Phosphorus under pressure: Ba-IV-type structure as a candidate for P-IV

Pressure-induced transformations in phosphorus are examined by means of ab initio density-functional calculations. Attempts are made to identify the structure of the P-IV high-pressure phase, and transformation paths between a self-hosting incommensurate Ba-IV-type structure and the body-centered cubic structure are discussed in this context. Total energy calculations for a series of commensurate approximations are used to optimize the Ba-IV-type structure with a body-centered tetragonal “guest.” This suggests that an optimized structure with a $c$-axis host-guest ratio $c_H∕c_G\sim 1.14$ is energetically favorable as a candidate for the structure of P-IV. The $c_H∕c_G$ found here differs significantly from the values determined experimentally for Ba-IV-type structures in Bi and Sb.

Figure 1

A Ba-IV-type structure shown in a $c$-axis projection. The open and gray circles denote host structure atoms with $z$ coordinates 0 and $1∕2$, respectively. In the figure, modulations of this substructure have been neglected and a “regular octagonal” host tube has been chosen. This term is used since the shown projection contains an octagon which is equilateral with the present choice of host structure $x$ parameter (see text for details). Different Ba-IV-type structures are distinguished by the choice of guest structure (black circles). The series of known guest structures are shown to the right in the figure. These structures are (from the top) $c$-face-centered monoclinic $\left(c\text{−}\mathrm{fcm}\right)$, body-centered tetragonal $\left(\mathrm{bct}\right)$, and $c$-face-centered tetragonal $\left(c\text{−}\mathrm{fct}\right)$. Apparently, the ratio $c_H∕c_G$ between the lengths of the host and guest unit cells along the $c$ axis can be irrational, i.e., the guest structure can be incommensurate with the host structure along this direction.

Figure 2

Calculated (GGA) enthalpy differences (relative to bcc) for P sc, ph, fcc, and two Ba-IV-type structures with $c_H∕c_G=4∕3$ as functions of pressure, in the range $80–200\mathrm{GPa}$. The transition from sc to ph (ph to bcc) occurs at $126\pm 3(194\pm 17)\mathrm{GPa}$ according to these calculations, where the uncertainty of these pressures has been obtained by defining a $\pm 0.5\mathrm{mRy}∕\text{atom}$ uncertainty of the calculated energies. The phase P-IV has been reported stable between 107 and $137\mathrm{GPa}$ in experiment (Ref. 5). Remaining structures are discussed in Sec. 3 (fcc) and Sec. 5 (Ba-IV type). The Ba-IV-type results shown in this figure correspond to structures which have not been optimized. The optimization (Sec. 5) yields energies $\approx 5\mathrm{mRy}∕\text{atom}$ below the bcc values.

Figure 3

Isothermal equations of state for P in the pressure range $0–300\mathrm{GPa}$ according to experiment $(T=300\mathrm{K})$ and theory $(T=0\mathrm{K})$. For further details, see the text and Table . The vertical dotted lines in the figure denote (from the left) the $\mathrm{A}7\to \mathrm{sc}$, $\mathrm{sc}\to \mathrm{P}\text{−}\mathrm{IV}$, $\mathrm{P}\text{−}\mathrm{IV}\to \mathrm{ph}$, and $\mathrm{ph}\to \mathrm{bcc}$ transition pressures reported in experiment (Refs. 5, 6) (see Sec. 1).

Figure 4

Axial ratio $c∕a$ of P ph as a function of pressure according to experiment (Ref. 6) and the GGA calculations of this work. The dotted curve in the figure approximates the “visual guide” to the complete set of 16 experimental data points.

Figure 5

Optimized (see text for details) host structure positional parameters $x_1$ (circles), $x_2$ $-1∕2$ (triangles), and $x_3$ (squares) at the atomic volume $V=8.93{\mathrm{\AA }}^3$ of the $I4cm$ Ba-IV model structure, describing P Ba-IV fct with host-guest ratio $c_H∕c_G=4∕3$, as functions of the relative position of the host and guest structures along the $c$ axis (specified by the parameter $z_4$). The three parameters specify the width $\left[\sqrt{1+8{\left(u_i\right)}^2-4u_i}\right]a$, $u_i=x_i$ modulo $1∕2$, of the host tube for this structure [a parameter defined here as the diagonal of the square of host atoms with equal $z$ coordinates (see Fig. 1)] at equidistant positions with separation $c_M∕12$ in the unit cell. The host tube width is smallest (i.e., $u_i$ is largest) when there is a guest atom at the center of the host square.

Figure 6

Ba-IV-type structure, with $c_H∕c_G=1$ and a regular octagonal host tube (compare, e.g., with Fig. 1), and bcc structure, shown in a projection onto the (001) plane. The bcc unit cell is marked by the dashed lines in the right figure. Filled (open) circles denote atoms with $z$ coordinate $1∕4$ $(3∕4)$, while the open squares denote the two guest atoms with no restrictions on the $z$ coordinates in the present discussion. The Ba-IV-type structure can be transformed into bcc in the following way: (i) The guest structure is changed to bct, with the $z$ coordinate of the atom at the center of the unit cell being equal to $3∕4$; (ii) the host square at $z=1∕4$ (dashed lines in the left figure) is truncated and rotated until it fits into the enlarged and equally rotated host square at $z=3∕4$ (dotted lines in the left figure), with all the atoms in the whole structure having equivalent surroundings; (iii) the axial ratio $c_M∕a$ of the Ba-IV-type unit cell is changed into $1∕\sqrt{5}$.

Figure 7

Total energy of P Ba-IV bct as a function of the host-guest ratio, at the atomic volume $V=8.93{\mathrm{\AA }}^3$. The optimized values of these parameters for this system have been tabulated in Table . The curve in the figure is a fit to the energies of the optimized Ba-IV bct structures with host-guest ratios of $2∕3$, 1, $4∕3$, and $3∕2$, described by the $P4∕n$ Ba-IV model (see text for details).

Figure 8

Same as in Fig. 7, but for $V=8.36{\mathrm{\AA }}^3$.