understanding spin orbit phenomena...g-factor: asymptotic analysis f up f up f up 0 10 20 30 40 50...

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

𝐻ext

1

IEEE Magnetics Society

ieeemagnetics.org

2

Outline

• Spin-Orbit Interaction

• Ferromagnetic Resonance Spectroscopy

• Anisotropy and Orbital Moments

• Damping and Spin-Pumping

• Spin-Orbit Torques

4

Spin-Orbit Interaction

e

+

I

e

effective B - field

+

𝐻𝑆𝑂 = 𝜉𝑆𝑂𝑳 ⋅ 𝑺

“Classical” electron spin orbiting nucleus

From the rest frame of the electron spin

• In solids the spin-orbit interaction is included as an addition to the Hamiltonian

• This effective B field leads to a splitting of energy levels ➔ Fine Structure

wikipedia

• In reality, Spin-Orbit is a relativistic effect

5

Spin-Orbit Phenomena

Spin HallEffect

Magnetic Anisotropy

Damping

Orbital Moments in Solids

Spin-lattice Coupling

Topological Surface States

MagnetizationDirection

Ener

gy

Dzyaloshinskii-MoriyaInteraction (DMI)

Rashba- Edelstein effect

Beaurepaire PRL 76 (1996)Science, 323, 915 (2009)

M Weiler

Spin-Orbit Coupling

Anomalous Hall Effect

𝐻𝑆𝑂 = 𝜉𝑆𝑂𝑳 ⋅ 𝑺

6

Outline






7

8

What happens when we excite a spin?

animations courtesy of Helmut Schultheiss

H

9

Collective Spin Excitations

animations courtesy of Claudia Mewes

Ferromagnetic Resonance (FMR) ModeAll spin precess in phase

Other Spinwave (Magnon) ModesNon-zero phase relationship

Landau-Lifshitz Equation

( ) ( ) HMMM

HMdt

Md

s

−−=0

0

Landau-Lifshitz equation


H

Larmor Term

Evgeny Lifshitz

Lev Landau

11

Landau-Lifshitz Equation

( ) ( ) HMMM

HMdt

Md

s

−−=0

0

Landau-Lifshitz equation

H

Larmor Term Damping Term


Evgeny Lifshitz

Lev Landau

12

Ferromagnetic Resonance (FMR)

Cavity Based ➔ single frequency!

Srivastava, JAP 85 7841 (1999)

A. Okada, PNAS, 114, 3815 (2017) 13

Constant Frequency, vary angle

Vector Network Analyzer FMR (VNA-FMR)

Vector-network-analyzer4/ 24/ 11 2:19 PMImages: E8363C PNA Microwave Network Analyzer| Agilent

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E8363C PNA Microwave Network Analyzer

© Agilent 2000-2011

Port 2Port 1

50 coplanar waveguide

H

2.85 2.86 2.87

0.22

0.24

0.26

0.28

0.30

0.32

2.85 2.86 2.87

0.10

0.12

0.14

0.16

0.18

0.20

Im[S

21 ]

Re

[S21]

0H (T)

0H (T)

𝑀eff: Effective magnetization = Ms - Hk

Hk: perpendicular anisotropy field𝑀S: Saturation magnetizationΔ𝐻: Linewidth

𝜒(𝐻, 𝑓) =𝑀𝑒𝑓𝑓(𝐻 −𝑀𝑒𝑓𝑓)

(𝐻 −𝑀𝑒𝑓𝑓)2 − 𝐻𝑒𝑓𝑓

2 − 𝑖Δ𝐻(𝐻 −𝑀𝑒𝑓𝑓)

𝑆21 = 𝐴 𝑒𝑖𝜑 𝜒 𝐻, 𝑓 + 𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑

Amplitude

Phase

Resonance Field, Linewidth

14

VNA-FMR

15

Outline






16

Magnetic Anisotropy

Why important?

Seagate

Magnetic Anisotropy gives a preferred orientation to the spontaneous magnetization

Simple model of Anisotropy Energy

∆𝐸 = 𝐴𝜉𝑆𝑂4𝜇𝐵

(𝜇𝐿⊥ − 𝜇𝐿

||)

P. Bruno, PRB 39, 865 (1989)

∆E magnetic anisotropy energy SO spin-orbit coupling constantA prefactor (< 0.2) - function of electronic structureµL orbital moment

Can originate from:• Magnetocrystalline• Interfaces • Strain (Magnetostriction)• Shape (Magnetostatic energy)

Duke U

Magnetization Direction

Ener

gy

M

M

M∆E

17

How do we measure anisotropy?

