Large differential attosecond delays in solid state photoemission

This study reveals that large differential attosecond delays in photoemission from Bi$_2$Te$_3$ and Bi$_2$Se$_3$ arise from strong energy-dependent variations in final-state dynamics caused by multiple surface scattering, rather than intra-atomic effects or ballistic transport.

The Gist

Here is an explanation of the paper, translated from complex physics jargon into a story about runners, tunnels, and traffic jams.

The Big Idea: Timing the "Exit" of an Electron

Imagine you are at a crowded concert. When the show ends, people rush for the exit. Usually, we assume that if two people leave at the same time, they will arrive at the street outside at roughly the same time, provided they walk at the same speed.

In the world of physics, scientists have been trying to measure exactly how long it takes for an electron (a tiny particle of electricity) to escape from a solid material (like a crystal) into the vacuum of space after being hit by a flash of light. This is called photoemission.

For a long time, scientists thought this was simple: The electron gets hit, it runs through the material like a bullet, and pops out. They assumed the time it took depended mostly on how fast it was running and how thick the material was.

This paper says: "No, that's not the whole story."

The researchers found that electrons don't just run in a straight line. When they hit the edge of the material (the surface), things get weird. Depending on their exact energy, some electrons get delayed by tiny amounts of time (attoseconds—billionths of a billionth of a second) because of how they interact with the "doorway" to the outside world.

The Experiment: The "Twin" Runners

To prove this, the scientists needed a very specific setup. They couldn't just compare two random electrons because they would be too different. Instead, they looked at "twin" electrons coming from the same atom.

1. The Setup: They used a special crystal (Bismuth Telluride or Bismuth Selenide). Inside these crystals, there are electrons in "spin-orbit split" states. Think of these as two identical twins, Twin A and Twin B.
2. The Difference: Twin A and Twin B are almost identical, but Twin A has a tiny bit more energy than Twin B. The difference is so small it's like the difference between a runner wearing a 10kg backpack and one wearing a 10kg + 10-gram backpack.
3. The Race: The scientists hit the crystal with a super-fast flash of light (an attosecond pulse). This knocks both twins out of the atom at the exact same moment.
4. The Measurement: They measured when Twin A and Twin B arrived at the "finish line" (the vacuum detector).

The Surprise: The "Ghost" Tunnel

If the old theory (the "bullet" theory) were true, the twins should arrive at almost the exact same time. Their energy difference is so small that the travel time difference should be negligible.

But they didn't.

The scientists found a delay of 30 to 100 attoseconds between the twins. That is a huge difference in the world of quantum physics.

Why did this happen?

The paper explains that the surface of the crystal isn't just a flat wall; it's a complex landscape. When the electrons try to leave, they don't just run out. They encounter a "traffic jam" caused by the atomic structure of the surface.

• The Propagating Wave: Some electrons act like runners on a track. They move forward smoothly.
• The Evanescent Wave: Other electrons act like ghosts trying to walk through a wall. In quantum mechanics, particles can "tunnel" or exist briefly in places they shouldn't be able to. These are called evanescent waves.

The researchers discovered that because Twin A and Twin B have slightly different energies, one of them hits a "traffic jam" (a band gap) at the surface, while the other finds a clear path. One electron gets stuck bouncing around the surface (multiple scattering) before finally escaping, while the other slips through a "ghost tunnel" and gets out faster (or slower, depending on the specific energy).

The Analogy: The Airport Exit

Imagine two travelers, Alice and Bob, arriving at an airport exit at the exact same time.

• The Old Theory: They both walk through the sliding glass doors at the same speed. They arrive outside together.
• The New Reality: The airport has a complex security checkpoint right at the door.
  • Alice (Twin A) has a boarding pass that matches the "Propagating" lane. She walks straight through the open door.
  • Bob (Twin B) has a boarding pass that matches the "Evanescent" lane. He gets stuck in a special, narrow tunnel that winds around the security guard. He has to bounce off the walls a few times before he can squeeze through.

Even though they started together, Bob arrives outside 30 attoseconds later than Alice. The delay isn't because Bob is slower; it's because the doorway itself treats them differently based on a tiny detail of their identity.

Why Does This Matter?

1. Breaking the "Bullet" Model: This proves that we can't treat electrons inside solids like simple bullets flying through space. The surface of the material plays a massive, active role in how electrons escape.
2. The "One-Step" Theory: The scientists used a complex mathematical model (called One-Step Photoemission Theory) that accounts for all the bouncing, tunneling, and scattering at the surface. This model perfectly predicted the delays they measured.
3. Future Tech: Understanding these tiny delays helps us design better solar cells, faster computer chips, and new materials. If we can control how electrons escape a material, we can control the flow of electricity and information with incredible precision.

The Bottom Line

This paper is like discovering that the "exit door" of a building is actually a magical, shifting portal that changes how fast you leave based on your shoe size. By measuring two nearly identical electrons, the scientists proved that the surface of a solid is a chaotic, complex place where electrons get delayed by "ghost tunnels" and "traffic jams," not just by how fast they are running.

Technical Summary

Here is a detailed technical summary of the paper "Large Differential Attosecond Delays in Solid State Photoemission."

1. Problem Statement

Time-resolved photoelectron spectroscopy allows for the investigation of electronic structure and non-equilibrium dynamics in matter. However, interpreting attosecond time delays in solids has been a subject of debate.

