Theoretical study of pressure-driven phase transitions in HgSe and HgTe

We present the results of a density-functional-theory study of the stability of high-pressure phases of HgSe and HgTe up to the so-called Cmcm phase, which is the highest-pressure phase that has been fully characterized in experiments in these compounds. We compare the results obtained using different exchange-correlation functionals for the energetics, stability, and structure of the high-pressure phases, finding that the qualitative picture is very similar and essentially in agreement with the experimental observations. We study the $\text{cinnabar}\to \text{NaCl}$ and the $\text{NaCl}\to \text{Cmcm}$ transitions in detail and provide the structural evolution under pressure of the cinnabar and Cmcm phases. We find that the mechanism for the $\text{NaCl}\to \text{Cmcm}$ transition is somewhat different from other related compounds exhibiting this transition, with cell deformation playing an important role in the stabilization of the Cmcm state in HgSe and HgTe. Our results support the view of a $\text{NaCl}\to \text{Cmcm}$ transition that is distinctly first order, in good agreement with the experimental observations and contrary to other materials where this transition is of second order.

Figure 1

The cubic zincblende structure is shown in perspective in a conventional crystallographic cubic cell on the left-hand side and in projection onto the $\mathit{xy}$ plane on the right-hand side. The black and white circles indicate anion and cation positions, respectively. The gray circles in the projection represent atomic positions in the orthorhombic C222${}_1$ structure, in which the atoms are displaced within the $\mathit{xy}$ plane with respect to the zincblende structure.

Figure 2

(Top) The cinnabar structure of HgTe at $~$3.6 GPa viewed along its threefold $c$ axis, with the contour of the projected unit cell 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 separated by a distance of $c/6$. The differences in height are represented in the projected figure by different sized atoms. (Bottom) The rocksalt or NaCl structure, which can be considered as a special case of the cinnabar structure, plotted in a similar manner. In this case the $c$ axis of the hexagonal cell in the cinnabar representation corresponds to the body diagonal of the cubic NaCl structure, which is also a threefold axis. Nearest-neighbor distances for the sixfold coordinated NaCl structure are shown by straight lines.

Figure 3

The cubic NaCl structure (left-hand side) and orthorhombic Cmcm structure (right-hand side), shown in perspective. The black and white circles represent anion and cation positions, respectively. The Cmcm structure can be obtained from the NaCl structure by shearing alternate (001) planes in the [010] direction, which in turn results in puckering of the [100] atomic rows and an orthorhombic deformation of the cell.

Figure 4

Energy-volume curves for the phases of HgSe calculated using the GGA-PBE. Both energies and volumes are given per formula unit. The energy is given with respect to that of the zero-pressure zincblende phase.

Figure 5

Comparison of the energy-volume curves for the observed phases of HgSe obtained using different XC functionals: (from left to right) LDA (blue), GGA-PBEsol (red), and GGA-PBE (black). Dashed curves correspond to data calculated including SOC, whereas solid curves and symbols correspond to data without SOC. In each case, the energy is given with respect to that of the zero-pressure zincblende phase.

Figure 6

Energy-volume curves for the phases of HgTe calculated using the GGA-PBE. Both energies and volumes are given per formula unit. The energy is given with respect to that of the respective zero-pressure zincblende phases.

Figure 7

Comparison of the energy-volume curves for the observed phases of HgTe obtained using different XC functionals: (from left to right) LDA (blue), GGA-PBEsol (red), and GGA-PBE (black). Dashed curves correspond to data calculated including SOC, whereas solid curves and symbols correspond to data without SOC. In each case, the energy is given with respect to that of the zero-pressure zincblende phase.

Figure 8

Calculated pressure-volume curves for the zincblende, cinnabar, NaCl, and Cmcm phases of HgSe, obtained using the LDA (black, solid curves), GGA-PBE (red, dashed), and GGA-PBEsol (blue, dotted) XC functionals.

Figure 9

Calculated pressure-volume curves for the zincblende, cinnabar, NaCl, and Cmcm phases of HgTe, obtained using the LDA (black, solid curves), GGA-PBE (red, dashed), and GGA-PBEsol (blue, dotted) XC functionals.

