Introduction to Magnetism : Scales and Nano-Magnetismqmmrc.net/winter-school-2009/panissod.pdf ·...
Transcript of Introduction to Magnetism : Scales and Nano-Magnetismqmmrc.net/winter-school-2009/panissod.pdf ·...
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
• 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 »
• 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
• 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 »
• 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
Summary - Energies into play
F = Fexch + Fani + Fdip +Fzee
j
jjuani kE 2uµ )(cos2MKuani KF
ji
JE ijexch,
µµ 20)cos( MJF MMexch
and kBT !Mean Field approximation
Uniaxial anisotropy
Simplest approximate expressions (continuous approximation)
Summary - Energies into play
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)
F = Fexch + Fani + Fdip +Fzee
j
jjuani kE 2uµ )(cos2MKuani KF
ji rE j
ij
iijijidip
,
34 3
0 µµuuµ
200
22MNMHF dddip
ji
JE ijexch,
µµ 20)cos( MJF MMexch
and kBT !Mean Field approximation
Uniaxial anisotropy
Uniform magnetisation
Simplest approximate expressions (continuous approximation)
)cos(0 MHzee MHF j
jzeeE Hµ0
The magnetising energy due to the applied field
Zeeman energy
Summary - Energies into play
F = Fexch + Fani + Fdip +Fzee
j
jzeeE Hµ0 )cos(0 MHzee MHF
j
jjuani kE 2uµ )(cos2MKuani KF
ji rE j
ij
iijijidip
,
34 3
0 µµuuµ
200
22MNMHF dddip
ji
JE ijexch,
µµ 20)cos( MJF MMexch
and kBT !Mean Field approximation
Uniaxial anisotropy
Uniform magnetisation
Simplest approximate expressions (continuous approximation)
Summary - Energies into play
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
F = Fexch + Fani + Fdip +Fzee
j
jzeeE Hµ0 )cos(0 MHzee MHF
j
jjuani kE 2uµ )(cos2MKuani KF
ji rE j
ij
iijijidip
,
34 3
0 µµuuµ
200
22MNMHF dddip
ji
JE ijexch,
µµ 20)cos( MJF MMexch
and kBT !Mean Field approximation
Uniaxial anisotropy
Uniform magnetisation
Simplest approximate expressions (continuous approximation)
Outline - Scales
• 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 »
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
• 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 »
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
• 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 »
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
• 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 »
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
Magnetic recording MediaBreaking the Superparamagnetic Limit
• 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
Magnetic recording MediaBreaking the Superparamagnetic Limit
• 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
Magnetic recording MediaBreaking the Superparamagnetic Limit
• 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
• 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 »
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
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
Anisotropy Zeeman Total
H = +3.0 K/M
M
H
H