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
Relevant Energy Densities
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
Domain walls separate one domain
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.
(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
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
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
Wall energy density ?
The wall energy density is obtained by
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.
The charge on a Neel wall can destabilize it and cause
it to degenerate into a more complex cross-tie wall
Once domains form, the orientation of
magnetization in each domain and the
domain size are determined by
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 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 = 10
ｕm, 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
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 ?
If using the parameters:
L/W=5, σdw≈ 0.1 mJ/m2,ｕoMs= 0.625 T; tc
Domain walls would not be expected in such a
film. It is for a typical thin film magnetoresistivity
(MR) read head.
Consider σ90 =σdw /2, the wall energy fdw
increases by the factor 1+0.41d/L; namely
Hence the energies change to
Geometry for estimation of equilibrium
closure domain size in thin slab of ferro-
magnetic material. If Δftot < 0, closure
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 Ku
Single domain partcle
σdw πr2 =4πr2(AK)1/2
△EMS≈ (1/3)ｕo Ms 2V=(4/9)ｕo Ms 2πr3
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
rc≈ 3nm for Fe
rc≈ 30nm for γFe2O3
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.
Spin configuration of stripe domains
Spin configuration in
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
Minimizing w respect to λ
Using eq.(1) we can get the condition for w>0,
Striple domains in 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
(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.
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 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 approach to saturation follows a
(2) There is no coecivity. Superpara-
paramagnetism magnetic demagnetization occurs without
coercivity because it is not the result of
the action of an applied field but rather of
Langevin function versus s;
M = NµmL(s); s = µmB/kBT
Some important parameters
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
(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)
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).
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
(a) Bitter Powder method;
(b) Lorentz Electron Microscopy;
(c) Scanning Electron Microscopy;
(d) X-ray topograhy;