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: ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mn>11</a:mn> <a:mover accent="true"> <a:mn>2</a:mn> <a:mo>¯</a:mo> </a:mover> <a:mn>0</a:mn> </a:mrow> </a:math> ) <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>a</c:mi> </c:math> plane, ( <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mn>1</d:mn> <d:mover accent="true"> <d:mn>1</d:mn> <d:mo>¯</d:mo> </d:mover> <d:mn>00</d:mn> </d:mrow> </d:math> ) <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>m</f:mi> </f:math> plane, ( <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mn>1</g:mn> <g:mover accent="true"> <g:mn>1</g:mn> <g:mo>¯</g:mo> </g:mover> <g:mn>02</g:mn> </g:mrow> </g:math> ) <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>r</i:mi> </i:math> plane, and (0001) <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mi>c</j:mi> </j:math> plane, encompassing nine distinct terminations. Our calculations reveal a clear energetic hierarchy with surface excess energies increasing in the order: <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mi>m</k:mi> </k:math> plane <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mo>&lt;</l:mo> <l:mo> </l:mo> <l:mi>a</l:mi> </l:math> plane <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>&lt;</m:mo> <m:mo> </m:mo> <m:mi>r</m:mi> </m:math> plane <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mo>&lt;</n:mo> <n:mo> </n:mo> <n:mi>c</n:mi> </n:math> 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.