precessional magnetization dynamics

PRECESSIONAL MAGNETIZATION DYNAMICS

(in the f- and t-domain)

Bert KoopmansEindhoven University of Technology, The Netherlands

©Zwampook

Bert Koopmans, September 2005 Summerschool Constanta 1


Hard disk: data rate road map

1 Gb/s1 bit read in 1 ns

http://www.research.ibm.com/journal/rd/443/thompson.html

Inte

rnal

dat

a ra

te (G

b/s)

1960 1970 1980 1990 2000 2010

1

10-1

10-2

10-3

10-4


Magnetic time scales

s

ms

ms

ns

ps

fs

hr

yr100yr Recording stability

Coherent dynamics

Creep

Superparamagnetic limit

Laser-induced (de)magnetizationSpin-lattice relaxationPrecessional switchingGilbert damping

Domain wall motionThermally assisted

Viscous

GHz data rate

Thermally activated

CoherentPhase change

Precession


This Lecture

Introduction

Local dynamics: “Macro-spin” behavior

From thermally-driven to precessional (LLG) dynamics

Precessional modes in thin films (Kittel relation)

Precessional switching

Measuring precessional dynamics

Nonlocal dynamics: Spin waves and confined structures

Summary


Statics (“macrospin”, small particle)Ha

Mx

H

MSMR

E

HC

θ


Dynamic CoercivityMx (T = 0, arbitray sweep time)Mx (T = finite, infinitely slow)

Mx (T = finite, fast)

HHc Hc

Thermal activation

Observed for: • Small particles• Domain wall “unpinning”

Ha


Spin precession

z ( = quantization axis) Ψ = (c↑, c↓) |c↑|2 + |c↓|2 = 1

y e.g.: (1, 0) : along +z (1, exp(iφ))/√2 : cos φ x + sin φ y(1, 1)/√2 : x (cos(θ/2), sin(θ/2)) : cos θ z + sin θ xx

γµ0H )sin,...(cos

)sin,cos()(

2/

2

2/

2/

θθ

θθ

h

hh

Eti

tiEtiE

e

eet∆=

=Ψ ↓↑

x

z

θ

h

HL

0γµω =

precessing spin at frequency:

γ ~ 10-4 eV/Th ~ 1 eV fsso, GHz-

Switching on a field along z:


( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛×+×=

dtMdM

MHM

dtMd

seff

rrrr

rαγµ0

30 GHz / T

Landau-Lifshitz-Gilbert Eq.

Spin Precession Damping

α ~ 0.01

effL Hf 02µ

πγ

=


Examples of Precessional Dynamics


A real experiment

Rietjens, Jozsa (TU/e)


The effective field =Applied field +

Shape anisotropy:

Crystalline anisotropy:

Exchange:

zMzMN zzzz ˆˆ 10−−=−= µMNHeff

rr⋅−= Thin film:

( )MEM

H aniseff

rrr∇−=

1

MMDHeff

rr2∇= 0Macro spin: =

Many shapes! (neglect it here)

Exchange stiffness


Damping of precessional modesHighly interesting and non-trivial…

… but let’s discuss it a next time…

L

MM = 0

N

S

De Haas &Einstein (1918)

I

But just let’s discuss spin-lattice relaxation in a “macroscopic” limit…

L = -M


Kittel equation – Thin films

H

M(t)

zx

Heff(t) = H x – Mz(t) zassumption: small amplitude

no dampingsolution:

My = cos(Ωt)Mz = ε sin(Ωt)

y

thin film system

^ ^

Just plug trial solution into LLG

ε2 = H / (H+Ms) < 1

Ω = γ √ H (H+Ms) Intermezzoderivation…rather than γH

zMxHH

zMyMxMM

zeff

zys

ˆˆ

ˆˆˆ1

0−−=

++=

µr

r

effHMdtMd rrr

×−= 0γµ

( )( ) yz

zsy

MHdtdM

MMHdtdM

−−=

+−= −

0

100

/

/

γµ

µγµ

( )H

MHii s

−⋅−=−+⋅−= −

0

100

γµεωµεγµω

)()(

ba

( )

