Strain-controlled spin splitting in the conduction band of monolayer ${\text{WS}}_2$

Spin splitting bands that arise in the conduction band minimum (CBM) of the ${\text{WS}}_2$ monolayer (ML) play an important role in the spin-orbit phenomena such as spin-valley coupled electronics. However, application of strain strongly modifies electronic properties of the ${\text{WS}}_2$ ML, which is expected to significantly affect the properties of the spin splitting bands. Here, by using fully-relativistic first-principles calculations based on density-functional theory, we show that a substantial spin splitting band observed in the CBM is effectively controlled and tuned by applying the biaxial strain. We also find that these spin splitting bands induce spin textures exhibiting fully out-of-plane spin polarization in the opposite direction between the $K$ and $Q$ points and their time reversals in the first Brillouin zone. Our study clarifies that the strain plays a significant role in the spin-orbit coupling of the ${\text{WS}}_2$ ML, which has very important implications in designing future spintronics devices.

Figure 1

(a) Side view of a $2H-{\text{WS}}_2$ unit cell, consisting of two ${\text{WS}}_2$ ML slab. (b) First Brillouin zone of monolayer ${\text{WS}}_2$ which is specified by the high symmetry points ($\mathrm{\Gamma },K,M$, and $Q$ points). Here, the $Q$ point is defined as a point located approximately midway between the $\mathrm{\Gamma }$ and $K$ point. (c) Orbital resolved of the electronic band structures of ${\text{WS}}_2$ ML. The radius of circles reflects the magnitudes of spectral weight of the specific orbitals to the band.

Figure 2

Electronic band structures of the strained ${\text{WS}}_2$ ML: (a) $\epsilon =-2\%$ (b) $\epsilon =0\%$, and (c) $\epsilon =2\%$. The black lines (pink lines) indicate the band structures without (with) spin-orbit interaction.

Figure 3

Relation between the structural parameters [bond length of the S-W atoms and the S-W-S angle], energy of spin splitting bands in the the conduction band minimum (CBM) and valence band maximum (VBM), and the energy minimum in the CBM as a function of strain. (a) The bond length $d_{\text{S-W}}$ (red circle-lines) and angle ${\theta }_{\text{S-W-S}}$(green square-lines) are given. (b) The energy of the spin splitting bands in the CBM calculated around the $K$ point ($\mathrm{\Delta }{E}_{K,\mathrm{CBM}}$, blue circle-lines] and the $Q$ point $[\mathrm{\Delta }{E}_{Q,\mathrm{CBM}}$, pink square-lines], and in the VBM around the $K$ point $[\mathrm{\Delta }{E}_{K,\mathrm{VBM}}$, blue solid-circle-lines] are shown. The energy minimum of the CBM around the $K$ point $[E_K$, blue square-lines) and $Q$ point $[E_Q$, pink circle-lines] calculated (c) without and (d) with spin-orbit coupling (SOC). Here, the energy minimum of the CBM is calculated with respect to the Fermi level, $E_F$.

Figure 4

Band structures of the CBM corresponding to the spin textures: (a) $\epsilon =-2\%$ (b) $\epsilon =0\%$, and (c) $\epsilon =2\%$. In the spin textures, the solid (dotted) lines represents up (down) spin states with out-of-plane orientations. The spin textures are calculated on the energy band of 225 meV above the CBM.