A Haldane-Anderson Hamiltonian Model for Hyperthermal Hydrogen Scattering from a Semiconductor Surface

TL;DR

The paper tackles bandgap-threshold nonadiabatic energy transfer in hyperthermal hydrogen scattering from Ge(111)c(2x8) by constructing a one-dimensional Haldane-Anderson Hamiltonian tuned to first-principles data. It compares independent-electron surface hopping (IESH), mean-field Ehrenfest dynamics, and numerically exact HEOM benchmarks, showing that IESH captures the experimentally observed threshold (nonadiabatic energy loss only when the initial energy exceeds the bandgap $E_{ ext{gap}}$) while Ehrenfest dynamics erroneously predict energy loss at all energies. HEOM benchmarks validate the IESH predictions and provide a rigorous reference for future MQCD developments. The work demonstrates that a simple, first-principles–parametrized NAH model can reproduce key features of nonadiabatic energy dissipation in semiconductor surfaces and offers a platform for systematic improvements with higher-dimensional parametrizations and ab initio inputs.

Abstract

Collisions of atoms and molecules with metal surfaces create electronic excitations in the metal, leading to nonadiabatic energy dissipation, inelastic scattering, and sticking. Mixed quantum-classical molecular dynamics simulation methods, such as molecular dynamics with electronic friction, are able to capture nonadiabatic energy loss during dynamics at metal surfaces. Hydrogen atom scattering from semiconductors, on the other hand, exhibits strong adsorbate-surface energy transfer only when the projectile kinetic energy exceeds the bandgap of the substrate. Electronic friction fails to describe this effect. Here, we report a first-principles parameterization of a simple Haldane-Anderson Hamiltonian model of hydrogen atom gas-surface scattering on Ge(111)$c(2\times8)$, for which hyperthermal scattering experiments have been reported. We subsequently perform independent-electron surface hopping and Ehrenfest dynamics simulations on this model, and validate these results through numerically exact quantum-dynamical simulations using the hierarchical equation of motion approach. While mean-field dynamics yield weak nonadiabatic energy loss that is independent of the initial kinetic energy, independent electron surface hopping simulations qualitatively agree with the experimental observation that nonadiabatic energy dissipation only occurs if the initial kinetic energy exceeds the bandgap of the surface.

Figures (8)

• Figure 1: Atomic hydrogen (grey) at Ge(111)$c(2\times8)$ above a rest-atom (green). Adatoms are shown in blue and represent the second kind of reconstructed Ge atoms of this respective surface reconstruction. The surface is shown in side view along with the definition of the spatial model variable $x$ (panel a) and in orthographic top view (panel b).
• Figure 2: Comparison of the molecule-surface couplings, $\Gamma(\varepsilon)$, used in the MQC (dashed line) and HEOM (solid line) methods. In the HEOM approach, the bandwidth is chosen as $W = 50$eV and the steepness parameter is $m = 3$, whereas the MQC calculations explicitly enforce the wide-band and infinite-$m$ limit.
• Figure 3: Panel a: The diabatic adsorbate states and coupling strength given by \ref{['eq:U0-hokseon']}-\ref{['eq:A-hokseon']} as a function of hydrogen atom height $x$ with corresponding conjugate parameters given in \ref{['tab:Hokseon-Parameters']}. Panel b: The adiabatic potential energy spectrum for the 2000 lowest energy many-electron states. The spectrum is produced with the gap-inserting Gauss-Legendre method using 150 states, 75 electrons and a bandwidth of $\Delta E =$ 50 eV. The inserted bandgap has width $E_{\text{gap}} = 0.49$ eV. The Fermi level is $E_\text{F}= 0$ eV. The rescaling parameter for the couplings is $\bar{a} = 0.479\,\text{eV}^{-1/2}$.
• Figure 4: Average kinetic energy loss of scattered hydrogen atoms obtained via IESH and Ehrenfest simulations with incidence energies $E_i$ ranging from 0.2 to 0.8 eV. Each IESH data point is averaged over 1000 trajectories (60,000 for $E_i = 0.5$ and $0.525$ eV), while the deterministic Ehrenfest method uses a single trajectory. The blue dashed line represents zero energy loss (elastic scattering), black solid line references an absolute constant energy loss, and the vertical black dashed line indicates the bandgap, $E_{\text{gap}} = 0.49$ eV.
• Figure 5: Ehrenfest adiabatic electronic state population and height of the hydrogen atom during scattering at an initial height of $x_0=$5 Å, with three initial kinetic energies $E_i=$ 0.7 eV(a), 0.4 eV(b) and 0.3 eV(c). The black curves are the adiabatic population and the red curve describes the hydrogen atom position (with respect to the right ordinate).