Rappels sur le magnétisme Quick reminder about...

Rappels sur le magnetismeQuick reminder about magnetism

Laurent RannoInstitut Neel CNRS-UJF

laurent.ranno@grenoble.cnrs.fr

Ecole Magnetisme et NeutronsJDN20 Seignosse 18-22 mai 2012

19 mai 2012

Laurent Ranno Institut Neel CNRS-UJF Rappels sur le magnetisme Quick reminder about magnetism

Magnetism : Why ?

What is special about magnetism ?Vast quantity of applications for magnetic materialsWhere are these materials ?

Hard Disk Drive (HDD)

Magnetism

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Discrete Components :TransformerFilterInductor

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Magnetism

Many different materials are used : oxides, elements, alloys, films,bulkMany different effects are exploited : high coercivity, evanescentanisotropy, magnetoresistance, antiferromagnetism, exchange bias...All need to be understood in detail

to improve the material properties

to imagine/discover new fields for applications and for research

to test ideas in great detail

You can build the model magnetic system to test your theoreticalideas.

Outline

In this lecture I will review some basic ideas to study magnetism

Origin of atomic magnetic moments

Magnetic moments in a solid

Interacting Magnetic moments

Ferromagnetism (time-permitting)

Atomic Magnetism

Electrons are charged particles (fermions) which have an intrinsicmagnetic moment (the spin).

Electronic magnetism is the most developped domain but nuclearmagnetism is not forgotten :because it’s a neutron schoolbecause nuclear magnetism is more difficult to study but has alsogiven birth to NMR, MRI which have a huge impact on society

Today I will restrict my lecture to electronic magnetism

Orbital Magnetic Moment

Electrons are charged particles :

one electron on its atomic orbit = current loop = coilIts orbital magnetic moment ~µorb is :

~µorb = S · I~n = πr2 · −ev2πr

~n = − rve

Its angular momentum is ~σl :

~σl = ~r ∧ ~p = ~r ∧m~v = mrv~n

orbital magnetic moment

the orbital magnetic moment and the angular momentum areproportional :

~µorb = − e

2m~σl = γ~σl

They point in opposite directions.γ is the gyromagnetic factor

First source of atomic magnetism : the orbital motion

Spin Magnetic Moment

The electron is a quantum particleA proper quantum calculation does not modify the previous result

However moments are quantified~L = ~r ∧ ~p gives L2|ψ〉 = l(l + 1)~2

To properly describe the electron one needs the relativistic Diracequation, which introduces a new quantity without classicalequivalent : the spin

Second origin for atomic magnetism : spin

From one electron magnetism to atom magnetism

To define an atomic electronic state 4 quantum numbers arerequired :ni : the main quantum numberEach ni corresponds to an electronic shell named K, L, M, N . . .

Quantum numbers

li is the orbital angular momentum quantum number

li can be chosen between 0 and ni − 1

li characterises the electronic sub-shell

li 0 1 2 3

s p d f

Quantum numbers

mi is the magnetic quantum number

−li 6 mi 6 li

Finallyσi is the spin quantum numberFor an electron (fermion s=1/2) it can be : σz |ψ〉 = ±1

2~|ψ〉

Quantum numbers

Each electronic state is characterised by its 4 quantum numbersTo describe the atom, the filling of each electronic state has to bedefined.

Pauli Exclusion Principle

How to fill the electronic states ?The lowest energy configuration must obey :Pauli principle(1925)Electrons are fermions, so one electron at most in each quantumstate (ni , li , mi , σi ).

Hund’s rules (1927)

One defines the atomic spin moment S

~S =∑

1st Hund’s rule : S maximum, taking Pauli principle into account.(=minimizing Coulomb interaction between electrons)

Hund’s rules

The atomic orbital momentum L

~L =∑

2nd Hund’s rule L maximum, keeping Pauli and 1st rule.

Hund’s rules

Tb+3 is 4f 8

mi -3 -2 -1 0 1 2 3

σ = 12 ↑ ↑ ↑ ↑ ↑ ↑ ↑

σ = −12 ↓

so L=3 and S=3Not enough to get the total atomic angular momentum~J = ~L + ~SThe 3rd Hund’s rule couples ~L and ~Sspin-orbit coupling H = λ~L.~S

~L and ~S are antiparallel, i.e.J = |L− S | for a filling < 50%

~L et ~S parallel, i.e.J = L + S for a filling > 50% (here J=6)

Multiplets

Cr3+ ≡ [Ar ]3d3

mi -2 -1 0 1 2

σ = 12 ↑ ↑ ↑

σ = −12

So S=3/2 et L=3J=3/2

Multiplets

[Ar] 3d3

L=3 S=3/2

L=1 S=3/2

L S=1/2

J=5/2J=7/2J=9/2

J=L+1/2J=L-1/2

J=5/2J=3/2J=1/2

The ground state is occupied, often the excited states are very/toohigh in energy.

