Hercules Magnetism Simonet

7/30/2019 Hercules Magnetism Simonet

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Refresher lecture in magnetismVirginie Simonet,

Institut Nel, CNRS-UJF, BP166, 38042 Grenoble Cedex 9

2/04/10

Outline :

IntroductionAtomic magnetic momentAssembly of non interacting magnetic moments

Magnetic moments in interactionFrom microscopic to macroscopic

Applications

Modern trends in research


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Introduction

Magnetic materials all around us : the earth, cars, audio, video, computertechnology, telecommunication, electric motors, medical imaging

Magnetism: science of cooperative effects of orbital and spin moments in matter-> Wide subject expanding over physics, chemistry, geophysics, life science.

Large variety of behaviours : dia/para/ferro/antiferro/ferrimagnetism, phasetransitions, spin liquid, spin glass, spin ice, magnetostriction, magnetoresistivity,magnetocaloric effect,in different materials : metals, insulators, semi-conductors, oxides, molecularmaterials

Inspiring or verifying lots of model systems : Ising 2D (Onsager)

Magnetism is a quantum phenomenon but phenomenological models commonly usedto treat classically matter as a continuum


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Atomic magnetic moment

An electric current is a source of a magnetic field

A magnetic moment m is equivalent to a current loop (Ampre)m=I.S (coil magnetic moment)creating a dipolar magnetic field

Biot Savart law

Note : magnetic monopoles so far undetected


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Magnetic moment is related to angular momentum :electrical current comes from the motion of electrons and is source ofmagnetism in matter

Example for a one turn coil : orbital magnetic moment

L = r p = mr2n

l =e

2mL = L

l = I.S=e

2

r2n =er2

2

n

gyromagnetic ratio

Consequences :magnetic moment and angular momentum are antiparallelCalculations with magnetic moment using formalism of angular momentumPrecession of magnetic moment in a magnetic field : Larmor precession

e- orbitingaround the nucleus

L = B0

angular momentum


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Electronic orbitals are eigenstates of and operatorsOrbital angular momentum and its projection are quantized in units of (Bohr)

lzl2

Quantum mechanics:

The component of the orbital angular momentumalong the z axis is

The magnitude of the orbital momentum isl(l + 1)

ml


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The component of the spin angular momentumalong the z axis is

The magnitude of the spin momentum is

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Quantum mechanics:S, spin angular momentum of pure quantum origin

Classical picture of e-rotating about itself

Two contributions to the atomic magnetic moment : spin and orbit

With s=1/2, ms=-1/2,+1/2 quantum numbers

with gs=2, gl=1s = gsB s

l = glB l

ms

s(s

+ 1)

B =e

2me

Magnetic moments

and the Bohr magneton


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1 : maximum

2: maximum in agreement with 1st rule

Spin-orbit coupling : relativistic expression of the magnetic induction effect

on the spin of the e- from its orbital motion

3 :

Several e- in an atom:Combination of the orbital and spin angular momenta of the different electrons :related to the filling of the electronic shells in order to minimizethe electrostatic energy and fulfil the exclusion Pauli principle

Hunds rules

L.

S

L =ne

l

S=

ne

s

S=ne

ms

L =

ne

ml

J= |L+ S|J= |L S|

J= L+ S

for more than filled shellfor less than filled shell

total angular momentum


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A given atomic shell (multiplet) is defined by 4 quantum numbers :L, S, J, MJ with -J


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Atomic magnetic momentMagnetism is a property of unfilled electronic shells :

Most atoms (bold) are concerned but only 22 magnetic in condensed matter


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Atom in matter:chemical bonding -> filled e- shells : no magnetic momentsExcept for :

Situation more complicated for 3d metals :magnetism due to delocalized 3d electrons

in insulatorsin insulator/metals


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Atom in matter:Influence of surrounding charges -> crystal field (CEF)

3d electronsLarge CEF>>spin-orbit :

angular distribution of 5 orbitals-> some favoured by CEF-> quenching of orbital momentum+ Spin-orbit coupling : g anisotropy

five 3d orbitals


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Atom in matter:Influence of surrounding charges -> crystal field (CEF)

4f electronsSpin-orbit>>CEF:4f charge distribution +CEF

-> selects some orbitalsSpin-orbit-> anisotropy J : alignement of magnetic moments along some directions

Charge distribution of rare earths


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Summary :Magnetism is a quantum phenomenon

Magnetic moments associated to angular momenta

Orbital magnetic moment and spin magnetic momentLocalized magnetic moment in 3d and 4f atoms : different behaviour

Orbital and spin moments can be strongly coupled (spin-orbit coupling in 4f)

Importance of environment, crystal field:

quenching of orbital moment in 3d and magnetocrystalline anisotropy in 4f


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WB = B(L + 2

S). B +

e2

8me

ie

(Ri B)

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One atomic moment in a magnetic field B

Energy:

