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.