Nonadiabatic H-Atom Scattering Channels on Ge(111) Elucidated by the Hierarchical Equations of Motion

TL;DR

This work addresses nonadiabatic H-atom scattering at a semiconductor interface, where the Born–Oppenheimer approximation fails due to dense surface states and electron–hole excitations across a band gap $E_g$. The authors develop a numerically exact HEOM framework in the matrix product state (MPS) representation, applied to a Newns–Anderson model of H–Ge(111) rest-site scattering, with adiabatic PESs fitted to DFT data. They demonstrate a robust bimodal kinetic energy distribution featuring an energy-loss channel governed by electronic excitations across the band gap, and show how coupling strength $oldeta$, incident energy $E_{in}$, and isotope substitution modulate channel weights and peak shapes. The results establish HEOM–MPS as a rigorous tool for quantum surface scattering beyond electronic friction, providing mechanistic insight into site-specific scattering and guiding future extensions to more complete surface environments including phonons and multi-site averaging.

Abstract

Atomic and molecular scattering at semiconductor interfaces plays a central role in surface chemistry and catalysis, yet predictive simulations remain challenging due to strong nonadiabatic effects causing the breakdown of the Born-Oppenheimer approximation. Here, we present fully quantum simulations of H-atom scattering from the Ge(111)c(2x8) rest site using the hierarchical equations of motion (HEOM) with matrix product states (MPS). The system is modeled by mapping a density functional theory (DFT) potential energy surface onto a Newns-Anderson Hamiltonian. Our simulations reproduce the experimentally observed bimodal kinetic energy distributions, capturing both elastic and energy-loss channels. By systematically examining atom-surface coupling, incident energy, and isotope substitution, we identify the strong-coupling regime required to recover the experimental energy-loss profile. This regime suppresses the elastic peak, implying additional site-specific scattering channels in the observed elastic peak. Deuterium substitution further produces a subtle shift in the energy-loss peak, consistent with experiment. These results establish HEOM as a rigorous framework for quantum surface scattering, capable of capturing nonadiabatic dynamics beyond electronic friction and perturbative approaches.

Figures (10)

• Figure 1: Accuracy of the BSD approximation for (a) the hybridization function $\Gamma(\epsilon)$ (normalized as $\Gamma(\epsilon)/\eta$) and (b) the Fermi distribution at $T=300$ K. The relative error of their product $\Gamma(\epsilon)f^\sigma(\epsilon)$ is below $10^{-3}$ across the full energy range.
• Figure 2: (a) Model potentials $U_0(x)$, $U_a(x)$, and position-dependent coupling $g(x)$ in the Newns--Anderson model, together with the computed adiabatic energy $E_{\rm adia}(x)$ and DFT reference $E_{\rm DFT}$. (b) Calculated occupation number $\langle n_H\rangle(x)$ compared with the spin magnetic moment $\mu_{H}$ from DFT results. All parameters are in Table \ref{['Tab:para_all']}.
• Figure 3: Time evolution of H-atom populations in states $\ket{0}$ and $\ket{a}$, denoted $P_0(t)$ and $P_a(t)$, respectively, during the scattering process.
• Figure 4: Spatial probability distributions of the H atom in the $|0\rangle$ and $|a\rangle$ states during the scattering process, denoted as $P_0(x)$ and $P_a(x)$, respectively. Panel (a) shows $P_0(x)$, and Panel (b) shows $P_a(x)$.
• Figure 5: Kinetic energy distribution of the H atom emitted in the $|0\rangle$ state at $t=154.8$ fs, projected using the technique in Sec. \ref{['sec:proj']} to exclude some near-surface contributions. For comparison, the initial kinetic energy distribution is also plotted (scaled by a factor of 1/3).