Introduction to Magnetism : Scales and Nano-Magnetismqmmrc.net/winter-school-2009/panissod.pdf · Introduction to Magnetism : Scales and Nano-Magnetism Pierre Panissod Institut de

Introduction to Magnetism :Scales and Nano-Magnetism

Pierre PanissodInstitut de Physique et Chimie des Matériaux de Strasbourg

Outline

• Context– Nano-composites and Mesoscopic scales– The market for magnetic materials

• Magnetism – Basics in a nutshell– Origin– Energy terms– Scales

• Nano-magnetism– Nano-crystalline permanent magnets– Nano-crystalline soft magnetic materials– Magnetic recording media– Magneto-electronics / « Spintronics »

Nanostructured Composite Materials

• Multilayers and superlattices

• Discontinuous multilayers, Nanowires

• Etched or self-assembled dots

• Clusters and grains

• Nanocrystalline materials

Context

• Mesoscopic (between micro- and macro-scopic)– Object size comparable to a characteristic length

scale of the phenomenon

• Applications– Magnetic materials engineering

• Permanent magnets (Hard magnets / high remanence : motors, holders)

• Flux concentrators (Soft magnets / high permeability : transformers, inductors, EM absorbers)

• Recording (Hard/Soft : recording media, read heads) – Device engineering

• magneto-electronics

Nano-composites and Mesoscopic scale

Typical sizeof the componentsMinimum size ofthe mean medium

Compares with the length scale of the physical phenomena

Examples :• Electronic wavelength (1-100 nm)

Level splitting, Quantum wells• Optical wavelength (100-1000 nm)

Diffusion, diffraction• Dislocation length (10-100 nm)

Elasticity-Ductility-Fragility

Market for Magnetic materials

World gross product of magnetic materials(1999 estimate - Total 30 B$)

Permanent magnetsRecording mediaFlux concentrators

MotorsActuatorsElectron tubesHolding devicesStatic/MRIMiscellaneousMass audiovisualProfessional audiovisualComputers hard driveComputers floppy driveMass storageMiscellaneousElectromagnetsMotors & actuatorsTransformers & generatorsHFapplicationsRF and microwave SensorsMiscellaneous

Applications of magnetic materials

Outline - Origin




• Magnetic Dipole Moment: an infinitely small current loop

• Electronic Magnetism: orbital (classical loop like) & spin (quantum) momenta

• Electron Hamiltonian in E&M Field: dia- and para-magnetism

• Atomic/Ionic Para-magnetism– N electron resultant magnetic moment: how do orbital and spin momenta combine– Hund's rule: how do electrons fill up a I shell (m,s), resultant angular & magnetic momenta

• Assembly of Non Interacting Atoms/Ions– Ensemble / Thermal average of angular & magnetic momenta : Boltzmann statistics– Magnetisation M(H) and Susceptibility dM/dH : Langevin (classic), Brillouin (quantum)

• Solid State Magnetism– Rare Earth Case : screened f shell looks as ion, no surprise– Transition Metal Case (Insulating state): mostly spin only moment, surprise ! – What's up in Solids vs Atomic State ? Answer: quenched orbital moment– Transition Metal Case (Metallic state): loosely related to spin only moment, surprise ! – What's up in Metals ? Answer: Fermi-Dirac statistics and Pauli paramagnetism

• Ferro-Magnetism– Spontaneous magnetisation (M>0 in H=0), why ? Early answer: Weiss molecular field– What is the interaction responsible for it ? Answer: exchange interaction

Magnetism – Basics(see Kittel's book)

Introduction : Magnetic Dipole Moment

700 104 HB

Far from the loop

rrrH 23

).(34

1),(rr

r

H: Magnetic field (Excitation)(A/m)

B: Magnetic flux density (Induction)(Tesla)

In CGS emu : (Oersted)0=1 B =H (Gauss)

1 Iemu = 10 A 1 Oe = 10/4 ~ 80 A/m1 emu = 10 Am2 1 G = 10 T

rrrH 23

).(31),(rr

In a vacuum

I

Macroscopic current loop

µ=I.Area(Am2)

Use Biot-Savart law and find :

Electronic Magnetism

! FORGET IT !

1 with222

)(2

)(2

. 22

lBlorb

orb

orb

ggm

qmq

mq

mrmqrqAreaI

lµ

llLµ

µ

Intuitive : Orbital moment Mysterious : Spin moment

2 with sBsspin gg sµ gs=2.0023 after higher order corrections

s : momentumAngular

Appears in the relativistic electronichamiltonian (Dirac)

1 system Gauss :2 00

mcq

BMagnetonBohr :

2 factor Landé :

mqg B

Both are ruled by Quantum Mechanicsl = 0(s), 1(p), 2(d), 3(f)… s = 1/2lz (m) = l,…,+l; l2 = l(l+1) sz = 1/2; s2 = 3/4

Electron Hamiltonian in E & M Fields

NotesSpin-orbit coupling : ~ Z/<r3>

Negligible for light sp elements. Weak but may play a part for 3d elements. Strong and essential for 4f rare earths

Langevin diamagnetism (microscopic counterpart of Lenz’s law) : Orbital moment precesses around B at = qB/2m, generates a current loop I = q(qB/4m), Area = Equivalent moment : µdia = (q2/4m)B = (µB

2/2)mBr using x+y = (2/3)r

Quantum Mechanical treatmentCoulomb and Lorentz forces on a moving electron (charge q) : F = q(E + vxB) Fields from potentials : E(r) = .V(r) and B(r) = xA(r)

Non relativistic Hamiltonian of an electron (kinetic, magnetic, electrostatic) H = (1/2m)[p + qA(r)]2 qV(r) In a uniform field : A = ½ Bxr and [A, p] = 0 H = [p2/2m qV] + (q/m)A.p + (q2/2m)A2

