Interplay of non-local transport and local scattering during electron thermalization and spatial equilibration in laser-excited metals

This paper employs a reformulated Boltzmann transport equation to demonstrate that while non-local transport accelerates apparent thermalization at the irradiated surface by removing athermal carriers, it simultaneously delays the complete equilibration of the entire electron system, revealing a complex interplay between transport and scattering that varies with position and energy.

The Gist

Imagine a crowded dance floor inside a metal block. Suddenly, a super-fast laser pulse hits the surface, like a DJ dropping a massive beat that only the dancers right at the front can hear. These "dancers" (electrons) get excited, jumping up and moving wildly, while the ones further back in the room are still sitting calmly.

This paper is about what happens next: How do these excited dancers calm down, and how does the energy spread through the whole room?

The Two Main Forces at Play

The researchers found that two main things are happening simultaneously, and they often work against each other:

1. The "Local Scattering" (The Dance Floor Bump): The excited dancers bump into each other and the walls of the room (the metal's atomic structure). This is like a chaotic mosh pit where everyone eventually slows down and starts dancing in a synchronized, calm rhythm. This is thermalization.
2. The "Non-Local Transport" (The Crowd Surge): Because the laser only hit the front, the excited dancers at the front are crowded and energetic, while the back is empty. Naturally, the energetic dancers from the front start running toward the back to fill the empty space. This is transport.

The Big Surprise: The "Fake-Out" Effect

The most interesting discovery in the paper is a bit of a trick of the light.

If you stand at the front door (the surface) and watch the dancers, you might think, "Wow, they calmed down really fast!" The researchers found that transport actually makes it look like the front of the metal cools down faster.

Why? Because the excited dancers are literally running away from the front door into the back of the room. They aren't necessarily calming down at the front; they are just leaving the front. So, if you only look at the surface, it seems like the system has reached a peaceful equilibrium very quickly.

However, the paper argues that the whole system isn't actually calm yet. The dancers are still running around, trying to fill the back of the room. The "peace" at the front is an illusion caused by the crowd moving away. The full system only becomes truly calm once the dancers have spread out evenly from front to back.

The "Energy Window" Analogy

The researchers also looked at specific groups of dancers based on how "wild" they are (their energy levels).

• The "Mildly Excited" Group (Low Energy): These are the dancers who are just slightly jittery. Their movement is mostly controlled by the crowd surge (transport). They are mostly just moving from the crowded front to the empty back.
• The "Wildly Excited" Group (High Energy): These are the dancers jumping on tables. Their behavior is mostly controlled by bumping into each other (scattering). They lose their wild energy by crashing into others very quickly, regardless of where they are in the room.

The Bottom Line

The paper concludes that you cannot understand what's happening in a laser-hit metal just by looking at the surface or assuming everything happens in one spot.

• At the surface: It looks like things calm down fast because the "hot" electrons are running away.
• Inside the metal: The system is actually still chaotic because those electrons are spreading out, creating a new kind of imbalance as they travel to the back.

The researchers built a new mathematical model (like a super-accurate simulation of the dance floor) that tracks both the bumping (scattering) and the running (transport) at the same time. This helps scientists understand that in thick metals, "cooling down" isn't just about slowing your feet; it's also about where you are standing in the room.

Technical Summary

Technical Summary: Interplay of Non-Local Transport and Local Scattering in Laser-Excited Metals

Problem Statement
Ultrafast laser excitation of metals induces electronic nonequilibrium both locally in energy distribution and non-locally in space, particularly when the sample thickness exceeds the optical penetration depth. While the subsequent dynamics are governed by the interplay between non-local transport and local scattering (electron-electron and electron-phonon), combined microscopic descriptions of these processes remain sparse. Existing theoretical approaches typically rely on simplifying assumptions: they either assume homogeneous heating to resolve scattering in detail or retain transport while employing approximate scattering models (e.g., relaxation time approximations). These approximations hinder the ability to disentangle the specific contributions of intrinsic thermalization and spatial transport to measured carrier dynamics, particularly in energy-resolved experiments like time-resolved photoemission spectroscopy (PES).

