exchangebias(eb): ii.fm partial wall models
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1. Semiclassical Justification of the phenomenological free energy formulaR L Stamps 2000 J. Phys. D: Appl. Phys. 33 R247
2. FM Partial (Incomplete) Domain Wall Model
M. Kiwi, J. Mejia-Lopez, R. D. Portugal and R. Ramirez, Europhys. Lett., 48 (5), pp. 573-579 (1999) M. Kiwi et al. / Solid State Communications 116 (2000) 315โ319 M.Kiwi et al. Appl. Phys. Lett. 75, 3995 (1999)J. Mejia-Lopez et al. Journal of Magnetism and Magnetic Materials 241 (2002) 364โ370
Spin Vector Array Model(SVAM)
(1) Magnetic moments are represented by an array of classical spin vectors ๐บ"
(2) ๐บ"๐Lattice structure is taken as simple cubic
(3) Equilibrium configurations are found by numerically integrating torque equations using a relaxation method: find zero torque configurations by numerically integrating the LandauโLifshitz equations of motion for each spin
Kim J-V, Wee L, Stamps R L and Street R 1999 IEEE Trans. Magn. 35 2994โ7
L. Wee, R. L. Stamps, and R. E. Camley ,Journal of Applied Physics 89, 6913 (2001)
LandauโLifshitz equations
Semiclassical Justification of the Phenomenological Model
Goal: Under a series of acceptable assumptions, SVAM is equivalent to phenomenological model
(2) Uncompensated interface: ๐ฝ% โซ ๐ฝ', while compensated interface: ๐ฝ% = 0, ๐ฝ' โ 0
(1) Justify the existence of biquadratic term, i.e. ๐ฝ' โ 0
(3) Suggest the potential existence of FM partial wall
R.L. Stamps, J. Phys. D: Appl. Phys. 33 (2000) R247
Semiclassical Justification of the Phenomenal Model
๐": lattice vectors of FM; ๐", ๐": lattice vectors of AFM;.๐0, .๐10: unit vectors of anisotropy easy axis of FM and AFM;
FM
AFM
Interface
๐ฟ
๐น ๏ฟฝโ๏ฟฝ + ๐ฟ โ ๐น ๏ฟฝโ๏ฟฝ + ๐ฟ 8 โ๐น ๏ฟฝโ๏ฟฝ + %'๐ฟ 8 โ
'๐น ๏ฟฝโ๏ฟฝ ๏ผ ๐น = ๐", ๐", ๐ก"
๐ท0 = ๐ง0 ๐ฝ0 ๐ฟ'/2,
๐ท10 = ๐ง10 ๐ฝ10 ๐ฟ'/2
Assumptions: (1) ) ๐ญ ๐ change slowly over lattice spacing length scales
(2) ๐ญ ๐ = ๐ญ ๐
(3) Nearest coordination neighbors
๐ฝยฑ =14 (๐ฝ1 ยฑ ๐ฝH)
๐๐๐๐๐๐ = ๐๐ ๐ฑR๐ 8 ๐ + ๐ฑU๐ 8 ๏ฟฝโ๏ฟฝ
z
Minimizing ๐ = ๐๐ + ๐๐๐ + ๐๐๐๐๐๐ by considering variations of ๐, ๐ and ๏ฟฝโ๏ฟฝwith constraints |๐| = |๐| = |๐| = ๐, i.e., |๐| = ๐, ๐๐ + ๐๐ = ๐
๐ซ๐๐ is large and so AFM two sublattice magnetizations remain nearly antiparallel
๐ is small and ๐ approaches to ๐ ๐
. Define ๐ท = ๐ ๐โ ๐, ๐ท โช ๐
๐ป
๐
๐ง๐
โฐ โ โซfRg โ๐ป๐0 cos ๐ โ ๐ + ๐ท0๐m' + ๐พ0๐๐๐ '๐ ๐๐ฆ
