Chapter 7 Micromagnetism, Domains and Hysteresis

Home , Coercivity, Remanence, Saturation (magnetic), Single domain (magnetic)

Chapter 7 Micromagnetism, domains and hysteresis

7.1 Micromagnetic energy 7.2 Domain theory 7.5 Reversal, pinning and nucleation

TCD March 2007 1 The hysteresis loop

coercivity virgin curve initial susceptibility

major loop

The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to a magnetic ﬁeld . It reﬂects the arrangement of the magnetization in ferromagnetic domains. The magnet cannot be in thermodynamic equilibrium anywhere around the open part of the curve! M and H have the same units (A m-1).

TCD March 2007 2 Domains form to minimize the dipolar energy Ed

TCD March 2007 3 TCD March 2007 4 Magnetostatics

Poisson’s equarion

Volume charge

Boundary condition

en 2. air + 1. solid + M +

M( r) ! H( r) BUT H( r) ! M( r)

Experimental information about the domain structure comes from observations at the surface. The interior is inscruatble.

TCD March 2007 5 7.1 Micromagnetic energy

TCD March 2007 6 1.1 Exchange

eM = M( r)/Ms (",#)

Exchange length A = kTC/2a 2 A = 2JS Zc/a0 A ~ 10 pJ m-1 Lex ~ 2 - 3 nm

Exchange energy of vortex 2 $Eex = JS ln (R/a)

TCD March 2007 7 1.2 Anisotropy

2 2 7 -3 EK = K1sin " Bulk K1 ~ 10 - 10 J m

-2 Surface Ksa ~ 0.1 - 1 mJ m .

-2 Interface Kea ~ 1 mJ m .

Exchange and anisotropy govern the width of the domain wall.

TCD March 2007 8 1.3 Demagnetizing field Demagnetizing ﬁeld governs the formation of the wall

(integral over all space) and B = µ0(H + M)

Hd is determined by the volume and surface charge distributions %.M and en.M 2 &m = qm/4'r; % &m= -(m H = - %&m

TCD March 2007 9 1.4 Stress

Magnetoelastic strain tensor

For isotropic material, uniaxial stress

Induced uniaxial anisotropy

TCD March 2007 10 1.5 Magnetosriction

Local stresses can be created by the magnetostriction of the ferromagnet itself:

Magnetostrictive stress

Deviation due to magnetostriction Elastic tensor

Usually this term is small < 1 kj m-3 , but it can inﬂuence the formation of closure domains.

TCD March 2007 11 1.6 Charge Avoidance

A guide to how nature minimizes the micromagnetic free energy is the charge avoidance principle.

Avoid forming bulk or surface chage, and keep charge of like sign as far apart as possible

e.g Keep magnetization parallel to the surface, wherever possible.

Toroid

Picture frame

TCD March 2007 12 General statement of the micromagnetic problem: No torque on the magnetization at any point.

Brown’s Micromagnetic equations

TCD March 2007 13 7.1 Domain Theory

A ~ 10-11 J m-1

5 -3 K1 ~ 10 J m

TCD March 2007 14 2.1 Bloch Wall

TCD March 2007 15 TCD March 2007 16 2.1 Néel Wall

Neel walls form in thin ﬁlms of soft material thinner than ~ 6 nm

TCD March 2007 17 2.3 Magnetization processes

There are two magnetization processes for a ferromagnet: 1) Domain-wall motion 2) Magnetization rotation H’

If the domain walls are perfectly free to move, they will do so until H =0; H’ = 1/N

TCD March 2007 18 7.3 Nucleation, reversal and pinning

TCD March 2007 19 Brown’s paradox

Brown’s theorem; for a homogeneous, uniformly- magnetized ellipse

TCD March 2007 20 TCD March 2007 21 A very small particle will be single-domain. Larger particles form domain walls to reduce demagnetizing energy

Single-domain particle size:

Cost of making two 90 degree walls is 2!R2(AK)1/2 should offset the N 2 gain in demagnetizing energy -(1/2) Ms

TCD March 2007 22 3.2 The Stoner Wohlfarth model Assume coherent rotation of the magnetization. H makes an angle ) with the axis of the particle.

NB R < Rsd does not guarantee coherent rotation.

When ) = 0, Hc=2Ku/µ0Ms

TCD March 2007 23 The energy landscape of a Stoner Wohlfarth particle

TCD March 2007 24 Hc = 0.479

Mr = 0.5 Area = 0.99

TCD March 2007 25 Preisach Model Model hysteresis loops with a distribution of elementary square loops. These are known as ‘hysterons’

TCD March 2007 26 Other reversal modes

TCD March 2007 27 3.3 Reversal in thin films and small elements

Consider a thin ﬁlm as a 2D S-W ‘particle’. The reversal The Stoner Wohlfarth asteroid. is assumed to be coherent Locus of points where a bifurcation of energy occurs Switching occurs on the surface, never within it. Take components of H along easy and hard directions, and

normalize them by the anisotropy ﬁeld 2Ku/Ms

dEtot/d"=0

TCD March 2007 28

TCD March 2007 29 TCD March 2007 30 3.4 The two-hemisphere model

A sphere made up of two halves with different anisotropy K) and K*

Exchange + dipole interactions Anisotropy + Zeeman interactions

If K1 = K) and K* = 0, Independent reveral of the soft hemisphere occurs when H ≈ (1/8) Ms l Except if R < ex , when the soft hemisphere cannot reverse independently.

TCD March 2007 31 Exchange stiffening operates on a length scale of up to ≈ 4lex ≈ 10 nm.

TCD March 2007 32 3.4 Switching dynamics;

Torque on a magnetic moment in a ﬁeld causes precession at the Larmor precession frequency i.e. 28 GHz/T when g=2 and + = -e/m

In the presence of unixial anisotropy:

Gilbert damping term M

TCD March 2007 33 3.5 Domain wall pinning

Barkhausen jumps

Domain wall velocity.

TCD March 2007 34 3.6 Real hysteresis loops

TCD March 2007 35 Kronmuller Equation

TCD March 2007 36 TCD March 2007 37 Approach to saturation

TCD March 2007 38 Time Dependence

Magnetic Viscosity M = M0 - S ln t

spontaneous magnetization

remanence

coercivity virgin curve initial susceptibility

major loop