Magnetism and Magnetic MaterialsMagnetism and Magnetic Materials

DTU (10313) – 10 ECTSDTU (10313) – 10 ECTSKU – 7.5 ECTSKU – 7.5 ECTS

Sub-atomic – pm-nm Mesoscale – nm-m Macro – m-mm

Module 1 – 01/02/2001 – Introduction

ContextContext

Magnetism is a physical phenomenon that intrigued scientists and laymen alike for centuries.

Some materials attract or repel each other, depending on their orientation.

Experimentally, it became soon clear that magnetism was related to the motion of charges.

But how exactly? And why? Classical physics gives us a basic

framework, but doesn’t help us much in developing a coherent and comprehensive bottom-up picture

The advent of QM provided some answers Relativity provided some additional answers How far can we go? Can we understand

what “magnetism” is, and how a magnet works? Yes we can…

Magnetism is a physical phenomenon that intrigued scientists and laymen alike for centuries.

Some materials attract or repel each other, depending on their orientation.

Experimentally, it became soon clear that magnetism was related to the motion of charges.

But how exactly? And why? Classical physics gives us a basic

framework, but doesn’t help us much in developing a coherent and comprehensive bottom-up picture

The advent of QM provided some answers Relativity provided some additional answers How far can we go? Can we understand

what “magnetism” is, and how a magnet works? Yes we can…

Magnetism:A tangible macroscopic

manifestation of the quantum world

Meet your teachersMeet your teachers

KL CF MB MFH BMA JBH LTK JOB

TimelineTimeline

Tuesday, Feb 1 13:00-16:45Introduction MB Friday, Feb 4 08:15-12:00Isolated magnetic moments MB Tuesday, Feb 8 13:00-16:45Crystal fields MB Friday, Feb 11 08:15-12:00Interactions MB

Tuesday, Feb 15 13:00-16:45Magnetic order MB

The basics

Friday, Feb 18 08:15-12:00Micromagnetism I MB Tuesday, Feb 22 13:00-16:45Micromagnetism II MB Friday, Feb 25 08:15-12:00Macroscopic magnets MB

The mesoscale

Friday, Mar 11 08:15-12:00Bulk measurements/dynamics KL Tuesday, Mar 15 13:00-16:45 Nanoparticles I CF

Friday, Mar 18 08:15-12:00 Nanoparticles II CF

Tuesday, Mar 22 13:00-16:45Magnetization measurements MFHHard disks MFH

Friday, Mar 25 08:15-12:00Imaging and characterization CF

Advanced topics

Tuesday, Mar 29 13:00-16:45Thermodynamics LTKGMR and spintronics JBH Friday, Apr 1 08:15-12:00Magnetism in metals BMA

Advanced topics

30/3 or 4/4Oral exam (KU students) 4/4-9/5Oral exam (DTU students)

26/5 or 27/5Oral exam (DTU students)

Evaluation

Experimental methodsand applications

Tuesday, Mar 8 13:00-16:45Order and broken symmetry KL

Feb 18 A1 Mar 11 A2Mar 26 A3May 9 PW (DTU only)

Assigments deadlines

WorkloadWorkload

Your homework will be:• Go through what you have learned in each Module,

and be prepared to present a “Flashback” at the beginning of the next Module

• Carry out home-assignments (3 of them)• Self-study the additional reading material given

throughout the course

Your group-work will be:• Follow classroom exercise sessions with Jonas• DTU only: project work

Your final exam will be:• Evaluation of the 3 home-assignments• Oral exam• DTU only: evaluation of the written report on the

project work

This course will be successful if…This course will be successful if…

•Macroscopic magnets, how they work (MB)• In depth (QM) explanation of bound currents (ODJ)• I know why some things are magnetic (JJ)• Know more about magnetic monopoles (ODJ)• Lorentz transformations of B and E (MB)• • .• .• .• .• .• .

