Magnetics- Domains and Domain Walls

Question 1

Magnetostatic energy, Em

Answer 1

Em=-μ0Hd•M=-μ0HdMcosθ
Where θ is angle between Hd and M
Hd is internal magnetic field within a magnet
M is magnetisation
Hd and M always in opposite directions so cosθ=-1 for uniform magnetisation.
Em is always positive

Question 2

When is Hd smallest?

Answer 2

When the magnetic poles are most separated

Question 3

When is magnetostatic energy, Em, reduced?

Answer 3

When M lies along the axis direction

Question 4

Relationship between Hd and magnetisation

Answer 4

Hd=-NdM
Where Nd is demagnetising factor
But also depends on shape of magnetic element and direction of magnetisation in that elements. So is demagnetising factor in each axis direction Nx, Ny and Nz where they sum to 1. They are demagnetising shape factors.

Question 5

Shape anisotropy

Answer 5

The preference for a particular direction due to shape

Question 6

Shape factors for sphere, thin film and long cylinder

Answer 6

Sphere: Nx=Ny=Nz=1/3 (no preference)
Thin film: Nx=Ny=0, Nz=1 (prefers in-plane direction)
Long cylinder: Nx=Ny=1/2, Nz=0 (for infinite length) (prefers long axis direction for magnetisation)

Question 7

Alternative formula for magnetostatic energy, Em

Answer 7

Em=1/2 μ0NdM^2

Question 8

How to reduce demagnetising field in a plate magnet

Answer 8

Introduce more domains where each adjacent one has a magnetisation in the opposite direction.

Question 9

Formula for magnetostatic energy density

Answer 9

Em=1.7(td/L^2)μ0Ms^2
Where t is thickness of plate
d is width of each domain
L is length of domain (one side of plate to other)
Ms is magnetisation

Question 10

Domain walls

Answer 10

Separate domains. Transition regions that separate continuous magnetisation in domains. See page 12 lecture 4 for diagram. They increase Em by changing magnetisation direction (increasing exchange energy) and causing magnetisation in non-easy directions increasing magnetocrystalline anisotropy energy.

Question 11

Formulae for exchange energy per unit area and anisotropy energy per unit area for domain walls

Answer 11

Exchange: σex=Aπ^2/δ
Where A is exchange stiffness and δ is domain wall width.
Favours wide domain walls
Anisotropy: σani=Kuδ
Where Ku is uniaxial magnetocrystalline anisotropy
Favours narrower domain walls

Question 12

Formula for domain wall thickness using sum of exchange and anisotropy energies per unit area

Answer 12

δ0=πrt(A/Ku)

Question 13

Formula for domain wall energy per unit area

Answer 13

σDW=σex+σani=Aπ^2/δ + Kuδ

Question 14

Domain wall energy per unit volume

Answer 14

EDW=σDW/d

Where d is width of domain walls.

Question 15

Total energy for magnetic domains

Answer 15

Etot=σDW+1.7(td/L^2)μ0Ms^2

=EDW+Em

Question 16

What happens with magnetic domains on the nanoscale?

Answer 16

A threshold exists below which a single domain has a lower energy than two domains. The magnetic structure at this point becomes single domain.

Question 17

Explain shape anisotropy in magnetics

Answer 17

For non-spherical object. Magnetisation orientated in different directions leads to demagnetising fields of different magnitudes. Interaction between demagnetising field and magnetisation results in different energies for different orientations. These differences depend on the object’s shape so is known as shape anisotropy.

Question 18

Why does a domain wall pass through a hard axis?

Answer 18

They are regions through which magnetisation gradually changes direction and usually separate magnetic domain aligned to the easy axis. The magnetisation of a domain wall must then pass through a hard axis.

Question 19

How does exchange and magnetocrystalline anisotropy energy depend on domain wall thickness?

Answer 19

Exchange energy is reduced when magnetisation changes gradually so this favours wide domains.
Magnetocrystalline anisotropy energy is reduced when fewer magnetic moments are in or close to a hard axis direction so favours narrow domain walls.

Question 20

Why do sufficiently small (nanoscale) magnetic elements become single domain?

Answer 20

As they reduce in size at the nanoscale, it becomes energetically unfavourable for domain walls to form. This is due to increasing changes in angle between adjacent magnetic moments which causes the exchange energy to increase

Question 21

Why do multiple magnetic domain form in materials?

Answer 21

A single domain causes a large demagnetising field to form in the material and a large stray field outside the material. Multiple domains (often adjacent ones oriented oppositely to each other) create an alternating magnetic charge pattern at surfaces which reduces the internal demagnetising field, reducing the magnetostatic energy of the system.