Band Model
1. Introduction
The Band Model is a theory for an understanding of the electronic structure of solids. The central concept
is the extension of the MO-theory: a solid is treated as an almost infinitely large molecule. Valence
electrons, which are supplied by the atoms, are delocalized and spread over the entire structure. This model
can be used to describe and explain the behavior of metallic and non-metallic solids.
2. Formation of bands
Via linear combination / overlap of N AO´s, the number of N MO´s arises, which are closely spaced in
energy and spread over the entire solid. Thus, these MO´s form energy bands with a finite width. The band
width depends on the strength of interaction between neighbouring atoms. The greater the degree of
interaction, the greater the degree of overlap which then leads to a greater energy separation of most
bonding and most antibonding MO. The bands are separated by band-gaps, i.e. values of energy for which
no MO´s are existent. The occupied band with highest energy is called the valence band, and the unoccupied
band with lowest energy is called the conduction band.
3. Density of states ρ
The density of states is not uniform across a band, i.e. at some energy levels the bands are packed together
more closely than at others. In 3D the density is greatest near the center of the band and lowest at its edges.
In a band gap the density has the value of zero.
4. Fermi level
At a temperature of 0 K the electrons occupy individual MO´s of the certain bands in accordance to the
Pauli principle and Hund´s rule. The energy corresponding to the highest occupied orbital at that
temperature is called the Fermi level.
5. Metallic conductors
Two different band constellations are possible for metals: The valence band can overlap with the
conduction band (Alkaline earth metals) or the valence band is not fully occupied (Alkaline metals). In
both cases electrons can be excited (thermally or by light) into unoccupied MO´s and move freely through
the solid what leads to conduction. But the conductivity decreases with increasing temperature due to
scattering of the electrons by atomic vibration. Special case: Semimetals (such as Si, Ge, As) with a
relatively small band gap between the populated valence- and the unpopulated conduction band.
6. Semiconductors
In opposite to the metals the conductivity of semiconductors increases with increasing the temperature. At
room temperature, their conductivity lies typically between those of electric conductors and insulators
(Differentiation is a matter of the band gap size).
a.) Intrinsic semiconductors (Band gap ~ 1 - 3 eV) : The band gap between the conduction- and the
valence band is very small, so that at room temperature a certain amount of electrons populate the
conduction band. By this, positively charged holes in the underlying valence band are created, which are
also mobile. (InP, Ge…)
b.) Extrinsic semiconductors : Semiconductor properties are due to added impurities
I.) n-type semiconductors
The dopant-, or here also called donor atom, introduces additional electrons to the solid, due to the
fact that it has more valence electrons than it´s host. The filled dopant band is energetically located
near the empty conduction band of the host so that dopant electrons are supplied to the conduction
band: Electron conduction. (Example: Si : As, As introduces one additional electron)
II.) p-type semiconductors
The host is doped with an element with a fewer number of valence electrons, so that positively
charged holes are introduced to the electronic structure of the solid. The dopant atoms form an
energetically narrow, empty acceptor band that lies very close to the filled valence band which can
be populated by excited electrons from the valence band. By this, holes in the valence band are
created and therefore the remaining electrons in the valence band get mobile and cause
conductivity: Hole conduction. (Example: Si : Ga)
7. Insulators
In insulators the valence band is completely filled with electrons and has a remarkable band gap to the
upper empty conduction band. An excitation of electrons to the conduction band is not possible or affords
a large input of energy. In general, the band gap has a value higher than 3 eV. (Example: NaCl with a band
gap of 7 eV)
Questions
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Think about the colours of typical conductors, semiconductors and isolators. How is their colour
related to the size of their band gap?
Point out the essential differences between the behavior of conductors, semiconductors and
isolators with increasing temperature and explain the reasons for these differences.
Literature:
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P. W. Atkins, T. Overton, J. Rourke, M. Weller, F. Armstrong, Inorganic Chemistry, 5th edition, 2010,
Oxford University Press, Oxford
P. W. Atkins, J. de Paula, Physikalische Chemie, 4. vollständig überarbeitete Auflage, 2006, WILEY-VCH
T. V. Ramakrishnan, Electrons in Condensed Matter, Resonance, 2, 12, 17-32, 1997
W. Tremel, R. Seshadri, E. W. Finckh, Metall oder Nichtmetall?, Chem. i. u. Z., 35, 1, 42-58, 2001 (german)