4 Lecture

H. Zabel 4. Lecture Magnetic domains and magnetization reversal

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


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


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.


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


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


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 =


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


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


Wall widthUsual expressions normalized by the spin quantum number:

KJE

KJw

⋅∝

∝

energy wallDomain

width wallDomain /

Low K material

High K material


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


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:


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


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


Magnetic hysteresis

M

H

Remanent magnetization

Coercive field

Saturation magnetization

Demagnetized state


Magnetic hysteresis

M

H

s-state at remanence


Basic reversal mechanisms

3. Domain formation:

2. Coherent Rotation:

1. Nucleation and domain wall movement:


How can we tell the difference?

90°

180°

Domain wall

M

HWall motion

Wall rotation

H H H


Pinning of domain walls

Pinning can cause Barkhausen noise when walls are unpinned and perform Barkhausen jumps in an external magnetic field.


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).


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)


Shape induced anisotropyFe(100)/GaAs:superposition of 4-fold and 2-fold anisotropy

Polycrystalline Fe film on sapphiresubstrate: no anisotropy

PolycrystallineFe – stripes: shapeinduced anisotropy


Nano-magnetsstripes

Theis-Bröhl et al., Bochum10 µm

bars

Temst et al., Leuven

Shinjo et al., Kyoto

disks

Klaeui et al., Cambridge

rings


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


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).


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


Modelling of stripe domains and domain propagation

B. Hausmanns, PhD Thesis, Duisburg-Essen, 2003, G. Nowak, Duisburg


Simulation of a magnetization reversal process

Single bar

Reversal of interacting bars

http://magnet.atp.tuwien.ac.at/scholz/gallery/werneranim.html


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


Magnetization reversal of a dot

http://magnet.atp.tuwien.ac.at/scholz/gallery/werneranim.html


Magnetization reversal in ring structures

MFM images Simulation of spin structure and magnetic divergence

D. Buntinx, PhD Thesis, Leuven, 2003


Switching processesin mesoscopic ferromagnetic rings

PEEM image of array of rings

M. Kläui, et al. Phys. Rev. B 66, 134426 (2003)


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


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 =−= ⊥


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:


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.


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.


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


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


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


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


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.

4 Lecture

Documents

lecture magnetic domains

exchange energy

field energy

magnetization reversal

superparamagnetic limit

domains

number

smaller