Strain-tuned spin polarization and optical conductivity in ${\mathrm{MoS}}_2/\mathrm{EuS}$ heterostructures

We investigate strain-tuned spin transport and optical conductivity in a monolayer (ML) $n$-type ${\mathrm{MoS}}_2/\mathrm{EuS}$ heterostructure under linearly polarized terahertz radiation using a low-energy effective Hamiltonian, which takes displacement of the Dirac point induced by strain into account. Through modifying the strain modulus, we find an adjustable relationship between the band edge energy and the strain. Thus the threshold and range of allowed angles are strongly dependent on the strength of strain $\epsilon$, whereas the angle $\theta$ along which the strain is applied only acts on the former. Due to spin-flipped scattering generated by Rashba spin-orbit coupling, the in-plane spin polarizations (i.e., $P_X$ and $P_Y$) can be achieved, and their intensity can be enhanced by increasing $\epsilon$. Interestingly, we find that the out-of-plane spin polarization (i.e., $P_Z$) is related to the synergy of $\epsilon$ and the Rashba parameter ${\lambda }_R$. Especially, $P_Z$ can approach $100\%$ when $\epsilon =9\%$ and ${\lambda }_R=2$ meV. Furthermore, the optical transition between the spin-splitting subbands of the conduction band under the strain is studied. It is shown that by varying $\epsilon$ we can control the magnitude of both absorption peaks and absorption valleys in optical conductivity. Moreover, the presence of ${\lambda }_R$ can be used to further enhance the absorption peaks and valleys because ${\lambda }_R$ can tune the energy difference between the spin-splitting subbands within a specific valley and thus affect the optical transition channels. Our findings demonstrate that the strain can influence the tunneling behavior and valley Hall conductivity in a ML $n$-type ${\mathrm{MoS}}_2/\mathrm{EuS}$ heterostructure, and these features are applicable to other two-dimensional ML transition metal dichalcogenides, which is promising for terahertz and valleytronic devices.

Figure 1

(a) Schematic of the proposed model. Regions $\mathrm{I}$ and $\mathrm{III}$ represent the pure monolayer (ML) ${\mathrm{MoS}}_2$ regions. Region $\mathrm{II}$ corresponds to the strained ${\mathrm{MoS}}_2/\mathrm{EuS}$ region, in which a vertical electric field and a relatively weak terahertz (THz) radiation field linearly polarized along the 2D ${\mathrm{MoS}}_2$ plane are applied. The selected $x$ axis is parallel to the zigzag direction of the ${\mathrm{MoS}}_2$ sheet, and the direction of the uniform planar tension is $\theta$. (b) Energy potential profile of the strained ${\mathrm{MoS}}_2/\mathrm{EuS}$ heterostructure with gate voltage modulation, which is sandwiched between two pure ${\mathrm{MoS}}_2$ MLs. $\varphi$ is the incident angle, and the region of positive (negative) incidence angles is defined as the area in the clockwise (counterclockwise) direction relative to the normal incidence.

Figure 2

The band edge energy as a function of (a) $\epsilon$ when $\theta =0\pi$ and (b) $\theta$ when $\epsilon =3\%$.

Figure 3

(a)–(d) Isoenergy surfaces (ISs) of the strained (colored circles) and the unstrained (US) regions (black circles for the $K$ valley and gray ones for the $K{}^{\prime }$ valley) under various configurations of $(\epsilon ,U\text{(meV)})$. The colored shaded parts represent the range of allowed angles in which significant transmission may occur. (e)–(h) Band structures under different $\epsilon$ with $U=100$ meV. The coordinates of point A are $(0.116,0.05375)$ ${\text{Å}}^{-1}$, and those of point B are $(0.125,-0.0283)$ ${\text{Å}}^{-1}$. The parameters are $B_c=10$ meV, $B_v=8$ meV, ${\lambda }_R=2$ meV, and $\theta =0.25\pi$.

Figure 4

(a)–(d) Angular distributions of $T_{ss{}^{\prime }}$ in the two valleys with different configurations of $(\epsilon ,U\text{(meV)})$. The incident energy is $E=950$ meV, $B_c=10$ meV, $B_v=8$ meV, ${\lambda }_R=2$ meV, $L=10$ nm, and $\theta =0.25\pi$.

Figure 5

$T_{\uparrow \uparrow }$ as a function of the incident angle $\varphi$ and the incident energy for different configurations of $(\epsilon ,U\text{(meV)})$ in the $K$ valley [(a), (c), and (e)] and $K{}^{\prime }$ valley [(b), (d), and (f)]. $B_c=10$ meV, $B_v=8$ meV, ${\lambda }_R=2$ meV, $L=10$ nm, and $\theta =0.25\pi$.