Students’ feedback to be gathered in the classroom – 01/02/2011

Intended Learning Outcomes (ILO)Intended Learning Outcomes (ILO)

(for today’s module)(for today’s module)

1. Describe the logic and structure of this course, and what will be learned2. List the electron’s characteristics: charge, mass, spin, magnetic moment3. Predict the main features of electron motion in presence of an applied field4. Calculate the expression and values of Larmor and cyclotron frequencies5. Define the canonical momentum, and explain its usefulness6. Describe the connections between magnetism and i) QM, ii) Relativity7. Write down simple spin Hamiltonians, and solve them in simple cases8. Manipulate consistently spin states (spinors) with spin operators

Meet the electronMeet the electron

Mass: me=9.10938215(45) 10-31 KgCharge: e=-1.602176487(40) 10-19 CSpin: 1/2Magnetic moment: ~1 B

Size: <10-22 m (from scattering)Classical radius: 2.8 fm (little meaning)

Calculate the classical electron radius

Electron in motion and magnetic momentElectron in motion and magnetic moment

L

v

I

S €

=I dS∫

€

I = −ev

2πR

€

=γL = −lμ Bˆ L

€

γ=− e2me

μ B =eh

2me

Calculate the classical electron velocity for some hypothetical l=1 state with R=a0.

PrecessionPrecession

Since magnetic moment is linked with angular momentum…

Einstein-De Haas

Barnett

coil

Ferromagnetic rod

Calculate the Larmor precession frequency

B, z

€

E = −μ ⋅B

€

τ =×B =dLdt

€

dμdt

= γμ × B

Electron motion in applied fieldElectron motion in applied field

The Lorentz force

Calculate the cyclotron frequency

B, z€

F = q(E + v × B)

y

x

-x

yz

Left or right?

B

More in general: canonical momentumMore in general: canonical momentum

€

F = q(E + v × B)

€

E = −∇V −∂tA

€

F = q −∇V −∂ tA + v × ∇× A( )[ ] = mdvdt

To account for the influence of a magnetic field in the motion of a point charge, we “just” need to replace the momentum with the canonical momentum in the Hamiltonian

€

v × ∇× A( ) =∇(v ⋅A) − (v ⋅∇)A

€

ddt

mv + qA( ) = −q∇ V − v ⋅A( )

€

p = mv + qA

€

p → p − qA

€

p → −ih∇ − qA

Classical Quantum mechanical

€

T =pi

2

2mi

∑ →pi + eA i (ri )[ ]

2

2mi

∑

Connection with Quantum MechanicsConnection with Quantum Mechanics

€

F = qv × B A classical system of charges at thermal equilibrium has no net magnetization.

The Bohr-van Leeuwen theorem

“It is interesting to realize that essentially everything that we find in our studies of magnetism is a pure quantum effect. We may be wondering where is the point where the h=/=0 makes itself felt; after all, the classical and quantum Hamiltonians look exactly the same! It can be shown […] that the appearance of a finite equilibrium value of M can be traced back to the fact that p and A do not commute. Another essential ingredient is the electron spin, which is a purely quantum phenomenon.”

P. Fazekas, Lecture notes on electron correlation and magnetism

Perpendicular to velocity

€

F ⋅dl∫ = F ⋅vdt = q (v × B) ⋅vdt =∫∫ 0 No work

No work = no change in energy = no magnetization

€

H =pi + eA i (ri )[ ]

2

2mi

∑ + other terms

€

Z = dridpi exp −βH ({ri ,pi})[ ]i

∏∫

€

M = −∂F∂B

⎛ ⎝ ⎜

⎞ ⎠ ⎟T ,V

= −1β

∂ log Z∂B

⎛ ⎝ ⎜

⎞ ⎠ ⎟T ,V

Z independent of B, ergo M=0€

F = −1β

log Z

Connection with RelativityConnection with Relativity

B and E, two sides of the same coin. No surprise we always talk about Electromagnetism as a single branch of physics.