0 10 20 30 40 50 60-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Data

Fit

0H

res (

T)

Frequency (GHz)

𝑓(𝐻𝑟𝑒𝑠) =𝑔||𝜇0𝜇𝐵2𝜋ℏ

𝐻𝑟𝑒𝑠 + 𝐻𝐾 𝐻𝑟𝑒𝑠 +𝑀𝑒𝑓𝑓 +𝐻𝐾

𝑀𝑒𝑓𝑓 = 𝑀𝑠 − 𝐻𝐾⊥

H

18

20 nm Ni80Fe20 (Permalloy)

• Angular Dependence of Resonance Field (Hres)

Can include anisotropy terms of any symmetry (see Farle, RPP (1998) for review)

Effective Magnetization:

• Frequency (f) Dependence of Resonance Field (Hres)

Kittel Equation:

A. Okada, PNAS, 114, 3815 (2017)

In-plane anisotropy field

Perpendicular anisotropy field

magnetmagnetsample

How do we measure orbital moments?

0 10 20 30 40 50 60-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Data

Fit

0H

res (

T)

Frequency (GHz)


𝐻𝑟𝑒𝑠 + 𝐻𝐾 𝐻𝑟𝑒𝑠 +𝑀𝑒𝑓𝑓 +𝐻𝐾

𝑀𝑒𝑓𝑓 = 𝑀𝑠 − 𝐻𝐾⊥

H

19

20 nm Ni80Fe20 (Permalloy)

Effective Magnetization:

• Frequency (f) Dependence of Resonance Field (Hres)

Spectroscopic g-factor

𝑔 = 2 +2𝜇𝐿𝜇𝑆

g-factor relates the spin (μS) and orbital (μL) moments:

Since μS >> μL , g is typically slightly larger than 2 in ferromagnets

g-factor and orbital moments

Permalloy

Thickness

g−factor Method Frequency

range

Reference

5 –50 nm

(extrapolated to

bulk value)

2.109 ± 0.003 VNA-FMR with

asymptotic

analysis

4—60 GHz This work

50 nm 2.20 ± 0.12 Einstein-de-Haas — [27]

50 nm 2.0—2.1 PIMM 0—2 GHz [28]

50 nm 2.1 PIMM 0—3 GHz [29]

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]

bulk 2.12 ± 0.03 FMR 24 GHz [9]

𝑔 = 2 +2𝜇𝐿𝜇𝑆

g-factor relates the spin (μS) and orbital (μL) moments:

Since μS >> μL , g is typically slightly larger than 2 in ferromagnets

Consider g = 2.05 +/- 0.02 (1% error) ➔ 40% error in μL

PROBLEM: Reported values of g-factors can vary considerably

PROBLEM: A small error in g leads to large error in μL

20

g-factor: Asymptotic Analysis

fup

fup

fup

0 10 20 30 40 50 601.98

2.00

2.02

2.04

2.06

2.08

2.10

2.12

Fitte

d g

-facto

r valu

e

Upper Fitting Frequency, fup

(GHz)

flow

flow

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Data

Fit

0Hres (

T)

Frequency (GHz)

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Data

Fit0Hres (

T)

Frequency (GHz)

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Data

Fit

0Hres (

T)

Frequency (GHz)

flow

gfit asymptotically approaches a value as fup➔∞

Shaw, JAP 114 243906 (2013)

0 10 20 30 40 50 60-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Data

Fit

0H

res (

T)

Frequency (GHz)

𝑀𝑒𝑓𝑓 = 𝑀𝑠 −2𝐾

𝜇0𝑀𝑠


𝐻𝑟𝑒𝑠 𝐻𝑟𝑒𝑠 +𝑀𝑒𝑓𝑓

20 nm Ni80Fe20

21

We make the assumption that the

y-intercept (limit of fup➔∞) is the intrinsic value of g and Meff.

