---
title: "Screening, Plasmons, and the Limits of Free Electrons"
module: Free-Electron Fermi Gas
moduleNumber: 5
lessonNumber: 4
order: 504
summary: >
  A mobile electron gas rearranges to screen any foreign charge, turning the bare
  Coulomb potential into a short-ranged Yukawa form over the Thomas–Fermi length.
  Displaced collectively, the gas rings at the plasma frequency, whose quantum is
  the plasmon and whose value sets the reflectivity edge of metals. A ledger of
  free-electron successes and failures then motivates band theory.
topics: [Free-Electron Fermi Gas]
draft: false
sources:
  - book: Ashcroft & Mermin
    ref: "Ch. 3 — Failures of the Free-Electron Model; Ch. 17 — Beyond the Independent-Electron Approximation (screening)"
  - book: Kittel
    ref: "Ch. 6 — Free Electron Fermi Gas; Ch. 14 — Plasmons, Polaritons, and Polarons"
  - book: Hook & Hall
    ref: "Ch. 3 — Electrons in Metals"
---

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
$\phi(\vec r)$ 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:

$$
\mu = E_F\big[n(\vec r)\big] - e\phi(\vec r) = \text{const}.
$$

For a small potential the density change is linear in $\phi$. Writing
$E_F[n] = E_F(n_0) + (E - E_F$ correction$)$ and expanding, $\delta n =
\dfrac{\partial n}{\partial E_F}\,e\phi = g(E_F)\,e\phi$, where $g(E_F)$ is now the
density of states **per unit volume** at the Fermi surface. The induced charge
density is $\rho_{\text{ind}} = -e\,\delta n = -e^2 g(E_F)\,\phi$. Feeding this
into Poisson's equation $\nabla^2\phi = -\rho/\varepsilon_0$ with the induced
charge as the source,

$$
\nabla^2\phi = \frac{e^2 g(E_F)}{\varepsilon_0}\,\phi \equiv k_{\text{TF}}^2\,\phi,
\qquad
k_{\text{TF}}^2 = \frac{e^2 g(E_F)}{\varepsilon_0} = \frac{3n e^2}{2\varepsilon_0 E_F}.
$$

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

$$
\phi(r) = \frac{Q}{4\pi\varepsilon_0 r}\,e^{-k_{\text{TF}} r},
\qquad
\lambda_{\text{TF}} = \frac{1}{k_{\text{TF}}}.
$$

> **Definition (Thomas–Fermi screening length).** The distance $\lambda_{\text{TF}}
> = k_{\text{TF}}^{-1}$ over which the electron gas screens a static charge, with
> $k_{\text{TF}}^2 = e^2 g(E_F)/\varepsilon_0$. For a typical metal $\lambda_{\text{TF}}
> \approx 0.5$–$1\ \text{Å}$, comparable to the interatomic spacing, so the bare
> Coulomb tail is essentially gone beyond a single atomic shell.

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.

$$
% caption: The bare Coulomb potential (dashed) falls off slowly as one over r.
% The Thomas–Fermi-screened potential (solid) is the same near the charge but is
% cut off exponentially beyond the screening length lambda-TF, marked on the axis.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.2,0) node[below] {$r$};
  \draw[->, black] (0,0) -- (0,3.7) node[left] {potential};
  % bare Coulomb 1/r
  \draw[black, thick, dashed, domain=0.55:6.8, samples=100, variable=\x]
    plot ({\x},{1.9/\x});
  \node[black, anchor=west, font=\scriptsize] at (3.8,0.75) {bare Coulomb};
  % screened: (1/r) exp(-r/lambda), lambda ~ 1.4
  \draw[acc, very thick, domain=0.55:6.0, samples=120, variable=\x]
    plot ({\x},{1.9/\x*exp(-\x/1.4)});
  \node[acc, anchor=west, font=\scriptsize] at (2.3,1.55) {screened (Yukawa)};
  % screening length marker
  \draw[black, dashed] (1.4,0) -- (1.4,0.9);
  \node[black, anchor=north, font=\scriptsize] at (1.4,0) {screening length};
\end{tikzpicture}
$$

## 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 $\pm neu$ appear on the
two faces, producing a uniform restoring field $\mathcal{E} = neu/\varepsilon_0$
inside, like a parallel-plate capacitor. Each electron feels a force $-e\mathcal{E}$,
so Newton's law for the slab is

$$
m\ddot u = -e\mathcal{E} = -\frac{ne^2}{\varepsilon_0}u,
$$

simple harmonic motion at the **plasma frequency**

$$
\omega_p = \sqrt{\frac{ne^2}{\varepsilon_0 m}}.
$$

For a metallic density $n\sim 10^{29}\ \text{m}^{-3}$, $\omega_p\sim 10^{16}\
\text{rad/s}$ and $\hbar\omega_p\sim 5$–$15\ \text{eV}$ — an energy in the
ultraviolet. Quantized, this collective mode is the **plasmon**, a quantum of
plasma oscillation carrying energy $\hbar\omega_p$. Fast electrons fired through a
thin metal film lose energy in discrete lumps of $\hbar\omega_p$ (and its
multiples), the standard experimental signature seen in electron energy-loss
spectra.

