H = 2 J S S j over the neighbor atoms ij i j - - PDF document

SLIDE 1

1

Basic properties 2

1

Magnetism, why bother?

• Up to now only terribly weak effects: χ = 10-4 << 1
• Collective coupling of spins leads to strong magnetic

moments that often can be detected directly but even more commonly leads to hidden order: antiferromagnets Ferromagnets Antiferromagnets Ferrimagnets

Elements: Cr, Oxides: MnO, NiO, HTSc Oxides: Fe2O3, Gd3Fe5O14 RE/TM: ErCo5, Dy2Fe14B Elements: Fe, Co, Ni, Gd Alloys: Permalloy FeNi

Basic properties 2

2

• Central for understanding magnetic interactions in solids
• Arises from Coulomb electrostatic interaction and

the Pauli exclusion principle The Exchange Interaction Coulomb repulsion-energy high Coulomb repulsion-energy lowered J r e UC

18 2 2

10 ~ 4

−

= πε (105 K !)

Basic properties 2

3

Exchange energy

Wave function of two electrons must be antisymmetric for exchange of particles Chance that two electrons with same spin are at the same place is zero. Pauli principle takes care that parallel spins avoid each other.

) , : , ( ) , : , (

1 1 2 2 2 2 1 1

s r s r s r s r ψ ψ − =

Exchange correlation gap Exchange correlation gap radius: r=2/ radius: r=2/k kF

F ~ 1

~ 1-

• 2Å

2Å

Electron density around each electron in free electron gas (Ibach and Lüth)

• Basic properties 2

4

Exchange energy

Consequences from exchange correlation: Antiparallel spins (singlet state) have lower Coulomb energy than parallel spins (triplet state) 2 1

2 s s E exchange ⋅ − = Δ J J

Exchange energy Exchange energy Singlet Triplet

J S A CS E E

s t

− ≡ − − = −

2

1 2

See Ibach&Luth

Basic properties 2

5

Heisenberg Hamiltonian Heisenberg Hamiltonian for lattice: Ferromagnetic coupling for positive J Antiferromagnetic coupling for negative J

∑ ∑

≠

⋅ − =

i j j i i

H S S

ij

J 2

What does this mean?

• Electrons on same atom preferring parallel spins

(Hund’s rules/atomic correlations): J>0

• Binding often results in antiparallel spins on neighbor atoms: J<0

(1 reason why not so many ferromagnetic materials exist)

i over all atoms j over the neighbor atoms

Heisenberg Hamiltonian is a good starting point for theories for magnetic materials with predominant neighbor interactions (e.g. isolator) Basic properties 2

6

Fe spin resolved

Schematic: Magnetic moment: 4.8-2.6=2.2 μB

SLIDE 2

2

Basic properties 2

7

Properties of ferromagnets

Not only whole or half integer moments

Basic properties 2

8

Heisenberg Hamiltonian Heisenberg model Hamiltonian Ferromagnetic coupling for positive J Antiferromagnetic for negative J

Singlet Triplet

J S A CS E E

s t

− ≡ − − = −

2

1 2

∑ ∑

≠

⋅ − =

i j j i i

H S S

ij

J 2

• Electrons on the same atom can have parallel spins

(Hund’s rules/atomic correlations)

• Bonding tends to lead to antiparallel moments on neighboring atoms

(one reason why there are not many magnetic compounds)

• But if J is positive neighboring atom moments couple parallel
• Heisenberg Hamiltonian is good starting point for many theories on

magnetism for systems with only pair wise interactions Basic properties 2

9

I&L8.3 Exchange interaction between Free Electrons

• Free electron wavefunction
• Two free electron pair wavefunction
• space antisymmetric:

r i

e V

k

r 1 ) ( = ψ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − =

i j j i j j i i j i ij

i e i e i e i e V r k r k r k r k r r 2 1 ) , ( ψ

j j i i ij

i e i e V r k r k r 1 ) ( = ψ

( ) ( )( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − + = −

j i j i j j i i j i ij

i e i e V r r k k r k r k r r 1 2 1 ) ( ψ

Basic properties 2

10

Exchange interaction between Free Electrons

• Probability of finding electron 1 in dr1 and electron 2 in dr2:

Two electrons with same spin cannot be at same position Ionic charge of a spin-up electron is not screened by other spin-up electrons. This lowers the energy and leads to a

collective exchange interaction with positive sign

( )( ) [ ]

j i j i j i j i j i ij

d d V d d r r r r k k r r r r − − − = − cos 1 1 ) (

2 2

ψ

( ) ( )( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − + = −

j i j i j j i i j i ij

i e i e V r r k k r k r k r r 1 2 1 ) ( ψ Basic properties 2

11

Exchange hole

Exchange interaction causes each spin to push away other spins local charge density modified: Exchange hole Use this density in Schrödinger equation: Hartree-Fock approximation. Full theory Density Functional theory in Local Density Approximation (DF-LDA) ( ) ( ) ( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − =

