Magnetism

Electromagnetic Fields in a Solid SI units cgs (Gaussian) units

Total magnetic field: B = μ0 (H + M) = μ μ0 H B = H + 4π M = μ H

Total electric field: E = 1/ε0 (D − P) = 1/εε0 D E = D − 4π P = 1/ε 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/ε versus μ 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 ε, μ > 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 = 104 G (Gauss, cgs);

H: A/m (SI) = 4π ⋅10−3 Oe (Oerstedt, cgs); 1 Oe ≡ 1 G

Susceptibility: χ = μ0 ⋅ M/B ≈ M/H (SI); Typically: χ ≈ −10−5,…,+10−3

χ describes the (linear) magnetic response of a solid to an external B-field.

Energy of a dipole in a magnetic field: U = − μ⋅B = mj ⋅ g ⋅ μB ⋅ B

μ (vector) = magnetic dipole moment (not to be confused with the permeability μ).

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 μB 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.

μB = 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/ε0 (D − P) = 1/εε0 D E = D − 4π P = 1/ε D Electric Fields

ε = D / ε0E ε = D / E Dielectric Constant

− e2 / 4πε0ε r − e2 / ε r Coulomb Potential

e−2 m−1

e4 m e4 m / 2 (4πε0ε)2 ћ2 e4m / 2 ε2 ћ2 Effective Rydberg = Ry/ε2

(4πε0ε) ћ2 / e2 m ε ћ2

/ e2 m Effective Bohr Radius = a0 ⋅ ε

(m = Electron Mass; Effective Mass Omitted)

ωP = (n e2 / ε0 m)1/2 ωP = (4π n e2 / m)1/2 Plasmon Frequency

(n = Electron Density)

B = μ0 (H + M) = μ μ0 H B = H + 4π M = μ H Magnetic Fields

µ = B / µ0H µ = B / H Permeability

e/m µB = e ћ / 2 m µB = e ћ / 2 m c Bohr Magneton

ωc = e BB / m ωc = e B / m c Cyclotron Frequency

α = e2 / 4πε0 ћc α = e2 / ћ c Fine Structure Constant e2

e−2

G0 = 2 e2/ h = 2 R0−1 G0 = 2 e2/ h Quantum Conductance (Spin ↑↓)

R0 = h / e2 R0 = h / e2 Quantum Hall Resistance

e1

e−1 Φ0 = h / 2 e Φ0 = h c / 2 e Supercond. Flux Quantum (Pairs)

KJ = 2 e / h = Φ0−1 KJ = 2 e / h c AC Josephson Effect (Hz/V)

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Moments for B=0 :

Types of Magnetism

Diamagnetism: M opposes B (χ < 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 (χ > 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 χ ~ 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 χ ~ 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

A) Bound Electrons (Energy Levels in Atoms) Curie Law: χ ~ 1 / T 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⋅μB⋅B = ± μB⋅B (ms = ±½, g = 2)

The probabilities p± are proportional to the Boltzmann factor:

p± ~ exp[ −U / kBT ] = exp[ ±μBBB / kBT ] = e ≈ 1±x with x = μ±xBB B / kBT <<1

The normalization condition p++ p−=1 gives 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 ±μB and probabilities p± :

M = N ⋅ (p+⋅μB − p−⋅μB) = N ⋅ μB ⋅ (e+x − e−x) / (e+x + e−x) ≈ N⋅μB⋅x = N ⋅ μB ⋅ (μBB/kT)

χ = μ0 ⋅ M/B ≈ μ0 ⋅ N ⋅ μB2/kBT

The 1/T law originates from 1/kBT in the Boltzmann factor exp(−U/kBT).

B) Free Electrons (Energy Bands in Metals)

Pauli Paramagnetism: χ independent of T determined by D(EFermi)

EFermi

2μBB

−μBB

+μBB

D↑(E) D↓(E)

Excess Spins

s=½ −½ +μB p+ ≈ ½e+x

+½ −μB p− ≈ ½e−x

ms μ Probability

E D(EFermi)

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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” ↑ (μ || B, ms= −½), the other for “spin down” ↓ (μ || −B, ms= +½).

They are separated by the magnetic exchange splitting δEex , 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.

3d-bands

s,p-band

EF=

k

Experiment

Stoner Criterion for Ferromagnetism: I ⋅ D̃(EF) > 1

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):

Ψ(r1,r2) = [ ψa(r1) ⋅ ψb(r2) − ψa(r2) ⋅ ψb(r1) ] / √2 Two-electron wave function

for parallel spins

ψa and ψb are two different wave functions. Two electrons cannot have the same wave

functions ψa = ψb , because Ψ 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⟩ = ⟨Ψ*|V|Ψ⟩ = ∫ Ψ*(r1,r2) ⋅ V ⋅ Ψ(r1,r2) dr1dr2

= + ∫∫ ψa*(r1) ψa(r1) ⋅ V ⋅ ψb*(r2) ψb(r2) dr1dr2 Coulomb repulsion

− ∫∫ ψa*(r1) ψa(r2) ⋅ V ⋅ ψb*(r2) ψb(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 μ relative the magnetic field B: U = −μ⋅B = mj ⋅ g ⋅ μB ⋅ B (p.1)

The magnetic moment μ 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 g-factor (p. 1) determines the ratio of the magnetic moment to the angular momentum.

Spin S Orbital Angular Momentum L

g=2 g=1 Zeeman Levels

ms μ ml μ E

+ ½ − μB +1 − μB

2 ћωL 0 0

− ½ +μB −1 + μB

ћωL

ћωL

Larmor frequency ωL = eB/2m 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 = ΔU ( = 2 ћωL 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 ωL),

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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