Zeeman energy : coupling of total magnetic moment with field

Diamagnetic term : induced orbital moment by the external field

Assembly of non-interacting magnetic moments

M=

E

B

Magnetization : derivative of energy wrt magnetic fieldsusceptibility: derivative of magnetization wrt magnetic fieldor ratio in the linear regime

= MB

=

MB

lin


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WB = B(L + 2

S). B + e

2

8me

ie

(Ri B)2

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Energy:

N atomic moments in a magnetic field B:Boltzmann statistics + perturbation theory


M =N

V

j

Ej

B

exp(Ej)j exp(Ej)

Diamagnetic term:

Diamagnetic magnetization due to induced moment by magnetic field :negative weak susceptibility, concerns all e- of the atom, T independent

= N

V0

e2

4me< R2

>


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WB = B(L + 2

S). B + e

2

8me

ie

(Ri B)2

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Energy:


Paramagnetic term:

and the Brillouin function

M=N

VgJJBBJ(x) x =

gJJBB

kBT

BJ(x) =2 + 1

2Jcoth

2 + 12J

x

1

2Jcoth

x

2J

with


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Paramagnetic term

Brillouin functions compared to Langevin functions from classical calculation

Limit x>>1 i.e. H>>kBTSaturation magnetization:

M=N

VgJJB


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Paramagnetic term

Limit xHCurie law:

with the effective moment

peff = gJ

J(J+ 1)B

=N

V

(BgJ)2J(J+ 1)

3kBT=

C

T=

N

V

p2eff

3kBT

Works well for magnetic moments without interactions,negligible CEF : ex. Gd3+, Fe3+ or Mn2+ (L=0)


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In metals :Pauli paramagnetism (>0, weak, T-independent)


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Magnetic moments in interaction

Dipolar interaction :

electrostatic origin + Pauli exclusion principle

E= 0

4r3 [1.2

3r2 (

1.r)(2.r)]

much too weak to account for ordering of most magnetic materials

Exchange interaction :

Heisenberg Hamiltonian

2 electrons cannot be in the same quantum statemany-electrons wavefunctions are antisymmetricwith respect to the exchange of 2 electrons

: Exchange coupling constant> 0 ferromagnetic< 0 antiferromagnetic coupling

J

H =

ijJij

Si.

Sj


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Exchange interaction :

Direct exchange usually weak -> small orbital overlap between magnetic orbitals

Superexchange : mediated by the non-magnetic ions between the magnetic ones

Most often antiferromagneticExplains the magnetism in transition metal oxides


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Exchange in metals

In 3d metals

In rare-earth metals

The interaction between 4f localized momentsis mediated by 5d and 6s itinerant electrons :Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction

Hij = J(Rij)Si.Sj

The magnetic arrangement determined by kF, the Fermi wave-vector

with J(r) cos(2kFr)r3

r >>1

2kFfor

Interaction via overlap of the 3d wavefunctions :its sign depends on the filling of the bands


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Magnetic moments in interactionFrom paramagnetic state at high temperature to ordered state at low temperature

kBT>>exchange interactions

All moments //

Several sublattices: directionsof magnetic moments-> compensate

Several sublattices: directions

of magnetic moments-> do not compensate


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Treatment of interacting magnetic moments : Molecular field

Interactions represented by a fictitious field originating from neighbouring moments

Ferromagnetic case :

with

Bmf = M With positive

At low temperature, the moments can be alignedby the internal molecular field without external B

H = gB

i

Si.( B + Bmf) Bmf = 2

gB

j

JijSj

H =

ij

JijSi.Sj + gB

j

Sj . B

=2zJ

ng22

B


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Magnetic susceptibility

M= (gJB)

2

J(J+ 1)3kBT

(B + M) = CT

(B + M)

In the low field, high temperature limit

TC=C Curie temperatureAt Tc, becomes infinite : the system becomes spontaneously magnetized

=C

T C=

C

T TC


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Magnetization below TC

M= gJBJBJ(x)

Solve simultaneously 2 equations x =gJBJ(B + M)

kBT

For B=0

M/Ms

y

No solution for T>TCOne solution for T


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Antiferromagnetism : same analysis but for each of the 2 sublattices

Spontaneous magnetization below the Nel temperature TN on each sublattice

TN = ||C =C

T+ TN

Susceptibility

More complicated below TN :depend of field orientation


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Generalization:

Curie-Weiss law

1/

=TC=0Ferromagnets

TC

Fe 1043 KCo 1394 KNi 631 KGd 293 K

AntiferromagnetsTN

CoO 293 KNiO 523 KMnO 116 K

Shull 1951Neutron diffraction

=-TN

T>TN

T


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Other types of magnetic orders

Helimagnetism : helical order of momentsEx. Rare earths crystals

case of a J1/J2 chain

Ferrimagnetism, =-TN but spontaneous magnetization ;Spontaneous magnetization on each sublattice may have T dependence->compensation temperatureEx. Ferrites, garnets

Solutions =0 (ferro), = (antiferro) or

helixcos() = J1

4J2

E= 2NS2(J1 cos() + J2 cos(2)

J2

J1


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Other types of magnetic ordersComplex magnetic structures often due to frustration of interactions

?

