Ferromagnetic Resonance Spectroscopy: the “Swiss Army Knife” for Understanding Spin–Orbit Phenomena Justin M

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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

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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?

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 − 𝑖Δ퐻(퐻 − 푀푒푓푓)

푀eff: Effective magnetization = Ms - Hk Hk: perpendicular anisotropy field 푀S: Saturation magnetization Phase Δ퐻: Linewidth Port 1 Port 2 𝑖휑 푆21 = 퐴 푒 휒 퐻, 푓 + 푏푎푐푘푔푟표푢푛푑 Close Window

] 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

We1.8 make the assumption that the 2.10 f f 1.6 2.08 low up y-intercept1.4 20 nm (limit Ni80Fe20 of f ➔ ∞) is the 1.6 up 2.06 1.4 1.2 1.2 1.0

1.0intrinsic value of g and M . (T)

(T) eff 2.04 0.8

-factor value -factor

res g res 0.6

0.8 H 0 H 0.4 Data 0 2.02  0.6 0.2  Data Fit 0.4 0.0 Fit Fitted 2.00 0 10 20 30 40 50 60 Can0.2 achieve a precision better than Frequency (GHz) 0.0 1.98 0 10 20 30 40 50 60 -0.2 0.2 % in permalloy Upper Fitting Frequency, f (GHz) 0 10 20 30 40 50 60 up Frequency (GHz) Shaw, JAP 114 243906 (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 ) OrbitalMoment 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 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. Czechoslovak Journal of Physics B 34, 1111–1124 (1984) Kamberský, V. Czechoslovak Journal of Physics B 26, 1366–1383 (1976) Kambersky, V. & Patton, C. E. Phys. Rev. B 11, 2668–2672 (1975) • Generalized torque correlation model: Gilmore, K., Idzerda, Y. U. & Stiles, M. D. Phys. Rev. Lett. 99, 027204 (2007) Thonig, D. & Henk, J. New J. Phys. 16, 013032 (2014) • Scattering theory: Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Phys. Rev. Lett. 101, 037207 (2008) Liu, Y., Starikov, A. A., Yuan, Z. & Kelly, P. J. Phys. Rev. B 84, 014412 (2011) • Numerical realization of torque correlation via linear response: Mankovsky, S., Ködderitzsch, D., Woltersdorf, G. & Ebert, H. Phys. Rev. B 87, 014430 (2013) • First priciples calculations: M. C. Hickey & J. S. Moodera, Phys. Rev. Lett. 102,137601 (2009) Ritwik Mondal,* Marco Berritta, and Peter M. Oppeneer, Phys. Rev. B 94, 144419 (2016) 28 Intraband versus Interband

Intraband Interband “Breathing Fermi Surface” t1 “Bubbling Fermi Surface” “Conductivity-like damping” t2 “Resistivity-like damping”

Inverse scattering time (1/τ) Temperature ➔ Defect density ➔ Gilmore and Stiles, PRB 81, 174414 (2010) 29 Intrinsic Damping: Critical Parameters

uniform mode dissipation in lattice Kamberský’s Torque Correlation model: spin-orbit damping Sum over all electronic states, bands

− 2 Courtesy of Claudia Mewes 훼 ∝ ෍ Γ푚푘,푛푘 푊푚푘,푛푘 푚,푛 푘 Torque between spin and spectral overlap function of Bloch state lifetime orbital moments  SO Critical Parameters affecting intrinsic damping:

N(EF): Density of states at the Fermi energy, EF

SO : Spin-Orbit Interaction τ: Spin scattering times 30 Origin of damping

푑푀 휶 훾 휇0 = − 훾 휇0 푀 × 퐻 − 푀 × 푀 × 퐻 푑푡 푀푠

• Originally just a phenomenological Magnon- Spin-currents parameter electron scattering Electrodynamics • Theory has worked to calculate damping parameter α from first principles Intrinsic Coherent Spin Excitation Damping (Magnon) Defect scattering In reality, α includes many intrinsic Magnon- and extrinsic sources and is not phonon Inhomogeneity necessarily a constant scattering 31 How Do We Measure Damping?

