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Anisotropy and Magnetization ReversalMagnetic anisotropy (a)
Magnetic crystalline anisotropy (b) Single ion anisotropy (c)
Exchange anisotropy
2. Magnetization reversal (a) H parallel and normal the
anisotropy axis, respectively (b) Coherent rotation
(Stoner-Wohlfarth model) (c) Micromagnetics: dynamic simulation;
solving LLG equation
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Magnetocrystalline anisotropyCrystal structure showing easy and
hard magnetization direction for Fe (a),Ni (b), and Co (c), above.
Respective magnetization curves, below.
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The Defination of Field Ha A quantitative measure of the
strength of the magnetocrystalline anisotropy is the field, Ha,
needed to saturate the magnetization in the hard direction. The
energy per unit volume needed to saturate a material in a
particular direction is given by a generation: The uniaxial
anisotropy in Co,Ku = 1400 x 7000/2 Oe emu/cm3 =4.9 x 106
erg/cm3.
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How is L coupled to the lattice ? If the local crystal field
seen by an atom is of low symmetry and if the bonding electrons of
that atom have an asymmetric charge distribution (Lz 0), then the
atomic orbits interact anisotropically with the crystal field. In
other words, certain orientation for the bonding electron charge
distribution are energetically preferred.
The coupling of the spin part of the magnetic moment to
theelectronic orbital shape and orientation (spin-orbit coupling)
ona given atom generates the crystalline anisotropy
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Physical Origin of Magnetocrystalline anisotropy Simple
representation of the role of orbital angularmomentum and
crystalline electric field in deter-mining the strength of magnetic
anisotropy.
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Uniaxial Anisotropy Careful analysis of the
magnetization-orientation curves indicates that for most purpose it
is sufficient to keep only the first three terms: where Kuo is
independent of the oreintation of M. Ku1>0implies an easy
axis.
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Uniaxial Anisotropy
Pt/Co or Pd/Co multilayers from interfaceCoCr films from shape
Single crystal Co in c axis from (magneto-crystal anisotropy)MnBi
(hcp structure)Amorphous GdCo filmFeNi film
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Single-Ion Model of Magnetic Anisotropy In a cubic crystal
field, the orbital states of 3d electrons are split into two
groups: one is the triply degenerate d orbits and the other the
doubly one d . dd
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Energy levels of dand d d electrons in(a) octahedral and (b)
tetrahedral sites.
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Table: The ground state and degeneracy of transition metal
ions
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Distribution of surrounding ions about the octahedral site of
spinel structure.Oxygen ions Cationsd electrons forFe2+ in
octahe-dral site.Co2+ ions
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(1) As for the Fe2+ ion, the sixth electron should occupy the
lowest singlet, so that the ground state is degenerate.
(2) Co2+ ion has seven electrons, so that the last one should
occupy the doublet. In such a case the orbit has the freedom to
change its state in plane which is normal to the trigonal axis, so
that it has an angular momentum parallel to the trigonal axis.
Since this angular momentum is fixed in direction, it tends to
align the spin magnetic moment parallel to the trigonal axis
through the spin-orbit interaction. Conclusion :Slonczewski
expalain the stronger anisotropy of Co2+ relative the Fe2+ ionsin
spinel ferrites ( in Magnetism Vol.3, G.Rado and H.Suhl,eds.)
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Perpendicular anisoyropy energy per RE atom substitution in
Gd19Co81films preparedby RF sputtering (Suzuki at el., IEEE
Trans.Magn. 23(1987)2275. Single ion model:
Ku = 2J J(J-1/2)A2,
Where A2 is the uniaxial anisotropy of the crystal field around
4f electrons, J Steven factor, J total anglar momentum quantum
numbee and the average of the square of the orbital radius of 4f
electrons.
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(1) J.J.Rhyne 1972 Magnetic Properties Rare earth matals ed by
R.J.elliott p156(2) Z.S.Shan, D.J.Sellmayer, S.S.Jaswal, Y.J.Wang,
and J.X.Shen, Magnetism of rare-earth tansition metal nanoscale
multilayers, Phys.Rev.Lett., 63(1989)449;(3) Y. Suzuki and N. Ohta,
Single ion model for magneto-striction in rare-earth transition
metal amorphous films, J.Appl.Phys., 63(1988)3633;(4) Y.J.Wang and
W.Kleemann, Magnetization and perpendicular anisotropy in Tb/Fe
multilayer films, Phys.Rev.B, 44 (1991)5132.References (single ion
anisotropy)
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Exchange Anisotropy Schematic representation of effect of
exchange coupling on M-H loop for a material with antiferromagnetic
(A) surface layer and a soft ferro-magnetic layer (F). The
anisotropy field is defined on a hard-axis loop, right ( Meiklejohn
and Bean, Phys. Rev. 102(1956)3047 ).
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Above, the interfacial moment configuration in zero field.
