Specific Heat in Solids (Debye’s Model)

At low temperatures the phonon

heat capacity is proportional to T3.

o The Debye model assumes that atoms in materials move in a collective fashion,
described by quantized normal modes with a dispersion relation, 𝜔 = vs∣Ԧ𝐤∣.
o The total energy and heat capacity are obtained by integrating the contribution of
the individual modes over k-space.
1
Anharmonic Effects

In the harmonic approximation:

o Phonons do not interact with each other; they do

not change with time
o The heat capacity becomes T independent for
T > 𝜃𝐷
o There is no thermal expansion of solids.
o Thermal conductivity of solids is infinite

𝑈 𝑥 = 𝑈ℎ𝑎𝑟𝑚 𝑥 + 𝑈𝑎𝑛ℎ𝑎𝑟𝑚 𝑥 = 𝑐𝑥 2 − 𝑔𝑥 3 − 𝑓𝑥 4

Thermal expansion is an example to the anharmonic effect.

In anharmonic effect phonons collide with each other and these collisions limit thermal
conductivity which is due to the flow of phonons.

2
Anharmonic Effects (Phonon-phonon coupling)

𝑈 𝑥 = 𝑈ℎ𝑎𝑟𝑚 𝑥 + 𝑈𝑎𝑛ℎ𝑎𝑟𝑚 𝑥 = 𝑐𝑥 2 − 𝑔𝑥 3 − 𝑓𝑥 4

3
Origin of magnetic moments

4
Space quantization of atoms (Larmor’s precession)

𝑒
𝝎L = 2𝑚
𝐵

5
Quantization of Orbital Angular Momentum

Number of possible orientations of L in a magnetic field is (2ℓ + 1), which

means orbital angular momentum is space quantized
6
Electron Spin
“Every electron has an intrinsic angular momentum, called spin, whose
magnitude is the same for all electrons. Associated with this angular
momentum is a magnetic moment.”

The quantum number s describes the spin angular momentum of the

electron. The only value s can have, s = 1/2
7
Electron spin and the Stern-Gerlach experiment

Classically, all orientations should be present in a beam of Ag atoms. Instead, the

initial beam split into two distinct parts that correspond to the two opposite spin
orientations in the magnetic field permitted by space quantization
8
Vector model of atom

9
 Vector model of atom (L-S coupling)

o In the presence of magnetic field B, J precesses about the direction of B while L and
S continue precessing about J.
o The precession of J about B gives rise to the anomalous Zeeman effect, since
different orientations of J involve slightly different energies in the presence of B.
10
Atomic magnetic moments
Orbital magnetic moment of the electron:
Orbital angular momentum

e
 orb =− L
2me
An orbiting electron is equivalent to a magnetic dipole moment morb.

Spin magnetic moment of the electron:

Intrinsic angular momentum
e
spin =− S
me
The spin magnetic moment processes about an external
magnetic field along z and has an average value of mz along z.

11
Atomic magnetic moments

𝑒ℏ
The quantity 𝜇B = 2𝑚𝑒
is called the Bohr magneton and has the value 9.27
× 10-24 A m2 or J T-1.

12
Magnetizing Field or Magnetic Field Intensity H

Magnetic field in a magnetized medium

B = B o + o M
Definition of the magnetizing field

1 1
H= B−M Or H= Bo
o o

13
Magnetic Permeability and magnetic susceptibility

Magnetic field
B
Definition of magnetic permeability =
H
Magnetizing field
B B
Definition of relative permeability r = =
Bo  o H

Definition of magnetic susceptibility M = mH

Relative permeability and susceptibility r = 1 +  m

14
Classification of magnetic materials

15
Diamagnetism vs. Paramagnetism

16
Superconductors: Perfect diamagnets

A magnet over a superconductor becomes levitated. The superconductor is a

perfect diamagnet which means that there can be no magnetic field inside the
superconductor.

17
Temperature-dependent magnetism

18
 Diamagnetism

o A diamagnetic material placed in a

non-uniform magnetic field
experiences a force towards smaller
fields. This repels the diamagnetic
material away from a permanent
magnet.

o In a superconductor or in an electron
orbit within an atom, the induced
current persists as long as the field is
present. The magnetic field of the
induced current is opposite to the
applied field, and the magnetic
moment associated with the current is
a diamagnetic moment.

