Angle-resolved hard X-ray photoelectron spectroscopy (AR-HAXPES) can probe buried chemical states nondestructively. Hard X-rays yield photoelectrons with long inelastic mean free paths. Hence, by measuring the spectra at multiple take-off angles, the depth sensitivity changes and thin-film thickness can be inferred for laminated materials. However, least-squares peak fitting is sensitive to initial values and analyst choices, and the signal-to-noise ratios vary by angle; therefore, uncertainty-aware integration is essential.

We introduced a Bayesian pipeline that connects spectral decomposition and film thickness estimation within a probabilistic model. For each take-off angle, the spectrum was modeled as a Ru overlayer contribution plus a MgO substrate contribution, with an offset and Shirley background under additive Gaussian noise. The Ru 4s peak has a Doniach–Sunjic line shape suitable for metals, whereas the MgO 2s peak has a Voigt profile suitable for insulators. The Ru 4s and Mg 2s binding energies are close, so their similar kinetic energies simplify the conversion from peak area to thickness.

To avoid local optimum trapping in spectral decomposition, we sampled the posterior spectral parameters using the Replica Exchange Monte Carlo (REMC) method. Unlike point estimates, posterior samples provide the credible intervals and parameter correlations required for principled uncertainty propagation. REMC runs multiple replicas at different temperatures and exchanges states, improving exploration when the posterior is multimodal. From posterior samples, we computed the Ru-to-MgO peak area ratio at each angle and deterministically mapped it to an angle-specific thickness using a standard attenuation relation. The mapping depends on physically defined quantities such as number densities, photoionization cross-sections, and inelastic mean free paths.

We then integrated all angles with a hierarchical Bayesian model that treated angle-specific thicknesses as noisy draws around a single true thickness and estimated the between-angle standard deviation. This integration effectively performed reliability-based weighting, emphasizing angles with tighter posteriors. Using realistic synthetic AR-HAXPES data at 7.94 keV for 13 take-off angles from 5° to 65° with a ground-truth thickness of 2.0 nm, the integrated posterior is sharply centered near the true value. The maximum marginal posterior is 1.997 nm (0.15% relative error), and the 95% highest-posterior-density interval is [1.986, 2.006] nm, which is narrower than the angle-by-angle inferences. The inferred between-angle standard deviation is nearly zero, consistent with the strong cross-angle coherence in the simulated Ru/MgO system. By modifying the likelihood and adding hierarchical parameters, the framework can be extended to non-Gaussian counting statistics and depth inhomogeneities, such as interface roughness.

(Written by Rikito Ota on behalf of all authors.)

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