+ โซUgf โ๐ป๐10๐ฝ cos ๐ โ ๐ + 2๐ท10 1 โ 2๐ฝ' (๐ฝm
'โ๐m') โ 2๐พ10(๐ฝ'๐๐๐ '๐ + (1 โ ๐ฝ')๐ ๐๐'๐) ๐๐ฆ+ 2๐[๐ฝR๐ฝy cos ๐y โ ๐y โ ๐ฝU(1 โ
%'๐ฝy
') sin ๐y โ ๐y ]
where ๐y, ๐y, ๐ฝy denotes the values at ๐ฆ = 0
AFM (๐(๐)) Direction: }โฐ}~= 0 Magnitude: }โฐ
}๏ฟฝ= 0
Equations
Boundary Condition at ๐ = ๐
๐๐ ๐๐
FM: neglecting any deformation of the ferromagnet order(๐ฝ๐ = ๐) and FM anisotropy(๐ฒ๐ = ๐) ๐ฟโฐ๐ฟ๐ = 0
Assumptions: (1) canting of the antiferromagnet at the interface is small (๐ท, ๐ท๐, ๐ โช ๐)(2)there is no significant twist in the ferromagnet
Interface Exchange Energy
๐ท = ๐ช๐๐๐
๐ถ = โ ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝU~๏ฟฝ๏ฟฝ๏ฟฝR๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝU~๏ฟฝ
= โ ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝU~๏ฟฝ๏ฟฝ๏ฟฝ
%
%R๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
= โ ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝU~๏ฟฝ๏ฟฝ๏ฟฝ
[1 โ ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝU~๏ฟฝ๏ฟฝ๏ฟฝ
+ ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝU~๏ฟฝ๏ฟฝ๏ฟฝ
'โ ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝU~๏ฟฝ
๏ฟฝ๏ฟฝ
๏ฟฝ+ โฏ ]
๐ฑ๏ฟฝ๐๐ทโช ๐ (interlayer exchange relatively weak)
Boundary Condition at ๐ = ๐
Where ๐ผ = โ๐; ๐1 = 2 2๐H;๐ยฑ = 2 2๐๐ฝยฑ
๐ป
๐
๐ง๐(1) Uncompensated case: ๐ฝ1 = 0, ๐๐ ๐ฝH = 0, ๐ฝ% โ 0, ๐ฝ' โ 0
(2) compensated case: ๐ฝ1 = ๐ฝH, ๐ฝ% = 0, ๐ฝ' โ 0
You can also derive these two equations from phenomenological model !
๐ผ
FM Partial Wall Models
๐ฟโฐ๐ฟ๐ = 0
โฐ โ ยกf
Rgโ๐ป๐0 cos ๐ โ ๐ + ๐ท0๐m' + ๐พ0๐๐๐ '๐ ๐๐ฆ
+ โซUgf ยข
ยฃโ๐ป๐10๐ฝ cos ๐ โ ๐ + 2๐ท10 1 โ 2๐ฝ' (๐ฝm
'โ๐m') โ2๐พ10(๐ฝ'๐๐๐ '๐ + (1 โ ๐ฝ')๐ ๐๐'๐) ๐๐ฆ+ 2๐[๐ฝR๐ฝy cos ๐y โ ๐y โ ๐ฝU(1 โ
%'๐ฝy') sin ๐y โ ๐y ]
FM partial wall could exist!
Neglecting FM anisotropy(๐ฒ๐ = ๐),
FM Partial Wall Models in FeF2/Fe
M. Kiwi, J. Mejia-Lopez, R. D. Portugal and R. Ramirez, Europhys. Lett., 48 (5), pp. 573-579 (1999)
J. Noguรฉs, D. Lederman, T. J. Moran, Ivan K. Schuller, and K. V. Rao, Appl. Phys. Leb. 68, 3186 (1996)
sequential e-beam evaporations
MgO (100)
FeF2 (110) 90nmFe polycrystalline (110) (100) 6.7nm, 13nm, 130nm
Compensated AFM interface
Prototype system
Kirkpatrick S., Gelatt C. D. and Vechi M. P., Science, 220 (1983) 671
FM Partial Wall Models in FeF2/Fe
Camley R. E., Phys. Rev. B, 35 (1987) 3608; Camley R. E. and Tilley D. R., Phys. Rev. B, 37 (1988) 3413.
