Coupled dynamical Boltzmann transport equations with long-range electron-phonon and electron-electron interactions in 2D materials

This paper presents a general theoretical framework of coupled dynamical Boltzmann transport equations to demonstrate that dynamical screening effects are fundamental for accurately describing electronic transport in electrostatically doped 2D semiconductors, particularly regarding scattering from long-range electron-phonon and electron-electron interactions.

The Gist

Imagine a bustling city made of a single, ultra-thin layer of atoms (a 2D material like graphene or a special type of boron nitride). In this city, there are two main groups of residents: Electrons (the commuters trying to get to work) and Phonons (the vibrations of the city streets themselves, like the rumble of traffic or the shaking of the pavement).

The goal of this paper is to figure out how fast these commuters can travel (electrical conductivity) and what slows them down.

The Old Way of Thinking: The "Static" City

For a long time, scientists treated this city like a rigid, unchanging stage. They assumed:

1. The Streets are Static: The vibrations (phonons) were always in a perfect, calm rhythm, never getting disturbed by the commuters.
2. The Commuters are Alone: They ignored the fact that commuters bump into each other (electron-electron interactions).
3. The Screen is Blank: They didn't account for how the crowd of commuters changes the atmosphere of the city.

In this old view, if a commuter hits a pothole (a phonon), they bounce off. Simple. But this model often gave wrong answers about how fast electricity could flow, especially in these ultra-thin 2D materials.

The New Discovery: The "Living, Breathing" City

The authors of this paper, Francesco and Thibault, realized that the city is actually dynamic. It's a chaotic, living system where everything affects everything else. They built a new set of rules (equations) to describe this chaos.

Here are the three big ideas they introduced, explained with analogies:

1. The "Crowd Effect" (Dynamic Screening)

Imagine a single person walking down a quiet street. If they trip, they fall hard. But if that same person is walking through a dense crowd, the crowd might catch them or push them back up. The "crowd" (the other electrons) screens or shields the person from the full impact of the pothole.

• The Paper's Insight: In 2D materials, the electrons are so close together that they form a "cloud" that changes shape instantly. When a phonon (a street vibration) tries to slow an electron down, the electron cloud rearranges itself to soften the blow.
• The Result: If you ignore this cloud (as old models did), you think the commute is much slower than it actually is. The authors show that you must calculate how the crowd reacts in real-time to get the speed right.

2. The "Traffic Jam" (Electron-Electron Interactions)

In the old models, commuters were treated as ghosts who didn't touch each other. In reality, they are bumper-to-bumper.

• The Paper's Insight: When one electron hits a phonon and slows down, it doesn't just stop; it bumps into its neighbor, who bumps into the next one. This "traffic jam" redistributes the energy.
• The Result: Instead of one person getting stuck, the whole group adjusts. Sometimes this helps the flow (by smoothing out the traffic), and sometimes it creates new bottlenecks. The authors found that ignoring these interactions leads to big errors in predicting how well the material conducts electricity.

3. The "Shaky Street" (Anharmonicity)

This is the most technical part, but here's the simple version:
Usually, scientists assume the streets vibrate in a perfect, predictable sine wave (like a metronome). But in reality, the streets are "shaky" and irregular (anharmonic).

• The Problem: In 2D materials, the "street vibrations" (phonons) get so mixed up with the "commuters" (electrons) that they stop looking like neat waves. They become a messy blur.
• The Paper's Solution: The authors developed a way to measure exactly how much of that messy blur is actually a "street vibration" versus just "electron noise." They call this the "Phonon Content."
• Why it matters: You can only calculate how fast electricity flows if you know exactly how much the streets are shaking. If you guess wrong about the "phonon content," your speed calculation is garbage.

The "Coupled" Dance

The biggest breakthrough is that the authors didn't just write one rulebook for the electrons and one for the phonons. They wrote two rulebooks that talk to each other.

• Book A (Electrons): "I am moving, and I am changing the street vibrations."
• Book B (Phonons): "I am vibrating, and I am changing how the electrons move."

They solved these two books simultaneously. It's like choreographing a dance where the dancers (electrons) and the music (phonons) are constantly improvising based on each other's moves.