Bohr magneton

Using classical mechanics

~µorb = − e

2m~σl

Quantification of the momentum :

~σl = ~~l

Leads to quantification of the orbital magnetic moment :

~µl = − e~2m~l = −µB

µB is the Bohr magneton, it is the smallest possible value for anelectron magnetic moment.

⇒ 1µB = 9.27 10−24 Am2

It is the relevant unit to measure atomic magnetic moments.

Spin magnetic moment

Orbital :

~σl = ~~l

~µl = −µB~l

Spin :

~σs = ~~s

~µs = −2µB~σs

Spin and orbit do not have the same gyromagnetic factor

Atomic magnetic moments

For an atom (L, S, J), there is :an atomic orbital moment ~mL = −µB

~Lan atomic spin magnetic moment ~mS = −2µB

Atomic magnetic moments

Lande factor

~mL + ~mS = −µB(~L + 2~S)~mL + ~mS is not colinear with ~J = ~L + ~SThe total atomic magnetic moment is not proportional to ~JJ is a good quantum number for the Hamiltonian.The magnetic moment measurable value is its projection along ~JOne can define the Lande factor gJ

~µJ = −gJµB~J

gJ = 1 +J(J + 1) + S(S + 1)− L(L + 1)

2J(J + 1)

Degeneracy of moment

Possible value for µJz = gJMJµB are :

g J(J+1) m

MJ between -J and +Jdegeneracy = 2J+1

Atom in a magnetic field

To lift the degeneracy, a magnetic field can be applied.When an electromagnetic field is applied, the electron Hamiltonianhas to be modifiedH = p2

2m becomes H = 12m (~p − q~A)2 + qφ

In a uniform field (//z) we can choose ~A = −12~r ∧ ~B

H = kinetic energy + Zeeman term + diamagnetic termZeeman HZeeman = −q

2m~L.~B = −~µ.~B + similar expression for spin.

and diamagnetic Hdia = q2

2mA2 = q2B2

8m (x2 + y2)The diamagnetic part explains the magnetic response of nonmagnetic materials (e.g. silicon) (χ < 0, proportional to electronnumber and size of the orbit, T-independent).

Special atomic configurations

maximum moment that can be measured : gJJµB

For a filled sub-shell :S=0, L=0 so J=0No magnetic moment

For a half-filled subshell :L=0, J=SNo orbital moment

From the isolated atom to the solid

Magnetic atoms :Partially unoccupied sub-shells : 79 out of 103 elements (singleatomic)

Atom to Solid

In a solid, chemical bonds appearhybridisation of neighboring orbitals (binding and non-bindingstates).

Magnétique Non magnétique

Atom to solid

Two series are mostly studied : 3d transition metals and 4frare-earths but also 4d (Pd, Ru), 5d (Pt,Re) , 5f (U) ...

Magnetic Atoms in a Solid

For a sphere of magnetic material (so isotropic), experimentally,there exists : easy axes to magnetise the sample and hard axes

Crystal Electric Field

Partially filled electronic shell –> non spherical shell.

Effect of the crystalline environment (electrostatic) :- prefered crystallographic axes to align the magnetic moment

Because ~L and ~S are coupled : spin-orbit coupling- modify the configuration (lift degeneracy)

Atomic orbitals

3d orbitalsSpherical harmonics is a good base for isotropic environmentseg − t2g base is more suitable for cubic environments

Atomic orbitals

Degeneracy is lifted in a cubic charged environment.High spin or Low spin configuration depending on CEF vs U.Since 〈eg |Lz |eg 〉 = 〈t2g |Lz |t2g 〉 = 0, there is quenching of the

orbital momentTrue for 3d atoms (not for 4f rare earths because 4f is an inner

subshell, less influence from neighbours)

Magneto-crystalline Anisotropy

Magnetocrystaline energy is described using a formalism adaptedto the crystallographic symmetry.Cubic symmetry

Ean.mc. = K1(α21α

22 + α2

2α23 + α2

3α21) + K2α

Uniaxial symmetry

Ean.mc.(θ) = K1 sin2 θ + K2 sin4 θ

K are temperature-dependent, which can lead to spin reorientation

transition.