( ) MieM

MeMkzti

z

kztiy

δε

δω

ω

−

−

=

=


( ) ( )

( ) ( )s

s

MHiHi

MHiHi

1000

10

20

−

−

+⋅−=−

+−=

µεγµγµε

µω

γµω ( )

s

s

MHH

MHH

10

2

100

−

−

+=

+=

µε

µγµω


(Reversal by) Dampingwith damping

solution:My = cos(ωt) exp(-t/τ)Mz = ε sin(ωt+φ) exp(-t/τ)

H

M(t)

zx

y

derive it yourself!

( )( )sMH 1

00

2

212

−++

=µαγµ

ατ

sMαγταωτ/2

1

== −

::

0

0

s

s

MHMH

<<>>

µµ

indep. of H α

πα/ps 10

periods 2/1≈≈

Switching: τs >> 1 ns for α = 0.01


Precessional switching

H

M(t)

−Ms

τs ≈ πγµ0 √(H(H+Ms)

~ 150 ps

τs = =π

ωL

πγµ0H

Ms = 1 Tγµ0H = 0.01 T ~ 1.5 ns


Switching/NoSwitching diagrams

y

x

z

Hpulse

Mi

Mf

experimentexperiment

macrospinmacrospintheoryntheoryn

2 x 4 µm PermalloySchumacher et al., PRL 90, 017201 (2003)


Switching a real device

0 ps

200 ps

Applied field500 nm25

0 nm

5 nm Permalloy element (J. Miltat)

Homogeneous excitation, still strongly non-homogeneous response !!!

Due to:• Non-homogeneous groundstate

• Excitation of spin waves

Thereby a slow relaxation…


Where are we…

Introduction



Frequency domain

Time domain

All-optical techniques


Outlook & Summary


Probing spin dynamics

Frequency domain techniques• Ferromagnetic Resonance• Brillouin Light Scattering

Time-domain techniques• Using fast electronics (> 100 ps)

• Real-time scheme• Using short laser pulses (down to fs)

• Stroboscopic techniques• Scanning approaches

• Specific case: Pulsed excitation

∆ω

ωresω

t

ωn τn


Ferromagnetic Resonance

res

res HdH

d

ωωα

ωω

∆=

∆=∆

Damping as well:


Brillouin Light Scattering

ω

ω ± ωM

ωM

hω ~ 1 eVhωΜ ~ 10-4…10-6 eV

Hillebrands, U. Kaiserslautern, website


Time-Domain Techniques: Excitation

Magnetic field pulses (50 ps)

Electron bunches (~ ps)

Laser pulses (30 fs)

Or combinations thereof?• Photo switches, Breaking Schottky barrier, …

I Hpulse

strip linestrip line

e- e-e-

ω


Real time: MR detection

free

C1 C2

NM pinned

Hpulse

Jpulse

Jsense

5 x 2.3 µm Py/CoFeSchumacher et al., PRL 90, 017204 (2003)

resolution ~ 100 ps


Stroboscopic: Pump-probe Optics

MO rotation

pump

probe

spin system

laser pulses~ 50 fs

polarizedlight

time delayjitter < fs

pulses < 30 fs


Strob.: Pump-probe “Hybrid”

Electrically generated magnetic field pulses (100 ps)

Capabilities• Vectorial resolution (4-quadrant detector) • Time resolution 100 ps (“no limit” for fully optical)• Spatial resolution (~ 400 nm, diffraction limit)

I Ti/Au

PyPy Hext

Hpulse

BS

AB

L

M

strip linestrip line

MOMOprobeprobe


PIMM (“real time FMR”)

Silva et al., JAP 85, 7849 (1999)


