ENERGY BANDS IN SOLIDS
Energy Bands
When a number of atoms are brought close
together to form a crystal, each atom will exert an
electric force on its neighbors. As a result of this
interatomic coupling, the crystal forms a single
electronic system obeying Pauli’s exclusion
principle. Therefore, each energy level of the
isolated atom splits into as many energy levels as
there are atoms in the crystal, so that Pauli’s
exclusion principle is satisfied. The separation
between the split-off energy levels is very small.
This large number of discrete and closely spaced
energy levels form an energy band. Energy bands
are represented schematically by shaded regions
in Fig.1.2
Fig 1.1. Discrete energy levels
Fig. 1.2 Splitting of energy levels of isolated atoms into energy bands as these
atoms are brought close together to produce a crystal.
The width of a band is independent of the number of atoms in the crystal, but
the number of energy levels in a band is equal to the number of atoms in the
solid. Consequently, as the number of atoms in the crystal increases, the
separation between the energy levels in a band decreases. As the crystal
contains a large number of atoms (≈ 1029 m–3), the spacing between the
discrete levels in a band is so small that the band can be treated as continuous.
The lower energy bands are normally completely filled by the
electrons since the electrons always tend to occupy the lowest
available energy states. The higher energy bands may be
completely empty or may be partly filled by the electrons.
Pauli’s exclusion principle restricts the number of electrons that a
band can accommodate. A partly filled band appears when a
partly filled energy level produces an energy band or when a
totally filled band and a totally empty band overlap.
As the allowed energy levels of a single atom expand into energy
bands in a crystal, the electrons in a crystal cannot have energies
in the region between two successive bands. In other words, the
energy bands are separated by gaps of forbidden energy.
1. The last completely filled (at leastat T = 0
K) band is called the Valence Band
2. The next band with higher energy is the
Conduction Band The Conduction Band
can be empty or partially filed
3. The energy difference between the bottom
of the CB and the top of the VB is called
the Band Gap (or Forbidden Gap)
Thermal excitation of electrons
• The removal of an electron from the
valence band leaves behind a gap in the
electrons forming the bonds
• These act like positively charged carriers,
zand are called holes
• You can view their behavior as
resembling bubbles moving in a liquid
On the basis of the band structure, crystals can be
classified into metals, insulators, and semiconductors
Metal
A solid which contains a partly filled band structure
is called a metal. Under the influence of an applied
electric field the electrons may acquire additional
energy and move into higher states. Since these
mobile electrons constitute a current, this substance
is a good conductor of electricity and the partly
filled region is the conduction band. The electrons in
the conduction band are known as free electrons or
conduction electrons. One example of the band
structure of a metal is given in Fig. 1-3a, which
shows overlapping valence and conduction bands.
Fig. 1-3a,
Energy band diagram: METALS
Monovalent metals Divalent metals
Monovalent metals: Ag, Cu, Au → 1 e in the outermost orbital
outermost energy band is only half filled
Divalent metals: Mg, Be → overlapping conduction and valence bands
they conduct even if the valence band is full
Trivalent metals: Al → similar to monovalent metals!!!
outermost energy band is only half filled !!!
Insulator
In some crystalline solids, the forbidden
energy gap between the uppermost filled
band, calledthe valence band, and the
lowermost empty band, called the
conduction band, is very large. In such
solids, at ordinary temperatures only a few
electrons can acquire enough thermal
energy to move from the valence band into
the conduction band. Such solids are
known as insulators. Since only a few free
electrons are available in the conduction
band, an insulator is a bad conductor of
electricity. Diamond having a forbidden
gap of 6 eV is a good example of an
insulator. The energy band structure of an
insulator is schematically shown in Fig.
1.3(b). Fig. 1-3b
Semiconductor
A substance for which the width of the
forbidden energy region is relatively small (-1
eV) is called a semiconductor. Graphite, a
crystalline form of carbon but having a crystal
symmetry which is different from diamond,
has such a small value of E G , and it is a
semiconductor. The most important practical
semiconductor materials are germanium and
silicon, which have values of EG of 0.785 and
1.21 eV, respectively, at O°K. Energies of this
magnitude normally cannot be acquired from
an applied field. Hence the valence band
remains full, the conduction band empty, and
these materials are insulators at low
temperatures. However, the conductivity
increases temperature, as we explain below.
These substances are known as intrinsic (pure)
semiconductors.
Energy band diagram: SEMICONDUCTORS
Semiconductor
Conduction Band
2-3 eV
Valence Band
Elements of the 4th column (C, Si, Ge, Sn, Pb) → valence band full but no overlap of
valence and conduction bands
Diamond → PE as strong function of the position in the crystal
Band gap is 5.4 eV
Down the 4th column the outermost orbital is farther away from the nucleus and less bound
the electron is less strong a function of the position in the crystal reducing band gap
down the column
Since the band-gap energy of a crystal is s function of interatomic
spacing it is not surprising that EG depends somewhat on
temperature. I t has been determined experimentally that EG decreases
with temperature,
