Plasmon Dispersion in 3D Electron Gas: RPA Approach

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to the sixth lesson in our module on many-body formalism.

In our last lesson, we derived the effective interaction within the Random Phase Approximation, $W_{R P A}$ , by summing an infinite series of polarization bubble diagrams. This gave us a crucial result:

W_{R P A} (q, ω) = \frac{V ( q )}{1 - V ( q ) χ _{0} ( q , ω )} = \frac{V ( q )}{ϵ _{R P A} ( q , ω )}

We understood this expression as the bare Coulomb interaction, $V (q)$ , screened by the dielectric function of the interacting electron gas, $ϵ_{R P A} (q, ω)$ .

Today, we will uncover the profound physical implications of this result. The mathematical structure of $W_{R P A}$ , specifically its poles, reveals the possible collective excitations of the electron gas. Our goal is to analyze these poles and derive the dispersion relation for the most famous of these excitations: the plasmon, a quantum of collective charge density oscillation. This will provide a direct and satisfying physical payoff for the formalism we've developed.

1. Collective Excitations and Zeros of the Dielectric Function

A fundamental principle in linear response theory is that poles in response functions correspond to the natural oscillatory modes, or elementary excitations, of a system. A pole at a specific frequency $ω$ and momentum $q$ signifies that the system can sustain an excitation with that energy ( $ℏ ω$ ) and momentum ( $ℏ q$ ) without any external driving force.

Looking at the expression for $W_{R P A} (q, ω)$ , we can see that its poles arise from two sources:

Poles of $χ_{0} (q, ω)$ : The non-interacting susceptibility $χ_{0}$ describes the creation of electron-hole pairs. Its poles correspond to a continuum of single-particle excitations. These are also present in the non-interacting system.
Zeros of the denominator: A pole can emerge if the denominator vanishes, even where $χ_{0}$ itself is smooth. This occurs when $ϵ_{R P A} (q, ω) = 1 - V (q) χ_{0} (q, ω) = 0$ .

This second condition is the signature of a collective excitation. It describes a self-sustaining oscillation where the induced potential created by the density fluctuation is exactly strong enough to create the fluctuation in the first place. The particles are cooperating, or moving collectively. Our task is to solve the equation $ϵ_{R P A} (q, ω) = 0$ for the electron gas.

2. Deriving the Plasmon Frequency

To solve $ϵ_{R P A} (q, ω) = 0$ , we first need an explicit form for the non-interacting susceptibility, $χ_{0} (q, ω)$ . For a 3D electron gas, this is given by the famous Lindhard function. While its full form is complex, its behavior in certain limits is quite simple and physically illuminating.

Let's focus on the long-wavelength limit ( $q \to 0$ ) at frequencies $ω$ that are large compared to the characteristic energy of single-particle excitations ( $ℏ ω ≫ ε_{F}$ ). In this regime, it can be shown that the real part of the Lindhard function has the following asymptotic form:

Re χ_{0} (q, ω) \approx - \frac{n q ^{2}}{m ω ^{2}}

where $n$ is the electron density and $m$ is the electron mass. Notice the $q^{2}$ dependence, which reflects the fact that at long wavelengths, it becomes difficult to create low-energy particle-hole pairs.

Now, we can substitute this into our condition for a collective mode, $ϵ_{R P A} = 0$ . We also need the Fourier transform of the 3D Coulomb interaction, $V (q) = \frac{e ^{2}}{ϵ _{0} q ^{2}}$ .

7 A Perspective on Screened Coulomb Interactions ...

The paper 'A Perspective on Screened Coulomb Interactions...' provides a concise derivation of the plasma frequency by analyzing the RPA susceptibility in the long-wavelength limit. This will guide our calculation.

Please read Section 2.5, 'Properties of the plasmon'. Focus on the derivation from equation (36) to (42). The author uses a slightly different sign convention for the dielectric function (\epsilon = 1+V\chi_0), but the core physics and the final steps are identical to our approach once the correct asymptotic form of \chi_0 is used. Your goal is to see how the q^2 dependence in both V(q) and \chi_0(q, \omega) conspires to produce a finite frequency.

Let's perform the calculation. The condition $ϵ_{R P A} (q, ω) = 0$ becomes:

1 - V (q) Re χ_{0} (q, ω) = 0

1 - (\frac{e ^{2}}{ϵ _{0} q ^{2}}) (- \frac{n q ^{2}}{m ω ^{2}}) = 0

The $q^{2}$ terms remarkably cancel out:

1 + \frac{n e ^{2}}{ϵ _{0} m ω ^{2}} = 0

This equation can't be satisfied for any real $ω$ . There is a subtlety here related to sign conventions in the definition of the response function. The standard convention used in many modern texts (and consistent with our previous lesson) leads to $ϵ_{R P A} = 1 - V χ_{0}$ . However, the polarization function $Π_{0} (q, ω)$ is often used, where $χ_{0} = - Π_{0}$ . This leads to $ϵ_{R P A} = 1 + V Π_{0}$ . To avoid confusion, let's follow the derivation in the resource LINK, which explicitly derives the dielectric function.

Longitudinal Response Functions in the Random Phase ...

The notes 'Longitudinal Response Functions in the Random Phase...' derive the RPA dielectric function using time-dependent perturbation theory. It arrives at the correct form for analyzing plasmons.

