1

Basics of magnetic materials

We will briefly review magnetism and magnetic materials as necessary background for understanding magnetoelectronics, particularly the data storage industry.

• Definitions in SI

• Types of magnetic materials

• Origins of magnetic properties

• Ferromagnetism

• Domains

• Hysteresis loops

Definitions in SI

Units are the worst part of dealing with magnetic systems!

B:

• magnetic induction (usually what physicists mean by magnetic field).

• most physically significant quantity - what shows up in Lorentz force law, in determining NMR frequencies, etc.

• Unit is the Tesla. Earth’s magnetic field = 6 x 10-5 T.

• More convenient cgs unit is the Gauss = 10-4 T.

• Important boundary condition: 0=⋅∇ B

2

Definitions in SIH:

• the magnetic field.

• Caused by currents of free charge.

• Unit is the Amp/m.

• Important relation:

• With no magnetic materials around,

JH =×∇

M:

• the magnetization.

• magnetic moment per unit volume of a material.

• Unit is the Amp/m.

• For a material with a permanent magnetization M0,

0MHM += χ

HB 0µ=

“permeability of free space” = 4π x 10-7 Tm/A

magnetic susceptibility

Susceptibility and permeability

)(0 MHB += µCombining effects of external currents and material response,

We define thepermeability by: HB µ=

So, for a material with a magnetic susceptibility χ,

)1(

)1(

0

0

χµµχµµ

+=→+== HHB

Note that for real materials χ and µ are tensorial.

Relative permeability is defined as µr = µ/µ0.

Susceptibility is most useful when discussing diamagnetic (χ < 0) and paramagnetic (χ > 0) materials, rather than systems with nonzero M0.

HB rµµ0=→

3

Magnetic dipoles and magnetization

A magnetic dipole can be modeled as a loop of current: nm IA=

m

A group of identical dipoles in a plane can be replaced by one big dipole with the same current circulating around the perimeter.

Units: Am2 = J/T

Convenient number:

A 3d stack of dipoles can be replaced by thinking about a sheet current running around the perimeter.

J/T1027.92

24−×=≡m

eB

hµ

Forces, torques, and fields

m

BTorque on a magnetic dipole:

Bmτ ×=

Force on a magnetic dipole:

)( BmF ∇⋅=

What is B at the surface of a long uniformly magnetized body with magnetization M in the absence of other fields?

Answer: just µ0M, since normal component of B must be continuous. M

Energy of magnetic dipole:Bm ⋅−=U

4

Field from a dipole

5

2

0 4

)(3)(

r

r

πµ mrrm

rB−⋅=

Note that B(r) ~ r -3

Current loop only approximates a dipole field in far field limit (r>> loop radius).

Image from Purcell.

Diamagnetism

Some materials develop a magnetization that is antialignedwith the applied external field H.

Such materials are diamagnetic, and have χ < 0.

Simple classical picture for diamagnetism: Lenz’s Law

mB

Try ramping up B = µ0H.

Result is a circumferential electric field that opposes the direction of the current in the loop.

This would act to reduce the dipole moment along H, and would be diamagnetic.

Correct quantum treatment involves 2nd order perturbation theory - can end up with either sign, depending on particulars of atoms. Larmor diamagnetism or Van Vleck paramagnetism.

5

Paramagnetism

Also common is paramagnetism, when χ > 0.

Two common origins of paramagnetism:

• Curie paramagnetism - localized moments free to flip.

• Pauli paramagnetism - requires “free” electrons in a metal.

Start w/ Curie case, spin J and gyromagnetic ratio g, in magnetic flux density B.

Bg µ/)/( Jm≡TkB

1≡β

Do statistical physics here. Alignment of spin with external field lowers spin energy.

Can write down partition function and solve to find equilibrium magnetization.

Curie paramagnetism − ++≡

=

J

x

Jx

J

J

J

Jx

gJBgJnM

J

BJB

2coth

2

1

2

12coth

2

12)(

)(

B

B βµµResult:

Bj is called the Brillouin function.

As long as µBB << kBT, can expand the above to get:

3

)1()( 022 +≈ JJ

Tk

HngM

BB

µµ

where we’ve also assumed that M’s contribution to B is small, valid for dilute systems of spins.

