Electron-phonon coupling in surface electronic states on Be(10$\overline{1}$0)

We present an ab initio study of the electron-phonon interaction in surface electronic states on Be(10$\overline{1}$0). The calculations based on density-functional theory were carried out using a linear-response approach and a mixed-basis pseudopotential method. It is shown that the strength of the electron-phonon coupling is sensitive to the energy and momentum position of an electronic state and varies from 0.16 to 0.54. The difference in the electron-phonon interaction of two Shockley-type surface states at $\mathrm{Ā}$ can be understood on the basis of their different energy positions in the projected band gap. Previous experiments addressing the electron-phonon coupling strength of these two surface states are discussed in light of our theoretical findings.

Figure 1

Calculated surface electronic states (black lines) for a 24-layer Be(10$\overline{1}$0) slab. The gray background gives bulk Be bands projected onto the two-dimensional Brillouin zone. Phonon-mediated transitions for the surface electronic states at $\mathrm{Ā}$ are shown by arrows and dashed lines. The right panel shows the local electronic density of states for the surface (solid line) and central (dashed line) layers of the slab.

Figure 2

Calculated phonon-dispersion curves for a 100-layer Be(10$\overline{1}$0) slab. The phonon spectrum contains surface modes related to both relaxed (black dots) and ideal bulk-truncated (gray dots) surfaces.

Figure 3

Electron-phonon Eliashberg spectral function ${\alpha }^{2}F(\omega )$ for M1 (solid line) and M2 (dashed line) surface electronic states.

Figure 4

(a,b) Electron-phonon Eliashberg spectral function $\alpha \hspace{0.5em}{}^{2}F(\omega )$ (solid line) for A1 (a) and A2 (b) surface electronic states on Be(10$\overline{1}$0). The contribution corresponding to the intraband scattering is shown by a shaded area. Dashed and dotted lines show the behavior of the spectral function at small energies when the Debye model for phonons is used, $\alpha \hspace{0.5em}{}^{2}F(\omega )=\lambda (\omega /{\omega }_m){}^2$, with $\hslash {\omega }_m=60$ meV from LEED measurements[40] and $\lambda$ extracted from the fitting of the experimental linewidth vs temperature data. The dashed lines correspond to $\lambda \approx 0.48$ (a) and $\lambda \approx 0.49$ (b) obtained by fitting[21] over a full temperature range of 45–700 K. The dotted line (a) corresponds to $\lambda \approx 0.66$ from fitting[20, 21] over a limited temperature range of 300–700 K. (c) Local densities of phonon states, $F(\omega )$, for the surface (solid line) and central (dashed line) layers of a 24-layer slab.

Figure 5

Electron-phonon coupling parameter $\lambda (\mathbf{k})$ as a function of binding energy for electronic states in the A1 and A2 surface bands along two symmetry directions, $\stackrel{̲}{\mathrm{A}\mathrm{\Gamma }}$ (solid lines with full circles) and $\stackrel{̲}{\mathrm{AL}}$ (dashed lines with open circles). The inset in the upper panel shows ${\lambda }_{\mathrm{A}1}({\epsilon }_F)$ as a function of electron momentum along a surface Fermi line from the $\stackrel{̲}{\mathrm{A}\mathrm{\Gamma }}$ to $\stackrel{̲}{\mathrm{AL}}$ direction.

Figure 6

Electron-phonon Eliashberg spectral function for the A1 surface electronic state at ${\epsilon }_F$ in the $\stackrel{̲}{\mathrm{A}\mathrm{\Gamma }}$ (a) and $\stackrel{̲}{\mathrm{AL}}$ (b) symmetry directions: the calculated $\alpha \hspace{0.5em}{}^{2}F(\omega )$ (solid line) with the contribution from the intraband scattering shown by a shaded area and the function extracted from photoemission measurements[24] (dashed line). The small energy behavior of the constraint function used to obtain $\alpha \hspace{0.5em}{}^{2}F(\omega )$ from experimental data is also shown by a dotted line in panel (a).