1. • Bohr quantized the atom…
•An atom has a set of
energy levels
•Some (but not all)
occupied by electrons
2. Band Theory of Solids
• What happens in crystalline solids when we bring
atoms so close together that their valence electrons
constitute a single system of electrons?
• The energy levels of the overlapping electron shells
are all slightly altered.
• The energy differences are very small, but enough so
that a large number of electrons can be in close
proximity and still satisfy the Pauli’s Exclusion
Principle.
• The result is the formation of energy bands,
consisting of many states close together but slightly
split in energy.
3. Band Theory of Solids
• Each atom in the system produces another energy level in
the band structure
• This formation of bands is mostly a feature of the
outermost electrons (valence electrons) in the
atom, which are the ones responsible for chemical
bonding and electrical conductivity.
• A solid has an infinite number of allowed bands,
just as an atom has infinitely many energy levels.
4. Band Theory of Solids
• The energy levels are so close together that for all
practical purposes we can consider bands as a
continuum of states, rather than discrete
energy levels as we have in isolated atoms.
5. When you bring two sodium
atoms together, the 3s energy
level splits into two separate
energy levels.
Things to note: 4 quantum
states but only 2 electrons.
You could minimize electron energy by putting both 3s electrons
in the lower energy level, one spin up and the other spin down.
There is an internuclear separation which minimizes electron
energy. If you bring the nuclei closer together, energy
increases.
6. When you bring five sodium
atoms together, the 3s
energy level splits into five
separate energy levels.
The three new energy
levels fall in between the
two for 2 sodiums.
There are now 5 electrons occupying these energy levels.
Notice how the sodium-sodium internuclear distance must
increase slightly.
7. When you bring N (some
big number) sodium atoms
together, the 3s energy
level splits into N separate
energy levels.
The result is an energy
band, containing N very
closely-spaced energy
levels.
There are now N electrons occupying this 3s band. They go
into the lowest energy levels they can find.
The shaded area represents available states, not filled states.
At the selected separation, these are the available states.
8. Take a closer look at the energy
levels in solid sodium.
Remember, the 3s is the
outermost occupied level.
When sodium atoms are
brought within about 1 nm of
each other, the 3s levels in the
individual atoms overlap
enough to begin the formation
of the 3s band.
The 3s band broadens as the
separation further decreases.
3s band
begins to
form
9. As an aid to visualization, we often represent energy bands like this (using
sodium as an example):
1s
2s
2p
3s
This is highly schematic. Real bands
aren't boxes or lines.
Sodium has a single 3s electron, so
the 3s energy band contains twice as
many states as there are electrons.
The band is half full.
At T=0 the band is filled exactly halfway up, and the Fermi level, εF, is right
in the middle of the band.
εF
Sodium is a metal because an applied field can easily give energy to and
accelerate an electron.
10. Showing how electronic band structure comes
about by the hypothetical example of a large
number of carbon atoms being brought together
to form a diamond crystal.
11. Band Theory of Solids
• The most important bands and band gaps—those
relevant for electronics and optoelectronics—are
those with energies near the Fermi level.
• An important parameter in the band theory is the
Fermi level, the top of the available electron
energy levels at low temperatures.
• The position of the Fermi level with the relation to
the conduction band is a crucial factor in
determining electrical properties.
12.
13.
14. Valence band
• The energy band which is formed by grouping the range of
energy levels of the valence electrons or outermost orbit
electrons is called as valence band.
• Electrons in the valence band have lower energy than the
electrons in conduction band.
• The electrons present in the valence band are loosely bound
to the nucleus of atom.
15. Conduction band
• The energy band which is formed by grouping the range
of energy levels of the free electrons is called as
conduction band.
• Generally, the conduction band is empty but when
external energy is applied the electrons in the valence
band jumps in to the conduction band and becomes free
electrons. Electrons in the conduction band have higher
energy than the electrons in valence band.
• The conduction band electrons are not bound to the
nucleus of atom.
16. Forbidden gap
• The energy gap which is present between the valence
band and conduction band by separating these two
energy bands is called as forbidden band or forbidden
gap.
• In solids, electrons cannot stay in forbidden gap because
there is no allowed energy state in this region.
Forbidden gap is the major factor for determining the
electrical conductivity of a solid.
• The energy associated with forbidden band is called
energy gap and it is measured in unit electron volt (eV).