Methodology
The authors present a spatially resolved Boltzmann framework that simultaneously incorporates local scattering and non-local transport. The core of the method is a reformulation of the Boltzmann transport equation in energy space that utilizes full microscopic collision integrals. Key methodological features include:

• Two-Branch Description: To resolve spatial motion without full momentum resolution, the electron momentum is projected onto the surface normal ($z$-direction). The dispersion is divided into two branches: electrons moving into the bulk ($v_z > 0$) and electrons moving toward the surface ($v_z < 0$).
• Full Collision Integrals (FCI): The model employs full FCI for both electron-electron and electron-phonon scattering within the random-$k$ approximation. This avoids the inaccuracies of relaxation time approximations while remaining computationally feasible compared to full momentum-resolved calculations.
• Scattering and Transport Coupling: The framework explicitly accounts for scattering events that may change the direction of motion (branch switching) and treats transport as the spatial evolution of these two branches. Boundary conditions at the front and rear surfaces are handled by reflecting electrons between branches.
• Simulation Setup: The model is applied to an aluminum-like free-electron metal (matching first-principles parameters for Fermi velocity and absorption) with a 50 nm thickness. Simulations cover a range of laser fluences ($0.05$ to $15 \text{ J m}^{-2}$) and compare homogeneous excitation against inhomogeneous Beer-Lambert excitation at identical absorbed energy densities.

Key Contributions
The primary contribution of this work is the development of a consistent microscopic framework that bridges the gap between spatial equilibration and energy-space scattering. By directly combining scattering and electron motion, the model implicitly covers the entire range from ballistic to diffusive transport. This allows for the direct quantification of how transport processes modify apparent thermalization dynamics, a distinction previously difficult to isolate due to the reliance on simplified models.

Results
The study yields several critical insights into the dynamics of hot electrons:

1. Accelerated Surface Thermalization: Transport accelerates the apparent thermalization observed at the irradiated surface. By removing athermal carriers from the surface region into the bulk, transport acts as an additional relaxation channel. This effect is most pronounced at low fluences where intrinsic electron-electron scattering is slower.
2. Delayed Global Equilibration: While surface thermalization appears faster, the spatial redistribution of energy delays the complete equilibration of the full electron system. The presence of spatial gradients leads to a persistent athermal carrier population across the sample depth for longer durations compared to homogeneous excitation.
3. Depth-Dependent Dynamics: The dominant mechanism driving dynamics varies significantly with position and energy:
   • Front Surface: Dynamics are driven by a competition between electron-electron scattering and transport (removing carriers).
   • Back Surface: Dynamics are dominated by transport processes, as direct laser excitation is negligible. Athermal electrons accumulate transiently at the back due to reflection.
   • Energy Dependence: Near the Fermi edge, transport is the primary driver of spectral density changes. At higher energies (e.g., 1.0 eV above the Fermi level), electron-electron scattering dominates the decay, occurring on the timescale of the laser pulse.
4. Spectral Density Evolution: The analysis of spectral densities (mimicking PES observables) reveals that the interpretation of relaxation times is highly dependent on the probed energy window and depth. For instance, at the back surface, spectral density changes are almost entirely driven by transport rather than local scattering.

Significance and Claims
The authors claim that their work improves the understanding of electronic nonequilibrium in optically thick, laser-driven systems. They emphasize that electronic thermalization in such systems cannot be interpreted as a purely local process. A key finding is the distinction between surface-sensitive observables and the global state of the electron system: surface measurements may indicate rapid thermalization because transport removes athermal carriers from the probe volume, even while the full electron system remains in nonequilibrium due to spatial redistribution.

The paper concludes that this distinction is particularly critical for energy-resolved ultrafast measurements, where the dominant relaxation channel depends on both the probed energy window and the probing depth. The presented framework provides a microscopic route to connect transport, scattering, and experimentally accessible hot-carrier dynamics, offering a more accurate basis for interpreting data relevant to future electronic applications involving hot carriers.