KIE 1006 : Electronic Physics
Week 3:
Energy Band, Imperfections in Solids
Lecturer : Prof. Norhayati Soin (G1), Dr. Sharifah
Fatmadiana Wan Muhd Hatta (G2), Dr Mohd Faiz
Mohd Salleh (G3)
7 April 2023
Lecture 2
1
What did we learnt last week?
volume density
= no. or atoms per unit cell / volume of unit cell
surface density
= no. of atoms per lattice plane / area of lattice plane
Space Lattices
(unit cell/primitive cell)
Lattice Points
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Crystal Planes
Lecture 2
Miller indices
2
Let’s recap what we did last week…
1. Basic crystal structures
Simple cubic
Body-centered cubic
Face-centered cubic
2. No. of atoms
Simple Cubic :
No of atoms present in this unit cell = 8 x 1/8 = 1
( Each Corner atom contributes 1/8th portion to the unit cell)
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Lecture 2
3
Let’s recap what we did last week…
Body-centered cubic
No of atoms present in this unit cell = (8 x 1/8) + 1 = 2
(Each Corner atom contributes 1/8th portion to the unit cell)
( Body centered atom is 1)
Face-centered cubic
No.of atoms present in this unit cell = (8 x 1/8) + (6 x ½) = 1 + 3 = 4
(Each Corner atom contributes 1/8th portion to the unit cell)
( Each face centered atom contributes ½ portion to the unit cell)
3. How to find Miler indices
a. Write reciprocal
b. Multiply by the lowest common denominator
c. Find the Miller plane (hkl)
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Lecture 2
4
4. Crystal Planes
• Eg. (100) plane
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(110) plane
p=1
q =1
s=
Lecture 2
(111) plane
p=1
q =1
s= 1
5
Band Theory of Solids
(Energy Bands)
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Lecture 2
6
Energy Gap and Material Classification
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Filled bands and empty bands do not allow
current flow
• Insulators have large Eg
• Semiconductors have small Eg
• Metals have no band gap
Lecture 2
– conduction band is partially
filled
7
Energy Band Diagram
Electron
Increasing electron energy
Eg
Increasing coulomb force
Hole
EC à Conduction band
à Lowest energy state for a free electron
Band Diagram Representation
Energy plotted as a function of position
EV à Valence band
à Highest energy state for filled outer shells
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EG à Band gap
à Difference in energy levels between EC and EV
à No electrons (e-) in the bandgap (only
above EC or below EV)
Lecture 2
à EG = 1.12eV in Silicon
8
Energy Band Diagram
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Lecture 2
9
Energy Band Diagram
Energy band diagram representation to show movement of
electron and holes in Si lattice:
Free
electron
Conduction
band
Heat
Energy
Energy gap
Valence
band
Hole
Electron-hole pair
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In intrinsic silicon, a few electrons can jump the energy gap
between the valence and conduction band. Having moved
into the conduction band, a “hole” (vacancy) is left in the
Lecture 2
crystal structure.
10
This week we will be learning on
7 April 2023
q Atomic Bonding
q Imperfections in Solids
q Energy band theory (Allowed and Forbidden Energy Bands, How
the bands are formed?)
q Energy Quantization, Energy-band Splitting (a bit complicated!)
q Electrical Conduction in Solids
q Density of States (DoS)
Lecture 2
11
Continuation on Crystal Solids
q Atomic Bonding
q Imperfections in Solids
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Lecture 2
12
Know your Periodic Table!
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Lecture 2
13
Atomic Bonding
• The interaction that occurs between
atoms to form solid depends on atoms
involved.
• They must be strong bond to create
solid.
• The interaction between atoms can be
described by quantum mechanics.
• Interactions between atoms can also be
described by considering the valence
electrons.
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Lecture 2
14
Atomic Bonding
Quantum Mechanics (Interaction between atoms)
• A. Ionic Bonding
• B. Covalent Bonding
• C. Metallic Bonding
• D. Van der Waals Bonding
https://www.youtube.com/watch?v=OTgpN62ou24
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Lecture 2
15
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Lecture 2
16
Metallic Bond
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Lecture 2
17
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Lecture 2
18
Atomic Bonding
1. Metal with non-metal
electron transfer and ionic
bonding
2. Non-metal with non-metal
electron sharing and covalent
bonding
3. Metal with metal
electron pooling and metallic
bonding
4. Van der wall bonding
Weakest of the chemical bonds. Low melting
temperature.
