Stoner Wohlfarth Model for Magneto Anisotropy Xiaoshan Xu 2016/10/20 Levels of details for ferromagnets • Atomic level: • Exchange interaction that aligns atomic moments J푖푗푆푖 ⋅ 푆푗 • 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 N

S S N Repulsion, S high energy

Attraction, low energy Polarize Depolarize each other, each other, Depolarization factor: low energy high energy 1 훼 퐷 = ln 훼 + 훼2 − 1 − 1 푧 훼2−1 훼2−1 퐷푥 + 퐷푦 + 퐷푧 = 1 퐿 훼 > 1 is the aspect ratio 푧 퐿푥 Spheroidal model for anisotropy Mathematically, both magneto-crystalline anisotropy and magnetic shape anisotropy can be described using the anisotropy tensor (symmetric matrix): 퐷푥푥 퐷푥푦 퐷푥푧 푫 = 퐷푥푦 퐷푦푦 퐷푦푧 퐷푥푧 퐷푦푧 퐷푧푧 The combined matrix can be diagonalized and along the principle axis, the tensor looks like 퐷11 0 0 푫 = 0 퐷22 0 . 0 0 퐷33

Geometrically, these matrices can be described using ellipsoids.

A simplified case assumes the ellipsoid is spherioid. 퐷11 = 퐷22

Anisotropy energy: 2 2 2 2 퐸퐴 = −푀 ⋅ 푫 ⋅ 푀 = −M sin 휃퐷11 − 푀 cos 휃 퐷33 2 2 2 = −푀 sin 휃 퐷11 − 퐷33 − 푀 퐷22

2 2 퐸퐴 = 퐾 sin 휃 , 퐾 = −푀 퐷11 − 퐷33 Stoner Wohlfarth model: single domain, homogeneous magnetization

Anisotropy energy: 푧Ԧ 2 2 푀 퐸퐴 = 퐾 sin 휃 , 퐾 = −푀 퐷11 − 퐷22

휃 퐻 휙 Zeeman energy:

퐸푍 = −퐻푀푐표푠(휙 − 휃)

Total energy:

퐸 = 퐾 sin2 휃 + 퐻푀푐표푠(휙 − 휃)

• 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 푀 on 퐻 and 휙. Stoner Wohlfarth model: critical field

Total energy:

퐸 = 퐾 sin2 휃 + 퐻푀푐표푠(휙 − 휃) 푧Ԧ • Assuming a nonzero 휙 (here 15˚), we can plot the angular 푀 dependence of total energy.

휃 퐻 휙

퐻푀 = 퐸

• 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: 퐸 = 퐾 sin2 휃 + 퐻푀 cos(휃 − 휙)

푧Ԧ • At high enough field one minimum disappears. 푀 Mathematically, this corresponds to: 휕퐸 휕2퐸 퐻 = 0, = 0 휃 휕휃 휕2휃 휙

퐾 sin 2휃 + 퐻푀 sin 휙 − 휃 = 0 2퐾 cos 2휃 − 퐻푀 cos 휙 − 휃 = 0

Expand: 2퐾 퐻푀 sin 휃 cos 휃 + 퐻 cos 휃 − 퐻 sin 휃 = 0 = 푀 푥 푧 퐸 2퐾 cos2 휃 − sin2 휃 − (퐻 cos 휃 + 퐻 sin 휃) = 0 푀 푧 푥 퐻푥 = 퐻 sin 휃; 퐻푧 = 퐻 cos 휃 Stoner Wohlfarth model: random field angle example Total energy: 퐸 = 퐾 sin2 휃 + 퐻푀 cos(휃 − 휙)

푧Ԧ Expand: 2퐾 푀 sin 휃 cos 휃 + 퐻 cos 휃 − 퐻 sin 휃 = 0 푀 푥 푧 퐻 2퐾 휃 cos2 휃 − sin2 휃 − (퐻 cos 휃 + 퐻 sin 휃) = 0 푀 푧 푥 휙 2퐾 sin 휃 cos 휃 + 퐻 cos 휃 − 퐻 sin 휃 = 0 × sin 휃 푀 푥 푧 2퐾 cos2 휃 − sin2 휃 − (퐻 cos 휃 + 퐻 sin 휃) = 0 × cos 휃 푀 푧 푥

2퐾 0.51.5퐻 ≈0.35 퐻 퐻 = cos3 휃 퐶 퐶 푧 푀 2퐾 퐻 = − sin3 휃 푥 푀

2 2 2 2퐾 3 2퐾 퐻3 + 퐻3 = 퐻푐 = 푥 푧 푀 푀 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.