Surface properties of hexagonal Si and Ge: First-principles study of energetics, atomic relaxation, and electronic structure

Low-index surfaces of hexagonal (lonsdaleite) silicon and germanium polytypes, 2H-Si and 2H-Ge, are investigated using density functional theory calculations for structural optimization and approximate quasiparticle methods for electronic structure analysis. We examine the relaxed geometries of four surface orientations: ( 11 2 ¯ 0 ) a plane, ( 1 1 ¯ 00 ) m plane, ( 1 1 ¯ 02 ) r plane, and (0001) c plane, encompassing nine distinct terminations. Our calculations reveal a clear energetic hierarchy with surface excess energies increasing in the order: m plane < a plane < r plane < c plane. Surface atomic relaxations are driven by the tendency to eliminate dangling bonds through either pairing or complete filling/emptying with electrons, effectively rendering the surfaces insulating. This behavior is evidenced by the formation of energy gaps between surface-bound states within the fundamental band gap region when projected onto the bulk band structure. We apply approximate quasiparticle methods such as the hybrid functional HSE06 and the shell-LDA-1/2 approach for an accurate description of the band structures. We calculate surface-dependent ionization potentials and electron affinities, and compare our findings with previous results for diamond-structure Si and Ge crystals. These results provide fundamental insights into the surface physics of hexagonal group-IV semiconductors and their potential applications in nanoelectronics.