We see that Curie paramagnets have susceptibilities like:

T

JJ

Tkng

BB

1~

3

)1()( 022 +≈ µµχ

6

Pauli paramagnetism

E

Starting from unpolarized electrons, applying a field B shifts the Fermi level for spin-up and spin-down electrons oppositely!

)(2

)(BfdVN Bup µεενε += ∫ )(

2

)(BfdVN Bdown µεενε −= ∫

Taylor expanding to find the difference,

HENNV

M FBdownupB

02 )()( µυµµ ⋅≈−=

Only good for metals - can’t have gap at Fermi surface.

Paramagnetic and diamagnetic plots

H

MH

Pauli or Van Vleck paramagnet

Larmor diamagnet

Curie paramagnet

1~0

Tk

Hg

B

Bµµ

7

Some comments

The literature on all this stuff can be terribly confusing!

People use H even when there are no “free currents” around, and often use a mishmash of SI and cgs units.

• Most insulators are weakly diamagnetic.

• Metals can be either.

• Susceptibilities are usually quoted per molar volume rather than in their dimensionless form, for experimental reasons.

• Strictly speaking, susceptibilities and permeabilities are defined as derivatives.

Some numbers

Image from Marder.

8

Ferromagnetism - toy model

Start from Curie paramagnetism picture - local moments (spins), but now allow them to respond to the local magnetic field at their position.

Assume M = χ0 (H + Hm) “molecular field”, = ηM

HM 00)1( χηχ =−

)1( 0

0

ηχχχ

−=effAt high temperatures,

Recall that χ0 ~ 1/T, so we find at high temperatures

ceff TT −

1~χ Curie-Weiss law

At Tc, the Curie temperature, the susceptibility diverges! Spontaneous magnetization = ferromagnetism.

Ferromagnetism

• Preferred direction of M can depend on crystallographic properties of material and material shape, stresses.

FM occurs because of the same sort of exchange interactionthat leads to Hund’s rules.

Effective spin-spin interaction because the spin-ordered state tends to reduce total Coulomb energy of system.

Result: In metals, different spin directions have different densities of states at the Fermi energy.

Image from Coey, Trinity College, Dublin.

9

Other types of magnetic order

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑

↑

↑

↑

↑

↑

↑

↑

↑

↑

↑

↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

canted ferromagnet

antiferromagnet

canted antiferromagnet

ferrimagnet

Ferromagnetism and equilibrium configurations

One contribution (the total dipole moment of the system interacting with the applied external field) to a system’s energy will be decreased if M lies along H.

However, the field produced by the ferromagnet itself contributes to the total energy.

Competition between “external field” contribution and self-field contributions leads to complicated behavior.

As a result, ferromagnetic properties are often characterized bysweeping H along a particular direction, and measuring M.

What is seen is hysteresis - the dynamics of M at a given applied field H depend on the direction (history) of M.

10

Ferromagnetism: M vs H hysteresis

H

MH

Hc coercive field

Mr remanent magnetization

Ms saturation magnetization

slope = initial χ

slope = max. χ

“Hard” ferromagnetic materials: large values of Hc, large Mr.

“Soft” ferromagnetic materials: small values of Hc, small Mr.

Crystalline anisotropy

Because of band structure origins of FM, there can be certain crystallographic directions along which it’s energetically favorable for M to lie.

Define m as the unit vector pointing along M; u as energy density due to anisotropy.

Most common cases:

• cubic anisotropy

)()( 23

22

212

23

22

23

21

22

211 mmmKmmmmmmKu cc +++=

• uniaxial anisotropy

θθ 42

21 sinsin uu KKu += where θ is angle between m and

anisotropy axis.

11

“Stray field” energies

Energy bookkeeping for FM calculations requires keeping track ofenergy cost of the field produced by the FM itself.

Remember 0)(0 =+⋅∇=⋅∇ MHB µ

Therefore, can write MH ⋅−∇=⋅∇ d Looks like magnetic charge density.

Turns out that the energy associated with these stray fields can be written as:

dVdVUsample

d

allspace

dd ∫∫ ⋅−== MHH 02

0 2

1

2

1 µµ

For a particular sample geometry, one can compute to find a fictitious magnetic charge density. Then one can find Hd everywhere, and compute the energy contribution from these stray fields.

