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

8/25/11 Outline 1. Introduction Exchange and ordering 2. Exchange 3. Superexchange Stephen Blundell University of Oxford 4. Orbitals 2011 School - Time-dependent phenomena in magnetism Targoviste, August 2011 Part 1 1 2 Outline 1. Introduction 2. Exchange a problem: 3. Superexchange so define: 4. Orbitals but: 3 4 5 6 1

8/25/11 7 8 Dipolar fields Louis
Néel
 (1904‐2000)
 |B| 9 10 Energy terms: • Kinetic energy eV • Coulomb energy eV • Size of atom given by balance of these two terms • Spin-orbit ~ meV • Magnetocrystalline anisotropy ~ µ eV 11 12 2

8/25/11 13 14 Why
do
you
get
H 2 
and
not
He 2 ?
 Eigenfunc8ons:


 σ 
=
(| ψ A >
+
| ψ B >)/√2


(Symmetric,
bonding)
 



















 σ * 
=
(| ψ A >
‐
| ψ B >)/√2


(An8symmetric,
an8bonding)
 15 16 Outline 1. Introduction 2. Exchange 3. Superexchange 4. Orbitals 17 18 3

8/25/11 • 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 of two between different conventions of the definition of J . 19 20 • 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, La 2 CuO 4 . 21 22 Superexchange • Case I: AF ordering Outline 1. Introduction 2. Exchange 3. Superexchange • Case II: F ordering 4. Orbitals • KE advantage for AF ordering 23 24 4

8/25/11 Toy model for superexchange Toy model for superexchange 25 26 Toy model for superexchange Toy model for superexchange 27 28 K. A. Müller J. G. Bednorz 5

8/25/11 Outline s‐orbitals
 1. Introduction 2. Exchange p‐orbitals
 3. Superexchange 4. Orbitals 31 32 Magne8c
elements
and
ions
 d‐orbitals
 tradi8onal
technology
uses
Fe,
Co,
Ni
and
alloys
–
plus
the
physics
of
 metals
 33 34 Cu II = 3d 9 Partially filled 3d shell gives Partially filled 3d shell gives rise to a magnetic moment rise to a magnetic moment 35 36 S.J. Blundell, Contemp. Phys. 48, 275 (2007) S.J. Blundell, Contemp. Phys. 48, 275 (2007) 6

8/25/11 A parable: p-orbitals A parable: p-orbitals imaginary, eigenfunctions note that these contain the eigenfunctions and in equal mixtures real eigenfunctions, for V(r) which is real 37 38 39 40 Goodenough-Kanamori-Anderson (GKA) rules 1. Exchange interaction of two half-filled orbitals is strong and antiferromagnetic 2. If this overlap is at 90 o , exchange interaction is weak and ferromagnetic 3. Exchange interaction of half-filled with empty (or doubly-occupied) orbital is weak and ferromagnetic 41 42 7

8/25/11 e g orbitals e g orbitals 43 44 t 2g orbitals weak, FM strong, AF 45 46 In the next lecture: • spin waves • frustration • phase transitions • metallic magnets 47 8