Tunable intrinsic spin Hall conductivity in bilayer ${\mathrm{PtTe}}_2$ by controlling the stacking mode

We report a systematic study on the intrinsic spin Hall conductivity (ISHC) of bilayer ${\mathrm{PtTe}}_2$ and explore the connection between the stacking order and ISHC. We find that by changing the stacking mode, ISHC can be manipulated from positive to negative values. Such strong stacking-dependent ISHC originates from the interlayer coupling, in which Te atoms in the upper and lower layers can form either van der Waals or covalentlike quasibonding depending on the stacking modes. Thus ISHC can be effectively tuned by changing the stacking order. These results not only allow us to establish fundamental understanding of ISHC in bilayer ${\mathrm{PtTe}}_2$ dependent on the stacking mode but also provide guidelines for the application of bilayer ${\mathrm{PtTe}}_2$ in next-generation spintronic devices.

Figure 1

(a)–(c) Side and (d)–(f) top views of bilayer ${\mathrm{PtTe}}_2$ for AA, AB, and ${\mathrm{AB}}^{\prime }$ stacking configurations. Blue and orange circles indicate Pt and Te atoms, respectively. The dashed circles represent atoms in the bottom layer. The ISHC as a function of the Fermi energy for (g) AA, (h) AB, and (i) ${\mathrm{AB}}^{\prime }$ stacking configurations. Green and red dashed lines indicate the Fermi levels $E-E_f=0$ and $E-E_f=-0.5$ eV, respectively. The dashed black line represents the zero value of ISHC.

Figure 2

Band structures projected by taking the logarithm of spin Berry curvature for (a) AA and (b) AB stacking configurations. Green and red dashed lines indicate the energy levels at $E-E_F=0$ eV and $E-E_F=-0.5$ eV, respectively. $k$-resolved spin Berry curvature after taking the logarithm of Eq. (2) [defined by Eq. (4)] in the 2D Brillouin zone ($k_z=0$) at the energy level position of (c) and (d) $E-E_f=0$ eV and (e) and (f) $E-E_f=-0.5$ eV for AA and AB stacking configurations. Red and blue colors represent positive and negative contributions, respectively.

Figure 3

Band-decomposed charge densities for (a) AA and (b) AB stacking configurations. Interlayer differential charge densities and the simplified models for (c) AA and (d) AB stacking configurations. The light blue and yellow isosurfaces indicate charge depletion and accumulation, respectively. Spin textures for (e) AA and (f) AB stacking configurations.

Figure 4

(a) Interlayer distance, (b) binding energy, and (c) ISHC for the full space of lateral shifts.