1
Course Teacher
Dr. Muhammad Anisuzzaman Talukder
Professor
Department of EEE, BUET
Contact Info
Email: anis@eee.buet.ac.bd
Website: https://anis.buet.ac.bd/
Phone: 55167100 Ext. 6582
Office: ECE 527
2
1
Course Website
Please check
https://anis.buet.ac.bd/properties_of_materials.html/
for EEE 307 lecture notes, grades, and updates.
3
4
2
Crystal Structure
5
Electrical and Thermal Conductivity
6
3
Quantum Mechanics
7
Band Theory
8
4
Theory of Metals
• Quantum theory of metal
• Free electron model
• Conduction in metals
9
Dielectric
Properties
10
5
Magnetic Properties
Magnetic moment, permeability
Ferromagnetism and magnetic domains
11
Superconductivity
•
Zero resistance and Meissner effect
•
Type I and type II superconductors
12
6
Metamaterial
13
Time Line
• Crystal structures – 1 week
• Electrical and thermal conduction – 2 weeks
• Quantum mechanics – 2 weeks
• Band theory – 2 weeks
• Theory of metals – 1 week
• Dielectric properties – 2 week
• Magnetic properties – 2 week
• Superconductivity – 1 week
• Metamaterial – 1 week
14
7
Continuous Assessment
• Four class tests at roughly regular intervals.
• If the policy changes, you will be notified.
15
Text Book
16
8
Good News!
• Materials covered will be helpful for
− In-depth understanding of
behavior of many many devices
− Research
− Graduate studies
17
Bad News!
Be Serious About Class Tests
18
9
No Phones, No Laptops in Class
19
First Assignment
• Email to
anis@eee.buet.ac.bd with
─ Subject: EEE 307
─ Body: “Your Name, Student Number” <email address>
• It will help me to keep you posted on the course updates.
20
10
1
General Bonding Principle
• Net force, FN = FA + FR , at equilibrium:
• Potential Energy E(r) is related as
=
=
+
= 0.
( )
2
1
Covalently Bonded Solid: Diamond
• Due to the strong Coulombic attraction between the shared electrons and the
positive nuclei, the covalent bond energy is usually the highest for all bond
types, leading to very high melting temperatures and very hard solids: diamond
is one of the hardest known materials.
C-C bond: Diamond
C-H bond: Methane
3
Metallic Bonding
The free electrons inside a metal can drift
under the action of the applied field and give
rise to a current. These free electrons in the
metallic bond are called conduction electrons.
• Metals are good electrical conductors
• Metals are also good thermal
conductors
Silver
© McGraw-Hill Education/Mark Dierker
(mhhe022468.jpg)
Copper
© McGraw-Hill Education/Stephen
Frisch
(MHHE005312.JPG)
Aluminum (98.5%)
© McGraw-Hill Education/Ken Cavanagh
(MHED9000878.JPG)
4
2
Crystalline Solids
• A crystalline solid is a solid object in which atoms bond with each other in a
regular pattern to form a periodic collection or array of atoms.
• Most important property of a crystal is periodicity leading to long-range order
(location of each atom is well-known by virtue of periodicity).
• Examples of periodic array/crystalline solids: nearly all metals, many ceramics,
semiconductors, various polymers.
5
Crystalline Solids
• Crystal = Lattice + Basis
• Lattice  an infinite periodic array of geometric points in space (without
atoms)
• Basis  an identical group of atoms
• By putting the basis at each lattice point, we obtain the actual crystal.
• Thus the crystal becomes essentially a periodic repetition of a small volume,
called the unit cell.
• The length of the cubic unit cell is called lattice constant.
• The repetition of the unit cell in 3D will generation the whole crystal.
6
3
Lattice and Basis
7
Packing of Solids
• Crystals can be
• Face centered cubic, FCC
• Body centered cubic, BCC
• Hexagonal close packed, HCP
8
4
BCC
• Atom in the corner and one atom at the center
• In one BCC unit cell, there is one atom in the center, 1/8th atom in 8 corners 
Total of 2 atoms in BCC unit cell
• Coordination number: 8
• Example: Fe, Cr, Mo, W
• Close packed structure with a packing density of 68%
9
FCC
• Atoms in the corners and in the centers of the faces
• In one FCC unit cell, 1/8th atom in 8 corners and ½ atom in each of 6planes  Total of 4 atoms in FCC unit cell
• Coordination number: 12
• Examples: Ag, Au, Cu, Pt
• Close packed structure with a packing density of 74%.
10
5
HCP
• On top of layer a, we can place an identical layer b, with the spheres taking
up the voids on layer a. The third layer can be placed on top of b and lined
up with layer a.
• The stacking sequence is abab …
• Number of atoms per unit cell: 6, 2
• Coordination number: 12
• Examples: Many metals  Co, Mg, Ti, Zn
11
Atomic Packing Factor
• Atomic packing factor (APF), packing efficiency or packing fraction is
the fraction of volume in a crystal that is occupied by constituent particles 
a dimensionless quantity and always less than unity.
• APF is determined by assuming that atoms are rigid spheres. Radius of spheres is
taken to be the maximum (atoms do not overlap). For crystals with only one type
of particle, the APF is
APF =
NAPF
particle=: number of particles in the unit cell; Vparticle : volume of each particle;
and Vunit cell : volume occupied by the unit cell.
• For one-component structures, the maximum APF is about 0.74  close-packed
structures. For multiple-component structures, the APF can exceed 0.74.
• APF explains many properties of materials. Metals with a high APF will have a
higher "workability" (malleability or ductility).
12
6
APF of BCC
APF =
• Simple cubic:
=2
APF =
4
3
(2 )
1∙
=
6
≈ 0.5236
• Body centered cubic:
=
APF =
4
3
2∙
4
3
4
3
=
3
8
≈ 0.6802
13
APF of FCC
APF =
=2 2
APF =
4∙
4
3
2 2
=
18
≈ 0.7405
a
a
14
7
APF of HCP
APF =
=?
=?
=
3 3
2
=?
c =?
=2
15
APF of HCP
=?
16
8
APF of HCP
APF =
=?
=
3 3
2
=2
=
2
∙4
3
4
3
APF =
3 3
2
6∙
=
=
18
6∙
3 3
2
2
4
3
2
∙4
3
≈ 0.7405
17
9
1
Diamond Cubic Crystal Structure
• Covalently bonded solids have diamond structure due to the directional nature
of the covalent bonds  tetrahedral configuration.
• Examples: Si and Ge
• If we place 2 Si-atoms at each site of an FCC unit cell appropriately, one right
at the lattice point, and the other displaced from it by a/4 along the cube edges,
we can generate the diamond cell  8 atoms per unit cell
2
1
Zinc-Blende Structure
• In Zinc-Blende structure (e.g. ZnS, GaAs), there are binary compounds involved.
• Thus it looks like a diamond cubic, but Zn and S-atoms alternating positions
• Examples: ZnS, AlAs, GaAs, GaP, GaSb, InAs, InP, InSb, ZnTe.
3
APF of Diamond Cubic
• Neighboring atoms are shifted by
• Radius of atoms
AFP =
8∙
=
2
4
3
=
=
≈ 0.34
8∙
4
3
= 3
= 3
4
8
3
8
4
2
Ionic Solid: NaCl
• The crystal structure depends on the relative charge and size per ion.
• Na+ ions are about half of Cl- ions, which results 6 nearest neighbors.
• The crystal can be described as two interpenetrating FCC unit cells,
each having oppositely charged ions at the corners and face centers.
5
Ionic Solid: CsCl
• When the cations and anions have equal charges and are about the same
size, as in the CsCl crystal, the unit cell is called the CsCl structure.
• Each cation is surrounded by eight anions (and vice versa), which are at the
corners of a cube.
• This is not a true BCC unit cell because the atoms at various BCC lattice
points are different.
6
3
Crystal Directions
• Crystal properties – such as elastic modulus, electrical resistivity, magnetic
susceptibility – are different in different directions and planes.
• Therefore, we need to specify directions and planes in a crystal.
• Crystal unit cell geometry  parallelepiped with sides a, b, c and angles ,  ,  known as crystal parameters.
• For BCC and FCC, = = , = =
= ≠ , = = 90° , and = 120° .
= 90° , and cubic symmetry. For HCP,
7
Finding Directions
• A point P on the vector can be expressed by the projections
point P onto the , , and axes.
•
• If
,
and
,
and
,
and
from
can be expressed in terms of lattice parameters a, b, and c.
are
, ,
, then P is at
,
,
 ½, 1, ½.
• These numbers are converted to the smallest integers 
= 121 .
• If any integer is a negative number, we use a bar on the top of that integer.
8
4
Important Directions
• Some directions are equivalent since the coordinate system xyz is arbitrary.
• Directions [100] and [010] are equivalent.
9
Family of Directions
• A set of directions considered to be equivalent is called a family of
directions.
100 = [100], [010], [001], [100], [010], [001]
•
Family of 111 directions
10
5
1
Crystal Planes
• We need to describe a particular plane of a crystal  Miller indices.
• Take any plane, note the x, y, z intercepts in terms of lattice parameters a, b, c.
• Invert the numbers – you get the Miller indices  (hkl) = (a-1b-1c-1)
• A bar is used for a negative integer due to a negative intercept.
• [hkl] direction is always perpendicular to the (hkl) plane.
2
1
Miller Indices
3
Family of Planes
• Family of planes is represented by curly braces as {100}: (100), (010),
(001), 100 , 010 , (001)
4
2
Family of Planes
• Family of planes {110}: (110), (101), (011), 110 , 101 , (011),
110 , 101 , (011), 110 , 101 , (011)
5
Equivalent Planes
• Planes can have the same (hkl) only if they are separated by a
multiple of lattice parameters.
• (010) plane is not identical to the (020) plane, even though they
are geometrically parallel
6
3
Planar Concentration
•
Frequently, we need to know the number of atoms per unit area (planar
concentration) on a given plane (hkl).
•
For example, if the surface concentration of atoms is high on one plane,
then that plane may encourage more rapid oxide growth.
7
Planar Concentration
Among the {100}, {110}, and {111} planes in FCC crystals which one is the
most densely packed?
8
4
FCC: (100) Plane
Consider the Cu FCC crystal with a = 0.3620 nm
Area,
=
Number of atoms in
= (4 corners) × ( atom) + 1 atom at face center = 2
Planar concentration
=
of (100) is
=
.
×
= 15.3 atoms nm-2
9
FCC: (110) Plane
Area,
= ( )(
2)
Number of atoms in = (4 corners) × ( atom) + (2 face diagonals)× ( atom
at diagonal center) = 2
Planar concentration
=
of (110) is
=
.
×
= 10.8 atoms nm-2
10
5
FCC: (111) Plane
(
) =?
2
2
Area,
= (
)(
2)
Number of atoms in = (3 corners) × ( atom) + (3 face diagonals)× ( atom
at diagonal center) = 2
Planar concentration
=
of (111) is
/
=
×
.
×
= 17.6 atoms nm-2
11
Class Test – 1
Day: 15 May 2019
Syllabus: Lectures 2-4
12
6
1
REVISED SCHEDULE
Class Test – 1
Day: 15 May 2019
Syllabus: Lectures 2-4
2
1
Electrical Conduction
•
Electrical conduction involves the motion of charges in a material under
the influence of an applied electric field.
•
Metals  the valence electrons from form a sea of electrons that are free
to move within the metal  conduction electrons.
Drude model describes electrical conduction in solids.
•
• The Drude model of electrical conduction was proposed
in 1900 by Paul Drude to explain the transport
properties of electrons in materials (especially metals).
• Drude model assumes the microscopic behavior of
electrons in a solid classically and looks much like
a pinball machine, with a sea of constantly jittering
electrons bouncing and re-bouncing off heavier,
relatively immobile positive ions.
3
The Drude Model
•
The electric current density:
=
∆
∆
q: net quantity of charge flowing through an
area A in time t.
•
Conduction electrons move around randomly in the metal  no net flow of
charge.
•
When an electric field Ex is applied, conduction electrons acquire a net
velocity in the x direction.
4
2
Drift Velocity
•
The average velocity of electrons in the x direction or the drift velocity, vdx:
=
1
+
+
+ ⋯+
vxi : x direction velocity of the
ith electron
N : number of conduction
electrons
•
Assume n = N∕V  number of electrons per unit volume
•
In time Δt, electrons move a distance Δx = vdx Δt, and Δq crossing A is enA Δx.
•
The current density in the x direction:
•
Time-dependent current:
=
=
∆
=
∆
∆
∆
=
( )
5
Random Motion of Conduction Electrons
• The kinetic energy originates from the electrostatic interaction of these
electrons with the positive metal ions and also with each other.
•
Conduction electrons move about randomly (with a mean speed u) being
frequently and randomly scattered by thermal vibrations of the atoms.
•
In the absence of an applied field there is no net drift in any direction.
6
3
Under Applied Electric Field
•
Conduction electrons experience a force of eEx in the opposite direction of Ex.
•
A net drift along the x direction is superimposed on the random motion of the
electron.
•
The electron accelerates along the x direction under the action of the force eEx,
and then it suddenly collides with a vibrating atom and loses the gained
velocity  there is an average velocity in the x direction
7
Drift Velocity
Let uxi be the velocity of electron
i in the x direction just after the
collision (initial velocity). Since
eEx/me is the acceleration of the
electron, the velocity vxi in the x
direction at time t will be
=
=
+
1
( − )
+
=
+
+ ⋯+
( − )
( − ) : average free time between collisions.
8
4
Mean Free Time
−
≡
: Mean free time, mean time between collisions, or mean scattering time
=
•
τ is directly related to the microscopic processes that cause the scattering of
the electrons in the metal — lattice vibrations, crystal imperfections, and
impurities, to name a few.
•
1∕τ represents the mean frequency of collisions or scattering events. During a
small time interval δt, the probability of scattering will be δt∕τ.
9
Drift Mobility
=
: drift mobility
=
•
represents the ease of electron conduction under an electric
field.
• If the electron is not highly scattered, then the mean free time
between collisions will be long, τ will be large, and
will also
be large; the electrons will therefore be highly mobile and be
able to “respond” to the field.
10
5
Ohm’s Law and Conductivity
=
• Using the expression for drift velocity vdx :
• Ohm’s Law:
•
=
=
=
 Conductivity
• A large
does not necessarily imply high conductivity,
because σ also depends on the concentration of conduction
electrons n.
11
6
1
Mean Free Time
•
Conduction electrons are scattered from the thermal vibrations of the atoms.
•
= 1  One scatterer.
•
The mean free time between collisions:
1
=
• u  slightly depends on temperature.
• The temperature dependence of
essentially arises from =
.
∝
1
2
1
Temperature Dependence of Resistivity
•
•
Thermal vibrations of atoms can be considered to be simple harmonic
motion, like a mass M attached to a spring.
K. E.
=
K. E.
=
1
1
2
=
1
1
2
1
4
Kinetic molecular theory  K. E.
1
4
≈
∝
1
cos
∝
1
2
1
1
2
∝