Interacting moments

Experimentally, magnetic order appears in some solidsmagnetic moments interact

What are the possible interactions ?dipolar (magnetic field created by one moment, acting on the otherones : typically one Bohr magneton at one atomic distance isequivalent to one Kelvin)exchangeinsulators (localised magnetic moments = localised magnetism)will not be modelled the same way as metals (band magnetism =itinerant magnetism)Exchange can be direct or indirect : oxygen (superexchange) atomsFe-O-Fe, conduction band (RKKY) to couple localised spins inrare-earth metals)

Magnetic order

Microscopic mechanism (Hubbard model)

H = −∑

tij .c+iσcjσ + U

∑ni↑ni↓

hopping t (∆x .∆p > h, more delocalised = less kinetic energy)and Coulomb repulsion One can then build an Effective Heisenberg

Hamiltonian : H = J∑

Superexchange

In oxides indirect coupling between two magnetic ions will dependon the geometry on the M-O-M bond.

Superexchange at 180 degrees is usually strong andantiferomagnetic. At 90 deg it is weaker and can be ferromagnetic.Double exchange involves 2 ions with different valences and leadsto increased hopping of the extra electron when the configurationis ferromagentic(it induces a metallic character (manganites)).Laurent Ranno Institut Neel CNRS-UJF Rappels sur le magnetisme Quick reminder about magnetism

Non magnetic band structure

For a non magnetic metal

Magnetic band structure

∆m = 2µBN (EF )δE

Pauli Paramagnetism

∆m = 2µBN (EF )δE

If δE is created by applying a magnetic field H0 then the magneticsusceptibility is :

χ =∆m

H= 2µ0µ

2BN (EF )

Such a susceptibility does not depend on temperature but on thedensity of conduction electrons (Pauli paramagnetism).

Electron interactions

A spin-dependent electron interaction can also create δE .

If the interaction gain is larger than the kinetic energy loss thenthe metal is magnetic.

Band structure : nickel

Band ferromagnetism creates a non integer magnetic moment.

Ni up : 5 electrons and Ni down 4.4 electrons ⇒ 0.6µB

When one band is full : strong ferromagnet

Interlayer coupling : RKKY

Indirect echange coupling in a metallic systemspin-polarised conduction band will propagate the spin informationto a second spin or layer through a non magnetic medium.

J =cos(2.kF .r)

oscillating coupling through a metallic spacer. lengthscale kF

finite, discrete thickness effect, period becomes a few monolayers.

Interlayer coupling : RKKY

Oscilating spin polarisation will create undirect exchange betweenspins.It was used to create AF coupling in the first GiantMagnetoresistance multilayers(Fe/Cr/Fe, 1988))It can lead to noncolinear orders. It is the exchange mechanism inmetallic rare-earths.Used also to create artifical antiferromagnet : Co/Ru/Co

Mean Field Model : Paramagnetism

Independent localised spins at temperature T (Brillouinparamagnetism)When applying a magnetic field, the probability to fill the differentmagnetic states follows a Maxwell distribution given by the ratioZeeman energy/Thermal energy (or using partition function).

M = MsatBJ(x) avec Msat = NgJJµB and x =gJJµ0µBH

BJ(x) =2J + 1

2Jcoth(

2J + 1

2Jx)− 1

2Jcoth

Brillouin function

Mean Field Model : Brillouin Paramagnetism

0 0.5 1 1.5 2 2.5 3 3.5 4

B/T (T/K)

H0= Ng2

J J(J + 1)µ2

Curie law, C is the Curie constantLaurent Ranno Institut Neel CNRS-UJF Rappels sur le magnetisme Quick reminder about magnetism

Mean Field Model : Ferromagnetism

With interactions : Exchange (Tc can reach 1000K) is equivalentto a huge magnetic field. In the mean field approximation, theinteraction from the other atomic moments is proportional to theirmagnetisation mean value.

