Extraction of the self energy and Eliashberg function from angle resolved photoemission spectroscopy using the xARPES code

This paper introduces the xARPES Python code, which utilizes an extended maximum-entropy method with Bayesian inference to consistently extract electron self-energies and Eliashberg functions from curved dispersions in angle-resolved photoemission spectroscopy data, demonstrating superior accuracy on both model and experimental datasets compared to existing linearization-based approaches.

The Gist

Imagine you are trying to understand how a crowded dance floor works. You have a group of dancers (electrons) moving to music (energy). Sometimes, the dancers bump into each other, or they get distracted by the bass vibrating the floor (phonons). These interactions change how fast they move and how long they stay on the floor.

In the world of physics, scientists use a technique called Angle-Resolved Photoemission Spectroscopy (ARPES) to take "snapshots" of these dancers. They shoot light at a material, knock electrons out, and measure their speed and direction. This creates a map of the dance floor.

However, reading this map is tricky. The raw data is a blurry, noisy picture where the dancers' paths are curved and tangled. To understand the rules of the dance (the physics), scientists need to separate the "natural" path of a dancer from the "disturbances" caused by the music and other dancers. This separation is called extracting the self-energy and the Eliashberg function.

Here is what this paper does, explained simply:

1. The Problem: Trying to Draw a Straight Line on a Curved Road

Previously, scientists tried to analyze these dance maps by assuming the dancers moved in perfectly straight lines. They would draw a straight line through the data and say, "The difference between the straight line and the actual path is the disturbance."

The authors of this paper say: "That doesn't work well when the road is curved."
In many materials, the natural path of an electron isn't a straight line; it's a curve (like a parabola). If you try to fit a straight ruler to a curved road, you get a bad measurement of the disturbances. It's like trying to measure the wind resistance on a rollercoaster by pretending the track is flat.

2. The Solution: The "xARPES" Code

The team created a new computer program called xARPES. Think of this program as a super-smart GPS for the dance floor. Instead of forcing the data into a straight line, xARPES allows the "road" to be curved (parabolic) or even more complex shapes.

It does three main things:

• Fits the Curve: It finds the best possible curved path that represents the electrons when they aren't interacting with anything.
• Separates the Noise: It mathematically peels away the "noise" (disturbances) to reveal exactly how much the electrons are being slowed down or sped up by the music (phonons) or by bumping into other electrons.
• Reveals the Music Sheet: It reconstructs the Eliashberg function. If the self-energy is the "disturbance," the Eliashberg function is the sheet music of the vibrations. It tells you exactly which notes (frequencies) the floor is vibrating at and how loudly they are playing.

3. The "Bayesian" Detective Work

One of the paper's biggest innovations is how it handles uncertainty. Usually, scientists have to guess the starting parameters for their analysis (like guessing the speed of the dancers before they start). This is subjective and can lead to bias.

The authors use a method called Bayesian Inference. Imagine a detective who doesn't just guess; they constantly update their theory based on new clues.

• The code starts with a guess.
• It checks the data.
• It asks, "Given this data, what is the most probable truth?"
• It repeats this loop until the answer stabilizes.

This removes the "human guesswork" and ensures that the result is the most statistically likely explanation of the data, rather than just what the scientist hoped to see.

4. Real-World Tests

The authors didn't just build the tool; they tested it on two real "dance floors":

• Strontium Titanate (SrTiO3): They looked at a thin layer of electrons on this material. They found that if you ignore the specific way light hits the electrons (called "matrix elements"), your measurements can be off by a factor of two. It's like measuring a shadow without accounting for the angle of the sun. xARPES corrected this, giving a much clearer picture of the vibrations.
• Lithium-Doped Graphene: They analyzed graphene (a single layer of carbon atoms). They took data from two different sides of the same band. In the past, these two sides gave slightly different, conflicting results. Using xARPES, they found that the results were unprecedentedly similar, proving that the tool can extract consistent, reliable data even from complex, curved paths.