=
=
3
Lattice-Scattering-Limited Conductivity
=
So, the resistivity
of a pure metal is
=
=
=
1
=
1
=
 lattice-scattering-limited conductivity
4
2
Matthiessen’s Rule
=
•
 works well with pure metals
 Fails for metallic alloys.
Strained region by impurity exerts a
scattering force = − ( )/
5
Effective Mean Free Time
• Two different types of scattering processes: (1) from impurities alone and
(2) from thermal vibrations alone.
• Two types of mean free times between collisions: (1) T for scattering from
thermal vibrations only, and (2) I for scattering from impurities only.
1
=
1
1
=
+
1
1
+
  is smaller than both T and I.
1
: drift mobility
: lattice-scattering-limited mobility
: impurity-scattering-limited mobility
=
1
=
1
+
1
=
+
6
3
Residual Resistivity
There may also be electrons scattering from dislocations and other crystal
defects, as well as from grain boundaries. All of these scattering processes add to
the resistivity of a metal, just as the scattering process from impurities. We can
therefore write the effective resistivity of a metal as
=
+
∶ residual resistivity due to the scattering of electrons by impurities,
dislocations, interstitial atoms, vacancies, grain boundaries, etc.
Residual resistivity shows very little temperature dependence.
ρ ≈ AT + B
where A and B are temperature-independent constants.
7
Temperature Coefficient
•
The temperature coefficient of resistivity (TCR) α0 is defined as the fractional
change in the resistivity per unit temperature increase at the reference
temperature T0
1
=
•
T0  usually 273 K (0 °C) or 293 K (20 °C)
δρ = ρ − ρ0
δT = T − T0
when α0 is constant over the temperature range of interest
ρ = ρ0[1 + α0 (T − T0)]
•
For metals:
=
1
 For T0 = 273 K,
= 1/273.
8
4
0 at 273 K
Metal
Aluminum, Al
1/233
Antimony, Sb
1/196
Copper, Cu
1/232
Gold, Au
1/251
Indium, In
1/196
Platinum, Pt
1/255
Silver, Ag
1/244
Tantalum, Ta
1/294
Tin, Sn
1/217
Tungsten, W
1/202
Iron, Fe
1/152
Nickel, Ni
1/125
9
Resistivity vs. Temperature
•
ρ ∝ T is approximately obeyed
except for the magnetic materials.
•
For alloys, such as nichrome (NiCr), ρ is relatively temperature
insensitive, with a very small TCR.
•
Empirical relation between  and
T for pure metals:
=
• For nonmagnetic metals, n  1,
whereas it is closer to 2 than 1 for
the magnetic metals Fe and Ni.
10
5
Resistivity vs. Temperature
ρ = AT + B
≲
•
The number of atoms that vibrate
with sufficient energy to scatter
the conduction electrons starts to
decrease rapidly with decreasing
temperature  becomes more
strongly temperature dependent

