Broadband Ferromagnetic Resonance Spectroscopy: The “Swiss Army Knife” for Understanding Spin–Orbit Phenomena Justin M. Shaw National Institute of Standards and Technology, Boulder, CO 80305, USA 퐻ext 1 IEEE Magnetics Society ieeemagnetics.org 2 3 Outline • Spin-Orbit Interaction • Ferromagnetic Resonance Spectroscopy • Anisotropy and Orbital Moments • Damping and Spin-Pumping • Spin-Orbit Torques 4 Spin-Orbit Interaction “Classical” electron spin orbiting nucleus • In reality, Spin-Orbit is a relativistic effect e • This effective B field leads to a splitting of + energy levels ➔ Fine Structure From the rest frame of the electron spin effective B - field wikipedia e + • In solids the spin-orbit interaction is included as an addition to the Hamiltonian I 퐻 = 휉 푳 ⋅ 푺 푆푂 푆푂 5 Spin-Orbit Phenomena Anomalous Hall Topological Effect Surface States Rashba- Edelstein effect Energy Magnetic Anisotropy Magnetization Direction Spin-Orbit Coupling Orbital Moments in Solids Spin Hall 퐻푆푂 = 휉푆푂푳 ⋅ 푺 Effect Dzyaloshinskii-Moriya Interaction (DMI) Spin-lattice M Weiler Damping Coupling Science, 323, 915 (2009) Beaurepaire PRL 76 (1996) 6 Outline • Spin-Orbit Interaction • Ferromagnetic Resonance Spectroscopy • Anisotropy and Orbital Moments • Damping and Spin-Pumping • Spin-Orbit Torques 7 What happens when we excite a spin? H animations courtesy of Helmut Schultheiss 8 Collective Spin Excitations Ferromagnetic Resonance (FMR) Mode Other Spinwave (Magnon) Modes All spin precess in phase Non-zero phase relationship animations courtesy of Claudia Mewes 9 Landau-Lifshitz Equation Lev Landau Landau-Lifshitz equation dM 0 = − 0 (M H )− M (M H ) dt M s Larmor Term H Evgeny Lifshitz animations courtesy of Helmut Schultheiss 11 Landau-Lifshitz Equation Lev Landau Landau-Lifshitz equation dM 0 = − 0 (M H )− M (M H ) dt M s Larmor Term Damping Term H Evgeny Lifshitz animations courtesy of Helmut Schultheiss 12 Ferromagnetic Resonance (FMR) Cavity Based ➔ single frequency! Constant Frequency, vary angle Srivastava, JAP 85 7841 (1999) A. Okada, PNAS, 114, 3815 (2017) 13 Vector Network Analyzer FMR (VNA-FMR) Images: E8 363C PNA Microwave Network Analyzer| Agilent 4/24/11 2:19 PM 푀푒푓푓(퐻 − 푀푒푓푓) E8363C PNA Microwave Network Analyzer Vector-network-analyzerClose Window 휒(퐻, 푓) = 2 2 (퐻 − 푀푒푓푓) − 퐻푒푓푓 − Δ퐻(퐻 − 푀푒푓푓) 푀eff: Effective magnetization = Ms - Hk Hk: perpendicular anisotropy field 푀S: Saturation magnetization Phase Δ퐻: Linewidth Port 1 Port 2 휑 푆21 = 퐴 푒 휒 퐻, 푓 + 푏푎푐푘푔푟표푢푛푑 Close Window © Agilent 2000-2011 Amplitude Resonance Field, Linewidth 0.32 0.20 ] 0.30 0.18 21 H Im[S 0.28 0.16 http:/ / www.home.agilent.com/ agilent/ product.jspx?cc= US&lc= eng&nid= - 536902643.761694&imageindex= 1 Pa ge 1 of 1 21 Re[S 0.26 0.14 ] 0.24 0.12 50 coplanar waveguide 0.22 0.10 2.85 2.86 2.87 2.85 2.86 2.87 14 0H (T) 0H (T) VNA-FMR 15 