Solid oxygen $\zeta$ phase and its transition from $\varepsilon$ phase at extremely high pressure: A first-principles analysis

The $\varepsilon$ and $\zeta$ phases of solid oxygen are studied with periodic hybrid Kohn-Sham calculations and atomic basis sets. The evolution of the lattice parameters and transition pressure are in excellent agreement with experiment. The greater stability of the $\zeta$ phase at high pressure is mainly due to thermal contributions. Phonon calculations indicate that the experimental $C2/m {\left({\mathrm{O}}_2\right)}_4$ unit cell for the $\zeta$ phase actually arises as a time average of $P1$ reduced-symmetry ${\left({\mathrm{O}}_2\right)}_{16}$ unit cells. The structural distortion of the reduced-symmetry $\zeta$ structure gives rise to a mechanism that could explain the superconductivity of the $\zeta$ phase.

Figure 1

Schematic representation of the C2/m arrangement for the $\varepsilon$ and $\zeta$ phases of solid oxygen. (a) View along the $ab$ plane; (b) perpendicular one. Intermolecular ${\mathrm{O}}_2-{\mathrm{O}}_2{d}_1$ [intracluster (orange)] and $d_2$ [intercluster (blue)] distances are shown.

Figure 2

(a) Optimized $a$ (top), $b$ (middle), and $c$ (bottom) lattice parameters with the B3LYP/TZP method as a function of pressure. $\varepsilon$ (green circles) and $\zeta$ phase (red diamonds); (b) $\varepsilon$ and $\zeta$ phase optimized vs experimental $\beta$ lattice parameter; (c) $\varepsilon$ and $\zeta$ phase optimized vs experimental $d{}_1$ (intracluster) and $d{}_2$ (intercluster) distances. Experimental data (blue squares) from Ref. [7].

Figure 3

Enthalpy difference ($\mathrm{\Delta }H$) between $\zeta$ and $\varepsilon$ phases in the pressure range in which both solutions coexist. The estimated free energy difference ($\mathrm{\Delta }G$) for both the C2/m and P1 $\zeta$ structures in the coexistence range are also depicted (stars and lines below; see details in text).

Figure 4

Phonon dispersion diagram of the C2/m $\zeta$ phase at 110 GPa. Frequencies below zero refer to imaginary values. The right panel corresponds to the line $(1/4,1/4,\lambda )$ connecting the $p=(1/4,1/4,0)$ and $q=(1/4,1/4,1/4)$ points.

Figure 5

Three perspectives of the imaginary phononic mode(blue arrows) that leads to the optimized reduced-symmetry P1 structure of the $\zeta$ phase at 110 GPa. The ${\left({\mathrm{O}}_2\right)}_4$ clusters are indicated with dashed lines.

Figure 6

Interatomic distances for the $\zeta -P1$ phase. Top and bottom panels: view along the $XY$ planes above and below $Z=0$, respectively. The blue line shows the propagation direction of the imaginary frequency phonon. The shortest intercluster distance in this direction is equal to the shortest intracluster distance at 110 GPa.

Figure 7

Enthalpy profile of the $\zeta$ phase along the imaginary phononic mode.