Can achieve a precision better than 0.2 % in permalloy

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

0.00 0.05 0.10 0.152.14

2.15

2.16

2.17

2.18

2.19

2.20

Out-of-plane

In-plane

g -

facto

r

1 / thickness (nm-1)

g┴

g||

J.M. Shaw, et al., Phys. Rev. B, 87, 054416 (2013)

CoFeNi

CoFeNi

CoFeNi

CoFeNi

CoFeNi

CoFeNi

CoFeNi

CoFeNi

3 nm Ta

5 nm Cu

3 nm Cu

3 nm Ta

CoFe/NiMultilayer

•Anisotropy is tuned by varying the thickness.

23

H

H

0.00 0.05 0.10 0.15-1

0

1

2

3

4

5

6

Anis

otr

opy,

K (

10

5 J

/m3)


Application of g-factor analysis: CoFe/Ni

K anisotropy energy density spin-orbit coupling constantA prefactor due to electronic structureN number atoms per unit cellV volume of unit cell

𝐾 = − 𝐴𝜉𝑆𝑂𝑁

4𝑉

(𝜇𝐿⊥ − 𝜇𝐿

||)

𝜇𝐵

A test of Bruno’s theory:

P. Bruno, PRB 39, 865 (1989)

0 1 2 3 4 5 60.000

0.005

0.010

0.015

0.020

Orb

ital M

om

ent A

sym

metr

y

(L

| -

L

||)

/

B

Anisotropy Energy, K (105J/m3)

Fit to the data: A = 0.097 ± 0.007

J.M. Shaw, et al., Phys. Rev. B, 87, 054416 (2013)

Compare to previous reports:A = 0.2 at RT, Au/Co/Au

D. Weller, et al. PRL 75, 3752 (1995)

A = 0.1 at near 0 K, Ni/Pt MLs F. Wilhelm, et al. PRL 75, 3752 (1995)

A = 0.05 at near 0 K, Fe/V MLsA.Anisimov, et al. PRL 82, 2390 (1999)

24

Outline






25

Why is damping important?

Data Storage & Magnetic Memory (STT-RAM)

Spin-torque nano-oscillators

13.10 13.15 13.20

0

2000

4000

6000

8000

Spe

ctra

l Den

sity

(nV

2 /H

z)

Frequency (GHz)

500nm50 nm nano-oscillators

Word line

Bit linefixed layertunnel barrierfree layer

2-terminal Spin-Transfer Torque (STT)

Magnonics and Spin Logic

Spin current source

Nakayama, IEEE Trans Mag. (2010)

K.Wang UCLAH.Schultheiss, HZDR

effs

c Hpg

VMeI

)(

20

=

Critical current needed to switch an MRAM cell

26

Mangin APL (2009)

Intrinsic Damping Theories

• Breathing Fermi surface model:

• Generalized torque correlation model:

• Scattering theory:

• Numerical realization of torque correlation via linear response:

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)

Gilmore, K., Idzerda, Y. U. & Stiles, M. D. Phys. Rev. Lett. 99, 027204 (2007)Thonig, D. & Henk, J. New J. Phys. 16, 013032 (2014)

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)

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

𝑑𝑀

𝑑𝑡= − 𝛾 𝜇0 𝑀 ×𝐻 −

𝜶 𝛾 𝜇0𝑀𝑠

𝑀 × 𝑀 × 𝐻

Intraband versus Interband

Gilmore and Stiles, PRB 81, 174414 (2010)

Intraband“Breathing Fermi Surface”

“Conductivity-like damping”

Interband“Bubbling Fermi Surface”“Resistivity-like damping”

t1t2

Inverse scattering time (1/τ)Temperature ➔Defect density ➔

29

Intrinsic Damping: Critical Parameters

Torque between spin and

orbital moments SO

30

𝛼 ∝

𝑚,𝑛

Γ𝑚𝑘,𝑛𝑘− 2

𝑊𝑚𝑘,𝑛𝑘

𝑘

Kamberský’s Torque Correlation model:

Sum over all electronic states, bands

uniform mode dissipation in lattice

spin-orbit damping

spectral overlapfunction of Bloch state lifetime

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

SO : Spin-Orbit Interaction

τ: Spin scattering times

Courtesy of Claudia Mewes

Critical Parameters affecting intrinsic damping:

Origin of damping

𝑑𝑀

𝑑𝑡= − 𝛾 𝜇0 𝑀 ×𝐻 −

𝜶 𝛾 𝜇0𝑀𝑠

𝑀 × 𝑀 × 𝐻

• Originally just a phenomenological parameter

• Theory has worked to calculate damping parameter α from first principles Intrinsic

Damping

In reality, α includes many intrinsic and extrinsic sources and is not necessarily a constant

Magnon-electron scattering

Magnon-phonon scattering

Spin-currents

Inhomogeneity

Defect scattering

Coherent Spin Excitation

(Magnon)

Electrodynamics

31

How Do We Measure Damping?