•
•
K:
=
∝
+
At T  0,  is limited by
scattering from impurities and
crystal defects.
11
6
1
Effect of Alloying on Resistivity
=
1
=
1
+
1
=
+
In a binary alloy that forms a solid solution, we would expect the above equation
to apply, with the temperature-independent impurity contribution ρI increasing
with the concentration of solute atoms. This means that as the alloy concentration
increases, the resistivity ρ increases and becomes less temperature dependent as
ρI overwhelms ρT, leading to α ≪ 1∕273.
Resistivity at 20
oC (n m)
 at 20 oC
(1/K)
Nickel
69
0.0064
Chrome
129
0.0030
Nichrome
(80%Ni-20% Cr)
1100
0.0004
Material
2
1
Effect of Alloying on Resistivity
•
 of Cu–Ni alloy as a function of Ni content (at.%) at room temperature.
• Nichrome is widely used as a heater wire in household appliances and
industrial furnaces
3
Nordheim’s Rule
•
Relates the impurity resistivity ρI to the atomic fraction X of solute atoms in a
solid solution, as follows:
=
(1 − )
where C is the constant termed the Nordheim coefficient, which represents
the effectiveness of the solute atom in increasing the resistivity.
• For sufficiently small amounts of impurity, experiments show that the increase
in the resistivity ρI is nearly always simply proportional to the impurity
concentration X, that is, ρI ∝ X.
4
2
Heterogenous Mixture
•
Consider a material with two distinct phases α and β stacked in layers.
•
Effective resistivity (Reff) for current flow in the x direction?
•
 and  layers are in parallel  series rule of mixtures  resistivity
mixture rule
=
+
=
=
+
=
=
5
Heterogenous Mixture
•
Reff in the y-direction?
•
 and  layers are in parallel  parallel rule of mixtures  conductivity
mixture rule.
=
+
6
3
Heterogenous Mixture
• Reff for a random mixture of phase α and phase β?
• Series mixture rule  when c and d are not markedly different
• If the resistivity of one phase is appreciably different 
• Empirical relations
7
HF Resistance of a Conductor
8
4
Skin Depth
• For a given conductor, we can assume that most of the current flows in
a surface region of depth δ, called the skin depth.
• In the central region, the current will be negligibly small.
• We can imagine the central conductor as a resistance R in series with an
inductance L.
=
2
9
AC Resistance
• Effective cross-sectional area:
• AC resistance:
•
The skin effect limits the use of
solid-core conductors in highfrequency applications 
frequencies > 109 Hz range, the
transmission of the signal over a
long distance becomes almost
impossible through an ordinary,
solid metal conductor. We must
then resort to pipes (or
waveguides).
10
5
1
Hall Effect
• Voltmeter gives a reading VH when a transverse magnetic field Bz is applied.
• Lorentz force is applied on the electrons in the downward direction.
• Hall Field in the –y direction.
• In steady-state:
=
=
=
1
2
1
Hall Coefficient
•
Hall coefficient (RH) measures
the Hall field, along y, per unit
transverse applied current and
magnetic field.
=
•
For metals:
=−
1
=−
1
3
Hall Effect Wattmeter
∝
•
=
=
∝
∝
The voltmeter can be
calibrated to read
directly the power
dissipated in the load.
4
2
Hall Effect in Semiconductors
•
•
Both positive and negative charge carriers  holes and electrons.
Remember relationships between drift velocity, mobility and force.
=
•
=
Both electrons and holes experience a Lorentz force in the same direction.
5
Open-Circuit
=
+
=
+
=0
=−
•
Either the electron or hole drift velocity must be reversed from its usual
direction  holes drifting in the opposite direction.
6
3
Force Equations
• Holes:
• Electrons:
=
−
=
−
=
−
−
=
+
=
+
=
+
7
RH
−
=−
−
=
+
=
=
=
=
,
(
−
+
+
−
+
)
8
4
1
Thermal Conduction
•
The transport of heat in a metal is accomplished by the electron gas (conduction
electrons), whereas in nonmetals, the conduction is due to lattice vibrations.
2
1
Thermal Conductivity
• The thermal conductivity measures the ease with which heat, that is, thermal
energy, can be transported through the medium.
=−
•
κ is a material-dependent constant
of proportionality that we call the
thermal conductivity.
•
Negative sign ® heat flow direction
is that of decreasing temperature.
3
 Versus 
=−
=−
•
In metals, electrons
participate in the processes
of charge and heat transport
® σ, κ.
•
σ and κ are related by the
Wiedemann–Franz–Lorenz
law, which is
=
CWFL = π2k2∕3e2 = 2.44 × 10−8 W Ω K−2 is a constant called the Lorenz
number (or the Wiedemann–Franz–Lorenz coefficient).
4
2
Temperature Dependence of 
•
≳
K: κ is constant,
because heat conduction depends
essentially on the rate at which
the electron transfers energy from
one atomic vibration to another
as it collides with them.
•
This rate of energy transfer
depends on the mean speed of the
electron u, which increases only
fractionally with the temperature.
5
Heat Conduction in Non-metals
•
Nonmetals do not have free conduction electrons ® the energy transfer involves
atomic vibrations of the crystal.
•
The efficiency of heat transfer depends not only on the nature of interatomic
bonding, but also on how the vibrational waves propagate and are scattered.
•
The stronger the bonding, the greater will be the thermal conductivity.
•
Diamond has an exceptionally strong covalent bond ® κ ≈ 1000 W m−1 K−1.
•
Polymers have weak secondary bonding ® κ < 1 W m−1 K−1.
6
3
Thermal Resistance
•
•
The rate of heat flow:
=
Δ
=
Δ
•
In analogy with electrical circuits:
•
Thermal resistance:
Ohm’s law:
=
=
Δ
=
Δ
Δ
=
7
8
4
Wave-Particle Duality
•
Light: Interference and diffraction phenomena displayed by light can only be
explained by treating light as an electromagnetic wave. But light can also
exhibit particle-like properties in which it behaves as if it were a stream of
discrete entities (Photons), each carrying a linear momentum and each
interacting discretely with electrons in matter (just like a particle colliding
with another particle).
•
Electron: Classically, electrons are considered to be a particle, and hence,
they obey Newton’s second law (F = ma). However, they can also exhibit
wave-like properties quite contrary to our intuition. An electron beam can give
rise to diffraction patterns and interference fringes, just like a light wave.
9
Light as a Wave
•
A ray of light is considered to be an electromagnetic (EM) wave with a given
frequency.
• The electric and magnetic fields (Ey and Bz) of this wave are perpendicular to
each other and to the direction of propagation x.
,
=
sin(
−
)
10
5
Young’s Double-Slit Experiment
•
For constructive interference at point P, the path difference between the two
rays is an integer multiple of the wavelength λ
−
=
• For destructive interference, the two rays have a path difference of λ∕2
−
=
+
1
2
11
X-Ray Diffraction
•
Bragg’s Law: The condition for existence of a diffracted beam:
2 sin =
,
= 1, 2, 3, …
12
6
Photoelectric Effect
•
When the cathode is illuminated with light, if the frequency f of the light is
greater than a certain critical value f0, the ammeter registers a current I.
•
Applying a positive voltage to the anode helps to collect more of the
electrons and thus increases the current, until it saturates because all the
photoemitted electrons have been collected.
13
Photoelectric Current versus Voltage
When = − ® just
“extinguishes” I ® potential energy
“gained” by the electron is just the
kinetic energy lost by the electron.
=
1
2
= KE
v : Velocity
KEm : Kinetic energy of the
electron just after photoemission.
14
7
Kinetic Energy versus Frequency
KE
•
•
=ℎ −ℎ
h : slope of the straight line and is
independent of the type of metal;
Planck’s constant ® 6.610−34 J s.
f0 : depends on the electrode
material for the photocathode
15
8
1
Kinetic Energy versus Frequency
KE
•
•
=ℎ −ℎ
h : slope of the straight line and is
independent of the type of metal;
Planck’s constant  6.610−34 J s.
f0 : depends on the electrode
material for the photocathode
2
1
Interpretation of Photoelectric Effect
KE
=ℎ −ℎ
•
•
h : Planck’s constant  6.610−34 J s.
f0 : minimum frequency for emission
•
The successful interpretation of the
photoelectric effect was first given in
1905 by Einstein, who proposed that
light consists of “energy packets,” each
of which has the magnitude hf. We can
call these energy quanta photons.
•
Einstein was awarded the Nobel
Prize in 1921 for "his discovery of the
law of the photoelectric effect",
3
Emission due to Photons
•
Photoemission only occurs
when hf is greater than .
•
Some of the photon energy
hf goes toward overcoming
PE barrier. The energy that
is left ℎ − Φ gives the
electron its KE.
4
2
Meaning of Light Intensity
1
2
• Photon flux density (Γph): Number of photons crossing a unit area per unit time.
Δ
Γ =
Δ
•
=
Classically,
• Light intensity (I) : Product of photon flux density and the energy per photon,
=Γ ℎ
5
Black Body Radiation
Color of a black body from 800 K to 12200 K. This range of colors approximates
the range of colors of stars of different temperatures, as seen or photographed in
the night sky.
Temperature of lava flow can
be approximated by emitted
color.
6
3
Black Body Radiation
An object that absorbs all radiation falling on it, at all wavelengths, is called a
black body. When a black body is at a uniform temperature, its emission has a
characteristic frequency distribution that depends on the temperature. Its emission
is called black-body radiation.
Planck’s black body radiation formula:
7
8
4
De Broglie Relationship
Young’s double-slit experiment with an electron beam.
9
Electron Diffraction Ring
•
Electrons obey the Bragg diffraction condition 2 sin =
•
Since we know the interatomic spacing d and we can measure the angle of
diffraction 2θ, we can readily evaluate the wavelength λ associated with the
wave-like behavior of the electrons.
•
Wavelength of an object:
=
.
ℎ
10
5
Wave-Particle Duality of Nature
•
Usain Bolt weighs 94 kg and he runs at a
velocity of 12.42 m s−1. What is his
wavelength?
ℎ
=
•
=
6.63 × 10
= 56.79 × 10
94 × 12.42
m
Fizz throws a 160 gm cricket ball at a speed of
40 m s-1. What is the wavelength of the ball?
=
ℎ
=
6.63 × 10
Js
= 1.04 × 10
160 × 10 × 40
m
11
Wave-Particle Duality of Nature
•
Electron accelerated by 100 V. Wavelength?
KE =
=
ℎ
2
=
= 0.123 nm
 Since this is comparable to typical interatomic distances in solids
 We would see a diffraction pattern when an electron beam strikes a crystal!
12
6
1
1
Young’s Double Slit Experiments
Light
Intensity
Electron
Probability
2
2
1
Copenhagen Interpretation of Quantum Mechanics
 A system is completely described by a wave function ψ, representing an
observer's subjective knowledge of the system.
 The description of nature is essentially probabilistic, with the probability of
an event related to the square of the amplitude of the wave function related
to it.
 It is not possible to know the value of all the properties of the system at the
same time; those properties that are not known with precision must be
described by probabilities. (Heisenberg's uncertainty principle)
 Matter exhibits a wave–particle duality. An experiment can show the
particle-like properties of matter, or the wave-like properties; in some
experiments both of these complementary viewpoints must be invoked to
explain the results.
 Measuring devices are essentially classical devices, and measure only
classical properties such as position and momentum.
 The quantum mechanical description of large systems will closely
approximate the classical description.
3
3
Schrodinger’s Cat
“ It is typical of these cases that an
indeterminacy originally restricted to the
atomic domain becomes transformed
into macroscopic indeterminacy, which
can then be resolved by direct
observation. That prevents us from so
naively accepting as valid a "blurred
model" for representing reality. In itself,
it would not embody anything unclear or
contradictory. There is a difference
between a shaky or out-of-focus
photograph and a snapshot of clouds and
fog banks.”
-Erwin Schrodinger, 1935
4
4
2
Heisenberg’s Uncertainty Principle
•
In the world of very small particles, one cannot measure any property of a
particle without interacting with it in some way
•
This introduces an unavoidable uncertainty into the result ® One can
never measure all the properties exactly
uncertainty
in momentum
uncertainty
in position
•
The more accurately you know the position (i.e., the
smaller Δx is), the less accurately you know the
momentum (i.e., the larger Δp is); and vice versa
Werner Heisenberg (1901-1976)
5
5
Electronic Wavefunctions
Ψ( , )
•
A complex function of position x and time t.
•
Ψ( , , , ) : probability of finding the electron per unit volume at x, y, z at
time t.
•
If the potential energy of electron is time independent:
Ψ ,
•
( ) and
/
=
exp(−
)
must be continuous and single-valued.
6
6
3
Atomic & Molecular Wavefunctions
• Let’s assume the case
of hydrogen atoms.
Continuum of free
electron states
+e
-e
r
energy of -e
n=3
n=2
Let’s represent the
atom in space by its
Coulomb potential
centered on the proton
(+e):
+e
-e
Continuum of free
electron states
+e
r
energy of -e
Superposition of Coulomb
potentials H2:
n=1
+e
7
7
Time-Independent SchrÖdinger Equation
−
ℏ
2
+
+
+
2
ℏ
+
=
( − ) =0
+
2
ℏ
−
=0
•
total energy = kinetic energy + potential
energy
•
In classical mechanics,
•
V depends on the system
=
+ .
– e.g., gravitational potential energy,
electric potential energy
Erwin Schrödinger (1887–1961)
8
8
4
The Free Electron
=0 ®
•
+
=
exp
=0
,
or
Quantum mechanics: KE =
=
Classical physics: KE =
=
exp(−
(ℏ )
2
=
)
ℎ
2
The probability distribution for the electron is constant over the entire space.
( )
•
=0
Solution:
+
•
2
ℏ
=
exp(
)
=
Uncertainties in position and momentum?
9
9
Infinite Potential Well
•
ψ = 0 when x ≤ 0 and x ≥ a, and ψ
is determined by the Schrödinger
equation in 0 < x < a with V = 0.
•
Schrödinger equation:
+
2
ℏ
=0
=
•
Solution:
•
Boundary condition:
•
From S. E.:
−2
exp
+
exp(−
0 = 0 at
(sin
)
=0
=
exp
)+
2
ℏ
− exp −
2
sin
=2
=0 ®
=
sin
ℏ
2
10
10
5
Infinite Potential Well
•
=2
•
Solutions:
sin
=0
=2
=
sin
=
ℏ (
2
)
=
ℎ
8
11
11
Normalization
•
=?
• Normalization: The total probability of finding the electron in 0 < x < a is
unity.
12
12
6
Class Test – 2
Day: 8 July 2019
Syllabus: Lectures 9-11
13
13
7
1
1
Finite Potential Well
•
A finite potential energy well
has zero potential energy (V =
0) inside the well (0 ≤ x ≤ a) but
a finite potential energy (V =
V0) outside the well (x < 0 and
x > a).
2
2
1
Classically Allowed Region
>
•
:
d 2 2me ( E  V )

