Diamond has the highest number density (i.e., the number of atoms per unit volume) of all known substances and a remarkably high valence electron density (r ws = 0.697Å). Searching for possible superdense carbon allotropes, we have found three structures (hP3, tI12, and tP12) that have significantly greater density. The hP3 and tP12 phases have strong analogy with two polymorphs of silica (β-quartz and keatite), while the tI12 phase is related to the high-pressure SiS 2 polymorph. Furthermore,

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... e found a collection of other superdense structures based on the motifs of the aforementioned structures, but with different ways of packing carbon tetrahedra, and among these the hP3 and tI12 structures are the densest. At ambient conditions, the hP3 phase is a semiconductor with the GW band gap of 3.0 eV, tI12 is an insulator with the band gap of 5.5 eV, while tP12 is an insulator, the band gap of which is remarkably high (7.3 eV), making it the widest-gap carbon allotrope. These allotropes are metastable and have comparable to diamond or slightly higher bulk moduli; their Vickers hardnesses are calculated to be 87.6 GPa for hP3, 87.2 GPa for tI12, and 88.3 GPa for tP12, respectively, thus making these allotropes nearly as hard as diamond (for which the same model gives the hardness of 94.3 GPa). Superdense carbon allotropes are predicted to have remarkably high refractive indices and strong dispersion of light. Carbon is a unique element in that it adopts a wide range of structures, which range from superhard insulating (diamond and lonsdaleite) to ultrasoft semimetallic (graphite, an excellent lubricant) and even superconducting (doped diamond and fullerenes). 1-4 The number of all possible metastable phases is infinite, and much work both in experiment and theory has been done to search for the carbon phases with special properties (such as metallic conductivity, hardness, etc.). [5] [6] [7] [8] [9] [10] [11] Of all the physical characteristics, density is of fundamental interest because it could affect many other mechanical, electronic, and optical properties. Diamond is not only the hardest known material, but also has the highest number density of all known materials, 12 whereas the densest two-dimensional (2D) material is graphene. Such extremely high density, with uniquely high valence electron density (Wigner-Seitz radius r ws = 0.697Å) is a result of a compromise between electronic kinetic energy and exchange-correlation energy. Localizing electrons in such small volume is penalized by the kinetic energy, and to compensate for this penalty, extremely strong bonding (stemming from exchange correlation) is required. Although diamond is the densest known three-dimensional (3D) carbon allotrope in a wide range of pressures, theoretical studies proposed bc8 or R8 structures to be denser. [13] [14] [15] [16] [17] Whether there can be other carbon allotropes denser than diamond is still unknown, but the open topology of the diamond structure gives reasons for positive expectations. Here, we report three allotropes of carbon, which are denser than diamond and any previously proposed structures and possess remarkable physical properties. To search for the densest structures, evolutionary structure prediction was performed using the USPEX code 18,19 in conjunction with ab initio structure relaxations using density functional theory 20 (DFT) within the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), 21 as implemented in the VASP code. 22 We used the all-electron projectoraugmented wave (PAW) with [1s 2 ] core and plane-wave basis set with the 600-eV cutoff. Such calculations provide an excellent description of the density of tetrahedral phases of carbon; e.g., the computed densities are 3.504 g/cm 3 for diamond (3.52 g/cm 3 from experiment) and 3.496 g/cm 3 for lonsdaleite (3.52 g/cm 3 from experiment). The most significant feature of USPEX is the capability of searching for the global minimum according to the fitness function, given only the chemical composition. Here, we used the density as the fitness function, and all structures are fully relaxed at constant pressure. To ensure that the obtained superdense structures are dynamically stable, we calculated phonon frequencies across the Brillouin zone using the finite-displacement approach as implemented in the FROPHO code. 23 We have performed structure searches at 0 GPa with up to 12 atoms in the unit cell. Our simulations produced the already known structures of diamond, hexagonal diamond (lonsdaleite), and the bc8 structure, but the highest density was indicated for two hitherto unknown structures, tI12 (with the tetragonal I-42d symmetry and 12 atoms per unit cell) and hP3 (chiral hexagonal structure with the P6 2 22 symmetry and three atoms per unit cell). The two structures have nearly the same density and, at 1 atm, are 3.2% denser than diamond and 2.2% denser than bc8. By using the chemical analogy approach, we explored the possibility for carbon to adopt the same structure as recently discovered for the new allotrope of germanium (tP12), and this gave us yet another superdense carbon allotrope, which is 1.1% denser than diamond. 24 193410-1