The range of energies available to electrons is called a band. This section derives the relationship between band filling and electric current and explains why heating increases the number of electrons in the conduction band.
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Eg=1.0 eV。300 K→600 Kで伝導帯の電子数は何倍？
係数∝T3/2。300 K→600 Kで電子数は何倍か？
Band formation explains electric current and the effects of heating.
1The vibrations of a chain of atoms are revisited.ω(k)=2 √(Km) |sin(ka2)|……
2Electron waves are placed in a crystal.V(x + a)=V(x)⇒ψ=eikx u(x)……
3The band is derived from hopping between neighboring atoms.Tight bindingapproximation:E(k)=E₀ − 2t cos(ka)……
4Atoms with different orbital energies alternate along the chain.Band gap=2Δ……
5Electrons fill the states from the lowest energy upward.For a single k in the same band,up to 2 electrons are allowed.……
6Band filling determines the type of material.Metal，Semiconductor，Insulator……
7Expressing the electron occupation probability at temperature Tn=Nc e−(Ec − μ)/kBT，p=Nv e−(μ − Ev)/kBT……
8n is obtained from the equality of the numbers of conduction-band electrons and holes.n=√(Nc Nv) e−Eg/2kBT（n=p）……
Extension: The band gap and the color of emitted light
9The wavelength of light is calculated from the band gap energy.λ=hcEg……
What this chapter has shown
In copper, an electric field shifts the distribution of electrons in a partially filled band, producing an electric current. In diamond, the opposing flows of electrons in a filled band cancel. A band gap of 5.5 eV separates this band from the next, so almost no electric current flows at room temperature in a weak electric field.
A factor of 1.58 times ten thousandThe factor by which the number of conduction-band electrons increases
The semiconductor contains no impurities, the prefactors are constant, and the band gap is 1.0 eV. The increase from 300 K→600 K is a factor of about 1.58 times ten thousand.
In everyday lifeA thermistor measures temperature using the strong temperature dependence of a semiconductor's electrical resistance.
Detailed summary
When atoms form a regular array, the allowed electron energies form bands separated by gaps. Electrons fill the lowest energy states first. A partially filled band makes the solid a metal. A filled band makes it an insulator if the gap to the next band is large, or a semiconductor if the gap is small. The number of electrons in a semiconductor's conduction band is governed by e−Eg/2kBT and increases by orders of magnitude with even a small rise in temperature. The factor of 2 in the exponent arises because the numbers of electrons and holes are equal.