Kohn Anomaly: Electron-Phonon Coupling in Phonon Dispersion

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to the final lesson in our module on electron-phonon interactions.

In our last session, we explored the concept of the polaron, where an electron and its self-induced lattice distortion form a new quasiparticle with a renormalized mass. This was a prime example of how phonons affect the properties of electrons. Today, we will flip the script and investigate the reverse scenario: how the sea of electrons in a metal affects the properties of phonons.

Our goal is to analyze the Kohn anomaly, a fascinating and sharp feature that can appear in phonon dispersion curves. This anomaly is not just a curiosity; it is a direct and powerful experimental signature of strong electron-phonon coupling, providing a window into the intricate dance between a material's electrons and its lattice vibrations.

1. The Physical Picture: Electron Screening and Fermi Surface Nesting

Imagine a lattice vibration—a phonon—with wavevector $q$ propagating through a metal. The oscillating positive ions create a periodic potential that the conduction electrons will try to screen. The effectiveness of this screening depends on how easily the electrons can rearrange themselves in response to the phonon.

The key insight, first articulated by Walter Kohn, is that this screening can become exceptionally strong for specific phonon wavevectors. A phonon with wavevector $q$ and energy $ℏ ω_{q}$ can be absorbed by an electron in state $k$ , exciting it to state $k + q$ . For this process to happen efficiently, there must be an occupied state at $k$ and an empty state at $k + q$ .

In a metal at low temperatures, the most readily available states for such transitions are near the Fermi surface. Consider a phonon with a wavevector $q$ that connects two points on the Fermi surface, $k_{F}$ and $k_{F} + q$ . This is called Fermi surface nesting. If a large portion of the Fermi surface can be "nested" by a single vector $q$ , then this phonon mode can create a vast number of low-energy electron-hole pairs. This massive electronic response leads to a dramatic screening of that particular phonon, which in turn lowers its frequency.

This effect is most pronounced in lower-dimensional systems. In a simple 1D metal, the Fermi "surface" consists of just two points, at $+ k_{F}$ and $- k_{F}$ . The wavevector $q = 2 k_{F}$ perfectly nests these two points, leading to a very strong response.

The following image provides an excellent visualization of this dimensional dependence.

Electronic Susceptibility and Kohn Anomaly in Different Dimensions — This image illustrates the connection between the electronic response and the phonon dispersion. (a) The real part of the electronic susceptibility $\chi^0(\mathbf{q})$, which quantifies the electron gas's response to a perturbation of wavevector $\mathbf{q}$. Note the sharp logarithmic divergence for the 1D case at $q = 2k_F$. (b) The resulting renormalized phonon dispersion $\omega(\mathbf{q})$. The dip at $q = 2k_F$, known as the Kohn anomaly, is sharpest in 1D, reflecting the strong divergence in the electronic response.

2. Phonon Frequency Renormalization

Let's formalize this physical picture using the perturbative approach we've developed. We want to see how the electron-phonon interaction modifies the energy of a bare phonon. This is analogous to how we calculated the polaron self-energy, but now the initial state is a single phonon in a sea of electrons, and we are calculating the correction to the phonon's energy.

The derivation is rooted in second-order perturbation theory, considering processes where a phonon is virtually absorbed and re-emitted by the electron gas.

Electron-phonon interactions

The following lecture notes from the University of British Columbia provide a concise derivation of the renormalized phonon frequency, which gets right to the heart of the Kohn anomaly.

Please read Section 2.4, 'Phonon renormalization and the Kohn anomaly' (page 11). Focus on the derivation leading to the final expression for the renormalized phonon frequency, ħω̃_q.

As you saw in the notes, the calculation leads to a renormalized phonon frequency $ω ~_{q}$ :

ℏ ω ~_{q} \approx ℏ ω_{q}^{0} + ∣ g (q) ∣^{2} \frac{1}{N} k \sum \frac{n _{F} ( E _{k} ) - n _{F} ( E _{k + q} )}{E _{k} - E _{k + q}}

where $ω_{q}^{0}$ is the bare phonon frequency, $g (q)$ is the electron-phonon coupling matrix element, and $n_{F} (E)$ is the Fermi-Dirac distribution function.

The crucial term is the sum over the electron momentum $k$ . This is precisely the static Lindhard function, $χ^{0} (q)$ , which describes the linear response of the non-interacting electron gas. This confirms our physical intuition: the renormalization of the phonon frequency is directly proportional to the susceptibility of the electron system. The features seen in $χ^{0} (q)$ in panel (a) of the image above directly translate into features in the phonon dispersion $ω (q)$ . The logarithmic divergence of $χ^{0} (q)$ at $q = 2 k_{F}$ in 1D leads to a kink with a vertical tangent in the phonon dispersion, which is the hallmark of the Kohn anomaly.

3. The Decisive Role of Coupling Strength

While Fermi surface nesting provides the condition for an anomaly at a specific $q$ , it is not, by itself, sufficient. A pronounced Kohn anomaly is a signature of strong electron-phonon coupling. The magnitude of the frequency dip depends critically on the strength of the interaction, encapsulated in the matrix element $∣ g (q) ∣^{2}$ .

This next resource provides a more detailed discussion, emphasizing that nesting alone is a poor predictor of anomalies and highlighting the central role of the coupling strength.

First-principle investigation of Kohn anomalies and Peierls ...

This thesis by Marazzi offers a fantastic discussion on why electron-phonon coupling is the 'main actor' in the formation of Kohn anomalies. It clarifies common misconceptions and presents the modern understanding of the phenomenon.