6 2

cos sin 9 1 2 r k r k r k en

F F F ex r

ρ

( ) ( ) ( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − =

6 2

cos sin 2 9 1 r k r k r k en

F F F ex r

ρ

radius radius r=2/ r=2/k kF

F ~ 1

~ 1-

• 2Å

2Å up electrons only: All electrons Basic properties 2

12

Heisenberg model does not completely explain ferromagnetism in metals. Band (Stoner) Model of ferromagnetism

N n I k E k E N n I k E k E

S S ↓ ↓ ↑ ↑

− = − = ) ( ) ( ) ( ) (

Is is Stoner parameter and describes energy reduction due to electron spin correlation is density of up, down spins

↓ ↑ n

n ,

SLIDE 3

3

Basic properties 2

13

Band (Stoner) Model

N n n R

↓ ↑ −

=

(spin excess) R V N M

B

μ =

( ) [ ]

1 ,

/ 2 / ) ( ~ exp ) ( ) ( 1

− ↓ ↑ ↓ ↑

− = − =

∑

kT E R I k E f k f k f N R

F s k

m 2 / ) ( ~ ) ( 2 / ) ( ~ ) ( R I k E k E R I k E k E

S S

+ = − =

↓ ↑

Then

N n n I k E k E

s

2 ) ( ) ( ) ( ~

↓ ↑ +

− =

Spin excess given by Fermi statistics:

Basic properties 2

14

Band (Stoner) Model Let R be small, use Taylor expansion:

... ) ( ) ( ~ ) ( 24 1 ) ( ) ( ~ ) ( 1

3 3 3

+ ∂ ∂ − ∂ ∂ − =

∑ ∑

R I k E k f N R I k E k f N R

s k s k

( )

∫ ∫ ∑

− − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ → ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ )) ~ ( ( 2 ~ ) 2 ( ~

3 3 F k

E E dk N V E f dk N V E f δ π π

(at T=0) ... 2 ) ( ' ' ' ! 3 2 ) ( ' ) 2 / ( ) 2 / (

3

+ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ − Δ − ≈ Δ + − Δ − x x g x x g x x g x x g with

R I x

s

= Δ

f(E) E EF

) ( 2

F

E D V − =

D.O.S.: density of states at Fermi level

Basic properties 2

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Band (Stoner) Model

( )

) 3 (

~ O R I E D R

s F

− =

Density of states per atom per spin

( )

) ( 2 ~

F F

E D N V E D =

Then

( )

) 3 (

) ~ 1 ( O I E D R

s F

− = −

Third order terms

When is R> 0?

( )

~ 1 < −

s F I

E D

• r

( )

1 ~ >

s F I

E D

For Fe, Co, Ni this condition is true Doesn’t work for rare earths, though Stoner Condition for Ferromagnetism

Basic properties 2

16

g(E g(EF

F) and

) and Δ ΔE for a 3d E for a 3d ferromagnet ferromagnet

• 1. DOS at Fermi level is greater in a 3d TM:
• 1. DOS at Fermi level is greater in a 3d TM:

E E 3d 3d 4s 4s

bandwidth of 3d band smaller bandwidth of 3d band smaller ( (d d-

• d

d overlap smaller than

• verlap smaller than s

s-

• s

s) ) band also contains more electrons band also contains more electrons (can take 10 in total, vs. 2 (can take 10 in total, vs. 2 for 4s band) for 4s band)

E EF

F Basic properties 2

17

2.

• 2. Δ

ΔE from alignment with (or against) B E from alignment with (or against) B0

0 is greater in a 3d TM:

is greater in a 3d TM:

3d 3d 4s 4s E E E EF

F

N N↑ ↑ N N↓ ↓

Δ ΔE E

plus: plus: even a even a small change in small change in energy, energy, Δ ΔE, leads E, leads to a relatively to a relatively large no. of large no. of electrons electrons changing spin changing spin state state

Spin resolved DOS: 3d Spin resolved DOS: 3d ferromagnet ferromagnet

Basic properties 2

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Band structure

EF

Ẽ↑↓(k)

I ↓ ↑

Exchange interaction splits spin degenerated bands

SLIDE 4

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Basic properties 2

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Scandium

e eg

g

t t2g

2g

p p

s s

3d:e 3d:eg

g

3d: t 3d: t2g

2g

4p 4p

4s 4s EF

Copper

EF 4sp band free electrons 3d bands localized electrons

Basic properties 2

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Susceptibility of ferromagnet How can we connect the microscopic Heisenberg Hamiltonian with macroscopic properties? Remark: Magnetization generates non-locale effects elsewhere: Field Bloc at the atom site is the applied field B=μ0H + the field generated by all the atoms of the magnetic object.