 0
dx 2
2
•
Solution:
 ( x )  Aeikx  Be  ikx
 ( x)  A sin kx  B cos kx
k
•
2me ( E  V )
2
Note: In the classically allowed region, we have oscillating solutions.
3
3
Classically Forbidden Region
•
<
:
•
We can divide the problem into
three regions I, II, and III
•
Region II:
•
Schrodinger equation:
=
2
ℏ
+
•
=0
Solution:
=
•
=0
exp
+
exp(−
)
Constants B1 and B2 can be found from boundary conditions.
4
4
2
Classically Forbidden Region
•
Regions I and III:
•
For
=
•
≤
and
2
( − )
ℏ
=0
Solution:
=
=
•
≥ :
Schrodinger equation:
−
•
=
≤
exp
exp
:
+
+
= 0 and
exp −
exp(−
≥ :
)
=0
5
5
Bound Solutions
 A1e x

 ( x)   B1 cos kx  B2 sin kx
  x
C 2 e

2me V0  E 
2
k
2mE
2
for x  0
for 0  x  a
for x  a
6
6
3
Symmetric Wavefunctions
 A1e x

 ( x)   B1 cos kx
  x
C 2 e

2me V0  E 
2
k
2mE
2
for x  0
for 0  x  a
for x  a
7
7
Symmetric Wavefunction
At x = a:
 (a)  B1 cos(ka)  C2e  a
 (a)  kB1 sin(ka)  C2 e a
Dividing the second equation by the first gives
tan  ka    / k

tan 
2
V0  E
E 
 
EL 
E
EL : infinite well ground state energy
8
8
4
Anti-symmetric Wavefunctions
 A1e x

 ( x)   B2 sin kx
  x
C 2 e

2me V0  E 
2
k
2mE
2
for x  0
for 0  x  a
for x  a
9
9
Antisymmetric Wavefunction
At x = a:
 (a)  B2 sin(ka)  C2e  a
 (a)  kB2 cos(ka)  C2 a e a
Dividing the second equation by the first gives
cot  ka    / k

 cot 
2
V0  E
E 
 
EL 
E
EL : infinite well ground state energy
10
10
5
Solutions
There is always at least one bound solution no matter how
shallow is the well!
11
11
(a) Energy solutions
(b) Wavefunction solutions
E/EL
(x)
Solutions
x
x
Note that The higher the energy, the lower the effective confining
potential, and the greater the penetration into the barriers.
12
12
6
1
Bound Solutions
 A1e x

 ( x)   B1 cos kx  B2 sin kx
  x
C 2 e
•

2me V0  E 
2
k
2mE
2
for x  0
for 0  x  a
for x  a
Penetration depth ( ): The
quantity 1/ is a measure of the
extent of penetration of the
electron into the barrier.
2
1
Penetration Depth
2
=
=
( − )
ℏ
1
= 2 nm,
= 0.5 eV
= 0.057 eV
= 0.22 eV
= 0.45 eV
= 0.29 nm
= 0.37 nm
= 0.87 nm
3
Classical vs. Quantum Mechanical
4
2
Electronic Wavefunctions
•
We can divide the electron’s space into three regions, I, II, and III and solve the
Schrödinger equation for each region.
=
exp
=
exp(
= exp
=
+ exp −
) + exp −
+ exp −
•
Region I:
exp −

incident wave in +x,
exp
 reflected wave
in –x direction.
•
Region III:
= 0  no
reflected wave.
=
2
ℏ
2
( − )
ℏ
5
Tunneling Coefficient
•
A1, A2, B1, B2, and C1 are determined by applying the boundary conditions and
the normalization condition.
•
Transmission coefficient (T): The relative probability that the electron will
tunnel from region I through II to III.
=
=
( )
(incident)
4 (
=
=
1+
1
sinh (
)
− )
6
3
Tunneling Coefficient
• For a wide or high barrier, using αa ≫ 1 and sinh(αa) ≈ exp(αa)/2,
=
exp(−2
)
=
16 (
− )
• Reflection coefficient (R):
=
•
=1−
The wider or higher the
potential barrier, the
smaller the chance of the
electron tunneling.
7
Tunneling
• The wavefunction decays
exponentially as we move away
from the surface because the PE
outside the metal is Vo and the
energy of the electron, E < Vo.
• If we bring a second metal close
to the first metal, then the
wavefunction can penetrate into
the second metal  electron
tunneling.
8
4
Scanning Tunneling Microscope
STM was invented by Gerd
Binning (right) and Heinrich
Rohrer (left) at the IBM
Research Laboratory in Zurich,
for which they were awarded
the 1986 Nobel prize.
9
STM Images
•
STIM image of graphite
surface: Contours represent
electron concentrations
within the surface. Carbon
rings are clearly visible. The
scale is in 2 Å.
•
STM image of a
Ni (110) surface.
10
5
1
Potential Box
•
A particle confined in a three-dimensional potential box: V = 0 in 0 < x <
a, 0 < y < b, and 0 < z < c, and V infinite outside.
•
The electron essentially lives in the “box.”
•
What will the behavior of the electron be in this box?
2
1
Wavefunctions
•
Schrodinger’s equation:
+
•
+
2
ℏ
−
=0
Separation of variables:
, ,
•
+
=
( )
Wavefunction:
, ,
=
sin
sin
sin(
)
3
Wavefunctions
•
Boundary conditions:
ψ(x, y, z) = 0 at x = a, y = b and z = c
•
=

=
=

=
=

=
Three quantum numbers n1, n2, and n3 associated with ψx(x), ψy(y), and ψz(z).
( , , )=
•
Normalization:
sin
( , , )
sin
sin
=
2
/
4
2
Electronic Energies
• Substituting the wavefunction into the
Schrödinger:
=
,
=
•
,
ℎ
8
=
ℏ
2
+
+
+
+
For a square box for which a = b = c:
=
ℎ (
+
8
+
)
=
ℎ
8
5
Degenerate States
•
E111 is the lowest energy for the electron when n1 =1, n2 = 1, and n3 = 1.
•
The number of states that have the same energy is termed the degeneracy of
that energy level. The second energy level E211 is thus three-fold degenerate.
•
How many states (eigenfunctions) are there at energy level E443 for a square
potential energy box?
Energy
Degeneracy = 3
Non-degenerate ground state
6
3
7
Potential Energy
•
Consider the behavior of the electron in a hydrogenic (hydrogen-like) atom,
which has a nuclear charge of +Ze.
•
For the hydrogen atom, Z = 1, whereas for an ionized helium atom He+, Z = 2.
For a doubly ionized lithium atom Li++, Z = 3
•
Potential energy:
=
−
4
8
4
Wavefunctions
•
The problem has a spherical symmetry.
•
In an analogy with the three-dimensional potential well, there should be
three quantum numbers to characterize the wavefunction, energy, and
momentum of the electron.
•
The three quantum numbers are called the principal, orbital angular
momentum, and magnetic quantum numbers  n, l, and ml.
9
Wavefunctions
, ,
=
,
•
R(r): radial function  depends only on r.
•
Y(θ, ϕ): spherical harmonic  expresses the angular dependence of the
wavefunction.
•
These functions are characterized by the quantum numbers n, l, ml. The radial
part R(r) depends on n and l, whereas the spherical harmonic depends on l
and ml, so
, ,
=
,,
, ,
=
,
,
( , )
10
5
Quantum Numbers
Principal quantum number
n = 1, 2, 3, …
Orbital angular momentum
quantum number
l = 0, 1, 2, …, (n-1) < n
Magnetic quantum number
ml = -l, -(-l-1), …, 0, …, (l - 1), l or |ml|  l
11
R(r) and Y(,)
12
6
R(r) and Y(,)
13
7
1
Energy
Energy Bands
Insulator
Metal
Semiconductor Semiconductor
2
1
Atomic & Molecular Wavefunctions
+e
-e
-e
+e
r
r
energy of -e
n=3
n=2
n=3
n=2
n=1
n=1
+e
+e
-e
Superposition of Coulomb
potentials H2:
+e
+e
energy of -e
r
3
Hydrogen Atoms
•
As the two atoms
approach each other, ψ1s
wavefunctions overlap.
•
The overlap of
wavefunctions can either
be in phase or out of
phase  two molecular
orbitals are formed with
different energies.
∗
=
=
+
−
( )
4
2
Probability Distribution
•
The first molecular orbital ψσ is symmetric and has considerable magnitude
between the nuclei, whereas the second ψσ*, is antisymmetric and has a node
between the nuclei.
5
Hydrogen Atoms
•
As R decreases and the two H atoms
get closer, the energy of the ψσ
orbital state passes through a
minimum at R = a.
• The wavefunction ψσ corresponding
to the lowest electron energy is
called the bonding orbital, and ψσ* is
the antibonding orbital.
6
3
Hydrogen Atoms
•
The changes in the electron energy as two isolated H atoms, far left and far
right, come together to form a hydrogen molecule.
• E1s, splits into two, Eσ and Eσ*.
7
He Atoms
•
Consider what happens when two He atoms come together.
•
When He atoms come together, two of the electrons enter the Eσ level and
two the Eσ* level, so the overall energy is greater than two isolated He atoms.
•
Quantum mechanical calculations show that the antibonding energy level Eσ*
shifts higher than the bonding level Eσ shifts lower.
•
The overlap of full atomic orbital states does not lead to bonding.
•
We need an overlap of half-occupied orbitals, as in the H2 molecule, to form
a bond between two atoms.
8
4
3 Hydrogen Atoms
•
Three separate molecular orbital states, ψa, ψb, and ψc, from three ψ1s atomic
states.
•
Although H2 and H3 both have two electrons in the lowest energy level, H3
also has an extra electron at the higher energy level (Eb), which tends to
increase the net energy of the atom. Thus, the H3 molecule is much less
stable than the H2 molecule.
9
Li Atoms
•
Consider N Li (lithium) atoms are brought close together.
10
5
Bloch Theorem
• Bloch wave: A type of wavefunction for a particle in a periodically
repeating environment → consider an electron in a crystal.
• A wavefunction ψ is a Bloch wave if it has the form:

 


 
 k , r  eik r u k , r
 
u k , r : Periodic function with the same periodicity as the
 
crystal
  
 
u k,r  R  u k,r

  
A plane wave modulated by a function that has
periodicity of the potential.
11
Bloch Theorem
A Bloch wave (bottom) can be broken up into the product of a periodic function
(top) and a plane-wave (center). Blue is real part, red is imaginary part. The left
side and right side represent the same Bloch wave broken up in two different ways,
involving the wave vector k1 (left) or k2 (right). The difference (k1-k2) is a
reciprocal lattice vector.
12
6
1
Kronig-Penny Model: Periodic Potential
• One electron approximation: Each valence electron is considered
independent and only acted upon by the periodic positive ions.
2
2
1
Schrodinger’s Equation
V(x)
V0
b
c
0
x
 2 d 2
 V ( x) ( x)   ( x)
2m dx 2
d 2 2m

