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Magnetic Domain and Domain Walls. 1. Domain walls Bloch wall Neel wall Cross-Tie wall 2. Magnetic domains Uniaxial wall spacing Closure domain Stripe domains 3. Some methods for the domain observation SEMPA MFM Magneto-optical. Relevant Energy Densities. Exchange energy.

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magnetic domain and domain walls
Magnetic Domain and Domain Walls

1. Domain walls

  • Bloch wall
  • Neel wall
  • Cross-Tie wall

2. Magnetic domains

  • Uniaxial wall spacing
  • Closure domain
  • Stripe domains

3. Some methods for the domain observation

  • MFM
  • Magneto-optical
relevant energy densities
Relevant Energy Densities

Exchange energy





In 1907, Weiss Proposed that magnetic domains that are regions inside the material that are magnetized in different direction so that the net magnetization is nearly zero.

Domain walls separate one domain from another.

P. Weiss, J.Phys., 6(1907)401.


Schematic of ferromagnetic material containing a 180o domain wall (center).

Left, hypothetical wall structure if spins reverse direction over one atomic

distance. Right for over N atomic distance, a. In real materials, N: 40 to 104.


magnified sketch of the spin orientation within

  • a 180o Bloch wall in a uniaxial materials; (b) an appro-
  • ximation of the variation of θ with distance z through
  • the wall.

Bloch Wall Thickness ?

In the case of Bloch wall, there is significant cost

in exchange energy from site i to j across the domain

wall. For one pair of spins, the exchange energy is :


Surface energy density is


In the other hand, more spins are oriented in directions

of higher anisotropy energy. The anisotropy energy per unit area increases with N approximately as


The equilibrium wall thickness will be that which

minimizes the sum with respect to N

thus the wall thickness

The minimized value No

is of order

, where A is the exchange stiffness

constant. A=Js2/a ~10-11 J/m (10-6 erg/cm), thus the wall

thickness will be of order 0.2 micron-meter with small aniso-

tropy such as many soft magnetic materials

wall energy density
Wall energy density ?

The wall energy density is obtained by substituting into

To give

neel wall
Neel Wall

Comparision of Bloch wall, left, with charged surface on

the external surface of the sample and Neel wall, right,

with charged surface internal to the sample.


Energy per unit area and thickness of a Bloch wall and a

Neel wall as function of the film thickness. Parameters

used are A=10-11 J/m, Bs=1 T, and K=100 J/m3.

In the case of Neel wall, the free energy density can be approximated as

Minimization of this energy density with respect to δN


For t/δN ≤1, the limiting forms of the energy density

σN and wall thickness δN follow from above Eq.


Neel wall near surface

Calculated spin distribution in a thin sample containing a

180o domain wall. The wall is a Bloch wall in the interior,

but it is a Neel wall near the surface.

cross tie wall
Cross-tie wall

The charge on a Neel wall can destabilize it and cause

it to degenerate into a more complex cross-tie wall

magnetic domains
Magnetic Domains

Once domains form, the orientation of magnetization in each domain and the domain size are determined by

Magnetostatic energy

Crystal anisotropy

Magnetoelastic energy

Domain wall energy


Domain formation in a saturated magnetic material is driven by the magnetostatic (MS) energy of the single domain state (a). Introduction of 180o domain walls reduces the MS energy but raises the wall energy; 90o closure domains eliminate MS energy but increase anisotropy energy in uniaxial material

uniaxial wall spacing
Uniaxial Wall Spacing

The number of domains is W/d

and the number of walls is (W/d)-1. The area of single wall

is tL The total wall energy is


The wall energy per unit volume



Domain Size d ?

The equilibrium wall spacing

may be written as

Variation of MS energy density

and domain wall energy density

with wall spacing d.

For a macroscopic magnetic ribbon;

L=0.01 m, σw= 1mJ/m2, uoMs= 1 T and t = 10um, the wall spacing is a little over 0.1 mm.

The total energy density reduces to

According to the Eq.(for do) for thinner sample the

equilibriumwall spacing do increases and there are

fewer domains.

a critical thickness for single domain
A critical thickness for single domain

(The magnetostatic energy of single domain)

Single domain size

Variation of the critical thickness with

the ratio L/W for two Ms (σdw=0.1mJ/m2)

size of mr read heads for single domain
Size of MR read heads for single domain ?

If using the parameters:

L/W=5, σdw≈ 0.1 mJ/m2,uoMs= 0.625 T; tc ≈13.7 nm;

Domain walls would not be expected in such a film. It is for a typical thin film magnetoresistivity (MR) read head.

closure domains
Closure Domains

Consider σ90 =σdw /2, the wall energy fdw

increases by the factor 1+0.41d/L; namely

δfdw≈ 0.41σdw/L

Hence the energies change to

Geometry for estimation of equilibrium

closure domain size in thin slab of ferro-

magnetic material. If Δftot < 0, closure

domain appears.