LL equation predicts linewidth is proportional to frequency

12 2ℎ휶 ∆퐻 = 푓 10 8

푔μ0μB 6

Linewidth(mT) 2

0 0 10 20 30 40 Frequency (GHz)

However, that is not always observed

32 Linewidth Analysis

❑ Inhomogeneous Broadening ❑ Low-field losses ❑ 2-magnon scattering • Many relaxation mechanisms and ❑ Spin-pumping contributions to linewidth exist ❑ Radiative Damping ❑ k2 Damping ❑ Other Contributions • Careful separation and quantification of all ❑ Eddy Currents ❑ Slow relaxer mechanisms is needed to compare with ❑ Motional Narrowing theory ❑ Chiral Damping ❑ Intrinsic Damping Point of emphasis: Broadband measurements are critical.

33 Inhomogeneous Linewidth

❑ Inhomogeneous Broadening ❑ 2ℎ휶 Low-field losses ∆퐻 = 푓 + ∆푯 ❑ 2-magnon scattering 푔μ μ ퟎ ❑ Spin-pumping 0 B ❑ Radiative Damping Inhomogeneous Linewidth ❑ k2 Damping ❑ Other Contributions ❑ Eddy Currents ❑ Slow relaxer H Theoretical ideal… ❑ Motional Narrowing ❑ Chiral Damping ❑ Intrinsic Damping 2h ∆H0 … what we gusuallyB 0 see

f Heinrich, JAP, 57, 3690 (1985) 퐻 Farle, Rep. Prog. Phys. 61, 755 (1998) McMichael, PRL 90, 227601 (2003) 34 Inhomogeneous Linewidth

❑ Inhomogeneous Broadening ❑ 2ℎ휶 Low-field losses ∆퐻 = 푓 + ∆푯 ❑ 2-magnon scattering 푔μ μ ퟎ ❑ Spin-pumping 0 B ❑ Radiative Damping Inhomogeneous Linewidth ❑ k2 Damping ❑ Other Contributions ❑ Eddy Currents 250 55 nm Cu ❑ Slow relaxer 200 ❑ Motional Narrowing Ta 30 nm Cu ❑ Chiral Damping 200 nm 5 nm Cu 150 20 nm Cu ❑ Intrinsic Damping CoFe/Ni 55 nm x Cu 100 15 nm Cu

Ta Linewidth(mT) 50 200 nm 5 nm Cu 0 0 10 20 30 40 50 Heinrich, JAP, 57, 3690 (1985) Frequency (GHz) Farle, Rep. Prog. Phys. 61, 755 (1998) McMichael, PRL 90, 227601 (2003) 35 Low Field Loss

❑ Inhomogeneous Broadening • At low fields (frequency), sample may not be saturated. ❑ Low-field losses ➔ Linewidth Broadening  ❑ 2-magnon scattering H = ❑ low− field n Spin-pumping f ❑ Radiative Damping 10 Data 2 ❑ k Damping Combined fit •Described by a power law fit ❑ Other Contributions 8 Low-field loss contribution Damping + H0 ❑ Eddy Currents 6

❑ Slow relaxer •Effect can extend to higher

❑ Motional Narrowing 4 frequencies for some materials ❑ Chiral Damping Linewidth(mT) 2 ❑ Intrinsic Damping 16 Permalloy Co2MnGe 0 0 2 4 6 8 10 14 Frequency (GHz)

Linewidth(mT) 12

10 E. Schloemann: JAP 41, 204 (1970) 5 10 15 20 25 30 35 40 45 Frequency (GHz) 36 2-Magnon Scattering

❑ Inhomogeneous Broadening • Mediated by Defect Scattering ❑ Low-field losses In-plane geometry Out-of-plane geometry ❑ 2-magnon scattering ω ω ❑ Spin-pumping ❑ Radiative Damping ❑ k2 Damping FMR H H ❑ Other Contributions FMR ❑ Eddy Currents ❑ k k Slow relaxer • Can reduce/eliminate with out-of-plane geometry ❑ Motional Narrowing 25 25