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Lecture 2
19
Atomic Bonding
• Covalent Bonding:
ØEg. Atoms in Group IV, hydrogen molecule
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Lecture 2
20
Continuation on Crystal Solids
q Atomic Bonding
q Imperfections in Solids
7 April 2023
Lecture 2
21
Imperfections in Solids
• One type of imperfections that all crystal have
is atomic thermal vibration
• Imperfections tend to alter the electrical
properties of a material
• Electrical parameters can be dominated by the
impurities
• Imperfections: atomic thermal vibrant and
point defect (vacancy, interstitial, line
dislocation)
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Lecture 2
22
Imperfections in Solids
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Lecture 2
23
Impurities in Solids
• Foreign or impurity atoms may present in
lattice
• Impurity atoms may be located at normal
lattice, which called substitutional impurities
• The technique of adding impurity atoms to a
semiconductor in order to change its
conductivity called doping
• Two method of doping: impurity diffusion and
ion implantation
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Lecture 2
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Lecture 2
25
Imperfections and Impurities
1. Atomic thermal vibration
q lattice vibration
2. Point defect
qVacancy
qInterstitial
3. Line Defect
qLine Dislocation
4. Impurities in solids
qSubstitutional
qInterstitial
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5. Doping
Lecture 2
26
Topic : Energy Band Theory
q Energy band theory (Allowed and Forbidden Energy Bands,
How the bands are formed?)
q Energy Quantization, Energy-band Splitting (a bit
complicated!)
q Electrical Conduction in Solids
q Density of Functions
7 April 2023
Lecture 2
27
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7 April 2023
Lecture 4
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Condu
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28
Band Theory of Solids
• In order to account for decreasing resistivity with increasing
temperature as well as other properties of semiconductors, a
theory known as the band theory is introduced.
• The essential feature of the band theory is that the allowed
energy states for electrons are nearly continuous over certain
ranges, called energy bands, with forbidden energy gaps
between the bands.
7 April 2023
Lecture 2
29
Energy band theory
1. Electrons move about the nucleus in
circular orbit. Each orbit corresponds to
a discrete quantity of energy.
2. As the electron orbits, it does not radiate
energy.
3. Electron only emits energy only when an
electron jumps to a higher state of
energy to a lower state of energy
(Emission).
Orbit closest to the nucleus is the
ground state . Higher up, is called
the excited state.
4. Conversely, an electron gains energy
when it jumps from a lower state of
energy to a high state energy
(Absorption).
5. The energy that an electron has at a
particular orbit is the energy level or
energy state of the electron.
Energy bands, sublevels
https://www.youtube.com/watch?v=J-DjEIlynjE
On the topic of ….Energy Levels, Energy Sublevels,
Orbitals, & Pauli Exclusion Principle
Take note on these points:
q Energy levels, sublevels and orbitals
q Pauli Exclusion Principle
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Lecture 4
31
Energy levels / Principle
Quantum numbers.
Denoted by ‘n’ eg, n=1, n=2
etc..
Energy Sublevels.
Denoted by ‘s’, ‘p’, ‘d’ and ‘f’
Max of 2
electrons in
each
orbital
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Atomic Orbitals.
Lecture 4
32
7 April 2023
Lecture 4
33
Another vid on quantum numbers, and atomic
orbitals
https://www.youtube.com/watch?v=Aoi4j8es4gQ
quantum
Take note on these points: Principle
number
q Bohr Model
q Quantum Numbers (n, l and ml and ms)
q Orbitals (s, d,f,p)
Angular momentum
quantum number
Magnetic spin number
Magnetic quantum
number
Example,
7 April 2023
Example,
Lecture 4
34
• The l value describes the shape of the orbitals
(Contains 2 axis per energy level)
(Contains 5 axis per energy level)
How the spheres extend into (and out
of ) the axis determines the number
or orbitals per energy level. The table
below is from the previous vid.:
7 April 2023
(Contains 7 axis per energy level)
Lecture 4
35
• The ml value describes how many orbitals there
are of a type of energy level – ie. Describes a
specific orbital set.
(Describes 5 orbital energy level)
7 April 2023
(Describes 3 orbital energy level)
Lecture 4
(Describes 7 orbital energy level)
36
All this helps to conclude that every atom has a unique set
of quantum numbers.ie. no two electrons in an atom can
have precisely the same four quantum numbers as stated
by the Pauli
7 April 2023
Exclusion Principle.
Ø Since no two electrons can have the same quantum number, the
discrete energy must split into a band of energies in order that
each electron can occupy a distinct quantum state.
Ø In order to accommodate all of the electrons in a crystal, then, we
must have many energy levels Lecture
within
4
the allowed band.
37
Energy Band diagram of an atom
(in the influence of the field from the nucleus)
r
P.E of two particles having charges
q1
q2
of q1 and q2 respectively separated
E = potential energy of electrostatic point particles
by distance r:
E
1
4
0
q1q2
r
ke = the Coulomb constant, k = 8.99 x 109 N·m2/C2.
q1 = charge of one of the point particles
q2 = charge of the other point particle
r = distance between the two point charges
ε0= Vacuum permittivity = 8.85 x 1012 Fm-1
Energy of electron a distance r from nucleus :
E
1
4
0
q1q2
r
Quantum Mechanics in the formation of bonds
For the two point charges, their potential energy is described by:
P.E
•
For two ‘like’
charges, the PE
becomes more +ve
as they approach
one another.