1 eV = 1.6 × 10-19 J
17. • Empty bands do not contain electrons and therefore
are not expected to contribute to the electrical
conductivity of the material.
• Partially filled bands do contain electrons as well as
unoccupied energy levels which have a slightly higher
energy. These unoccupied energy levels enable
carriers to gain energy when moving in an applied
electric field. Electrons in a partially filled band
therefore do contribute to the electrical conductivity
of the material.
• Completely filled bands do contain plenty of
electrons but do not contribute to the conductivity of
the material. This is due to the fact that the electrons
can not gain energy since all energy levels are already
filled.
18. Classification of SOLIDS
• Conductors: material capable of carrying electric
current, i.e. material which has “mobile charge
carriers”
• Semiconductor: materials with conductivity between
that of conductors and insulators
• Insulators: materials with no or very few free charge
carriers
20. Conductors
• The materials which easily allow the flow of electric
current through them are called as conductors.
• Metals such as copper, silver, iron, aluminum etc. are
good conductors of electricity.
• In terms of the band theory of solids, metals are unique
as good conductors of electricity. In the band theory,
this is depicted as an overlap of the valence band and
the conduction band so that at least a fraction of the
valence electrons can move through the material.
21. Conductors
• In a conductor, valence band and conduction band
overlap. Therefore, there is no forbidden gap in a
conductor.
• A small amount of applied external energy provides
enough energy for the valence band electrons to move
in to conduction band.
• In conductors, large number of electrons are present
in conduction band at room temperature, I.e,
conduction band is almost full with electrons.
Whereas valence band is partially occupied with
electrons.
23. Semiconductors
• The material which has electrical conductivity between that of a
conductor and an insulator is called as semiconductor.
• Silicon, germanium and graphite are some examples of
semiconductors.
• In semiconductors, the forbidden gap between valence band and
conduction band is very small.
• It has a forbidden gap of about 1 electron volt (eV).
24. Semiconductors
• At low temperature, the valence band is
completely occupied with electrons and
conduction band is empty because the electrons
in the valence band does not have enough
energy to move in to conduction band.
Therefore, semiconductor behaves as an
insulator at low temperature.
• However, at room temperature some of the
electrons in valence band gains enough energy
in the form of heat and moves in to conduction
band.
25. • When the temperature increases, the number of valence
band electrons moving in to conduction band also
increases. This shows that electrical conductivity of the
semiconductor increases with increase in
temperature. I.e. a semiconductor has negative
temperature co-efficient of resistance.
• The resistance of semiconductor decreases with increase
in temperature.
Semiconductors
27. Insulators
• The materials which does not allow the flow of electric
current through them are called as insulators.
• Insulators are also called as poor conductors of
electricity.
• Rubber, wood, diamond, plastic are some examples of
insulators.
• Insulators such as plastics are used for coating of
electrical wires. These insulators prevent the flow of
electricity to unwanted points and protect us from
electric shocks.
28. Insulators
• Normally, in insulators the valence band is fully occupied
with electrons due to sharing of outer most orbit electrons
with the neighboring atoms. Where as conduction band is
empty, I.e, no electrons are present in conduction band.
• The forbidden gap between the valence band and conduction
band is very large in insulators. The energy gap of insulator
is approximately equal to 15 electron volts (eV).
30. Electron Mobility
• When an electrical field is applied, a force is brought to
bear on the free electrons; as a consequence, they all
experience an acceleration in a direction opposite to
that of the field, by virtue of their negative charge.
• According to quantum mechanics, there is no interaction
between an accelerating electron and atoms in a perfect
crystal lattice.
• Due to frictional forces, it counters the accelerating
electron from the external field.
31. Electron Mobility
• The friction forces result from scattering of electrons.
• Scattering causes an electron to lose kinetic energy and
to change its direction of motion.
• There is, however, some net electron motion in the
direction opposite to the field, and this flow of charge is
the Electric Current.
• The scattering phenomenon is manifested as a
resistance to the passage of an electric current.
32. Electron Mobility
• Parameters used to describe scattering:
• DRIFT VELOCITY, vd
- average electron velocity in the direction of the force imposed by
the applied field. It is directly proportional to the electric field
vd=ueE
• ELECTRON MOBILITY, ue
- the constant of proportionality, ue, is an indication of
frequency of scattering events; its units are square meters per
volt-second (m2/V-s)
