2.2.3 Energy bands
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Study Aids -
In this section:
2.2.3 Energy bands
Energy bands occur in solids where the discreet energy levels of the individual
atoms merge into bands which contain a large number of closely spaced energy levels.
In this section, we first discuss the crystal structure of common semiconductors
to illustrate the fact that most semiconductors have an ordered
structure in which atoms are placed in a periodic lattice. We then
consider the Kronig-Penney model. This one dimensional model
illustrates how a periodic potential yields a set of energy bands and
energy band gaps. It is the detailed band structure of a given material
which can be directly
linked to its conducting, insulating or semiconducting behavior.
18.104.22.168 Crystals and crystal structures
Semiconductors consist of atoms which are placed in an ordered
form which is called a crystal. Crystals are identified based on
lattice structure. For instance the crystal structure of silicon
is like that of diamond and refered to as the
shown in the figure below. Each atom in the diamond lattice has a
covalent bond with four adjacent atoms which together form a
tetrahedron. This lattice can also be formed from two
lattices which are displaced along the body diagonal of the
larger cube in the figure by one quarter of that body diagonal.
Fig.2.2.4 The diamond lattice of silicon and germanium
Compound semiconductors such as GaAs and InP have a
lattice structure which is similar to that of diamond. However the
lattice contains two different types of atoms. Each atom still has four
covalent bonds, but they are bonds with atoms of the other type. This
structure is refered to as the zinc-blende lattice as shown below.
Fig.2.2.5 The zinc-blende lattice of GaAs and InP
22.214.171.124 The Kronig-Penney model
The Kronig-Penney model demonstrates that a simple one-dimensional
periodic potential yields energy bands as well as energy band gaps.
The periodic potential is shown in the figure below:
Fig.2.2.6 The periodic potential assumed in the
Kronig-Penney model. The potential barriers with width b
are spaced by a distance a+b
The analysis requires the use of Bloch functions, traveling wave
solutions multiplied with a periodic function which has
the same periodicity as the potential.
Solutions are obtained only if the following equation is satisfied:
This transcendental equation can be solved graphically as illustrated
with the figure below. Solutions are only obtained if F is between
-1 and 1 since it has to equal cos(ka).
kronigp.xls - kronigf.gif
Fig.2.2.7 Graphical solution to the Kronig-Penney
model. Shown is the function F which has to equal cos(ka)
The corresponding band structure is shown below (black curve) as well
as the energy for a free electron (green curve).
kronigp.xls - kronigp.gif
Fig.2.2.8 Energy versus k corresponding to the previous
figure. Shown are the reduced band diagram of electrons in the
periodic potential with rectangular barriers (black curve) and that
of a free electron (green curve)
The corresponding wavefunctions are shown below.
kronigpw.xls - kronigpw.gif
Fig.2.2.9 Wavefunction of an electron in a periodic
potential consisting of delta functions (yellow curve). Shown are
the real part (red curve) and the imaginary part (green curve)
of the wavefunction as well as the square of the absolute value
Note that the wavefunction resembles that of a
traveling wave which is distorted at the delta functions.
126.96.36.199 Metals, insulators and semiconductors
Once we know the bandstructure of a given material we still need to find
out which energy levels are actually occupied and whether specific bands are
empty, partially filled or completely filled.
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
In order to find the filled and empty bands we must find out how many
electrons can be fit in one band and how many electrons are available: Since
one band is due to one ore more atomic energy levels we can conclude that
the minimum number of states in a band equals twice the number of atoms
in the material. The reason for the factor of two is that even a single
energy level can contain two electrons with opposite spin.
To further simplify the analysis we assume that only the valence
electrons (the electrons in the outer shell) are of interest, while the core electrons are assumed to be
tightly bound to the atom and are not allowed to wander around in the
Four different possible scenarios are shown in the figure below:
Fig.2.2.10 Possible energy band diagrams containing one filled
or partially filled band and one empty or partially empty band. Shown are
a) a half filled band, b) two overlapping bands,
c) an almost full band separated by a small bandgap from
an almost empty band and d) a full band separated by
a large bandgap from an empty band.
A half-filled band is shown in a). This situation occurs in materials
consisting of atoms which contain only one valence electron each. A lot
of highly conducting metals satisfy this condition. Materials consisting
of atoms which contain two valence electrons can still be highly conducting
provided that the resulting filled band overlaps with an empty band. This
scenario is shown in b). No conduction is expected for scenario d) where
a completely filled band is separated from the next higher empty band
by a larger energy gap. Such materials behave as insulators. Finally
scenario c) depicts the situation in a semiconductor. The completely
filled band is now close to the next higher empty band so that electrons
can make it into the next higher band yielding an almost full band below
an almost empty band. We will call the almost full band the valence
since it is occupied by valence electrons and the almost empty band
will be called the conduction band as electrons are free to move in
this band and contribute to the conduction of the material.
188.8.131.52 Energy bands of semiconductors
As semiconductors are of primary interest in this text, we now introduce
a simplified energy band diagram for semiconductors and define some key
parameters. The diagram is shown in the figure below:
Fig.2.2.11 A simplified energy band diagram used to
describe semiconductors. Shown are the valence and conduction band
as indicated by the valence band edge Ev and the conduction
band edge Ec. The vacuum level,
EVACUUM and the electron
are also indicated on the figure.
The diagram identifies the almost-empty conduction band simply by
a line which indicates the bottom of the conduction band and is
labeled Ec. Similarly the top of the valence band is indicated with
a line labeled Ev.The energy band gap is between the two lines which
are separated by the bandgap energy Eg. The distance between the
conduction band edge Ec and the energy of a free electron (called
the vacuum level labeled Evacuum) is
quantified by the electron affinity, c
multiplied with the
electronic charge q.
Note: The actual bandstructures of semiconductors is more complex than
the reader is lead to believe by the discussion above. More details regarding the
bandstructure of germanium, silicon and gallium arsenide together with
the effective masses for the different bands can be found in section
184.108.40.206 Effective mass of electrons and holes in semiconductors
© Bart J. Van Zeghbroeck, 1996, 1997