Outline 1. Introduction Exchange and ordering 2. Exchange 3. - - PDF document

8/25/11 1

Exchange and ordering

Stephen Blundell University of Oxford

2011 School - Time-dependent phenomena in magnetism Targoviste, August 2011

Part 1

1

• 1. Introduction
• 2. Exchange
• 3. Superexchange
• 4. Orbitals

Outline

2

• 1. Introduction
• 2. Exchange
• 3. Superexchange
• 4. Orbitals

Outline

3 4 a problem: so define: but: 5 6

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7 8

|B|

Dipolar fields

9

Louis
Néel
 (1904‐2000)


10 11

Energy terms:

• Kinetic energy eV
• Coulomb energy eV
• Size of atom given by balance of these two terms
• Spin-orbit ~ meV
• Magnetocrystalline anisotropy ~ µeV

12

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13 14 Eigenfunc8ons:


σ
=
(|ψA>
+
|ψB>)/√2


(Symmetric,
bonding)
 



















σ*
=
(|ψA>
‐
|ψB>)/√2


(An8symmetric,
an8bonding)
 15

Why
do
you
get
H2
and
not
He2?


16 17

• 1. Introduction
• 2. Exchange
• 3. Superexchange
• 4. Orbitals

Outline

18

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19

• Interaction between pair of spins motivates the

general form of the Heisenberg model:

• The quantity gives the exchange energy

between two spins. Be very careful on the factor

• f two between different conventions of the

definition of J.

20 21

• Interaction between pair of spins motivates the

general form of the Heisenberg model:

• Direct exchange: important in many metals such

as Fe, Co and Ni

• Superexchange: exchange interaction mediated

by oxygen. This leads to a very long exchange

• path. Important in many magnetic oxides, e.g.

MnO, La2CuO4.

22

• 1. Introduction
• 2. Exchange
• 3. Superexchange
• 4. Orbitals

Outline

23

Superexchange

• Case I: AF ordering
• Case II: F ordering
• KE advantage for AF ordering

24

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Toy model for superexchange

25

Toy model for superexchange

26

Toy model for superexchange

27

Toy model for superexchange

28

• K. A. Müller
• J. G. Bednorz

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• 1. Introduction
• 2. Exchange
• 3. Superexchange
• 4. Orbitals

Outline

31

s‐orbitals
 p‐orbitals


32

d‐orbitals


33 tradi8onal
technology
uses
Fe,
Co,
Ni
and
alloys
–
plus
the
physics
of
metals


Magne8c
elements
and
ions


34

S.J. Blundell, Contemp. Phys. 48, 275 (2007)

Partially filled 3d shell gives rise to a magnetic moment

35

S.J. Blundell, Contemp. Phys. 48, 275 (2007)

Partially filled 3d shell gives rise to a magnetic moment

CuII = 3d9

36

8/25/11 7 A parable: p-orbitals

real eigenfunctions, for V(r) which is real

37

A parable: p-orbitals

imaginary, eigenfunctions note that these contain the eigenfunctions and in equal mixtures

38 39 40 41

1. Exchange interaction of two half-filled orbitals is strong and antiferromagnetic 2. If this overlap is at 90o, exchange interaction is weak and ferromagnetic 3. Exchange interaction of half-filled with empty (or doubly-occupied)

• rbital is weak and ferromagnetic

Goodenough-Kanamori-Anderson (GKA) rules

42

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eg orbitals

43

eg orbitals

44

strong, AF weak, FM

45

t2g orbitals

46 47

In the next lecture:

• spin waves
• frustration
• phase transitions
• metallic magnets