 - V ( x) ( x)  0
dx 2  2
-
• Wavefunction will be of Bloch form:  ( x)  eikx u ( x)
 u(x) has the periodicity of the lattice.
3
3
Schrodinger’s Equation
• After substitution
d 2 ikx
2m
e u ( x)  2  - V ( x)  eikx u ( x)  0
2
dx

d  ikx
du ( x)  2m
ike u ( x)  eikx
 2  - V ( x)  eikx u ( x)  0


dx 
dx  
du ( x)
du ( x) ikx d 2u ( x)
2
 ikeikx
e
 ik  eikxu( x)  ikeikx
dx
dx
dx 2
2m
 2  - V ( x)  eikx u ( x)  0

2
d u ( x)
du ( x)
2m
 2ik
- k 2 u ( x )  2  - V ( x )  u ( x )  0
2
dx
dx

2
2m
d u ( x)
du ( x)  2
2mV ( x) 
2
 2ik
- k -  2 
u ( x)  0,   2
2
2


dx
dx



4
4
2
Potential Well and Barrier
d 2u ( x)
du ( x)  2
2mV ( x) 
 2ik
- k -  2 
u ( x)  0
2
dx
dx
 2 

V(x)
0 x b:
d 2u1 ( x)
du
 2ik
-  k 2 -  2  u1 ( x)  0
2
dx
dx
c
b
u2(x)
u1(x)
-c  x  0 :
d 2 u2 ( x )
du
 2ik
-  k 2 -  2  u2 ( x )  0
2
dx
dx
2 
x
0
a
2m( - V0 )
2
5
5
Solutions
0 x b:
u1 ( x)  Aei ( - k ) x  Be- i (  k ) x
V(x)
-c  x  0 :
u2 ( x)  Cei (  - k ) x  De -i (   k ) x
A, B, C, D are constants and must be
determined.
c
b
u2(x)
u1(x)
0
Apply Boundary Conditions:
x
a
Wavefunctions and their derivatives must be continuous at
all points.
6
6
3
Boundary Conditions
• At x = 0
u1 (0)  u2 (0)
V(x)
A B  C  D
du1 ( x)
du ( x )
 2
dx x 0
dx x 0
c
b
u2(x)
u1(x)
0
x
a
i ( - k ) A - i (  k ) B  i (  - k )C - i (   k ) D
7
7
Boundary Conditions
• At x = b
u1 (b)  u2 (-c)
Ae
i ( - k ) b
 Be
- i (  k ) b
 Ce
-i (  -k )c
 De
V(x)
i( k )c
du1 ( x)
du ( x)
 2
dx x b
dx x - c
i ( - k ) Aei ( - k ) b - i (  k ) Be -i (  k )b
c
b
u2(x)
u1(x)
0
x
a
 i (  - k )Ce -i (  -k ) c - i (   k ) Dei (   k ) c
8
8
4
Solutions
A B  C  D
i ( - k ) A - i (  k ) B  i (  - k )C - i (   k ) D
Aei ( - k ) b  Be - i (  k ) b  Ce - i (  - k ) c  Dei (   k ) c
i ( - k ) Aei ( - k ) b - i (  k ) Be -i (  k ) b  i (  - k )Ce - i (  - k ) c - i (   k ) Dei (   k ) c
• For meaningful solutions to exist, the determinant must be zero:
1
 -k
e
i ( - k ) b
1
1
1
- (  k )
 -k
- (  k )
e
- i (  k ) b
e
-i (  -k )c
ei (   k ) c
0
( - k )ei ( - k ) b - (  k )e - i (  k ) b (  - k )e- i (  - k ) c - (   k )ei (   k ) c
9
9
Dispersion Relation
2 
2m( - V0 )
2
 2  0   is real.
2  2
sin  b sin  c  cos  b cos  c  cos ka
2
 2  0   is imaginary.
 2 - 2
sin  b sinh  c  cos  b cosh  c  cos ka
2
  i
10
10
5
Dispersion Relation
• Using trigonometry


V02
1

sin 2  c 

 4   - V0 

1/ 2


V02
sinh 2  c 
1 
 4 V0 -  

where
tan   -
cos( b -  )  cos ka,
(  V0 )
1/ 2
cos( b -  )  cos ka,
(0    V0 )
2  2
 2 - 2
tan  c and tan   
tanh  c
2
2
11
11
Energy Bands
P
• In an infinite lattice, the
states within any allowed
band would form a
continuum.
ka  2
ka  0
1
0
a
-1
ka  
ka  
Forbidden energy range
2 
• For a lattice of N atoms,
there are N discrete states,
however, there are 2N
states for spin degeneracy.
• The energy gaps decrease
as electron energy
increases  free electron
behavior at high energies.
2m
2
12
12
6
Dispersion Relation
 (k)
-2 /a - /a
0
 /a 2 /a
k
• Energies close to zero are forbidden.
• At k = n /a, there are two possible values of energies.
• Bragg reflection at k =  n /a
13
13
Reduced Zone Representation
• Energy band diagram is plotted in the first Brillouin zone.

k
• How the energy bands (E-k relation) will change for a 3-D
case?
14
14
7
Dispersion Relation: 3D
 (k)
- /a
- /b
- /c
 /a
 /b
 /c
0
kx, ky, kz
a = b= c
15
15
Dispersion Relation: 3-D
• k-values at the zone boundaries along different crystal orientations
may be different  overlap of energy states at zone boundaries.

y
z
x
/a /b
/c
kx, ky, kz
16
16
8
1
Effective Mass
• A particle's effective mass (often denoted m*) is the mass that it
seems to have when responding to forces.
• The effective mass is usually stated in units of the true mass of
the electron me (9.11×10−31 kg). In these units it is usually in the
range 0.01 to 10, but can also be lower or higher.
• The electronic effective mass can be seen as an important basic
parameter that influences measurable properties of a solid,
including everything from the efficiency of a solar cell to the
speed of an integrated circuit.
2
2
1
Parabolic Isotropic Dispersion
• At the highest energies of the valence band in many semiconductors (Ge, Si,
GaAs), and the lowest energies of the conduction band in some semiconductors
(GaAs), the band structure E(k) can be locally approximated as:


2 k 2
E k  E0 
2m*
 
• It can be shown that the electrons placed in these bands behave as free
electrons except with a different mass, as long as their energy stays within the
range of validity of the approximation above. As a result, the electron mass in
models such as the Drude model must be replaced with the effective mass.
3
3
Parabolic Non-isotropic Dispersion
• In some important semiconductors (notably, silicon) the
lowest energies of the conduction band are not symmetrical
so that the band minimum can be approximated only by
E  k   E0 

2
2
k  k0, x 
*  x
2 mx
2
2
2
2
k

k

k  k0, z 
0, y 
*  y
*  z
2m y
2 mz
• The speed of an electron will depend on its direction, and it will accelerate to a
different degree depending on the direction of the force.
• For the purposes of calculating conductivity as in the Drude model, via the
harmonic mean
*
conductivity
m
 1
1
1 
 3 *  *  * 
 mx my mz 
1
4
4
2
General Case
• The dispersion relation:
  
• The electron’s acceleration:
d  d  d  d  dk 1 d 2 d (k )
 




dt
dt  dk  dk  dk  dt  2 dk 2 dt
dvg
• If force Fe accelerates electron, will do work dWe in time dt, while
electron’s energy and momentum change by d and dk.
dWe  Fe dx  Fe vg dt  Fe
dWe  d  
d
dt
dk
d
d
dk  
dk
dk
dk
5
5
Effective Mass
• Equating:
Fe 
• After substituting:
dvg
dt

m* 
Material
Si (4.2 K)
Ge
GaAs
InSb
ZnO
ZnSe
d ( k )
dt
F
1 d 2
Fe  e
2
2
 dk
m*
2
d 2 / dk 2
 Effective mass
Electron Effective Mass Hole Effective Mass
Group IV
1.08
0.56
0.555
0.37
Groups III-IV
0.067
0.45
0.013
0.60
Groups II-VI
0.,19
1.21
0.17
1.44
6
6
3
Free Electron
2k 2
d 2  2

 2 
2m
dk
m
• Effective mass:
2
m*  2
m
 /m
7
7
Effective Mass
• In parabolic sections: Near the top and bottom of the bands:
  C (k  n ) 2 
d 2
2

2
C

m
*

dk 2
2C
• In non-parabolic sections: d 2 / dk2 depends on energy  m* is
not constant.
 (k)
k
8
8
4
Parabolic Approximation
9
9
Effective Mass
10
10
5
Effective Mass
11
11
Negative Effective Mass
• Negative curvatures near the top of dispersion
relation  negative effective mass.
• External field E in the positive x-direction acting
on a negatively charged particle with a negative
mass produces a positive acceleration component
along the x-direction.
• The change in sign of m* can be thought of the
change of sign of charge q.
dvg
dt