Using A.p = ½ (Bxr).p = ½ B.(rxp) = ½ B.L H = [p2/2m qV] + µBB.l + ½(µB/)2m(Bxr)2

Spin-field and spin-orbit interactions introduced by Dirac in the relativistic Hamiltonian

Zeeman paramagnetism Langevin diamagnetism Spin-orbit interactionB(l+2s)B = µeB

Adds to B Opposes to B

l.srBB.sB.l

p2

022

2

)(21 2

)(2/

BBBB

ee

m

rm

HForce = q(E + vxB)

Atomic/Ionic Paramagnetism

Atomic/ionic angular momentum J :Vector sum* of Z electronic orbital and spin angular momenta

Atomic/ionic magnetic moment µat : Vector sum of the electronic orbital and spin magnetic moments

)( SL slBat gg

Z

ii

Z

ii

11 with* mLsSSLJ

• Closed shells do not contribute to paramagnetism but do have a diamagnetic moment (/ atom) = (q2/2m)B• Filling of the outer shell must obey :

Pauli principle : pp) 1 electron per (m, s) stateHund’s rules : i) maximise S {as much as pp permits (keeps electrons apart, minimises Coulomb energy)}

ii) maximise L {as much as pp and i permit (keeps electrons apart, minimises Coulomb energy)}*iii) L-S if shell less than half filled, L+S otherwise (spin-orbit interaction)

)1(2)1()1(

23with

JJ

LLSSg j

JBjg //(will be revisited in solid state)

z

L J

Sµat

µ//at and J are not parallel because gl=1 and gs=2 !• But only µ//J counts because S and hence at precess much faster around J

than J precesses around B

Examples : Atomic/Ionic Paramagnetism

Shell filling according to Hund’s rules

Ion : Cr2+ Mn2+ Fe2+ Ce3+ Gd3+ Ho3+

e Conf.: (Ar)3d4 (Ar)3d5 (Ar)3d6 (Xe)4f1 (Xe)4f7 (Xe)4f10

S :

L :

J :

Spec.: 2S+1LJ5D0

6S5/25D4

2F5/28S7/2

5I8

Assembly of Non Interacting Atoms/IonsEnsemble / Thermal average of Jz

Thermal population of the Zeeman split J states in a field B

BgmmUBmU

Bjjj

mj

)(

)(

Example: J=1

H>0, T>0Degeneracy lifted

Some atoms in excited states.

Bolzman's statistics

mJ

H=0, T=0Degenerate levels All atoms in the

ground state

H>0, T=0Degeneracy liftedAll atoms in the

ground state.

mJ

)exp()exp(

)(

jmj

jj xm

xmmP

)exp(

)exp(

j

j

mj

mjj

z xm

xmmJ

Ensemble/Thermal average of Jz

Boltzmann's partition function

TkHg

xB

Bj 0

Assembly of non interacting Atoms/IonsMagnetisation and Magnetic Susceptibility

Average thermal magnetisationzBj

ziziz Jg

VN

BE

VNm

VM

1

)(),( yJgVNHTM jBjzz B

yJJ

yJ

JJ

Jyj 21coth

21

212coth

212)(B

TkHg

JyB

Bj 0BJ(y) is the Brillouin function with

S states

84

21

1/2

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

Mz/

Ms

2 B/kBT

LimitsJ (classical moment µ) : (Langevin function)

y (T0 or H) : <Mz>Ms = (N/V)gjJµB = (N/V)µj (Saturation magnetisation)

HTk

TkH

VNHTM B

Bzz

0

0coth),(

TC

Tkp

VN

TkJJg

VNT j

B

eff

B

Bjj

33)1(

)(22

02

Magnetic susceptibility : obeys Curie law T=Cj = <M>/Hy0 (T or B0)

Solid State MagnetismRare Earth Case

Comparison of theoretical and experimental effective magnetic moments peff for rare earth metal ions

0 2 4 6 8 10 12 14La LuPm !

Eu ?P eff

=gj[J

(J+1

)]1/

2

0

1

2

3

4

5

6

7

8

9

10

11

Gd

Measured

Calculated(J of 3+ ions)

Rare Earth

Magnetic moment as expected from the atomic total

angular momentum

2 exceptionsLow lying excited statesSm ?

Solid State MagnetismTransition Metal Cases

Theoretical and experimental peff for transition metal ions

Transition metal ions

Magnetic moment is spin only

Orbital momentumis "quenched"

0 2 4 6 8 10

Electrons in 3d shell

P eff

in µ

B

01234567

Measured

Calculated(J of 2+ ions)Calculated(S of 2+ ions)

V4+ Mn ZnV3+

Temperature

Th. : Langevin paramagnetismExp.: Pauli paramagnetism

Transition metalsNot explained by Atomic Magnetism

is temperature independent(Pauli paramagnetism)

Paramagnetic susceptibility of transition metals (conductors)

Solid State MagnetismWhat's new vs Atomic State ?

Electric potential is no longer central becauseof lower than spherical symmetry Crystal FieldThe Hamiltonian is now H = H0 + Hso + Hcf + HZee+ Hdia

Typical magnitudes ofenergy terms (in K)

Crystal field favours orientation of the electron cloud along some crystallographic axes New eigenfunctions (see Annex 2)

Rare earth are much less affected because of shielding by 5p and 5d closed shells

1≈3 1021-5 1031-6 1054f

11 - 104102 -1031-5 1043d

HZ(1T)HcfHsoH0

Consequences :• New eigenfunctions may have no angular momentum :

Orbital angular momentum for 3d ions is quenched (often and, at least, partly)• Favours the orientation of the orbital moment along some crystallographic axes and, in turn, that of the spin moment because of the spin-orbit interaction :

Magnetocrystalline anisotropy appears easy directions of magnetisation

Solid State MagnetismWhat's new in metallic systems ?