Example for a triangle of magnetic moments

Antiferromagnetic interactions

Ising moments

Antiferromagnetic interactions

Heisenberg moments-> Non collinear

Example Ba3NbFe3Si2O14Helix + 120 arrangement


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Magnetic excitations

perfect order at T=0At T0, order disrupted by spin waves

Short rangeinteractions

Allows entropy gain without loosing too much in exchange energy


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Magnetic moments in interactionMagnetic excitations

Dispersion relationFor a cubic crystal

Ferromagnetic case Antiferromagnetic case

Bloch law :valid at small T,outside critical region

Ms(0)Ms(T)

Ms(0) T

3/2

E(k) = 4JS(1 cos(ka)) E(k) = 4JS| sin(ka)|


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From microscopic to macroscopicMacroscopic behaviour of magnetization, a compromise between 4 interactions:

Exchange interaction :favours uniform magnetization. Very strong but short-ranged

Dipolar interaction :tends to avoid formation of magnetic poles. Weak but long-ranged

Magnetocrystalline anisotropy :orients magnetic moments along privileged directions

Zeeman energy, interaction with an external magnetic field :alignment of magnetic moments along the field

For a homogeneous ferromagnetic material, minimization of free energy:

T = ex + dip + an + H


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From microscopic to macroscopic

-> magnetic moments will prefer to align along certain crystallographic directions(stronger for 4f than for 3d atoms)

Magnetocrystalline anisotropy

Ex. metamagnetic transitions in antiferromagnets

Weak anisotropy :spin-flop transition

Strong anisotropy :spin-flip transition


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From microscopic to macroscopicMagnetocrystalline anisotropy

Magnetization variation against anisotropy in ferromagnets

Uniaxial anisotropy

E= 0HappMs sin+Ksin

E

= 0 sin = 0HappMs

2K

sin = 1

easy axis

hard axis

Anisotropy field

Happ = HA =2K

0Msfor


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From microscopic to macroscopicMagnetocrystalline anisotropy

Magnetization variation against anisotropy in ferromagnets

Cubic symmetry

EA = K1(22 + 22 + 22) + K2

222 + ...

easy axis

easy axis

, , : cosines of the angles between magnetizationand the x, y, z directions// 4-fold axes


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minimising the demagnetising field produced by the material

-> formation of magnetic domainswith magnetization along the directions privileged by anisotropy

Dipolar energy E=

0

4r3 [1.2

3

r2 (1.r)(2.r)]

-> shape anisotropy

Explains zero macroscopic magnetization in ferromagnetic materials below TCif they have not been submitted to a magnetic field.


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Cost in exchange and anisotropy energiesat the boundaries between domains: domain walls


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From microscopic to macroscopicWidth of the wall : balance between exchange and anisotropy energy

Note : other types of domain walls in reduced dimension systems

EA = N K < sin2 >

K

2

Eexch = NJS2(1 cos ) JS2

=2aEexch

K

=

2

KEexchEnergy of the domain wall:

5-100 nm

Exchange energy lost:

Anisotropy energy lost

Total energy minimization

Domain wall width:


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Coercitivity represents the magnetization ability to resist

reversal against applied magnetic field

Coercive field for coherent rotation :Stoner-Wohlfarth model

E= Ksin2 + 0MsHcos

Energy minimization wrt :

As long as , =0 and are

two minima separated by a barrierWhen

the energy barrier flattens and the magnetization can rotate to the =minimum

uniaxial anisotropy Zeeman term

0

H= 2K/0Ms

H < 2K/0Ms


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Stoner-Wohlfarth model works well for nanoparticlesThe coercive field

In macroscopic materials, influence of defectsRotation occurs by nucleation on defectsand propagation of domain walls

But for most systems

Hc = 2K/0Ms

Hc


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Hysteresis cycle of a ferromagnet


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ApplicationsApplied research -> lots of applications, concerns mostly ferromagnetic materials

Hard magnetic materials (reasonable value of remanence, high coercitivity)Soft magnetic materials (high remanence, low coercitivity)Magnetic memory materials (high remanence, moderate coercitivity)Materials for electronics : operate at high frequencies

Recording and reading


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Research in magnetism : modern trends

Frustration : complex magnetic orders, spin liquid, spin ices Molecular magnetism : photoswitshable, molecular magnetsFrom quantum to classical: mesoscopic scale-> Quantum computerMultiferroism : coexistence of two ferroic orders (magnetic, electric, elastic)Low dimension systems: Haldane, BEC, Luttinger liquid

Quantum phase transitionsMagnetism and superconductivityNano materials : thin films, multilayers, nano particles->SpintronicsMagnetoscience

Hercules Magnetism Simonet

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