4In 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.
5Schematic of ferromagnetic material containing a 180o domain wall (center). Left, hypothetical wall structure if spins reverse direction over one atomicdistance. Right for over N atomic distance, a. In real materials, N: 40 to 104.
6magnified 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 throughthe wall.
7Bloch Wall Thickness ?In the case of Bloch wall, there is significant costin exchange energy from site i to j across the domainwall. For one pair of spins, the exchange energy is :,Surface energy density is,In the other hand, more spins are oriented in directionsof higher anisotropy energy. The anisotropy energy per unit area increases with N approximately as
8The equilibrium wall thickness will be that which minimizes the sum with respect to Nthus the wall thicknessThe minimized value Nois of order, where A is the exchange stiffnessconstant. 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
9Wall energy density ?The wall energy density is obtained by substituting intoTo give
10Neel 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.
11Energy per unit area and thickness of a Bloch wall and a Neel wall as function of the film thickness. Parametersused are A=10-11 J/m, Bs=1 T, and K=100 J/m3.
12In the case of Neel wall, the free energy density can be approximated as Minimization of this energy density with respect to δNgivesFor t/δN ≤1, the limiting forms of the energy densityσN and wall thickness δN follow from above Eq.
13Neel wall near surfaceCalculated 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.
14Cross-tie wall The charge on a Neel wall can destabilize it and cause it to degenerate into a more complex cross-tie wall
15Magnetic DomainsOnce domains form, the orientation of magnetization in each domain and the domain size are determined byMagnetostatic energyCrystal anisotropyMagnetoelastic energyDomain wall energy
16Domain 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
17Uniaxial Wall Spacing The number of domains is W/d and the number of walls is (W/d)-1. The area of single wallis tL The total wall energy is.The wall energy per unit volumeis
18Domain Size d ? The equilibrium wall spacing may be written as Variation of MS energy densityand domain wall energy densitywith wall spacing d.
19For 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 toAccording to the Eq.(for do) for thinner sample theequilibriumwall spacing do increases and there arefewer domains.
20A critical thickness for single domain (The magnetostatic energy of single domain)Single domain sizeVariation of the critical thickness withthe ratio L/W for two Ms (σdw=0.1mJ/m2)
21Size of MR read heads for single domain ? If using the parameters:L/W=5, σdw≈ 0.1 mJ/m2,ｕoMs= 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.
22Closure Domains δfdw≈ 0.41σdw/L Hence the energies change to Consider σ90 =σdw /2, the wall energy fdwincreases by the factor d/L; namelyδfdw≈ 0.41σdw/LHence the energies change toGeometry for estimation of equilibriumclosure domain size in thin slab of ferro-magnetic material. If Δftot < 0, closuredomain appears.
23Energy 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.
24Domains in fine particles for large Ku Single domain partcleσdw πr2 =4πr2(AK)1/2△EMS≈ (1/3)ｕo Ms 2V=(4/9)ｕo Ms 2πr3The 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 Ferc≈ 30nm for γFe2O3
25Domains in fine particles for small Ku If the anisotropy is not that strong, the magnetization will tend to follow the particle surfaceThe spin rotate by 2πradians over that radius(a)(b)(a) A domain wall similar tothat in bulk; (b) The magneti-zation conforms to the surface.
26Construction for calculating the exchange energy of a The exchange energy density can be determined over the volume of asphere by breaking the sphere into cylinders of radius r, each of whichhas spins with the same projection on the axis symmetry=2(R2-r2)1/2Construction for calculating the exchange energy of aparticle demagnetized by curling.
27If 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.
28Stripe DomainSpin configuration of stripe domains
29The slant angle of the spins is given as, θ = θo sin ( 2πx/λ ) Spin configuration instripe domainsThe slant angle of the spins is given as, θ = θo sin ( 2πx/λ )The total magnetic energy (unit wavelength);When w >0 the stripedomain appears.
30Minimizing w respect to λ (1)Using eq.(1) we can get the condition for w>0,
31Striple domains in 10Fe-90Ni alloys film observed by Bitter powder (b) After switch off a strongH along the direction normalto striple domain.(a) After switch of H alonghorizontal direction.(c) As the same as (b), butusing a very strong field.
32The stripe domain observed in 95Fe-5Ni alloys film with 120 nm thick by Lolentz electron microscopy.
33SuperparamagnetismProbability P per unit time for switching out of the metastablestate 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 radiifor stability over 1 year and 1 second, respectively, are
34Paramagnetism and Superparamagnetism Paramagnetism describes the behavior of materials thathave a local magnetic moments but no strong magneticinteraction between those moments, or. it is less than kBT.Superparamagnetism: the small particle shows ferromagneticbehavior, but it does not in paramagnete. Application of anexternal H results in a much larger magnetic response than wouldbe the case for paramagnet.
35Superparamagnetism (1) The approach to saturation follows a The M-H curves of superparamagnts can resemble those of ferromagnets but with two distinguishing features;(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.paramagnetismLangevin function versus s;M = NµmL(s); s = µmB/kBT
37Scanning 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 pcomponentThe horizantalp component(a)3.5x3.5 µm2(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.
38Below, structure of Fe film/ Cr wedge/ Fe whisker illustrating the Cr thickness dependence of Fe-Fe exchange. Above, SEMPAimage of domain pattern generated from top Fe film. (J. Unguris etal., PRL 67(1991)140.)
39Magnetic Force Microscopy (MFM) Geometry for description of MFMtechnique. A tip scanned to thesurface and it is magnetic or iscoated with a thin film of a hardor soft magnetic material.
40Domain 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) 10.0Nm Ni; (d) 12.5nm Ni (Bochi et al., PRB 53(1996)R1792).
41Magneto-optical Effect θ k is defined as the main polarization plans is tilted overa small angle;εk = arctan(b/a).
42(a) Assembly of apparatus (b) Rotation of polarizationof reflecting light.
43Domain on MnBi AlloysThe 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.)
44Other Observation Methods (a) Bitter Powder method;(b) Lorentz Electron Microscopy;(c) Scanning Electron Microscopy;(d) X-ray topograhy;(e) Holomicrography