Secondary-electron-emission spectroscopy of tungsten: Angular dependence and phenomenology

Angle-resolved energy-distribution measurements of secondary-electron emission (SEE) from metals reveal spectral fine structure that relates directly to the density distribution of the one-electron states throughout $E-\overrightarrow{\mathrm{K}}$ space located above the vacuum level $E_v$. The angular dependence of the SEE spectra from (100), (110), and (111) tungsten surfaces has been studied as a function of polar angle $0°\lesssim \Theta \lesssim 70°$ along azimuthal directions $\phi$ such that the energy- and angle-resolved SEE current $j_{\mathrm{SEE}}(E, \Omega )$ effectively scans states throughout the $\frac{1}{48\mathrm{th}}$ irreducible body-centered-cubic zone. Calculations have been carried out in both "reduced" and "extended" $\overrightarrow{\mathrm{K}}$ space in order to assess the relative contribution of elastic umklapp scattering to the density distribution of contributing states profiles. The results indicate that the overall secondary-electron yield may be represented as the sum of basically two contributions $J_{\mathrm{SEE}}^{\mathrm{total}}=\int 0^{\pi }d\Omega \int 0^{E_{max}}{j}_{\mathrm{SEE}} (E, \Omega )dE=J_{\mathrm{SEE}}^{\mathrm{bulk}}+J_{\mathrm{SEE}}^{\mathrm{surface}}$. The bulk contribution represents emission due to Bloch waves propagating out of states in the semi-infinite crystal; the surface contribution represents that part of the current due to evanescent waves at the metal-vacuum interface. In addition, transmission-induced spectral features are observed (transmission resonances), which are not related to the density-of-states fine structure, but are due to a quantum-mechanical enhancement in the escape probability arising from wave-function matching at the emitter-vacuum interface. Bulk and surface band-structure effects are concurrently manifest in the SEE spectra via the wave-matching conditions imposed at the solid-vacuum interface. The results are discussed within the general conceptual framework provided by "the (time-reversed) incoming final-state wave-function" approach to electron emission phenomenology of metal surfaces, thereby establishing a relationship with recently developed low-energy electron diffraction, photoemission, and field-emission formalism.