From: M. Fowler’s website, U. Virginia

(a)

(b) €

F =qμ 0λv2

2πrˆ z (a)

€

F = 0 ??(b)

Hint: Lorentz contraction

Restore relativity and show that the force experienced in (a) and (b) is the same, although in (b) the force is electric

Quantum mechanics of spinQuantum mechanics of spin

Quantum numbers: n,l,ml,s,ms

Orbital angular momentum: l,ml;l(l+1) is the eigenvalue of L2 (in hbar units)ml is the projection of L along an axis of choice (e.g. Lz)The resulting magnetic moment is 2=l(l+1)B and z=-mlB

Spin angular momentum: s,ms;s(s+1) is the eigenvalue of S2 (in hbar units)ms is the projection of S along an axis of choice (i.e. Sz)The resulting magnetic moment is 2=gs(s+1)B and z=-gmsB

Zeeman splitting: E=gmsBB (remember the Stern-Gerlach experiment)

The g-factor (with a value very close to 2) is one difference between oam and spinAnother difference is that l can only be integer, while s may be half-integerAlso, spin obeys a rather unique algebra (spinors instead of “normal” vectors)Other than that, they behave similarly.

But there are consequences… [exercise on EdH effect]

Pauli matrices and spin operatorsPauli matrices and spin operators

€

ˆ σ x =0 1

1 0

⎛

⎝ ⎜

⎞

⎠ ⎟ ˆ σ y =

0 −i

i 0

⎛

⎝ ⎜

⎞

⎠ ⎟ ˆ σ z =

1 0

0 −1

⎛

⎝ ⎜

⎞

⎠ ⎟

€

ˆ S = ˆ S x , ˆ S y, ˆ S z( ) =12

ˆ σ =12

ˆ σ x , ˆ σ y, ˆ σ z( )

s=1/2

Generic spin state

€

ˆ S 2 s,ms = s(s +1) s, ms

€

ˆ S z s,ms = ms s, ms

€

ˆ S ± s,ms = s(s +1) − ms (ms ±1) s,ms ±1

€

ˆ S i , ˆ S j[ ] = iε ijkˆ S k ˆ S 2 , ˆ S i[ ] = 0

€

ˆ S ± = ˆ S x ± i ˆ S y ˆ S + , ˆ S −[ ] = 2 ˆ S z

Commutators

Ladder operators€

ψ =a

b

⎛

⎝ ⎜

⎞

⎠ ⎟= a↑ +b↓

€

→ = 12

1

1

⎛

⎝ ⎜

⎞

⎠ ⎟

€

← = 12

1

−1

⎛

⎝ ⎜

⎞

⎠ ⎟

Stern GerlachStern Gerlach

What is the final state?Will the final beam split?

The simplest spin Hamiltonian: coupling of two spinsThe simplest spin Hamiltonian: coupling of two spins

€

H = A ˆ S a ⋅ ˆ S b

€

ˆ S tot = ˆ S a + ˆ S b

€

ˆ S tot( )

2= ˆ S a( )

2+ ˆ S b( )

2+ 2 ˆ S a ⋅ ˆ S b

Combining two s=1/2 particles gives an entity with s=0 or s=1.

The total S2 eigenvalue is then 0 or 2.

Hence, the energy levels are:

€

A ˆ S a ⋅ ˆ S b =A2

ˆ S tot( )

2− ˆ S a( )

2− ˆ S b( )

2

[ ] =A4 s = 0

− 3A4 s =1

⎧ ⎨ ⎩

Possible basis:

Consider symmetry of wave function for Fermions

Eigenstates: triplet and singlet€

↑↑, ↑↓ , ↓↑ , ↓↓

€

↑↑,↑↓ + ↓↑

2,↑↓ − ↓↑

2, ↓↓

Sneak peekSneak peek

Diamagnetism

Paramagnetism

Hund’s rules

Wrapping upWrapping up

Next lecture: Friday February 4, 8:15, KU

Isolated magnetic moments (MB)

•Magnetic moment•Electron motion under an applied field•Precession of magnetic moments•Magnetism as a quantum-relativistic phenomenon•Einstein-de Haas effect•Orbital and spin angular momentum•Spin behaves strangely•Stern-Gerlach•Coupling of spins