Optical injection of spin current into a zigzag nanoribbon of monolayer ${\mathrm{MoS}}_2$ with antiferromagnetic Kekule distortion

The Kekule pattern of the (anti)ferromagnetic exchange field on monolayer ${\text{MoS}}_2$ can be induced by proximity to the (111) surface of ${\text{BiFeO}}_3$ on both sides. The three-band tight binding model of the ${\text{MoS}}_2$ layer with Kekule patterned exchange field is applied to describe the heterostructures. The tight binding model is justified by the first principle calculations. The magnetization orientations of the substrates control the pattern of the exchange field, which then switches the band structures of the lowest zigzag edge states between being metallic and insulating. The lowest four zigzag edge bands provide conducting channels with a spin-polarized current. Optical excitation of carriers in these bands generates sizable spin and charge currents, which are theoretically modeled by the perturbation solution of the semiconductor Bloch equation. The injected spin currents have multiple resonant peaks at a few frequencies, which can be switched off by rotating the magnetization orientations of the substrates.

Figure 1

(a) Lattice configuration of a ${\text{MoS}}_2$ monolayer with antiferromagnetic Kekule distortion. The yellow dots represent the ${\text{S}}_2$ sites. The blue circular dots, upward-pointing triangles, and downward-pointing triangles represent the Mo sites with different magnetic exchange fields. The dashed line indicates the primitive unit cell. The dotted lines indicate the rectangular unit cells. (b) Three-dimensional atomic structure of the ${\text{BiFeO}}_3/{\text{MoS}}_2/{\text{BiFeO}}_3$ heterostructure within one rectangular unit cell. The dashed and dotted lines connect the nearest and next nearest Fe and Mo atoms, respectively. The arrows at the Fe atoms represent the local magnetic moment.

Figure 2

Orbital projected band structures for the ${\text{BiFeO}}_3/{\text{MoS}}_2/{\text{BiFeO}}_3$ heterostructure with the Y-Kekule configuration obtained from DFT calculations are plotted in (a)–(d). The Fermi energy is set to zero. The symbol size is proportional to its projection in the corresponding state. (a) Contribution from the three $\text{Mo}d$ orbitals $(d_{z^2},d_{xy},d_{x^2-y^2})$. (b) Contribution from the other orbitals of ${\text{MoS}}_2$. (c) Contribution from Fe atoms. In (a)–(c), spin up and down states are plotted as solid (blue) and empty (red) dots, respectively. (d) Contribution from (Bi, O, H) atoms, with both spins being plotted as empty (magenta) squares. The band structures are zoomed in on around the $\mathrm{\Gamma }$ point for the Y-Kekule and O-Kekule configurations in (e) and (f), respectively. Only contributions from the three $\text{Mo}d$ orbitals and (Bi, O, H) atoms are included in (e) and (f). The solid (blue) and dashed (red) lines in (e) and (f) are fittings of the spin up and down band structures by the tight-binding model, respectively.

Figure 3

Band structure of a zigzag nanoribbon with a width of $5\sqrt{3}{a}_0$. The edge states localized on the Mo- and S-edge terminations are plotted as solid (blue) and dashed (red) lines, respectively. The orientations of the magnetizations in the substrates are (a),(d) ${\mathbf{M}}_B={\mathbf{M}}_T=B_0\widehat{z}$, (b),(e) ${\mathbf{M}}_B=B_0\widehat{z}$ and ${\mathbf{M}}_T=B_0\widehat{x}$, (c),(f) ${\mathbf{M}}_B=-{\mathbf{M}}_T=B_0\widehat{z}$, (g) ${\mathbf{M}}_B={\mathbf{M}}_T=B_0\widehat{y}$, (h) ${\mathbf{M}}_B=B_0\widehat{y}$ and ${\mathbf{M}}_T=B_0\widehat{x}$, and (i) ${\mathbf{M}}_B=-{\mathbf{M}}_T=B_0\widehat{y}$. In (d)–(i), the possible band gaps above the first or second edge bands are indicated by the gray area.

Figure 4

Band structure of the zigzag nanoribbon with a width of $5\sqrt{3}{a}_0$ and a magnetization orientation of the substrates of ${\mathbf{M}}_B=-{\mathbf{M}}_T=B_0\widehat{z}$ [the same as that in Fig. 3]. In (a), the bulk states are plotted as thin black lines; the edge states of the Mo- and S-edge terminations are plotted as thin and thick lines with color. The color scale corresponds to the expectation value of the velocity operator in the unit of nm/fs, i.e., $\langle {\widehat{v}}_x\rangle$. In (b), (c), and (d), the energy range is zoomed in on; the color scale corresponds to the expectation value of the spin-velocity operator in the unit of nm/fs, i.e., $\langle {\widehat{s}}_{x,x}\rangle , \langle {\widehat{s}}_{x,y}\rangle$, and $\langle {\widehat{s}}_{x,z}\rangle$, respectively. The physical features for the quantum states marked by a square, diamond, pentagram, or hexagram dot in (b) are plotted in (e), (f), (g), and (h), respectively. Only the three unit cells near the Mo-edge termination are plotted. The probability density at each Mo site is indicated by the size of the marker. The local spin moment at each Mo site, i.e., $\langle {\sigma }_x\rangle \widehat{x}+\langle {\sigma }_y\rangle \widehat{y}+\langle {\sigma }_z\rangle \widehat{z}$, is indicated by the arrows. The velocity $\langle {\widehat{v}}_x\rangle$ is indicated by the horizontal thick arrow in the top view.

Figure 5

Optical injection of charge and spin currents into the nanoribbon with a magnetization orientation of the substrates of ${\mathbf{M}}_B=-{\mathbf{M}}_T=B_0\widehat{z}$ in (a)–(d) or ${\mathbf{M}}_B={\mathbf{M}}_T=B_0\widehat{z}$ in (e) and (f). The Fermi energy level is $E_{F1}$ in (a),(c),(e) and $E_{F2}$ in (b),(d),(f). The temperature in (a),(b) and (e),(f) is 300 K and that in (c),(d) is 70 K.