Please read the section 'Plasmons (the \omega \gg v_F q limit)'. Focus on the derivation leading to equation (58). You'll see that the dielectric function is found to be \epsilon(q, \omega) = 1 - \omega_p^2 / \omega^2. This is the expression we need.

Following the result from the resource, we have:

ϵ (q \to 0, ω) = 1 - \frac{n e ^{2}}{ϵ _{0} m ω ^{2}}

Setting $ϵ = 0$ to find the collective mode frequency gives:

1 - \frac{n e ^{2}}{ϵ _{0} m ω ^{2}} = 0 ⟹ ω^{2} = \frac{n e ^{2}}{ϵ _{0} m}

This yields the celebrated formula for the plasma frequency, $ω_{p}$ :

ω_{p} = \frac{n e ^{2}}{ϵ _{0} m}

This is a striking result. In the long-wavelength limit ( $q \to 0$ ), there exists a collective excitation with a finite energy, $ℏ ω_{p}$ . For typical metals like sodium or aluminum, this energy is in the range of 5-15 eV. The existence of this "plasmon gap" is a direct consequence of the long-range nature of the Coulomb interaction, whose $1/ q^{2}$ divergence perfectly cancels the $q^{2}$ dependence of the low-momentum density response.

3. The Plasmon Dispersion Relation

So far, we have found the plasmon frequency only at $q = 0$ . To find the full dispersion relation, $ω (q)$ , we would need to retain the next order terms in the expansion of $χ_{0} (q, ω)$ . This is a rather involved calculation, but the result for small $q$ is:

ω^{2} (q) \approx ω_{p}^{2} + \frac{3}{5} v_{F}^{2} q^{2}

where $v_{F}$ is the Fermi velocity. This tells us that the plasmon energy increases quadratically with momentum for small $q$ .

The plasmon is not an infinitely sharp excitation for all momenta. It exists as a well-defined mode only as long as its energy is outside the continuum of single-particle (particle-hole) excitations. At a critical momentum $q_{c}$ , the plasmon dispersion curve enters this continuum. For $q > q_{c}$ , the plasmon can decay into an electron-hole pair. This decay mechanism is a collisionless process known as Landau damping, and the plasmon ceases to be a long-lived, well-defined quasiparticle.

The full excitation spectrum of the electron gas within RPA is beautifully summarized in the following diagram.

Excitation spectrum of the electron gas within RPA

This diagram illustrates the full excitation spectrum of an electron gas in the RPA. The shaded region represents the particle-hole continuum, where single-particle excitations can occur. The solid line starting at \omega_p for q=0 is the plasmon dispersion curve, \omega_p(q). Note that the plasmon is a well-defined, "gapped" collective mode at low momentum, clearly separated from the single-particle excitations. It eventually merges with the continuum, at which point it becomes heavily damped (Landau damping).

Test your understanding!

Suppose the electrons in our model interacted via a short-range Yukawa potential, $V (q) = \frac{g ^{2}}{q ^{2} + κ ^{2}}$ , instead of the long-range Coulomb potential. What would happen to the plasmon frequency $ω_{p}$ in the $q \to 0$ limit?

Show answer

In the $q \to 0$ limit, the Yukawa potential is finite: $V (q \to 0) = g^{2} / κ^{2}$ . The condition for the collective mode would be $1 - V (0) χ_{0} (0, ω) = 0$ . Since $χ_{0} (q \to 0, ω) \propto q^{2}$ , the condition becomes $1 - (g^{2} / κ^{2}) \times (something \times q^{2}) = 0$ . As $q \to 0$ , this can only be satisfied if $ω$ also goes to zero. Therefore, a short-range interaction would lead to a "gapless" or "acoustic" collective mode ( $ω \to 0$ as $q \to 0$ ), not a gapped plasmon. The finite plasmon frequency is a unique feature of the long-range $1/ r$ Coulomb interaction.

Conclusion

In this lesson, we have reaped the rewards of our formal work on screening. By analyzing the poles of the RPA effective interaction, we have theoretically discovered the existence of plasmons and derived their fundamental properties. This demonstrates the power of the RPA framework not just to renormalize interactions, but to predict entirely new emergent phenomena.

Key Takeaways:

Collective excitations of the electron gas correspond to the zeros of the dielectric function, $ϵ_{R P A} (q, ω) = 0$ .
Solving this equation in the long-wavelength limit ( $q \to 0$ ) for a 3D electron gas yields a collective mode with a finite frequency, the plasma frequency $ω_{p} = n e^{2} / ϵ_{0} m$ .
The finite plasmon "gap" at $q = 0$ is a direct consequence of the long-range $1/ q^{2}$ nature of the Coulomb interaction.
The plasmon has a positive dispersion, with $ω (q)$ increasing with $q$ , until it enters the particle-hole continuum and becomes strongly damped (Landau damping).

Preview of the Next Lesson

We have now completed our introductory tour of screening and collective excitations within the RPA. We have defined the irreducible polarization bubble $χ_{0}$ , used it to build the dielectric function $ϵ_{R P A}$ and the effective interaction $W_{R P A}$ , and analyzed the poles of $W_{R P A}$ to find the plasmon.

We are now equipped to return to our ultimate goal: calculating the properties of a single electron moving through this interacting medium. In the next module, we will learn how to formally connect the quantities we've derived to experimentally observable properties, starting with the optical properties of solids. We will see how plasmons and the dielectric function directly govern how materials reflect and absorb light.