~Hm = λ · 〈S〉

and x =gJJµ0µBHi

kTwith Hi = H + Hm

so x =gJJµ0µB(H + λM)

Weiss model

At fixed temperature T and field H , M(H,T) is given by thesimultaneous resolution of both equations :

M = MsatBJ(x) (1)

λgJJµ0µBx − H

Graphical resolution

When H = 0, the line goes through zero. Depending on T(T < Tcritical ) it allows to get a spontaneous magnetisation and aferromagnetic order

0 0.5 1 1.5 2

Curie temperature

Identifying initial slopes, one gets Tc

Tc = λC

Tc is proportional to λ and to Curie constant CWhen T=0 one gets the saturation magnetisation

Msp(T = 0) = Msat = NgJJµB

Close to Tc

When H is not zero, even at high T, there is a non zeromagnetisation. Susceptibiliy is χ

T − Tc

Curie-Weiss lawIn this mean field model, Tc = θp

Close to Tc

Mean field models do not work close to transition Tc (nofluctuation in the model) and do not contain excitations such asspin-waves. However it is a good simple model to get a first idea offield, temperature dependences and the energies involved.

350 400 450 500 550

Température (°C)

Spontaneous Magnetisation

Mean field models can also be used to describe multi-sub latticecrystals

Antiferromagnetism : 2 identical lattices antiferromagneticallycoupled. No net magnetisation but AF long range order up to the

Neel temperature.Ferrimagnetism : 2 (or more) non equivalent lattices, AF-coupled

Excitations

Thermal excitations can disorder the magnetic saturated state andreduce the saturation magnetisation (similar to phonon disorderingthe periodic crystal lattice).These excitations can be modelled as spin waves / magnons andmeasured using neutron inelastic diffusion for example.

It will explain theM(T). Anisotropy can open a gap in the excitation spectrum.

GHz dynamics

Landau-Lifschitz-Gilbert equation

dt= γM×Heff + αM× (M×Heff) (3)

γ the gyromagnetic ratio (1 T = 30 GHz precession)α the damping constantHeff the effective field :

Heff = − 1

The effective field includes contributions from the applied field(Zeeman energy), the demagnetizing field (shape anisotropy),magnetocrystalline and exchange energies.Electron magnetism can be studied using these GHz spectroscopiesEPR, FMR. The gyromagnetic factor anisotropy can for examplebe studied using EPR.NMR nuclear magnetism is MHz spectroscopy

B,H,M,mB magnetic induction (Tesla)H magnetic field (A/m)

~B = µ0(~H + ~M)

(no magnetisation or weak magnetisation leads to = µ0~H and big

confusion between field and induction.m = magnetic moment (Am2)M = Magnetisation = magnetic moment / unit volume (A/m)

Maxwell equationsdiv B = 0rot H = j + dD/dt

S.I. vs c.g.s1 Tesla = 10 000 Gaussin vacuum :B=1 Gauss corresponds to H=1 Oersted~B = µ0

~H in air.Laurent Ranno Institut Neel CNRS-UJF Rappels sur le magnetisme Quick reminder about magnetism

Ferromagnetism

More details about the ferromagnetic state.

Magnetic configuration

For a given sample, its magnetic configuration ( ~M distribution)depends on the minimisation of the total energy which contains 4main contributions.

Exchange energy (saturated state is the equilibrium)

Zeeman energy (alignment with applied field is theequilibrium)

Magnetic Anisotropy energy (alignement with easy axes is theequilibrium)

Demagnetising energy (demagnetised state is the equilibrium(no magnetic poles,no stray field)).

Lengthscale : domain walls

At remanence (Zero applied field),A sample will demagnetise due to demagnetising field

Here GdCo film (perpendicular anisotropy) Kerr images

magnetisation reversal

out of equilibrium statehysteretic in most cases

characteristic fields and magnetisationremanent magnetisation Mr, spontaneous magnetisation Ms,coercive field HcApplied magnetic field Ha vs total magnetic field ~H = ~Ha + ~Hd

magnetisation reversal : uniform

Minimise : E = EZeeman +Edemag +Eanisotropy (no exchange energy)

Stoner-Wohlfahrt astroıd

magnetisation reversal mechanism

nucleation then domain wall propagation

Lengthscale : domain wall width

Between domains : a domain wallCompetition between anisotropy energy and exchange energy

Lengthscale : domain wall width

domain wall width δ and wall energy γ

δ(m) = π

Kand γ(J/m2) = 4

√AK (4)

Large anisotropy ⇒ narrow domain wallSmall anisotropy ⇒ wide domain walls

Interlayer coupling : Orange peel

Coupling may still exit in “uncoupled” systemorange peel=magnetostatic, pinholes

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