Vacuum Wannier Functions for First-Principles Scattering and Photoemission

Abstract

We establish a first-principles theory of vacuum Wannier functions unifying tight-binding and nearly-free-electron descriptions across solid-vacuum interfaces. Analytic solutions for canonical Wannier functions in arbitrary dimension and disentangled functions in 1D motivate a numerically verified 3D Wannier close-packing principle, enabling dense k-space construction of full Born-series scattering states at interfaces and thus predictive photoemission calculations without semiempirical vacuum potentials. Applications to graphene and h-BN reveal corrections beyond the first-Born approximation.

Figures (4)

• Figure 1: (a) Dispersions for an empty 1D lattice (black) and the two disentangled bands obtained from the analytic $\theta(k)$ interpolation (red). The dashed line marks the frozen window energy boundary. (b) and (c) compare the decay of the magnitudes of the sinc-like and analytically disentangled MLWFs, respectively.
• Figure 2: Stability tests of close-packed lattices for MLWF centers in vacuum: (a–d) projected view of fcc arrangement in a cubic supercell; (e–f) projected view of hcp arrangement in a hexagonal supercell. Blue $\times$'s mark initial randomized centers, and orange circles mark the relaxed positions. Percentages denote rms displacements (relative to the nearest-neighbor spacing) of the initial Gaussian-distributed displacements.
• Figure 3: Vacuum Wannier functions (VWFs): (a) disentangled VWF in the empty lattice and (b) numerically optimized VWF located $\sim 50$ Å above a graphene sheet. The close similarity demonstrates that the analytic theory and realistic slab calculations yield the same compact, regularized functions. Final Wannier-center locations after optimization projected along (c) the normal and (d) transverse planar directions. Yellow and blue surfaces in (a) and (b) represent equal‑but‑opposite sign contour levels; blue $\times$ symbols and red circles in (c) and (d) denote interfacial and deep-vacuum Wannier functions, respectively. The final optimized Wannier functions show striking regularity in location and spacing.
• Figure 4: Photoemission results for graphene (Gr) and hexagonal boron-nitride (h-BN). (a) rms widths of the longitudinal energy distributions. (b–d) Transverse momentum distributions for h-BN using (b) the full theory, (c) constant matrix elements, and (d) matrix elements based on plane‑wave final states. (e) Mean transverse energy (MTE) as a function of excess photon energy: monolayer graphene (blue), boron nitride (red), and the Dowell–Schmerge model (black). Explicit calculations are shown as circles (graphene) and $\times$ symbols (h-BN), with the effective‑mass model in Eq. \ref{['eqn:mte_estimate']} shown as dashed lines.