Magnetism and Magnetic Materials DTU (10313) – 10 ECTS KU – 7.5 ECTS

1. Magnetism and Magnetic Materials DTU (10313) – 10 ECTS KU – 7.5 ECTS Module 4 11/02/2001 Interactions Sub-atomic – pm-nm With some surrounding environment and a first step towards the nanoscale

2. Intended Learning Outcomes (ILO) (for today’s module) List the various forms of exchange interactions between spins Estimate the influence of dipolar interactions between spins Explain how exchange interactions can favor either FM or AFM alignment Describe the continuum-limit of exchange

3. Flashback • Arrange the electronic wave function so as to maximize S. In this way, the Coulomb energy is minimized because of the Pauli exclusion principle, which prevents electrons with parallel spins being in the same place, and this reduces Coulomb repulsion. • The next step is to maximize L. This also minimizes the energy and can be understood by imagining that electrons in orbits rotating in the same direction can avoid each other more effectively. • Finally, the value of J is found using J=|L-S| if the shell is less than half-filled, J=L+S is the shell is more than half-filled, J=S (L=0) if the shell is exactly half-filled (obviously). This third rule arises from an attempt to minimize the spin-orbit energy. Spin-orbit: -S and L not independent -Hund’s third rule Hund’s rules: -How to determine the ground state of an ion The fine structure of energy levels: -Apply Hund’s rules to given ions Co2+ ion: 3d7: S=3/2, L=3, J=9/2, gJ=5/3, 4F9/2

4. Data and comparison (4f and 3d) Hund’s rules seem to work well for 4f ions. Not so for many 3d ions. Why? How do we measure the effective moment?

5. Origin of crystal fields When an ion is part of a crystal, the surroundings (the crystal field) play a role in establishing the actual electronic structure (energy levels, degeneracy lifting, orbital “shapes” etc.). Not good any longer!

6. A new set of orbitals Octahedral Tetrahedral

7. Crystal field splitting; low/high spin states The crystal field results in a new set of orbitals where to distribute electrons. Occupancy, as usual, from the lowest to the highest energy. But, crystal field acts in competition with the remaining contributions to the Hamiltonian. This drives occupancy and may result in low-spin or high-spin states.

8. Orbital quenching Examine again the 3d ions. We notice a peculiar trend: the measured effective moment seems to be S-only. L is “quenched”. This is a consequence of the crystal field and its symmetry. Is real. No differential (momentum-related) operators. Hence, we need real eigenfunctions. Therefore, we need to combine ml states to yield real functions. This means, combining plus or minus ml, which gives zero net angular momentum. Examples

9. Jahn-Teller effect In some cases, it may be energetically favorable to shuffle things around than to squeeze electrons within degenerate levels.

10. Dipolar interaction Dipolar interaction is the key to explain most of the macroscopic features of magnetism, but on the atomic scale, it is almost always negligible (except at mK temperatures). Dipolar interaction energy m1 m2 Estimate the magnitude of the dipolar energy between two aligned moments (1 mB) separated by 0.1 nm. Now think of the moments as tiny magnetized spheres each carrying N Bohr magnetons and separated by 10 nm. How large is N if we want an energy of the order of 1000 K? r

11. Exchange symmetry This is, instead, the real thing underpinning long range magnetic ordering. Effectively, its strength is enormous. Singlet, antisymmetric Triplet, symmetric Singlet, total wave function (antisymmetric) Triplet, total wave function (antisymmetric) Singlet, energy Triplet, energy

12. Exchange Hamiltonian Key observation: even if H does not include “spin terms”, the energy levels depend on the alignment of spins via symmetry of the wave function. Remember this (and correct a mistake in M1) If we construct this operator with It happens to produce the same energy splitting of the real Hamiltonian. We take this, remove the constant, and use it as “spin Hamiltonian” the “exchange constant” (or “exchange integral”)

13. Generalization and general features A positive exchange constant favors parallel spins, while a negative value favors antiparallel alignment The “Heisemberg Hamiltonian” Exchange coupling between electrons belonging to the same atom can be interpreted as underpinning Hund’s first rule (with J>0) Suppose J is about 1000 K. How strong is the effective exchange field? Coupling between electrons in different atoms, where bonding and/or antibonding orbitals may exist. In this case, J>0 is more likely.

14. Indirect exchange: superexchange Oxygen mediated, typical of MnO and similar compounds, mainly antiferromagnetic • When two cations have loves of singly occupied 3d-orbitals which point towards each other giving a large overlap and hopping integrals, the exchange is strong and antiferromagnetic (J<0). This is the usual case for 120-180 degrees M-O-M bonds. • When two cations have an overlap integral between singly occupied 3d-orbitals which is zero by symmetry, the exchange is ferromagnetic and relatively weak. This is the case for about 90 degree M-O-M bonds • When to cations have an overlap between singly occupied 3d-orbitals and empty or doubly occupied orbitals of the same type, the exchange is also ferromagnetic, and relatively weak. t is the “hopping integral” and U is the Coulomb energy

15. Indirect exchange: double exchange Typical of mixed-valence compounds, like Mn3+/Mn4+ (manganites) or Fe2+/Fe3+ (magnetite). Double exchange is essentially ferromagnetic superexchange in an extended system.

16. The continuum approximation Is the “exchange stiffness”, with c a crystal-structure-dependent factor, and a the nearest-neighbour distance

17. Sneak peek Ferromagnetism (Weiss)

18. Wrapping up • Crystal fields (from last module) • Exchange (and dipolar) interaction • Spin Hamiltonian • Superexchange • Double exchange • The continuum limit Next lecture: Tuesday February 15, 13:15, KU [Auditorium 9] Magnetic order (MB)