Refresher lecture in magnetismneel.cnrs.fr/.../Elsa/Hercules_magnetism_simonet.pdf · 2010-09-30 · Refresher lecture in magnetism Virginie Simonet, Institut Néel, CNRS-UJF, BP166,

Refresher lecture in magnetism Virginie Simonet,

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

2/04/10

Outline : Introduction

Atomic magnetic moment Assembly of non interacting magnetic moments

Magnetic moments in interaction From microscopic to macroscopic

Applications Modern trends in research

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Introduction Magnetic materials all around us : the earth, cars, audio, video, computer technology, 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, phase transitions, spin liquid, spin glass, spin ice, magnetostriction, magnetoresistivity, magnetocaloric effect, in different materials : metals, insulators, semi-conductors, oxides, molecular materials…

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

Magnetism is a quantum phenomenon but phenomenological models commonly used to 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 (Ampère) 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 of magnetism in matter

Example for a one turn coil : orbital magnetic moment

!L = !r ! !p = mr2"!n

!µl =!e

2m!L = "!L

!µl = !I.!S =!e"

2##r2!n =

!e"r2

2!n

gyromagnetic ratio

Consequences : ✔magnetic moment and angular momentum are antiparallel ✔Calculations with magnetic moment using formalism of angular momentum ✔Precession of magnetic moment in a magnetic field : Larmor precession

e- orbiting around the nucleus

!L = "B0

angular momentum

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

lzl2

Quantum mechanics:

The component of the orbital angular momentum along the z axis is

The magnitude of the orbital momentum is !l(l + 1)!

ml!

The component of the spin angular momentum along the z axis is

The magnitude of the spin momentum is

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Atomic magnetic moment 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=1 µs = !gsµB s

µl = !glµB l

ms!!

s(s + 1)!

µB =!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 minimize the electrostatic energy and fulfil the exclusion Pauli principle

Hund’s 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 ½ filled shell for less than ½ 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<MJ<J

with the Lande g-factor Total magnetic moment

M = !µB(L + 2S)g = 1 +

J(J + 1) + S(S + 1)! L(L + 1)2J(J + 1)M = !gµB J

Application of Hund’s rules:

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Atomic magnetic moment Magnetism is a property of unfilled electronic shells : Most atoms (bold) are concerned but only 22 magnetic in condensed matter

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Atomic magnetic moment Atom in matter: ✔ chemical bonding -> filled e- shells : no magnetic moments

Except for :

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

in insulators in insulator/metals

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Atomic magnetic moment Atom in matter: ✔ Influence of surrounding charges -> crystal field (CEF)

3d electrons Large 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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Atomic magnetic moment Atom in matter: ✔ Influence of surrounding charges -> crystal field (CEF)

4f electrons Spin-orbit>>CEF: 4f charge distribution +CEF -> selects some orbitals Spin-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 moment

Localized 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

WB = µB(!L + 2 !S). !B +e2

8me

!

ie!( !Ri ! !B)2

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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 field susceptibility: derivative of magnetization wrt magnetic field or ratio in the linear regime

! ="M

"B=

!M

B

"

lin

WB = µB(!L + 2 !S). !B +e2

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

Vµ0

e2

4me< R2

! >

WB = µB(!L + 2 !S). !B +e2

8me

!

ie!( !Ri ! !B)2

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Energy:


Paramagnetic term:

and the Brillouin function

M =N

VgJJµBBJ(x) x =

gJJµBB

kBT

BJ(x) =2J + 1

2Jcoth

!2J + 12J

x"! 1

2Jcoth

! x

2J

"

with

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Assembly of non-interacting magnetic moments Paramagnetic term

Brillouin functions compared to Langevin functions from classical calculation

Limit x>>1 i.e. H>>kBT Saturation magnetization: M =

N

VgJJµB

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Assembly of non-interacting magnetic moments Paramagnetic term

Limit x<<1, i.e. kBT>>H Curie law:

with the effective moment

peff = gJ

!J(J + 1)µB

! =N

V

(µBgJ)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) <- spin of conduction e- Landau diamagnetism (<0, weak, T-independent) <- orbital moments of conduction e-

At small H/kBT : linear regime

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Magnetic moments in interaction

✔ Dipolar interaction :

electrostatic origin + Pauli exclusion principle

E =µ0

4!r3["µ1."µ2 !

3r2

("µ1."r)("µ2."r)]

much too weak to account for ordering of most magnetic materials

✔ Exchange interaction :

Heisenberg Hamiltonian

2 electrons cannot be in the same quantum state many-electrons wavefunctions are antisymmetric with respect to the exchange of 2 electrons

: Exchange coupling constant > 0 ferromagnetic < 0 antiferromagnetic coupling

J

H = !!

ij

Jij!Si.!Sj

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Magnetic moments in interaction ✔ 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 antiferromagnetic Explains the magnetism in transition metal oxides

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Magnetic moments in interaction ✔ Exchange in metals

In 3d metals

In rare-earth metals The interaction between 4f localized moments is 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(2kF r)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 interaction From paramagnetic state at high temperature to ordered state at low temperature

