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David Tong's electromagnetism lectures connect macroscopic fields with the response of matter. Their treatment of Thomas-Fermi screening derives how an electron gas rearranges around an electric potential and relates the screening length to the electronic density of states. Comparing this local approximation with Friedel oscillations shows why a useful long-wavelength theory need not reproduce the far-field oscillatory response. The example makes the assumptions behind an effective material model physically explicit.^[ref]

Thomas-Fermi screening treats the carrier density locally as that of a uniform electron gas whose chemical potential is shifted by the slowly varying electrostatic potential. Linearizing gives $\delta n=(\partial n/\partial \mu )e\phi$ for electrons. The resulting screened Poisson equation has inverse screening lengthFor a degenerate gas, $\partial n/\partial \mu =g_{\mathrm{vol}}(E_F)$ including both spins, giving Thomas-Fermi screening length $k_s^{-1}$. A nondegenerate semiconductor instead has $\partial n/\partial \mu =n/(k_{B}T)$, giving the Debye length $\sqrt{\epsilon k_{B}T/(n{e}^2)}$. Electrons and holes contribute additively to the charge susceptibility when both are present; the lattice permittivity replaces the vacuum value.

The approximation requires a potential varying slowly compared with microscopic electronic length scales and a small perturbation for the linearized result. It captures smooth electrostatic screening, but not the full nonlocal response or Friedel oscillations. Low carrier density or carrier freeze-out can make a semiconductor's screening much weaker than a metal's.

In a conductor, mobile electrons redistribute in response to an electric potential, producing an induced charge density that opposes its source. Finite electronic compressibility means a finite density change requires a change of local kinetic Fermi energy. Screening is therefore not perfect at every microscopic distance. On scales comparable to atomic structure or the Fermi wavelength, a continuum local-density model itself fails.

Thomas-Fermi screening treats electrons as a locally uniform degenerate free electron gas, replaces the ionic background by a smooth neutralizing charge, and neglects nonlocal response, exchange and correlation corrections. The electrochemical chemical potential is uniform in equilibrium. Because an electron has charge $-e$, its electrostatic energy is $-eV$, givingAt zero potential $\mu =E_{F0}$. Linearizing at fixed $\mu$, with $|eV|\ll E_{F0}$, givesInsert this in Poisson's equation to obtain $({\nabla }^2-{\xi }^{-2})V=-{\rho }_{\mathrm{free}}/{ϵ}_0$, where the Thomas-Fermi screening length isAway from a point source, direct radial differentiation shows ${\nabla }^2(e^{-r/\xi }/r)={\xi }^{-2}{e}^{-r/\xi }/r$. The decaying solution is therefore $V=C{e}^{-r/\xi }/r$. Integrating Poisson's equation over a small sphere containing charge $Q$ fixes $C=Q/(4\pi {ϵ}_0)$. For a sufficiently large source, the linearized formula is only a far-field approximation, not a justified description of its singular core.

For monovalent sodium, the electron number density is its atomic number density,This gives $E_{F0}=3.15\mathrm{eV}$ andRoom temperature is far below the corresponding Fermi temperature, so degeneracy is well justified, and sodium's simple conduction band makes a free-electron estimate useful. However $k_F\xi =0.615$ and the mean atomic spacing is $n_0^{-1/3}=0.340\mathrm{nm}$. The potential varies on a scale shorter than $k_F^{-1}$ and the atomic spacing, so the slowly varying local-density and continuum assumptions are not quantitatively controlled at the screening length itself. A nonlocal treatment includes Friedel oscillations, absent from the pure exponential.

Nearly unchanged interlayer force constants in two-layer and twenty-layer films are consistent with very local screening and bonding forces: reducing the distant bulk need not strongly change the force between adjacent layers. They also indicate that metallic electronic response can survive in the thin film. The observation does not by itself validate the numerical Thomas-Fermi approximation, establish an identical full phonon spectrum, or rule out surface-induced electronic changes; it directly constrains the measured local force constant.

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.

The static independent-electron density response of a uniform three-dimensional, two-spin Fermi gas, combined with self-consistent Poisson equation screening, gives the displayed dielectric function. For bare electron mass and vacuum background, ${\kappa }^2=m_e{e}^2{k}_F/({ϵ}_0{\pi }^2{\hslash }^2)$ andHere $F(0)=1$, $F(1)=1/2$ by continuous extension, and $F(x)\sim 1/(3x^2)$ at large $x$. A host dielectric background and band effective mass require replacing the vacuum parameters appropriately. This nonlocal response contains Friedel oscillations, which its long-wavelength Thomas-Fermi screening limit misses.

In the long-wavelength static response of a degenerate conductor, a small electric potential $\varphi$ changes the electron density by $\delta n=eg(E_F)\varphi$, where $g(E_F)$ is the spin-summed density of states per volume per energy. The induced charge is therefore $-e^{2}g(E_F)\varphi$. With background permittivity ${\epsilon }_b$, Poisson equation becomesA point charge $Q$ then has $\varphi (r)=Q{e}^{-k_{\mathrm{TF}}r}/(4\pi {\epsilon }_{b}r)$. Larger electronic density of states gives stronger screening. This local response approximation does not capture the microscopic oscillatory tail of Friedel oscillations.^[ref]