Small rings and amorphous tetrahedral carbon

We apply first-principles density-functional calculations to study strain in dense amorphous tetrahedral carbon $(a-\mathrm{tC}).\mathrm{}$ While the large strain present in small-ring structures, particularly three-member rings, could argue against their existence in $a-\mathrm{tC},$ we demonstrate, based on energetic arguments, that strained small (three- and four-member) rings are plausible topological microstructural elements. We present two bulk carbon structures made up entirely of fourfold-coordinated atoms: the first with every atom in one three-member ring, the second with every atom in one four-member ring. Calculations show these bulk ring structures are relatively low in energy, only 0.37 and 0.23 eV/atom above diamond, respectively. This computed strain energy is much less than that present in recent models for $a-\mathrm{tC}.$ We examine properties of these structures with the intention to provide benchmark calculations for more approximate models, and to investigate the impact small rings might have on the properties of $a-\mathrm{tC}.$ We use a recently developed linear-response algorithm to compute phonon spectra for these ring structures.