Energy density of △ftot versus sample length L

foruo Ms=0.625 T, σ=0.1 mJ/m2, Kud=1mJ/m2,

and td=10-14 m2.

domains in fine particles for large k u
Domains in fine particles for large Ku

Single domain partcle

σdw πr2 =4πr2(AK)1/2

△EMS≈ (1/3)uo Ms2V=(4/9)uo Ms2πr3

The critical radius of the sphere would be that which makes these two energies equal(the creation of a domain wall spanning a spherical particle and the magnetostatic energy, respectively).

rc≈ 3nm for Fe

rc≈ 30nm for γFe2O3

domains in fine particles for small k u
Domains in fine particles for small Ku

If the anisotropy is not that strong, the magnetization will tend to follow the particle surface

The spin rotate by 2π

radians over that radius



(a) A domain wall similar to

that in bulk; (b) The magneti-

zation conforms to the surface.


The exchange energy density can be determined over the volume of a

sphere by breaking the sphere into cylinders of radius r, each of which

has spins with the same projection on the axis symmetry


Construction for calculating the exchange energy of a

particle demagnetized by curling.

If this exchange energy density cost is equated to the magnetostatic energy density for a uniformly magnetizes sphere, (1/3)uoMs2, the critical radius for single-domain spherical particles results:

Critical radius for single-domain behavior versus saturation magnetization.

For spherical particles for large Ku, 106 J/m3 and small one.

stripe domain
Stripe Domain

Spin configuration of stripe domains


Spin configuration in

stripe domains

The slant angle of the spins is given as, θ = θo sin ( 2πx/λ )

The total magnetic energy (unit wavelength);

When w >0 the stripe

domain appears.


Minimizingwrespect toλ


Using eq.(1) we can get the condition for w>0,

striple domains in 10fe 90ni alloys film observed by bitter powder
Striple domainsin 10Fe-90Ni alloys film observed by Bitter powder

(b) After switch off a strong

H along the direction normal

to striple domain.

(a) After switch of H along

horizontal direction.

(c)As the same as (b), but

using a very strong field.


The stripe domain observed in 95Fe-5Ni alloys film

with 120 nm thick by Lolentz electron microscopy.


ProbabilityP per unit time for switching out of the metastable

state into the more stable demagnetzed state:

the first term in the right side is an attempt frequancy factor equal approxi- mately 109 s-1.

Δf is equal to ΔNµo Ms2 or Ku .

For a spherical particle with Ku = 105 J/m3 the superparamagnetic radii

for stability over 1 year and 1 second, respectively, are


Paramagnetism and Superparamagnetism

Paramagnetism describes the behavior of materials that

have a local magnetic moments but no strong magnetic

interaction between those moments, or. it is less than kBT.

Superparamagnetism:the small particle shows ferromagnetic

behavior, but it does not in paramagnete. Application of an

external H results in a much larger magnetic response than would

be the case for paramagnet.


The M-H curves of superparamagnts can resemble those of ferromagnets but with two distinguishing features;

(1)The approachto saturation follows a

Langevin behavior.

(2) There is no coecivity. Superpara-

magnetic demagnetization occurs without

coercivity because it is not the result of

the action of an applied field but rather of

thermal energy.


Langevin function versus s;

M = NµmL(s); s = µmB/kBT

scanning electron microscopy with spin polarization analysis sempa
Scanning Electron Microscopy with Spin Polarization Analysis (SEMPA)

Principle: when an energetic primary electron or photon enters a ferromagnetic material, electron can be excited and escape from the material surface. The secondary electrons collected from the small area on the surface are analyzed to determine the direction of magnetization at the surface from which they were emitted.

The vertical p


The horizantal

p component


3.5x3.5 µm2

(a) magnetic surface domain structure on Fe(100). The arrows indicate the measured

polarization orientation in the domains. The frame shows the area over which the polari-

zation sistribution of (b) is averaged.


Below, structure of Fe film/ Cr wedge/ Fe whisker illustrating the

Cr thickness dependence of Fe-Fe exchange. Above, SEMPA

image of domain pattern generated from top Fe film. (J. Unguris et

al., PRL 67(1991)140.)

magnetic force microscopy mfm
Magnetic Force Microscopy (MFM)

Geometry for description of MFM

technique. A tip scanned to the

surface and it is magnetic or is

coated with a thin film of a hard

or soft magnetic material.


Domain structure of epitaxial Cu/tNi /Cu(100) films imaged by

MFM over a 12 µm square: (a) 2nm Ni, (b) 8.5 nm Ni, (c) 10.0

Nm Ni; (d) 12.5nm Ni (Bochi et al., PRB 53(1996)R1792).


Magneto-optical Effect

θ k is defined as the main polarization plans is tilted over

a small angle;

εk= arctan(b/a).


(a)Assembly of apparatus

(b) Rotation of polarization

of reflecting light.


Domain on MnBi Alloys

The magnetic domains on the thin plate MnBi alloys observed by

Magneto-optical effect; (a) thicker plate (b) medium (c) thinner.

(Roberts et al., Phys. Rev., 96(1954)1494.)

other observation methods
Other Observation Methods

(a) Bitter Powder method;

(b) Lorentz Electron Microscopy;

(c) Scanning Electron Microscopy;

(d) X-ray topograhy;

(e) Holomicrography