Fully Ab Initio Approach to Inelastic Atom-Surface Scattering

We introduce a fully ab initio theory for inelastic scattering of any atom from any surface exciting single phonons, and apply the theory to helium scattering from Nb(100). The key aspect making our approach general is a direct first-principles evaluation of the scattering atom-electron vertex. By correcting misleading results from current state-of-the-art theories, this fully ab initio approach will be critical in guiding and interpreting experiments that adopt next-generation, nondestructive atomic beam scattering.

Figure 1

A helium atom (dotted lines) with initial and final momenta ${\mathbf{k}}_{i,f}$ indirectly excites a surface phonon (jagged line) mode $\nu$ with lateral momentum $\mathbf{Q}$ via an electron-hole pair excitation (solid lines) from bands $n$ and $m$ with momenta $\mathbf{k}$ and $\mathbf{k}+\mathbf{Q}$. Intersecting lines show the helium-electron vertex (dark magenta) and the electron-phonon vertex (light gray).

Figure 2

Contour color-map showing the perturbing potential in the plane of a helium atom at its estimated turning point $z_{\mathrm{t}}$ from Nb(100). The magenta dotted line shows the helium potential, averaged over planes, as a function of distance from the surface. Gray circles show the $z$ locations of niobium atoms, with the top plane of atoms defining $z=0$. The black dashed line shows the planar-averaged density of Fermi-level electrons at an electronic temperature of 900 K.

Figure 3

Top-left panel shows the interaction energy profile $\mathrm{\Delta }{E}_{\mathrm{He}}$ between a helium atom and the Nb(100) surface (black curve); the shaded region depicts typical energies for incoming helium atoms, and gray circles show the $z$ locations of niobium atoms, with the top plane of atoms defining $z=0$. Bottom panel shows the predicted HAS signal intensities $|M_{\mathbf{Q}\nu }(z^{\prime }){|}^2$ corresponding to helium atoms at $z$ locations given by the violet vertical lines, spaced uniformly at approximately 0.4 Å apart. The top-right panel shows the natural log of the total and maximum of each signal from the bottom panel as a function of distance from the Nb(100) surface and includes the result of a line of best fit to extract the exponential decay of the signal.

Figure 4

Top panel illustrates incoming and outgoing helium atom $z$ wave functions (red curves) from potential $V(z)=\mathrm{\Delta }{E}_{\mathrm{He}}(z)$. Bottom panel combines the wave functions with the exponential decay of the helium-surface interaction (blue curve). The violet vertical lines in the bottom panel give the same $z$ coordinates of the helium atoms shown in Fig. 3.

Figure 5

Color maps showing predicted inelastic HAS signals from Nb(100) at three levels of theory. Color-bar labels show the Feynman diagram elements considered at each level. The bottom panel considers only phonons to show the top-layer phonon density of states for shear vertical (SV) and longitudinal (L) polarizations. The middle panel incorporates the electron-phonon vertex to give the inelastic HAS predictions estimated from surface EPC strengths [45]. The top panel includes both the electron-phonon and helium-electron vertices, giving the final prediction of the fully ab initio HAS calculation from Eq. (5). Black circles in each panel correspond to measured helium-scattering data [44].