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The sharp Fermi surface gives a nonlocal oscillatory electronic response around an impurity. In a three-dimensional isotropic metal its far-field oscillations have characteristic wavevector $2k_F$ and algebraically decaying amplitude. They are absent from the purely exponential Thomas-Fermi screening potential. The local screening formula is a useful long-wavelength model, not an exact asymptotic expression at every distance.

Dielectric screening is the rearrangement of a medium's charges in response to an applied charge. In a metal, mobile Electrons accumulate near a positive perturbation and reduce its long-range potential. Thomas-Fermi screening treats the electron gas locally as an equilibrium gas whose density responds to the local electrostatic potential, ignoring gradients of the density on the microscopic wavelength scale.

Take $e>0$, background permittivity ${\epsilon }_b$, and electrostatic potential $\varphi$. An Electron's potential energy is $-e\varphi$, so the local kinetic Fermi energy becomes $\mu +e\varphi$. For a three-dimensional spin-degenerate parabolic band at zero temperature,where $g(E_F)=dn/d\mu$ is the spin-summed density of states per volume per energy. The induced charge is ${\rho }_{\mathrm{ind}}=-e\delta n=-e^{2}g(E_F)\varphi$. Combining it with Poisson equation givesFor a point external charge $Q$, spherical symmetry and decay at infinity selectThus the bare $1/r$ potential is cut off exponentially. Fourier transformation makes the same result transparent: $\varphi (\mathbf{q})={\rho }_{\mathrm{ext}}(\mathbf{q})/[{\epsilon }_b(q^2+k_{\mathrm{TF}}^2)]$, so the static longitudinal dielectric factor is $1+k_{\mathrm{TF}}^2/q^2$ relative to the background. For a free Electron band, $g(E_F)=3n/(2E_F)$ and ${\lambda }_{\mathrm{TF}}^2=2{\epsilon }_b{E}_F/(3n{e}^2)$.

The local linear approximation requires a weak perturbation and slowly varying fields on the relevant microscopic scale. It omits nonlocal Friedel oscillations and atom-scale structure; a nonlinear local-density treatment is needed near a sufficiently strong charge. Finite temperature replaces $g(E_F)$ by the appropriate compressibility derivative $\partial n/\partial \mu$. The nondegenerate limit gives classical screening rather than the degenerate Thomas-Fermi length. A larger density of states means that a smaller potential can move more carriers, hence stronger screening.

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 areHere ${\epsilon }_b$ 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 $g(E_F)$ is their total spin-summed density of states per volume and energy, $\delta n=eg(E_F)V$ and ${\rho }_{\mathrm{ind}}=-e^{2}g(E_F)V$. Inserting this charge into Poisson equation gives the Thomas-Fermi screening lengthA 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, $g(E_F)=3n/(2E_F)$, hence ${\lambda }_{\mathrm{TF}}^2=2{\epsilon }_b{E}_F/(3n{e}^2)$.

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.