Screening, Plasmons, and the Limits of Free Electrons

Two effects distinguish a gas of interacting electrons from the independent particles counted so far, and both are collective. Drop a foreign charge into the metal and the electrons swarm around it, cancelling its field beyond a fraction of a nanometre — screening. Displace the whole gas rigidly against the ion background and it oscillates as one, ringing at a frequency in the ultraviolet — the plasma oscillation, whose quantum is the plasmon. Both follow from the same electron density responding to an electrostatic potential, and together they explain why metals are opaque mirrors in the visible and transparent in the ultraviolet. Closing the module, they set up the ledger of what free electrons get right and wrong.

Thomas–Fermi screening

Place a fixed test charge in the metal, producing an electrostatic potential $ϕ (r)$ that varies slowly on the scale of the interelectron spacing. The electrons respond by shifting their local density. In equilibrium the electrochemical potential must be flat, so the local Fermi energy rises where the potential lowers the electron energy:

μ = E_{F} [n (r)] - e ϕ (r) = const .

For a small potential the density change is linear in $ϕ$ . Writing $E_{F} [n] = E_{F} (n_{0}) + (E - E_{F}$ correction $)$ and expanding, $δ n = \frac{\partial n}{\partial E _{F}} e ϕ = g (E_{F}) e ϕ$ , where $g (E_{F})$ is now the density of states per unit volume at the Fermi surface. The induced charge density is $ρ_{ind} = - e δ n = - e^{2} g (E_{F}) ϕ$ . Feeding this into Poisson's equation $\nabla^{2} ϕ = - ρ / ε_{0}$ with the induced charge as the source,

\nabla^{2} ϕ = \frac{e ^{2} g ( E _{F} )}{ε _{0}} ϕ \equiv k_{TF}^{2} ϕ, k_{TF}^{2} = \frac{e ^{2} g ( E _{F} )}{ε _{0}} = \frac{3 n e ^{2}}{2 ε _{0} E _{F}} .

The screened potential of a point charge $Q$ is the spherically symmetric solution that decays at infinity — the Yukawa (screened Coulomb) form:

ϕ (r) = \frac{Q}{4 π ε _{0} r} e^{- k_{TF} r}, λ_{TF} = \frac{1}{k _{TF}} .

Because $g (E_{F}) \propto n^{1/3}$ , denser electron gases screen more tightly. The consequence is far-reaching: the long-ranged $1/ r$ Coulomb interaction that would make the many-electron problem intractable is cut to a short-ranged potential, which is precisely why treating the electrons as independent — as the Sommerfeld model does — works as well as it does. Screening also explains why the residual scattering off charged impurities is weak, and why the ion–ion interaction that builds the crystal is effectively short-ranged.

The bare Coulomb potential (dashed) falls off slowly as one over r. The Thomas–Fermi-screened potential (solid) is the same near the charge but is cut off exponentially beyond the screening length lambda-TF, marked on the axis.

Plasma oscillations and the plasmon

Screening is the static response; the dynamic response is a collective oscillation. Imagine displacing the entire electron slab by a small distance $u$ relative to the fixed positive background. Surface charges $\pm n e u$ appear on the two faces, producing a uniform restoring field $E = n e u / ε_{0}$ inside, like a parallel-plate capacitor. Each electron feels a force $- e E$ , so Newton's law for the slab is

m \overset{u}{¨} = - e E = - \frac{n e ^{2}}{ε _{0}} u,

simple harmonic motion at the plasma frequency

ω_{p} = \frac{n e ^{2}}{ε _{0} m} .

For a metallic density $n \sim 1 0^{29} m^{- 3}$ , $ω_{p} \sim 1 0^{16} rad/s$ and $ℏ ω_{p} \sim 5$ – $15 eV$ — an energy in the ultraviolet. Quantized, this collective mode is the plasmon, a quantum of plasma oscillation carrying energy $ℏ ω_{p}$ . Fast electrons fired through a thin metal film lose energy in discrete lumps of $ℏ ω_{p}$ (and its multiples), the standard experimental signature seen in electron energy-loss spectra.

A plasma oscillation. The electron gas (shaded) is displaced rigidly by u against the fixed positive ion background, exposing surface charge that pulls it back. The restoring field grows with the displacement, giving simple harmonic motion at the plasma frequency.

The dielectric function and the reflectivity edge

The same physics governs how a metal responds to light. Neglecting collisions (valid at optical frequencies where $ω τ ≫ 1$ ), an electron driven by a field $E e^{- iω t}$ obeys $m \overset{x}{¨} = - e E$ , giving a displacement $x = e E / m ω^{2}$ and a polarization $P = - n e x$ . The resulting dielectric function is

ε (ω) = 1 - \frac{ω _{p}^{2}}{ω ^{2}} .

The free-electron dielectric function epsilon(omega) = 1 minus (omega_p over omega) squared. It is negative below the plasma frequency, where waves are evanescent and the metal reflects, passes through zero at omega_p, and rises toward one above it, where the metal transmits.

Its sign controls propagation. For $ω < ω_{p}$ , $ε (ω) < 0$ , the refractive index $ε$ is imaginary, and an incident wave decays evanescently: the metal reflects almost perfectly. For $ω > ω_{p}$ , $ε (ω) > 0$ is real and positive, and the metal becomes transparent. The crossover at $ω = ω_{p}$ , where $ε = 0$ , is the reflectivity edge (or plasma edge).

This prediction is one of the free-electron model's cleanest successes. The measured ultraviolet transparency thresholds of sodium, potassium, rubidium, and caesium match $ω_{p} = n e^{2} / ε_{0} m$ from their known densities. The same edge, in the infrared for doped semiconductors and in the visible for some conductors, underlies the colour of gold and copper and the design of plasma mirrors.

Metal reflectivity versus frequency. Below the plasma frequency omega-p the dielectric function is negative and the metal reflects nearly all incident light; above omega-p it turns transparent and the reflectivity collapses. The sharp drop is the plasma (reflectivity) edge.

The ledger: successes and failures

Assembled across the module, the free-electron Fermi gas has an impressive record and a set of pointed failures. The successes are everything that depends only on counting states at the Fermi surface and on a single relaxation time; the failures are everything that depends on which states those are — on the shape of the Fermi surface and the periodic potential that shapes it.

Free-electron result	Verdict
Ohm's law, $σ = n e^{2} τ / m$	Correct (form)
Linear-in- $T$ electronic heat capacity $γ T$	Correct
Wiedemann–Franz law, Lorenz number $L_{0}$	Correct for good metals
Ultraviolet transparency at $ω_{p}$	Quantitatively correct
Thermal / degeneracy pressure, compressibility	Right order of magnitude
Magnitude of $γ$ in transition metals	Wrong (needs $m^{*}$ )
Sign of the Hall coefficient (Be, Zn, Cd, Al)	Wrong (holes)
Magnetoresistance	Wrong (predicts none)
Why some solids are insulators / semiconductors	No answer
Band gaps, optical absorption edges	No answer

Every failure in the lower block traces to the one ingredient omitted so far: the periodic potential of the ions. It reshapes the free-electron parabola into energy bands separated by gaps, bends the Fermi surface where it meets the Brillouin-zone boundary, and turns some carriers into holes. That programme occupies the next module. The reciprocal lattice and Brillouin zones built in crystal structure provide the stage, and Bloch's theorem opens the analysis of electrons in a periodic potential — the theory that finally sorts metals from insulators.