Download Anisotropy and Magnetization Reversal

Anisotropy and Magnetization Reversal
1. Magnetic 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
Magnetocrystalline anisotropy
Crystal structure showing easy and hard magnetization direction for Fe (a),
Ni (b), and Co (c), above. Respective magnetization curves, below.
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.
How is µL 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 the
electronic orbital shape and orientation (spin-orbit coupling) on
a given atom generates the crystalline anisotropy
Physical Origin of
Magnetocrystalline anisotropy
Simple representation of the role of orbital angular
momentum <Lz> and crystalline electric field in determining the strength of magnetic anisotropy.
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>0
implies an easy axis.
Uniaxial Anisotropy
(1)Pt/Co or Pd/Co multilayers from interface
(2)CoCr films from shape
(3)Single crystal Co in c axis from (magneto-crystal anisotropy)
(4)MnBi (hcp structure)
(5)Amorphous GdCo film
(6)FeNi film
Single-Ion Model of Magnetic Anisotropy
dε
dγ
In a cubic crystal field, the orbital states of 3d
electrons are split into two groups: one is the
triply degenerate dε orbits and the other the
doubly one d γ.
Energy levels of dεand d dγ electrons in
(a) octahedral and (b) tetrahedral sites.
Table: The ground state and degeneracy of transition metal ions
d electrons for
Fe2+ in octahedral site.
Co2+ ions
Oxygen ions
Cations
Distribution of surrounding ions about
the octahedral site of spinel structure.
Conclusion :
(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.
Slonczewski expalain the stronger anisotropy of Co2+ relative the Fe2+ ions
in spinel ferrites ( in Magnetism Vol.3, G.Rado and H.Suhl,eds.)
Single ion model:
Ku = 2αJ J(J-1/2)A2<r2>,
Where A2 is the uniaxial anisotropy
of the crystal field around 4f
electrons, αJ Steven’ factor, J total
anglar momentum quantum
numbee and <r2> the average of
the square of the orbital radius of
4f electrons.
Perpendicular anisoyropy energy per RE
atom substitution in Gd19Co81films prepared
by RF sputtering (Suzuki at el., IEEE Trans.
Magn. 23(1987)2275.
References (single ion anisotropy)
(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.
Exchange Anisotropy
Schematic representation of effect of exchange coupling on M-H loop
for a material with antiferromagnetic (A) surface layer and a soft ferromagnetic layer (F). The anisotropy field is defined on a hard-axis loop,
right ( Meiklejohn and Bean, Phys. Rev. 102(1956)3047 ).
Above, the interfacial moment configuration in zero field. Below, left, the
weak-antiferromagnete limit, moments of both films respond in unison
to field. Below, right, in the strong-antiferromagnet limit, the A moment
far from the interface maintain their orientation.
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.
Exchange field and coecivity as function of FeMn
Thickness (Mauri JAP 62(1987)3047).
Mauri et al., (JAP 62(1987)3047) derived an expression for
M-H loop of the soft film in the exchange-coupled regime,
(tA>tAc)
There are stable solution at θ=0 and π
corresponding to ± MF.
H along z direction
Oscillation Exchange Coupling
Field needed to saturate the magnetization at 4.2 K versus Cr thickness
for 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).
Magnetization Process
The magnetization process describes
the response of material to applied field.
(1) What does an M-H curve look like ?
(2) why ?
For uniaxial anisotropy and domain walls
are parallel to the easy axis
Application of a field H transverse to the EA results in rotation of the
domain magnetization but no wall motion. Wall motion appears as H
is parallel to the EA.
Hard-Axis Magnetization
The energy density
(1)
(For zero torque condition)
(2)
(For stability condition)
θ= 0 for H > 2 Ku / Ms
θ the angle between H and M
θ= π for H <
(Ku >0 )
-2 Ku / Ms
(Ku <0)
The other solution fro eq.1 is given by
(2)
This is the equation of motion for the magnetization
in field below saturation -2Ku/Ms <H < 2Ku/Ms
Eq.(2) may be written as
HaMscosθ= MsH
Using cosθ=m=M/Ms , eq.3 gives
m=h,
( h=H/Ha)
(3)
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 hard
axis magnetization
process
M-H loop for easy-axis magnetization process
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 freedomain-wall limit, Hc=0 and in the single-domain
or pinned wall limit by rotational hysterisis,
Hc=2Ku/Ms.
Stoner-Wohlfarth Model
The free energy
f = -Kucos2 (θ- θo)+ HMscosθ
Minimizing with respect to θ, giving
Coordinate system for
magnetization reversal
process in single-domain
particle.
Kusin2 (θ- θo) –HMssin θ=0
Kusin2 (θ- θo) –HMsSin θ=0
(1)
∂2E/ ∂ θ2 =0 giving,
2KuCos (θ- θo)- Ho MsCos θ=0
(2)
Eq.(1) and (2) can be written as
sin2(θ- θo) =psinθ
(3)
cos (θ- θo) =(p/2)cosθ
(4)
with p=Ho Ms/Ku
From eq.(3) and (4) we obtain
(5)
Using Eq.(3-5) one gets
(6)
The relationship between p and θo
Sin2θo=(1/p2) [(4-p2)/3]3/2
p
θo is the angle between H
and the easy axis; p=Ho Ms/Ku.
θo =45o, Ho =Ku/Ms; θo =0 or 90o, Ho =2Ku/Ms
Stoner Wohlfarth model of coherent rotation
Hc [2Ku/Ms]
M/Ms
H [2Ku/Ms]
o
Wall motion coecivity Hc
H
The change of wall energy
per unit area is
∂εw /∂ s =2IsHcos θ
θ is the angle between H and Is
Ho={1/2Iscos θ } (∂εw/ ∂s)max
(1)
If the change of wall energy arises from interior stress
max
(2)
here δ is the wall thick. Substitution of (2) into (1) getting,
When ι ≈ δ
For common magnet, Homax =200 Oe.
(λ≈10-5, Is=1T, σo=100
KG /mm2.)
Micromagnetics-Dynamic
Simulation
(1) The film is divided into nx ∙ ny regular elements,
(2) Determining all the field on each element
(3) Solving Landau-Lifshith-Gilbert equation
Two dimension
Magnetic thin film modelded in two-dimensional
approximation. The film is divided into nx x ny elements for the simulation.
ΔM < 1.0 x10-7 G; The sum
torque T <102 erg/cc
Computation flow diagram for solving the magnetization
In the magnetic film.
Micromagnetics-dynamic simulation
Cross-tie wall in thin Permalloy film: simulated (a and b) and observed (c)
Nakatani et al., Japanese JAP 28(1989)2485.
Hysterisis Loop Simulation
(an example Co/Ru/Co and Co/Ru/Co/Ru/Co Films)
Co
Ru
Ru
Ru
Co
Co
Co
Co
Wang YJ et al., JAP 89(2001)6994;91(2002)9241.
Landau-Lifshitz-Gilbert Equation
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