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Energy Band Structure of Semiconductors
2.1 Electrons in energy bands
Some of the electrons from the atoms in a semiconductor can be freed from their chemical bonds
and can move through the crystal, either by being pulled by an electric field or by diffusion from
a region of high electron concentration, n measured in cm−3
, to low concentration. However,
the wave-like properties of the electrons and their collisions with atoms and other electrons are
important. The possible electron wave states that can exist in a crystal are described by a dispersion
diagram, which depicts how electron states of different momenta are dispersed over energy. In a
crystal, these allowed energies of electron waves lay within bands of energies due to the contructive
and destructive interferences of the electron waves within the periodic crystal lattice.
Band structure In a crystal, electron wave interference dictates which waves fit around the
atoms, and the energies of possible states disperse over wavevectors in complicated ways. Figure
2 gives a dispersion diagram of the electron states in GaAs. The states below E = 0 are
essentially all filled with electrons and correspond to electrons held in chemical bonds between
atoms, and these bands are referred to as the “valence band” of energies. The states that lay at
energies greater than Eg above the valence band maximum are all essentially unoccupied with
electrons, and this higher band is the “conduction band” of energies. No wave states are possible,
because of interference, in the energy range Eg, which is called the semiconductor bandgap.
For most of the semiconductor materials properties needed for devices, the states that make
a contribution to the value of the property, such as α, are those near the top of the valence band
edge EV and near the bottom of the conduction band EC, just below and above the bandgap.
Consequently, a simplified dispersion diagram is used, with the dispersion of the band edge states
fit with parabolas near these energies, as depicted in Fig. 3. An electron moving in the conduction
band occupies one of the states on the parabola. The curvature of the parabola is different from that
for an electron in free space, corresponding to a different mass, the electron effective mass mn.
Optical processes A semiconductor generally only absorbs light with photon energy hν ≥ Eg,
exciting an electron from a valence band state to a conduction band state. An electron from the
conduction band can move to an empty valence band state and emit a photon with an energy given
by the energy difference of these states. These processes are at the heart of solar cells and optical
detectors. An electron can also make this downwards transition by losing energy without emitting
a photon, for example by giving energy to a crystal lattice vibration (a phonon). Photon emission
is used for LEDs and lasers.
Band diagrams
The principle way a semiconductor material’s electronic structure is represented in a device is with
a band diagram, which plots the band edge energies EC and EV and other energy levels versus
position x (Fig. 4). Light absorption, with photon energy hν ≥ Eg, can be depicted as a valence
band electron making a transition to a conduction band state. Light with photon energy hν ≤ Eg
is not absorbed and passes through the material.
Electrons and holes Electrons tend to move to the lowest energy state available, which corresponds
to moving downwards in the band diagram. In the valence band, this means that the empty
states tend to bubble up. Because the negative charges of the valence band electrons are balanced
by the positive charges on the crystal atoms, the valence band empty state has a positive charge.
These empty states, which have a positive charge and can move around bubble-like, are called
“holes”, and they have an effective mass mp given by the valence band curvature. The concentration
of holes in a material is p. A hole moves to a lower energy state by moving upwards in a band
diagram.