In transition metals d electrons participate in the conduction bandDelocalised "free" electrons Fermi-Dirac statistics rather than Boltzmann's

The field raises/lowers the energy of the spin-down/spin-up bands by = µBµ0H.Difference of number of electrons between the up and down bands :n = n – n = (D(EF)/2) – ( )(D(EF)/2) = µBµ0HD(EF)

The result is an induced magnetisation:

And a magnetic susceptibility :

To which should be subtracted 1/3 for the diamagnetic response (Landau) :

HVED

VnM B

FBP 0

2)(

20

)(B

FP V

ED

E

D(E)

EF

H

2Bµ0H

FB

BB

FB

FP TkV

NEN

VVED 2

020

20 2

3132)(

32

Refined : (Pauli)

Quick estimation :Use T+TF TF instead of T independent of T

FB

B

FB

BsP TkV

NTTk

SSgVNT

20

20

2

)(3)1()(

Ferro-MagnetismSpontaneous magnetisation – no external B – is possible

H

M

T=2TC

MsBj[µ0 µH/kBT]H/

T=TC

T=TC/2

Some elements (Fe, Co, Ni, some Rare Earth)and some of their oxides or salts show

Magnetisation at 1000K even in the absence of external field

Early explanation : Weiss Molecular FieldSuppose M creates a field Hmol = M on the electrons in the matterThe total field experienced by the electrons is thus

Hint = Hext + HmolAnd at some temperature the magnetisation is

M = (Hext+ Hmol) = (N/V)µBj[µ0 µ(Hext+ Hmol)/kBT]Let Hext = 0. What are the self-consistent solutions for

M = (N/V)µBj[µ0 µM/kBT] ?1) M = 0 (trivial)2) M 0 below some critical temperature

Critical temperature TC (Curie temperature) such that 1/= Ms dBj(H,T)/dH (H=0)

i.e. (TC)=1

Ferro-MagnetismWhat is the physical interaction responsible for it ?

• Molecular Field : phenomenological, not identified• Spontaneous magnetisation means that atomic moments are

parallel and behave coherently (magnetic ordering)• Search for interactions ~1000K that favour moment alignment

Dipole-Dipole interaction (Dipolar field created by all the moments at one moment)• Real field and long ranged (1/r3) but :• Tricky angular dependence : 3cos(a,r)cos(b,r) – cos(a,b)• Strength µ0 µB

2/a3 ~ 1K/degree of freedom• Magnetic ordering possible but academic for atoms (macroscopic magnets do order)• Compares to EZeeman = µBB ~ 1K/degree of freedom for B=1T• Plays a significant part in macroscopic magnetic properties, though

Magneto-crystalline anisotropy (Crystal Field + Spin-Orbit Interactions)• At most 10K/degree of freedom • Even in cases where crystal field favours a single axis there are still two equally probable directions at 180° : <M> = 0• Useful to avoid arbitrary rotation of the spontaneous magnetisation, though (permanent magnets)

Ferro-MagnetismDue to the Quantum Mechanical Exchange Interaction

Interplay of Pauli principle and Coulomb interaction• Two electrons of opposite spin can share the same orbital and come close• Two electrons of same spin cannot further apart Lower Coulomb energy (see Annex 3)• Hidden in the potential e(r), not really a magnetic field due to the spins

Exchange interaction at work in an isolated atom• Responsible for the first Hund's rule

Exchange interaction at work in covalent bondsMore subtle : there are other considerations than spin orientation that minimize the Coulomb energy when forming the bond. Examples :• H2 : s-s () bond, the favoured bonding orbital is the S=0 (singlet state ) whereas the S=1 (triplet state ) is antibonding with a larger energy. H2 is diamagnetic• O2 : p-p () bond, parallel spins are favoured. O2 is paramagnetic.

Exchange interaction at work in solid state• The sign of the interaction (whether it favours parallel or antiparallel alignment of the moments) is hard to guess. Generally written as Eexch = JS1S2 :

J > 0 favours parallel spins and ferromagnetic order J < 0 favours antiparallel spins and antiferromagnetic order

• The strength of the interaction depends on the orbital overlap between neighbour atoms decreases fast with distance (exponentially or 1/r10)

Outline - Energies




• Indirect exchange via ligand covalent bonds• SuperExchange, often antiferromagnetic

Schematics of and superexchangep eg

t2g

peg

t2g

Ferro-MagnetismExchange Interaction takes various paths

Exchange interaction calculationTreatment of magnetism in condensed matter relies on model Hamiltonian like Heisenberg's

The sum is usually taken over the nearest neighbour sites i,j . Tough !jiij

ijexch JH SS

Exchange interaction at work in Transition Element oxides and saltsWeak overlap between 3d orbitals of the magnetic ions

Exchange interaction at work in Rare Earth metals• Virtually no overlap between magnetic 4f orbitals• 5p and 5d shells closed, Stoner criterion not fullfilled (see next)• No superexchange through ligands• Indirect exchange through 6s conduction electrons• Named RKKY (Ruderman-Kittel-Kasuya-Yoshida)

Long range, oscillatory Ferro, Antiferro or Helicoidal

depending on structure/distances Oscillatory response of the

spin density at the Fermi seato a local magnetic moment

3)2()2cos()(

rkrkrJ

F

FRKKY

Spin density

r

• Indirect exchange via electron hopping• DoubleExchange, ferromagnetic

Hopping electrons like to keep their spin state+ Hund's rule on the from and to ions Same orientation of the neighbour moments

Ferro-MagnetismCase of Transition Metals – Itinerant Magnetism

Exchange interaction at work in Transition Metals• Molecular Field description is even less plausible : no Brillouin thermal dependence of M• Fair overlap between neighbour d orbitals but not sufficient• Need also a large density of states at the Fermi level (large Pauli susceptibility)

