Superstructures of Cubic and Hexagonal Diamonds Comprising a Family of Novel $sp^3$ Superhard Carbon Allotropes

Superstructures of cubic and hexagonal diamonds (h- and c-diamond) comprising a family of stable diamond-like $sp^3$ hybridized novel carbon allotropes are proposed, which are referred to as U$_n$-carbon where $n geq 2$ denotes the number of structural layers in a unit cell. The conventional h- and c-diamond are included in this family as members with $n=2$ and 3, respectively. U$_n$-carbon ($n=4-6$), which are unveiled energetically and thermodynamically more stable than h-diamond and possess remarkable kinetic stabilities, are shown to be insulators with indirect gaps of $5.6 sim 5.8$ eV, densities of $ 3.5 sim 3.6$ g/cm$^3$, bulk modulus of $4.3 sim 4.4 times 10^{2}$ GPa, and Vickers hardness of $92.9 sim 97.5$ GPa even harder than h- and c-diamond. The simulated x-ray diffraction and Raman spectra are presented for experimental characterization. These new structures of carbon would have a compelling impact in physics, chemistry, materials science and geophysics.