Doping-induced persistent spin helix with a large spin splitting in monolayer SnSe

Finding a new class of materials supporting a long spin lifetime is essential in development of energy-saving spintronics, which is achievable by using persistent spin helix (PSH) materials. However, for spintronic devices, the PSH states with large spin splitting are required for operation at room temperature. By employing first-principles calculations, we show that the PSH states with large spin splitting are achieved in the SnSe monolayer functionalized by a substitutional halogen impurity. We find the PSH states in the Fermi level where $k$-space Fermi surface is characterized by the shifted two loops, dominated by out-of-plane spin orientations. We clarify the PSH states in terms of an effective $\vec{k}·\vec{p}$ Hamiltonian obtained from symmetry consideration. Finally, large spin-orbit strength in the PSH states with a substantially small wavelength is found, rendering that this system is promising for the development of efficient and high-density scalable spintronic devices operating at room temperatures.

Figure 1

(a) Structure of the pristine SnSe monolayer in the top view and side views is shown. The unit cell is indicated by the rectangle with a red line, which is characterized by $a$ and $b$ lattice parameters. The layers sit on the $x\text{−}y$ plane with the in-plane distortion ${\mathrm{\Delta }}_y$ along the $y$ axis. The bond lengths between Sn and Se in the in-plane ($d_1$) and out-of-plane ($d_2$) directions are indicated. ${\mathrm{\Delta }}_y$ represents in-plane distortion along the $y$ direction. (d) Various possible doping configurations such as (1) $X_{\text{Sn}}$, (2) $X_{\text{Se}}$, (3) $X_{\text{T-Sn}}$, (4) $X_{\text{T-Se}}$, (5) $X_{\text{B}}$, and (6) $X_{\text{H}}$ are shown. The optimized structures for (c) ${\mathrm{I}}_{\text{Se}}$ and (d) ${\mathrm{I}}_{\text{H}}$ as a representative example of the $X_{\text{Se}}$ and $X_{\text{H}}$, respectively. The red ball indicates the I atom. Mirror symmetry $M_{yz}$ operation in the optimized ${\mathrm{I}}_{\text{Se}}$ is indicated by the blue dashed lines. (e) First Brillouin zone of the SnSe ML is shown, where the high-symmetry points such as $\mathrm{\Gamma }$, $X$, $Y$, and $K$ points are indicated.

Figure 2

Electronic band structures and density of states (DOS) of the pristine SnSe ML. (a) The band structures calculated without the SOC corresponding to the DOS projected to the atom (b). Here, red and green colors represent s and p orbitals of Sn atom, respectively, while blue and black colors represent s and p orbitals of Se atom, respectively. (c) The band structures calculated with the SOC are shown.

Figure 3

Electronic band structures of $X_{\text{Se}}$ and the pristine systems. The calculated band structures without the SOC for (a) the pristine, (b) ${\mathrm{Cl}}_{\text{Se}}$, (c) ${\mathrm{Br}}_{\text{Se}}$, and (d) ${\mathrm{I}}_{\text{Se}}$, and with the SOC for (e) the pristine, (f) ${\mathrm{Cl}}_{\text{Se}}$, (g) ${\mathrm{Br}}_{\text{Se}}$, and (h) ${\mathrm{I}}_{\text{Se}}$. The dashed black lines show the Fermi level. The red lines indicate the localized impurity bands (LIB) around the Fermi level.

Figure 4

The band structures of the ${\mathrm{I}}_{\text{Se}}$ around the Fermi level calculated without (left) and with (right) the SOC. Here, the spin-splitting energy ${\mathrm{\Delta }}_{\text{SOC}}$ is indicated.

Figure 5

Orbital resolved of the spin-split impurity bands of the ${\mathrm{I}}_{\text{Se}}$ projected to the atoms near impurity site. The red, blue, and green colors indicate the Sn, Se, and I atoms, respectively. The circle radius reflects the spectral weight of the specific orbital to the band.

Figure 6

(a) The calculated spin textures projected to the 2D $\left(k_x\text{−}{k}_y\right)$ plane are shown. The blue and pink arrows indicate the spin textures for the upper and lower bands, respectively. The expectation values for (b) $〈S_x〉$, (c) $〈S_y〉$, and (d) $〈S_z〉$ are given. The blue and pink colors reflect the expectation value of spin calculated for the upper and lower bands, respectively.