Gavin W Morley
Department of Physics
University of Warwick
Diamond Science & Technology
Centre for Doctoral Training, MSc course
Module 2 – Properties and Characterization of Materials
Module 2 – (PX904)
Lectures 15 & 16 – Magnetic properties and characterization
2
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Overview
Lecture
15 & 16
Magnetic properties of materials
- Paramagnetism
- Diamagnetism
- Ferromagnetism
Magnetic characterization
- SQUID magnetometry
- Neutron scattering
- Magnetic resonance
- Electron paramagnetic resonance
- Nuclear magnetic resonance
3
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetism
Current loop has
magnetic moment, µ = I A
Area, A
Maxwell 4:
 × B = μ0 J
Current, I
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
4
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetism
Current loop has
magnetic moment, µ = I A
Area, A
Current, I
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
5
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
What happens if we try to cut a
magnet in half?
Area, A
Current, I
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
6
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
What happens if we try to cut a
magnet in half?
a)
b)
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
7
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetism
Magnetic
Monopoles
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
8
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Angular momentum and Magnetism
Current loop has
magnetic moment, µ = I A
µ=γL
(L is angular momentum,
γ is gyromagnetic ratio)
Area, A
Current, I
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
9
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
What happens if you put a magnetic
moment into a uniform magnetic field?
a)
b)
c)
d)
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
It moves
It lines up
It precesses
Hey, I thought
you were
supposed to
be teaching
me
10
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
What happens if you put a magnetic
moment into a uniform magnetic field?
a)
b)
c)
d)
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
It moves
It lines up
It precesses
Hey, I thought
you were
supposed to
be teaching
me
11
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Precession of a magnetic moment
Energy of the magnetic moment in a magnetic field, B:
E=-µ•B
Area, A
Joseph Larmor
(1857 – 1942)
Current, I
Larmor precession frequency = γ B
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
12
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnitude of magnetic moment
Electron has charge, e and mass, m, so
Current: I = -e/t
as speed, v = 2 π r/t
for radius, r.
v
e-
Magnetic moment,
μ = I A = I π r 2 = - e ℏ / 2m
r
≡ -μB
(as electron angular
momentum = mvr = ℏ)
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
13
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetization, M:
Magnetic moment per unit volume in a solid
In vacuum:
B = µ0 H
permeability of free space,
µ0 = 4π × 10-7 Hm-1
In a magnetic solid: B = µ0 (H + M)
For a linear material, M = χ H
for susceptibility, χ
So then
B = µ0 (1+ χ )H = µ0 µr H
For relative permeability, µr
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
14
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetization, M:
Magnetic moment per unit volume in a solid
In vacuum:
B = µ0 H
permeability of free space,
µ0 = 4π × 10-7 Hm-1
In a magnetic solid: B = µ0 (H + M)
For a linear material, M = χ H
for susceptibility, χ
So then
B = µ0 (1+ χ )H = µ0 µr H
For relative permeability, µr
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
15
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetization, M:
Magnetic moment per unit volume in a solid
In vacuum:
B = µ0 H
permeability of free space,
µ0 = 4π × 10-7 Hm-1
In a magnetic solid: B = µ0 (H + M)
For a linear material, M = χ H
So then
for susceptibility, χ
B = µ0 (1+ χ )H
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
16
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Relative permeability, µr
In vacuum:
B = µ0 H
of free space,
µr= 1+ permeability
χ
µ0 = 4π × 10-7 Hm-1
In a magnetic solid: B = µ0 (H + M)
For a linear material, M = χ H
So then
for susceptibility, χ
B = µ0 (1+ χ )H = µ0 µr H
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
17
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetization, M:
Magnetic moment per unit volume in a solid
In vacuum:
B = µ0 H
permeability of free space,
µ0 = 4π × 10-7 Hm-1
In a magnetic solid: B = µ0 (H + M)
For a linear material, M = χ H
So then
for susceptibility, χ
B = µ0 (1+ χ )H
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
18
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Susceptibility, χ
In vacuum:
B = µ0 H
from
permeability of freeTable
space,
& Laby
µ0 = 4π × 10-7 Hm-1Kaye
www.kayelaby.npl.co.uk
In a magnetic solid: B = µ0 (H + M)
For a linear material, M = χ H
for susceptibility, χ
So then
B = µ0 (1+ χ )H = µ0 µr H
For relative permeability, µr
Chapter 1, Blundell, Magnetism in
Condensed Matter, OUP 2001
19
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetism of an atom
1. From the electrons
1. Spin angular momentum
2. Orbital angular momentum
3. An applied magnetic field can change their
orbital angular momentum
2. From the nuclei
1. Spin angular momentum
20
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
A beam of atoms hits a screen
Classical
prediction
21
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Stern-Gerlach experiment