LL equation predicts linewidth is proportional to frequency

∆𝐻 =2ℎ𝜶

𝑔μ0μB𝑓

However, that is not always observed

32

0 10 20 30 400

2

4

6

8

10

12

14

Lin

ew

idth

(m

T)

Frequency (GHz)

Linewidth Analysis

Point of emphasis:Broadband measurements are critical.

❑ Inhomogeneous Broadening❑ Low-field losses❑ 2-magnon scattering❑ Spin-pumping❑ Radiative Damping❑ k2 Damping❑ Other Contributions

❑ Eddy Currents❑ Slow relaxer❑Motional Narrowing❑ Chiral Damping

❑ Intrinsic Damping

• Many relaxation mechanisms and contributions to linewidth exist

• Careful separation and quantification of all mechanisms is needed to compare with theory

33

Inhomogeneous Linewidth




∆𝐻 =2ℎ𝜶

𝑔μ0μB𝑓


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

34

H

f

0

2

B

h

g

Theoretical ideal…

… what we usually see

+ ∆𝑯𝟎

∆H0

𝐻


0 10 20 30 40 500

50

100

150

200

250 55 nm Cu

30 nm Cu

20 nm Cu

15 nm Cu

Lin

ew

idth

(m

T)

Frequency (GHz)

5 nm CuTa

x Cu

CoFe/Ni

TaCu5 nm

55 nm

200 nm

200 nm




Heinrich, JAP, 57, 3690 (1985)Farle, Rep. Prog. Phys. 61, 755 (1998)McMichael, PRL 90, 227601 (2003) 35

∆𝐻 =2ℎ𝜶

𝑔μ0μB𝑓


+ ∆𝑯𝟎

Low Field Loss

• At low fields (frequency), sample may not be saturated.➔ Linewidth Broadening

0 2 4 6 8 100

2

4

6

8

10 Data

Combined fit

Low-field loss contribution

Damping + H0

Lin

ew

idth

(m

T)

Frequency (GHz)

nfieldlowf

H

= −

•Described by a power law fit

E. Schloemann: JAP 41, 204 (1970) 5 10 15 20 25 30 35 40 4510

12

14

16

Lin

ew

idth

(m

T)

Frequency (GHz)

•Effect can extend to higher frequencies for some materials



❑ Intrinsic DampingPermalloy Co2MnGe

36

2-Magnon Scattering




k

ω

k

ω

FMRFMR

In-plane geometry Out-of-plane geometry

Sparks, Phys Rev. 122, 791 (1961)Hurben, JAP 83, 4344 (1989)McMichael, JAP 91, 8647 (2002)Lindner, PRB 68, 060102 (2003)Barsukov, PRB 84, 140410 (2011)

0 5 10 15 20 25 30 35 40 45 50 55

0

5

10

15

20

25

Perpendicular

In-plane

Lin

ew

idth

(m

T)

Frequency (GHz)

CoFe

Al (5 nm)

Ti(3 nm)

CuN (3 nm)

SiO2

0 5 10 15 20 25 30 35 40 45 50 55 60

0

5

10

15

20

25

Perpendicular

In-plane

Lin

ew

idth

(m

T)

Frequency (GHz)

CoFe

Al (5 nm)

