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Lecture 24 -

Lecture 24: Magnetism Outline • Magnetism is a purely quantum phenomenon! (Kittel Ch. 11-12) Totally at variance with the laws of classical physics (Bohr, 1911) • Diamagnetism • – (Pauli paramagnetism) Magnetism • Effects of -electron interactions Hund’s rules for atoms – examples: Mn, Fe Atoms in a – Curie Law Atomic-like local moments in Explains magnetism in transition , rare earths • Magnetic order and cooperative effects in solids Quantum Electron-Electron Transition temperature Tc Mechanics Interactions Curie-Weiss law • Magnetism: example of an “order parameter”

• (Kittel Ch. 11-12 – only selected parts) Physics 460 F 2006 Lect 24 1 Physics 460 F 2006 Lect 24 2

Magnetism and Definitions

• Why is magnetism a quantum effect? • B = µ0 ( H + M) (µ0 is the permeability of free space, • In classical physics the change in energy of a particle B is the field that causes forces on particles) per unit time is F . v (F = force, v = velocity vectors). • If the is proportional to field, M = χ H and B = µ H , µ = µ (1 + χ ) • In a magnetic field the force is always perpendicular to 0 velocity - therefore the energy of a system of particles cannot change in a magnetic field B • Diamagnetic material: χ < 0; µ < µ0 • Similarly the equilibrium free energy cannot change with applied B field • Paramagnetic material: χ > 0; µ > µ0

• Since the change in energy is dB . M , there must be no total M ! • Ferromagnetic material: M ≠ 0 even if H = 0

Physics 460 F 2006 Lect 24 3 Physics 460 F 2006 Lect 24 4

Diamagnetism “Classical” Theory of Diamagnetism • Consider a single “closed shell” atom in a magnetic • If the field B is small compared to the quantum energy field level separation, the closed shell atom may be (In a closed shell atom, spins are paired and the considered to rotate rigidly due to the field B are distributed spherically around the atom - there is no total angular momentum.) - • This is like a classical current - BUT it occurs only because the + • Diamagnetism results from current set up in atom due atom is in a quantized state to magnetic field • The entire electron system rotates together with the - • Like Lenz’s law - current acts to frequency ω = eB/2m oppose the external field and + “shield” the inside of the atom • Like Lenz’s law - the current acts to oppose the from the field (like a ) external field

Physics 460 F 2006 Lect 24 5 Physics 460 F 2006 Lect 24 6

1 Lecture 24 - Magnetism

“Classical” Theory of Diamagnetism “Classical” Theory of Diamagnetism • Total current = charge/time = I = (-Ze) (1/2π)(eB/2m) • From previous slide - for closed shell atoms - M/B = - NZe2/6m) • Magnetic moment = current times area = I x π<ρ2> χ = µ0 µ0 2 2 + = µ = (-Ze B/4m) <ρ > ρ (ρ = distance from axis) - Average value + • Results: VERY small diamagnetism! ρ For rare gasses in a , is • Susceptibility = only VERY slightly less than in

2 2 2 2 χ = µ0 M/B = - µ0NZe /4m <ρ > = - µ0NZe /6m • Similar results are found for typical “closed shell” insulators -- like Si, , NaCl, SiO2, …. because where M = Nµ, N = density of atoms, and for a they have paired spins and filled bands like a closed spherical atom = 2/3 <ρ2>, where r is the shell atom -- VERY weak diamagnetism radius in 3 dimensions

Physics 460 F 2006 Lect 24 7 Physics 460 F 2006 Lect 24 8

Spin Paramagnetism Spin Paramagnetism in a • What about spin? • What happens in a metal? • Spin up electrons E • Unpaired spins are affected by magnetic field! (parallel to field) µ • The energy in a field is given by are shifted opposite . U = - µ B = - m g µB B to spin down electrons where m = component of spin = ±1/2, (antiparallel to field) N↑ -N↓ g = “g factor” = 2, • Energies shift by and µB = = eh/2m ∆E = ±µB B • Magnetization 2 µBB • Any atom with an unpaired spin (e.g. and odd number M = µB (N↑ -N↓ ) of spins) must have this effect 2 = µB (1/2) D(EF) 2 µB B = µB D(EF) B Density of states for both spins • At temperature = 0, the spin will line up with the field in • Free electron (see previous notes + Kittel) a paramagnetic way - i.e. to increase the field 2 Physics 460 F 2006 Lect 24 9 M = (3/2) N µB B/(kBTF) Physics 460 F 2006 Lect 24 10