Outline • Spin-Orbit Interaction • Ferromagnetic Resonance Spectroscopy • Anisotropy and Orbital Moments • Damping and Spin-Pumping • Spin-Orbit Torques 16 Magnetic Anisotropy Magnetic Anisotropy gives a preferred orientation to the spontaneous magnetization Why important? Can originate from: M • Magnetocrystalline • Interfaces M • Strain (Magnetostriction) • Shape (Magnetostatic energy) ∆E M Energy Seagate Magnetization Direction Duke U Simple model of Anisotropy Energy 휉 ∆E magnetic anisotropy energy 푆푂 ⊥ || SO spin-orbit coupling constant ∆퐸 = 퐴 (휇퐿 − 휇퐿 ) A prefactor (< 0.2) - function of electronic structure 4휇퐵 µL orbital moment P. Bruno, PRB 39, 865 (1989) 17 How do we measure anisotropy? • Angular Dependence of Resonance Field (Hres) sample magnet magnet • Frequency (f) Dependence of Resonance Field (Hres) A. Okada, PNAS, 114, 3815 (2017) Kittel Equation: 1.8 1.6 20 nm Ni Fe (Permalloy) Can include anisotropy terms of any symmetry (see Farle, RPP (1998) for review) 80 20 1.4 H 1.2 || 1.0 푔 휇0휇퐵 (T) 푓(퐻 ) = 퐻 + 퐻 퐻 + 푀 + 퐻 res 0.8 푟푒푠 푟푒푠 퐾 푟푒푠 푒푓푓 퐾 H 0 0.6 2휋ℏ Data 0.4 Fit In-plane anisotropy field 0.2 0.0 Perpendicular anisotropy field -0.2 ⊥ 0 10 20 30 40 50 60 Effective Magnetization: 푀푒푓푓 = 푀푠 − 퐻퐾 Frequency (GHz) 18 How do we measure orbital moments? g-factor relates the spin (μS) and orbital (μL) moments: 2휇 푔 = 2 + 퐿 휇푆 Since μS >> μL , g is typically slightly larger than 2 in ferromagnets • Frequency (f) Dependence of Resonance Field (Hres) Spectroscopic g-factor 1.8 1.6 20 nm Ni80Fe20 (Permalloy) || 1.4 H 푔 휇0휇퐵 1.2 푓(퐻푟푒푠) = 퐻푟푒푠 + 퐻퐾 퐻푟푒푠 + 푀푒푓푓 + 퐻퐾 1.0 2휋ℏ (T) res 0.8 H 0 0.6 Data 0.4 Fit 0.2 0.0 ⊥ Effective Magnetization: 푀푒푓푓 = 푀푠 − 퐻퐾 -0.2 0 10 20 30 40 50 60 Frequency (GHz) 19 g-factor and orbital moments g-factor relates the spin (μS) and orbital (μL) moments: 2휇퐿 푔 = 2 + PROBLEM: Reported values of g-factors 휇 푆 can vary considerably Since μS >> μL , g is typically slightly larger than 2 in ferromagnets Permalloy g−factor Method Frequency Reference Thickness range 5 –50 nm 2.109 ± 0.003 VNA-FMR with 4—60 GHz This work (extrapolated to asymptotic bulk value) analysis PROBLEM: A small error in g leads to 50 nm 2.20 ± 0.12 Einstein-de-Haas — [27] 50 nm 2.0—2.1 PIMM 0—2 GHz [28] large error in μ 50 nm 2.1 PIMM 0—3 GHz [29] L 10 nm 2.05 PIMM 0—3 GHz [29] bulk 2.12 ± 0.02 FMR 19.5 & 26 GHz [30] bulk 2.12 Einstein-de-Haas — [31] 4—50 nm 2.08 ± 0.01 FMR not reported [32] bulk 2.08 ± 0.03 FMR 10 GHz [9] 9 Consider g = 2.05 +/- 0.02 (1% error) bulk 2.12 ± 0.03 FMR 24 GHz [ ] ➔ 40% error in μL 20 g-factor: Asymptotic Analysis f f low up f f low up || 1.6 푔 휇0휇퐵 1.4 1.6 1.2 1.4 푓(퐻푟푒푠) = 퐻푟푒푠 퐻푟푒푠 + 푀 1.2 푒푓푓 (T) 1.0 1.0 2휋ℏ res 0.8 (T) 0.8 H 0.6 res 0 0.6 0.4 Data H Data 0 0.4 0.2 Fit Fit 0.2 2퐾 0.0 푀푒푓푓 = 푀푠 − 0 10 20 30 40 50 60 0.0 휇 푀 0 10 20 30 40 50 60 0 푠 gfit asymptoticallyFrequency