EF

EEEF

Paramagnetic metal Ferromagnetic metal

Element Cr Mn Fe Co Ni PdJ.D(EF) 0.27 0.63 1.43 1.70 2.04 0.78J <0 <0 >0 >0 >0 >0

Stoner criterionExchange energy :Assume dn states are transferred from the band to the band

Exchange energy decreases by

But Kinetic energy increases by

The transfer takes place spontaneously if

nJn

dnED

dnEdn

dnJ

F )(

)( 2

J.D(EF) > 1

E

EF

)(ED )(ED

E

Dipolar Energy and Demagnetising Field

ji rE j

ij

iijijidip

,

34 3

0 µµuuµ

• Tough to sum up• Not absolutely convergent r2dr/r3= Log(r)• Converges slowly only thanks to the

angular term

Simpler approach (continuous approach)• Fictitious currents : Volume density : J(r) = xM(r) and Surface density : Js(r) = nxM(r)

and use of Biot-Savart to calculate the dipolar field at any point• Electrostatic analogue : Less academic (magnetic poles do not exist) but much easier (local)

Volume charge density (r) = .M(r) Surface charge density (r) = n.M(r)

Hd=M/3

Hd= M

M

M

Simplest case (uniform magnetisation)• No volume terms xM(r) = 0 or .M(r) = 0

• Surface pole density = M component perpendicular to the surface of the object• Examples : infinite slab, sphere

200

22MNMHF dddip

Demagnetising Field : The dipolar field arising from the external surface is always opposed to the magnetisationDemagnetising factor (shape dependent) : Nd = Hd/MDemagnetising energy density(shape anisotropy)

Magneto-Crystalline Anisotropy

)(cos 2MKani KF

Simplest case (uniaxial anisotropy : cylindrical symmetry)where uj is the unit vector of the easy axis

• Takes actually more complicated angular dependence (higher order terms)• This first order term is absent in a cubic crystal (3 equivalent easy axis)

Energy density : (K in J/m3, MK angle between M and easy axis)

Magnetic anisotropy energy vs Thermal energy

0° MK 180°

Hopping

Ener

gy

E

Individual moments :E=Ka3~1KHopping < M> = 0

Collective moment :E=KV~1024K/cm3

No hopping Mr 0

M/M

s

-1

0

1

H

RemanentMagnetisation

Mr

Coercive field Hc

Hysteresis loop M(H)

Exchange interaction (to align the moments between them) and magnetic anisotropy (to align the collective moment along one direction) are necessary to obtain a permanent magnet

2jjuj

ani kE uµ

vi

JE jiexch µµ 20)cos( MJF MMexch

The interaction responsible for the magnetic ordering

Short ranged (1st neighbours)

Summary - Energies into play

F = Fexch + Fani + Fdip +Fzee

jiJE ijexch

,µµ 2

0)cos( MJF MMexch

and kBT !Mean Field approximation

Simplest approximate expressions (continuous approximation)

)(cos2MKani KF

jjjuani kE 2uµ

The interaction responsible for the remanent magnetisation

Useful for permanent magnets, harmful for flux concentrators



j

jjuani kE 2uµ )(cos2MKuani KF

ji

JE ijexch,

µµ 20)cos( MJF MMexch


Uniaxial anisotropy



j

ij

iijijidip

ji rE µ

µuuµ

,

34 3

0

200

22MNMHF dddip

Harmful inside (demagnetising) this is the useful field outside of a magnet (named stray field)

Long ranged (the whole volume)


j


ji rE j

ij

iijijidip

,

34 3

0 µµuuµ

200

22MNMHF dddip

ji

JE ijexch,



Uniaxial anisotropy

Uniform magnetisation


)cos(0 MHzee MHF j

jzeeE Hµ0

The magnetising energy due to the applied field

Zeeman energy



j

jzeeE Hµ0 )cos(0 MHzee MHF

j


ji rE j

ij

iijijidip

,

34 3

0 µµuuµ

200

22MNMHF dddip

ji

JE ijexch,



Uniaxial anisotropy




Magnitude (Energy density)Ku ~ 10+5±2 J/m3 1mK-10K/at 10+6±2 erg/cm3

Kd=½µ0M2 ~ 10+6 J/m3 1K/at 10+7 erg/cm3

J ~ 10+9 J/m3 1000K/at (TC) 10+10 erg/cm3

A=Ja2 ~ 1011 J/m 106 erg/cm


j

jzeeE Hµ0 )cos(0 MHzee MHF

j


ji rE j

ij

iijijidip

,

34 3

0 µµuuµ

200

22MNMHF dddip

ji

JE ijexch,



Uniaxial anisotropy



Outline - Scales




Exchange vs Demagnetising EnergyInfinite Ku

2 domains

Hd2

Hs1Hd1

Hs2HdM

1 domain

E=FdemV E=F'demV+FwallSEdem NdKda3 EdemNdKda3

Eexch+Eani 0 Eexch+Eani 2Aa+0

Single domain if NdKda3<4Aa

i.e.True for the very most cases

21

21

21

nm 1.04

aNKA d

d

Scale #1: lexch=Crude definition : maximum sizewithin which atomic momentscan be considered as parallel

Typical value : 3 nm(Kd=10+6 J/m3, A=1011 J/m)(1000 atoms in the cube)

• Sets the boundary betweenatomic and continuous

• Sets the mesh size inmicromagnetic calculations

dKA

Exchange vs Anisotropy EnergyCreate walls at lower cost

.

