Pentadiamond: A Hard Carbon Allotrope of a Pentagonal Network of ${\mathrm{sp}}^2$ and ${\mathrm{sp}}^3$ C Atoms

A pentagonal covalent network consisting of ${\mathrm{sp}}^2$ and ${\mathrm{sp}}^3$ C atoms has been investigated based on the density functional theory. Our theoretical investigations clarified that the pentagonal covalent network is a metastable three-dimensional carbon allotrope with the $Fm\overline{3}m$ space group possessing remarkable mechanical properties: relatively high bulk modulus of 381 GPa together with a negative Poisson’s ratio of $-0.241$. Accordingly, the pentagonal covalent network possesses extremely high Young’s and shear moduli of 1691 and 1113 GPa, respectively, surpassing those of the diamond. The electronic structure of the pentagonal network is a semiconductor with an indirect band gap of 2.52 eV between $L$ and $X$ points for valence and conduction band edges, respectively, with the relatively small carrier masses.

Figure 1

(a) An optimized geometry of the pentadiamond under an optimum lattice constant of 9.195 Å and the $Fm\overline{3}m$ space group. Pale and dark gray balls indicate ${\mathrm{sp}}^3$ and ${\mathrm{sp}}^2$ carbon atoms, respectively. Indexes C1, C2, and C3 indicate the independent atomic site under the $Fm\overline{3}m$ space group. (b) Spiro[4.4]nona-2,7-diene and [5.5.5.5]fenestratetraene are alternately arranged at the vertexes of the cubic lattice. White and gray balls indicate H and C atoms, respectively.

Figure 2

Phonon dispersion and phonon density of states of pentadiamond.

Figure 3

(a) Calculated Young’s modulus, (b) shear modulus, and (c) Poisson’s ratio of the pentadiamond as a function of the lattice orientations. Solid and dotted lines indicate the positive and negative values, respectively.

Figure 4

The electronic band structure and density of states of the pentadiamond along the high symmetry points and axes. The energies are measured from that of the valence band top.

Figure 5

Isosurfaces of the squared wave function of the pentadiamond of the highest branch of the valence band at (a) $\mathrm{\Gamma }$ and (b) $L$ points, and those of the lowest branch of the conduction band at (c) $\mathrm{\Gamma }$ and (d) $X$ points. Pale and dark gray balls indicate ${\mathrm{sp}}^3$ and ${\mathrm{sp}}^2$ carbon atoms, respectively. The isosurfaces correspond with the electron density of $3.5e/(\mathrm{a}.\mathrm{u}.{)}^3$.