Where are we…

Introduction



Frequency domain

Time domain

All-optical techniques


Outlook & Summary


A surprising experiment

Heating ferromagnetic Nickel with a 50 fs laser pulse

Mag

neto

-Opt

ical

Con

tras

t

Beaurepaire et al., PRL 76, 4250 (1996)


Laser-induced Demagnetization

latticeelectronslaser spins

0,0 0,5 1,00,0

0,5

1,0

mag

netiz

atio

n

temperature

Τ0

∆T

laser h

eating ∆M


All-Optical Probing of Spin Precession

-3 0 3 200 400 600

-1,0

-0,5

0,0II IIII

f = 9.98 GHzα = 0.05

ind.

Mz [

%]

∆t [ps]

IIbθc′

IIIz

θc

IMH

IIa

θc

θc′

van Kampen, Phys. Rev. Lett. 2002

∆ t

probepump

H

poly-Ni 7nm


Frequency vs. Time-Domain

0,0 0,1 0,2 0,30

3

6

9

12 microwave all-optical

9.45 GHz

frequ

ency

[GH

z]

field [Tesla]

Ω = γ √ H (H+Ms)

van Kampen (TU/e), Phys. Rev. Lett. 2002

7 nm Nickel film

0,16 0,18 0,2005101520253035

FMR9.45 GHz

abs

orpt

ion

[a.u

.]

field [Tesla]


Where are we…

Introduction




Exchange-driven: Perpendicular spin waves in thin films

Dipole-driven: Lateral spin waves

Laterally confined structures

Outlook & Summary


Sources of Non-homogeneous Response

Exchange field + finite k

Dipole field + finite k

22 DkMMDHeff ∝∇=

rrλ = 2π/k

ji SSJrr

⋅−

d = λ/2Hdip(k)k

dHeff ∝∝

1r

ω

k00

ω

k00

L

Lnk π

=

ω

k

nfinement


Spin Waves - Exchange driven

Using:

we find:

i.e., mode stiffening, independent of direction of wave vector

λ = 2π/k

ji SSJrr

⋅−

( )

MMDkHM

MDHH

eMMM

applappleff

trki

rrrrr

rrr rr

δ

δ ω

22

0

+=∇+=

⋅+= −⋅

ω

k0

J

π/a

2

2

eVA 1 2

≈

= JSaD

( )( )sMDkHDkH 10

220

−+++= µγµω


Standing Spin Wavesd

Free surface: General:

n = 1n = 0 n = 2

2

0

int

0

⎟⎠⎞

⎜⎝⎛+=

=

dnD

dzMd

πωω

δr

( ) 2

0

intint

0

⎟⎠⎞

⎜⎝⎛ +

+=

=+

dnD

MDK

dzMd s

πφωω

δδ rr

ω

k0

0

1

2

All-Optically Probing Standing Spin-WavesInhomogeneous excitation/detection

30 40 50 60 705

10

15

n = 0

n = 1

f = f0 + a / t 2

frequ

ency

, f [G

hz]

thickness, t [nm]

0 200 400 600-3

-2

-1 n = 1

n = 0

∆MZ [%

]

delay [ps]

poly-Ni

thin film

40 nm nickel



Optically Probing Spin Waves: Analysis

Observed:2

0 Dk+= ωω

Conclusions:

Boundary conditions: Free surface

D = 0.44 eVA2 (as wexpected)


And the amplitudes… (why no n = 2?)

