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Magnetism
Electromagnetic Fields in a Solid
SI units
cgs (Gaussian) units
Total magnetic field:
B = 0 (H + M) =  0 H
B = H + 4 M =  H
Total electric field:
E = 1/0 (D  P) = 1/0 D
E = D  4 P = 1/ D
B, E are the total fields that appear in the Lorentz force:
F = q (E + vB)
M, P are the dipole densities in a solid.
H, D are defined by the equations above (averaged, “macroscopic” fields)
The different signs of M and P, and the choice of 1/ versus  are related to the
fact that most solids are dielectric, i.e., the polarization opposes the external field, but
paramagnetic, i.e., the magnetization reinforces the external field. In dielectric
(diamagnetic) solids, the dipoles are created by the external field and oppose it (Lenz’s
law). In paramagnetic solids, dipoles already exist and become reoriented by the external
field. In most solids, P and M are positive with ,  > 1.
Units: In the following, the standard SI units are used, although cgs units are
more popular in the magnetism community (for conversions and constants see next page).
B:
T (Tesla, SI) = Vs/m2 =
H:
A/m
Susceptibility:
(SI)
104 G
(Gauss,
cgs);
= 4 103 Oe (Oerstedt, cgs);
 = 0  M/B  M/H
(SI);
1 Oe  1 G
Typically:  105,…,+103
 describes the (linear) magnetic response of a solid to an external B-field.
Energy of a dipole in a magnetic field:
U =  B
= mj  g  B  B
 (vector) = magnetic dipole moment (not to be confused with the permeability ).
mj = quantum number for the z-component of the total angular momentum (B || z).
g = g-factor = ratio of magnetic moment to angular momentum in units of B and ћ.
g = 2 for pure spin angular momentum, g = 1 for pure orbital angular momentum.
The sign reversal is due to the negative charge of the electron.
B = eћ/2me = 5.8 105 eV/T = Bohr magneton  U << meV << kBT at room temp.
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Fundamental Electromagnetic Constants
SI units
cgs (Gaussian) units
E = 1/0 (D  P) = 1/0 D
E = D  4 P = 1/ D
 = D / 0E
 = D/E
Dielectric Constant
 e2 / 40 r
 e2 /  r
Coulomb Potential
e4 m
e4 m / 2 (40)2 ћ2
e4m / 2 2 ћ2
e2 m1
(40) ћ2 / e2 m
 ћ2 / e2 m
Electric Fields
Effective Rydberg = Ry/2
Effective Bohr Radius = a0  
(m = Electron Mass; Effective Mass Omitted)
P = (n e2 / 0 m)1/2
P = (4 n e2 / m)1/2
Plasmon Frequency
(n = Electron Density)
B = 0 (H + M) =  0 H
B = H + 4 M =  H
µ = B / µ0H
µ = B/H
µB = e ћ / 2 m
µB = e ћ / 2 m c
c = e B / m
c = e B / m c
 = e2 / 40 ћc
 = e2 / ћ c
e
G0 = 2 e2/ h = 2 R01
G0 = 2 e2/ h
Quantum Conductance (Spin )
e2
R0 = h / e2
R0 = h / e2
Quantum Hall Resistance
e1
0 = h / 2 e
0 = h c / 2 e
Supercond. Flux Quantum (Pairs)
e1
KJ = 2 e / h = 01
KJ = 2 e / h c
AC Josephson Effect (Hz/V)
e/m
2
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Magnetic Fields
Permeability
Bohr Magneton
Cyclotron Frequency
Fine Structure Constant
Moments
for B=0 :
Types of Magnetism
Diamagnetism: M opposes B ( < 0). Caused by induced currents that generate a field
. . . . .
opposing to the inducing field (Lenz’s law). Characteristic of atoms without magnetic
moment, e.g., rare gases (filled shell), carbon, silicon (paired electrons), superconductors.
Paramagnetism: M reinforces B ( > 0). Caused by existing magnetic moments being
aligned by the external field. Characteristic of unpaired electrons, e,g. in alkali metals.