❑ Chiral Damping Perpendicular Perpendicular Al (5 nm) In-plane In-plane ❑ Intrinsic Damping 20 20 CoFe Al (5 nm) CuN (3 nm) 15 15 CoFe Ti(3 nm) SiO SiO 2

10 2 10 Linewidth(mT) Sparks, Phys Rev. 122, 791 (1961) Linewidth(mT) 2-magnon scattering 5 5 Hurben, JAP 83, 4344 (1989)

McMichael, JAP 91, 8647 (2002) 0 0 Lindner, PRB 68, 060102 (2003) 0 5 10 15 20 25 30 35 40 45 50 55 60 0 5 10 15 20 25 30 35 40 45 50 55 Frequency (GHz) Frequency (GHz) Barsukov, PRB 84, 140410 (2011) arXiv:1804.0878637 Spin-Pumping & Spin-Memory Loss

❑ Inhomogeneous Broadening Loss of angular momentum at or across an interface ❑ Low-field losses Ferromagnet Normal Metal ❑ 2-magnon scattering  ❑ Spin-pumping s ❑ Radiative Damping H y-intercept is the damping without ❑ k2 Damping Flow of spin current spin-pumping contribution ❑ Other Contributions Spin-Pumping

❑ Eddy Currents  0.004 ❑ Slow relaxer Spin-Memory ❑ Motional Narrowing Loss 0.003 ❑ Chiral Damping Can be phenomenologically treated as an 5nm Ta ❑ Intrinsic Damping 0.002 interfacial damping term Co2MnGe 0.001 5nm Ta ↑↓ Oxidized Si 푔 1 Damping Parameter, 푒푓푓 0.000 훼 푡 = 훼𝑖푛푡 + 푔휇퐵 0.00 0.05 0.10 0.15 0.20 0.25 0.30 4휋푀 푡 -1 푠 1 / thickness (nm ) t = thickness Y. Tserkovnyak, PRL 88, 117601 (2002) g = g-factor ↑↓ O.Mosendz, PRB 82, 214403 (2010) g = effective real part spin-mixing conductance 38 Radiative Damping

❑ Inhomogeneous Broadening Damping contribution due to inductive coupling of the sample to the waveguide ❑ Low-field losses ❑ 2-magnon scattering 30 nm Co25Fe75 Sample ❑ Spin-pumping ❑ Radiative Damping CoFe ❑ k2 Damping 200 μm Glass CoFe ❑ Other Contributions ❑ Eddy Currents ❑ Slow relaxer 4 ❑ Motional Narrowing no spacer ❑ Chiral Damping 200 um glass spacer 3 ❑ Intrinsic Damping

2  = 0.0021 δ = thickness Z0 = waveguide impedance Linewidth(mT) 1  = 0.0017

0 M.Schoen, PRB, 92, 184417 (2015) 0 2 4 6 8 10 12 14 16 18 20 22 Frequency (GHz) 39 Wavevector Dependent Damping

❑ Inhomogeneous Broadening Theory predicts a wavevector (k) ❑ Low-field losses 2 ❑ 2-magnon scattering dependence of damping  |k| ❑ Spin-pumping Y. Tserkovnyak Phys. Rev. B 79, 094415 (2009) S. Zhang Phys. Rev. Lett., 102, 086601 (2009) 0.011 ❑ Radiative Damping  ❑ k2 Damping End Mode ❑ Other Contributions 0.010 ❑ Eddy Currents 100 nm

❑ Slow relaxer 0.009 ❑ Motional Narrowing ❑ Chiral Damping ➔Generation of intralayer Center Mode ❑ Intrinsic Damping spin currents 0.008 NiFe film

Spin- Parameter, Damping current Spin- 0.007 current 100 200 300 400 Nanomagnet Width (nm)

