1. Topics in Magnetism III. Hysteresis and Domains
Anne Reilly
Department of Physics
College of William and Mary

2. After reviewing this lecture, you should be familiar with:
1. General features of ferromagnetic hysteresis curves
2. Affects of anisotropy
3. Affects of domains
Material from this lecture is taken from Physics of Magnetism by Chikazumi, Chapters

3. In ferromagnetic materials, exchange interaction leads to an alignment of atomic spins. When a magnetic field is applied, these spins are reoriented, leading to hysteresis. H M H H
M=magnetization along direction of H

4. Features of Hysteresis Curve:M
Saturation magnetization (Ms)
Remnant magnetization (Mr) H
Coercivity (Hc)
M=magnetization along direction of H

5. What determines shape of hysteresis loop?
Coherent rotation determined mainly by Anisotropy
Domain formation and domain wall motion
Important principle:
Magnetization will lie in direction which is an energy minimum
Consider a simple example: M f H q
“easy axis”

6. (Stoner-Wohlfarth model)
Simple example: H
“easy axis” M q f
Zeeman energy
Uniaxial anisotropy
Find M (q) by condition:
See:

7. f=00 (along easy axis) M f=900 (along hard axis) M H H M H For 00:
Coherent rotation of magnetization considering only uniaxial anisotropy::
f=00 (along easy axis) M H M
f=900 (along hard axis) H H M
For 00:
Hc=2K1/Ms
Note: Hysteresis shown above is the component of M in the direction of H

8. Magnetic Anisotropy
Anisotropy: preferred (easy axes) and unfavorable (hard axes) directions of magnetization
Due to coupling of electronic spins to electronic charge density
For this rotation, as long as spins remain parallel, exchange energy does not change, but dipolar and LS coupling energy will change.

9. Magnetic Anisotropy Example: hcp Co easy M c-axis (hard) hard (easy)
Anisotropy: preferred (easy axes) and unfavorable (hard axes) directions of magnetization
Due to coupling of electronic spins to electronic charge density
Example: hcp Co
easy M
c-axis (hard)
hard
(easy)
8000
H (G)

10. Note: cubic lattices can have several easy and hard axes
Magnetic Anisotropy
Two major types of anisotropy, written in terms of empirical anisotropy coefficients:
Uniaxial:
Cubic:
(e.g., Co)
(e.g., Fe, Ni)
Note: cubic lattices can have several easy and hard axes

11. Domains
In ferromagnetic materials, exchange interaction leads to an alignment of atomic spins
However, this leads to a large external and dipolar magnetic fields which will tend to demagnetize the material. Domains are formed to minimize this effect.
Domain wall
From

12. Domains
Domain size and wall size determined by energy cost,
dependent on material and geometry.
Ni thin film

13. Energy is minimized by having a wall of finite width
Domain Walls
Energy is minimized by having a wall of finite width
N spins
Energy cost (exchange)
Energy cost (exchange + anisotropy)
(per unit area)
K = anisotropy constant
a = lattice constant

14. Energy is minimized by having a wall of finite width
Domain Walls
Energy is minimized by having a wall of finite width
N spins over d
For iron, J=2.16x10-21, S=1, K=4.2x104 and a=2.86x10-10
d=42 nm (150 lattice constants)
domain size will depend on sample geometry (see Chikazumi, Chp. 16)

15. Domain Walls
Domains have different shapes and orientations
Two examples of thin film domain walls:
Neel wall (rotation in plane)
Bloch wall (rotation out of plane)

16. Domains and Hysteresis
Domain formation and domain wall motion affects the shape of hysteresis loop: M H

17. Domains and Hysteresis
Barkhausen noise: Tiny steps of domain walls M H

18. Domains and Hysteresis
Domain walls move across energy landscape (determined by film morphology) Uw
irreversible motion
reversible motion x

19. Domains and Hysteresis
Coercivity can be increased over that for single domain system because domain walls can become pinned (hard to move).
Pinning on lattice defects (dislocations, voids, etc.) , impurities. Walls move between pinning points.
Defects and stress in thin film can increase number of pinning sites and thus coercivity.