Stoner Wohlfarth Model for Magneto Anisotropy
Xiaoshan Xu
2016/10/20
Levels of details for ferromagnets• Atomic level:
• Exchange interaction that aligns
atomic moments J𝑖𝑗𝑆𝑖 ⋅ 𝑆𝑗
• Micromagnetic level• Smear the individual atoms into
continuum, see magnetization as a function of position (domain wall)
• Domain level• Domains are separated by walls
of zero thickness
• Nonlinear level• Average magnetization of the
entire magnet
Magnetic anisotropy
• Magneto-crystalline anisotropy• Microscopic
• Single-ion
• Symmetry of the atomic local environment
• Shape anisotropy• Macrocopic
• Shape of the magnet
• Depolarization field
Magnetic shape anisotropy
N
S
N
S
N
S
N
S
Attraction, low energy
Repulsion, high energy
Polarize each other, low energy
Depolarize each other, high energyDepolarization factor:
𝐷𝑧 =1
𝛼2−1
𝛼
𝛼2−1ln 𝛼 + 𝛼2 − 1 − 1
𝐷𝑥 + 𝐷𝑦 + 𝐷𝑧 = 1
𝛼 > 1 is the aspect ratio 𝐿𝑧
𝐿𝑥
Spheroidal model for anisotropyMathematically, both magneto-crystalline anisotropy and magnetic shape anisotropy can be described using the anisotropy tensor (symmetric matrix):
𝑫 =
𝐷𝑥𝑥 𝐷𝑥𝑦 𝐷𝑥𝑧
𝐷𝑥𝑦 𝐷𝑦𝑦 𝐷𝑦𝑧
𝐷𝑥𝑧 𝐷𝑦𝑧 𝐷𝑧𝑧
The combined matrix can be diagonalized and along the principle axis, the tensor looks like
𝑫 =
𝐷11 0 00 𝐷22 00 0 𝐷33
.
Geometrically, these matrices can be described using ellipsoids.
A simplified case assumes the ellipsoid is spherioid.𝐷11 = 𝐷22
Anisotropy energy:
𝐸𝐴 = −𝑀 ⋅ 𝑫 ⋅ 𝑀 = −M2 sin2𝜃𝐷11 −𝑀2 cos2 𝜃 𝐷33
= −𝑀2 sin2 𝜃 𝐷11 − 𝐷33 −𝑀2𝐷22
𝐸𝐴 = 𝐾 sin2 𝜃 , 𝐾 = −𝑀2 𝐷11 − 𝐷33
Stoner Wohlfarth model: single domain, homogeneous magnetization
Ԧ𝑧
𝑀
𝐻𝜃
𝜙
Total energy:
𝐸 = 𝐾 sin2 𝜃 + 𝐻𝑀𝑐𝑜𝑠(𝜙 − 𝜃)
Anisotropy energy:
𝐸𝐴 = 𝐾 sin2 𝜃 , 𝐾 = −𝑀2 𝐷11 − 𝐷22
Zeeman energy:
𝐸𝑍 = −𝐻𝑀𝑐𝑜𝑠(𝜙 − 𝜃)
• The direction of the magnetization is a result of competition between the anisotropy energy and the Zeeman energy.
• Since the magnetic anisotropy is fixed for a sample, we will study the dependence of
𝑀 on 𝐻 and 𝜙.
Stoner Wohlfarth model: critical field
Ԧ𝑧
𝑀
𝐻𝜃
𝜙
Total energy:
𝐸 = 𝐾 sin2 𝜃 + 𝐻𝑀𝑐𝑜𝑠(𝜙 − 𝜃)
• Assuming a nonzero 𝜙 (here 15˚), we can plot the angular dependence of total energy.
𝐻𝑀
𝐸=
• As field increases, the energy of the two minima become different. • One minimum eventually disappears at high field, corresponding to saturation
magnetization.
Stoner Wohlfarth model: random field angle example Total energy: 𝐸 = 𝐾 sin2 𝜃 + 𝐻𝑀 cos(𝜃 − 𝜙)
Ԧ𝑧
𝑀
𝐻𝜃
𝜙
• At high enough field one minimum disappears.Mathematically, this corresponds to:
𝜕𝐸
𝜕𝜃= 0,
𝜕2𝐸
𝜕2𝜃= 0
𝐻𝑀
𝐸=
𝐾 sin 2𝜃 + 𝐻𝑀 sin 𝜙 − 𝜃 = 02𝐾 cos 2𝜃 − 𝐻𝑀 cos 𝜙 − 𝜃 = 0
Expand:2𝐾
𝑀sin 𝜃 cos 𝜃 + 𝐻𝑥 cos 𝜃 − 𝐻𝑧 sin 𝜃 = 0
2𝐾
𝑀cos2 𝜃 − sin2 𝜃 − (𝐻𝑧 cos 𝜃 + 𝐻𝑥 sin 𝜃) = 0
𝐻𝑥 = 𝐻 sin 𝜃; 𝐻𝑧 = 𝐻 cos 𝜃
Stoner Wohlfarth model: random field angle example Total energy: 𝐸 = 𝐾 sin2 𝜃 + 𝐻𝑀 cos(𝜃 − 𝜙)
Ԧ𝑧
𝑀
𝐻𝜃
𝜙
Expand:2𝐾
𝑀sin 𝜃 cos 𝜃 + 𝐻𝑥 cos 𝜃 − 𝐻𝑧 sin 𝜃 = 0
2𝐾
𝑀cos2 𝜃 − sin2 𝜃 − (𝐻𝑧 cos 𝜃 + 𝐻𝑥 sin 𝜃) = 0
2𝐾
𝑀sin 𝜃 cos 𝜃 + 𝐻𝑥 cos 𝜃 − 𝐻𝑧 sin 𝜃 = 0 × sin 𝜃
2𝐾
𝑀cos2 𝜃 − sin2 𝜃 − (𝐻𝑧 cos 𝜃 + 𝐻𝑥 sin 𝜃) = 0 × cos 𝜃
𝐻𝑧 =2𝐾
𝑀cos3 𝜃
𝐻𝑥 = −2𝐾
𝑀sin3 𝜃
𝐻𝑥
23 + 𝐻𝑧
23 =
2𝐾
𝑀
23
𝐻𝑐 =2𝐾
𝑀
0.51.5𝐻𝐶 ≈0.35 𝐻𝐶
Slonczewski asteroid
Hysteresis
Angular dependence (anisotropic MR)
Conclusion
• Stoner Wohlfarth model describes the magnetic anisotropy problem of a single domain particle
• It can explain the ideal case hysteresis and angular dependence of magnetizations.
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