qE
m*
• Quasi-particle  positive charge and positive mass  holes.
• Without lattice, holes cannot exist.
12
12
6
1
Density of States
• How many states per unit energy per unit volume?
• N atoms ® N electron wavefunctions.
=
+
+
+
+⋯
=
−
+
−
+⋯
•
: Density of states
•
® number of states
(i.e., wavefunctions) in the
energy interval E to (E + dE)
per unit volume of the sample.
2
2
1
Density of States
• The number of states per unit volume up to some energy E′
=
•
The energy of an electron in a cubic PE well of size L is given by
=
ℎ
8
+
+
= ′
•
The spatial dimension L of the well now refers to the size of the entire solid,
as the electron is confined to be somewhere inside that solid. Thus, L is very
large compared to atomic dimensions, which means that the separation
between the energy levels is very small.
•
Enumerate all possible choices of integers for n1, n2, and n3 that satisfy
+
+
≤ ′ .
3
3
2D Material
•
+
≤ ′
Area =
1
4
4
4
2
3D Material
•
•
+
+
≤ ′
The number of orbital states Sorb(n′)
within this volume is given by
=
1 4
8 3
′
=
1
6
′
• The number of states, including spin:
1
=2
=
′
3
• Substitute ′ :
=
ℎ
′ ® ′ =
8
8
ℎ
5
5
3D Material
•
The number of states per unit volume:
=
•
) /
(8
3ℎ
=
•
≤ ′:
The number of states with
′) /
(8
3ℎ
Density of states per unit volume:
/
=
= (8 2 )
/
ℎ
6
6
3
Carrier Statistics
• If E is the electron energy and f (E) is the probability that a state with energy E
is occupied, then
=
.
• We need to know
7
7
Boltzmann Probability Function
• If E is the electron energy and f (E) is the probability that a state with energy E
is occupied, then
=
exp −
•
( ) decreases exponentially with energy.
•
Any number of particles may have a given
energy E. There is no restriction to permit
only one particle per state at an energy E,
as in the Pauli exclusion principle.
= exp −
•
If
≫
, then
−
≪
.
8
8
4
Boltzmann Probability Function
•
Classical particles obey the Boltzmann
statistics.
•
If there are many more states than the
number of particles, the likelihood of two
particles having the same set of quantum
numbers is negligible and we do not have
to worry about the Pauli exclusion principle
® We can use the Boltzmann statistics.
•
Example: Electrons in the conduction band
of a semiconductor, where usually there are
many more states than electrons.
9
9
Fermi-Dirac Statistics
• If E is the electron energy and f (E) is the probability that a state with energy E
is occupied, then
•
−
≫
:
10
10
5
Carrier Density
=
( )
11
11
Carrier Density
=
( )
•
In the small energy range E
to (E + dE), there are nE dE
electrons per unit volume.
•
Total electrons in the band:
=
=
• f → 0 when E ≫ EF
=
8 2
ℎ
/
/
1 + exp
−
12
12
6
Fermi Energy
•
At T = 0 K
=
=
8 2
ℎ
( )
/
/
−
1 + exp
=
ℎ
8
3
/
13
13
7
1
Capacitance
=
•
If there is a material medium between the plates, then the capacitance, the
charge storage ability per unit voltage, increases by a factor of εr, where εr is
called the dielectric constant of the medium or its relative permittivity.
•
The increase in the capacitance is due to the polarization of the medium in
which positive and negative charges are displaced with respect to their
equilibrium positions.
•
The opposite surfaces of the dielectric medium acquire opposite surface
charge densities that are related to the amount of polarization in the material.
2
2
1
Relative Permittivity
=
=
>
=
=
• Note that electric field remains same when dielectric medium is inserted.
3
3
Dipole Moment
• Dipole moment a measure of the electrostatic effects of a pair of opposite
charges +Q and −Q separated by a finite distance a
=
• Although the net charge is zero, this entity still gives rise to an electric field
in space and also interacts with an electric field from other sources.
• An atom is said to be polarized if it possesses an effective dipole moment.
• The induced dipole moment depends on the electric field causing it.
=
: polarizability
4
4
2
Polarization
• When a material is placed in an electric field, the atoms and the molecules of
the material become polarized, so we have a distribution of dipole moments in
the material.
5
5
Polarization
• Polarization P: If there are N atoms ®
=
• If ⃗
1
⃗ + ⃗ + ⋯+ ⃗
is the average dipole moment ®
=
⃗
N : number of atoms per unit volume.
• Considering +
and −
surface charges separated by a distance d
=
=
=
=
6
6
3
Polarization
=
=
•
The direction of P is normal to the surface. For +σp (right face), it comes out
from the surface and for -σp (left face), it is directed into the surface.
•
Generally:
=
7
7
Susceptibility
•
The dependence of the induced dipole moment on the electric field:
=
•
The dependence of the polarization on the electric field:
=
: Electric susceptibility
•
relates the effect ( ) to its cause ( ).
•
may depend on E ® P may nonlinearly depend on E.
=
=
=
1
8
8
4
Susceptibility
=
=
=
=
9
9
Susceptibility
=
+
=
+
=
+
=
+
=
1+
=
®
=1+
®
=1+
10
10
5
Class Test – 3
Day: 31 July 2019
Syllabus: Lectures 19-
11
11
6
1
Local Field
•
As soon as the dielectric becomes polarized, the field at some arbitrary point
depends not only on the charges on the plates (Q) but also on the orientations
of all the other dipoles around this point in the dielectric  E changes.
•
Lorentz field:
=
+
1
3
=
2
2
1
Electronic Polarization
• Electron clouds within each atom become shifted by the field  quite small.
• Valence electrons in covalent bonds can easily tunnel from bond to bond 
significant shift under an electric field.
• Large dielectric constants in covalent crystals  εr = 11.9 for Si and εr = 16 for
Ge.
3
3
Ionic Polarization
• This type of polarization occurs in ionic
crystals such as NaCl, KCl, and LiBr.
=0
=
−
=0
−
≠0
≠0
=
=
=
=
4
4
2
Orientational Polarization
• Certain molecules possess permanent dipole moments  HCl molecule has a
permanent dipole moment po from the Cl- ion to the H+ ion.
• When = 0, dipole moments are randomly oriented as a result of thermal
agitation.
5
5
Oriental Polarization
• When ≠ 0: Cl- and H+ experience forces
in opposite direction.
• The molecule experiences a torque  about
its center of mass.
• For full alignment 
=
• The collisions due to thermal energy destroy
dipole alignments.
• Net average dipole moment depends on
temperature:
=
1
3
=
1
3
6
6
3
Interfacial Polarization
• Interfacial polarization occurs whenever there is an accumulation of charge
at an interface between two materials or between two regions within a material.
7
7
Interfacial Polarization
• Interfacial polarization occurs whenever there is an accumulation of charge
at an interface between two materials or between two regions within a material.
8
8
4
Total Polarization
• In the presence of electronic, ionic, and dipolar polarization mechanisms, the
average induced dipole moment per molecule will be the sum of all the
contributions in terms of the local field
=
+
+
• Interfacial polarization cannot be simply added to the above equation as αif Eloc
because it occurs at interfaces and cannot be put into an average polarization
per molecule in the bulk.
9
9
Class Test – 3
Day: 31 July 2019
Syllabus: Lectures 19-21
10
10
5
1
Time-Varying Field
•
When the applied field varies with time, the
polarization is generally different than the
static case.
•
Let us consider orientational polarization
involving dipolar molecules.
•
The time-varying field changes magnitude and
direction continuously, and it tries to line up
the dipoles one way and then the other way
and so on.
• If the instantaneous induced dipole moment p per molecule
can instantaneously follow the field variations, then at any
instant
=
2
2
1
Alignment
• There are two factors opposing the
immediate alignment of the dipoles with the
field.
• First: Thermal agitation randomizes the
dipole orientations
• Second: Interactions with neighbors.
• As a result, dipoles cannot respond
instantaneously to the changes in the applied
field.
3
3
Polarizability vs. Frequency
•
If the field changes too rapidly, then the
dipoles cannot follow the field.
•
At high frequencies, therefore, αd will be zero
as the field cannot induce a dipole moment.
•
At low frequencies, of course, the dipoles can
respond rapidly to follow the field and αd has
its maximum value.
•
We need to find the behavior of αd as a
function of frequency ω.
4
4
2
Change in Field
• Suppose that after a prolonged
application, corresponding to
dc conditions, the applied field
across the dipolar gaseous
medium is suddenly decreased
from Eo to E at a time we
define as zero,
5
5
Relaxation
• τ: Average time , called the
relaxation time, between
molecular collisions, then
this is the mean time it takes
per molecule to randomize
the induced dipole moment.
6
6
3
Sinusoidal Electric Field
•
αd(ω) is a complex number that
indicates that p and E are out of phase
•
At low frequencies, ωτ ≪ 1, αd(ω) is
nearly αd(0), and p is in phase with E.
7
7
Phase Delay
8
8
4
Complex Dielectric Constant
=1+
=1+
=1+
=1+
(0)
1+
(0)
1+
(0)
1+
= ′ −
×
1−
1−
−
(0)
1+
"
9
9
Frequency Dependence
= ′ −
′ =1+
" =
"
(0)
1+
(0)
1+
•
ε′r decreases from its maximum
value ε′r(0) to 1 at high
frequencies.
•
ε″r(ω) is zero at low and high
frequencies but peaks when ωτ = 1 or
when ω = 1∕τ.
10
10
5
Lossless Capacitor & Parallel Conductance
• Admittance:

11
11
Dielectric Resonance
When ω = 1∕τ, energy is being transferred to heat most efficiently.
12
12
6
Piezoelectricity
• Mechanical stress creates polarization and hence potential across the surfaces.
• Examples: Quartz (crystalline SiO2), BaTiO3
• Electric field creates mechanical strain/deformation: A complementary effect.
• The direction of deformation depends on the direction of applied field.
2
1
Why Piezoelectricity
•
Consider a NaCl-type cubic unit cell that has center of symmetry.
•
When unstressed, the center of mass of the negative charges at the corners of
the unit cell coincides with the positive charge at the center: P = 0.
•
Under stress, the unit cell becomes strained, but the center of mass of the
negative charges still coincides with the positive charge: P = 0.
3
Why Piezoelectricity
• Piezoelectric crystals have no center of symmetry.
• When unstressed, the center of mass of the negative charges coincides with the
center of mass of the positive charges, both at O: P = 0.
• When stressed, the positive charge at A and the negative charge at B both
become displaced inwards to A′ and B′, respectively: P  0.
4
2
Stress vs. Induced Polarization
•
Generally, an applied stress in one direction can give rise to induced
polarization in other crystal directions.
•
Suppose that Tj is the applied mechanical stress along j direction and Pi is the
induced polarization along some i direction; then the two are linearly related by
Pi = dijTj ; dij: piezoelectric coefficients.
• The converse piezoelectric effect is that between an induced strain Sj along j
and an applied electric field Ei along i, Sj = dijEi
5
Piezoelectric Transducers
6
3
Electromechanical Coupling Factor
=
Electrical energy converted to mechanical energy
Input of electrical energy
=
Mechanical energy converted to electrical energy
Input of mechanical energy
Crystal
d (m V-1)
k
Quartz (crystal SiO2)
2.310-12
0.1
Rochelle salt (NaKC4H4O6  4H2O)
35010-12
0.78
Barium titanate (BaTiO3)
19010-12
0.49
PZT, lead zirconate titanate
48010-12
0.72
7
Piezoelectric Spark Generator
•
The piezoelectric spark generator, as used in various applications such as
lighters and car ignitions, operates by stressing a piezoelectric crystal to
generate a high voltage which is discharged through a spark gap in air.
•
The energy can increase by using two piezoelectric crystals back to back.
8
4
Force Needed for Spark
Suppose that the piezoelectric coefficient d = 250 × 10−12 m V−1 and εr = 1000.
The piezoelectric cylinder has a length of 10 mm and a diameter of 3 mm. The
spark gap is in air and has a breakdown voltage of about 3.5 kV. What is the
force required to spark the gap?
•
=
=
=
=
=
•
=
=
=
= 87.6 N
This force can be applied by squeezing by hand an appropriate lever
arrangement; it is the weight of 9 kg.
9
Quartz Oscillators
10
5
Ferroelectricity
• Certain crystals are permanently polarized even in the absence of an applied
field. The crystal already possesses a finite polarization vector due to the
separation of positive and negative charges in the crystal. These crystals are
called ferroelectric.
• BaTiO3 cubic crystal above 130 oC.
•
P = 0 above 130 oC  Not ferroelectric.
12
6
Ferroelectricity
•
Below 130 °C, the structure of barium titanate is tetragonal.
•
•
P  0  Ferroelectric.
Curie temperature (Tc): Above which ferroelectricity is lost.
• The development of the permanent dipole moment below Tc involves longrange interactions between the ions outside the simple unit cell. The energy of
the crystal is lower when the Ti4+ ion in each unit cell is slightly displaced
along the c direction, which generates a dipole moment in each unit cell.
13
Poling
•
If we apply a temporary field E and let the crystal cool to below 130 °C, we can
induce the spontaneous polarization P to develop along the field direction.
•
In other words, we would define the c axis by imposing a temporary external
field.
•
At < Tc: =
− 1 cannot be used
 Already has permanent polarization
•
Rather, ∆ =
used.
E
− 1 ∆ should be
14
7
Ferroelectricity  Piezoelectricity
• All ferroelectric crystals are also piezoelectric, but the reverse is not true: not all
piezoelectric crystals are ferroelectric.
15
8
Pyroelectricity
• Pyroelectricity: When a temperature change δT induces a change δP in the
polarization of the crystal.
• Example: BaTiO3
• Pyroelectric coefficient:
=
2
1
Microscopic Picture
•
Spontaneous polarization depends
on temperature.
•
Current sign depends whether
temperature increases or decreases.
•
Pyroelectric effect is observed only
when temperature changes.
3
Pyroelectric Crystals
Material
Pyroelectric
Coefficient
(10-6 C m-2 K-1)
Curie Temperature
(oC)
BaTiO3, barium titanate
20
130
LiTaO3, lithium tantalate
230
610
PZT, lead zirconate titanate
380
230
•
Very small temperature changes, even in thousandths of degrees, can
develop voltages that can be readily measured.
•
If δT = 10−3 K for a PZT material, we find δP = 3.8 × 10−7 C m−2.
=
−1
•
with εr = 290, we find: δE = 148 V m−1
•
If the distance between the faces of the ceramic where the charges are
developed is 0.1 mm, then δV = 0.0148 V or 15 mV
4
2
Pyroelectric Radiation Detector
•
Pyroelectric crystals are widely used as infrared detectors (PID). Any infrared
radiation that can raise the temperature of the crystal even by a thousandth of a
degree can be detected.
5
PID Components
6
3
Pyroelectric Infrared Detector
Working Principle:
7
Infrared/Thermal Imaging
8
4
Pyroelectric Sensor
9
Pyroelectric Current & Voltage
•
Pyroelectric applications use both pyroelectric current and voltage that are
generated by the change in temperature.
=
• Induced current density:
=
• Current responsivity:
=
Pyroelectric current generated
=
Input radiation power
• Voltage responsivity:
=
Pyroelectric output voltage generated
Input radiation power
10
5
Magnetic Dipole Moment
•
The magnetic moment is the magnetic strength and orientation of a magnet or
other object that produces a magnetic field.
•
Examples of objects that have magnetic moments include: loops of electric
current (such as electromagnets), permanent magnets, elementary particles
(such as electrons), various molecules, and many astronomical objects (such as
many planets, some moons, stars, etc).
•
Consider a current loop:
=
2
1
Magnetic Dipole
3
In External Field
•
A magnetic field, applies a rotating torque on the magnetic moment to align its
axis with the magnetic field.
•
The magnetic dipole moment is defined by the torque that object experiences.
•
The strength (and direction) of this torque depends on both its magnitude and
on orientation relative to the direction of the magnetic field.
4
2
Atomic Magnetic Moments
Orbital Magnetic Moment
= Charge flowing per unit time = −
=
=−
=
=
=−
•
Period
=−
2
2
2
Magnetic moment is proportional to the orbital angular momentum through
the factor e ∕ 2me  gyromagnetic ratio.
The negative sign in Equation indicates that μorb is in the opposite direction to
L and is due to the negative charge of the electron.
•
5
Atomic Magnetic Moments
Spin Magnetic Moment
•
The electron also has an intrinsic angular momentum S, that is, spin. The
spin of the electron has a spin magnetic moment, denoted by μspin.
=−
•
The overall magnetic moment of the
electron consists of μorb and μspin
vectorially added.
•
The overall magnetic moment μatom
depends on the orbital motions and
spins of all the electrons.
•
Electrons in closed subshells do not
contribute to the overall magnetic
moment.
6
3
Precession
• Consider an atom that has closed inner shells and a single electron in an s
orbital (ℓ = 0).
•
The spin magnetic moment precesses about an external magnetic field along z
and has a value μz along z.
=−
=−
ℏ=
ℏ
2
=
7
Magnetization
• Consider a tightly wound long solenoid, ideally infinitely long, with free space
(or vacuum) as the medium inside the solenoid.
=
=
′
: Current per unit length
μ0: Permeability of free space
8
4
Magnetization
•
Let us now place a cylindrical material to fill the inside of this solenoid.
•
The magnetic field has changed from B0 to B.
•
B0 to be the applied magnetic field into which the material medium is placed.
9
Magnetic Dipole Moment
•
Let us now place a cylindrical material to fill the inside of this solenoid.
•
Each atom of the material responds to the applied field B0 and develops, or
acquires, a net magnetic moment μm along the applied field.
• Magnetic dipole moment per unit volume (M): If there are N atoms in a small
volume ΔV and each atom i has a magnetic moment μmi 
=
1
∆
=
10
5
Surface Currents
• All neighboring atomic current loops in
the bulk have adjacent currents in
opposite directions that cancel each other.
• However, the currents at the surface in
the surface loops cannot be cancelled and
this leads to a net surface current.
• Total M = M  Volume = Mal
• Total M = Total current  Cross-sectional area = ImlA
• M = Im  Surface current per unit length
11
Magnetic Field
=
+
=
=
+
+
• Magnetizing field or magnetic field intensity (H):
=
=
1
−
=
= Total conduction current per unit length
12
6
Magnetic Permeability
•
Magnetic permeability: Magnetic field (B) per unit magnetizing field (H)
=
•
relates the effect B to the cause H at
the same point inside a material.
•
Qualitatively, μ represents to what
extent a medium is permeable by
magnetic fields.
•
Relative permeability:
=
=
→
=
13
Magnetic Susceptibility
•
Magnetic susceptibility (
):
=
=
=
( +
)
+
=
+
=
1+
=1+
14
7
Classification
• Materials that does not have permanent magnetic dipoles: Diamagnetic
• Materials that have permanent magnetic dipoles.
• Paramagnetic
• Ferromagnetic
• Antiferromagnetic
• Ferrimagnetic
2
1
Diamagnetic
m
m versus Temp.
Comments and Examples
• Negative and
small (-10-6)
• T independent
• Negative and
large (-1)
• Below a
critical temp.
• Weakly magnetized when placed in an external
magnetic field, in a direction opposite to the
applied field.
• Atoms of the materials have closed shells.
• Organic materials, covalent solids, e.g., Si, Ge;
some ionic solids, e.g., alkalihalides; some
metals, e.g., Cu, Ag, Au
• Superconductors
• When placed in a nonuniform
magnetic field experiences a
force toward smaller fields.
3
Paramagnetic
m
m versus Temperature
Positive and
T independent
small (10-5-10-4)
Comments and Examples
• Alignment of spins of conduction
electros.
• Alkali and transition metals.
4
2
Ferromagnetic
m
• Positive and
very large
m versus Temperature
Comments and Examples
• Ferromagnetic below • May possess a large permanent
and paramagnetic
magnetization even in the absence of
above Curie
an applied field.
temperature
• Some transition and rare earth
materials, Fe, Co, Ni, Gd, Dy.
5
Antiferromagnetic
m
Positive and small
m versus Temperature
Comments and Examples
Antiferromagnetic
• Cannot possess any magnetization in
below and paramagnetic
the absence of an applied field.
above Néel temperature • Mainly salts and oxides of transition
metals, e.g., MnO, NiO, MnF2, and
some transition metals, -Cr, Mn
6
3
Ferrimagnetic
m versus Temperature
m
Positive and very
large
Comments and Examples
Ferrimagnetic below
• May possess a large permanent
and paramagnetic above
magnetization even in the absence of
Curie temperature
an applied field.
• Ferrites: Fe3O4
7
Origin of Ferromagnetism
•
Magnetized iron crystal: atomic
magnetic moments are aligned in the
same direction  net magnetization
along this direction.
•
However, this is not alignment of bar
magnets in an SNSN . . . fashion as the
magnetic potential energy of
interaction is small, indeed smaller than
the thermal energy.
•
•
•
Hund’s rule:
An isolated Fe atom has four
unpaired spins and a spin magnetic
moment of 4.
= ℏ/(2
)  Bohr magneton
8
4
Temperature Effect on Ferromagnetism
•
•
As temperature increases,
lattice vibrations become
more energetic and leads to
a frequent disruption of the
alignments of the spins.
The ferromagnetic behavior
disappears at a critical
temperature called the Curie
temperature:
Ferromagnetic  Paramagnetic
9
Magnetic Domains
• If a magnetized of iron is heated to above Curie temperature and then allowed to
cool in the absence of a magnetic field, it will possess no net magnetization.
• The reason for the absence of net magnetization is due to the formation of
magnetic domains that effectively cancel each other.
• A magnetic domain is a region of the crystal in which all the spin magnetic
moments are aligned to produce a magnetic moment in one direction only.
10
5
Externally Applied Field
• Rotation of orientation and increase in size of magnetic domains in response
to an externally applied field.
11
Easy Directions & Anisotropy
• The magnetization of each domain is normally along one of the preferred
directions in which the atomic spin alignments are easiest  Easy Directions.
• For iron, the magnetization is easiest along any one of six ⟨100⟩ directions
(along cube edges).
• Ferromagnetic crystals exhibit magnetic
anisotropy: magnetic properties are
different along different crystal
directions.
• Iron (BCC): the spins in a domain are
most easily aligned in any of the six
[100] directions  six edges of the
cubic unit cell.
12
6
Class Test – 4
Day: 9 September 2019
Syllabus: Lectures 24-25
13
7
Superconductivity
•
Superconductivity is a
phenomenon in which the
resistance of the material to the
electric current flow is zero.
•
Kamerlingh Onnes made the first
discovery of the phenomenon in
1911 in mercury (Hg).
•
Superconductivity is not relatable to periodic table, such as atomic
