Download - 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
Bert Koopmans, September 2005 Summerschool Constanta 2
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
Bert Koopmans, September 2005 Summerschool Constanta 3
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
Bert Koopmans, September 2005 Summerschool Constanta 4
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
Bert Koopmans, September 2005 Summerschool Constanta 5
Statics (“macrospin”, small particle)Ha
Mx
H
MSMR
E
HC
θ
Bert Koopmans, September 2005 Summerschool Constanta 6
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
Bert Koopmans, September 2005 Summerschool Constanta 7
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:
Bert Koopmans, September 2005 Summerschool Constanta 8
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛×+×=
dtMdM
MHM
dtMd
seff
rrrr
rαγµ0
30 GHz / T
Landau-Lifshitz-Gilbert Eq.
Spin Precession Damping
α ~ 0.01
effL Hf 02µ
πγ
=
Bert Koopmans, September 2005 Summerschool Constanta 9
Examples of Precessional Dynamics
Bert Koopmans, September 2005 Summerschool Constanta 10
A real experiment
Rietjens, Jozsa (TU/e)
Bert Koopmans, September 2005 Summerschool Constanta 11
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
Bert Koopmans, September 2005 Summerschool Constanta 12
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
Bert Koopmans, September 2005 Summerschool Constanta 13
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
−
−
=
=
Bert Koopmans, September 2005 Summerschool Constanta 14
( ) ( )
( ) ( )s
s
MHiHi
MHiHi
1000
10
20
−
−
+⋅−=−
+−=
µεγµγµε
µω
γµω ( )
s
s
MHH
MHH
10
2
100
−
−
+=
+=
µε
µγµω
Bert Koopmans, September 2005 Summerschool Constanta 15
(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
Bert Koopmans, September 2005 Summerschool Constanta 16
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
Bert Koopmans, September 2005 Summerschool Constanta 17
Switching/NoSwitching diagrams
y
x
z
Hpulse
Mi
Mf
experimentexperiment
macrospinmacrospintheoryntheoryn
2 x 4 µm PermalloySchumacher et al., PRL 90, 017201 (2003)
Bert Koopmans, September 2005 Summerschool Constanta 18
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…
Bert Koopmans, September 2005 Summerschool Constanta 19
Bert Koopmans, September 2005 Summerschool Constanta 20
Where are we…
Introduction
Local dynamics: “Macro-spin” behavior
Measuring precessional dynamics
Frequency domain
Time domain
All-optical techniques
Nonlocal dynamics: Spin waves and confined structures
Outlook & Summary
Bert Koopmans, September 2005 Summerschool Constanta 21
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
Bert Koopmans, September 2005 Summerschool Constanta 22
Ferromagnetic Resonance
res
res HdH
d
ωωα
ωω
∆=
∆=∆
Damping as well:
Bert Koopmans, September 2005 Summerschool Constanta 23
Brillouin Light Scattering
ω
ω ± ωM
ωM
hω ~ 1 eVhωΜ ~ 10-4…10-6 eV
Hillebrands, U. Kaiserslautern, website
Bert Koopmans, September 2005 Summerschool Constanta 24
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
Bert Koopmans, September 2005 Summerschool Constanta 25
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
Bert Koopmans, September 2005 Summerschool Constanta 26
Stroboscopic: Pump-probe Optics
MO rotation
pump
probe
spin system
laser pulses~ 50 fs
polarizedlight
time delayjitter < fs
pulses < 30 fs
Bert Koopmans, September 2005 Summerschool Constanta 27
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
Bert Koopmans, September 2005 Summerschool Constanta 28
PIMM (“real time FMR”)
Silva et al., JAP 85, 7849 (1999)
Bert Koopmans, September 2005 Summerschool Constanta 29
Where are we…
Introduction
Local dynamics: “Macro-spin” behavior
Measuring precessional dynamics
Frequency domain
Time domain
All-optical techniques
Nonlocal dynamics: Spin waves and confined structures
Outlook & Summary
Bert Koopmans, September 2005 Summerschool Constanta 30
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)
Bert Koopmans, September 2005 Summerschool Constanta 31
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
Bert Koopmans, September 2005 Summerschool Constanta 32
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
Bert Koopmans, September 2005 Summerschool Constanta 33
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]
Bert Koopmans, September 2005 Summerschool Constanta 34
Where are we…
Introduction
Local dynamics: “Macro-spin” behavior
Measuring precessional dynamics
Nonlocal dynamics: Spin waves and confined structures
Exchange-driven: Perpendicular spin waves in thin films
Dipole-driven: Lateral spin waves
Laterally confined structures
Outlook & Summary
Bert Koopmans, September 2005 Summerschool Constanta 35
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
Bert Koopmans, September 2005 Summerschool Constanta 36
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
Bert Koopmans, September 2005 Summerschool Constanta 37
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
Bert Koopmans, September 2005 Summerschool Constanta 38
Bert Koopmans, September 2005 Summerschool Constanta 39
Optically Probing Spin Waves: Analysis
Observed:2
0 Dk+= ωω
Conclusions:
Boundary conditions: Free surface
D = 0.44 eVA2 (as wexpected)
Bert Koopmans, September 2005 Summerschool Constanta 40
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
Bert Koopmans, September 2005 Summerschool Constanta 41
Artificial Spin-Chains: Basic Results
300 nm pillars
1+⋅−= ii SSJHrr
Hstray
1d-Heisenberg system
Bert Koopmans, September 2005 Summerschool Constanta 42
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…
Bert Koopmans, September 2005 Summerschool Constanta 43
Spin waves – Dipole driven
Three sorts
k
M
k k
M M
( )sMHH 100−+= µγµω H0γµω =
Bert Koopmans, September 2005 Summerschool Constanta 44
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 =
Bert Koopmans, September 2005 Summerschool Constanta 45
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
Bert Koopmans, September 2005 Summerschool Constanta 46
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…
Bert Koopmans, September 2005 Summerschool Constanta 47
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
Bert Koopmans, September 2005 Summerschool Constanta 48
Damping by Emission of Spin Waves
Strip line
Current pulse
Excitation
MSSM
Hpulse
MSSM
MSBVM
Excitation
Bert Koopmans, September 2005 Summerschool Constanta 49
Observation of localized modes
TR-MOKE
µ-BLS
TR-MOKEµ-BLS
Perzlmaier, PRL 94, 057202 (2005)
4 µm
Vortex state
Bert Koopmans, September 2005 Summerschool Constanta 50
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
Bert Koopmans, September 2005 Summerschool Constanta 51
Bert Koopmans, September 2005 Summerschool Constanta 52
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
Bert Koopmans, September 2005 Summerschool Constanta 53
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
Bert Koopmans, September 2005 Summerschool Constanta 54
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
Bert Koopmans, September 2005 Summerschool Constanta 55
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
Bert Koopmans, September 2005 Summerschool Constanta 56
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
Bert Koopmans, September 2005 Summerschool Constanta 57
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