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 Incoming links: Thomas-Fermi screening length

Use Thomas-Fermi screening, treating a homogeneous three-dimensional conductor in static linear response. For a positive test charge the screened electric potential and bare Coulomb potential are

V (r) = \frac{Q}{4 π ε _{b} r} e^{- r / λ_{TF}}, V_{bare} (r) = \frac{Q}{4 π ε _{b} r} .

Here

ε_{b}

is the background permittivity before the mobile carriers screen; it is

ϵ_{0}

in the simplest free-electron model. Both curves have the same leading

1/ r

divergence near the origin, while the screened curve lies below the bare one and decays much faster.

A positive potential attracts Electrons. If

g (E_{F})

δ n = e g (E_{F}) V

and

ρ_{ind} = - e^{2} g (E_{F}) V

λ_{TF} = \frac{ε _{b}}{e ^{2} g ( E _{F} )} .

A larger density of states lets the conductor rearrange more charge for the same potential change and shortens the screening length. For a parabolic free electron model,

g (E_{F}) = 3 n / (2 E_{F})

, hence

λ_{TF}^{2} = 2 ε_{b} E_{F} / (3 n e^{2})

The exponential form is the intended local screening model. A microscopic zero-temperature metal also has Friedel oscillations, so the exponential alone is not an exact description at every length scale.

The Thomas-Fermi screening length is the exponential decay scale of a statically screened potential in the local degenerate-electron approximation. For a three-dimensional parabolic free electron model,

g (E_{F}) = 3 n / (2 E_{F})

, so

λ_{TF}^{2} = 2 ε_{b} E_{F} / (3 n e^{2})

. At fixed mass and background permittivity it scales as

n^{- 1/6}

, rather than the classical Debye density power at fixed temperature.