…

kBT>>exchange interactions

All moments //

Several sublattices: ≠ directions of magnetic moments -> compensate

Several sublattices: ≠ directions of magnetic moments -> do not compensate

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Magnetic moments in interaction 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 aligned by the internal molecular field without external B

H = gµB

!

i

!Si.( !B + !Bmf ) !Bmf = ! 2gµB

!

j

Jij!Sj

H = !!

ij

Jij!Si.!Sj + gµB

!

j

!Sj . !B

! =2zJ

ng2µ2B

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Magnetic susceptibility

M =(gJµB)2J(J + 1)

3kBT(B + !M) =

C

T(B + !M)

In the low field, high temperature limit

TC=λC Curie temperature At Tc, becomes infinite : the system becomes spontaneously magnetized

! =C

T ! "C=

C

T ! TC

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Magnetization below TC

M = gJµBJBJ(x)

Solve simultaneously 2 equations x =gJµBJ(B + !M)

kBT

For B=0

M/Ms

y

No solution for T>TC One solution for T<TC : spontaneous magnetization 2nd order transition at TC

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Antiferromagnetism : same analysis but for each of the 2 sublattices

Spontaneous magnetization below the Néel temperature TN on each sublattice

TN = |!|C! =C

T + TNSusceptibility

More complicated below TN : depend of field orientation

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Generalization:

Curie-Weiss law

1/!

θ=TC θ=0

Ferromagnets TC

Fe 1043 K Co 1394 K Ni 631 K Gd 293 K

Antiferromagnets TN

CoO 293 K NiO 523 K MnO 116 K

Shull 1951 Neutron diffraction

θ=-TN

T>TN

T<TN

! =C

T ! "

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Magnetic moments in interaction Other types of magnetic orders

Helimagnetism : helical order of moments Ex. Rare earths crystals

case of a J1/J2 chain

Ferrimagnetism, θ=-TN but spontaneous magnetization ; Spontaneous magnetization on each sublattice may have ±T dependence ->compensation temperature Ex. Ferrites, garnets …

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

helix cos(!) = ! J1

4J2

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

J1

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Magnetic moments in interaction

Other types of magnetic orders Complex 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 Ba3NbFe3Si2O14 Helix + 120° arrangement

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Magnetic moments in interaction Magnetic excitations

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

Short range interactions

Allows entropy gain without loosing too much in exchange energy

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Magnetic moments in interaction Magnetic excitations

Dispersion relation For 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 macroscopic Macroscopic 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:

FT = Fex + Fdip + Fan + FH

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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 macroscopic Magnetocrystalline anisotropy

Magnetization variation against anisotropy in ferromagnets

Uniaxial anisotropy

E = !µ0HappMs sin! + K sin2 !

!E

!"= 0 sin ! = !µ0HappMs

2K

sin ! = 1

easy axis

hard axis

Anisotropy field

Happ = HA =2K

µ0Ms

for

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From microscopic to macroscopic Magnetocrystalline anisotropy

Magnetization variation against anisotropy in ferromagnets

Cubic symmetry

EA = K1(!2"2 + "2#2 + !2#2) + K2!2"2#2 + ...

easy axis <100> easy axis <111>

α, β, γ : cosines of the angles between magnetization and the x, y, z directions// 4-fold axes

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minimising the demagnetising field produced by the material

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

Dipolar energy E =µ0

4!r3["µ1."µ2 !

3r2

("µ1."r)("µ2."r)]

-> shape anisotropy

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

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

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

Note : other types of domain walls in reduced dimension systems

!EA = NK < sin2 ! >! K"

2!

!Eexch = NJS2(1! cos !) " "JS2!

! =!

2"a

!Eexch

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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From microscopic to macroscopic Coercitivity represents the magnetization ability to resist reversal against applied magnetic field

Coercive field for coherent rotation : Stoner-Wohlfarth model

E = K sin2 ! + µ0MsH cos !

Energy minimization wrt θ :

As long as , θ=0 and π are

two minima separated by a barrier

When

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

uniaxial anisotropy Zeeman term

θ 0 π

H = 2K/µ0Ms

H < 2K/µ0Ms

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From microscopic to macroscopic Stoner-Wohlfarth model works well for nanoparticles

The coercive field

In macroscopic materials, influence of defects Rotation occurs by nucleation on defects and propagation of domain walls

But for most systems

Hc = 2K/µ0Ms

Hc << 2K/µ0Ms

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From microscopic to macroscopic Hysteresis cycle of a ferromagnet

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Applications Applied 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 magnets From quantum to classical: mesoscopic scale -> Quantum computer Multiferroism : coexistence of two ferroic orders (magnetic, electric, elastic) Low dimension systems: Haldane, BEC, Luttinger liquid Quantum phase transitions Magnetism and superconductivity Nano materials : thin films, multilayers, nano particles ->Spintronics Magnetoscience …

Documents

Refresher lecture in magnetismneel.cnrs.fr/.../Elsa/Hercules_magnetism_simonet.pdf · 2010-09-30 · Refresher lecture in magnetism Virginie Simonet, Institut Néel, CNRS-UJF, BP166,