Eexch=A2/n0 but Eani=nKu/2

Eani=0 but Eexch=A

Scale #2 : wall , exchCrude definition: Bloch domain wall width, Bloch length, exchange correlation length

Typical values (Kd=10+6 J/m3, A=1011 J/m)

Soft material (Ku=10+3 J/m3) : wall , exch ~ 300 nmMedium (Ku=10+5 J/m3) : wall , exch ~ 30 nmHard material (Ku=10+7 J/m3) : wall , exch ~ 3 nm

uKA /

Energy and domain wall widthApproximation : d/dx/na, par= naEexch=Jna(2/2) =Ja22/2na =A2/2par

Eani= Kuna(1/2) = Kupar/2Ewall minimum for Eexch = Eani

uwall KA / uwall AK

Note : Energies are per unit wall areaNote : Bloch walls do not generateDemagnetising energy (.M=0)

Demagnetising vs Bloch wall energyMultidomains at lower cost

HdM

s

s

s

1 domain 2 domains

Hd2

Hs1Hd1

Hs2

Scale #3 : dsing

Crude definition: minimum size above which multidomain patterns can develop

Typical values(Kd=10+6 J/m3, A=1011 J/m)

Soft material : dsing ~ 2 nm(Ku=10+3 J/m3)

Medium : dsing ~ 20 nm(Ku=10+5 J/m3)

Hard material : dsing ~ 200 nm(Ku=10+7 J/m3)

du KAK /6

E=FdemV E=F'demV+FwallSEdem NdKds3 EdemNdKds3

Eexch+Eani Eexch+Eani=

Single domain if NdKds3<2

i.e. s <

2sAKu

2sAKu

d

u

d KAK

N2

Domains, Walls and Others

Inconsistency of previous coarse description/evaluationfor soft materials dsing ~ 2 nm << wall ~ 300 nm !e.g. a 30 nm sided cube cannot be single domain but a Bloch wall would be much wider, so what ?

Schematic distribution of M in a cube as a function of its size (in units of lexch) and the magnetic anisotropy of the material (in units of the dipolar energy)

More complex distribution of M (from micromagnetic simulations

A. Hubert, R. Schaeffer, "Magnetic Domains",Springer Verlag, Berlin, 2000

Stadium domains

Vortex state

Flower state

Soft Hard

SuperParaMagnetismA fourth length scale

T=300K, Ku=10+5 J/m3 and tm=1 s to 10 years Scale #4 : dblock~ 7 to 30 nm

u

Bcrit K

TkV

B

ubloc k

VKT

< tm > tmtmMeasurement

time

V increases, T decreases

=log(tm/0) =25 for tm=100 s

BlockedSuperpara

Reversal (thermal, tunnel)

- Magnetisation direction +

Tunnel

Thermal

Ener

gyKV

V< kBT/Ku

Multi Domain Single DomainV< Vsing

SuperparamagnetismM=M0exp(-t/)

0exp(KuV/kT)0~108-1010 s

Outline – Hard Magnetics




Magnetic Nanocomposites

2 Sizes : Dgrain,, dg-g

5 Scale lengths: dsing , dblock , exch1, exch2 , exch>

Permanent magnets

-2 -1 0 1 2-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

M/M

S

H/Hc

Maximise Mr

Optimisation

Maximise BHMaximise Hc

Wanted :

• K : large

• Ms : large

• No domain walls

Permanent magnetsHow to ?

H

H

M

Hc=2K/Ms

Single Domain Single Crystal bloc :Ideal ...BUT

Demagnetising dipolar energy :

Domain & wall formation

2dd KN

Early solution : minimise Nd

Mr decreases muchHc decreases much

Nd1 Nd~1/3 Nd0Nd1 Nd~1/3 Nd0Nd1 Nd~1/3 Nd0Nd1 Nd~1/3 Nd0

Decoupled Single Domain Grains Stoner-Wohlfarth

HMKF s )cos()(cos 02

: angle between moment and field: angle between easy axis and field

E.C. Stoner, E.P. Wohlfarth, Philos. Trans. Roy. Soc. London A 240 (1948) 599

TkVTK

TMTKTH

Bsc

)(251)(

)()(0

Effect of superparamagnetism :dsing ~ dblock(10 years)

Random axis :p()=sin() ; p(cos()) uniform

sc M

KH0

sr MM21

dg-g<exch>; Dgrain< dsing

Hard axis (=90°) (See Annex 5)

Easy axis (=0°) (See Annex 4)

Nanocrystalline Permanent MagnetsWeak coupling – Non or weakly magnetic matrix

H. Zhang, S. Zhang, B. Shena, H. Kronmuller, J. Magn. Magn. Mat. (2003)

)]()[cos()(cos 02 HMNHMKF ms

Weak coupling expressed as a mean field NmM

CouplingNm>0 : ExchangeNm<0 : Dipolar

Gain :Mr enhanced if Nm>0

Hc unchanged

dg-g~<exch>; Dgrain< dsing

Nanocrystalline Permanent MagnetsHard magnetic phase in Soft magnetic matrix

Km<<Kg ; Mm=Mg ; Vm=Vg (f=0.5)

H. Zhang, S. Zhang, B. Shena, H. Kronmuller, J. Magn. Magn. Mat. (2003)

Compromise

• Strong coupling exch> dg-g : very weak Hc(not shown – see later)

Dgrain~exch1; dg-g~exch2

• Weak coupling exch> dg-g :2 coercive fields, small BH

• Medium coupling (Nm~1):1 large Hc, increased MrMr = f.0.5Mg+(1-f)MmMr~0.8Ms for Mg=MmNote : Nm~1 : Fexch~M2 << J

Outline – Soft Magnetics




Soft Nanocrystalline FerromagnetsStrong Coupling between grains

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

M/M

s

H [K/0Ms]SI ou [K/Ms]emu

Optimisation

Maximise Ms

Maximise µr

Minimise Hc

Wanted :

• K : weak

• Ms : large

Inside the material :B = 0(H+M) = 0H(1+) = 0rH

The larger r the better the flux concentration

H

Soft Magnetic MaterialsHow to ?