19
Langevin theory of diamagnetism
𝑒𝐵
Larmor frequency: 𝜔= 2𝑚
1 𝑒𝐵
I = Charge × revolutions per unit time = (–Ze) 2𝜋
× 2𝑚
𝑍𝑒 2 𝐵
𝜇= − 4𝑚 𝜌2 , 𝜌2 = 𝑥 2 + 𝑦 2
(Mean square distance from field axis)

𝑟2 = 𝑥2 + 𝑦2 + 𝑧2 (Mean square distance from nucleus)

3
For spherically symmetrical distribution, 𝑥 2 = 𝑦 2 = 𝑧 2 , so that 𝑟 2 = 𝜌2
2

𝜇0 𝑁𝜇 𝜇0 𝑁𝑍𝑒 2 2
𝜒= =− 𝑟
𝐵 6𝑚
20
Quantum theory of diamagnetism

1
Ԧ
𝐵 = ∇ × 𝐴, Ԧ
𝐴 = 2 𝐵 × 𝑟Ԧ

21
Paramagnetism

o In a paramagnetic material each individual atom possesses a permanent magnetic moment

due to thermal agitation, there is no average moment per atom and M = 0.
o In the presence of an applied field, individual magnetic moments take alignments along the
applied field and M is finite and along B.
o A paramagnetic material placed in a non-uniform magnetic field experiences a force towards
greater fields. This attracts the paramagnetic material (e.g. liquid oxygen) towards a
permanent magnet.
22
Quantum theory of Paramagnetism
The magnetic moment of an atom or ion in free space is given by

𝛾 = gyromagnetic ratio, g = spectroscopic splitting factor

For an electron spin, g = 2. For a free atom, the g factor is given by the Landé
equation

1
For a single spin with no orbital moment, mJ = ± and
2
g = 2, and therefore, U = ±𝜇BB.
23
Quantum theory of Paramagnetism

In a magnetic field an atom with angular momentum quantum number J has 2J + 1

equally spaced energy levels.

24
Quantum theory of Paramagnetism

BJ (x) = Brillouin function

Describes the dependency of magnetization

M on the applied magnetic field B

25
Quantum theory of Paramagnetism

Curie Law

BJ (x) = Brillouin function

26
Determination of J (Hund’s rule)
The Hund rules as applied to electrons in a given shell of an atom affirm that electrons
will occupy orbitals in such a way that the ground state is characterized by the
following:

1. The maximum value of the total spin S allowed by the Pauli’s exclusion principle;
2. The maximum value of the orbital angular momentum L consistent with this value
of S
3. The value of the total angular momentum J is equal to |L − S| when the shell is
less than half full and to L + S when more than half full. When the shell is just half
full, the application of the first rule gives L = 0, so that J = S.

o The first Hund rule has its origin in the exclusion principle and the coulomb
repulsion between electrons.
o The second Hund rule is best approached by model calculations.
o The third Hund rule is a consequence of the sign of the spin-orbit interaction: For a
single electron the energy is lowest when the spin is antiparallel to the orbital
angular momentum.

Example: The ion Ce3+ has a single f electron with L = 3 and S = 1/2 The J value by
the preceding rule is |L−S| = L−1/2 = 5/2.
27
Ferromagnetism

TC = Curie temperature
For Fe, TC = 1043 K

In a magnetized region of a ferromagnetic material such as iron, all the magnetic

moments are spontaneously aligned in the same direction. There is a strong
magnetization vector M even in the absence of an applied field.
28
Quantum theory of Ferromagnetism
In the mean-field approximation, each magnetic atom experiences a field
proportional to the magnetization, 𝐵𝐸 = 𝜆𝑀 (Weiss field or Exchange field)
For Iron, 𝐵𝐸 ≈ 1000 T

𝑀 = 𝜒𝑝 (𝐵𝑎 + 𝐵𝐸 )
𝐶
The paramagnetic susceptibility, 𝜒𝑝 = (Curie law)
𝑇

𝑀 𝐶
MT = C (𝐵𝑎 + 𝜆𝑀) ⇒ 𝜒 = =
𝐵𝑎 𝑇−𝑐𝜆

𝐶
𝜒= , 𝑇𝐶 = 𝑐𝜆 (Curie-Weiss law)
𝑇−𝑇𝐶

29
Origin of ferromagnetism (Weiss’s theory)

Fe: [Ar]3d64s2

The isolated Fe atom has 4

unpaired spins and a spin
magnetic moment of 4 Bohr
magneton
Exchange energy
(Heisenberg Model):

Eex = – 2Je 𝑆1 ∙ 𝑆2
Hund’s rule for an atom with many electrons is based on the
exchange interaction.
30
Quantum Mechanical Exchange: J

