Plasmons and Collective Charge Modes

A plasmon is a collective oscillation of charge density, identified mathematically by a pole of the retarded density response or screened interaction. Long-range Coulomb forces lift this mode above the neutral-fluid sound scale. The pole, its residue, and its damping must be distinguished from a merely prominent maximum in a real-frequency loss spectrum. The collective-mode and individual-excitation separation is developed in Pines and Bohm 1952, pp. 338–345.

Required background. Coulomb Screening, Dielectric Response, and RPA defines $\epsilon$ and $W$.

Helpful background. Spectral Moments and Many-Body Sum Rules supplies the oscillator-strength checks.

Collective poles of the dielectric response

Within RPA,

$$W^R(q,z)=\frac{v_q}{\epsilon(q,z)}, \qquad \epsilon(q,z)=1-v_q\Pi_0^R(q,z).$$

An undamped mode at real $\omega_q$ satisfies $\epsilon(q,\omega_q)=0$ in a region where $\operatorname{Im}\Pi_0^R=0$. Near a simple zero,

$$W^R(q,\omega)\simeq \frac{v_q}{(\omega-\omega_q)\,\partial_\omega\epsilon(q,\omega_q)+i0^+},$$

so the derivative fixes the residue. If the mode overlaps a continuum, its pole moves to the lower half of the analytically continued frequency plane, $z_q=\Omega_q-i\gamma_q$. Solving only $\operatorname{Re}\epsilon=0$ on the real axis then gives at most an estimate.

Three-dimensional plasma gap

At frequencies above the long-wavelength particle–hole continuum, number conservation and the $f$-sum rule give

$$\Pi_0^R(q,\omega)=\frac{nq^2}{m\omega^2}+O(q^4).$$

For $v_q=4\pi e^2/(\epsilon_bq^2)$,

$$\epsilon(q,\omega)=1-\frac{\omega_p^2}{\omega^2}+O(q^2), \qquad \omega_p^2=\frac{4\pi ne^2}{m\epsilon_b}.$$

Thus the three-dimensional plasmon remains gapped as $q\to0$. For a parabolic band the next RPA term is

$$\omega_{\mathrm{pl}}^2(q)=\omega_p^2+\frac35v_F^2q^2+O(q^4).$$

The coefficient beyond the gap is model dependent once band structure, correlations, or collisions are included.

Two dimensions and environmental screening

Charges confined to a plane but interacting through a three-dimensional Coulomb field have $v_q=2\pi e^2/(\epsilon_bq)$. The same high-frequency polarization gives

$$\omega_{\mathrm{pl}}^2(q)=\frac{2\pi n e^2 q}{m\epsilon_b}+O(q^2),$$

so $\omega_{\mathrm{pl}}\propto\sqrt q$. Nearby gates, dielectric interfaces, finite thickness, and multicomponent bands alter $v_q$ and can change the long-wavelength power law. The electromagnetic environment is part of the mode definition, not a secondary correction.

Landau damping and the particle–hole continuum

At zero temperature a sharp RPA plasmon is possible outside the particle–hole continuum. When its dispersion enters the continuum, decay into particle–hole pairs gives Landau damping. Collisions, disorder, interband absorption, and finite temperature create additional damping channels even before that crossing.

For weak damping, expanding $\epsilon(q,z)$ about a real root $\Omega_q$ gives

$$\gamma_q\simeq \frac{\operatorname{Im}\epsilon(q,\Omega_q)} {\partial_\omega\operatorname{Re}\epsilon(q,\Omega_q)},$$

with signs evaluated so that the retarded pole has $\gamma_q>0$. This approximation fails for a broad, asymmetric feature or near a branch point; then the complex pole must be found directly.

Spectral-weight checks

At small $q$ in three dimensions, the plasmon carries the leading $f$-sum weight because its finite gap lies above a continuum whose width shrinks as $v_Fq$. A model loss function must nevertheless satisfy the complete sum rule, including background continua and interband contributions in a solid. Pole position alone does not determine oscillator strength.

Common pitfalls

Equating a loss peak with a pole. A continuum edge can produce a maximum in $-\operatorname{Im}\epsilon^{-1}$. Inspect the analytic zero and its residue.

Using the three-dimensional formula in a layer. The $q$ dependence follows from the Coulomb kernel of the actual electromagnetic geometry.

Ignoring the order of limits. The plasma gap comes from the dynamic $q\to0$ expansion. Inserting the static polarization instead produces Thomas–Fermi screening, not a mode.

Exercises

Derive the two-dimensional dispersion

Use $v_q=2\pi e^2/(\epsilon_bq)$ and $\Pi_0\simeq nq^2/(m\omega^2)$ to solve $\epsilon=0$.

Solution

$0=1-v_q\Pi_0=1-2\pi ne^2q/(m\epsilon_b\omega^2)$. Therefore $\omega^2=2\pi ne^2q/(m\epsilon_b)$ and $\omega\propto q^{1/2}$.

Estimate weak damping

Let $\epsilon(\omega)=\epsilon_R(\omega)+i\epsilon_I(\omega)$ with $\epsilon_R(\Omega)=0$ and small $\epsilon_I$. Find the nearby retarded zero.

Solution

Put $z=\Omega-i\gamma$ and expand: $0\simeq-i\gamma\,\partial_\omega\epsilon_R(\Omega)+i\epsilon_I(\Omega)$. Hence $\gamma=\epsilon_I/\partial_\omega\epsilon_R$, with physical passivity selecting the positive result. A non-small $\gamma$ invalidates the linear expansion.

Continue

Spectral Moments and Many-Body Sum Rules checks the mode weight. Current Vertices and Ward-Consistent Response shows how conservation protects the long-wavelength structure. Hedin Equations, Screened Interactions, and GW incorporates screened interactions into the electronic self-energy.

References

• Pines, David, and David Bohm. “A Collective Description of Electron Interactions: II. Collective vs Individual Particle Aspects of the Interactions.” Physical Review 85 (1952): 338–353. DOI.

Further reading

• Giuliani, Gabriele F., and Giovanni Vignale. Quantum Theory of the Electron Liquid. Cambridge: Cambridge University Press, 2005. DOI.
• Nozières, Philippe, and David Pines. “Electron Interaction in Solids. Characteristic Energy Loss Spectrum.” Physical Review 113 (1959): 1254–1267. DOI.