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Anisotropy, Reversal and Micro-magnetics

Anisotropy, Reversal and Micro-magnetics. 1. Magnetic anisotropy (a) Magnetic crystalline anisotropy (b) Single ion anisotropy and atom pairs model (c) Exchange energy (anisotropy) (d) Interface anisotropy (e) Interlayer anti-ferromagnetic coupling

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Anisotropy, Reversal and Micro-magnetics

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  1. Anisotropy, Reversal and Micro-magnetics 1. Magnetic anisotropy (a) Magnetic crystalline anisotropy (b) Single ion anisotropy and atom pairs model (c) Exchange energy (anisotropy) (d) Interface anisotropy (e) Interlayer anti-ferromagnetic coupling 2. Magnetization reversal (a) H parallel and normal the anisotropy axis, respectively (b) Coherent rotation (Stoner-Wohlfarth model) (c) Spin torque (Current induced switching) (d) For votex 3. Micromagnetics dynamic simulation; solving LLG equation

  2. Magnetocrystalline anisotropy Crystal structure showing easy and hard magnetization direction for Fe (a), Ni (b), and Co (c), above. Respective magnetization curves, below.

  3. 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.

  4. 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

  5. Physical Origin of Magnetocrystalline anisotropy Simple representation of the role of orbital angular momentum <Lz> and crystalline electric field in deter- mining the strength of magnetic anisotropy.

  6. 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 orientation of M.Ku1>0 implies an easy axis.

  7. Uniaxial Anisotropy • Pt/Co or Pd/Co multilayers from interface • CoCr films from shape • Single crystal Co in c axis from (magneto-crystal anisotropy) • MnBi (hcp structure) • Amorphous GdCo film • FeNi film

  8. 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.

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

  10. Table: The ground state and degeneracy of transition metal ions

  11. d electrons for Fe2+ in octahe- dral site. Co2+ ions Oxygen ions Cations Distribution of surrounding ions about the octahedral site of spinel structure.

  12. 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.)

  13. Single ion model: Ku = 2αJ J(J-1/2)A2<r2>, Where A2 is the uniaxial anisotropy of the crystal field around 4f electrons, αJSteven’ factor, J total anglar momentum quantum numbee and <r2> the average of the square of the orbital radius of 4f electrons. Perpendicular anisotropy energy per RE atom substitution in Gd19Co81films prepared by RF sputtering (Suzuki at el., IEEE Trans. Magn. 23(1987)2275.

  14. over the nearest-neighbor ions j.

  15. Y.J.Wang and W.KleemannPRB 44(1991)5132.

  16. 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/Femultilayer films, Phys.Rev.B, 44 (1991)5132.

  17. Exchange Anisotropy Co particle 2r=20nm 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 ).

  18. FeMn NiFe strong-antiferromagnet weak-antiferromagnete 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. (Mauri JAP 62(1987)3047)

  19. NiFe/FeMn 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).

  20. 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. θ Hex along z direction

  21. Interface anisotropy Carcia et al., APL 47(1985)178

  22. (Si substrate) Effective anisotropy times Co thickness versus cobalt thickness for [Co/Pt] multilayers (Engle PRL 67(1990)1910).

  23. The effective anisotropy energy measured for a film of thickness d may be described as (1) , or writing as (2) (3) = Keff d = 2ks + (kV -2πMs2)d

  24. Surface Magnetic Anisotropy ? • The reduced symmetry at the surface (Neel 1954); • The ratio of Lz2 / (Lx2 + Ly2) is increased near the surface • Interface anisotropy (LS coupling) [1]J.G.Gay and Roy Richter, PRL 56(1986)2728, [2] G.H.O. Daalderop et al., PRB 41(1990)11919, [3] D.S.Wang et al., PRL 70(1993)869.

  25. Interlayer AF coupling Grunberg et al., PRL 57(1986)2442 Fe/Cr/Fe Fe/Au/Fe Fig.2 Spectra from Cr 8 and Au 20 with Bo along the easy axis. The arrow indicate the suggested magnetization direction on the two Fe layers where Bo is supposed to point up. Observed spin-wave propagation then is along a horizontal line.

  26. 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).

  27. Interlayer exchange coupling strength J12 for coupling of Ni80Co20 layers through a Ru spacer layer. The solid line corresponds to a fit to the data of RKKY form. Parkin et al., PRB 44(1991)7131.

  28. Parkin et al., PRL 66(1991)2152

  29. Bruno, Chappert PRL 67(1991)1602 The spin polarization of the conduction electrons gives rise to an indirect exchange interaction Hij = J(Rij) S i·S j . The interlayer coupling is obtained by summing Hij over all the pairs ij, i and j running respectively on F1 and F2.

  30. Co/Au(111)/Co Dependence of the exchange coupling J between Co layers vs the thickness tAu of the Au(111) interlayer. Line: theoretical fit of experimental data to RKKY model, with I33.8 erg/cm2, Λ=4.5 AL, Ψ=0.11 rad, tc=5AL and m*/m=0.16 PRL 71(1993)3023

  31. RKKY theory

  32. Fermi spanning vector

  33. Magnetization Process The magnetization process describes the response of material to applied field. (1) What does an M-H curve look like ? (2) why ?

  34. 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.

  35. Hard-Axis Magnetization The energy density (1) (For zero torque condition) (2) (For stability condition) θ= 0 for H > 2 Ku / Ms (Ku >0 ) θ the angle between H and M θ= π for H < -2 Ku / Ms (Ku <0)

  36. (2) The other solution from eq.1 is given by 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 (3) Using cosθ=m=M/Ms , eq.3 gives m=h, ( h=H/Ha)

  37. It is the general equatiuon for the magnetization processs with the field applied in hard direction for an uniaxial material, m = h, ( m = M/Ms ; h = H/Ha ) M-H loop for hard axis magnetization process

  38. M-H loop for easy-axis magnetization process

  39. Stoner-Wohlfarth Model The free energy f = -Kucos2 (θ- θo)+ HMscosθ Minimizing with respect to θ, giving Kusin2 (θ- θo)– HMssin θ=0 Coordinate system for magnetization reversal process in single-domain particle.

  40. Kusin2 (θ- θo) – HoMsSin 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

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

  42. The relationship between p and θo Sin2θo=(1/p2) [(4-p2)/3]3/2 θois the angle between H and the easy axis; p=Ho Ms/Ku. p θo=45o, Ho =Ku/Ms; θo =0 or 90o, Ho =2Ku/Ms

  43. Stoner Wohlfarth model of coherent rotation Hc [2Ku/Ms] M/Ms o H [2Ku/Ms]

  44. Wall motion coecivity Hc The change of wall energy per unit area is H ∂εw /∂ s =2IsHcos θ θ is the angle between H and Is Ho={1/ (2Iscos θ) } (∂εw/ ∂s)max (1)

  45. 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.)

  46. Homax= πλσo/2Iscosθ Dependence of the coercive force on the magnitude for of internal stress nickel (a) hard-drawn in various stress; (b) a hard-worked Ni specimen which was an- nealed to release the internal stress.

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