# Underlying topological Dirac nodal line mechanism of anomalously large electron-phonon coupling strength on Be (0001) surface

## TL;DR

This study uses first-principles calculations to reveal that Dirac nodal line-induced surface states in beryllium (0001) significantly enhance electron-phonon coupling, explaining its anomalously large surface coupling strength.

## Contribution

It uncovers the mechanism by which Dirac nodal line surface states cause strong electron-phonon coupling on Be (0001) surface, a novel insight into topological surface state effects.

## Key findings

- DNSSs strongly couple with phonons, contributing over 80% to the surface electron-phonon coupling.
- The calculated electron-phonon coupling coefficient matches experimental observations.
- Decomposition of Eliashberg function links DNSSs directly to enhanced coupling.

## Abstract

Beryllium was recently discovered to harbor a Dirac nodal line (DNL) in its bulk phase and the DNL-induced non-trivial drumhead-like surface states (DNSSs) on its (0001) surface, rationalizing several already-existing historic puzzles [Phys. Rev. Lett., \textbf{117}, 096401 (2016)]. However, to date the underlying mechanism, as to why its (0001) surface exhibits an anomalously large electron-phonon coupling effect ($\lambda_{e-ph}^s$ $\approx$ 1.0), remains unresolved. Here, by means of first-principles calculations we have evidenced that the coupling of the DNSSs with the phononic states mainly contributes to its novel surface \emph{e-ph} enhancement. Besides that the experimentally observed $\lambda_{e-ph}^s$ and the main Eliashberg coupling function (ECF) peaks have been reproduced well, we have decomposed the ECF, $\alpha^{2}$$F$(\emph{k},\textbf{\emph{q}};\emph{v}), and the \emph{e-ph} coupling strength $\lambda(\emph{k},\textbf{\emph{q}};\emph{v})$ as a function of each electron momentum (\emph{k}), each phonon momentum (\textbf{\emph{q}}) and each phonon mode ($v$), evidencing the robust connection between the DNSSs and both $\alpha^{2}$$F$(\emph{k},\textbf{\emph{q}};\emph{v}) and $\lambda(\emph{k},\textbf{\emph{q}};\emph{v})$. The results reveal the strong \emph{e-ph} coupling between the DNSSs and the phonon modes, which contributes over 80$\%$ of the $\lambda_{e-ph}^s$ coefficient on the Be (0001) surface. It highlights that the anomalously large \emph{e-ph} coefficient on the Be (0001) surface can be attributed to the presence of its DNL-induced DNSSs, clarifying the long-term debated mechanism.