Nembach PRL 110, 117201 (2013) 40 Other Sources of Linewidth and Damping

❑ Inhomogeneous Broadening Slow Relaxer (Resonant Damping) Eddy Currents ❑ Low-field losses ❑ 2-magnon scattering ❑ Spin-pumping δ = thickness ρ = resistivity ❑ Radiative Damping C = correction factor ❑ k2 Damping ❑ Other Contributions Generally not significant for metallic thin films < 20 nm ❑ Eddy Currents Van Vleck, PRL 11, 65 (1963). J. M. Lock, British JAP 17, 1645 (1966) ❑ Slow relaxer Woltersdorf, PRL 102, 257602 (2009) C. Scheck, APL 88, 252510 (2006) ❑ Motional Narrowing Nembach, PRB, 84, 054424 (2011) ❑ Chiral Damping Motional Narrowing Chiral Damping ❑ Intrinsic Damping

E. Jue, Nature Materials 15, 272 (2015) A. Okada, PNAS, 114, 3815 (2017) 41 Intrinsic and Total Damping

• To compare with theory, we must account for or eliminate additional damping terms. Broadband Measurement Thickness series Measure in What we Required needed to quantify measure perpendicular contribution geometry Thickness series, or 휶푻풐풕풂풍 풇 = 훼𝑖푛푡 + ∆훼𝑖푛ℎ표푚표 + Δ훼2−푚푎푔 + Δ훼푠푝𝑖푛−푝푢푚푝𝑖푛푔 restrict to ≲ 20 nm, or insulators

Parameter of interest +Δ훼푙표푤−푓𝑖푒푙푑 + Δ훼푟푎푑 + Δ훼푒푑푑푦 푐푢푟푟푒푛푡 + Δ훼표푡ℎ푒푟

Broadband measurements to Decrease inductive coupling Broadband measurements and/or exclude non-linear, with spacer between CPW sample series needed to quantify low field/freq data and sample 42 Why is LW analysis so important?

Prediction of ultra-low damping (α ≈ 0.0005) in CoFe

bcc mixed fcc

0.005

0.004

 tot 0.003

Mankovsky, PRB 87, 014430 (2013) Damping  0.002 int  sp 0.001  rad

0.000 0.0 0.2 0.4 0.6 0.8 1.0 Co Concentration Schoen, Nature Physics 12, 839 (2016) Turek, PRB 92, 214407 (2015) 43 Why is LW analysis so important?

Damping is strongly controlled by the DOS at EF

bcc mixed fcc int

 1.5 0.003 b

n(E 1.2

F 0.002 (eV )

0.9

-1

0.001 ) Theory Mankovsky et al. 0.6 IntrinsicDamping, 0.000 0.0 0.2 0.4 0.6 0.8 1.0 Co Concentration Schoen, Nature Physics 12, 839 (2016)

44 DOS and damping

 0.008 Alloying in Ge in Co Fe 0.007 A2 --> B2 50 50 0.006 B2 --> L21 enhances the “pseudo-gap” a 0.005 Co MnGe 0.004 2 Co2MnSi moves it to the Fermi level 0.003 0.002 0.001 0.000

Calculated Damping Parameter, 0 10 20 30 40 50 60 70

n(EF) (states/eV) J.Shaw, PRB, 97, 094420 (2018) Pradines, PRB, 95, 094425 (2017)

Co50Fe50Gex

Co2FeAl

H.Lee, APL 95, 082502 (2009) J.Chico, PRB, 93, 214439 (2016) Mizukami, JAP, 105, 07D306 (2009) 45 How can we control Damping?

How do we reduce spin pumping?

Ta (3 nm) Replace Ta with low atomic number Cu (3 nm) CoFe elements or insulators 0.006 Cu (3 nm) Ta(3 nm) 0.005 SiO2 0.004 Al (5 nm) 11, 054036 (2019) 0.003

CoFe total  0.002 Cu (3 nm) Ti(3 nm)

0.001 SiO2

0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 2 nm film with a total 1 / thickness (nm-1) damping of 0.0018 !!! E. Edwards, Phys. Rev. Applied 11, 054036 (2019) 46 Why is LW analysis so important?