) (

nonlocal applied

• loc

H H B + = μ

Basic properties 2

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Weiss molecular field Pierre Weiss: Exchange interaction of neighboring moments produces molecular field proportional to magnetization Weiss field enhances Bloc What is connection between macroscopic λ and microscopic exchange interaction Jij?

M B B

• loc

eff

μ

λ

+ = M B

• Weiss

μ

λ

=

∑ ∑

≠

⋅ − =

i j j i i

H S S

ij

J 2

Basic properties 2

22

Mean field theory

approximate operator in Hamiltonian with its mean (average) value

Replace Sj by mean value <S> <S> can be expressed in magnetization: From previous chapter: μeff=-gLμBS Thus magnetization of N atoms

S M

B

Ngμ − =

∑ ∑

≠

⋅ − =

i j j i i

H S S

ij

J 2 > < − =

∑ ∑

≠

S S

i j i i

H

ij

J 2

M B

• Weiss

μ

λ

=

2 2

2

B i j ij

g N J μ μ λ

∑ ≠

=

BWeiss

After some operations see gray box p.222 & problem 8.2

Basic properties 2

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Mean Field Theory for disordered phase T>Tc

Brillouin function with Beff Simple case: L=0, J=S=1/2, g=2 High temperature: x<< 1: tanh x x Because M is small: Bloc=μ0H Curie-Weiss law C=N μ0 μB

2 /kB ,

Curie temperature: TC=λ C

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = T k J B g B gJ N T B M

B eff B J B loc

μ μ ) , ( ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = T k B N T B M

B eff B B loc

μ μ tanh ) , (

( )

M B T k N B T k N T B M

loc B B eff B B loc ο

λμ μ μ + = =

2 2

) , (

C m

T T C H M T − = = ) ( χ

) (T

m

χ

Curie Curie-

• Weiss

Weiss

T

Tc Basic properties 2

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Estimation of exchange interaction

2 2 2 2

2 2

B B i j ij

g N zJ g N J μ μ μ μ λ ≈ = ∑ ≠ z=6 neighbors J~30 meV

C m

T T C H M T − = = ) ( χ

Curie-Weiss law C=N μ0 μB

2 /kB ,

Curie temperatuur: TC=λ C For Fe: N=9x1028 m-3 C ~ 1 K TC =1043 K λ ~ 1000

SLIDE 5

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Basic properties 2

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Mean Field Theory for ordered phase T<Tc T<Tc : Spontaneous magnetization (also in zero field H=0) Assume Bloc=0 (field generated by rest of sample small)

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = T k B N M

B eff B B

μ μ tanh N M B B

• loc

eff

μ μ

λ λ

≈ + = y T T T k B x

c B eff B

= = μ x N M y

B

tanh = = μ

Two simultaneous equations for y. Can only be solved graphically. Basic properties 2

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Mean Field Theory for ordered phase T<Tc

Ni Tc= 627K x N M y

B

tanh = = μ y T T T k B x

c B eff B

= = μ

Basic properties 2

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T-Tc (K)

ln χ

( )

γ

χ

−

− =

C

T T

Curie m

T T C T − = ) ( χ

Mean field theories exclude fluctuations fails around phase transition: critical behavior

Failure of Mean Field T>Tc

Basic properties 2

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Ni Tc= 627K

( )

β

T T M

C −

=

Failure of Mean Field T<Tc

See SSS Ising

Basic properties 2

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Beyond mean field

Mean field γ = 1 and β = 0.5 Basic properties 2

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Ni Tc= 627K Failure of Mean Field T0

At low T: Spinwaves

Magnon

www.physics.colostate.edu/groups/maglab

SLIDE 6

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Basic properties 2

31

Spin waves

Similar to phonons linear chain Energy Effective field Equation of motion

( )

1 1

2

+ − +

⋅ − =

n n n n

E S S S J

( )

1 1

2

+ − +

⋅ − =

n n B n

g S S B μ J

( )

1 1

2

+ − +

× = × =

n n n n n n

dt d S S S B μ S J h

Basic properties 2

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Spin waves

Low excitations linearize chain Equation of motion n n

σ z S + − = ˆ S ( ) ( )