•
•
For two ‘unlike’ charges, the PE
becomes more -ve as they
approach one another. ie. as
they get closer, it requires more
energy to separate them.
•
•
E
If the nucleus is held fixed at the origin and the
electron allowed to move relative to it, the
potential energy would vary in the manner
indicated the RHS fig.
The potential energy is independent of the
direction in space and depends only on the
distance r between the electron and the nucleus.
Whether the electron moves to the right or to the
left the potential energy varies in the same
manner.
The potential energy is zero when the two
particles are very far apart (r = ¥ ), and equals
minus infinity when r equals zero. We shall take
the energy for r = ¥ as our zero of energy.
When a stable atom is formed, the electron is
attracted to the nucleus, r is less than infinity, and
the energy will be negative.
A negative value for the energy implies that
energy must be supplied to the system if the
electron is to overcome the attractive force of the
nucleus and escape from the atom
1
4
0
q1q2
r
Fig: The potential
energy of
interaction
between a
nucleus (at the
origin) and an
electron as a
function of the
distance r betwee
n them.
Ø An electron found in the ground state is assigned a negative
energy.
Ø To move an electron further from the nucleus, energy must be
applied.
Ø Hence energy becomes less negative as electron is moved
away.
Ø The minimum energy needed to move electron infinitely far from
atom is the ionization energy.
https://www.youtube.com/watch?v=_vK5KPycEvA
Bohr’s Hydrogen Model :
Potential energy of an electron in a hydrogen atom is quantized. This
means that an electron can only inhabit certain energy levels that are at
fixed distances from the nucleus.
Each atoms has energy levels at different values due to its unique no. of
protons in the nucleus and an electron will transition from one energy
level to another when a photon of a very specific energy is either
absorbed or emitted by the electron.
The energy of the photon will correspond to the difference between the
two energy levels so if the electron in a hydrogen atom goes from n=3 to
n=2 energy level, a photon will be emitted to that specific energy gap.
Rydberg constant (RH) = 2.179 X 10-18 J
7 April 2023
Change of energy in an electron during a transition.
Using this equation, we can predict the wavelength of
photon associatedLecture
with 2any possible transition for the
hydrogen atom.
41
Examples
Rydberg constant (RH) = 2.179 X 10-18 J
1. How much energy is released when an electron falls from n=4 to n=2 energy level
inside a hydrogen atom? Find the frequency of this photon and calculate the
wavelength in nm.
7 April 2023
Lecture 2
42
2. An electron in the n=3 state absorbs a photon with a wavelength of
1283.45nm. Into what energy level will the electron jump to?
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Lecture 2
43
3. Which electron transition involves the greatest release of energy?
(a) n= 2 to n=5
(b) n= 6 to n=4
(c) n=1 to n=7
(d) n=3 to n=1
(e) n=7 to n=2
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Lecture 2
44
4. Which of the following electron transitions will emit a photon in the visible
light spectrum?
(a) n= 5 to n=3
Ultraviolet
Infrared light
Visible
(b) n= 5 to n=1
(c) n=4 to n=2
(d) n=6 to n=4
(e) n=7 to n=5
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Lecture 2
45
Formation of Energy Bands
•
•
•
•
band.
7 April 2023
Lecture 4
Figure below shows radial probability density function
for the lowest electron energy state of single hydrogen
atom.
Then the next fig. shows the same probability curves for
two atoms that ate in close proximity to each other
The wave functions of the two atom electrons overlap,
which means the two electron will interact
This interaction results in discrete quantized energy ie
the energy level splitting in two discrete level (refer to
Pauli exclusion principle)
46
Eg. formation of covalent bond (H2)
E
r
1
4
0
q1q2
r
2. If atoms get closer, the electron density is
squeezed out of the centre the nuclei. Protons
expose to one another .Repulsive forces.
r
1. As atoms approach one
another, electron density
of one atoms attracted to
the nucleus of the second.
PE decreases.
3. Max .overlap of electron clouds. Most stable state
ie. Lowest in energy. Strong attraction forces.
7 April 2023
Lecture 2
48
Contents of this week
q Atomic Bonding
q Imperfections in Solids
q Energy band theory (Allowed and Forbidden Energy Bands,
How the bands are formed?)
q Energy Quantization, Energy-band Splitting (a bit
complicated!)
q Electrical Conduction in Solids
7 April 2023
Lecture 2
49