$$
% caption: A plasma oscillation. The electron gas (shaded) is displaced rigidly
% by u against the fixed positive ion background, exposing surface charge that
% pulls it back. The restoring field grows with the displacement, giving simple
% harmonic motion at the plasma frequency.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % fixed positive background box
  \draw[black, thick] (0,0) rectangle (6.0,2.4);
  \node[black, anchor=south, font=\scriptsize] at (3.0,2.42) {f\/ixed positive background};
  % displaced electron slab (shifted right by u)
  \fill[acc!14] (0.7,0.15) rectangle (6.7,2.25);
  \draw[acc, very thick] (0.7,0.15) rectangle (6.7,2.25);
  \node[acc, anchor=north, font=\scriptsize] at (3.7,2.2) {electron gas};
  % displacement arrow
  \draw[black, ->, thick] (0,-0.5) -- (0.7,-0.5) node[right, font=\scriptsize] {$u$};
  % exposed charge labels
  \node[black, anchor=east, font=\scriptsize] at (-0.05,1.2) {net pos};
  \node[black, anchor=west, font=\scriptsize] at (6.75,1.2) {net neg};
  % restoring field
  \draw[acc, ->, thick] (5.4,0.6) -- (4.4,0.6) node[left, black, font=\scriptsize] {f\/ield};
\end{tikzpicture}
$$

## The dielectric function and the reflectivity edge

The same physics governs how a metal responds to light. Neglecting collisions
(valid at optical frequencies where $\omega\tau\gg 1$), an electron driven by a
field $\mathcal{E}e^{-i\omega t}$ obeys $m\ddot x = -e\mathcal{E}$, giving a
displacement $x = e\mathcal{E}/m\omega^2$ and a polarization $P = -nex$. The
resulting dielectric function is

$$
\varepsilon(\omega) = 1 - \frac{\omega_p^2}{\omega^2}.
$$

$$
% caption: The free-electron dielectric function epsilon(omega) = 1 minus
% (omega_p over omega) squared. It is negative below the plasma frequency, where
% waves are evanescent and the metal reflects, passes through zero at omega_p,
% and rises toward one above it, where the metal transmits.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.0,0) node[below right] {frequency};
  \draw[->, black] (0,-2.4) -- (0,1.8) node[left] {dielectric};
  % asymptote at 1
  \draw[black, dashed] (0,1.0) -- (6.8,1.0);
  \node[black, anchor=east, font=\scriptsize] at (6.7,1.22) {approaches 1};
  % curve 1 - 9/x^2 with plasma freq at x=3
  \draw[acc, very thick, domain=1.95:6.6, samples=120, variable=\x]
    plot ({\x},{1 - 9/(\x*\x)});
  % zero crossing
  \fill[black] (3.0,0) circle (1.8pt);
  \draw[black, dashed] (3.0,0) -- (3.0,-1.55);
  \node[black, anchor=north, font=\scriptsize] at (3.0,-1.6) {plasma freq};
  \node[acc, anchor=west, font=\scriptsize] at (1.25,-2.0) {negative};
  \node[acc, anchor=west, font=\scriptsize] at (4.6,0.2) {positive};
\end{tikzpicture}
$$

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

> **Theorem (Metallic reflectivity edge).** A collisionless electron gas has
> dielectric function $\varepsilon(\omega) = 1 - \omega_p^2/\omega^2$. It totally
> reflects electromagnetic radiation below the plasma frequency and transmits it
> above. The alkali metals accordingly turn transparent in the ultraviolet, at
> wavelengths quantitatively predicted by their electron densities.

This prediction is one of the free-electron model's cleanest successes. The
measured ultraviolet transparency thresholds of sodium, potassium, rubidium, and
caesium match $\omega_p = \sqrt{ne^2/\varepsilon_0 m}$ 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.

$$
% caption: Metal reflectivity versus frequency. Below the plasma frequency omega-p
% the dielectric function is negative and the metal reflects nearly all incident
% light; above omega-p it turns transparent and the reflectivity collapses. The
% sharp drop is the plasma (reflectivity) edge.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.2,0) node[below] {frequency};
  \draw[->, black] (0,0) -- (0,3.4) node[left] {ref\/lectivity};
  % high reflectivity plateau then drop at omega_p (x=4)
  \draw[acc, very thick] (0.15,3.0) -- (3.5,3.0);
  \draw[acc, very thick, domain=3.5:4.5, samples=60, variable=\x]
    plot ({\x},{1.5 + 1.5*tanh((4.0-\x)*3.2)});
  \draw[acc, very thick, domain=4.5:6.8, samples=40, variable=\x]
    plot ({\x},{0.55/(\x-3.4)});
  \draw[black, dashed] (4.0,0) -- (4.0,1.5);
  \node[black, anchor=north, font=\scriptsize] at (4.0,0) {plasma freq};
  \node[acc, anchor=south, font=\scriptsize] at (1.7,3.0) {ref\/lecting};
  \node[black, anchor=west, font=\scriptsize] at (4.9,0.9) {transparent};
\end{tikzpicture}
$$

## 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, $\sigma = ne^2\tau/m$ | Correct (form) |
| Linear-in-$T$ electronic heat capacity $\gamma T$ | Correct |
| Wiedemann–Franz law, Lorenz number $L_0$ | Correct for good metals |
| Ultraviolet transparency at $\omega_p$ | Quantitatively correct |
| Thermal / degeneracy pressure, compressibility | Right order of magnitude |
| Magnitude of $\gamma$ in transition metals | Wrong (needs $m^\ast$) |
| 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](/condensed-matter/crystal-structure/reciprocal-lattice-and-brillouin-zones)
provide the stage, and
[Bloch's theorem](/condensed-matter/band-theory/blochs-theorem-and-energy-bands)
opens the analysis of electrons in a periodic potential — the theory that finally
sorts metals from insulators.