Thomas–Fermi screening

Thomas–Fermi screening is a theoretical approach to calculating the effects of electric field screening by electrons in a solid. It is a special case of the more general Lindhard theory; in particular, Thomas–Fermi screening is the limit of the Lindhard formula when the wavevector (the reciprocal of the length-scale of interest) is much smaller than the fermi wavevector, i.e. the long-distance limit.

The Thomas-Fermi wavevector (in Gaussian-cgs units) is

where μ is the chemical potential (fermi level), n is the electron concentration and e is the elementary charge.

Under many circumstances, including semiconductors that are not too heavily doped, n∝e^μ/k_BT, where k_B is Boltzmann constant and T is temperature. In this case,

i.e. 1/k₀ is given by the familiar formula for Debye length.

For more details and discussion, including the one-dimensional and two-dimensional cases, see the article: Lindhard theory.

The internal chemical potential (closely related to fermi level, see below) of a system of electrons describes how much energy is required to put an extra electron into the system, neglecting electrical potential energy. A basic fact is: As the number of electrons in the system increases (other things equal), the internal chemical potential increases. This is largely because electrons satisfy the Pauli exclusion principle: Lower-energy electron states are already full, so the new electrons must occupy higher- and higher-energy states. (However, this fact is true quite generally, regardless of the Pauli exclusion principle.)

The relationship is described by a function ${\displaystyle n(\mu )}$, where n, the electron density, is a function of μ, the internal chemical potential. The exact functional form depends on the system. For example, for a three-dimensional noninteracting electron gas at absolute zero temperature, the relation is ${\displaystyle n(\mu )\propto \mu ^{3/2}}$. Proof: Including spin degeneracy,

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