Eliminate domain wall pinning (need wall size of defects)

Weakest possible anisotropy Multidomain state, magnetizes by domain growthSuppress magnetostriction

Early materials :

• Soft iron (pure Fe)but still K and Magnetostriction

• Iron-Silicon (FeSi~6%)possibly oriented

• Permalloy or Mumetal (Ni80Fe20)Very weak anisotropyNull magnetostrictionµ up to 10,000ButLow MUneasy implementation

H

M

Pinning

Pinning

Soft materials : Random anisotropy1 magnetic phase (hypothetical

1effN

KfK

R. Alben, J. J. Becker, and M. C. Chi, J. Appl. Phys. 49 (1978) 653 G. Herzer, IEEE Trans. Magn. 25 (1989) 332 ; ibid. 26 (1990)1397

21

eff

eff

KA

exch

Happ

Happ

Happ

Dg

exchfN3

1 Phase : f=1, Aeff=A1

)()(

)(0

, TMTK

THs

effcplc

VKTkC CB )0(/25 1

Cn

ndecc T

TTm

CTmTh)(

1)()( 1,

)(10)(3

)()0()()()0()(

)0(/)()(2

cubicnuniaxialn

TmKTKTmATAMTMTm

n

ss

)()( 74, TmTh ncplc

TkVTK

TMTKTH

Bsdecc

)(251)(

)()( 1

0

1,

<exch> Dgrain>dg-g

6

1

eff

exchf 2

K

K Dg

N

fVirtual if single phased

f: magnetic volume fraction

Random AnisotropySoft magnetic phase in Soft magnetic matrix

T/TC1

Coe

rciv

e Fi

eld

(arb

.)

But :TTC2 A20

J. Arcas et al. Phys.Rev. B58 (1998) 5193

Aeff =0 decoupling

21 AAAeff

exch1Dgrain; exch2dg-g Finemet ®Fe50Co25Si12B9CuNb3Maximises MMinimises K1AmorphisesNucleates grainsInhibits grain growthSoft crystalline grains: FeCoSiAmorphous matrix : FeCoNbCuB

Coe

rciv

e Fi

eld

(A/m

)

Temperature [°C]

Tanneal

ExperimentSoft amorphous partly crystallised

Outline – Recording Media




Magnetic Recording Media

Tra

ck w

idth

in µ

m

Bit length in nm

Cell / Bit size

1988

Source : www.almaden.ibm.com

2002

Magnetic Recording Media- Where to

Advanced Magnetic

Technologies

Future Technologies

Source : www.almaden.ibm.com

SuperparamagnetismA

real

Den

sity

Gbi

tspe

r squ

are

inch

85 90 95 2000 05 10 15Year

10000

1000

100

10

1

0.1

0.01

Magnetic recording MediaBreaking the Superparamagnetic Limit

• High Coercivity MediumProblem :

Writing field not strong enough

Synchronised heating laser pulses

Solution : Thermally assisted writingQuestion : Writing speed

0.1 0.2 0.3 0.4 0.6 0.8 1µ0Hc [Tesla]

1000

100

10

1

0.1

Are

al d

ensi

ty G

bits

/in2


• High Coercivity Medium

• Reduction of stray field between bits

• Perpendicular Recording

Perpendicular Recording

MR Read

Flux return pole

Soft underlayer for flux closure

Advantages :• Larger writing field• Reduced demagnetising field• Abrupt 0/1 Transitions

Drawback :• More complex head geometry

Writepole


• High Coercivity Medium

• Reduction of stray field between bits

• Nano-structure Medium : Isolated dots - 1 bit / dot• Perpendicular Recording

Nano-structured Medium

AdvantagesNo exchange interaction between bitsWeak stray fieldsDrawbackLithography

Hundreds of studies of the magnetisation processExperiment and simulations


• High coercivity medium

• Reduce stray field between adjacent bits

• Trilayer medium with antiferromagnetic coupling• Nano-structured medium : isolated magnetic dots - 1 bit per dot• Perpendicular recording

Trilayer with antiferromagnetic coupling

Two magnetic layers antiferromagnetically coupled through the Ru layer :

Density Gain ~ 2 : 100 Gbits/in2 reached

CoCrPtRu 0.6 nmCoCrPt

Reduced stray fieldbetween bits

Outline - Spintronics




Magnetic Coupling between separated layers

100

110 11

1ex: ex: cfc ((100) Growth) Growth

These qq vvectors arestationary with respectto qq

P.Bruno: Phys.Rev. B52 (1995) 411 J.C.Slonczewski: J.Mag.Mag.Mat (1995)

Typical periods (scale #5): 1 atomic distance and 1-1.5 nm(depending on the Fermi surface shape of the spacer metal and the crystallographic orientation)

"RKKY" like in 1 dimension : Magnetic/Normal/Magnetic layers• Coupling between two magnetic layers separated by a non magnetic

metal of thickness t• The susceptibility (q) of the spacer metal shows singularities for

some stationary q vectors (2kF for a 3D free electron gas)

Oscillating antiferro/ferromagnetic coupling between the magnetic layers as a function of t

Magnetic Coupling Mag/NMag/Mag

P.Grünberg et al: Phys. Rev. Lett. 57 (1986) 2442; S.S.P. Parkin, N. More and K.P. Roche, Phys. Rev. Lett. 64 (1990) 2304; J.Unguris et al: Phys.Rev. B49 (1994) 14 et 564

Coupling oscillations between 2 magnetic layers through a non magnetic metal spacer