M⋅∇−

Demagnetizing factors and fields

The resulting Hd is often called a demagnetizing field.

Generally one can relate Hd to the magnetization M by the demagnetizing factor tensor:

MNH ⋅−= ˆd

One can calculate N for given sample geometries.

Special case: for ellipsoids of revolution, Hd || M.

Image from Bertram, Th. of Mag. Recording

12

Demagnetizing factors and fields

Why the name?

Inside a magnetized object, Hd opposes Hext.

Total magnetic field inside sample is Htot = Hext - Hd.

NN

N

NNS

S

S

S

S

Hext

Geometric or shape anisotropy

The demagnetizing field energy contribution leads to “geometric” or “shape” anisotropy: it costs less, energetically, to have the magnetization pointed along certain directions because that minimizes the stray fields.

Example: uniformly magnetized infinite (x-y) plate.

NNNNNNNNNNNNNNNNNNNNNNNNNN

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

Effective surface magnetic charge density ~ Msat cos θ

angle btwM and z-axis

Resulting Hd = - Msatcosθ; θµ 220 cos2 sd Mu =

So, we find it energetically unfavorable for a thin plate to have its magnetization lying perpendicular to its surface.

13

Other energetic considerations

• FM depends strongly on the crystal structure of the materials in question.

• Particularly important in insulators, where exchange interaction between local moments is not mediated by conduction electrons.

• Result: strong coupling between deformation and magnetization: magnetostriction.

• Can be used intentionally: built-in stresses can be used to “force” easy directions to lie where desired, or to “pin” magnetization.

DomainsSo, if Tc for iron is 1073 K, and its µ0Ms = 0.17 T, why don’t we see massive magnetic effects from iron bars all the time?

• FMs can lower their total energy by spontaneously breaking up their magnetization into domains.

• Total energy must include contribution from stray fields extending all over all space.

• Domains can greatly reduce these stray fields:

NNNNN

SSSSS

closure domains

14

Domain walls

Boundary between two domains is called a domain wall.

• Because of the FM exchange interaction, it’s very energetically costly for the direction of M to change very sharply.

• Result is that local magnetization spreads out the change oversome distance = domain wall thickness.

• Two types of domain walls:

Bloch wall Neel wall

Domain wall thickness

What sets domain wall thickness(often called “width”)?

Competition between exchange interaction (wants to keep nearest neighbor spins aligned) and crystalline (or shape) anisotropy(wants to keep M oriented only along “easy” directions).

Exchange energy for pair of spins:21

2 2)2( SS ⋅−= JJSU

For small change in angle φ between neighboring spins,

Energy is minimized ( = 0) when neighboring spins are aligned.

φ2JSU ≈

For a total rotation of π spread out over a chain of N+1 spins, the total energy cost is 22 )/( NNJSU π≈

For lattice spacing a, energy density per domain wall area for Natom thick wall ~ )/( 222 NaJSπ

15

Domain wall thickness

If anisotropy energy per unit volume is given by θ2sinK

where θ is angle between M and easy axis, then averaging over the thickness of an N atom thick domain wall gives an average anisotropy energy density ofKNa

Total energy density is then KNaNaJS +)/( 222π

Can find Nm that minimizes this, and wall thickness = Nma.

)/( 322 KaJSNm π≈ )/(22 KaJSW π≈

Plugging in numbers for iron, for example, gives W ~ 100 nm.

Note that it’s easy to make structures on size scales comparableto that of domain wall thicknesses….

Domains and hysteresis

What happens to domains when an external fieldis swept?

Domains continuously rearrange themselves to minimize the total energy of the whole system.

Relative sizes of different domains can change (at right).

Individual domains can also reorient their magnetizations.

Domain walls cannot always move freely!

16

Domain wall motion

Often domain walls can be “pinned” by disorder, grain boundaries, etc.

At large enough fields, pinning energy can be overcome.

Also, kBT can push domain walls “over barriers”.

Effect of pinning: Barkhausen noise.

H

MH

Next time:

Nanoscale issues in magnetic materials