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 H a A quantitative measure of the strength of the magnetocrystalline anisotropy is the field, H a, 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,K u = 1400 x 7000/2 Oe emu/cm 3 =4.9 x 10 6 erg/cm 3.

How is µ 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 (L z ≠ 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 and crystalline electric field in deter- mining 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 K uo is independent of the oreintation of M. K u1 >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 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. dε dγ

Energy levels of dεand d dγ electrons in (a) octahedral and (b) tetrahedral sites.

Table: The ground state and degeneracy of transition metal ions

Distribution of surrounding ions about the octahedral site of spinel structure. Oxygen ions Cations d electrons for Fe 2+ in octahe- dral site. Co 2+ ions

(1) As for the Fe 2+ ion, the sixth electron should occupy the lowest singlet, so that the ground state is degenerate. (2) Co 2+ 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 Co 2+ relative the Fe 2+ ions in spinel ferrites ( in Magnetism Vol.3, G.Rado and H.Suhl,eds.)

Perpendicular anisoyropy energy per RE atom substitution in Gd 19 Co 81 films prepared by RF sputtering (Suzuki at el., IEEE Trans. Magn. 23(1987)2275. Single ion model: K u = 2α J J(J-1/2)A 2, Where A 2 is the uniaxial anisotropy of the crystal field around 4f electrons, α J Steven’ factor, J total anglar momentum quantum numbee and the average of the square of the orbital radius of 4f electrons.

(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)

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 ).

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.

Exchange field and coecivity as function of FeMn Thickness (Mauri JAP 62(1987)3047). In the weak-antiferromagnet limit, K A t A ＜＜ J, t A ≦ j / K A = t A c, For FeMn system, t A c ≈ 5 0 (A) for j ≈ 0.1 mJ/m 2 and K A ≈ 2x10 4 mJ/m 3.

Mauri et al., (JAP 62(1987)3047) derived an expression for M-H loop of the soft film in the exchange-coupled regime, (t A >t A c ) There are stable solution at θ=0 and π corresponding to ± M F. 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 / t Cr Cr ] n /50A Cr, deposited at T=40 o C ( solid circle, N=30); at T=125 o C (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 stability condition) θ= 0 for H > 2 K u / M s (K u >0 ) θ= π for H <-2 K u / Ms (K u <0) θ the angle between H and M (2) (For zero torque condition)

The other solution fro eq.1 is given by This is the equation of motion for the magnetization in field below saturation -2K u /M s <H < 2K u /M s Eq.(2) may be written as H a M s cosθ= M s H (3) Using cosθ=m=M/M s, eq.3 gives m=h, ( h=H/H a ) (2)

m = h, ( m = M/M s ; 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 free- domain-wall limit, H c =0 and in the single-domain or pinned wall limit by rotational hysterisis, H c =2K u /M s.

Stoner-Wohlfarth Model f = -K u cos 2 (θ- θ o )+ HM s cos θ Minimizing with respect to θ, giving The free energy Coordinate system for magnetization reversal process in single-domain particle. K u sin2 (θ- θ o ) –HM s sin θ=0

K u sin2 (θ- θ o ) –HM s Sin θ=0 ∂ 2 E/ ∂ θ 2 =0 giving, 2K u Cos (θ- θ o )- H o M s Cos θ=0 (2) Eq.(1) and (2) can be written as sin2(θ- θ o ) =psinθ (3) cos (θ- θ o ) =(p/2)cosθ (4) (1) with p=H o M s /K u

From eq.(3) and (4) we obtain (5) Using Eq.(3-5) one gets (6)

The relationship between p and θo θo =45 o, H o =K u /M s; θo =0 or 90 o, H o =2K u /M s p Sin2θo=(1/p 2 ) [(4-p 2 )/3] 3/2 θo is the angle between H and the easy axis; p=H o M s /K u.

Stoner Wohlfarth model of coherent rotation H [2K u /M s ] M/M s H c [2K u /M s ] o

Wall motion coecivity H c H The change of wall energy per unit area is ∂ ε w / ∂ s =2I s Hcos θ θ is the angle between H and I s H o ={1/2I s cos θ } (∂ε w / ∂s) max (1)

max If the change of wall energy arises from interior stress (2) here δ is the wall thick. Substitution of (2) into (1) getting, When ι ≈ δ For common magnet, H o max =200 Oe. (λ≈10 -5, I s =1T, σ o =100 KG /mm 2.)

Micromagnetics-Dynamic Simulation (3) Solving Landau-Lifshith-Gilbert equation (1) The film is divided into n x ∙ n y regular elements, (2) Determining all the field on each element

Magnetic thin film modelded in two-dimensional approximation. The film is divided into n x x n y ele- ments for the simulation. Two dimension

Computation flow diagram for solving the magnetization In the magnetic film. ΔM < 1.0 x10 -7 G; The sum torque T <10 2 erg/cc

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 Co Ru Wang YJ et al., JAP 89(2001)6994;91(2002)9241.

Landau-Lifshitz-Gilbert Equation

The other fields (1) Radom anisotropy field : h a = ( m ∙ e ) ∙ h K, m = M/M s, and e denotes the unit vector along the easy axis in the cell; (2) Exchange energy fild: h ex = (3) Demagnetizing field (dipole-dipole interaction) h mag i = - ∑ (1/r ij 3 ) [3(m j ∙ r ij )/r ij –m j ] (4) The applied field h app = h ∙ m