Summary

This paper introduces xARPES, a new software tool that acts like a high-precision lens for studying how electrons interact with vibrations in materials.

• Old way: Tried to force curved data into straight lines, leading to blurry, biased results.
• New way: Uses curved math and a "detective" algorithm (Bayesian inference) to automatically find the most accurate path and the exact "sheet music" of the vibrations.
• Result: Scientists can now trust their measurements of electron interactions much more, especially in materials where the electron paths are curved.

The authors have released this code as open-source software so other scientists can use it to decode the "dance floors" of new materials.

Technical Summary

1. Problem Statement

Angle-resolved photoemission spectroscopy (ARPES) is a primary tool for studying electron-boson interactions, particularly electron-phonon coupling (EPC), by analyzing the electron spectral function. However, extracting quantitative many-body parameters (specifically the electron self-energy $\Sigma(E)$ and the Eliashberg function $\alpha^2F(\omega)$) from experimental data faces several challenges:

• Non-linear Dispersions: Traditional methods often approximate non-interacting band dispersions ($\varepsilon_n(k)$) as linear. This leads to biased self-energy extraction when bands are curved (parabolic or higher-order), as the spectral function is not a simple Lorentzian in momentum space.
• Manual Parameter Assignment: Decomposing the total self-energy into phonon ($\Sigma_{ph}$), electron-electron ($\Sigma_{el}$), and impurity ($\Sigma_{imp}$) contributions typically relies on manual visual inspection or arbitrary assumptions, introducing human bias.
• Matrix Element Effects: Ignoring photoemission matrix elements (PMEs) can significantly distort the extracted self-energy, sometimes by a factor of two.
• Inverse Problem Instability: Extracting $\alpha^2F(\omega)$ from $\Sigma(E)$ is an ill-posed inverse problem. Standard Maximum-Entropy Method (MEM) approaches require careful tuning of hyperparameters and prior knowledge, often lacking a rigorous statistical framework for parameter optimization.

2. Methodology

The authors introduce xARPES, a Python code (GPL-v3 licensed) that implements a rigorous, automated workflow to extract self-energies and Eliashberg functions. The methodology consists of three main stages:

A. Advanced MDC Fitting with Curved Dispersions

• Instead of assuming linear bands, xARPES fits Momentum-Distribution Curves (MDCs) using polynomial non-interacting dispersions (e.g., parabolic).
• The code maps detector angles to crystal momenta, accounting for experimental geometry and sample rotation.
• It incorporates Photoemission Matrix Element (PME) corrections, allowing the user to specify angular-dependent matrix elements $|M_n(\eta)|^2$ to correct for spectral weight variations caused by orbital character and light polarization.
• The fitting extracts dimensionless peak positions ($\tilde{r}_n$) and widths ($\tilde{\gamma}_n$), which are then converted into real and imaginary parts of the self-energy, $\tilde{\Sigma}'_n(E)$ and $-\tilde{\Sigma}''_n(E)$.

B. Self-Energy Decomposition

The total self-energy is decomposed according to Matthiessen's rule:
$$\Sigma_n(E) = \Sigma_{ph,n}(E) + \Sigma_{el,n}(E) + \Sigma_{imp,n}(E)$$

• $\Sigma_{imp}$: Modeled as a static, imaginary decay rate ($-i\Gamma_{imp}$).
• $\Sigma_{el}$: Modeled using a Kramers-Kronig consistent phenomenological function that satisfies Fermi-liquid behavior at low energies and decays as $|E|^{-2}$ at high energies to avoid ultraviolet divergence.
• $\Sigma_{ph}$: Dominated by the dynamical Fan-Migdal term, which is related to the Eliashberg function via a kernel integral equation.

