Band Theory of Solids

A 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. Crucial to the conduction process is whether or not there are electrons in the
conduction band. In insulators the electrons in the valence band are separated by a large
gap from the conduction band, in conductors like metals the valence band overlaps the
conduction band, and in semiconductors there is a small enough gap between the valence
and conduction bands that thermal or other excitations can bridge the gap. With such a
small gap, the presence of a small percentage of a doping material can increase
conductivity dramatically.

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.

Energy Bands for Solids

Energy Bands Comments

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).

Glass is an insulating material which may be transparent to visible

light for reasons closely correlated with its nature as an electrical
insulator. The visible light photons do not have enough quantum
energy to bridge the band gap and get the electrons up to an available
energy level in the conduction band. The visible properties of glass can
also give some insight into the effects of "doping" on the properties of
solids. A very small percentage of impurity atoms in the glass can give
it color by providing specific available energy levels which absorb
certain colors of visible light. The ruby mineral (corundum) is
aluminum oxide with a small amount (about 0.05%) of chromium
which gives it its characteristic pink or red color by absorbing green
and blue light.

While the doping of insulators can dramatically change their optical

properties, it is not enough to overcome the large band gap to make
them good conductors of electricity. However, the doping of
semiconductors has a much more dramatic effect on their electrical
conductivity and is the basis for solid state electronics.
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.

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.

Silicon Energy Bands

At finite temperatures, the number of electrons which reach the conduction band and
contribute to current can be modeled by the Fermi function. That current is small
compared to that in doped semiconductors under the same conditions.
Germanium Energy Bands
At finite temperatures, the number of electrons which reach the conduction band and
contribute to current can be modeled by the Fermi function. That current is small
compared to that in doped semiconductors under the same conditions.

Intrinsic Semiconductor
A silicon crystal is different from an insulator because at any temperature above absolute
zero temperature, there is a finite probability that an electron in the lattice will be
knocked loose from its position, leaving behind an electron deficiency called a "hole".

If a voltage is applied, then both the electron and the hole can contribute to a small
current flow.

The conductivity of a semiconductor can be

modeled in terms of the band theory of solids.
The band model of a semiconductor suggests
that at ordinary temperatures there is a finite
possibility that electrons can reach the
conduction band and contribute to electrical
conduction.

The term intrinsic here distinguishes between

the properties of pure "intrinsic" silicon and the
dramatically different properties of doped n-
type or p-type semiconductors.

Semiconductor Current
Both electrons and holes contribute to current flow in an intrinsic semiconductor.
The current which will flow in an intrinsic semiconductor consists of both electron and
hole current. That is, the electrons which have been freed from their lattice positions into
the conduction band can move through the material.

In addition, other electrons can hop

between lattice positions to fill the
vacancies left by the freed electrons.
This additional mechanism is called
hole conduction because it is as if the
holes are migrating across the
material in the direction opposite to
the free electron movement.

The current flow in an intrinsic semiconductor is influenced by the density of energy

states which in turn influences the electron density in the conduction band. This current is
highly temperature dependent.

Electrons and Holes

In an intrinsic semiconductor like silicon at temperatures above absolute zero, there will
be some electrons which are excited across the band gap into the conduction band and
which can produce current. When the electron in pure silicon crosses the gap, it leaves
behind an electron vacancy or "hole" in the regular silicon lattice. Under the influence of
an external voltage, both the electron and the hole can move across the material. In an n-
type semiconductor, the dopant contributes extra electrons, dramatically increasing the
conductivity. In a p-type semiconductor, the dopant produces extra vacancies or holes,
which likewise increase the conductivity. It is however the behavior of the p-n junction
which is the key to the enormous variety of solid-state electronic devices.

The Doping of Semiconductors

The addition of a small percentage of foreign atoms in the regular crystal lattice of silicon
or germanium produces dramatic changes in their electrical properties, producing n-type
and p-type semiconductors.

Pentavalent impurities
Impurity atomw with 5 valence electrons produce n-type semiconductors by contributing
extra electrons.

Trivalent impurities
Impurity atoms with 3 valence electrons produce p-type semiconductors by producing a
"hole" or electron deficiency.

P- and N- Type Semiconductors

N-Type Semiconductor
The addition of pentavalent impurities such as
antimony, arsenic or phosphorous contributes
free electrons, greatly increasing the
conductivity of the intrinsic semiconductor.
Phosphorous may be added by diffusion of
phosphine gas (PH3).

P-Type Semiconductor
The addition of trivalent impurities such as boron,
aluminum or gallium to an intrinsic semiconductor
creates deficiencies of valence electrons,called
"holes". It is typical to use B2H6 diborane gas to
diffuse boron into the silicon material.

Bands for Doped Semiconductors

The application of band theory to n-type and p-type semiconductors shows that extra
levels have been added by the impurities. In n-type material there are electron energy
levels near the top of the band gap so that they can be easily excited into the conduction
band. In p-type material, extra holes in the band gap allow excitation of valence band
electrons, leaving mobile holes in the valence band.

