Origin of incommensurate modulations in the high-pressure phosphorus IV phase

We present results from x-ray diffraction experiments and density functional theory calculations which provide a fully consistent picture of the high-pressure phosphorus-IV phase (P-IV). P-IV has an incommensurately-modulated crystal structure described by the four-dimensional superspace group $Cmmm\left(00\gamma \right)s00$. Electronic structure calculations using a three-dimensional commensurate approximant to this structure give excellent agreement with experiment for the structural parameters and their variation with pressure. Density functional perturbation theory shows a phonon instability at the incommensurate wave vector, related to the opening of a pseudogap at the Fermi surface, showing that the atomic motions comprising the incommensuration occur to eliminate Fermi-surface nesting. Unusually, the pseudogap opens not at the incommensurate wave vector $\left(00\gamma \right)$ itself, but at two larger wave vectors corresponding to strong reflections in the diffraction pattern.

Figure 1

Diffraction profiles from (a) simple cubic P-III at 108 GPa, (b) intermediate phase P-IV at 124 GPa and (c) simple hexagonal P-V at 136 GPa. The background arising from Compton scattering in the diamond anvils has been subtracted from all three profiles. The additional sharp peak in profile (b) is from the ${\text{H}}_2$ pressure transmitting medium.

Figure 2

Rietveld refinement of P-IV at 124 GPa. The upper and lower tick marks below the profile show the peak positions of the main and satellite reflections, respectively. The indices of some of the Bragg peaks are given. The Bragg peak from the ${\text{H}}_2$ pressure medium is also indicated.

Figure 3

Pressure dependence of the atomic volume of SC P-III, incommensurate P-IV and SH P-V, showing the first-order nature of both the III–IV and IV–V transitions. The inset shows the pressure dependence of the incommensurate wave vector $\left(00\gamma \right)$.

Figure 4

(a) Four unit cells of the incommensurately-modulated structure of P-IV as seen looking down the $b$ axis. The dotted line shows the modulation wave. (b) Structure of the $Cmcm$ and (c) $C2mm$ approximants shown in $a$-axis and $b$-axis projections, respectively. Two unit cells of the $C2mm$ approximant are shown for ease of comparison with the $Cmcm$ approximant. In all three structures, the dark gray atoms are located at a height of (a and c) $y=\frac{1}{2}$ or (b) $x=\frac{1}{2}$.

Figure 5

(Color online). Calculated cell parameters of the SC, SH, and $Cmcm$ approximant phases, all calculated using the $Cmcm$ structure (lines). The solid and open symbols show our, and previous (Ref. 30), experimental data, respectively. Also shown are the calculated cell parameters at selected pressures (open triangles) obtained using the $C2mm$ approximant of Mikhaylushkin et al. (Ref. 10).

Figure 6

Calculated pressure dependence of the modulation amplitude in P-IV. The open and closed symbols show our calculated and experimental data, respectively.

Figure 7

(Color online) Calculated pressure dependence of the enthalpies of the $Cmcm$ and SH phases relative to that of the SC phase.

Figure 8

(Color online) Calculated phonon dispersion curves for (a) the SC and (b) the SH structures versus pressure.

Figure 9

(Color online) (a) Calculated folded phonon dispersion of unmodulated $Cmcm$ and (b) phonon dispersion of the $Cmmm$ structure along the $\Gamma \text{−}Z$ direction.

Figure 10

Calculated phonon dispersions of the $Cmmm$ structure along (a) the $\left[20\gamma \right]$ and (b) the $\left[11\gamma \right]$ directions with $\gamma =0.29$. Figure 10c shows the soft phonon along the $\left[11\gamma \right]$ direction for several values of $\gamma$.

Figure 11

(Color online) Calculated density of electronic states for SC, SH, and modulated phosphorus at 150 GPa, showing the opening of a pseudogap at $E_F$ by the modulation.

Figure 12

Electronic band structure of (a) the SC, and (b) the $Cmcm$ commensurate approximant of P-IV, at 125 GPa. Both calculations were performed using the $Cmcm$ structure.

Figure 13

(Color online) Isosurfaces and ELF profiles of (a) the SC, (b) the SH and (c and d) the $Cmcm$ structures. The represented isosurfaces in (a), (b) and (c) correspond to the values 0.6 (yellow) and 0.8 (blue), and the ELF scale varies between 0 (red) and 1 (dark blue). (a) Isosurfaces and ELF across the (001) plane of the SC phase. (b) Isosurfaces of SH phase. (c) Isosurfaces of the $Cmcm$ structure. The magenta spheres show the location of the phosphorus atoms. (d) ELF modulation along the $c$ axis of the $Cmcm$ phase.