First-principles study of HgTe/CdTe heterostructures under perturbations preserving time-reversal symmetry

The observation of the quantum spin Hall effect without the need of an external magnetic field in HgTe/CdTe heterostructures triggered the study of materials exhibiting persistent spin-polarized electronic currents at their interfaces. These Dirac-like spin states are predicted to be topologically protected against perturbations preserving time-reversal symmetry. However, nonmagnetic (time-reversal preserving) perturbations will certainly affect these interface states. In this work, the density functional theory is used to characterize the topologically protected states of the (001) HgTe/CdTe heterostructure as well as to understand the influence of external adiabatic parameters as pressure and electric fields on these states. The intrasite Hubbard U term is seen to be important to correctly describe the HgTe bulk band structure. It is shown that, differently from the three-dimensional topological insulators, the HgTe/CdTe interface states present fully in-plane Rashba-like spin texture. Further, biaxial external pressures and electric fields perpendicular to the interfaces are seen to change the energetics and dispersion of the protected states, modifying the energy ordering of the crossing of the polarized interface states inside the band structure, and altering their Fermi velocities while not changing the topological quantum phase. These adiabatic variables can then be used to tune the topologically protected states with respect to the Fermi level.

Figure 1

Band structures for the zinc-blende (a) CdTe, and (b) HgTe bulk materials with DFT/GGA approach. In (c) the band structure for the zinc-blende HgTe bulk material calculated using the DFT/GGA+U approach, with $U-J=9.4$ eV on the Hg $5d$ level.

Figure 2

Calculated projected density of states (PDOS) for the HgTe bulk using GGA (solid line) and GGA+U (dotted line) DFT approaches.

Figure 3

Calculated band structure around $\Gamma$ point for HgTe/CdTe heterostructure with fixed HgTe thickness (64.65 Å), and variable CdTe thickness. The value of the CdTe thickness appears at the top of each band structure. The red line segments show levels for which the major contributions have a ${\Gamma }_6$ $s$-like character.

Figure 4

Calculated band structure around $\Gamma$ point for HgTe/CdTe heterostructure with fixed CdTe thickness (64.65 Å), and variable HgTe thickness. The value of the HgTe thickness appears at the top of each band structure. The red line segments show levels for which the major contributions have a ${\Gamma }_6$ $s$-like character.

Figure 5

The figure at the left side illustrates the unit cell of the HgTe/CdTe heterostructure. The middle panel shows the calculated band structures for the $\Gamma \to X$ direction of the planar Brillouin zone and their respective spin textures (${\mathrm{S}}_x$, ${\mathrm{S}}_y$, and ${\mathrm{S}}_z$) for half the band structure for (a) the interface at the top, and (b) the interface at the bottom of the illustrative picture. The right panel shows in (c) the band structure and spin textures of the top interface for the $\Gamma \to M$ direction, and in (d) a schematic representation of the spin configuration at the planar Brillouin zone. The green and red points represent the levels with resulting spin-up and spin-down polarizations, respectively. The points in black designate the states whose charge densities are concentrated at the interface regions, i.e., the interface states. The spin textures were calculated by considering the layers inside the black boxes of the illustrative picture.

Figure 6

The calculated spin-resolved energy contours for three selected spin-polarized interface states, one above and two below TIS crossing point. The green, red, violet, and gold colors represent the $S_y\uparrow$, $S_y\downarrow$, $S_x\uparrow$, and $S_x\downarrow$, respectively. The intermediate colors mean that the direction of the spin polarization is gradually changing between its values on the coordinate axes.

Figure 7

The right panel shows the calculated charge densities of specific levels along the heterostructure HgTe/CdTe. The picture below the charge densities represents the HgTe/CdTe heterostructure unit cell (the colors follow the same pattern of Fig. 5), and allows one to identify the charge density values with the different regions of the heterostructure. The arrows coming from the HgTe/CdTe band structure indicate which levels are being considered to the charge density analysis. The green and red circles represent the levels with resulting up and down spin polarization, respectively. The size of the circles are directly related to the magnitude of the spin polarization. The points in black designate the states whose charge densities are concentrated at the interface regions, i.e., the interface states.

Figure 8

Calculated band structures for one interface of the HgTe/CdTe heterostructures subjected to external hydrostatic pressures. The values on top of each band structure represent the variation of the lattice constant relatively to the stress free configuration. Positive (negative) percentage values result in tensile (compressive) stress, respectively. The green and red circles represent the levels with resulting up and down spin polarization, respectively. The size of the circles are directly related to the magnitude of the spin polarization. The points in black designate the states whose charge densities are concentrated at the interface regions, i.e., the interface states.

Figure 9

Calculated band structures for one interface of the HgTe/CdTe heterostructures subjected to external biaxial pressures. The values on top of each band structure represent the variation of the lattice constant relatively to the stress free configuration. Positive (negative) percentage values result in tensile (compressive) stress, respectively. The green and red circles represent the levels with resulting up and down spin polarization, respectively. The size of the circles are directly related to the magnitude of the spin polarization. The points in black designate the states whose charge densities are concentrated at the interface regions, i.e., the interface states.

Figure 10

The picture at the right side illustrates the CdTe/HgTe/CdTe quantum well unit cell and the direction of the applied electric field. Also shown are the calculated band structures for the interfaces at (a) the top, and (b) the bottom part of the quantum well. The values of the applied external electric fields appear on top of each band structure. The green and red circles represent the levels with resulting up and down spin polarization, respectively. The size of the circles are directly related to the magnitude of the spin polarization. The points in black designate the states whose charge densities are concentrated at the interface regions, i.e., the interface states.

Figure 11

The calculated average electrostatic potential, ${\overline{V}}_{el}$, along the direction perpendicular to the interface planes of the (001) HgTe/CdTe heterostructure. The solid black line shows the ${\overline{V}}_{el}$ without external electric field, while the dashed red line shows the ${\overline{V}}_{el}$ for an applied external electric field of $5\times {10}^{-3}$ eV/Å. The vertical dotted lines show the position of the interfaces.