Please read Section 1.6, 'Electron-phonon coupling' (pages 12-14). Pay close attention to Eq. (1.16), which explicitly shows how the 'dressed' phonon frequency depends on both the nesting (via the logarithm term, which comes from the Lindhard function) and the electron-phonon matrix element |g|².

The key formula from the reading, which gives the renormalized frequency squared, makes this explicit:

(ω_{ν, q})^{2} = (ω_{ν, q}^{0})^{2} + C n, m \sum ∣ g_{nm, ν} (k, q) ∣^{2} χ_{nm}^{0} (q)

(This is a schematic representation of Eq. 1.15 and 1.16 from the resource). Here, $ν$ is the phonon mode index, and $n, m$ are electron band indices. This equation tells us several important things:

Mode Selectivity: The anomaly will only be strong for phonon modes $ν$ that have a large coupling matrix element $g$ . This is why, even with good nesting, some phonon branches might show a strong anomaly while others show none. The coupling strength is dictated by symmetries and the specific nature of the electronic and vibrational states.
A Measure of Coupling: For a given nesting condition, the depth of the anomaly is a direct measure of $∣ g ∣^{2}$ . Observing a deep Kohn anomaly is experimental proof of strong coupling.
Lattice Instability (CDW): If the coupling is extremely strong, the second term can become so large and negative that $(ω_{ν, q})^{2}$ becomes negative. A negative squared frequency implies an imaginary frequency, which signals an exponential growth of that particular lattice distortion. The crystal is unstable and will spontaneously distort to a new, lower-energy ground state. This is a Peierls transition, and the new state is a Charge Density Wave (CDW).

Test your understanding!

Imagine two hypothetical 1D metals.

Material A has a perfectly nested Fermi surface at $q = 2 k_{F}$ , but the electron-phonon coupling matrix element $g (q = 2 k_{F})$ for the longitudinal acoustic phonon mode is very small.
Material B has a slightly warped Fermi surface, leading to imperfect nesting. However, the coupling matrix element $g (q \approx 2 k_{F})$ for its longitudinal acoustic mode is exceptionally large.

Which material is more likely to exhibit a pronounced Kohn anomaly in its phonon spectrum, and why?

Show answer

Material B is more likely to show a pronounced Kohn anomaly. While perfect nesting (Material A) creates a divergence in the electronic susceptibility $χ^{0} (q)$ , the overall effect on the phonon frequency is the product of the susceptibility and the coupling strength $∣ g ∣^{2}$ . If $∣ g ∣^{2}$ is very small, the anomaly will be negligible.

In contrast, Material B's large coupling strength $∣ g ∣^{2}$ can amplify even the weaker electronic response from imperfect nesting, leading to a significant and easily observable frequency dip. This illustrates the principle that strong coupling is often more critical than perfect nesting.

4. Experimental Observation and Temperature Dependence

Kohn anomalies are not just a theoretical construct; they are directly observable. The primary experimental technique for measuring phonon dispersion curves is inelastic scattering, using either neutrons (INS) or X-rays (IXS). In these experiments, the energy and momentum lost by the scattered particle are transferred to create or annihilate a phonon, allowing a point-by-point mapping of the $ω (q)$ relation.

The sharpness of the anomaly is also highly dependent on temperature. At high temperatures, the Fermi-Dirac distribution is smeared out, meaning there are already many thermally excited electrons above $E_{F}$ and holes below it. This "blurs" the Fermi surface and washes out the nesting condition, causing the anomaly to become broader and shallower. As the temperature is lowered, the anomaly deepens.

Temperature Evolution of the Kohn Anomaly — Experimental data showing the temperature evolution of a Kohn anomaly. Left: Phonon dispersion measured at different temperatures. Notice how the dip at the anomaly wavevector (q_CDW) becomes progressively deeper ('softens') as the temperature is lowered from 300K to 120K. Right: The squared frequency of this 'soft mode' approaches zero at a critical temperature, signaling the onset of a lattice instability (a charge density wave transition).

This "softening" of a phonon mode upon cooling is a critical concept and a major area of research, as it can be a precursor to phase transitions like the CDW formation mentioned earlier.

Conclusion

Today we've seen how the presence of a Fermi sea of electrons can leave a dramatic imprint on the vibrational spectrum of a metal.

Key Takeaways:

The Kohn anomaly is a sharp dip or kink in a phonon dispersion curve $ω (q)$ .
It occurs at a wavevector $q$ that nests the Fermi surface, most simply $q = 2 k_{F}$ in 1D, allowing for efficient creation of low-energy electron-hole pairs.
Its physical origin is the enhanced screening of lattice vibrations by the responsive electron gas at the nesting vector.
The magnitude of the anomaly is proportional to $∣ g (q) ∣^{2}$ , the squared electron-phonon coupling matrix element, making it a key experimental signature of strong coupling.
An extremely strong anomaly where $ω (q) \to 0$ signals a lattice instability, often leading to a Charge Density Wave (CDW) phase transition.

This lesson concludes our module on Electron-Phonon Interactions. We've journeyed from seeing how phonons cause resistance, to how they can "dress" an electron to form a polaron, and finally, how electrons can re-sculpt the phonon dispersion itself.

Preview of the Next Lesson

We will now move to a new topic that directly leverages your interest in spectroscopy and connects to many concepts we've built, from band structure to screening: the Optical Properties of Solids. In our first lesson of the new module, we will derive the frequency-dependent dielectric function, $ϵ (ω)$ , for metals and doped semiconductors using the Drude model. This function is the master key to understanding how materials reflect, absorb, and transmit light.