Figure 6

(a) and (b) Isoenergy surfaces in the strained (colored circles) and the unstrained (black and gray circles) regions under different configurations of $(\theta ,{\lambda }_R\text{(meV)})$ in the $K$ and $K{}^{\prime }$ valleys with spin up, respectively. The shaded regions represent the range of incident angles in which transmission can occur. (c) and (d) Angular distributions of $T_{\uparrow \uparrow }$ under various $(\theta ,{\lambda }_R\text{(meV)})$ in the $K$ and $K{}^{\prime }$ valleys, respectively. The incident energy is $E=950$ meV, $B_c=10$ meV, $B_v=8$ meV, $U=100$ meV, and $\epsilon =3\%$.

Figure 7

$T_{\uparrow \uparrow }$ as a function of the incident angle $\varphi$ and the incident energy under different configurations of $(\theta ,{\lambda }_R\text{(meV)})$ in the $K$ valley [(a), (c), and (e)] and $K{}^{\prime }$ valley [(b), (d), and (f)]. $B_c=10$ meV, $B_v=8$ meV, $U=100$ meV, $\epsilon =3\%$, and $L=10$ nm.

Figure 8

Valley polarization as a function of the incident angle $\varphi$ and the incident energy for (a) $\theta =0$ and (b) $\theta =\pi /2$. $B_c=10$ meV, $B_v=8$ meV, ${\lambda }_R=2$ meV, $\epsilon =3\%$, and $L=10$ nm.

Figure 9

The components of the spin polarization vector with variable $\epsilon$. $B_c=10$ meV, $B_v=8$ meV, $\theta =0$, and $U=100$ meV. ${\lambda }_R=2$ meV for (a), (c), and (e), and ${\lambda }_R=50$ meV for (b), (d), and (f).

Figure 10

The components of the spin polarization vector with variable $\epsilon$. $B_c=10$ meV, $B_v=8$ meV, $\theta =0.25\pi$, and $U=100$ meV. ${\lambda }_R=2$ meV for (a), (c), and (e), and ${\lambda }_R=50$ meV for (b), (d), and (f).

Figure 11

The components of the spin polarization vector as a function of $\epsilon$. $B_c=10$ meV, $B_v=8$ meV, $\theta =0$, $E_F=932$ meV, and $U=100$ meV. ${\lambda }_R=2$ meV for (a) and (c), and ${\lambda }_R=50$ meV for (b) and (d).

Figure 12

(a) and (b) $G_{xy}$ and (c) and (d) $G_{ss}$ as a function of $\epsilon$. $B_c=10$ meV, $B_v=8$ meV, $\theta =0$, $E_F=932$ meV, and $U=100$ meV. The insets in (c) and (d) are the total conductance vs $\epsilon$. ${\lambda }_R=2$ meV for (a) and (c), and ${\lambda }_R=50$ meV for (b) and (d).

Figure 13

Real parts of the optical conductivities as a function of radiation frequency under variable $\epsilon$. (a1) and (a2) $E_F=830$ meV, (b1) and (b2) $E_F=870$ meV, (c1) and (c2) $E_F=910$ meV, and (d1) and (d2) $E_F=950$ meV. $B_c=10$ meV, $B_v=8$ meV, ${\lambda }_R=2$ meV, and $T=5$ K.

Figure 14

(a) and (b) Band structures of the conduction band and possible transition channels under different strain moduli. The solid and dotted colored lines indicate the spin-up and spin-down subbands, respectively. The solid and dashed black lines are the positions of $E_F=830$ meV and $E_F=950$ meV, respectively. (c) and (d) Radiation frequency range within which excitation of the spin-flip interband transition is possible when $E_F=830$ meV; the shaded areas indicate the range of wave vectors $k_x$ and the corresponding frequency interval where the transition can occur. For the red shading, $\epsilon =-20\%$; for the blue shading, $\epsilon =0\%$; and for the green shading, $\epsilon =20\%$. $B_v=10$ meV, $B_c=8$ meV, ${\lambda }_R=2$ meV, and $T=5$ K. (a) and (b) are for the $K$ valley, and (b) and (d) are for the $K{}^{\prime }$ valley.

Figure 15

Real parts of the optical conductivities as a function of radiation frequency with variable ${\lambda }_R$. (a1) and (a2) $T=5$ K, (b1) and (b2) $T=25$ K, and (c1) and (c2) $T=50$ K. $B_c=10$ meV, $B_v=8$ meV, $\epsilon =-20\%$, and $E_F=830$ meV.

Figure 16

(a) and (b) Band structures of conduction band and possible transition channels under different ${\lambda }_R$. The solid and dotted colored lines indicate the spin-up and spin-down subbands, respectively. The solid black line is the position of $E_F=830$ meV. (c) and (d) Radiation frequency range within which excitation of the spin-flip interband transition is possible; the yellow shaded area indicates the range of wave vectors $k_x$ and the corresponding frequency interval where the transition can occur when $E_F=830$ meV. $B_c=10$ meV, $B_v=8$ meV, and $\epsilon =-20\%$. (a) and (c) are for the $K$ valley, and (b) and (d) are for the $K{}^{\prime }$ valley.