SiO2

• Mediated by Defect Scattering

2-magnon scattering

HH

• Can reduce/eliminate with out-of-plane geometry

arXiv:1804.0878637

Spin-Pumping & Spin-Memory Loss




Y. Tserkovnyak, PRL 88, 117601 (2002)O.Mosendz, PRB 82, 214403 (2010)

t = thickness

g = g-factor

g↑↓ = effective real part spin-mixing conductance

𝛼 𝑡 = 𝛼𝑖𝑛𝑡 + 𝑔𝜇𝐵𝑔𝑒𝑓𝑓↑↓

4𝜋𝑀𝑠

1

𝑡

Can be phenomenologically treated as an interfacial damping term

0.00 0.05 0.10 0.15 0.20 0.25 0.300.000

0.001

0.002

0.003

0.004


Da

mp

ing

Pa

ram

ete

r,

Oxidized Si

5nm Ta

Co2MnGe

5nm Ta

y-intercept is the damping without spin-pumping contribution

38

Ferromagnet

s

Normal Metal

Flow of spin current

H

Spin-Pumping

Spin-Memory Loss

Loss of angular momentum at or across an interface

Radiative Damping

Damping contribution due to inductive coupling of the sample to the waveguide

δ = thicknessZ0 = waveguide impedance

M.Schoen, PRB, 92, 184417 (2015)

CoFe

CoFe

200 μm Glass

30 nm Co25Fe75 Sample

0 2 4 6 8 10 12 14 16 18 20 220

1

2

3

4

= 0.0017

no spacer

200 um glass spacer

Lin

ew

idth

(m

T)

Frequency (GHz)

= 0

.0021




39

Wavevector Dependent Damping




100 200 300 4000.007

0.008

0.009

0.010

0.011

NiFe film

Center Mode

Da

mp

ing

Pa

ram

ete

r,

Nanomagnet Width (nm)

End Mode

Nembach PRL 110, 117201 (2013)

Theory predicts a wavevector (k) dependence of damping |k|2

Y. Tserkovnyak Phys. Rev. B 79, 094415 (2009)S. Zhang Phys. Rev. Lett., 102, 086601 (2009)

Spin-current Spin-

current

➔Generation of intralayerspin currents

40

100 nm

Other Sources of Linewidth and Damping




Slow Relaxer (Resonant Damping)

Motional Narrowing

Eddy Currents

Chiral Damping

Van Vleck, PRL 11, 65 (1963).Woltersdorf, PRL 102, 257602 (2009)Nembach, PRB, 84, 054424 (2011)

δ = thicknessρ = resistivityC = correction factor

Generally not significant for metallic thin films < 20 nm

J. M. Lock, British JAP 17, 1645 (1966)C. Scheck, APL 88, 252510 (2006)

A. Okada, PNAS, 114, 3815 (2017)E. Jue, Nature Materials 15, 272 (2015)

41

Intrinsic and Total Damping

𝜶𝑻𝒐𝒕𝒂𝒍 𝒇 = 𝛼𝑖𝑛𝑡 + ∆𝛼𝑖𝑛ℎ𝑜𝑚𝑜 + Δ𝛼2−𝑚𝑎𝑔 + Δ𝛼𝑠𝑝𝑖𝑛−𝑝𝑢𝑚𝑝𝑖𝑛𝑔

Measure in perpendicular geometry

Broadband measurements to exclude non-linear, low field/freq data

Thickness series, or restrict to ≲ 20 nm, or insulators

Decrease inductive coupling with spacer between CPW and sample

Thickness series needed to quantify contribution

Broadband measurements and/or sample series needed to quantify

What we measure

• To compare with theory, we must account for or eliminate additional damping terms.

42

Parameter of interest +Δ𝛼𝑙𝑜𝑤−𝑓𝑖𝑒𝑙𝑑 + Δ𝛼𝑟𝑎𝑑 + Δ𝛼𝑒𝑑𝑑𝑦 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 + Δ𝛼𝑜𝑡ℎ𝑒𝑟

Broadband MeasurementRequired

Why is LW analysis so important?

Schoen, Nature Physics 12, 839 (2016)

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

Mankovsky, PRB 87, 014430 (2013)

Turek, PRB 92, 214407 (2015)

0.0 0.2 0.4 0.6 0.8 1.00.000

0.001

0.002

0.003

0.004

0.005

mixed fcc

rad

sp

int

Da

mpin

g

Co Concentration

bcc

tot

43


Schoen, Nature Physics 12, 839 (2016)

Damping is strongly controlled by the DOS at EF

0.0 0.2 0.4 0.6 0.8 1.00.000

0.001

0.002

0.003

0.6

0.9

1.2

1.5

bcc mixed fcc

Theory

Mankovsky et al.

b

n(E

F ) (eV

-1)

