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H. Zabel 4. Lecture Magnetic domains and magnetization reversal
4. Lecture
Magnetic domains and magnetization reversal
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Content
I. Introduction and overview
II. Magnetic Domain Walls
III. Shape and size effects
IV. Stoner Wohlfarth model
IV. Superparamagnetism
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Free energy of a ferromagnetFree energy of a ferromagnet at T < Tc has two minima:
M)T(M− )T(M
In order to go from one magnetization direction to theother, an energy barrier has be overcome.
Verknüpfung mit ISING.EXE.lnk
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Thermal fluctuationsIn an infinite system, thermodynamics decides for onedomain or the other. Only close to TC, when the potential barrier is low, thermal fluctuations may be responsiblefor a spontaneous domain reversal. Therefore, below Tcthere must be another physical reason for the generationof magnetic domains.....
Ni81Fe19Initial magnetization distribution in a square 50µmx50µm Permalloy element. R. Schäfer and A. deSimoneHysteresis in soft ferromagnetic films: experimental observation and micromagnetic analysisSubmitted to IEEE Trans. Magn.
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Why do magnetic domains form?
SSSS
NNNN
Magnetic field energy in vacuum of a magnetic dipole:
dVHEDipole ∫= 20
2µ
SS
NN
NN
SS
In case of two domains, the field energyis reduced to roughly ½ of its original value
= ∫ dVHEDipole
20
221 µ
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Domain walls
Introducing more domains will reduce the field energyfurther. However, it increases the wall energy. Finding a compromise requres a finite number of domains.
D. Buntnix, PhD Thesis, Leuven, 2003
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Magnetic domain walls
For a magnetizationreversal in N steps, an exchange energy per unitwall area is reduced bythe number of steps N:
22
2
π=N
JSaNEex
180° in N=5 steps180° in one step
For a 180° magnetizationreversal in one step, an exchange energy (per wall area) has to beovercome:
222 JSa
Eex =
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Exchange versus anisotropyWithout crystal anisotropy, the domain wall width wouldbecome infinitely thick. However, with crystal anisotropythe rotation away from the easy axis costs extra energyEani = NK1a.
Thus the total energy is (w=Na):Easyaxis
Kwwa
JSEEE aniextot +=+=2
2 π
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Wall energyThe total energy with respect to the number of latticeplanes N becomes minimal if
KJ
KaJSw
Kaw
JSwEtot
∝=
=+−=∂∂
22
2
22
0
π
π
or
The total wall energy is therefore:
JKaJKSEtot ∝= /2 22π
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Wall widthUsual expressions normalized by the spin quantum number:
KJE
KJw
⋅∝
∝
energy wallDomain
width wallDomain /
Low K material
High K material
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Literature values
infinite100015-2042
Domain wall width[nm]
0Py0.042Ni0.85Co
1,23°0.05Fe
Angle betweenspins in adjacentplanes (180°/N)
Magneto-crystallineanisotropy K[MJ/m3]
Close to Tc the anisotropy energy K drops, which leads to an increase of the domain wall thickness
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Domains and domain wallsin thin films
180° Wand
90° Wand
Ideal Landau domain structure for soft magneticmaterials:
Raute pattern in case of high crystal anísotropy:
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Domain wall orientationA Bloch wall in a thin films generates stray fields in the outside region, which is unfavourable.
Néel walls become more favourable when the film thickness t becomes smaller than the wall width w: t<w
In both cases a 180° domain wall is shown with a wall width stretching over the box size.
Pictures from D. Buntnix, PhD Thesis, Leuven, 2003
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
General shape of magnetic domains
Magnetic domains in an Fe-wisker,Flux closure domains:
Magnetic domains in a thin NiFe-stripe
Perpendiculardomains in garnetfilms
R.J. Celotta
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Magnetic hysteresis
M
H
Remanent magnetization
Coercive field
Saturation magnetization
Demagnetized state
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Magnetic hysteresis
M
H
s-state at remanence
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Basic reversal mechanisms
3. Domain formation:
2. Coherent Rotation:
1. Nucleation and domain wall movement:
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
How can we tell the difference?
90°
180°
Domain wall
M
HWall motion
Wall rotation
H H H
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Pinning of domain walls
Pinning can cause Barkhausen noise when walls are unpinned and perform Barkhausen jumps in an external magnetic field.
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Domain wall propagation
Domain propagation along the easy axis, coherent rotation and propagation along the hard axis. Very small coercivity indicates high quality film with few pinning centers. K. Theis-Bröhl et al. Phys. Rev. B 53, 11613 (1996).
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
How fast do domain walls propagate?
Use the GMR effect to determine, when reversal takes place during field sweep at 20 Oe/s. Resistance measured at a rate of 10 ms.
Time variation of the resistance during the M reversal of the 400-Å NiFe layer at 77 K, which was collected at 40-ns intervals. Velocity depends linearly on applied field.