Range: A unit interface magnetic cell extends up to 65 F, and many AF monolayers;
Methods: Stimulated annealing and Camleyโs method;
Results and Conclusions: (0) ๐ยคยฅยฆ~100๐๐, ๐ยคยฉยฅยฆ~a few monolayers(1) When measurement field points opposite to the cooling field, a long-rangeFM partial wall develops;
(2) Interface layer AF spins deviate significantly relative to the AF bulk, only a tiny deviation develops in the second AF layer and no appreciable cantingin the third;
(3) The presence of canted spins at the AF interface layer is no guarantee for ๐ปยช โ 0 in the absence of a symmetry-breaking mechanism
Pre-theoretical Simulations
Note for (3) โwe performed simulations with a fraction of AF interface magnetic moments clamped,
๐ยค~ ๐ด/๐พ
FM Partial Wall Models in FeF2/FeAssumptions:(1) Perfect, flat, compensated two-sublahce AFM interface;(2) Strong AFM anisotropy(3)Symmetry-breaking Mechanism: During field cooling, the first AF interface layer freezes into the canted
spin configuraXon when ๐ป โ ๐ป๐ต, and remains frozen in a metastable state during the cycling of ๐ฏ, where |๐ฏ| < ๐ฏ๐๐For a single magnetic cell
Where ๐ = |๐|; ๐ยด: Bohr magneton; ๐: Fe gyromagnetic ratio๐ป:external magnetic field; ๐ฝยถ: Heisenberg exchange parameter; ๐พยท: uniaxial anisotropy; ๏ฟฝฬ๏ฟฝยฉยฅ: unit vector along AF uniaxial anisotropy direction;๐(ยบ), ๐(๏ฟฝ): canted spin vectors in the AF interface, belonging to ๐ผ- and ฮฒ- sublattices; ๐ยผ: spin vectors of the ๐-th FM layer, with 1 โค ๐ โค ๐, ๐=1 labeling FM interface;
FM Partial Wall Models in FeF2/Fe
Part I : field cool process
E =โรร +โร/รร
S' =
๏ฟฝฬ๏ฟฝยฉยฅ
๐ปร 0
๐(ยบ)
๐(๏ฟฝ)
๐(ยบ)
๐(๏ฟฝ)
๐ = ๐(ยบ) = โ ๐(๏ฟฝ)
describes the behavior of the system during the cooling process andthe removal of the cooling field
}ยช}๏ฟฝ= 0,
}รยช}๏ฟฝร
> 0๐ = ๐ร
This provides the symmetry breaking required for EB to develop !
(1) |๐ปร 0| < 2J โร ยฉยฅ โ ๐ยด๐, ๐ = ๐ร >ร'
, Negative EB(2)|๐ปร 0| > 2J โร ยฉยฅ โ ๐ยด๐, ๐ = ๐ร <
ร'
, Positive EB
(3) |๐ปร 0| = 2J โร ยฉยฅ โ ๐ยด๐, ๐ = ๐ร =ร'
, No EB
Analytical Results:
Define ๐ = ร'+ ๐พ, and expand ๐ธ ๐ relative to ๐พ, then
minimizing ๐ธ ๐ , to the second order of ๐พ, we get
FM Partial Wall Models in FeF2/FePart II : measurement process
ฯต =โรร +โร/รร
๐ฝยฅS'=
๐ยผ๐%๐ปร 0
๐ยผ
FM
= 0
If ๐ = 0, ๐ = 0, ๐
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