Intrinsic persistent spin helix state in two-dimensional group-IV monochalcogenide $MX$ monolayers ($M=\mathrm{Sn}$ or Ge and $X=\mathrm{S}$, Se, or Te)

Energy-saving spintronics are believed to be implementable on systems hosting the persistent spin helix (PSH) since they support an extraordinarily long spin lifetime of carriers. However, achieving the PSH requires a unidirectional spin configuration in the momentum space, which is practically nontrivial due to the stringent conditions for fine-tuning the Rashba and Dresselhaus spin-orbit couplings. Here, we predict that the PSH can be intrinsically achieved on a two-dimensional (2D) group-IV monochalcogenide $MX$ monolayer, a new class of the noncentrosymmetric 2D materials having in-plane ferroelectricity. Due to the $C_{2v}$ point-group symmetry in the $MX$ monolayer, a unidirectional spin configuration is preserved in the out-of-plane direction and thus maintains the PSH that is similar to the [110] Dresselhaus model in the [110]-oriented quantum well. Our first-principle calculations on various $MX$ ($M=$ Sn, Ge; $X=$ S, Se, Te) monolayers confirmed that such typical spin configuration is observed, in particular, at near the valence-band maximum where a sizable spin splitting and a substantially small wavelength of the spin polarization are achieved. Importantly, we observe reversible out-of-plane spin orientation under opposite in-plane ferroelectric polarization, indicating that an electrically controllable PSH for spintronic applications is plausible.

Figure 1

(a) Atomic structure of the $MX$ monolayer corresponding to it symmetry operations. Black and green balls represent the $M$ ($M=$ Sn, Ge) and $X$ ($X=$ S, Se, Te) atoms, respectively. The unit cell of the crystal is indicated by red lines characterized by $a$ and $b$ lattice parameters in the $x$ and $y$ directions. $d_1$ and $d_2$ represent bond length between the $M$ ($M=$ Sn, Ge) and $X$ ($X=$ S, Se, Te) atoms in the in-plane and out-of-plane directions, respectively. (b) First Brillouin zone of the $MX$ monolayer characterized by high-symmetry $\vec{k}$ points ($\mathrm{\Gamma }$, Y, M, X) are shown. (c) Spin-split bands induced by the SOC and $C_{2v}$ point-group symmetry and (d) the corresponding Fermi contours in the momentum space are schematically shown. Here, the Fermi contours are characterized by two Fermi loops shifted by the wave vector $\vec{D}$, exhibiting a unidirectional spin configuration in the out-of-plane direction. The red and blue lines (or arrows) represent positive and negative spins, respectively, in the out-of-plane directions.

Figure 2

(a) Electronic band structure of the $MX$ monolayers corresponding to the density of state projected to the atomic orbitals for (a) SnS, (b) SnSe, (c) SnTe, (d) GeS, (e) GeSe, and (f) GeTe. The black and red lines show the calculated band structures without and with the SOC, respectively.

Figure 3

(a) Energy-band dispersion of SnTe monolayer along the $M\text{−}Y\text{−}\mathrm{\Gamma }$ lines calculated without (black lines) and with (red lines) the SOC. (b) Zoom-in on the energy dispersion near the VBM close to the $Y$ point along $M\text{−}Y$ and $Y\text{−}\mathrm{\Gamma }$ lines, as highlighted by the blue lines in panel (a). (c) Spin-splitting properties of the bands around the $Y$ point along the $M\text{−}Y\text{−}M$ lines characterized by (i) splitting energy ($\mathrm{\Delta }E$), i.e., the energy difference between the VBM along the $Y\text{−}M$ line and the energy at the $Y$ point, and (ii) momentum offset ($k_0$).

Figure 4

Schematic view of the energy level around the $Y$ point near the VBM. The SOC splits the states into two doublets with eigenvalues of $M_{yz}=\pm 1$, which are further split into a singlet with sign-reversed expectation values of spin.

Figure 5

Energy profiles of the spin textures calculated around the $Y$ point for (a) outer and (b) inner branches in the spin-split bands near the VBM. The color scale in panels (a) and (b) indicate the energy band. Constant-energy contours corresponding to a cut 1 meV below the VBM characterized by (c) $S_x$, (d) $S_y$, and (e) $S_z$ components of the spin distribution are shown. The color scale in panels (c)–(e) show the modulus of the spin polarization.

Figure 6

(a) Nudged elastic band calculation for the polarization switching process through centrosymmetric (paraelectric) structures in SnTe monolayer. Two ferroelectric structures (FE) in the ground state with opposite direction of the in-plane electric polarization and a paraelectric structure with zero electric polarization (NP) are shown. $E_b$ is the barrier energy defined as the difference between the total energy of the ferroelectric ($E_{\mathsf{\text{T}}}$) and paraelectric ($E_{\mathsf{\text{NP}}}$) structures. Reversible out-of-plane spin orientation in SnTe monolayer calculated at 1 meV below the VBM for the ferroelectric structure with opposite in-plane polarization: (b) $-P$ and (c) $P$.