Soft-phonon instability in zincblende HgSe and HgTe under moderate pressure: Ab initio pseudopotential calculations

The mercury chalcogenides HgSe and HgTe show peculiar behavior under moderate compression such as the unusual observation of a “hidden” transition from their ambient-pressure zincblende phase to an orthorhombic metastable phase with space group $C{222}_1$. We present here the results of an ab initio pseudopotential study within the framework of the density-functional theory of the stability of the low-pressure zincblende phases of HgSe and HgTe and the transition to the $\text{C}{222}_1$ structure. We relate the observation of the $\text{C}{222}_1$ phase to an instability in the transverse-acoustic phonon branches of the zincblende phases of these materials. We have studied in detail the mechanism of the $\text{zincblende}\to \text{C}{222}_1$ transition as well as the structural evolution of the $\text{C}{222}_1$ phase under pressure.

Figure 1

The zincblende structure, plotted in perspective using a conventional crystallographic cubic cell (left) and in projection onto the $xy$ plane (right). The black and white circles represent anion and cation positions, respectively. The gray circles in the projected plot represent the atomic positions in the orthorhombic $\text{C}{222}_1$ structure, in which the atoms are displaced within the $xy$ plane with respect to the zincblende structure.

Figure 2

The cinnabar structure of HgTe at $\sim 3.6\text{GPa}$, viewed along its threefold main $c$ axis. (For reference, the outline of the projected unit cell is shown as a dashed line.) The black and white circles represent anion and cation positions, respectively. Atomic planes of like atoms perpendicular to the main axis are uniformly separated by a distance of $c/6$. The differences in height are represented in the projected figure by the different sizes of the atoms. This structure is sometimes described as formed of spirals of atoms along the $c$ axis in which cations and anions alternate. In our drawing the projected spirals are represented by solid lines.

Figure 3

(Color online) Calculated energy-volume curves for the zincblende, cinnabar, and $\text{C}{222}_1$ phases of HgSe, obtained within the LDA. Both magnitudes are given per formula unit. The energy is given with respect to that of the equilibrium volume of the zincblende phase. The experimental equilibrium volume of the zincblende phase is marked by an arrow.

Figure 4

(Color online) Same as Fig. 3 but for HgTe.

Figure 5

(Color online) Calculated phonon frequencies of zincblende HgSe plotted along symmetry lines of the Brillouin zone of the zincblende structure. [${\mathbf{k}}_{\Gamma }=0$, ${\mathbf{k}}_X=\left(0,0,1\right)$, ${\mathbf{k}}_L=\left(1/2,1/2,1/2\right)$, ${\mathbf{k}}_W=\left(0,1/2,1\right)$, in units of $2\pi /a$.] Full lines and circles correspond to zero pressure whereas dotted lines correspond to a pressure slightly above 3 GPa.

Figure 6

(Color online) Calculated phonon frequencies of zincblende HgTe plotted along symmetry lines of the Brillouin zone of the zincblende structure. Full lines and circles correspond to zero pressure whereas dotted lines correspond to a pressure slightly above 3 GPa.

Figure 7

Evolution with pressure of the calculated phonon frequencies at the $X$ (circles) and $\Gamma$ (triangles) points of the Brillouin zone of zb-HgSe.

Figure 8

Same as Fig. 7 but for zb-HgTe.

Figure 9

Contour plot of the energy $E$ of $\text{C}{222}_1\text{-HgSe}$ as a function of its internal parameters $\left(x,y\right)$. This plot is for $V=56.3{\text{Å}}^3$ $\left(p\sim 0\text{GPa}\right)$, that is, before the onset of the zb instability. The energy is given in meV and is measured with respect to that of the reference zb structure [which in the plot corresponds to the point (0.25,0.25)]. At each $\left(x,y\right)$ point the shape of the cell was fully relaxed. The dashed line indicates the path of minimum-energy change from the zb configuration.

Figure 10

Same as Fig. 9 but for a volume $V=52.7{\text{Å}}^3$ $\left(p\sim 3\text{GPa}\right)$ at which the zb structure is unstable against the $\text{C}{222}_1$ distortion. The calculated minimum-energy $\text{C}{222}_1$ configuration is marked by a circle.

Figure 11

The change in energy per formula unit along the minimal path shown in Fig. 9 (left panel) and Fig. 10 (right panel). (Note the difference in the energy scales between the panels.) We use the value of the internal parameter $x$ to indicate the position along the path. The solid curve and symbols (which correspond to calculated values) are for full relaxation of the cell shape at fixed volume and each $\left(x,y\right)$ point. The dashed curve corresponds to keeping the shape fixed to that of the zb configuration.

Figure 12

(Color online) Evolution of the calculated structural parameters of $\text{C}{222}_1\text{-HgSe}$ with pressure: lattice parameters $a$, $b$, and $c$ (upper panel) and internal parameters $x$ and $y$ (lower panel). We have denoted by open symbols and dashed lines those data in the region of instability of the $\text{C}{222}_1$ phase. The dotted lines correspond to values of the parameters for the zincblende phase at the same pressure.

Figure 13

(Color online) Same as Fig. 12 but for $\text{C}{222}_1\text{-HgTe}$.

Figure 14

(Color online) Evolution with pressure of the calculated nearer neighbor distances ($d_1$ and $d_2$) of $\text{C}{222}_1\text{-HgSe}$ (upper panel) and $\text{C}{222}_1\text{-HgSe}$ (lower panel). Data in the region of instability of $\text{C}{222}_1$ are denoted by open symbols and dashed lines. The nearest-neighbor distance of the zincblende phases, $d\left(\text{zb}\right)$, is also given (dotted lines), for reference.

Figure 15

(Color online) The calculated Kohn-Sham electronic band structure of the $\text{C}{222}_1$ phases of HgSe (left panel) and HgTe (right panel) along two symmetry lines from the $\Gamma$ point (circles). [${\mathbf{k}}_{\Gamma }=0$, ${\mathbf{k}}_Z=\left(0,0,1/2\right)$, and ${\mathbf{k}}_L=\left(1/4,1/4,1/4\right)$ in units of $\left(4\pi /a,4\pi /b,4\pi /c\right)$.] The zero of the energy scale is set at the middle of the small gap at $\Gamma$ (which is hardly discernible in the scale of the figures). These figures are for volumes of $53.2{\text{Å}}^3$ (HgSe) and $62.5{\text{Å}}^3$ (HgTe) for which structural data have been reported experimentally (Ref. 10). The calculated band structures of the respective zincblende phases at the same volumes are also given for reference (full lines).