31
The exchange force

32
Origin of ferromagnetism

33
Origin of ferromagnetism (Heisenberg Picture)

Eex = – 2Je 𝑆1 ∙ 𝑆2
r = interatomic distance
rd = radius of the d-orbit (or the
average d-subshell radius)

For Gd, the x-axis is r/rf where rf is

the radius of the f-orbit.

o Exchange interaction is electrostatic (strong) and non-magnetic in nature

o For majority of the solids, Je is negative, so Eex is negative if 𝑆1 and 𝑆2 are
antiparallel (as we find in covalent bonding): Antiferromagnetic state
o For Fe, Co and Ni, Je is positive, Eex is then negative if 𝑆1 and 𝑆2 are parallel:
Ferromagnetic state

34
Magnetic Domains

(a) Magnetized bar of ferromagnet in which there is only one domain and hence an external
magnetic field.
(b) Formation of two domains with opposite magnetizations reduces the magnetostatic energy.
There are, however, field lines at the ends.
(c) Domains of closure fitting at the ends eliminates the external fields at the ends.
(d) A specimen with several domains and closure domains. There is not external Magnetic field
and the specimen appear unmagnetized.
35
Magnetocrystalline anisotropy

There is an energy in a ferromagnetic

crystal which directs the magnetization
along certain crystallographic axes called
directions of easy magnetization. This
energy is called the magnetocrystalline
or anisotropy energy.

Magnetocrystalline anisotropy in a
single iron crystal. M vs. H depends on
the crystal direction and is easiest along
[100] and hardest along [111]

36
Magnetic Domains

o In a Bloch wall, the neighboring spin magnetic moments rotate gradually and
takes several hundred atomic spacings to rotate the magnetic moment by 180°.
o Potential energy of a Bloch wall depends on its thickness 𝛿

 2 Eex
U wall  + K
2a
37
Magnetic Domains

o Schematic illustration of magnetic domains in the grains of an unmagnetized

polycrystalline iron sample. Very small grains have single domains.
o Potential energy of a domain wall depends on the exchange and anisotropy
energies

38
Magnetic Hysteresis behavior
M vs. H behavior of a unmagnetized
polycrystalline iron specimen. An
example grain in the unmagnetized
specimen is that at O.

Soft and hard magnetic materials

39
Magnetic Domains

(a) An unmagnetized crystal of iron in the absence of an applied magnetic field. Domains A and
B are the same size and have opposite magnetizations.
(b) When an external field is applied the domain wall migrates into domain B which enlarges A
and B. The result is that the specimen now acquires net magnetization.

40
Antiferromagnetism

In this antiferromagnetic BCC crystal

(Cr) the magnetic moment of the
center atom is cancelled by the
magnetic moments of the corner
atoms (an eighth of the corner atom
belongs to the unit cell).

41
Superconductivity
Kammerlingh Onnes (1913 Nobel prize in Physics)

A superconductor such as Pb
undergoes a transition to zero
resistivity at a critical temperature
Tc (7.2 K for Pb) whereas a
normal conductor such as Ag
does not and exhibits residual
resistivity at the lowest
temperatures.

o As low temperatures, the scattering of electrons by atoms vibrating about their

lattice position decreases.
o However, the contributions of defects and impurities to resistivity remain non-zero
at low temperatures.
42
Superconductivity progress

43
Penetration depth and coherence length

o The penetration of magnetic field

inside the superconductor results in 𝑥
surface currents, known as 𝐻𝑒 𝑥 = 𝐻0 𝑥 exp −
supercurrents 𝜆𝐿
o The penetration depth λL is the
characteristic depth of the
supercurrents on the surface of the
material.
o The density of superconducting
electrons ns decreases to zero near a
superconducting /normal interface,
with a characteristic length ξ
(coherence length).
o The coherence length is proportional
to the mean free path of conduction
electrons; e.g. for pure metals it is
quite large, but for alloys (and
ceramics…) it is often very small.

44
The Meissner effect

A superconductor cooled below its critical temperature expels all magnetic field lines
from the bulk by setting up a surface current. A perfect conductor (σ = ∞) shows no
Meissner effect.

45
Superconductivity

A magnet over a superconductor becomes levitated. The superconductor is a

perfect diamagnet which means that there can be no magnetic field inside the
superconductor.

46
Type I superconductors

The critical field vs. temperature in Type I superconductors.