Spin Paramagnetism in a metal Spin Paramagnetism in a metal Density of states Density of states • Result for a metal: • Early success of 2 E quantum mechanics E •M = µB D(EF) B or 2 χ = µB D(EF) µ • Explained by Pauli µ • This is a way to measure the density of states! • The magnitude is greatly (Note: There are corrections reduced by the factor from the electron-electron µBB/(kBTF) interactions. ) due to the fact that the Fermi 2 µ B 2 µ B B energy EF = kBTF >> µBB B • Paramagnetic for any reasonable B

Tends to align with the field • The same reason that the heat to increase the magnetization capacity is very small compared

Physics 460 F 2006 Lect 24 11 to the classical result Physics 460 F 2006 Lect 24 12

2 Lecture 24 - Magnetism

Magnetic materials Questions for understanding materials: • What causes some materials (e.g. Fe) to be • Why are most magnetic materials composed of the ferromagnetic? 3d transition and 4f rare earth elements • Others like Cr are antiferromagnetic (what is this?) • Magnetic materials tend to be in particular places in the periodic table: transition metals, rare earths Why?

• Starting point for understanding: the fact that open shell atoms have moments. Why?

• us to a re-analysis of our picture of electron bands in materials. The band picture is not the whole story!

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The first step in understanding When are atoms magnetic? magnetic materials • An atom MUST have a magnetic moment if there are and odd number of electrons – spin ½ (at least) • “Open shell” atoms have moments – Hund’s rules – • Magnetic moments of atoms 1st rule: maximum spin for electrons in a given shell • In most magnetic materials (Fe, Ni, ….) the first step in 2nd rule: maximum angular momentum possible for the understanding magnetism is the consider the material given spin orientation as a collection of atoms • Of course the atoms change in the solid, but this gives • Example: Mn2+ - 5 d electrons a good starting point – qualitatively correct A d shell has L=2, mL = -2,-1,0,1,2

Stotal = 5/2, Ltotal = 0 mL = -2 -1 0 1 2

•Fe2+ - 6 d electrons

Stotal = 4/2, Ltotal = 2 Physics 460 F 2006 Lect 24 15 Physics 460 F 2006 Lect 24 16 mL = -2 -1 0 1 2

Hund’s Rules & Electron Interactions Magnetic atoms in free space Hund’s rules – • Curie Law (Kittel p 305) st 1 rule: maximum spin for electrons in a given shell • Consider N isolated atoms, each with two states (spin Reason – parallel-spin electrons are kept apart 1/2) that have the same energy with no magnetic because they must obey the exclusion principle – thus field, but are split in a field into E1 = -µB, E2 = µB the repulsive interaction between electrons is reduced for parallel spins!

2nd rule: maximum angular momentum possible for the given spin orientation • In the field B, the populations are: N1 / N = exp(µB/kBT) / [exp(-µB/kBT) + exp(µB/kBT) ] Reason – maximum angulat momentum means N / N = exp(-µB/k T) / [exp(-µB/k T) + exp(µB/k T) ] electrons are going the same direction around the 2 B B B nucleus – stay apart – lower energy! • So the magnetization M is Electron-Electron Interactions! M = µ (N1 -N2) = µ N tanh(x), x = µB/kBT Physics 460 F 2006 Lect 24 17 Physics 460 F 2006 Lect 24 18

3 Lecture 24 - Magnetism

Magnetic atoms in free space When do solids act like an array • Curie Law -- continued of magnetic moments? • Similar laws hold for any spin • Consider a solid made from atoms with magnetic moments M = gJµ N B (x), x = gJµ B/k T B J B B • If the atoms are widely spaced, they retain their atomic where B (x) = Brillouin Function (Kittel p 304) character -- insulators because electron-electron J interactions prevent electrons from moving freely • Key point: For small x (small B or large T) • Thus the material can be magnetic and insulating! the susceptibility has the form Curie Law χ = M/B = C/T, where C = Curie constant • OPPOSITE to what we said before! Real materials can be metallic and non-magnetic (like Na) or • For large x (large B or T small compared to gJµBB/kB) M saturates and χ→constant magnetic insulators (see later) Physics 460 F 2006 Lect 24 19 Physics 460 F 2006 Lect 24 20

When are atoms magnetic? Questions for understanding materials: • Which atoms are most likely to keep their atomic like • Why are most magnetic materials composed of the properties in a solid? 3d transition and 4f rare earth elements Transition metals and rare earths Why? Because the electrons act like they have partially filled shells even in the solid! This is why they have a special place in the periodic table - the elements in the transition series have similar chemical properties as the electrons fill the 3d or 4f shell