Consider Relationship “...a linear relation between the perpendicular between Perpendicular magnetic anisotropy and α is established” Anisotropy (PMA) and Damping “No correlation of g with the increase in α... is enhanced locally at the interface between the multilayer and buffer (capping) layer... may be attributed to the spin-pumping effect”

“The estimated α was found to be independent both on total thickness and anisotropy field of the multilayer...”

...the intrinsic Gilbert damping constant was found to be independent of the perpendicular magnetic anisotropy...”

“We find that the magneto-crystalline anisotropy and the Gilbert damping constant show a linear relationship.“ 47 Both anisotropy and damping are proportional to SO to some power Relationship between PMA and α?

0.020

• No relationship between anisotropy  and PMA 0.015 0.010

➔spin-orbit parameter does not Out-of-plane In-plane Damping, Damping, vary 0.005 0.000

30 25 20 Co Fe /Ni Multilayers (mT) 90 10 0 15

H Anisotropy is tuned by varying 

0 10 • Anisotropy results from orbital  5 the total thickness. Inhomogenous LW Inhomogenous 0

moment asymmetry NOT increased 0.020

spin-orbit B 0.015

 /

➔ Independent of damping. L 0.010  0.005 휉 푁 Δ휇 0.000 퐾 = − 퐴 푆푂 퐿 -0.5 0.0 0.5 1.0 Total Anisotropy Field (T) 4푉 휇퐵 J.M. Shaw, APL 105, 062406 (2014) 48 Outline • Spin-Orbit Interaction • Ferromagnetic Resonance Spectroscopy • Anisotropy and Orbital Moments • Damping and Spin-Pumping • Spin-Orbit Torques

49 Spin-Orbit Torques (SOT)

Spin-Orbit Torques ➔ transfer of angular momentum from lattice to carriers to magnetization

• Consider Ferromagnet (FM)/Heavy Metal interface Heavy Metal • Heavy metals have large spin-orbit coupling • A charge current through the heavy metal can generate a transverse pure spin current FM

inverse Spin-Hall Effect inverse Rashba-Edelstein Effect

Charge Spin

Spin-Hall Effect Rashba-Edelstein Effect

50 Spin-Orbit Torques (SOT)

Spin-Hall Effect Rashba-Edelstein Effect (intefacial) “Damping-like Torque” “Field Like Torque”

푀 푴(푡) റ 푆 퐸 charge current I NM

푆푂푇 ∝ 푀 × (푀 × 푆റ) 푆푂푇 ∝ 푀 × 푆റ

In reality, neither SHE or REE are purely “damping-like” or “field-like,” but can be mixed.

V. Amin and M.Stiles PRB 94, 104420 (2016) V. Amin and M.Stiles PRB 94, 104419 (2016) 51 Why are SOT important?

Efficient switching of magnetic memory

2-terminal Spin-Transfer Torque (STT) 3-terminal Spin-Orbit Torque (SOT)

Bit line fixed layer fixed layer tunnel barrier tunnel barrier free layer free layer

Word line Word line

Shared read/write path: Separate read and write paths  Large current density for write ✓ Can apply large current density operations can damage tunnel through write line without barrier damaging MTJ  Read operations have finite probability of flipping the bit 52 SOT Measurements Methods

Harmonic Method with Hall Cross Inverse SHE DC devices

Mosendz et al., PRB, 82, 214403 (2010) Hayashi et al. PRB 89, 144425 (2014) Weiler, IEEE Magn. Lett. 5, 3700104 (2014)

Spin-Torque Inverse SHE AC devices Ferromagnetic Resonance

M. Weiler, PRL 113, 157204 (2014) …and now VNA-FMR...at the film level Liu et al., PRL, 106 (2011) 53 Measured Inductances in VNA-FMR

Measured signal ~0 𝑖휔 Δ푆21 ≈ − 퐿푁𝑖퐹푒 + 퐿퐶푢 푍0 푚푦 휕 푚ෝ 휇0푙푑FM NiFe 푡 퐿FM = 휒푦푦 4푊wg Cu ~ Constant signal as a function of frequency