number, atomic weight, electro-negativity, ionization potential etc.
•
In fact, superconductivity does not even correlate with normal conductivity.
In some cases, a superconducting compound may be formed from nonsuperconducting elements.
2
1
Critical Temperature
• The quest for a near-roomtemperature superconductor goes on, with many
scientists around the world trying different materials, or synthesizing
them, to raise Tc even higher.
•
Silver, gold and copper do not show conductivity  at low temperature,
resistivity is limited by scattering and crystal defects.
3
Critical Temperature
4
2
Meissner Effect
• A superconductor below Tc expels all the magnetic field from the bulk of the
sample  perfectly diamagnetic substance  Meissner effect.
• Below Tc a superconductor is a perfectly diamagnetic substance (χm = −1).
• A superconductor with little or no magnetic field within it is in the Meissner state.
5
Levitating Magnet
• The “no magnetic field inside a
superconductor” levitates a
magnet over a superconductor.
• A magnet levitating above a
superconductor immersed in
liquid nitrogen (77 K).
6
3
Critical Field vs. Temperature
7
Penetration Depth
•
The field at a distance x from the surface:
=
−
: Penetration depth
•
At the critical temperature, the penetration length is infinite and any
magnetic field can penetrate the sample and destroy the superconducting
state.
•
Near absolute zero of temperature, however, typical penetration depths
are 10–100 nm.
8
4
Type I Superconductors
•
•
Meissner state breaks down abruptly.
In Type I superconductors, as the applied magnetic field B increases, so does the
opposing magnetization M until the field reaches the critical field Bc, whereupon
the superconductivity disappears.
• Most pure elemental superconductors (aluminium, lead, and mercury.),
except niobium and carbon nanotubes, are type I.
9
Type II Superconductors
•
The transition does not occur sharply from the Meissner state to the normal
state but goes through an intermediate phase in which the applied field is
able to pierce through certain local regions of the sample.
•
The mixed state is also called a vortex state.
• Type II superconductors therefore have two critical fields Bc1 and Bc2.
10
5
Type II Superconductors
11
High Temperature Superconductors
• A HTS (Bi223) power cable for use at 10 kV and 2,300 A in Germany since 2014.
12
6
Critical Current Density
•
When the current density through the sample exceeds a critical value Jc, it
is found that superconductivity disappears.
•
Jc is very high for type II superconductors.
13
Superconductivity Origin
• Bardeen, Cooper, and Schrieffer (BCS) Theory:
•
At sufficiently low temperatures,
two oppositely spinning and
oppositely traveling electrons can
attract each other indirectly
through the deformation of the
crystal lattice of positive metal
ions.
• This indirect interaction at
sufficiently low temperatures is able
to overcome the mutual Coulombic
repulsion between the electrons and
hence bind the two electrons to each
other.
• The two electrons are called a
Cooper pair.
14
7
Critical Temperature
• BCS theory applies.
• Accurate theory is still challenging.
15
8
Anisotropic Magnetoresistance
•
The change in the resistivity due to the applied magnetic field is anisotropic
(depends on the direction) and is called anisotropic magnetoresistance (AMR).
2
1
Giant Magnetoresistance
•
GMR: Certain special multilayer structures exhibit substantial changes in the
resistance (>10%) when a magnetic field is applied
•
Widely used in the read heads of hard disk drives, and also in various sensors.
•
In the absence of an external field, magnetizations are antiparallel.
•
In the presence of an external field, magnetizations are parallel.
3
Giant Magnetoresistance
•
RP < RAP
• The difference in RP and RAP in this simple trilayer is roughly 10% or less.
• Angular dependence:
4
2
Giant Magnetoresistance
• Multilayered structures: A series of alternating magnetic and nonmagnetic
layers can change the resistance >100% at low temperature and 60–80% at
room temperature.
• GMR effect in trilayers and multilayers
Sample
Structure and Layer Thicknesses
∆ /
(%)
Temp.
(K)
CoFe/CAgCu/CoFe
Trilayer
4–7
300
NiFe/Cu/Co
Trilayer, 10/2.5/2.2 nm (spin valve)
4.6
300
Co90Fe10/Cu/Co90Fe10
Trilayer, 4/2.5/0.8 nm (spin valve)
7
300
[Co/Cu]100
100 layers of Co/Cu 1 nm / 1 nm
80
300
[Co/Cu]60
60 layers Co/Cu 0.8 nm / 0.83 nm
115
4.2
5
Spin Valve
6
3
Longitudinal Magnetic Recording
•
Magnetic materials in magnetic recording: (1) magnetic heads to write
(record), (2) read sensors, (3) storage of information either permanently or until
the next write requirement.
•
Example: Recording of digital data on a magnetic disk in a hard disk drive 
Electrical signal is stored as a spatial magnetic pattern in the magnetic film in
circular tracks.
8
4
Write and Read Heads
• Both the write and the read heads are in a single compact assembly that moves
radially across the rotating disk
• The read-write head is on an air bearing and the head to thin magnetic film
separation is roughly 10 nm or less.
9
Perpendicular Magnetic Recording
• Induced M is either up or down.
•
The write operation is distinctly different than that in longitudinal storage.
•
There is an inductive write head with a narrow “write pole” that brings the
magnetic flux onto the film.
10
5
Information Boundary
•
The magnetic thin film has small crystalline grains and a region between
the grains that is nonmagnetic
11
Bit Length
• How small can we make the magnetized
region for one bit?
• λw should be as small as possible to increase
the storage capacity.
•
SNR
≈ 20 log
= 10 log
• A 20 dB SNR implies that a bit-volume should
be 100 grains.
• In perpendicular recording, we can reduce w
and increase t, keeping the bit volume the
same.
12
6
Materials for Magnetic Storage
• Magnetic thin film: Typical film
thicknesses are less than 30 nm. The film is
not a single phase homogenous magnetic
medium  roughly 10 nm grains separated
by an oxide nonmagnetic amorphous region.
•
The film must be able to retain the spatial
magnetization pattern  high remanent
magnetization.
•
However, Hc cannot be too high because,
otherwise, the inductive head will not be
able to change the magnetization M of a
magnetic grain; we need a semihard
magnetic medium.
• Example: CoCrPt-SiO2
13
Materials for Magnetic Storage
•
Soft underlayer: Should be sufficiently
soft to be easily magnetized  magnetic
field lines restricted to this layer.
•
Interlayer: Separates storage thin film and
the SUL.
•
Top overcoat layer: Usually carbon 
protects the magnetic thin film’s surface.
•
Thin film inductive recording head:
must be able to produce a strong
magnetizing field. The thin film coil is
normally Cu and the oxide insulation
maybe SiO2 or Al2O3. The magnetic core
is a soft medium that can be easily
magnetized
14
7
Bit Patterned Recording Medium
•
•
•
A recording medium in which a bit is a well-defined nanostructure, and these
nanostructures are patterned to form a periodic array of “isolated magnetic
islands.”
Clearly, we no longer have a granular medium  an N-fold increase in storage
capacity, breaking through the limits of multigrain media.
Further, we can develop write and read techniques that involve synchronization
with the periodicity in the bit-pattern; obviously such new techniques would
require a more demanding technology for the inductive write head.
15
8
Josephson Junction
• The Josephson junction is a junction between two superconductors that are
separated by a thin insulator (a few nanometers thick).
• If the insulating barrier is sufficiently thin, then there is a probability that the
Cooper pairs can tunnel across the junction.
2
1
What is Metamaterial?
• A metamaterial (Greek word μετά meta means "beyond“) is a material
engineered to have a property that is not found in naturally occurring materials.
• They are made from assemblies of multiple elements fashioned from
composite materials such as metals and plastics.
• The materials are usually arranged in repeating patterns, at scales that are
smaller than the wavelengths of the phenomena they influence.
• Metamaterials derive their properties not from the properties of the base
materials, but from their newly designed structures.
• Their precise shape, geometry, size, orientation and arrangement gives them
their smart properties capable of manipulating electromagnetic waves: by
blocking, absorbing, enhancing, or bending waves, to achieve benefits that go
beyond what is possible with conventional materials.
4
2
Structured Electromagnetic Materials
Nature:
Historical artificial:
5
Early History: J.C. Bose
•
Explorations of artificial materials for manipulating
electromagnetic waves began at the end of the 19th
century.
•
In 1898, J. C. Bose researched substances
with chiral properties.
Polarization control using
composite media: (a) Linear
polarizer consisting of metal foil in
between the pages of a book. (b)
Bundle of twisted jute which rotates
the polarization state.
6
3
World War II and Stealth Technology
•
The history of metamaterials begins with artificial dielectrics in microwave
engineering as it developed just after World War II. Yet, there are seminal
explorations of artificial materials for manipulating electromagnetic waves at
the end of the 19th century.
• Stealth attack plane
•
Stealth ground vehicle
• French stealth frigate
7
Modern History: John Pendry
• Conventionally, the function or behavior of
materials can be altered through their chemistry.
• During 1990s, while studying radiationabsorbing carbon for stealth technology, Pendry
discovered that the radiation absorption property
did not come from the molecular or chemical
structure of the material, i.e. the carbon per se.
This property came from the long and thin,
physical shape of the carbon fibers.
• He realized rather than conventionally altering a
material through its chemistry, as lead does with
glass, the behavior of a material can be altered
by changing a material’s internal structure on a
very fine scale.
8
4
Split-Ring Resonance
A split-ring resonator (SRR) is an artificially produced structure common to
metamaterials. Their purpose is to produce the desired magnetic susceptibility
(magnetic response) in various types of metamaterials up to 200 terahertz.
9
Propagation of EM Waves
10
5
Single Negative Metamaterials
•
Single negative (SNG) metamaterials have either negative relative
permittivity (εr) or negative relative permeability (µr), but not both.
•
Epsilon negative media (ENG) display a negative εr while µr is positive.
Noble metals such as gold or silver are ENG in the infrared and visible
spectrums.
•
Mu-negative media (MNG) display a positive εr and negative µr. Gyrotropic
or gyromagnetic materials exhibit this characteristic. A gyrotropic material is
one that has been altered by the presence of a quasistatic magnetic field,
enabling a magneto-optic effect.
•
SNGs act as metamaterials when ENGs are combined with a MNGs and
jointly acting as a Double negative (DNG) medium.
•
Metamaterials are innately dispersive, so their εr, µr and refraction index n, are
a function of frequency.
11
Negative Index Metamaterials
12
6
Negative Index Metamaterials
(a) First metamaterial with simultaneously negative permittivity and permeability. (b) The
metamaterial used for the first demonstration of negative refraction. (c) Negative refraction
in the optical part of the spectrum has been demonstrated for layered fishnet structures.
(d) SEM image of a fishnet structure with a negative refractive index at 780 nm.
13
Super Lens
• A superlens, or super lens, is a lens which uses metamaterials to go beyond
the diffraction limit. The diffraction limit is a feature of conventional lenses
and microscopes that limits the fineness of their resolution.
a) When a wave strikes a positive refraction index material from a vacuum. b) When a
wave strikes a negative-refraction-index material from a vacuum. c) When an object is
placed in front of an object with n=−1, light from it is refracted so it focuses once inside
the lens and once outside. This allows subwavelength imaging.
14
7
Invisible Cloaking
• An invisibility device should guide light around an object as if
nothing were there.
Left: The cross section of a PEC cylinder
subject to a plane wave (only the electric
field component of the wave is shown). The
field is scattered.
Right: a circular cloak, designed using
transformation optics methods, is used to
cloak the cylinder. In this case the field
remains unchanged outside the cloak and
the cylinder is invisible electromagnetically.
Note the special distortion pattern of the
field inside the cloak.
15
Harry Potter Film  Reality
16
8
Harry Potter vs. Invisible Man
• Harry Potter’s Invisible Cloak: "most of the techniques used to develop invisibility
cloaks take advantage of the extraordinary properties of certain materials so that light
dodges the object to be invisible."
• The Invisible Man: This model to make things invisible precludes the use of potions,
since light is forced to interact with the object when exposed to light. "We have used
another technique, plasmonic invisibility, which causes the object and the potion to
become invisible together ," Serna describes.
17
EEE 307
• We have finished EEE 307!
• You can email me your questions/concerns at anis@eee.buet.ac.bd.
Please use “EEE 307” in the subject line.
• You can also meet me in my office if you have something to
discuss.
• Good luck!
18
9