Magnetism and Magnetic MaterialsMagnetism and Magnetic MaterialsDTU (10313) – 10 ECTSDTU (10313) – 10 ECTS

KU – 7.5 ECTSKU – 7.5 ECTS

Module 5

15/02/2011

Magnetic order

Sub-atomic – pm-nm Mesoscale – nm-m

Macro – m-mm

Intended Learning Outcomes (ILO)Intended Learning Outcomes (ILO)

(for today’s module)(for today’s module)

1. List the various forms of magnetic order in magnetic materials2. Calculate the room-T magnetization of a given ferromagnet3. Relate exchange interactions with the ”molecular field” in Weiss models4. Explain the peak in magnetic susceptibility at the Neel temperature in antiferromagnets

FlashbackFlashback

A new set of orbitals

The spin Hamiltonian

Crystal field splitting

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ˆ H spin = −2JS1 ⋅S2

Superexchange

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J= ψa* (r1)ψb

*(r2 ) ˆ H ∫ ψ a (r2 )ψb (r1)dr1dr2

The exchange integral

FerromagnetismFerromagnetism

In the Weiss model for ferromagnetism, exchange interactions are responsible for the huge “molecular field” that keeps moments aligned.

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ˆ H = −2 J ijSi ⋅S ji< j∑ + gμ B Si ⋅B

i∑

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Bmf = − 2gμ B

J ijS jj

∑

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ˆ H = gμ B Si ⋅ B + Bmf( )i

∑

We define an effective field acting upon each spin due to exchange interactions

The Hamiltonian now looks just like the paramagnetic Hamiltonian, except there’s a field even with no applied field

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Bmf = λM We relate the molecular field with the “order parameter”, i.e. the magnetization

Review Brillouin paramagnetismReview Brillouin paramagnetism

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mJ =mJ exp mJ x( )

mJ =−J

J

∑

exp mJ x( )mJ =−J

J

∑, x = gJμ BB

kBT

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M = ngJμ B mJ = ngJμ BJBJ (y) = M S BJ (y)

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BJ (y) = 2J +12J

coth 2J +12J

y ⎛ ⎝ ⎜ ⎞

⎠ ⎟− 12J

coth y2J ⎛ ⎝ ⎜ ⎞

⎠ ⎟, y = gJμ BJBkBT

J=1/2 J=3/2

J=5

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eff = gJμ B J(J +1)

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s = gJμ BJ

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gJ = 32

+ S(S +1) − L(L +1)2J(J +1)

The spontaneous magnetizationThe spontaneous magnetization

By solving numerically the two equations, we determine the spontaneous magnetization (in zero applied field) at a given temperature

Re-estimate the effective molecular field Bmf=MS if TC is 1000 K and J=S=1/2.

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M = M S BJ (y)

y = gJμ BJ(B + λM )kBT

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TC = gJμ B(J +1)λM S

3kB

=nλμ eff

2

3kB

T=TC T<TCT>TC

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BJ (y) ≈ J +13J

y

The temperature dependence M(T)The temperature dependence M(T)

Estimate the room-T M/Ms of Fe (J=S=3/2, Tc=1043 K)

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M ∝ (TC −T )1/2

M ∝1Near TC (mean-field critical exponent)

Low T (as required by thermodynamics)

The case of Nickel (S=1/2)

Ferromagnet and applied fieldFerromagnet and applied field

T=TC T<TCT>TC

Increasing B

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M ∝ B1/3 T = TC

Origin of the molecular fieldOrigin of the molecular field

When L is involved (e.g. 4f ions), only a part of S contributes to the spin Hamiltonian: de Gennes factor

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= 2zJng2μ B

2

If we assume that exchange interactions are effective over z nearest-neighbours, we find:

So that we reveal the proportionality between Tc and the exchange constant

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TC = 2zJ(J +1)3kB

J

J

LS

L+2S=J+S

(gJ-1)J

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=2zJ(gJ −1)2

ngJ2μ B

2

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TC = 2zJ(J +1) gJ −1( )2

3kB

JThis is valid when L is quenched (3d ions) and, therefore, J=S

AntiferromagnetismAntiferromagnetism

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B+ = −λ M− −ΓM+( )B− = −λ M+ −ΓM−( )

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M+ = M− = M

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M± = M S BJ −gJμ BJ λ M m

kBT ⎛ ⎝ ⎜

⎞ ⎠ ⎟

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TN = gJμ B(J +1)λ M S

3kB

=n λ μ eff

2

3kB

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M+ − M− = M Staggered magnetization (order parameter)

Neglect those for now (but they are important for a realistic theory)

The magnetic susceptibilitiesThe magnetic susceptibilities

Paramagnet Ferromagnet Antiferromagnet

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χ ∝ 1T

, θ = 0

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χ ∝ 1T −TC

, θ = TC

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χ ∝ 1T +TN

, θ = −TN

AFM with a strong magnetic fieldAFM with a strong magnetic field

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E = −MBcosθ − MBcosφ + AM 2 cos(θ +φ) − Δ2

cos2 θ + cos2 φ( )

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θ =arccos MB2AM 2 − Δ ⎛ ⎝ ⎜ ⎞

⎠ ⎟

Types of antiferromagnetic orderTypes of antiferromagnetic order

Simple cubic

BCC

Ferrimagnetism and helical orderFerrimagnetism and helical order

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E = −2NS2 (J1 cosθ + J 2 sin2θ )

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J1 + 4J 2 cosθ( )sinθ = 0

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cosθ = − J1

4J 2

Ferrimagnets: important technologically for their non-metallic nature and flexible magnetic response

Sneak peekSneak peek

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B = μ 0 (M + Hd )

B

M

M

Shape effects and magnetic domains

Wrapping upWrapping up

Next lecture: Friday February 18, 8:15, KU room 411D

Micromagnetics I (MB)

•Ferromagnetism•Spontaneous magnetization•Ferromagnetic-to-Paramagnetic transition at Tc•Antiferromagnetism•Susceptibilities and Curie-Weiss laws•Ferrimagnetism•Helical order