Why Should You Care?

This isn't just abstract math. These 2D materials are the future of electronics.

• Faster Phones: If we can accurately predict how electrons move in these materials, we can design chips that are faster and cooler.
• Better Sensors: Understanding these interactions helps us build better sensors for detecting light or heat.
• Superconductors: The same physics that slows down electrons here is also the key to making materials that conduct electricity with zero resistance (superconductivity).

The Bottom Line

The paper says: "Stop treating 2D materials like static, rigid grids. They are fluid, dynamic, and chaotic. To understand them, you have to watch the electrons and the vibrations dance together in real-time."

By doing this, they found that the speed of electricity in these materials depends heavily on the "crowd" of electrons and the specific "shakiness" of the atomic streets, correcting many previous mistakes in the field.

Technical Summary

1. Problem Statement

The paper addresses a critical gap in the theoretical description of electronic transport in two-dimensional (2D) materials and van der Waals (vdW) heterostructures. Traditional approaches to calculating carrier mobility often suffer from two main limitations:

1. Neglect of Dynamical Screening: Most ab initio calculations treat electron-phonon (e-ph) interactions using static screening (assuming electrons respond instantaneously to phonons) or ignore the dynamical generation of electron-hole pairs entirely. This fails to capture the complex interplay between long-range Fröhlich interactions and the dynamic response of free carriers.
2. The Bloch Ansatz: Standard treatments often assume phonons remain in thermal equilibrium even when an electric field is applied (Bloch's ansatz). This ignores the out-of-equilibrium dynamics of the phonon population and the feedback loop where non-equilibrium electrons drive phonons out of equilibrium, which in turn affects electron scattering.
3. Quasi-particle Definition: In 2D systems, polar phonons interacting with the electron-hole continuum do not always behave as well-defined Lorentzian quasi-particles. Their spectral shapes are heavily modified by interference with electronic excitations, making it difficult to define where anharmonic dissipation (momentum degradation) occurs.

The authors aim to develop a unified, general theory that treats electrons and electrodynamic excitations (phonons coupled with light and electron-hole pairs) on equal footing, incorporating both long-range e-ph and electron-electron (e-e) interactions with full dynamical screening.

2. Methodology

The authors propose a framework based on two coupled dynamical Boltzmann Transport Equations (BTEs): one for the electron distribution function and one for the system's electrodynamic excitations.

A. Theoretical Framework

• Coupled BTEs: The equations describe the time evolution of the electron distribution ($F_{nk}$) and the excitation distribution ($N(q, \omega)$). The collision integrals account for scattering between electrons and excitations, as well as electron-electron scattering mediated by the screened Coulomb interaction.
• Scattering Rate Derivation: The inverse lifetime ($\tau^{-1}$) is derived from the imaginary part of the electronic self-energy ($\Sigma_{el}$) within the Random Phase Approximation (RPA). The self-energy is expressed in terms of the inverse dielectric function ($\epsilon^{-1}_{tot}$), which contains the spectral weight of all electrodynamic excitations (plasmons, phonons, and electron-hole pairs).
• Peierls' Argument and Momentum Conservation: The authors invoke Peierls' argument to show that in a closed system without Umklapp scattering, momentum is conserved, leading to infinite conductivity. To obtain finite conductivity, they introduce anharmonic decay as the mechanism for momentum dissipation.
• Phonon Content ($F$): A key innovation is the definition of a "phonon content" function, $F(q, \omega)$. Since polar phonons in 2D are not always sharp Lorentzians due to strong interference with the electron-hole continuum, the authors define a frequency region where anharmonic dissipation is physically meaningful. This function quantifies the "phononic" nature of an excitation, allowing the anharmonic decay rate to be applied only where phonon excitations exist.
• Local Fields: The theory separates long-range (macroscopic) and short-range (local field) interactions. Long-range interactions are treated dynamically, while short-range interactions are treated via Matthiessen's rule, assuming they are less sensitive to dynamical screening.