Intr

insic

Da

mp

ing

,

int

Co Concentration

44

DOS and damping

Alloying in Ge in Co50Fe50

enhances the “pseudo-gap” a moves it to the Fermi level

H.Lee, APL 95, 082502 (2009) J.Chico, PRB, 93, 214439 (2016) Mizukami, JAP, 105, 07D306 (2009)

Co2FeAl

0 10 20 30 40 50 60 700.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

A2 --> B2

B2 --> L21

Ca

lcu

late

d D

am

pin

g P

ara

me

ter,

n(EF) (states/eV)

Co2MnGe

J.Shaw, PRB, 97, 094420 (2018) Pradines, PRB, 95, 094425 (2017)

Co2MnSi

Co50Fe50Gex

45

How can we control Damping?

How do we reduce spin pumping?

0.0 0.1 0.2 0.3 0.4 0.5 0.60.000

0.001

0.002

0.003

0.004

0.005

0.006

to

tal


CoFe

Ta (3 nm)Cu (3 nm)

Ta(3 nm)Cu (3 nm)

SiO2

CoFe

Al (5 nm)

Ti(3 nm)Cu (3 nm)

SiO2

Replace Ta with low atomic number elements or insulators

2 nm film with a totaldamping of 0.0018 !!!

46

11, 054036 (2019)

E. Edwards, Phys. Rev. Applied 11, 054036 (2019)


“We find that the magneto-crystalline anisotropy and the Gilbert damping constant show a linear relationship.“

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

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

“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”

“...a linear relation between the perpendicular magnetic anisotropy and α is established”

Consider Relationship between Perpendicular Anisotropy (PMA) and Damping

47

Relationship between PMA and α?

Both anisotropy and damping are proportional to SO to some power

Co90Fe10/Ni MultilayersAnisotropy is tuned by varying the total thickness.

J.M. Shaw, APL 105, 062406 (2014)

• No relationship between anisotropy and PMA➔spin-orbit parameter does not

vary

• Anisotropy results from orbital moment asymmetry NOT increased spin-orbit ➔ Independent of damping.

𝐾 = − 𝐴𝜉𝑆𝑂𝑁

4𝑉

Δ𝜇𝐿𝜇𝐵

48

0.000

0.005

0.010

0.015

0.020

0

5

10

15

20

25

30

-0.5 0.0 0.5 1.00.000

0.005

0.010

0.015

0.020

Da

mp

ing,

Inh

om

og

en

ou

s L

W

0

H0 (

mT

)

In-plane

L/

B

Total Anisotropy Field (T)

Out-of-plane

Outline






49

Spin-Orbit Torques (SOT)

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

Charge Spin

FM

Heavy Metal

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

Spin-Hall EffectRashba-Edelstein Effect

inverse Spin-Hall Effectinverse Rashba-Edelstein Effect

50

Spin-Hall Effect“Damping-like Torque”

Rashba-Edelstein Effect (intefacial)“Field Like Torque”

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

Spin-Orbit Torques (SOT)

𝐸

I

𝑀

NM

charge current

FM

𝑴(𝑡)

V. Amin and M.Stiles PRB 94, 104420 (2016)V. Amin and M.Stiles PRB 94, 104419 (2016) 51

𝑆𝑂𝑇 ∝ 𝑀 × (𝑀 × റ𝑆) 𝑆𝑂𝑇 ∝ 𝑀 × റ𝑆

റ𝑆

Why are SOT important?

Efficient switching of magnetic memory

Word line

Bit linefixed layertunnel barrierfree layer

2-terminal Spin-Transfer Torque (STT)

Shared read/write path: Large current density for write

operations can damage tunnel barrier

Read operations have finite probability of flipping the bit

Separate read and write paths✓ Can apply large current density

through write line without damaging MTJ

Word line

fixed layertunnel barrierfree layer

3-terminal Spin-Orbit Torque (SOT)

52

SOT Measurements Methods

…and now VNA-FMR...at the film level

Harmonic Method with Hall Cross

Spin-TorqueFerromagnetic Resonance

Inverse SHE DC devices

Inverse SHE AC devices

Hayashi et al. PRB 89, 144425 (2014)

Liu et al., PRL, 106 (2011)

M. Weiler, PRL 113, 157204 (2014)

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

53

Measured Inductances in VNA-FMR

Cu

𝐻dipole

NiFe𝜕𝑡 ෝ𝑚

𝑚𝑦

𝑊wg

Berger, et al. PRB 97 094407 (2018)