Fe/Cr/Fe

Observations by SEM with spin polarisation analysis

Co/Ru/Co

Observations by Brillouin light scattering

Discovery of the Giant MagnetoresistanceMag/NonMag/Mag multilayers

Antiferromagnetic coupling between Fe layers in Fe/Cr/Fe

and

Thickness smaller than electronic mean free path and/or spin diffusion length (5-200 nm)

Magnetic

Magnetic

Nonmagnetic

Fe

Cr

Fe

M. Baibich et al., Phys.rev.Lett. 61 (1988) 2472

Tunnel and Giant Magnetoresistance2 Current Model

~R/2 > ~2rH=0 < Hsat

>

Current in plane

(GMR only)

Current perpendicular to

plane(Spin filter)e

e e

ee e

M. Baibich et al., Phys.rev.Lett. 61 (1988) 2472; A.Fert et al: Physics World Nov.1994 34

"Giant" MagnetoresistanceOrigins

• Spin dependent scattering by impurities

A.Fert et al / S.Maekawa, J.Inoue / P.M. Levy, S.Zhang: Mat. Sci. & Ing. B31 (1995) 1 / 31 / 157

Conductivity

• Spin dependent transmission at interfaces

Transmission

e

e

e

e

J

FF

FFJ

"Tunnel" MagnetoresistanceOrigin

Parallel

P 2 N1N2(1+P1P2)

Antiparallel

AP 2 N1N2(1-P1P2)

• TMR : R/R=2 P1P2/ (1-P1P2)

Magnetic

Magnetic

Non MagneticInsulator

• Typical polarisation of a ferromagnetictransition metal : 40% R/R35- 40%

)()()()(

FF

FF

ENENENENP

M.Jullière, Phys. Lett. 54A (1975) 225; J. C. Slonczewski, Phys. Rev. B 39 (1989) 6995;J. S. Moodera et al. Phys. Rev. Lett. 74 (1995) 3273

FM1 FM2I

EF

FM1 FM2I

EF

• Possible improvement : half-metal (P=100%) or symmetry filtering

EF

Spintronics : Scale #6

Spin diffusion length :

Average distance between two scattering events with spin flip (inversion)

Spintronics :• Distinguishes 2 carrier kinds e and e• Over distances < the two channels are independent

2 currents flowing parallel with n n and

Scale #6: ~ 5 nm (ferro alloy) 50 nm (ferro metal) 150 nm (non mag. metal)T. Valet, A. Fert, Phys. Rev. B 48 (1993) 7099; A. Fert, T. Valet, J. Barnas, J. Appl. Phys. 75 (1994) 6693A. Fert, L. Piraux, J. Mag. Mag. Mat. 200 (1999) 338

Ferromagnetic Non MagneticMaterial Material

e

e

e

e

P

Non Magnetic FerromagneticMaterial Material

e

e

e

eP

Magnetoresistive Device - GMR / TMR

>

Soft magneticmetal

Metal or OxideGMR TMRnon magnetic

Hard magneticmetal

~R/2~R/2 ~2r~2r

GMR and TMR at work

• Differents configurations• Antiparallel coupled multilayers• Asymmetric hard/soft multilayer• Pinned layer / Free layer• Discontinuous multilayer• Magnetic clusters in a non magnetic matrix

R / R

- H 0 + H - H 0 + H - H 0 + H - H 0 + H - H 0 + H

• Applications: sensors, reading head, memory, ...

Magnetic multilayersArtificial Nanocomposites and Devices

Protection layers

Soft layerTunnel barrierHard layer (Artificial AF)

Buffer layers10 nm

CrCuFeCo

Al 2O3Co50Fe50

RuCo50Fe50

Cu

Fe

CrSubstrate

Si (111)

Magnetoresistive Device - Cycles

M

R

Hc2 H

H

Hc1

2 Pinned1 Free

Work CycleMain Cycle

-600 -300 0 300 6000

10

20

30

40

TM

R (%

)

Applied field (Oe)

Angular Sensor

External magnet

Freemagnetic layer

Nonmagnetic layer

Pinnedmagnetic layer

Rés

ista

nce

0 180 360Angle

R=Rp + [(Rap-Rp)(1-cos)/2]

MRAM – Magnetoresistive Memory

1 cm

1960 – Magnetic Tore Memory

Reading a bit

Writing a bit "0"or"1"

Diode

DMR

State"1"

State"0"

Annex 1

Units

CGS-emu MKSA-SI• B(G) = µ0=1)(H(Oe) + 4M(emu)) B(T) = µ0(H(A/m) + M(A/m))

• Induction 1 G 10-4 T

• Field 1 Oe 1000/4 A/m

• Magnetization 1 emu(M), 1 Erg.G1.cm3, 4Oe 1000 A/m

• Moment 1 emu(m), 1 Erg .G1 10-3 A.m2

• Intensity 1 emu(I) 10 A

• Demagnetizing Field Hd = NM

• Homogeneous Sphere 4/3)M(emu) Oe /3)M(A/m) A/m

• B(G) = µ0=1)(1+4(uem))H(Oe) = µ0=1)µrH(Oe) B(T) = µ0 (1+)H(A/m) = µ0µrH(A/m)

• Susceptibility 1 emu() 4 Unity

• Total energy(densty) E(erg/cm3) = (1/4B(G).H(Oe) E(J/m3) = B (T).H(A/m)

• Matter only (densty) E(erg/cm3) = µ0=1)(emu).H(Oe) E(J/m3) = µ0M(A/m).H(A/m)

• 1 Erg.cm3 1 G.emu(M) 1/10 J/m3

• 1 G.Oe 1/4 Erg.cm3 1/40 J/m3

• Demagnetizing Energy (density) (1/2)NM2

• Sphere µ0=1)(2/3)M2(emu) Erg.cm3 (µ0/6)M2

(A/m) J/m3

• Energy W(erg) = µ0=1)m(emu).H(Oe) W(J) = µ0 m(Am2).H(A/m)

