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Band Theory & Optical Properties in solids
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Band Theory
Useful way to visualize the difference between conductors, insulators and
semiconductors is to plot the available energies for electrons in the materials.
Instead of having discrete energies as in the case of free atoms, the available
energy states form bands
where m* is called the effective mass
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Electronic properties
Energy Bands models for solids
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)
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Energy Bands for Solids
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.
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Insulator Energy Bands
Most solid substances are insulators, and in terms of the band theory of
solids this implies that there is a large forbidden gap between the energies of
the valence electrons and the energy at which the electrons can move freely
through the material (the conduction band).
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Conductor Energy Bands
In terms of the band theory of solids, metals are unique as good conductors of
electricity. This can be seen to be a result of their valence electrons being
essentially free. 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.
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Semiconductor
A semiconductor material is one whose electrical properties lie in
between those of insulators and good conductors. Examples are:
germanium and silicon. In terms of energy bands, semiconductors can be
defined as those materials which have almost an empty conduction
band and almost filled valence band with a very narrow energy gap (of
the order of 1 eV) separating the two.
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Types of Semiconductors:
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Intrinsic Semiconductors
An intrinsic semiconductor is one which is made of the semiconductor material in its extremely pure form.
Examples of such semiconductors are: pure germanium and silicon which have forbidden energy gaps of 0.72
eV and 1.1 eV respectively. The energy gap is so small that even at ordinary room temperature; there are many
electrons which possess sufficient energy to jump across the small energy gap between the valence and the
conduction bands Alternatively, an intrinsic semiconductor may be defined as one in which the number of
conduction electrons is equal to the number of holes. Schematic energy band diagram of an intrinsic
semiconductor at room temperature is shown in Fig. below.
Fermi level" is the term used to describe the top of the collection of electron
energy levels at absolute zero temperature. the highest energy level which an
electron can occupy the valance band at 0k is called fermi energy.
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b. Extrinsic Semiconductors:
Those intrinsic semiconductors to which some suitable impurity or doping
agent or doping has been added in extremely small amounts (about 1 part in
108) are called extrinsic or impurity semiconductors. Depending on the type of
doping material used, extrinsic semiconductors can be sub-divided into two
classes:
(i) N-type semiconductors and
(ii) P-type semiconductors.
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N-type semiconductor:
When a small amount of pentavalent donor atoms (e.g., phosphorus (P) and Arsenic
(As)) is added, a silicon atom in the lattice may be replaced by a donor atom with
four of its valence electrons forming the covalent bounds and one extra free
electron. This is an N-type semiconductor whose conductivity is much improved
compared to the intrinsic semiconductors, due to the extra free electrons in the
lattice, which are called predominant or majority current carriers. There also exist
some tiny number of holes called minority carriers.
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P-type semiconductor:
When a small amount of trivalent acceptor atoms (e.g., boron (B) and
aluminum (Al)) is added, a silicon atom in the lattice may be replaced by an
acceptor atom with only three valence electrons forming three covalent bounds
and a hole in the lattice. This is a P-type semiconductor whose conductivity is
also much improved compared to the intrinsic semiconductors, due to the holes
in the lattice, which are called predominant or majority current carriers. There
also exist some tiny number of free electrons called minority carriers.
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Comparison of P & N type Semiconductors
N-type Semiconductor
1) Doped with pentavalent
atoms. Electrons are majority
charge carriers. Holes are
minority charge carriers.
P-type Semiconductor
1) Doped with trivalent atoms.
Holes are majority charge
carriers. Electrons are minority
charge carriers.
2) It gives out electrons, hence, 2) It accepts electrons, hence,
known as donor atoms.
known as accepter atoms.
3) Bismuth, Antimony, Arsenic
and Phosphorus are the
elements used for doping.
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3) Aluminum, Gallium, Indium
and Boron are the elements
used for doping.
PN-Junction
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P-N Junction Diode
•Construction ; It is two terminal devices consisting of a P-N junction formed either in Ge or Si
crystal. It is circuit symbol is shown in fig. (3.2-a). The P and N type regions are referred to as
anode and cathode respectively. In fig. (1-b) arrowhead indicates the conventional direction of
current flow when forward
biased. It is the same direction in which hole flow takes place.
Fig.3.2
•Working; A P-N junction diode is a one way device offering low resistance when forward biased and behaving almost
as an insulator when reverse biased. Hence such diodes are mostly used as rectifiers for converting alternating current
into direct current.
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Semiconductor Energy Bands
For intrinsic semiconductors like silicon and germanium, the Fermi level is essentially halfway
between the valence and conduction bands. Although no conduction occurs at 0 K, at higher
temperatures a finite number of electrons can reach the conduction band and provide some
current. In doped semiconductors, extra energy levels are added.
The increase in conductivity with temperature can be modeled in terms of the Fermi function,
which allows one to calculate the population of the conduction band.
There are two cases;
· Direct gap;
· Indirect gap;
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
Temperature dependence of the energy
bandgap
The energy bandgap of semiconductors tends to decrease as the temperature
is increased. The temperature dependence of the energy bandgap, Eg, has
been experimentally determined yielding the following expression for Eg as a
function of the temperature, T:
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