefunction and the radial probability density of ψ′state\n 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5\n 0 0.5 1 1.5 2 2.5 3 3.5 4Radial Wave Function ( χc)\nr (fm)T=2 Tc\n Pure Thermal\n eB=10 mπ2\n eB=25 mπ2\n 0 0.1 0.2 0.3 0.4 0.5\n 0 0.5 1 1.5 2 2.5 3 3.5 4Probability Density ( χc)\nr (fm)T=2 Tc\n Pure Thermal\n eB=10 mπ2\n eB=25 mπ2\nFigure 5: The wavefunction and the radial probability density of χcstate\n21 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5\n 1 1.2 1.4 1.6 1.8 2 2.2 2.4√ () (fm)\nT/Tc Pure Thermal\nJ/ψ\nψ(2S)\nχc\n 0.5 1 1.5 2 2.5 3 3.5 4\n 1 1.2 1.4 1.6 1.8 2 2.2 2.4√ () (fm)\nT/Tc eB=25 mπ2\nJ/ψ\nψ(2S)\nχc\nFigure 6: The average size (√\nr2) of quarkonia in pure thermal medium (left) and then thermal\nmedium in presence of strong magnetic field (right)\n 1 1.5 2 2.5 3 3.5 4 4.5\n 10 15 20 25 30 35 40√ () (fm)\neB (mπ2)T=2 Tc\nJ/ψ\nψ(2S)\nχc\nFigure 7: Variation of the size of quarkonia with the magnetic field at a fixed temperature.\n22 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35\n 1 1.2 1.4 1.6 1.8 2 2.2 2.4Binding Energy (GeV)\nT/Tc Pure Thermal\nJ/ψ\nψ(2S)\nχc\n 0 0.05 0.1 0.15 0.2 0.25 0.3\n 1 1.2 1.4 1.6 1.8 2 2.2 2.4Binding Energy (GeV)\nT/Tc eB=25 mπ2\nJ/ψ\nψ(2S)\nχc\nFigure 8: The binding energies of quarkonia in pure thermal medium (le ft) and thermal medium\nin presence of magnetic field (right).\n 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16\n 10 15 20 25 30 35 40Binding Energy (GeV)\neB (mπ2)T=2 TcPure Thermal\nJ/ψ\nψ(2S)\nχc\nFigure 9: Variation of binding energies with the magnetic field\n234.2 Thermal Width and Dissociation of Quarkonia\nUsing the first-order perturbation theory, the width (Γ) has bee n evaluated numerically by folding\ntheeigenstatesofaspecificquarkoniumstateinthedeconfinedme diuminthepresenceofmagnetic\nfield\nΓ =−2/integraldisplay∞\n0Im V(r;B,T)|Φi(r)|2dτ. (83)\nWe have thus computed the width as a function of temperature in ab sence (presence) of magnetic\nfield in the left (right) panel of Figure 10, respectively. We have fou nd that in pure thermal\nmedium (left panel) the width increases with the temperature faste r than in the presence of\nstrong magnetic field (right panel). However, the magnetic field alwa ys enhances the width of the\nresonances (Figure 11).\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8\n 1 1.2 1.4 1.6 1.8 2 2.2 2.4Γ (GeV)\nT/Tc Pure Thermal\nJ/ψ\nψ(2S)\nχc\n 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\n 1 1.2 1.4 1.6 1.8 2 2.2 2.4Γ (GeV)\nT/Tc eB=25 mπ2\nJ/ψ\nψ(2S)\nχc\nFigure 10: Variation of the thermal widths with the temperature of the medium in absence (left)\nas well as presence (right) of magnetic field\nHaving studied the change of properties of quarkonia in the presen ce of magnetic field, we\ninvestigate now the effect of strong magnetic field on the dissociatio n of quarkonia from the\nconservative criterion on the width of the resonance in Γ ≥2Re B.E.[29]. So we have first\nestimated the dissociation temperatures of quarkonia in absence o f magnetic field in the Table 1\nand then did the same in presence of magnetic fields in Table 2. We foun d that the dissociation\ntemperatures increase due to the presence of strong magnetic fi eld but with the further increase of\nmagnetic field the dissociation temperatures decrease. For example ,J/ψ’s andχc’s are dissociated\n24 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8\n 10 15 20 25 30 35 40Γ (GeV)\neB (mπ2)T=2 Tc\nJ/ψ\nψ(2S)\nχc\nFigure 11: Thermal widths of quarkonia is plotted as a function of ma gnetic field\nat higher temperatures at 2 Tcand 1.1Tcat a magnetic field eB≈6 and 4m2\nπ, respectively,\ncompared to the values 1.60 Tcand 0.80Tcin the absence of magnetic field, respectively. However,\ntheJ/ψis dissociated at smaller temperatures, 1.8 Tcand 1.5Tcfor higher magnetic fields,\neB= 27 and 68 m2\nπ, respectively. Similarly for higher magnetic field, eB= 12m2\nπ,χcgets\ndissociated at the critical temperature.\nStateDissociation Temperature\nTD(inTc)\nJ/ψ1.60\nχc0.80\nψ(2S)0.70\nTable 1: Dissociation temperature for thermal medium in absence of magnetic field\n5 Conclusion\nThe noncentral events in ultra-relativistic heavy-ion collisions prov ide an opportunity to probe\nthe properties of heavy quarkonia in the presence of a strong mag netic field. So we utilize this by\ncalculating the bound state radii, binding energy, thermal width etc . of quarkonia by resummed\nperturbative thermal QCD in the presence of strong magnetic field , thereby studying the dissocia-\ntion of quarkonia due to the Landau damping. For that purpose, us ing the Keldysh representation\nin real-time formalism, we have first calculated the real and imaginary parts of retarded gluon\n25StateDissociation Temperature (Magnetic field)\nTD/Tc(eB(m2\nπ))\nJ/ψ2.0 (6.50)\n1.8 (27.0)\n1.5 (68.0)\nχc1.1 (3.7)\n1.0 (12)\nψ(2S)<1(