Ferromagnetism: M exists spontaneously, even without external B. The exchange
interaction between electrons on neighboring atoms aligns their spins parallel. Typical
ferromagnets are transition metals (partially-filled 3d shell, Fe, Co, Ni) and rare earths
(partially-filled 4f shell, Gd). Ferromagnets lose their magnetic order above the Curie
temperature TC and become paramagnets with  ~ 1/(TTC) .
Antiferromagnetism: M exists spontaneously, but alternates in sign between adjacent
atoms. Typical antiferromagnets are again transition metals (partially-filled 3d shell, Cr)
and rare earths (partially-filled 4f shell), but their interatomic distances are smaller, such
that the orbitals on adjacent atoms overlap too much and Pauli’s principle forces the spins
antiparallel. Antiferromagnets lose their magnetic order above the Néel temperature TN
and become paramagnets with  ~ 1/(TTN) .
Ferrimagnetism: M exists spontaneously and alternates in sign between adjacent atoms.
In contrast to antiferromagnets, however, the magnetic moments on adjacent atoms are
different in magnitude, such that a net magnetization remains. Typical ferrimagnets
contain two sub-lattices of inequivalent atoms, e.g., Fe3O4 = FeO+Fe2O3 with Fe in two
different oxidation states.
Ferroelectricity: P exists spontaneously, even without external E. Electrical analog to
ferromagnetism.
Multiferroic: Both ferromagnetic and ferroelectric.
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Paramagnetic Susceptibility
ms  Probability
A) Bound Electrons in Atoms
Curie Law:
+½ B p ½ex
s=½
~1/T
½ +B p+  ½e+x
With increasing temperature T the alignment of the magnetic moments in a B field is less
effective, due to thermal spin flips. Calculation for an electron (spin ½, negative charge):
Energy of an electron in a magnetic field: U = msgBB =  BB
(ms = ½, g = 2)
The probabilities p are proportional to the Boltzmann factor:
p ~ exp[ U / kBT ] = exp[ BB / kBT ] = ex  1x
The normalization condition
p++ p=1
gives
with
x = BB / kBT <<1
p= ex / (e+x + ex)  (1  x) / 2
The magnetization M is obtained by separating the electron density N into opposite spins
with magnetic moments B and probabilities p :
M = N  (p+B  pB) = N  B  (e+x  ex) / (e+x + ex)  NBx = N  B  (BB/kT)
 = 0  M/B  0  N  B2/kBT
The 1/T law originates from 1/kBT in the Boltzmann factor exp(U/kBT).
B) Free Electrons in a Metal
Pauli Paramagnetism:
 independent of T,   D(EF) .
D(EFermi)
Compare cV (Lect. 15).
E
EFermi
2BB
Excess Spins
+BB
D(E)
BB
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D(E)
Energy Bands of Ferromagnets
The band structure plot E(k) combines the quantum numbers of electrons in a solid
(energy E and momentum p = ћk). Ferromagnets have two sets of bands, one for
electrons with “spin up”  (|| B, ms= ½), the other for “spin down”  (|| B, ms= +½).
They are separated by the magnetic exchange splitting Eex , which ranges from 0.3 eV
in Ni to 2 eV in Fe. These bands can be measured using angle-resolved photoemission, as
shown below (high intensity is dark). Magnetism is carried mainly by the 3d bands.
Experiment
EF=
3d-bands
k
s,p-band
I  D̃ (EF) > 1
Stoner Criterion for Ferromagnetism:
I = Exchange integral,
D̃ (EF) = Density of states at the Fermi level (per atom, spin)
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The Exchange Interaction
The exchange interaction is responsible for both ferromagnetism and antiferromagnetism.
It originates from the antisymmetry* of the electron wave function with respect to
exchange of two electrons 1 and 2 with coordinates r1 and r2 (see double-arrow):
(r1,r2) = [ a(r1) b(r2)  a(r2)  b(r1) ] / 2
Two-electron wave function for
parallel spins (see Lect. 2, p. 5)
a and b are two different wave functions. Two electrons cannot have the same wave
functions a = b , because  would be zero If the spins are the same, the spatial wave
functions have to be different. That causes the electrons with parallel spins to be farther
apart. This is the mathematical version of the Pauli principle (“two electrons with the
same spin cannot be at the same place”). Being farther apart which reduces the Coulomb
repulsion between electrons with parallel spins.
