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

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 15 - 18

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

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

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”

(Stoner-Wohlfarth model) Simple example: H “easy axis” M q f Zeeman energy Uniaxial anisotropy Find M (q) by condition: See: http://www.student.uni-kl.de/~mewes/magnet.e.html

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

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.

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)

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

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 http://www.aacg.bham.ac.uk/magnetic_materials

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

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

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)

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)

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

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

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

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.