푊wg 퐻dipole

H0 microwave out 푧 푦 푙 푥 sample 7.5 GHz 35 GHz microwave in

Berger, et al. PRB 97 094407 (2018) 54 Measured Inductances in NiFe/Pt

Total Inductance H0 microwave out 푧 푦 푙 푥 푧

sample 푥 푦 퐻eff 푚푦 휕푡푚ෝ microwave in 푠 NiFePy The inductance (L) of the sample arises 퐽Rashba−Edelstein from anything that produces a flux 푄푠Ƹ 퐽spin Hall around the CPW Pt 퐽Faraday Φ 퐿 = = 퐿 + 퐿 (휔) 퐼 NiFe 푃푡 CPW CPW 퐻 NM 퐻dipole

1. Dipolar inductive coupling, ~ frequency-independent 퐿NiFe 2. Second-order inductive coupling ~ 𝑖휔 (Faraday-induced currents in the NM) 3. Damping-like ~ 휔 (e.g. spin Hall effect) Spin-charge transduction, 퐿푃푡(휔) 4. Field-like ~ 𝑖휔 (e.g. Rashba-Edelstein effect) a.k.a Spin-Orbit Torque (SOT) Berger, PRB 97 094407 (2018)55 Measurement of SOT in NiFe/Pt Frequency Dependent 𝑖휔 Δ푆21 ≈ − 퐿푁𝑖퐹푒 + 퐿푃푡(휔) 푍0 Field-like terms ℏ휔 Re 퐿 ∝ 휇 푙푑 + 퐿 휎SOT − 휎F 0 퐹푀 12 푒 푓푙 푓푙 Damping-like terms ℏ휔 Im(퐿) ∝ −퐿 휎SOT 12 푒 푑푙

Change in both Amplitude and Phase 7.5 GHz 35 GHz

Berger, et al. PRB 97 094407 (2018) 56 Inconsistency in spin diffusion length

From damping measurements

From spin Hall measurements

Nakayama, Feng, PRB 85, 144408 (2012) PRB 85, 214423 (2012)

If the pumped spin current is generating the charge current via the SHE, how can this be?

57 Application: Spin-Diffusion Length

VNA-FMR ➔self-consistent, simultaneous measurement PROBLEM: Inconsistency in Spin of spin-pumping and iSHE diffusion length in Pt

Spin-pumping ➔ < 2 nm

iSHE measurements ➔ 4-8 nm Nakayama, PRB 85, 144408 (2012) Feng, PRB 85, 214423 (2012)

x Pt

Berger, et al. PRB 97 094407 (2018) 58 Advantages of VNA-FMR Method

• No rectification artifacts

• No need for sample patterning

• Phase sensitive separation of damping-like and field-like contributions

• Simultaneous extraction of spin pumping contribution to damping

• Identical method for both perpendicular and in-plane anisotropy FM materials

59 Acknowledgements

NIST (Boulder) Uppsala University Hans Nembach Olle Eriksson Tom Silva Danny Thonig Martin Schoen (U.Manitoba) Erna Delczeg-Czirjak Mathias Weiler (WMI, Munich) Yaroslav Kvashnin Eric Edwards (IBM) Olof “Charlie” Karis Carl Boone Mike Schneider Matt Pufall Emilie Jue Andy Berger (SRS) T.U. Munich Eindhoven University Christian Back Juriaan Lucassen Bert Koopmans

60 Summary

• Spin-orbit (SO) coupling generates a wealth of phenomena in magnetic systems

• Recent advances in VNA-FMR techniques enables access to measure many of these SO phenomena.

• Improved precision of VNA-FMR measurements advanced our understanding of many phenomena: • Precise measurement of orbital moment to compare with theory • Determination of intrinsic damping and quantification of other sources • Better understanding of factors that control damping • Quantification of SOT without device fabrication • Self-consistent separation of spin-pumping and spin-memory losses

0.020 bcc mixed fcc

0.015 int  b 1.5

B 0.003  n(E

/ / 1.2 0.010 F

) (eV L 0.002

  0.9 0.005 -1 0.001 ) Theory Mankovsky et al. 0.6 0.000 IntrinsicDamping, 0.000 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 5 3 K (10 J/m ) Co Concentration 61