T. Ono et al. Science, 284, 468 , (1999)
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Shape induced anisotropyFe(100)/GaAs:superposition of 4-fold and 2-fold anisotropy
Polycrystalline Fe film on sapphiresubstrate: no anisotropy
PolycrystallineFe – stripes: shapeinduced anisotropy
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Nano-magnetsstripes
Theis-Bröhl et al., Bochum10 µm
bars
Temst et al., Leuven
Shinjo et al., Kyoto
disks
Klaeui et al., Cambridge
rings
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Domains in stripes as a function of aspect ratio
Diploma Thesis, Thorsten Last, RUB, 2992
Ni – stripes, MFM images Co-stripes, Kerr microscopy
B. Hausmanns, PhD Thesis, Duisburg-Essen, 2003
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Demagnetized state of different stripe arrays (Kerr-images)
Co0.7Fe0.3 stripes, w=2.4 mm, D=3 mm, thickness 80 nm,ripple domains
Fe stripesW=2.5 µmLandau domains
Co0.7Fe0.3 stripes, w=1.2 µm, D=3 µm, thickness 90 nm,head-to-head domains
H ext0°
T. Schmitte et al. JAP, 92, 4524 (2002)K. Theis-Bröhl et al. Phys. Rev B B 68, 184415 (2003).
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Coercivity of stripes
tM
wt
Coercivity of magnetic stripes is inversely proportional to the stripe width w:
wHH sia π+=
B. Hausmanns, PhD Thesis, Duisburg-Essen, 2003
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Modelling of stripe domains and domain propagation
B. Hausmanns, PhD Thesis, Duisburg-Essen, 2003, G. Nowak, Duisburg
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Simulation of a magnetization reversal process
Single bar
Reversal of interacting bars
http://magnet.atp.tuwien.ac.at/scholz/gallery/werneranim.html
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Single-Domain Circular Nanomagnetsd= diameter, t=thickness
d=300nm, t=10nm
d=100nm, t=10nm
vortex
Single domain
R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, and M. E. Welland, D. M. Tricker, PRL. 83 (1999) 1042
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Magnetization reversal of a dot
http://magnet.atp.tuwien.ac.at/scholz/gallery/werneranim.html
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Magnetization reversal in ring structures
MFM images Simulation of spin structure and magnetic divergence
D. Buntinx, PhD Thesis, Leuven, 2003
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Switching processesin mesoscopic ferromagnetic rings
PEEM image of array of rings
M. Kläui, et al. Phys. Rev. B 66, 134426 (2003)
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
From stable to unstable domains
Thermal effectsreduce coercitity
Domain wall motionDomain rotation
Particle size = domain wall width
Super-paramagnetic limitin fine particles
Multi-domain Stoner – Wohlfarthlimit for single stabledomains
Hc
unstable
1/D
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Singel domain reversal
Simple model for the magnetic energy density for a particle with a single uniaxial anisotropy:
φµφµφ sincossin 0||02
sstot MHMHKf ⊥−−=Find stable solution as a function of φ:
0;0 2
2
=∂∂
=∂∂
φφff
Yielding:
φµ
φµ
3
0
3
0|| sin2;cos2
ss MKH
MKH =−= ⊥
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Stoner-Wohlfarth asteroidThe solution describes a hypocycloid via the condition:
sK
KK MKH
HH
HH 2;1
2/32/3|| ==
+
⊥
( ) ( )kHHhhh ==+ ⊥ ;12/32/3
|| 1Mr2M
r
The magnetization direction follows from a tangent stretching from the asteroid to the tip of the field direction.
Which can be reduced to:
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Stable magnetization
1Mr2M
rIf h is inside the asteroid, two magnetization directions are possible. The one realized depends on the history of the sample magnetization.
1Mr
If h is outside, only one magnetization direction can be realized.
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Superparamagnetic limitBelow a certain size (blocking volume VB), islands behavein a superparamagnetic fashion. M is homogeneous butfluctuates with the period:
BuKTkE VKEe BK == ,0ττ
EK is the stored crystal anisotropy in a particle. For T<TB, the spin blocks freeze out, for T>TB , theremanent magnetization MR vanishes. For magnetic recording, a particle energy of EK = KuVB > 55 kBT is required for a 10 year stability.
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Thermal fluctuationsAnimation of a thermally activated magnetization reversal process of a small cubic particle, which has been discretized with eight magnetization vectors. A finite difference and finite element micromagnetics code, which solves the stochastic differential equation in the sense of Stratonovich, has been developed to perform temperature dependent simulations. Werner Scholz: werner.scholz (at) tuwien.ac.athttp://magnet.atp.tuwien.ac.at/scholz/gallery/werneranim.html
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Example: What is the critical cluster size for the superparamagneticlimit at room temperature?Parameters: τ0≈10-10 s, desirable: τ≈10j = 3×108 s at 300K
spins 56004412510103ln
/2.025ln 10
8
0
=×=
×=
= −atommeV
meVKTkVU
BB τ
τ
This corresponds to roughly a cluster size of 150Å×150Å.
With equal size and distance,this corresponds to 700Gb/inch2
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Hysteresis as a function of cluster size
Fe clusters in a Ag matrix from a cluster source
8.1 nm
11.7 nm
H. Meiwes-Broer, Rostock
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
In media development, experts discuss the benefits of anti-ferromagnetic-coupled exchange media as an approach to delay the effects of superparamagnetism. The superparamagnetic limit is a fundamental physical constraint beyond which conventional hard drives can no longer reliably store data, due to signal-to-noise effects.
Superparamagnetic limit in the recording industry
~40 nm
~250 grains/bit
8 nm
H. Zabel 4. Lecture Magnetic domains and magnetization reversal
Summary
• Domains are formed to reduce the stray field energy
• Domains depend on anisotropy and shape• In islands and rings, vortex and onion shape
domains occur• Single domains for particles smaller than the
domain wall width• Superparamagnetism occurs if crystal
anisotropy energy stored becomes smaller than thermal energy.