• 1 Erg 1 G.emu(m) 10-7 J

• µ0 = 1 G/Oe µ0 = 7 Tm/A or N/A2

Some Universal constants

Planck’s constanth = 6.6226 10-34 Js, = h/2π = 1.055 10-34 Js

Boltzmann's constantkB = 1.381 10-23 J/K

Electron chargeq = 1.602 10-19 CElectron rest massm = 9.109 10-31 kg

Bohr magnetonµB = 9.274 10-24 Am2

Permittivity of a vacuumµ0 = 4 10-7 N/A2

Annex 2

Orbital Moment

Quenching of orbital angular momentumExample : p case, l = 1, ml = 0, ±1

Spherical harmonics are the eigenstates of a central potentialY0 = R(r)cos(Y±1 = R(r)sin(e±i

But they are not eigenstates of the hamiltonian in the non spherical crystal fielde.g. the potential created by an octahedron of charges q and side a (cubic field cf) is

Hcf = (eq/a6) D [(x4+y4+z4) 3(y2z2+z2x2+x2y2)]

Y±1 are not eigenfunctions of Hcf : <Yi|Hcf|Yj>≠ Ei.ij

Linear combinations of Yi that are eigenfunctions of Hcf :

Y0 = R(r)cos() = zR(r) = px

(1/√2)(Y+1 Y-1) = R’(r)sin()cos() = xR(r) = py

(1/√2)(Y+1 Y-1) = R’(r)sin()sin() = yR(r) = pzThe z-component of angular momentum; lz = i∂/∂ is zero for these wavefunctions.

The orbital angular momentum is quenched !

Orbital quenching of transition metal ions• Contrary to f electrons that are well screened by 5s2,5p6,5d0,1,6s2 electronsd electrons are subject to the strong crystal electric field (CEF) of the neighbour ions• The CEF lifts the 2L+1 degeneracy of the dn - electrons

eg

t2g

spherical symmetry octhahedral symmetry• Orbital angular momenta of non-degenerate levels have no fixed phase relationship• Therefore the time average expectation value of the orbital moment is <Lz>=0• Lz is not a good quantum number.

In general, for an anisotropic field, Lz is not an integral of the motion.Depending on the filling of the orbitals and their degeneracy the time average will lead to a total or partial cancellation of the orbital angular momentum.

Note: If crystal field splitting << LS spin-orbit coupling the phase coherence of the angular momenta is preserved.

Annex 3

Exchange

H12 acts only on the spatial part of the wave function (1,2)YET the energies of the singlet and triplet states are different

Two electron system and exchange interactionConsider two electrons (1,2), total wave function (1,2), on two atoms (a,b)

(1,2) can be factorized into spatial () and spin () functions : (1,2) = (1,2) (1,2)

The antisymmetry (Pauli principle) of (1,2) can be achieved in two ways :

• Singlet state (S=0) with symmetric and antisymmetric

s(1,2) = (1/√2)[a(1) b(2) a(2) b(1)] (1/√2)(> – >)

• Triplet state (S=1) with antisymmetric and symmetric

( ) mS=+1t(1,2) = (1/√2)[a(1)b(2) – a(2) b(1)] (1/√2)( ) mS= 0

( ) mS=1

If the atoms are close to each other the hamiltonian corresponding to the mutual Coulomb interaction can be written as : H12 = V(a,b) V(1,b) V(2,a) V(1,2)

The corresponding energies are Es,t = s,t*H12s,t dV Es = K12 J12 and Et = K12 – J12

K12 the Coulomb integral : K12 = a(1)b

(2)H12a(1)b(2)dV1dV2

J12 the exchange integral : J12 = a(1)b

(2)H12a(2)b(1)dV1dV2

The effective spin interaction and its hamiltonian(Heisenberg)

A pair of electrons can assume two spin states :

• Singlet non magnetic state (S=0) : (1/√2) (> – >)

( ) mS=+1• Triplet magnetic state (S=1) : (1/√2) ( ) mS= 0

( ) mS=1

Because of the combined effect of the Coulomb interaction and the Pauli principle the energies of these to states are different if the electrons can exchange positions :

Es = K12 J12 and Et = K12 – J12 Es - Et = 2J12

One can devise a hamiltonian Hspin that acts only on the spin part of the wave function and that yields the same eigenvalues Es,t

Consider the spin only operator 2S1S2

2S1S2 = S122 – S1

2 – S22 (S12 is the total spin operator S1+S2)

Eigenvalues of 2S1S2 : s12(s12 1) – s1(s1 1) – s2(s2 1) = s12(s12 1) – 2s(s 1)Singlet (S=0) : eigenvalue = –3/2Triplet (S=1) : eigenvalue = +1/2

Hspin = K12 – J12/2 – J12.2S1S2

Annex 4

Anisotropy (Easy)

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle

Anisotropy Zeeman Total

H = +3.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +1.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +1.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +0.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +0.0 K/M

M

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −0.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −1.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −1.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −3.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −1.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −1.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −0.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +0.0 K/M

M

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +0.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +1.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +1.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +3.0 K/M

M

H

H

Annex 5

Anisotropy (Hard)

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +3.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +1.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +1.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +0.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +0.0 K/M

M

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −0.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −1.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −1.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −3.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −1.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −1.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = −0.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +0.0 K/M

M

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +0.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +1.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +1.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.0 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +2.5 K/M

M

H

H

-180 -135 -90 -45 0 45 90 135 180

-4

-3

-2

-1

0

1

2

3

4

Ene

rgy

Moment angle


H = +3.0 K/M

M

H

H

Documents

Introduction to Magnetism : Scales and Nano-Magnetismqmmrc.net/winter-school-2009/panissod.pdf · Introduction to Magnetism : Scales and Nano-Magnetism Pierre Panissod Institut de