Below, left, theweak-antiferromagnete limit, moments of both films
respond in unisonto field. Below, right, in the
strong-antiferromagnet limit, the A moment far from the interface
maintain their orientation.
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Exchange field and coecivity as function of FeMnThickness (Mauri
JAP 62(1987)3047).In the weak-antiferromagnet limit, KA tA J,
tA j / KA= tAc,
For FeMn system, tAc 5 0 (A) for j 0.1 mJ/m2 and KA 2x104
mJ/m3.
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Mauri et al., (JAP 62(1987)3047) derived an expression forM-H
loop of the soft film in the exchange-coupled regime, (tA>tAc)
There are stable solution at =0 and corresponding to MF. H along z
direction
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Oscillation Exchange CouplingField needed to saturate the
magnetization at 4.2 K versus Cr thicknessfor Si(111) / 100ACr /
[20AFe / tCr Cr ]n /50A Cr, deposited at T=40oC (solid circle,
N=30); at T=125oC (open circle, N=20) (Parkin PRL 64
(1990)2304).
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Magnetization Process The magnetization process describes the
response of material to applied field.
(1) What does an M-H curve look like ? (2) why ?
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For uniaxial anisotropy and domain walls are parallel to the
easy axisApplication of a field H transverse to the EA results in
rotation of the domain magnetization but no wall motion. Wall
motion appears as His parallel to the EA.
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Hard-Axis Magnetization The energy density
(1)(For stability condition)= 0 for H > 2 Ku / Ms (Ku >0
)
= for H < -2 Ku / Ms (Ku
- The other solution fro eq.1 is given by This is the equation of
motion for the magnetizationin field below saturation -2Ku/Ms
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m = h, ( m = M/Ms ; h = H/Ha )
It is the general equatiuon for the magnetization processs with
the field applied in hard direction for an uniaxial material,M-H
loop for hardaxis magnetizationprocess
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M-H loop for easy-axis magnetization process
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In summary A purely hard-axis, uniaxial magnetization process
involves rotation of the domain magnetization into the field
direction. This results in a linear m-h characteristic.
An easy-axis magnetization process results in a square m-h loop.
It is chracterized in the free-domain-wall limit, Hc=0 and in the
single-domain or pinned wall limit by rotational hysterisis,
Hc=2Ku/Ms.
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Stoner-Wohlfarth Model f = -Kucos2 (- o)+ HMscosMinimizing with
respect to , giving The free energy Coordinate system for
magnetization reversal process in single-domain particle. Kusin2 (-
o) HMssin =0
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Kusin2 (- o) HMsSin =02E/ 2 =0 giving,2KuCos (- o)- Ho MsCos =0
(2) Eq.(1) and (2) can be written as
sin2(- o) =psin (3)
cos (- o) =(p/2)cos (4) (1)with p=Ho Ms/Ku
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From eq.(3) and (4) we obtain
(5)Using Eq.(3-5) one gets(6)
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The relationship between p and o o =45o, Ho =Ku/Ms; o =0 or 90o,
Ho =2Ku/MspSin2o=(1/p2) [(4-p2)/3]3/2 o is the angle between H and
the easy axis; p=Ho Ms/Ku.
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Stoner Wohlfarth model of coherent rotationH [2Ku/Ms]M/MsHc
[2Ku/Ms]o
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Wall motion coecivity HcHThe change of wall energy per unit area
is w / s =2IsHcos is the angle between H and Is Ho={1/2Iscos } (w/
s)max(1)
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maxIf the change of wall energy arises from interior stress(2)
here is the wall thick. Substitution of (2) into (1) getting,When
For common magnet, Homax =200 Oe. (10-5, Is=1T, o=100 KG /mm2.)
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Micromagnetics-Dynamic Simulation(3) Solving
Landau-Lifshith-Gilbert equation (1) The film is divided into nx ny
regular elements,
(2) Determining all the field on each element
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Magnetic thin film modelded in two-dimensional approximation.
The film is divided into nx x ny ele-ments for the simulation.Two
dimension
- Computation flow diagram for solving the magnetizationIn the
magnetic film.M < 1.0 x10-7 G; The sum torque T
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Micromagnetics-dynamic simulationCross-tie wall in thin
Permalloy film: simulated (a and b) and observed (c)Nakatani et
al., Japanese JAP 28(1989)2485.
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Hysterisis Loop Simulation(an example Co/Ru/Co and
Co/Ru/Co/Ru/Co Films)CoCoRuCoCoCoRuRuWang YJ et al., JAP
89(2001)6994;91(2002)9241.
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Landau-Lifshitz-Gilbert Equation
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The other fields (1) Radom anisotropy field : ha = ( m e ) hK ,
m = M/Ms , and e denotes the unit vector along the easy axis in the
cell;(2) Exchange energy fild: hex =
(3) Demagnetizing field (dipole-dipole interaction)
hmagi = - (1/rij3) [3(mj rij)/rij mj] (4) The applied field happ
= h m