The document discusses band theory of solids and key concepts like the Fermi level, band gaps, and conduction in different materials. It explains that the Fermi level describes the highest occupied electron energy level at absolute zero. Materials are classified based on their band gap as conductors, semiconductors, or insulators. Conductors have a small or overlapping band gap allowing conduction, while insulators have a large gap preventing conduction except at very high temperatures. Semiconductors have an intermediate gap size and can conduct at higher temperatures when electrons are thermally excited across the gap.

1. Band Theory of Solids
-BY KHAN RAHEELA
MSC PART I

2. FERMI LEVEL:
• Fermi level" is the term used to describe the top of the collection of electron
energy levels at absolute zero temperature. This concept comes from Fermi
direc statisticss.
• Electrons are fermions and by the pauli exclusion principle cannot exist in
identical energy states. So at absolute zero they pack into the lowest
available energy states and build up a "Fermi sea" of electron energy states.
• The Fermi level is the surface of that sea at absolute zero where no
electrons will have enough energy to rise above the surface.
• The concept of the Fermi energy is a crucially important concept for the
understanding of the electrical and thermal properties of solids. Both
ordinary electrical and thermal processes involve energies of a small
fraction of an electron volt. But the Fermi energies of metals are on the
order of electron volts.
• This implies that the vast majority of the electrons cannot receive energy
from those processes because there are no available energy states for them
to go to within a fraction of an electron volt of their present energy. Limited
to a tiny depth of energy, these interactions are limited to "ripples on the
Fermi sea".

4. Band Theory of Solids
• The sum of them is shown in the
figure
• These two possible combinations
represent two possible states of
two atoms system with different
energies
• Once the atoms are brought together the
wavefunctions begin to overlap
– There are two possibilities
1. Overlapping wavefunctions are the same (e.g., ψs
+
(r))
2. Overlapping wavefunctions are different

5. WHAT IS BAND GAP :
• The last completely filled (at least at T = 0
K) band is called the Valence Band
• The next band with higher energy is the
Conduction Band
– The Conduction Band can be empty or partially
filed
• The energy difference between the bottom
of the CB and the top of the VB is called the
Band Gap (or Forbidden Gap)

6. ON THE BASIS OF THESE
SOLID CAN BE DIVIDED IN 3
TYPES :
CONDUCTOR , SEMI-
CONDUCTOS &
INSULATORS .

7. IN CONTEXT , THERE IS DIFFEENT
BAND GAP IN DIFFEENT SOLIDS .
FOR EXAMPLE : IN CONDUCTORS
THERE IS VERY NEGLIGIBLE BAND
GAP , IN SEMI CONDUCTOR ITS MORE
THAN CONDUCTOR & IN INSULATORS
ITS MUCH MUCH GREATER SO NO
CONDUCTION OCCURS .

8. ENERGY CONDUCTION IN DIFFERENT
SOLIDS .

9. INSULATORS:
• Consider a solid with the empty
Conduction Band
• If apply electric field
to this solid, the
electrons in the
valence band (VB)
cannot participate in
transport (no current)

10. • The electrons in the VB do not
participate in the current,
since
– Classically, electrons in the electric
field accelerate, so they acquire
[kinetic] energy
– In QM this means they must
acquire slightly higher energy
and jump to another quantum
state
– Such states must be available, i.e.
empty allowed states
– But no such state are available in the
This solid
would behave
as an insulator

11. • Consider a solid with the half filled
Conduction Band (T = 0K)
• If an electric field is
applied to this solid,
electrons in the CB do
participate in transport,
since there are plenty of
empty allowed states with
energies just above the
Fermi energy
• This solid would behave as a
conductor (metal)

12. Band Overlap
• Many materials are
conductors (metals) due to
the “band overlap”
phenomenon
• Often the higher energy
bands become so wide that
they overlap with the lower
bands
– additional electron energy
levels are then available

13. Metals:
• There is a qualitative difference between
metals, semiconductors and insulators .
– the highest energy band “containing” electrons
is only partially filled for Metals (sometimes due
to the overlap)
•Thus they are good conductors even at very low
temperatures
•The resisitvity arises from the electron scattering
from lattice vibrations and lattice defects
•Vibrations increases with temperature ⇒ higher
resistivity
•The concentration of carriers does not change
appreciably with temperature

14. Insulators, Semiconductors :
• The difference between Insulators and
Semiconductors is “quantitative”
– The difference in the magnitude of the band gap
• Semiconductors are “Insulators” with a
relatively small band gap
– At high enough temperatures a fraction of
electrons can be found in the conduction band
and therefore participate in transport

15. Insulators vs Semiconductors
• There is no difference between Insulators and
Semiconductors at very low temperatures
• In neither material are there any electrons in the
conduction band – and so conductivity vanishes in
the low temperature limit

16. Insulators vs Semiconductors
• Differences arises at high temperatures
– A small fraction of the electrons is thermally
excited into the conduction band. These
electrons carry current just as in metals
– The smaller the gap the more electrons in the
conduction band at a given temperature
– Resistivity decreases with temperature due
to higher concentration of electrons in the
conduction band
*
2
1
m
nq τ
σ
ρ
==

17. Holes
• We can “replace” electrons at the top of the
band which have “negative” mass (and
travel in opposite to the “normal”
direction) by positively charged particles
with a positive mass, and consider all
phenomena using such particles
• Such particles are called Holes
• Holes are positively charged and move in
the same direction as electrons “they
replace”

18. Hole Conduction
• To understand hole motion, one requires another
view of the holes, which represent them as
electrons with negative effective mass
• To imagine the movement of the hole think
of a row of chairs occupied by people with
one chair empty
• To move all people rise all together and move
in one direction, so the empty spot moves in
the same direction

19. Conduction
Electrical current for holes and electrons in the same direction

20.  SEMI CONDUCTORSCAN BE MADE
CONDUCTOR BY DOPING OR BY
TEMPERATURE INCREASE .
 LIKE SI AT HIGH TEMPBEHAVE AS
CONDUCTOR ASIT ISKNOWN AS
INTRINSIC SEMICONDUCTOR .
 WHILE SOME CAN BE MADE BY
ADDING OTHER ELEMENTSIN THEM
SUCH ARE KNOWN ASEXTRINSIC
SEMICONDUCTORS.
 THERE ARE TWO TYPESOF EXTRINSIC
SEMI CONDUCTORS: PTYPE AND

21. P TYPE :
THERE ARE MORE NO
OF HOLES .
THEY ARE
POSITIVELY
CHARGED .
THEY
ARE ACCEPTOR.
SEMICONDUCTORS
EG: BORON DOPED
BY SILICON .
N TYPE :
THERE ARE MORE
NO OF ELECTRONS .
THEY ARE
NEGATIVELY
CHARGED .
THEY ARE DONOR
SEMICONDUCTOS .
EG:PHOSPOROUS
DOPED IN SILICON .

22. EXAMPLE OF
SEMICONDUCTORS:

23. REFERENCES:
• http://hyperphysics.phy-
astr.gsu.edu/hbase/Solids/Fermi.html .
• https://en.wikipedia.org/wiki/Band_gap
.
• https://aip.scitation.org/doi/10.1063/1.1
663501

24. THANK YOUTHANK YOU