Pressure- and temperature-driven phase transitions in HgTe quantum wells

We present theoretical investigations of pressure- and temperature-driven phase transitions in HgTe quantum wells grown on a CdTe buffer. Using the eight-band $\mathbf{k}·\mathbf{p}$ Hamiltonian we calculate evolution of energy-band structure at different quantum well widths with hydrostatic pressure up to 20 kbars and temperature ranging up to 300 K. In particular, we show that, in addition to temperature, tuning of hydrostatic pressure allows us to drive transitions between semimetal, band insulator, and topological insulator phases. Our realistic band-structure calculations reveal that the band inversion under hydrostatic pressure and temperature may be accompanied by nonlocal overlapping between conduction and valence bands. The pressure and temperature phase diagrams are presented.

Figure 1

Typical band structure of (001)-oriented HgTe QWs at zero temperature and at different QW width: (a) BI phase, $d<d_c$, (b) Dirac cone, $d=d_c$, (c) TI phase $d>d_c$, (d) gapless state, $d=d_{\text{SM}}$, and (d) SM phase, $d>d_{\text{SM}}$. The electronlike E1 sub-band is shown in blue, while red curves correspond to the heavy-hole sub-bands. In the panels (d) and (e), the E1 sub-band lies significantly lower in energy. Here, we do not differ the TI state, in which the band gap is defined by the gap between H1 and E1 sub-bands, and TI states, in which the E1 sub-band lies below several heavy-hole-like sub-bands [19].

Figure 2

(a,b) Band edge of the electron-like E1; the heavy-hole-like H1, H2, and H3; and the light-hole-like LH1 sub-bands at the $\mathrm{\Gamma }$ point in 8-nm HgTe/${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\mathrm{Te}$ QWs as a function of hydrostatic pressure (a) and temperature (b). (c,d) Dependence of critical pressure $P_c$ and temperature $T_c$, at which the Dirac cone in the $\mathrm{\Gamma }$ point arises [see Fig. 1], as a function of $d$ for HgTe/${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\mathrm{Te}$ QWs. In all the panels, solid curves represent the calculations for (001)-oriented QWs, while the dashed curves correspond to the structures grown on the [013] plane.

Figure 3

(a) Landau levels for (001)-oriented HgTe/${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\mathrm{Te}$ QWs of 8-nm thickness at $T=0$ K and $P=0$ kbar. A pair of zero-mode LLs is shown by red curves. (b) A critical field $B_c$ as a function of QW width $d$ at zero pressure and temperature. (c,d) A critical field $B_c$ as a function of hydrostatic pressure and temperature for 8- and 12-nm QWs. The solid curves in the panels (b–d) display the calculations for (001)-oriented QWs, while the dashed curves correspond to the (013)-oriented structures.

Figure 4

Band structure and contour lines for HgTe/${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\mathrm{Te}$ QWs of 8-nm width, grown on (a) [001] and (b) [013] planes for $T=0$ and $P=0$. The E1 sub-band is shown in blue, and the red surface corresponds to the H1 sub-band. The $x$ and $y$ axes for the (001) QW are oriented along (100) and (010) crystallographic directions, while for the (013) QW, the axes correspond to the directions of (100) and ($03\overline{1}$), respectively.

Figure 5

The band gap in HgTe/${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\mathrm{Te}$ QWs grown on a (001) CdTe buffer as a function of (a) $P$ and (b) $T$, calculated for different QW widths. The negative band-gap values correspond to inverted band structure. The dashed curve shows the evolution of the gap at the $\mathrm{\Gamma }$ point.

Figure 6

(a,b) Overlapping between conduction and valence sub-bands in 20-nm HgTe/${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\mathrm{Te}$ QWs, grown on a (001) CdTe buffer as a function of (a) hydrostatic pressure and (b) temperature. (c,d) Overlapping (black curves) and band gap at the $\mathrm{\Gamma }$ point (red curves): (c) as a function of $P$ for $T=200$ K and (d) as a function of $T$ for $P=10$ kbars. The negative overlapping values in the panels (c,d) define the indirect band gap. The white-open regions are connected with the BI phase, the striped regions correspond to the SM phase, while the gray regions define the range of $P$ and $T$ with the inverted band structure.

Figure 7

Evolution of the band structure in 20-nm HgTe/${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\mathrm{Te}$ QWs, grown on a (001) CdTe buffer, at $T=200$ K with hydrostatic pressure: (a) $P=7.5$ kbars (the SM phase with inverted band structure); (b) $P=8.45$ kbars (the SM phase with the Dirac cone in the $\mathrm{\Gamma }$ point); and (c) $P=9.5$ kbars (the SM phase with direct band ordering). Electron-like and heavy-hole-like sub-bands are shown in blue and red, respectively.

Figure 8

(a) Pressure and (b) temperature phase diagram for (001) HgTe/${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\mathrm{Te}$ QWs, grown on a CdTe buffer. The solid curves correspond to the arising of the Dirac cone at the $\mathrm{\Gamma }$ point. The dotted curves conform to a formation of the gapless states, shown in Fig. 1.