Physical Properties of Group 14 in $P6_{2}$22 Phase: First-Principles Calculations *

Two new Group IV element allotropes Si$_{3}$ and Ge$_{3}$ in P6$_{2}$22 phase are predicted in this work and their physical properties are investigated using the density functional theory. Each of the newly predicted allotropes has a super dense structure, which is mechanically, dynamically, and thermodynamically stable, as verified by elastic constants, phonon dispersion spectra and relative enthalpies, respectively. The mechanical anisotropy propertiesare studied in detail by illustrating the directional dependence of Young's modulus, discussing the universal anisotropic index, and calculating shear anisotropy factors together with bulk moduli. It shows that P6$_{2}$22-Si$_{3}$ exhibits the greater anisotropy than P6$_{2}$22-Ge$_{3}$,and interestingly both of the newly predicted crystals appear to be isotropic in the (001) plane. Additionally, the Debye temperature, sound velocities, and the minimum thermal conductivity are examined to evaluate the thermodynamic properties of C$_{3}$, Si$_{3}$, and Ge$_{3}$ in P6$_{2}$22 phase, and the electronic band structures are achieved by HSE06 hybrid functional, which indicate that P6$_{2}$22-C$_{3}$ and -Si$_{3}$ are indirect band gap semiconductors and P6$_{2}$22-Ge$_{3}$ exhibits the metallic feature.