Classical
prediction
S
N
See also:
http://commons.wikimedia.org/w/index.php?title=
File%3AQuantum_spin_and_the_SternGerlach_experiment.ogv
22
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetism of an atom
1. From the electrons
1. Spin angular momentum
2. Orbital angular momentum
3. An applied magnetic field can change their
orbital angular momentum
2. From the nuclei
1. Spin angular momentum
23
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Solve Schrödinger’s equation
for an electron in a box:
→ Discrete energy levels
Erwin
Schrödinger
(1887 – 1961)
Page 240, Eisberg and Resnick,
Quantum Physics of Atoms, Molecules,
Solids, Nuclei, and Particles, Wiley 1985
24
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Solve Schrödinger’s equation for electron in
Coulomb potential and include spin
n
1
l
0
0
1
0
ml
0
0
-1,0,+1
0
+½,-½
+½,-½
+½,-½
+½,-½
ms
2
+½,-½ +½,-½
3
1
2
-1,0,+1 -2,-1,0,+1,+2
Number of
degenerate
eignenfunctions
for each l
2
2
6
2
6
10
Subshell name
1s
2s
2p
3s
3p
3d
Page 241, Eisberg and Resnick,
Quantum Physics of Atoms, Molecules,
Solids, Nuclei, and Particles, Wiley 1985
25
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Atomic orbitals
1s
2s
2px
2py
2pz
26
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetism of an atom
1. From the electrons
1. Spin angular momentum
2. Orbital angular momentum
3. An applied magnetic field can change their
orbital angular momentum
2. From the nuclei
1. Spin angular momentum
27
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Sample magnetization
1. From the electrons
1. Spin
2. Orbital
Magnetic Field
Paramagnetic
For spin ½,
Magnetization is
M = Ms tanh(µBB/kBT)
28
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Sample magnetization
1. From the electrons
1. Spin
2. Orbital
Magnetic Field
Paramagnetic
Paramagnetic susceptibility
follows the Curie Law:
χ = CCurie/T
29
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Page 20, Blundell, Magnetism in
Condensed Matter, OUP 2001
30
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Atomic orbitals
1s
2s
2px
2py
2pz
31
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Page 20, Blundell, Magnetism in
Condensed Matter, OUP 2001
32
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Conduction electrons have
“Pauli paramagnetism”
(Chapter 7 of Blundell’s book)
Fermi-Dirac distribution function, Page 9,
Singleton, Band Theory and Electronic Properties
of Solids, OUP 2001
33
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetism of an atom
1. From the electrons
1. Spin angular momentum
2. Orbital angular momentum
3. An applied magnetic field can change their
orbital angular momentum
2. From the nuclei
1. Spin angular momentum
34
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Sample magnetization
From change in orbital
angular momentum-
Diamagnetic
Paramagnetic
- From spin and orbital angular momentum
Magnetic Field
35
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Sample magnetization
From change in orbital
angular momentum -
Diamagnetic
Paramagnetic
- From spin and
orbital angular momentum
Magnetic Field
36
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Page 20, Blundell, Magnetism in
Condensed Matter, OUP 2001
37
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetism of an atom
1. From the electrons
1. Spin angular momentum
2. Orbital angular momentum
3. An applied magnetic field can change their
orbital angular momentum
2. From the nuclei
1. Spin angular momentum
38
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Interactions → Ferromagnetism
1. From the electrons
1. Spin angular momentum
2. Orbital angular momentum
3. An applied magnetic field can change their
orbital angular momentum
Ferromagnet in zero
2. From the nuclei
applied magnet field
1. Spin angular momentum
( J > 0 ):
39
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Sample magnetization
Diamagnetic
Paramagnetic
Ferromagnetic
Magnetic Field
40
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Sample magnetization
Saturation magnetization
Remanent magnetization
Magnetic Field
Coercive field
Ferromagnetic
41
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Ferromagnetic domains
Page 131, Blundell, Magnetism in
Condensed Matter, OUP 2001
42
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Interactions → Antiferromagnetism
1. From the electrons
1. Spin angular momentum
2. Orbital angular momentum
3. An applied magnetic field can change their
orbital angular momentum
Antiferromagnet in
2. From the nuclei
zero applied magnet
1. Spin angular momentum
field ( J < 0 ):
43
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Diamond Superconductivity
→ perfect diamagnetism
In vacuum:
B = µ0 H
permeability of free space,
µ0 = 4π × 10-7 Hm-1
In a magnetic solid: B = µ0 (H + M)
For a linear material, M = χ H
So then
for susceptibility, χ
B = µ0 (1+ χ )H = µ0 µr H
Page 202, Singleton, Band Theory and Electronic
Properties of Solids, OUP 2001
44
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Boron-doped Diamond: Superconductivity
E Bustarret et al, Dependence of the
Superconducting Transition Temperature
on the Doping Level in Single-Crystalline
Diamond Films, Physical Review Letters,
93, 237005 (2004)
45
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Diamond Magnetic characterization
In a magnetic solid: B = µ0 (H + M)
For a linear material, M = χ H
Measure magnetization, M which could be a function of
temperature, magnetic field, orientation etc.