47
Type I and Type II superconductors

Characteristics of Type I and Type II superconductors. B = µoH is the applied field

and M is the overall magnetization of the sample. Field inside the sample, Binside =
µoH + µoM, which is zero only for B < Bc (Type I) and B < Bc1 (Type II).
48
Type II superconductors

The critical surface for a Nb3-Sn alloy which

The mixed or vortex state in a is a Type II superconductor.
Type II superconductor.

49
Type I and Type II superconductors

o Pure specimens of many materials exhibit this behavior. The values of Hc are always too low
for type I superconductors to have application in coils for superconducting magnets.

o Type II or hard superconductors, usually alloys, have superconducting electrical properties

up to a field denoted by Hc2. Between the lower critical field Hc1 and the upper critical field
Hc2 the flux density B ≠ 0 and the superconductor is threaded by flux lines and is said to be
in the vortex state.

50
Thermodynamic Properties: Entropy

The entropy is lower in the superconducting state because the electrons

are more ordered here than in the normal state.
51
Thermodynamic Properties: Specific Heat

Δ
𝐶𝑒𝑠 = exp −
𝑘𝐵 𝑇

𝐶𝑛 = 𝛾𝑇 + 𝛽𝑇 3

52
Thermodynamics: Energy Gap

o Electrons in excited states above the gap behave as normal electrons in rf fields:
they cause resistance; at dc they are shorted out by the superconducting electrons.
The gap is typically, 𝐸𝑔 ~ 10−4 𝜖𝐹
o The temperature dependence of electronic specific heat indicates the existence of
an energy gap Eg in a superconductor
𝐸𝑔 = 2∆
For Ga, 𝐸𝑔 = 2 × 1.4 𝑘𝐵 𝑇𝑐 ~ 10−4 eV

53
BCS Theory of Superconductivity
A pictorial and intuitive view of an indirect attraction between two oppositely
traveling electrons via a lattice distortion and vibration.

A cooper pair is a bound pair of electrons formed by the interaction between

electrons through lattice distortion (phonon).
54
BCS Theory of Superconductivity

o Electrons are modelled as non-interacting, ballistic quasiparticles and

the Coulomb interaction is neglected

o Formation of Cooper pairs is caused by electron–phonon interactions at

low temperatures.

o Cooper pairs follow Bose–Einstein statistics, which enables the pairs to

condense into a stable ground state below the Fermi level and become
phase coherent.

o The balance between electron-phonon interaction forces and Coulomb

(electrostatic) forces determines whether the material is superconducting

o The two electrons which pair up have opposite momentum and their
correlation persists over lengths as large as 10-6 m. The binding energy
of a Cooper pair, called the energy gap Eg, is of order 10-3 eV

55
High TC Superconductivity (Phase Diagram)

56
High TC Superconductivity (Qualitative Picture)

o Phonons are not the main players: In high TC superconductors,

phonons have little to no role; instead, spin-density waves dominate the
pairing mechanism.
o Proximity to magnetism: High- TC materials exist near magnetic
transitions (e.g., to antiferromagnetism) and exhibit strong spin-density
wave fluctuations, much like conventional superconductors rely on strong
phonon interactions.
o Mechanism of pairing: As an electron moves, its spin disturbs the local
spin environment, creating a spin-density depression that attracts
another electron.
o Temperature dependence: Upon lowering the temperature, more spin-
density waves and Cooper pairs form, eventually leading to
superconductivity.
o Coulomb repulsion and d-wave pairing: Strong Coulomb repulsion
prevents electron pairing on the same lattice site. Instead, pairing occurs
between nearest neighbours, resulting in d-wave symmetry, where the
superconducting gap has nodes (zero amplitude) at specific points.

57
High TC Superconductivity (Other Mechanisms)

o Spin Fluctuation Mediated Pairing (Unconventional Mechanism):

Antiferromagnetic spin fluctuations act as the "glue" for Cooper pairing.

o Strong Electronic Correlation/Mott Physics: Superconductivity

emerges from a doped Mott insulator with strong on-site Coulomb
repulsion.

o RVB (Resonating Valence Bond) Theory: Electrons from neighbouring

atoms can form valence bonds to form pairs that can be doped to form a
stable superconducting ground state.

o Multiband Superconductivity with Interband Pairing: Multiple Fermi

surfaces interact, and pairing can occur between different bands with
sign-changing gap functions (like s± symmetry).

58
Superconductivity

Superconducting electromagnets used on MRI. Operates with liquid He,

providing a magnetic field 0.5–1.5 T.

59