Semiconductor-Band Structure
Semiconductor-Band Structure
Semiconductor-Band Structure
Density of electrons is equal to the density Density of electrons is not equal to the density of holes
of holes
The bond model of electrons in silicon of valency 4 is shown below. Here, when one of the free
electrons (blue dots) leaves the lattice position, it creates a hole (grey dots). This hole thus
created takes the opposite charge of the electron and can be imagined as positive charge carriers
moving in the lattice.
where k is the wave vector and me* the effective mass of the electron. The energy Eg represents
the energy gap. The zero-energy level is chosen to lie at the top of the VB. We have used the
standard band form to describe the CB, because we are primarily interested in the energy range
close to the bottom of the band, since it is this range which contains most of the electrons. The
energy of the VB may be written as
where mh* is the effective mass of the hole. The VB is again represented by the standard inverted
form because we are interested only in the region close to the top of the band, where most of the
holes lie. The primary band-structure parameters are thus the electron and hole masses me and mh
(the asterisks have been dropped for convenience), and the band gap Ec. Note that the masses
differ considerably from-and are often much smaller than-the freeelectron mass, and that the
energy gaps range from 0.18 eV in InSb to 3.7 eV in ZnS.
The energy gap for a semiconductor varies with temperature, but the variation is usually slight.
That a variation with temperature should exist at all can be appreciated from the fact that the
crystal, when it is heated, experiences a volume expansion, and hence a change in its lattice
constant. This, in turn, affects the band structure, whichis a sensitive function of the lattice
constant. It also follows that the gap may be varied by applying pressure, as this too induces a
change in the lattice constant. Studies of semiconductors under high pressure have, in fact,
proved very helpful in elucidating some of their properties. The conduction and valence bands in
semiconductors are related to the atomic states. When two hydrogen atoms are brought together
to form a molecule, the atomic ls state splits into two states: a low-energy bonding state and a
high-energy antibonding state. In solid hydrogen, these states broaden into bonding and
antibonding energy bands, respectively. In like fashion, the valence and conduction bands in
semiconductors are, respectively, the bonding and antibonding bands of the corresponding
atomic valence states. Thus the VB and CB in Si, for example, result from the bonding and
antibonding states of the hybrid 3s13p3. Similar remarks apply to the bands in Ge, C, and other
semiconductors.
(a) Intrinsic Semiconductor at T = 0 Kelvin, behaves like an insulator (b) At t>0, four
thermally generated electron pairs