C. Bayesian Inference and Maximum-Entropy Method (MEM)

• One-Shot Extraction: The code uses MEM to invert the integral equation relating $\Sigma_{ph}(E)$ to $\alpha^2F(\omega)$. This incorporates prior knowledge (e.g., positive semidefiniteness of $\alpha^2F$) via a generalized Shannon-Jaynes entropy term.
• Bilevel Optimization: A novel contribution is the integration of MEM within a Bayesian inference loop.
  • The inner loop optimizes $\alpha^2F(\omega)$ for a fixed set of model parameters.
  • The outer loop optimizes the model parameters (non-interacting mass $m_b$, Fermi wavevector $k_F$, impurity strength $\Gamma_{imp}$, electron-electron coupling $\lambda_{el}$, and model function height $h$) to maximize the posterior probability.
  • This eliminates manual parameter tuning, providing a statistically robust determination of the most probable parameters.

3. Key Contributions

1. xARPES Software: Release of a comprehensive, open-source Python package that automates the extraction of many-body parameters from ARPES data, supporting both linear and curved (parabolic) dispersions.
2. Curved Dispersion Handling: A formalism that correctly handles non-linear band dispersions, removing the bias inherent in traditional Lorentzian fitting of curved bands.
3. Automated Bayesian Optimization: A new framework that simultaneously optimizes the non-interacting dispersion parameters and the magnitude of scattering channels (electron-electron, impurity) alongside the Eliashberg function, removing human bias.
4. Matrix Element Correction: Explicit implementation of PME corrections, demonstrating their critical importance for accurate self-energy extraction.
5. Quantitative Comparison Metric: Introduction of a Euclidean distance measure to quantitatively compare Eliashberg functions extracted from different branches or datasets.

4. Results and Validation

A. Synthetic Data Verification

• Using artificial data with known ground-truth parameters, xARPES was shown to recover the true self-energy and Eliashberg function with high accuracy.
• The method demonstrated that curved dispersion fitting yields unbiased results (95% confidence interval overlap) even with finite energy resolution, whereas linear/Lorentzian fitting introduces significant bias at higher binding energies.
• The Bayesian loop successfully recovered model parameters (mass, $k_F$, coupling strengths) within ~5% of true values for realistic noise levels.

B. Case Study 1: TiO$_2$-terminated SrTiO$_3$

• System: A 2D electron liquid (2DEL) on Nb-doped SrTiO$_3$.
• Findings:
  • Extracted Eliashberg functions from the inner left and inner right branches of the $d_{xy}$ band showed high similarity ($M \approx 0.01$), suggesting underlying symmetry.
  • PME Impact: Neglecting matrix element corrections resulted in self-energies approximately twice as large as the corrected values, highlighting the necessity of PME in quantitative analysis.
  • Identified phonon modes (TO4, LO4) consistent with previous studies but with improved resolution and reduced artifacts.

C. Case Study 2: Li-doped Graphene

• System: Quasi-freestanding Li-doped graphene on Co(0001).
• Findings:
  • Applied to linear Dirac cone dispersions.
  • Extracted Eliashberg functions from two symmetry-equivalent kinks (L and R branches) showed unprecedented agreement ($M \approx 0.0067$), significantly better than previous experimental comparisons.
  • The results confirmed that experimental asymmetries (e.g., Fermi-edge shifts) were the primary source of minor discrepancies, validating the robustness of the extraction method.
  • Identified dominant phonon modes around 166 meV (A'$_1$ and E$_{2g}$ modes).

5. Significance

• Bridging Experiment and Theory: xARPES provides a standardized, unbiased method to extract Eliashberg functions, enabling direct and fair comparison with first-principles calculations (e.g., DFPT, GW, DMFT).
• Reproducibility: By automating parameter selection via Bayesian inference, the code removes the "black box" nature of manual fitting, ensuring that results are reproducible and objective.
• Community Standard: The release of xARPES establishes a new benchmark for analyzing ARPES data, particularly for systems with complex band structures or strong many-body interactions.
• Future Directions: The modular code structure allows for future extensions, such as higher-order dispersion terms, inclusion of plasmon coupling, and direct comparison with theoretical non-interacting dispersions.

In summary, this work presents a major advancement in the quantitative analysis of ARPES data, moving from qualitative visual inspection to rigorous, automated, and statistically sound extraction of electron-boson coupling parameters.