approaches (GHz) a valueFrequency as (GHz)fup ➔ ∞ 2.12 We make the assumption that the 2.10 f f 2.08 low up y-intercept20 nm (limit Ni80Fe20 of f ➔ ∞) is the 1.8 1.6 up 2.06 1.4 1.6 1.2 1.4 1.0 intrinsic value of g and M . (T) eff 2.04 0.8 -factor value -factor 1.2 res g 0.6 H 1.0 0 0.4 2.02 (T) Data 0.2 Fit res 0.8 H 0.0 0 Fitted Fitted 2.00 0.6 0 10 20 30 40 50 60 Data 0.4 Frequency (GHz) Can achieve a precision better than Fit 1.98 0.2 0 10 20 30 40 50 60 0.0 0.2 % in permalloy Upper Fitting Frequency, f (GHz) -0.2 up 0 10 20 30 40 50 60 Shaw, JAPFrequency 114 243906 (GHz) (2013) 21 What can cause errors in g-factors • Field misalignment • Higher order anisotropy terms • Rotatable (history dependent) anisotropy • Pinning of local spins • Defects • Interfaces • Excessive fitting parameters • Any missing or field dependent term in Kittel eqn. 22 Application: Co90Fe10/Ni Superlattices H 2.20 ┴ 2.19 g Ni 2.18 3 nm Ta CoFe Ni 2.17 || H 3 nm Cu CoFe g Ni - factor 2.16 CoFe g Ni 2.15 Out-of-plane CoFe/Ni CoFe In-plane Ni 2.14 Multilayer CoFe 0.00 0.05 0.10 0.15 Ni -1 ) 1 / thickness (nm ) CoFe 3 5 nm Cu Ni 6 CoFe J/m 5 3 nm Ta Ni 5 CoFe 4 (10 K 3 •Anisotropy is tuned by 2 varying the thickness. 1 0 Anisotropy, -1 J.M. Shaw, et al., Phys. Rev. B, 87, 054416 (2013) 0.00 0.05 0.10 0.15 -1 1 / thickness (nm ) 23 Application of g-factor analysis: CoFe/Ni A test of Bruno’s theory: K anisotropy energy density || ⊥ spin-orbit coupling constant 휉푆푂푁 (휇퐿 − 휇퐿 ) 퐾 = − 퐴 A prefactor due to electronic structure 4푉 휇퐵 N number atoms per unit cell V volume of unit cell P. Bruno, PRB 39, 865 (1989) 0.020 Fit to the data: B 0.015 A = 0.097 ± 0.007 ) / || 0.010 L Compare to previous reports: A = 0.2 at RT, Au/Co/Au - | | 0.005 D. Weller, et al. PRL 75, 3752 (1995) L ( A = 0.1 at near 0 K, Ni/Pt MLs 0.000 F. Wilhelm, et al. PRL 75, 3752 (1995) 0 1 2 3 4 5 6 5 3 A = 0.05 at near 0 K, Fe/V MLs Anisotropy Energy, K (10 J/m ) Orbital Moment Asymmetry A.Anisimov, et al. PRL 82, 2390 (1999) J.M. Shaw, et al., Phys. Rev. B, 87, 054416 (2013) 24 Outline • Spin-Orbit Interaction • Ferromagnetic Resonance Spectroscopy • Anisotropy and Orbital Moments • Damping and Spin-Pumping • Spin-Orbit Torques 25 Why is damping important? Data Storage & Magnetic Memory (STT-RAM) Spin current source 2-terminal Spin-Transfer Torque (STT) Bit line fixed layer Critical current needed to tunnel barrier switch an MRAM cell free layer 2e M sV Ic0 = H eff g( ) p Nakayama, IEEE Trans Mag. (2010) Mangin APL (2009) Word line Magnonics and Spin Logic Spin-torque nano-oscillators ) 8000 /Hz 2 6000 nV ( 4000 50 nm nano- 500nm 2000 oscillators 0 Spectral Density Spectral 13.10 13.15 13.20 Frequency (GHz) H.Schultheiss, HZDR K.Wang UCLA26 Intrinsic Damping Theories 푑푀 휶 훾 휇0 = − 훾 휇0 푀 × 퐻 − 푀 × 푀 × 퐻 푑푡 푀푠 • Breathing Fermi surface model: Kamberský, V.