Band Theory for Solids

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Band Theory of SolidsA useful way to visualize the difference between conductors,insulators and semiconductors is to plot the available energies forelectrons in the materials. Instead of having discrete energies as in thecase of free atoms, the available energy states form bands. Crucial tothe conduction process is whether or not there are electrons in theconduction band. In insulators the electrons in the valence band areseparated by a large gap from the conduction band, in conductors likemetals the valence band overlaps the conduction band, and insemiconductors there is a small enough gap between the valence andconduction bands that thermal or other excitations can bridge the gap.With such a small gap, the presence of a small percentage of a dopingmaterial can increase conductivity dramatically.

An important parameter in the band theory is the Fermi level, the topof the available electron energy levels at low temperatures. Theposition of the Fermi level with the relation to the conduction band isa crucial factor in determining electrical properties.

Closer look at bands Band gap dependence on interatomic spacing

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Energy Bands for Solids

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Band Theory for Solids

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Click on any part of the illustration for further information.

Band theory More details on diagram

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Energy Bands Comments

Band gap dependence on interatomic spacing

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Band Theory for Solids

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Insulator Energy BandsMost solid substances are insulators, and in termsof the band theory of solids this implies that thereis a large forbidden gap between the energies ofthe valence electrons and the energy at which theelectrons can move freely through the material(the conduction band).

Glass is an insulating material which may betransparent to visible light for reasons closelycorrelated with its nature as an electrical insulator.The visible light photons do not have enoughquantum energy to bridge the band gap and getthe electrons up to an available energy level in theconduction band. The visible properties of glasscan also give some insight into the effects of"doping" on the properties of solids. A very smallpercentage of impurity atoms in the glass can giveit color by providing specific available energylevels which absorb certain colors of visible light.The ruby mineral (corundum) is aluminum oxidewith a small amount (about 0.05%) of chromiumwhich gives it its characteristic pink or red colorby absorbing green and blue light.

While the doping of insulators can dramaticallychange their optical properties, it is not enough toovercome the large band gap to make them goodconductors of electricity. However, the doping ofsemiconductors has a much more dramatic effecton their electrical conductivity and is the basis forsolid state electronics.

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Semiconductor Energy BandsFor intrinsic semiconductors like siliconand germanium, the Fermi level isessentially halfway between the valence

Band Theory for Solids

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and conduction bands. Although noconduction occurs at 0 K, at highertemperatures a finite number of electronscan reach the conduction band andprovide some current. In dopedsemiconductors, extra energy levels areadded.

The increase in conductivity withtemperature can be modeled in terms ofthe Fermi function, which allows one tocalculate the population of the conductionband.

Band theory of solids

Show the effect ofimpurities

Silicon energybands

Germanium energybands

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Conductor Energy BandsIn terms of the band theory of solids,metals are unique as good conductors ofelectricity. This can be seen to be a resultof their valence electrons beingessentially free. In the band theory, thisis depicted as an overlap of the valenceband and the conduction band so that atleast a fraction of the valence electronscan move through the material.

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Band Theory for Solids

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Silicon Energy BandsAt finite temperatures, the number of electrons which reach the conductionband and contribute to current can be modeled by the Fermi function. Thatcurrent is small compared to that in doped semiconductors under the sameconditions.

Compare to germanium

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Germanium Energy BandsAt finite temperatures, the number of electrons which reach the conductionband and contribute to current can be modeled by the Fermi function. Thatcurrent is small compared to that in doped semiconductors under the sameconditions.

Compare to silicon

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Band Theory for Solids

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