# Evolution of the topologically protected surface states in superconductor $\beta$-Bi$_{2}$Pd from the three-dimensional to the two-dimensional limit

## TL;DR

This study uses first-principles calculations to explore how topologically protected surface states in $eta$-Bi$_{2}$Pd evolve from three-dimensional bulk to two-dimensional thin films, highlighting the role of spin-orbit coupling and film thickness.

## Contribution

It provides a detailed analysis of the evolution of topological surface states in $eta$-Bi$_{2}$Pd as a function of film thickness using first-principles methods.

## Key findings

- Topological surface states emerge at 9 triple-layers.
- Rashba-type surface states appear at 11 triple-layers.
- Van der Waals corrections are essential for accurate structural modeling.

## Abstract

The recent discovery of topologically protected surface states in the noncentrosymmetric $\alpha$-BiPd and the centrosymmetric $\beta$-Bi$_{2}$Pd has renewed the interest in the Bi-Pd family of superconductors. Here, we employ first-principles calculations to investigate the structure, electronic, and topological features of $\beta$-Bi$_{2}$Pd, in bulk and in thin films of various thicknesses. We find that the Van der Waals dispersion corrections are important for reproducing the experimental structural parameters, while the spin-orbit interaction is critical for properly describing the appearance of topological electronic states. By increasing the thickness of the slab, the Dirac-cone surface states and the Rashba-type surface states gradually emerge at 9 and 11 triple-layers.