• The Ambiguity: Previous studies measured relative photoemission delays between channels with significantly different final state energies (e.g., core levels of different elements or core vs. valence bands). These large energy differences obscure the complex fine structure of the emission process, making it difficult to disentangle contributions from intra-atomic effects, ballistic transport, and surface scattering.
• The Limitation of Existing Models: The prevailing "ballistic propagation" model assumes photoelectrons travel through the crystal with a group velocity ($v_g$) and escape after a time determined by the inelastic mean free path ($\Lambda$). This model predicts that for small energy differences, relative delays should be negligible.
• The Gap: There was a lack of experimental evidence probing the energy dependence of photoelectron escape times ($t_{esc}$) with sufficient resolution to distinguish between ballistic transport and complex surface scattering effects, particularly near band gaps or Cooper minima where the ballistic model may fail.

2. Methodology

The authors employed a novel approach combining high-precision experimental measurement with rigorous theoretical modeling.

Experimental Approach:

• Technique: Attosecond RABBITT (Reconstruction of Attosecond Beating By Interference of Two-photon Transitions) spectroscopy.
• Samples: Topological insulators $\text{Bi}_2\text{Te}_3$ and $\text{Bi}_2\text{Se}_3$.
• Strategy (Differential Attosecond Delays): Instead of comparing widely separated energy levels, the team measured the relative delay between spin-orbit split initial states ($d_{5/2}$ and $d_{3/2}$) of the same element (Bi 5d, Te 4d, Se 3d).
  • These states have very small energy differences ($\sim$1–3 eV) and originate from the same emitter atom with identical spatial symmetry.
  • This setup effectively cancels out contributions from semi-classical ballistic transport and intra-atomic delays, isolating the energy dependence of the escape time $t_{esc}(E_{kin})$.
• Setup:
  • Generated an Attosecond Pulse Train (APT) using the 57th and 59th harmonics of a NIR laser ($\sim$90–94 eV).
  • Overlapped the APT with a NIR probe field on the sample surface.
  • Measured the phase oscillation of sidebands in the photoelectron spectrum to extract relative delays.
  • Used a complex fit method to disentangle overlapping spectral features caused by the small energy spacing of the spin-orbit components.

Theoretical Approach:

• Model: One-step photoemission theory based on the Eisenbud-Wigner-Smith (EWS) time delay formalism (OSTEWS).
• Calculation:
  • Used Density Functional Theory (DFT) for initial and final states.
  • Calculated Time-Reversed Low-Energy Electron Diffraction (TR LEED) states to account for elastic and inelastic scattering at the bulk-vacuum interface.
  • Included an optical potential ($V_{opt}$) to model inelastic scattering.
• Prediction: The theory predicts that $t_{esc}$ varies strongly near band gaps and band edges due to the interplay between propagating Bloch waves and evanescent waves.

3. Key Contributions

1. Novel Measurement Concept: Introduction of "Differential Attosecond Delays" ($\tau_{DAD}$) using spin-orbit split states to probe the derivative of the escape time with high temporal and energy resolution, eliminating ambiguities associated with different atomic species or large energy separations.
2. Experimental Validation of Non-Ballistic Dynamics: Provided direct experimental evidence that the simple ballistic propagation model is insufficient for describing solid-state photoemission, even for small energy differences.
3. Quantitative Agreement: Demonstrated that the rigorous one-step photoemission theory, which accounts for surface-induced multiple scattering and the mixing of evanescent and propagating states, quantitatively reproduces the observed large delays.

4. Key Results

• Measured Delays: The experiment revealed significant differential attosecond delays for energetically close-lying states:
  • Bi 5d ($\text{Bi}_2\text{Te}_3$ & $\text{Bi}_2\text{Se}_3$): $\tau_{DAD} \approx +30$ to $+31$ as (electrons from $d_{5/2}$ are delayed relative to $d_{3/2}$).
  • Te 4d ($\text{Bi}_2\text{Te}_3$): $\tau_{DAD} \approx -39$ as.
  • Se 3d ($\text{Bi}_2\text{Se}_3$): $\tau_{DAD} \approx -93$ as (with larger uncertainty).
• Exclusion of Alternative Mechanisms:
  • Ballistic Transport: Ruled out, as ballistic models predict delays $<3$ as for such small energy differences.
  • Intra-atomic Delays: Ruled out via relativistic ab initio calculations for gas-phase atoms, which predicted delays $<5$ as for the relevant photon energies.
• Theoretical Correlation: The OSTEWS calculations matched the experimental data quantitatively. The theory showed that the large delays arise because the two spin-orbit components probe different parts of the final state continuum:
  • One component may couple to a propagating state (standard delay).
  • The other may couple to an evanescent state (associated with band gaps), which exhibits an "advanced" emission time (negative delay) or significant delay depending on the proximity to the band edge.
• Mechanism: The large $\tau_{DAD}$ is driven by the strong energy dependence of the photoelectron escape time near band gaps and band edges, caused by the interference and varying relative weights of evanescent and propagating Bloch waves at the surface.

5. Significance

• Paradigm Shift: The findings necessitate abandoning the picture of photoelectrons as nearly-free particles traveling with a well-defined group velocity inside a solid. Instead, the emission process is dominated by the complex multiple scattering of the outgoing electron at the surface.
• Surface Sensitivity: The study highlights that the breaking of lattice periodicity at the surface creates a structured final state continuum (mixing of propagating and evanescent waves) that fundamentally dictates the timing of photoemission.
• Future Applications: This work establishes a robust framework for interpreting attosecond delays in solids. It suggests that differential delays can serve as a sensitive probe for the electronic band structure, specifically for identifying band gaps, band edges, and surface scattering dynamics with unprecedented resolution.
• Theoretical Validation: It confirms the validity of the one-step photoemission theory combined with the EWS formalism as the correct tool for describing ultrafast electron dynamics in condensed matter.