Δ𝑆21 ≈ −𝑖𝜔

𝑍0𝐿𝑁𝑖𝐹𝑒 + 𝐿𝐶𝑢

𝐿FM =𝜇0𝑙𝑑FM4𝑊wg

𝜒𝑦𝑦

~0

~ Constant signal as a function of frequency

Measured signal

H0 microwave out

microwave in

sample

𝑧

𝑥

𝑦 𝑙

54

7.5 GHz 35 GHz

Measured Inductances in NiFe/Pt

Total Inductance

1. Dipolar inductive coupling, ~ frequency-independent2. Second-order inductive coupling ~ 𝑖𝜔 (Faraday-induced currents in the NM)3. Damping-like ~ 𝜔 (e.g. spin Hall effect)4. Field-like ~ 𝑖𝜔 (e.g. Rashba-Edelstein effect)

The inductance (L) of the sample arises from anything that produces a flux around the CPW

𝐿 =Φ

𝐼CPW= 𝐿NiFe + 𝐿𝑃𝑡(𝜔)

Spin-charge transduction, a.k.a Spin-Orbit Torque (SOT)

H0 microwave out

microwave in

sample

𝑧

𝑥

𝑦 𝑙

𝐿NiFe

𝐿𝑃𝑡(𝜔)

𝐻dipole𝐻NM

𝐽spin Hall𝐽Rashba−Edelstein

𝐽FaradayPt

𝑄 Ƹ𝑠

Py

𝜕𝑡 ෝ𝑚𝑚𝑦𝐻eff

CPW

𝑧

𝑥 𝑦

𝑠 NiFe

Berger, PRB 97 094407 (2018)55

Measurement of SOT in NiFe/Pt

Re 𝐿 ∝ 𝜇0𝑙𝑑𝐹𝑀 + 𝐿12ℏ𝜔

𝑒𝜎𝑓𝑙SOT − 𝜎𝑓𝑙

F

Im(𝐿) ∝ −𝐿12ℏ𝜔

𝑒𝜎𝑑𝑙SOT

7.5 GHz 35 GHz

Δ𝑆21 ≈ −𝑖𝜔

𝑍0𝐿𝑁𝑖𝐹𝑒 + 𝐿𝑃𝑡(𝜔)

Field-like terms

Damping-like terms

Change in both Amplitude and Phase

Frequency Dependent

Berger, et al. PRB 97 094407 (2018)56

57

Inconsistency in spin diffusion length

From damping measurements

From spin Hall measurements

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

Nakayama, PRB 85, 144408 (2012)

Feng, PRB 85, 214423 (2012)

Application: Spin-Diffusion Length

PROBLEM: Inconsistency in Spin diffusion length in Pt

Berger, et al. PRB 97 094407 (2018)

Spin-pumping ➔ < 2 nm

iSHE measurements ➔ 4-8 nm

Nakayama, PRB 85, 144408 (2012)

Feng, PRB 85, 214423 (2012)

VNA-FMR ➔self-consistent, simultaneous measurement of spin-pumping and iSHE

FM

x Pt

58

59

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

Acknowledgements

NIST (Boulder)Hans NembachTom SilvaMartin Schoen (U.Manitoba)Mathias Weiler (WMI, Munich)Eric Edwards (IBM)Carl BooneMike SchneiderMatt PufallEmilie JueAndy Berger (SRS)

Uppsala UniversityOlle ErikssonDanny ThonigErna Delczeg-CzirjakYaroslav KvashninOlof “Charlie” Karis

Juriaan LucassenBert Koopmans

Eindhoven UniversityT.U. Munich

Christian Back

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

0.0 0.2 0.4 0.6 0.8 1.00.000

0.001

0.002

0.003

0.6

0.9

1.2

1.5

bcc mixed fcc

Theory

Mankovsky et al.

b

n(E

F ) (eV

-1)In

trin

sic

Da

mpin

g,

int

Co Concentration

0 1 2 3 4 5 60.000

0.005

0.010

0.015

0.020

L /

B

K (105J/m

3)

61

understanding spin orbit phenomena...g-factor: asymptotic analysis f up f up f up 0 10 20 30 40 50...

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