Atomically Thin Pythagorean Tilings in Two Dimensions

We perform a theoretical study of an atomically thin, two-dimensional layer obtained by positioning atoms at the vertices of the classical Pythagorean tiling. This leads to an unusual geometrical pattern that is only stable for the three halogens Cl, Br, and I. In this Pythagorean structure, halogen atoms are arranged in strongly bound diatomic units that bind together by weaker electrostatic bonds. The energy of these phases is competitive with those of the low-temperature phase of the halogens and the two-dimensional layer obtained by exfoliating it. The Pythagorean layers are semiconducting, with an unusual band structure composed of very mobile holes and extremely heavy electrons. They are also soft, exhibiting small values of the elastic constants and a very low energy flexural mode. Analysis of the allowed Raman transitions reveals breathing-like modes that might be used to fingerprint, experimentally, the Pythagorean structure. Finally, we present a series of substrates that, due to lattice matching and compatible symmetry, can be used to stabilize these peculiar two-dimensional layers.