Magnetism and Magnetic Materials

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Magnetism and Magnetic Materials DTU (10313)
10 ECTS KU 7.5 ECTS
Module 4 11/02/2001 Interactions
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Intended Learning Outcomes (ILO)
(for todays module)

1. List the various forms of exchange interactions
   between spins
2. Estimate the influence of dipolar interactions
   between spins
3. Explain how exchange interactions can favor
   either FM or AFM alignment
4. Describe the continuum-limit of exchange

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Flashback

1. 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.
2. 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.
3. Finally, the value of J is found using JL-S if
   the shell is less than half-filled, JLS is the
   shell is more than half-filled, JS (L0) 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 -Hunds
third rule
Hunds rules -How to determine the ground state
of an ion
The fine structure of energy levels -Apply
Hunds rules to given ions
Co2 ion 3d7 S3/2, L3, J9/2, gJ5/3, 4F9/2
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Data and comparison (4f and 3d)
Hunds rules seem to work well for 4f ions. Not
so for many 3d ions. Why?
How do we measure the effective moment?
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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!
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A new set of orbitals
Octahedral
Tetrahedral
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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.
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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
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Jahn-Teller effect
In some cases, it may be energetically favorable
to shuffle things around than to squeeze
electrons within degenerate levels.
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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

1. Estimate the magnitude of the dipolar energy
   between two aligned moments (1 mB) separated by
   0.1 nm.
2. 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
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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
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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)
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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
Hunds first rule (with Jgt0)
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, Jgt0 is more likely.
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Indirect exchange superexchange
Oxygen mediated, typical of MnO and similar
compounds, mainly antiferromagnetic

1. 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 (Jlt0).
   This is the usual case for 120-180 degrees M-O-M
   bonds.
2. 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
3. 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
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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.
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The continuum approximation
Is the exchange stiffness, with c a
crystal-structure-dependent factor, and a the
nearest-neighbour distance
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Sneak peek
Ferromagnetism (Weiss)
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Wrapping up

• Crystal fields (from last module)
• Exchange (and dipolar) interaction
• Spin Hamiltonian
• Superexchange
• Double exchange
• The continuum limit

Next lecture Tuesday February 15, 1315, KU
Auditorium 9 Magnetic order (MB)