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

Relevant Energy Densities

Exchange energy

Magnetocrystalline

Magnetoelastic

Zeeman

Magnetostatic

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.

(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.

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,,

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 wallthickness 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

Wall energy density ?

The wall energy density is obtained by substituting into

To give

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

gives

For t/δN ≤1, the limiting forms of the energy densityσN and wall thickness δN follow from above Eq.

Calculated spin distribution in a thin sample containing a180o domain wall. The wall is a Bloch wall in the interior, but it is a Neel wall near the surface.

Neel wall near surface

Cross-tie wall

The charge on a Neel wall can destabilize it and causeit to degenerate into a more complex cross-tie wall

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

The number of domains is W/dand 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

Domain Size d ?

Variation of MS energy density and domain wall energy density with wall spacing d.

The equilibrium wall spacingmay be written as

For a macroscopic magnetic ribbon; L=0.01 m, σw= 1mJ/m2, ｕ oMs= 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 fewer domains.

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)

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; t

c ≈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

Geometry for estimation of equilibriumclosure 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

Energy density of f△ tot versus sample length Lforｕ o Ms=0.625 T, σ=0.1 mJ/m2, Kud=1mJ/m2,

and td=10-14 m2.

Domains in fine particles for large Ku

σdw πr2 =4πr2(AK)1/2

△EMS≈ (1/3)ｕ o Ms 2V=(4/9)ｕ o Ms 2πr3

The critical radius of the sphere would be that which makes these t

wo energies equal (the creation of a domain wall spanning a spheric

al particle and the magnetostatic energy, respectively).

rc≈ 3nm for Fe

rc≈ 30nm for γFe2O3

Single domain partcle

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 tothat in bulk; (b) The magneti-

zation conforms to the surface.

The spin rotate by 2πradians over that radius

The exchange energy density can be determined over the volume of a sphere by breaking the sphere into cylinders of radius r, each of whichhas spins with the same projection on the axis symmetry

Construction for calculating the exchange energy of aparticle demagnetized by curling.

=2(R2-r2)1/2

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.

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.

Minimizing w respect to λ

(1)

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

Striple domains in 10Fe-90Ni alloys film

observed by Bitter powder

(a) After switch of H alonghorizontal direction.

(b) After switch off a strong H along the direction normalto striple domain.

(c) As the same as (b), butusing a very strong field.

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

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

Paramagnetism describes the behavior of materials thathave a local magnetic moments but no strong magneticinteraction 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.

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 aLangevin behavior.

(2) There is no coecivity. Superpara-magnetic demagnetization occurs withoutcoercivity because it is not the result ofthe action of an applied field but rather ofthermal energy.

paramagnetism

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.

(a)

(a) magnetic surface domain structure on Fe(100). The arrows indicate the measuredpolarization orientation in the domains. The frame shows the area over which the polari-zation sistribution of (b) is averaged.

The vertical pcomponent

The horizantal p component

3.5x3.5 µm2

Below, structure of Fe film/ Cr wedge/ Fe whisker illustrating theCr thickness dependence of Fe-Fe exchange. Above, SEMPA image of domain pattern generated from top Fe film. (J. Unguris etal., PRL 67(1991)140.)

Magnetic Force Microscopy (MFM)

Geometry for description of MFM technique. A tip scanned to thesurface and it is magnetic or iscoated with a thin film of a hard or soft magnetic material.

Domain structure of epitaxial Cu/tNi /Cu(100) films imaged byMFM over a 12 µm square: (a) 2nm Ni, (b) 8.5 nm Ni, (c) 10.0Nm 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.

The magnetic domains on the thin plate MnBi alloys observed byMagneto-optical effect; (a) thicker plate (b) medium (c) thinner. (Roberts et al., Phys. Rev., 96(1954)1494.)

Domain on MnBi Alloys

Other Observation Methods

(a) Bitter Powder method;

(b) Lorentz Electron Microscopy;

(c) Scanning Electron Microscopy;

(d) X-ray topograhy;

(e) Holomicrography