B. Computational Implementation

The authors apply this theory to two cases:

1. Parabolic Band Model: A simplified model for h-BN and 2H-MoS2 with a single parabolic band and a single longitudinal optical (LO) phonon.
2. Van der Waals Electrodynamics (VED) Framework: A realistic ab initio approach applied to BN-encapsulated graphene. This utilizes layer-resolved response functions computed via Density Functional Theory (DFT) to capture the coupling between graphene electrons and remote phonons from the BN layers (both LO and out-of-plane optical ZO modes).

The solution involves an iterative process:

1. Calculate the dynamical dielectric response ($\epsilon^{-1}_{tot}$).
2. Solve the electronic BTE to find the non-equilibrium distribution ($\Phi$).
3. Solve the excitation BTE to find the non-equilibrium phonon population ($G$).
4. Iterate until convergence.

3. Key Contributions

• Unified Dynamical Treatment: The paper provides the first general ab initio methodology that simultaneously treats long-range e-ph and e-e interactions with full dynamical screening, moving beyond the static approximation.
• Non-Equilibrium Phonon Dynamics: It explicitly solves for the deviation of the phonon population from equilibrium, rejecting the standard Bloch ansatz.
• Spectral Shape Handling: The authors develop a rigorous method to handle the non-Lorentzian spectral shapes of polar phonons in 2D materials by defining a "phonon content" function. This allows for the correct application of anharmonic dissipation even when phonons are strongly hybridized with electron-hole pairs.
• Interplay of Mechanisms: The study elucidates two distinct mechanisms by which e-e interactions affect transport:
  1. Direct: e-e interactions contribute to momentum dissipation in frequency ranges where phonon content is non-zero.
  2. Indirect: The continuum of electron-hole excitations redistributes energy among electrons, flattening the non-equilibrium distribution function and altering the scattering integrals.

4. Results

The authors present results for simplified parabolic models (h-BN, MoS2) and discuss the application to BN-encapsulated graphene (detailed in an accompanying paper).

• Impact of Dynamical Screening:
  • In the intermediate doping regime (typical for experiments), dynamical screening leads to non-trivial trends in mobility that differ significantly from both "unscreened" and "statically screened" approximations.
  • For MoS2, the dynamically screened mobility is closer to the static result because the phonon frequency is low, making the perturbation appear quasi-static to electrons.
  • For h-BN, the dynamically screened mobility falls between the unscreened and static limits.
• Role of Electron-Electron Interactions:
  • At high doping, e-e interactions significantly impact mobility. The indirect mechanism (flattening of the transport lifetime $\tau_{tr}(\epsilon)$) becomes dominant, making the lifetime profile resemble that of a system with only e-e interactions rather than e-ph interactions.
  • The "direct" mechanism (momentum dissipation via e-e scattering dressed by phonons) is most effective in the intermediate doping regime where screening is weak but carrier density is sufficient.
• Phonon Content Analysis:
  • The study reveals that in 2D, the spectral shape of polar phonons is heavily distorted by the electron-hole continuum. The "phonon content" function successfully isolates the regions where anharmonic decay can occur, preventing unphysical contributions to conductivity from low-energy electronic tails.

5. Significance

This work represents a major advancement in the theoretical understanding of transport in low-dimensional materials.

• Correct Quantification: It demonstrates that neglecting dynamical screening and e-e interactions leads to inaccurate predictions of carrier mobility, particularly in the doping ranges relevant for device operation.
• Beyond Quasi-particles: By moving away from the assumption of sharp phonon quasi-particles, the theory provides a more robust framework for describing transport in materials where electronic and vibrational responses are strongly coupled.
• Broader Implications: The methodology is generalizable to complex vdW heterostructures. The findings suggest that dynamical screening effects are crucial not only for transport but also for understanding Raman scattering, hot carrier relaxation, and potentially superconductivity in 2D systems.
• Open Source: The authors have implemented this methodology in an open-source code, facilitating its adoption by the broader community for realistic material simulations.

In conclusion, the paper establishes that a correct description of electronic transport in 2D materials requires a self-consistent, dynamical treatment of both electrons and excitations, acknowledging that the "phonon" is a dressed entity whose properties and dissipation mechanisms are fundamentally altered by the presence of free carriers.