Physics 460 F 2006 Lect 24 21 Physics 460 F 2006 Lect 24 22

Magnetic solid How do magnetic atoms • “Localized” magnetic moments on the atoms affect each other? • Curie-Weiss Law (Kittel p 324) • The simplest approximation is to assume each atom acts like it is in an effective magnetic field BE due to the neighbors

• One expects BE = λ M where λ is some factor - see next slide • At high temperature we do not expect any net order, i.e., BE = 0 and M = 0 unless one applies an external • How do the atoms decide to order? field BA . • What happens as the temperature is lowered? Physics 460 F 2006 Lect 24 23 Physics 460 F 2006 Lect 24 24

4 Lecture 24 - Magnetism

How do magnetic atoms Curie-Weiss Law affect each other? • At high temperature we have M = χ (BE + BA) or M (1- λχ ) = χ BA • Note: The “effective magnetic field BE” is NOT really a “magnetic field” as in Maxwell’s Equations or M = BA χ / (1- λχ ) Approximate form valid at high T - sufficient • Using the form of χ = C/T, for present purposes

χ / (1- λχ ) = 1/ (T/C - λ) Diverges as T is reduced to T = Tc = λC • The “effective magnetic field BE” is due to the or exclusion principle and electron-electron interactions χeff = χ / (1- λχ ) = C/ (T - Tc), Tc = C λ that depend upon the relative spin of nearby electrons

• The “effective magnetic field BE” can favor parallel • What does this mean? The magnetic moments all spins or antiparallel spins - depends upon many allign to make a ferromagnet without any external field details - simplest approximation is BE = λ M below a critical temperature Tc Physics 460 F 2006 Lect 24 25 Physics 460 F 2006 Lect 24 26

Ferromagnetic solid Example of a transition • “Localized” magnetic moments on the atoms aligned to a state of new order together to give a net magnetic moment • At high temperature, the material is paramagnetic Magnetic moments on each atom are disordered

• At a critical temperature Tc the moments order Total magnetization M is an “Order Parameter” • Transition temperatures: Tc = 1043 K in Fe, 627 K in Ni, 292 K in Gd

M

• Although there is some thermal disorder, there is a net moment at finite temperature. T Physics 460 F 2006 Lect 24 27 Tc Physics 460 F 2006 Lect 24 28

Real Magnetic materials • Domains and Hysteresis • Whenever there is an order, there can be variations in • A usual breaks up into domains unless it is the order as function of position “poled” - an external field applied to allign the • “Magnons” are quanta of magnetic vibrations very domains much like “” • A real magnet has “hysteresis” - it does not change the direction of its magnetization unless a large enough field is applied - irreversibility M Remnant Saturation magnetization magnetization • Can be observed directly by neutron scattering B

Physics 460 F 2006 Lect 24 29 Physics 460 F 2006 Lect 24 30

5 Lecture 24 - Magnetism

Antiferromagnetic solid Summary • Magnetic moments can also order to give no net • Magnetism is a purely quantum phenomenon! moment - antiferromagnet Totally at variance with the laws of classical physics - (Bohr, 1911)

• B = µ0 ( H + M) , M = χ H and B = µ H • Diamagnetism -Mopposite to H Closed shell atoms Insulators like Si, NaCl, ... Very weak • Spin paramagnetism -Madds to H Example of metal - measures density of states • How does happen? Other forms of magnetism? • Transition temperature T =T transition Neel • Why does magnetism occur in transition metals, (named for Louis Neel) rare earths? Physics 460 F 2006 Lect 24 31 Physics 460 F 2006 Lect 24 32

Summary Next time • Open shell atoms have magnetic moments • Special presentation – Raffi Budakian Controlled by electron-electron interactions Magnetic Resonance Force Spectroscopy Hund’s Rules – see Kittel, p. 356 • Curie Law for atoms in a magnetic field • Atomic-like effects (local magnetic moments) can • Start Surfaces and Scanning Tunneling Microsope occur in solids – transition metals, rare earths (STM) • Magnetism is cooperative phenomenon whereby all the moments together go through a to form an ordered state Curie-Weiss Law Ferromagnetism Only Magnetism as an “order parameter” mentioned Magnons briefly Domains, irreversibility • (Kittel - parts of Ch 11-12 ) Physics 460 F 2006 Lect 24 33 Physics 460 F 2006 Lect 24 34

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