20 30 40 50 60 700,0

0,3

0,6

0,9

1,2

1,5 A1/A0 [ measured] A2/A0

A n2 /A02

L [nm]

Laser extinction depth ~ 15 nm


Artificial Spin-Chains: Basic Results

300 nm pillars

1+⋅−= ii SSJHrr

Hstray

1d-Heisenberg system


Artificial Spin Chains: Analysis

J = 60 mT

Surprising! Mode with more nodes has lower frequency…Negative dispersion…

Yes! Syst

em likes to

be in ant

i-phase…


Spin waves – Dipole driven

Three sorts

k

M

k k

M M

( )sMHH 100−+= µγµω H0γµω =


Magnetostatic Backward Volume Mode

δHdip(k)Hdip(0)

out-of-plane phase

in-plane phase

No stray field⇒ No k-dep. restoring field

• Just replace:

then:•Limit of kd >> 1:

MBVMss AkdMM ⋅−→ −− 10

10 µµ

( )MBVMs AkdMHH ⋅−+= −100 µγµω

H0γµω =


Magnetostatic Forward Volume Mode

shear phase

No stray field

perpendicular phase

Stray field!!!⇒ k-dep. restoring field

• Now we get a stiffening, rather than a softening!

• Limit of kd >> 1: ( )sMHH 100−+= µγµω


Magnetostatic surface mode (Damon-Eshbach)

δHdip(k)Hdip(0)

out-of-plane phase

in-plane phase

+ stray field⇒ k-dep. restoring field

• Now it gets complicated:• Softening during out-of-plane phase• Hardening during in-plane phase

• The latter is known to win…


Dipolar modes - Summary

MSFWM

MSBWM

MSSM

( )sMHH 100−+ µγµ

H0γµ

0 ~ 2/dk

large k: exchange, k2

ω

Negative group velocity!

( )

m/s 10nm1/10

T 5.0 GHz/T 180

/20

3

100

≈≈

≈→−

dMH

kv sg

µγµ


Damping by Emission of Spin Waves

Strip line

Current pulse

Excitation

MSSM

Hpulse

MSSM

MSBVM

Excitation


Observation of localized modes

TR-MOKE

µ-BLS

TR-MOKEµ-BLS

Perzlmaier, PRL 94, 057202 (2005)

4 µm

Vortex state


Link with lateral spin waves4 µm

7.1 GHz

2.4 GHz

Positive dispersion:“MSSM”

Negative dispersion:“MSBVM”

k M

M

k

TR-MOKE

µ-BLS

2.1 GHz

4.0 GHz 5.5 GHz

Dynamics of Real Devices

13 µm

NiFe

CoFeAl2O3

Copper

IrMn

Ta

Seed layers

Contact padCurrent line

Stray field

Ferromagnetic coupling

Rietjens (TU/e) – Boeve (Philips Research) et al., APL submitted



Dephasing after homogeneous excitation

Raster scans at fixed time delay (50 ps steps)

Mz ~ 0.02

Mz ~ -0.02

Different frequency and damping at edges


Results: Time domain

-6 -4 -2 0 2 4 60

1

2

3

4

Position [µm]

Tim

e [n

s]

-6 -4 -2 0 2 4 60

1

2

3

4

Position [µm]

Tim

e [n

s]

Positions oftime scans

Hfm , M

Hpulse

Lower frequency, higher damping

Same frequency, higher dampingEDGES


Results: Frequency domain

-4 -2 0 2 40

50123

x-position [µm]

|Hin

tx

| [kA

/m]

Freq

uenc

y [G

Hz]

e)d)strayfieldno strayfield

strayfield

c)b)

no strayfield

a)

01234

Comparing simulations with experiment


Final analysis

Bias field dependence of uniform and localized mode

-2 -1 0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

Freq

uenc

y [G

Hz]

ExperimentSimulationNo strayfield

Hbias [kA/m]

mode

modeLocalized

Uniform


Summary

Local dynamics: “Macro-spin”• LLG equation• Kittel relation for thin films• Precessional Switching

Measuring precessional dynamics • In the f- and t-domain (including “all-optical”)

Nonlocal dynamics: Spin waves and confined structure• Exchange modes• Dipolar modes (positive and negative dispersions!)• Manifestation in confined structures• Complicated dynamics in “real” devices


©Zwampook

precessional magnetization dynamics

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