This argument leads to Hund’s first rule, which says that a configuration of
electrons with parallel spins has the lowest energy. That favors maximum total spin in
isolated atoms, where the electrons have enough room to get away from each other. In a
solid, the wave functions become squeezed by neighbor atoms, and some of the spins
have to become antiparallel to satisfy the Pauli principle. Consequently, the total spin is
reduced. Surfaces are between atoms and solids. They can support larger total spin than
the bulk. That leads to enhanced surface magnetism.
The exchange energy term in the Schrödinger equation is part of the electrostatic
energy between two electrons. Take the expectation value of the Coulomb potential V 
1/|r1r2| using the antisymmetric two-electron wave function from above:
V = *|V| =  *(r1,r2)  V  (r1,r2) dr1dr2
= +  a*(r1) a(r1)  V  b*(r2) b(r2) dr1dr2
Coulomb repulsion
  a*(r1) a(r2)  V  b*(r2) b(r1) dr1dr2
Exchange attraction
________________________________________________________________________
* Electrons are indistinguishable in quantum mechanics. They cannot be “painted red or
blue”. The wave function for two electrons has to change its sign when interchanging
their coordinates r1 and r2 , because they are fermions. Bosons, such as photons, are also
indistinguishable, but their wave function has to remain the same.
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Magnetic Data Storage
Storage Media: Each bit is stored in a collection of 102 magnetic particles,
each 10nm in size. That is just above the superparamagnetic size limit, where thermal
energy begins to spontaneously reverse the magnetization of a particle at room
temperature. Storage densities of >200 Gbit/inch2 have been demonstrated, 100 million
times denser than the first hard disk. A typical storage medium is a CoPtCr alloy, which
segregates into CoPt grains that are magnetically separated by Cr.
Sensors: The reading head detects the magnetic field lines escaping between two
magnetic domains of opposite orientation. The effect of magnetoresistance is used, i.e. a
change of the resistance in a magnetic field. A spin valve contains a Co/Cu/NiFe
sandwich which exhibits “giant magnetoresistance” (GMR), a 10% effect. It can be
explained by considering the interfaces as spin filters. There are two extreme
configurations, one with the magnetization of the Co and NiFe layers parallel, the other
opposite. The first has low resistance (parallel spin filters), the second high resistance
(opposing spin filters). The magnetic field emerging from a stored bit rotates the
magnetization of the NiFe layer, whereas that of the Co layer is pinned by proximity to an
antiferromagnetic MnFe or NiO layer (magnetic bias). A similar effect exists in magnetic
tunnel junctions, which have an insulating spacer instead of the Cu layer.
These structures are also used for a magnetic random access memory (MRAM).
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Level Splitting in a Magnetic Field
A magnetic field splits electron energy levels depending on the orientation of the
magnetic moment  relative the magnetic field B: U = B = mj  g  B  B (p.1)
The magnetic moment  is proportional to the angular momentum L, S, or J = L+S ,
whose component along B is determined by the quantum number ml , ms , or mj . The gfactor (p. 1) determines the ratio of the magnetic moment to the angular momentum.
Spin S
Orbital Angular Momentum L
g=2
E
Zeeman
Levels
g=1
ms

+ ½  B
ml

+1
 B
ћL
2 ћL
0
½
Larmor frequency
0
ћL
1
+B
L = eB/2m
+ B
SI units
Magnetic Resonance
Transitions between these energy levels are induced by microwave photons whose
frequency f is given by Planck’s formula h f = U ( = 2 ћL for spin).
NMR:
Nuclear Magnetic Resonance
(nuclear moment)
EPR (ESR): Electron Paramagnetic (Spin) Resonance
(electron moment)
Electrons have 103 the magnetic moment of nuclei (due to the mass ratio me/mp in L),
giving rise to transitions in the GHz regime, whereas nuclei resonate in the MHz regime:
Electrons: f = 28.0  B  GHz/T
Protons:
f = 42.6  B  MHz/T
NMR is widely used in chemistry for the analysis of compounds.
MRI (Magnetic Resonance Imaging) makes internal organs visible via NMR from
protons in water. A gradient in the magnetic field shifts the resonance frequency between
different locations.
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