Figure 10

Evolution of the structural parameters of cin-HgSe under compression calculated using different XC functionals: LDA (blue), GGA-PBEsol (red), and GGA-PBE (black). The horizontal dotted lines give the values of the structural parameters corresponding to the NaCl structure (see Sec. 3).

Figure 11

Evolution of the structural parameters of cin-HgTe under compression calculated using different XC functionals: LDA (blue), GGA-PBEsol (red), and GGA-PBE (black). The horizontal dotted lines give the values of the structural parameters corresponding to the NaCl structure (see Sec. 3).

Figure 12

Calculated frequencies (within the GGA-PBE) of the vibrational modes $\omega (\mathbf{k})$ of cin-HgSe plotted along high-symmetry lines in the Brillouin zone, for $p=3.8$ GPa (black circles) and $p=7.7$ GPa (red lines). [${\mathbf{k}}_{\Gamma }=0$, ${\mathbf{k}}_M=(1/2,0,0)$, ${\mathbf{k}}_K=(1/3,1/3,1/3)$, ${\mathbf{k}}_A=(0,0,1/2)$, ${\mathbf{k}}_L=(1/2,0,1/2)$, ${\mathbf{k}}_H=(1/3,1/3,1/2)$ in terms of the conventional reciprocal generators of the hexagonal lattice.]

Figure 13

Calculated frequencies (within the GGA-PBE) of the vibrational modes $\omega (\mathbf{k})$ of cin-HgTe plotted along high-symmetry lines in the Brillouin zone, for $p=2.4$ GPa (black circles) and $p=5.6$ GPa (red lines).

Figure 14

Evolution of the structural parameters of Cmcm-HgSe under compression calculated using different XC functionals: LDA (blue), GGA-PBEsol (red), and GGA-PBE (black).

Figure 15

Evolution of the structural parameters of Cmcm-HgTe under compression calculated using different XC functionals: LDA (blue), GGA-PBEsol (red), and GGA-PBE (black).

Figure 16

Change in energy per formula unit of NaCl-HgSe along the path of its Cmcm distortion (see text). We use the value of the internal parameter $y$ of the cation to indicate the position of the internal configuration along the path of distortion, with $y=0.75$ representing the undistorted NaCl configuration (see Sec. 3). At each of the three compressions depicted, the solid curve and the circles are for full relaxation of the cell shape at each point on the path of distortion, whereas the dashed curves correspond to keeping the shape fixed to that of the NaCl configuration.

Figure 17

Calculated frequencies (within the GGA-PBE) of the vibrational modes $\omega (\mathbf{k})$ of NaCl-HgSe plotted along high-symmetry lines in the Brillouin zone at three different compressions. Full black lines and circles correspond to a pressure $p~17.6$ GPa ($V=40.47$ Å${}^3$), red dashed lines correspond to a larger pressure of $~$44.5 GPa ($V=35.15$ Å${}^3$), and blue dotted lines correspond to a smaller pressure of $~$10.8 GPa ($V=42.74$ Å${}^3$). [${\mathbf{k}}_{\Gamma }=0$, ${\mathbf{k}}_X=(0,0,1)$, ${\mathbf{k}}_L=(1/2,1/2,1/2)$, ${\mathbf{k}}_W=(0,1/2,1)$, in units of $2\pi /a$.]

Figure 18

Calculated frequencies (within the GGA-PBE) of the vibrational modes $\omega (\mathbf{k})$ of NaCl-HgTe plotted along high-symmetry lines in the Brillouin at three different compressions. Full black lines and circles correspond to a pressure $p~9.0$ GPa ($V=50.69$ Å${}^3$), red dashed lines correspond to a larger pressure of $~$15.8 GPa ($V=47.53$ Å${}^3$) and blue dotted lines correspond to a smaller pressure of $~$2.4 GPa ($V=55.36$ Å${}^3$).

Figure 19

Calculated evolution with pressure of the phonon frequencies of NaCl-HgSe at the $\Gamma$ (circles), $X$ (triangles), and $\Lambda$ (squares) points, calculated using the GGA-PBE (solid curves, filled symbols) and LDA (dashed curves, empty symbols).

Figure 20

Calculated evolution with pressure of the phonon frequencies of NaCl-HgTe at the $\Gamma$ (circles), $X$ (triangles), and $\Lambda$ (squares) points, calculated using the GGA-PBE (solid curves, filled symbols) and LDA (dashed curves, empty symbols).