46
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Diamond Magnetic characterization
In a magnetic solid: B = µ0 (H + M)
For a linear material, M = χ H
Measure magnetization, M which could be a function of
temperature, magnetic field, orientation etc.
Extraction magnetometer:
V=
>0
V
47
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Diamond Magnetic characterization
Vibrating sample
magnetometer (VSM):
VV
> 00
ac =
V
48
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
SQUID magnetometer
Vibrating sample
magnetometer (VSM)
with SQUID detection:
VV
> 00
ac =
Bias
current
V
SQUID = superconducting
quantum interference device
49
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
SQUID magnetometer
Vibrating sample
magnetometer (VSM)
with SQUID detection
in an applied magnetic field
→ susceptibility
VV
> 00
ac =
Bias
current
V
M = χ H for susceptibility χ
50
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Neutron Scattering
Analogous to X-ray diffraction with neutrons instead of X-rays.
Neutrons have no charge but spin ½
Page 104, Blundell, Magnetism in
Condensed Matter, OUP 2001
51
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Neutron Scattering
Analogous to X-ray diffraction with neutrons instead of X-rays.
Neutrons have no charge but spin ½
Page 106, Blundell, Magnetism in
Condensed Matter, OUP 2001
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetic resonance
Energy of the magnetic moment in a magnetic field, B:
E=-µ•B
Energy
of a spin
0
Magnetic field, B
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Magnetic resonance
Energy of the magnetic moment in a magnetic field, B:
E=-µ•B
Energy
of a spin
0
Photon energy = h f
Magnetic field, B
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Electron paramagnetic resonance
…NMR for electrons
The crucial difference is that the electron magnetic
moment is 660 times larger than that of a proton
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Electron paramagnetic resonance
Bridge
source
detector
Circulator
Modulation coils
Main
magnet
S
Microwave resonator
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Pulsed magnetic resonance
Felix Bloch (19051983)
Photo courtesy Stanford
News Service
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Rotating frame
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Rotating frame
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Spin echo
In rotating frame
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Spin echo
In rotating frame
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Nuclear magnetic resonance
Probe nuclear
paramagnetism
Main
magnet
with a
vertical
field
S
RF coil provides a horizontal
magnetic field which is oscillating
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Nuclear Magnetic Resonance
NMR periodic table from
Philip Grandinetti
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Diamond NMR
99% of carbon is 12C with
zero nuclear spin. 1% is 13C
with nuclear spin I = ½
L. H. Merwin, C. E. Johnson
and W. A. Weimer, 13C NMR
investigation of CVD
diamond: Correlation of NMR
and Raman spectral
linewidths, Journal of
Materials Research 9, 631
(1994).
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Pure 13C Diamond NMR
K. Lefmann et al.,
NMR spectra of pure
13C diamond, Physical
Review B 50, 15623
(1994).
Module 2 – Properties and Characterization of Materials
- Lectures 15 & 16 – Magnetic properties & characterization
Lecture Summary
Lecture
15 & 16
Magnetic properties of materials
- Paramagnetism
- Diamagnetism
- Ferromagnetism
Magnetic characterization
- SQUID magnetometry
- Neutron scattering
- Magnetic resonance
- Electron paramagnetic resonance
- Nuclear magnetic resonance
Module 2 – Properties and Characterization of Materials
- Summary
Lectures 15 & 16 – Magnetic properties & characterization
Module Summary
Module 2 – Properties and Characterization of Materials
- Summary
Lectures 15 & 16 – Magnetic properties & characterization
Lectures
Lecturer
1-3
Philip Martineau
Crystallography
4-6
Gavin Morley
Electronic properties
7-8
Stephen Lynch
Optical
9
Gavin Morley
Electronic characterization
10
Richard Beanland
Electron microscopy
11-12
Claire Dancer
Mechanical
13-14
Martin Kuball
Thermal
15-16
Gavin Morley
Magnetic