Magnetics- Domains and Domain Walls Flashcards
Magnetostatic energy, Em
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
When is Hd smallest?
When the magnetic poles are most separated
When is magnetostatic energy, Em, reduced?
When M lies along the axis direction
Relationship between Hd and magnetisation
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.
Shape anisotropy
The preference for a particular direction due to shape
Shape factors for sphere, thin film and long cylinder
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)
Alternative formula for magnetostatic energy, Em
Em=1/2 μ0NdM^2
How to reduce demagnetising field in a plate magnet
Introduce more domains where each adjacent one has a magnetisation in the opposite direction.
Formula for magnetostatic energy density
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
Domain walls
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.
Formulae for exchange energy per unit area and anisotropy energy per unit area for domain walls
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
Formula for domain wall thickness using sum of exchange and anisotropy energies per unit area
δ0=πrt(A/Ku)
Formula for domain wall energy per unit area
σDW=σex+σani=Aπ^2/δ + Kuδ
Domain wall energy per unit volume
EDW=σDW/d
Where d is width of domain walls.
Total energy for magnetic domains
Etot=σDW+1.7(td/L^2)μ0Ms^2
=EDW+Em
What happens with magnetic domains on the nanoscale?
A threshold exists below which a single domain has a lower energy than two domains. The magnetic structure at this point becomes single domain.
Explain shape anisotropy in magnetics
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.
Why does a domain wall pass through a hard axis?
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.
How does exchange and magnetocrystalline anisotropy energy depend on domain wall thickness?
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.
Why do sufficiently small (nanoscale) magnetic elements become single domain?
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
Why do multiple magnetic domain form in materials?
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.