1 1 1 1

2 S 2 S 4 S 2

+ − + −

+ − × − = × − + × − =

n n n n n n n

dt d σ σ σ z z σ σ σ z S ˆ ˆ ˆ J J J h

Basic properties 2

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Spin waves

Wave like solutions Dispersion relation

( ) wt kna i n

Ae

−

= σ

( )

( ) [ ] ka e e

ika ika

cos − = + − − =

−

1 S 4 2 S 2 J J ω h

Spin waves in the antiferromagnet perovskite LaMnO3: A neutron-scattering study ", Phys. Rev B 54, 15149 (1996)

2 2

S 2 k a k J ≈ = → ω ε h

Basic properties 2

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Spin waves

Energetics of excitations Number of states Specific heat and magnetization from excitations

( ) dk Vk dk k g

2 2

2π =

2 3

AT dT dE C = =

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = − =

2 3

1 BT M N g M M

s B s

μ

Basic properties 2

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Tight binding revisited

single electron Try linear combination of atomic orbitals Perturbation theory gives E’ minimize w.r.t. cA & cB

+

• +

R rA rB B B A A

c c φ φ + = Ψ

R e r e r e m

B A

+ − − Δ − = πε πε πε 4 4 4 2

2 2 2 2

h H H H2

2+ + ion

ion

r r d d E

∫ ∫

= ψ ψ ψ ψ

* *H

'

Basic properties 2

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Tight binding revisited Original energy integral Interaction energy integral Overlap integral

∫

=

1 *

r d H H

A A AA

φ φ r d e S

B Aϕ

ϕ πε

* 2

4

∫

= Bonding Antibonding

S H H E

AB AA

± ± ≈

±

1

AA

H

BB

H

H H2

2+ + ion

ion

∫

= r d H H

B A B

φ φ *

r r d d E

∫ ∫

= ψ ψ ψ ψ

* *H

'

SLIDE 7

7

Basic properties 2

37

Exchange Interaction and Bonding

two electrons neglect Hint and try product of atomic wavefunctions: Reasonable simplification: neglect terms with both electrons on same atom Heitler London wavefunction:

( ) ( ) ( ) [ ] ( ) ( ) [ ]

2 2 1 1 2 , 1

B A B A

ϕ ϕ ϕ ϕ + + = Ψ

( ) ( ) ( ) ( )

1,2 H 2 H 1 H 1,2 H

int

+ + =

( ) ( ) ( ) ( ) ( )

2 1 2 1 2 , 1

A B B A

ϕ ϕ ϕ ϕ + = Ψ

• +
• +

R rA rB

H H2

2 molecule

molecule

Basic properties 2

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Symmetry requirement of wavefunction

Space part symmetric for exchange of (1) and (2) spin part antisymmetric: spin singlet S=0 Space part asymmetric for exchange of (1) and (2) spin part symmetric: spin triplet S=1

( ) ( ) ( ) ( ) ( )

2 1 2 1 2 , 1

A B B A

ϕ ϕ ϕ ϕ + = Ψ

( ) ( ) ( ) ( ) ( )

2 1 2 1 2 , 1

A B B A

ϕ ϕ ϕ ϕ − = Ψ

| ↑↓> -| ↓↑> | ↑↑> | ↑↓> + | ↓↑> | ↓↓> H H2

2 molecule

molecule

Basic properties 2

39

Exchange Interaction

Perturbation theory gives C: Coulomb integral A: Exchange integral S: Overlap integral S A C E E

I

± ± + = 1 2 ( ) ( ) ( ) ( )

2 1 * * 2 1 12 2

2 1 2 1 1 1 1 1 4 r r d d r r r R e A

B B A A B A AB

ϕ ϕ ϕ ϕ πε ∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − + =

( ) ( )

2 1 2 2 1 2 12 2

2 1 1 1 1 1 4 r r d d r r r R e C

B A B A AB

ϕ ϕ πε ∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − + =

( ) ( ) ( ) ( )

2 1 * * 2

2 1 2 1 4 r r d d e S

B B A A

ϕ ϕ ϕ ϕ πε ∫ =

Singlet Triplet

+ for singlet

• for triplet

EI EI In simple molecules bonding leads to singlet ground states

H H2

2 molecule

molecule

Basic properties 2

40

Size of splitting defined as exchange constant J Exchange Interaction and Bonding J S A CS E E

s t

− ≡ − − = −

2

1 2

S A C E E

I

± ± + = 1 2

+ for singlet

• for triplet

Singlet Triplet EI EI

∑ ∑

≠

⋅ =

i j j i i

H S S

ij

J 2

Generalization to many atoms: Heisenberg Hamiltonian Ferromagnetic coupling for positive J Antiferromagnetic for negative J