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
Intended Learning Outcomes (ILO)
(for today’s module)
1.
2.
3.
4.
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
Flashback
H so  S  L
Spin-orbit:
-S and L not independent
-Hund’s third rule
(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 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.
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
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?
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.).
Hˆ  Hˆ 0  Hˆ so  Hˆ cf  Hˆ Z
Vcf 
1
4  0
Hˆ cf 
 (r)
 r  r' d r'
3
3

(r)
V
(r)d
r
 0 cf
Not good any longer!
A new set of orbitals
Octahedral
Tetrahedral
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.
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.
Vcf 
1
4  0
 (r)
 r  r'
d 3r'
p x  l  1, ml  1  l  1, ml  1
d xy  l  2, ml  2  l  2, ml  2
d x 2 y 2  l  2, ml  2  l  2, ml  2 
dz 2  l  2, ml  0
Examples
Is real. No differential (momentumrelated) 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.
Jahn-Teller effect
In some cases, it may be
energetically favorable to
shuffle things around than to
squeeze electrons within
degenerate levels.
EJT  A  B 2

Dipolar interaction
Dipolar interaction energy
1  r 2  r
0 1  2

E
3


3

4   r
r5
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).
1
2
r
1. Estimate the magnitude of the dipolar energy
between two aligned moments (1 B)
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?
Exchange symmetry
  
2
  
T   ,
, 
2
S 
Singlet, antisymmetric
Triplet, symmetric
1
S 
a (r1 )b (r2 )  a (r2 )b (r1 ) S
2
1
T 
a (r1 )b (r2 )  a (r2 )b (r1 )T
2

   Hˆ  dr dr
ES 
ET
S* Hˆ S dr1dr2
*
T
T
This is, instead, the real thing
underpinning long range
magnetic ordering. Effectively,
its strength is enormous.
1
2
Singlet, total wave function (antisymmetric)
Triplet, total wave function (antisymmetric)
Singlet, energy
Triplet, energy
Exchange Hamiltonian
ES  ET  2   (r1 ) (r2 ) Hˆ a (r2 )b (r1 )dr1dr2
*
a
*
b
Remember this (and correct a mistake in M1)
1

2
2
2
1
Sˆ a  Sˆ b  Sˆ tot   Sˆ a   Sˆ b    4 3
2
 4

s 1
s0

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.
If we construct this operator
1
eff
ˆ
H  ES  3ET   ES  ET S1  S2
4
It happens to produce the
same energy splitting of
the real Hamiltonian. We
take this, remove the
constant, and use it as
“spin Hamiltonian”
Hˆ spin  2JS1  S2
with
J
*
*

(r
)

 a 1 b (r2 )Hˆ a (r2 )b (r1 )dr1dr2

the “exchange constant” (or “exchange integral”)

Generalization and general features
Hˆ  2 Jij Si  S j
i j
A positive exchange constant favors
parallel spins, while a negative value
favors antiparallel alignment
The “Heisemberg Hamiltonian”
Suppose J is about 1000 K.
How strong is the effective
exchange field?
Exchange coupling between electrons
belonging to the same atom can be
interpreted as underpinning Hund’s
first rule (with J>0)
Coupling between electrons
in different atoms, where
bonding and/or antibonding
orbitals may exist. In this
case, J>0 is more likely.
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 (J<0). 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 3dorbitals and empty or doubly occupied orbitals of the same type,
the exchange is also ferromagnetic, and relatively weak.
J  -t 2 /U
t is the “hopping
integral” and U is the
Coulomb energy
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.
The continuum approximation
2
JS
Hˆ  J Si  S j  JS 2  cos ij  
ij2

2 i, j
i, j
i, j
ij  mi  m j  rij  m




E  A  m x   m y   mz  d 3r
JS 2
Ac
a

M
m
MS
2
2
2
Is the “exchange stiffness”, with c a
crystal-structure-dependent factor,
and a the nearest-neighbour distance
Sneak peek
Ferromagnetism (Weiss)
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)