P-N Junction
One of the crucial keys to solid state electronics is the nature of the P-N junction. When
p-type and n-type materials are placed in contact with each other, the junction behaves
very differently than either type of material alone. Specifically, current will flow readily
in one direction (forward biased) but not in the other (reverse biased), creating the basic
diode. This non-reversing behavior arises from the nature of the charge transport process
in the two types of materials.
The open circles on the left side of the junction above represent "holes" or deficiencies of
electrons in the lattice which can act like positive charge carriers. The solid circles on the
right of the junction represent the available electrons from the n-type dopant. Near the
junction, electrons diffuse across to combine with holes, creating a "depletion region".
The energy level sketch above right is a way to visualize the equilibrium condition of the
P-N junction. The upward direction in the diagram represents increasing electron energy.

Depletion Region
When a p-n junction is formed, some of the free electrons in the n-region diffuse across
the junction and combine with holes to form negative ions. In so doing they leave behind
positive ions at the donor impurity sites.

Depletion Region Details

In the p-type region there are holes from the acceptor impurities
and in the n-type region there are extra electrons.
 When a p-n junction is formed, some of the electrons from the n-
region which have reached the conduction band are free to diffuse
across the junction and combine with holes.

Filling a hole makes a negative ion and leaves behind a positive ion
on the n-side. A space charge builds up, creating a depletion region
which inhibits any further electron transfer unless it is helped by
putting a forward bias on the junction.

Bias effect on electrons in depletion zone

Equilibrium of junction

Coulomb force from ions prevents further

migration across the p-n junction. The electrons
which had migrated across from the N to the P
region in the forming of the depletion layer
have now reached equilibrium. Other electrons
from the N region cannot migrate because they
are repelled by the negative ions in the P region
and attracted by the positive ions in the N
region.

Reverse bias

An applied voltage with the indicated polarity

further impedes the flow of electrons across the
junction. For conduction in the device,
electrons from the N region must move to the
junction and combine with holes in the P
region. A reverse voltage drives the electrons
away from the junction, preventing conduction.
 Forward bias

An applied voltage in the forward direction as

indicated assists electrons in overcoming the
coulomb barrier of the space charge in
depletion region. Electrons will flow with very
small resistance in the forward direction.

Forward Biased P-N Junction

Forward biasing the p-n junction drives holes to the junction from the p-
type material and electrons to the junction from the n-type material. At the
junction the electrons and holes combine so that a continuous current can
be maintained.

Reverse Biased P-N Junction

The application of a reverse voltage to the p-n junction will cause a
transient current to flow as both electrons and holes are pulled away from
the junction. When the potential formed by the widened depletion layer
equals the applied voltage, the current will cease except for the small
thermal current.
The P-N Junction Diode
The nature of the p-n junction is that it will conduct current in the forward
direction but not in the reverse direction. It is therefore a basic tool for
rectification in the building of DC power supplies.

Varactor
A varactor diode uses a p-n junction in reverse bias
and has a structure such that the capacitance of the
diode varies with the reverse voltage. A voltage
controlled capacitance is useful for tuning
applications.

The capacitance is controlled by the method of

doping in the depletion layer. Typical values are from
tens to hundreds of picofarads

A popular application of the varactor is in electronic tuning circuits, as in television

tuners (see Fig). The DC control voltage varies the capacitance of the varactor, retuning
the resonant circuit.
Schottky Diode

The junction of a doped semiconductor (usually

n-type) with a metal electrode can produce a very
fast-switching diode which is mainly used in high
frequency circuits or high speed digital circuits.
Under forward bias, the electrons move from the
n-type material to the metal and give up their
energy quickly. There are no holes (minority
carriers), so the conduction quickly stops upon
change to reverse bias. Schottky diodes find
application as rectifiers for high frequency
Junction of lightly doped n-type
signals.
semiconductor with a metal electrode.

The Zener Effect

With the application of sufficient reverse voltage, a p-n junction will experience a rapid
avalanche breakdown and conduct current in the reverse direction. Valence electrons
which break free under the influence of the applied electric field can be accelerated
enough that they can knock loose other electrons and the subsequent collisions quickly
become an avalanche. When this process is taking place, very small changes in voltage
can cause very large changes in current. The breakdown process depends upon the
applied electric field, so by changing the thickness of the layer to which the voltage is
applied, zener diodes can be formed which break down at voltages from about 4 volts to
several hundred volts.
Zener Diode

The zener diode uses a p-n junction in reverse bias to

make use of the zener effect, which is a breakdown
phenomenon which holds the voltage close to a
constant value called the zener voltage. It is useful in
zener regulators to provide a more constant voltage,
for improvement of regulated power supplies, and for
limiter applications.

The Zener effect as embodied in the zener diode has many applications for control and
regulation.

Zener Regulator
The constant reverse voltage of the zener diode makes it a valuable component for the
regulation of the output voltage against both variations in the input voltage from an
unregulated power supply or variations in the load resistance. The current through the
zener will change to keep the voltage at within the limits of the threshold of zener action
and the maximum power it can dissipate.