Magnetic materials form magnetic domains to minimize their magnetostatic energy. Domain walls separate domains with different magnetization orientations. Bloch walls have spins rotating continuously across the wall, while Neel walls have spins rotating in the plane of the wall. The equilibrium domain size and wall thickness are determined by a balance of exchange, anisotropy, magnetostatic, and wall energies. Various techniques like SEMPA, MFM, and magneto-optical imaging are used to observe domain structures with high resolution.

1. Magnetic Domain and Domain Walls
1. Domain walls
(a) Bloch wall
(b) Neel wall
(c) Cross-Tie wall
2. Magnetic domains
(a) Uniaxial wall spacing
(b) Closure domain
(c) Stripe domains
3. Some methods for the domain observation
(a) SEMPA
(b) MFM
(c) Magneto-optical

2. Relevant Energy Densities
Exchange energy
Magnetocrystalline
Magnetoelastic
Zeeman
Magnetostatic

4. 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.

5. 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
.

6. (a) 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.

7. 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 :
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
Surface energy density is,
,

8. The minimized value No
thus the wall thickness
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
The equilibrium wall thickness will be that which
minimizes the sum with respect to N

9. Wall energy density ?
The wall energy density is obtained by
substituting into
To give

10. 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.

11. 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
.

12. In the case of Neel wall, the free energy density
can be approximated as
Minimization of this energy density with respect to δN
gives
For t/δN ≤1, the limiting forms of the energy density
σN and wall thickness δN follow from above Eq.

13. 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.
Neel wall near surface

14. Cross-tie wall
The charge on a Neel wall can destabilize it and cause
it to degenerate into a more complex cross-tie wall

15. 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

16. 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

17. 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
is

18. Domain Size d ?
Variation of MS energy density
and domain wall energy density
with wall spacing d.
The equilibrium wall spacing
may be written as

19. For a macroscopic magnetic ribbon;
L=0.01 m, σw= 1mJ/m2
, ｕ oMs= 1 T and t = 10
ｕ m, 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.

20. A critical thickness for single domain
Single domain size
Variation of the critical thickness with
the ratio L/W for two Ms (σdw=0.1mJ/m2
)
(The magnetostatic energy of single domain)

21. Size of MR read heads for single domain ?
If using the parameters:
L/W=5, σdw≈ 0.1 mJ/m2
, ｕ oMs= 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.

22. Closure Domains
Geometry for estimation of equilibrium
closure domain size in thin slab of ferro-
magnetic material. If Δftot < 0, closure
domain appears.
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

23. Energy density of f△ tot versus sample length L
for ｕ o Ms=0.625 T, σ=0.1 mJ/m2
, Kud=1mJ/m2
,
and td=10-14
m2
.

24. Domains in fine particles for large Ku
σdw πr2
=4πr2
(AK)1/2
△EMS≈ (1/3) ｕ o Ms
2
V=(4/9) ｕ o Ms
2
πr3
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
Single domain partcle

25. Domains in fine particles for small Ku
If the anisotropy is not that strong, the magnetization will
tend to follow the particle surface
(a)
(b)
(a) A domain wall similar to
that in bulk; (b) The magneti-
zation conforms to the surface.
The spin rotate by 2π
radians over that radius

26. 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.
=2(R2
-r2
)1/2

27. If this exchange energy density cost is equated to the
magnetostatic energy density for a uniformly magnetizes
sphere, (1/3) ｕ oMs
2
, 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.

28. Stripe Domain
Spin configuration of stripe domains

29. 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.

30. Minimizing w respect to λ
(1)
Using eq.(1) we can get the condition for w>0,

31. Striple domains in 10Fe-90Ni alloys film
observed by Bitter powder
(a) After switch of H along
horizontal direction.
(b) After switch off a strong
H along the direction normal
to striple domain.
(c) As the same as (b), but
using a very strong field.

32. The stripe domain observed in 95Fe-5Ni alloys film
with 120 nm thick by Lolentz electron microscopy.

33. Superparamagnetism
Probability P 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 Ms
2
or Ku .
For a spherical particle with Ku = 105
J/m3
the superparamagnetic radii
for stability over 1 year and 1 second, respectively, are

34. 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.
Paramagnetism and Superparamagnetism
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.

35. Superparamagnetism
The M-H curves of superparamagnts
can resemble those of ferromagnets
but with two distinguishing features;
Langevin function versus s;
M = NµmL(s); s = µmB/kBT
(1) The approach to 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.
paramagnetism

36. Some important parameters
]

37. 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.
(a)
(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.
The vertical p
component
The horizantal
p component
3.5x3.5 µm2

38. 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.)

39. 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.

40. 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).

41. Magneto-optical Effect
θ k is defined as the main polarization plans is tilted over
a small angle;
εk = arctan(b/a).

42. (a) Assembly of apparatus
(b) Rotation of polarization
of reflecting light.

43. 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.)
Domain on MnBi Alloys

44. Other Observation Methods
(a) Bitter Powder method;
(b) Lorentz Electron Microscopy;
(c) Scanning Electron Microscopy;
(d) X-ray topograhy;
(e) Holomicrography