Electric field and strain-induced band-gap engineering and manipulation of the Rashba spin splitting in Janus van der Waals heterostructures

The compositional as well as structural asymmetries in Janus transition metal dichalcogenides (J-TMDs) and their van der Waals heterostructures (vdW HSs) induce an intrinsic Rashba spin splitting. We investigate the variation of band gaps and the Rashba parameter in three different Janus heterostructures having AB-stacked $\mathrm{Mo}XY/\text{W}XY$ $(X, Y=\mathrm{S},\mathrm{Se},\mathrm{Te};X\ne Y)$ geometry with a $Y\text{−}Y$ interface, using first-principles calculations. We consider the effect of external electric field and in-plane biaxial strain in tuning the strength of the intrinsic electric field, which leads to remarkable modifications of the band gap and the Rashba spin splitting. In particular, it is found that the positive applied field and compressive in-plane biaxial strain can lead to a notable increase in the Rashba spin splitting of the valence bands about the $\mathrm{\Gamma }$ point. Moreover, our ab initio density functional theory (DFT) calculations reveal the existence of a type-II band alignment in these heterostructures, which remains robust under positive external field and biaxial strain. These suggest novel ways of engineering the electronic, optical, and spin properties of J-TMD van der Waals heterostructures holding a huge promise in spintronic and optoelectronic devices. Detailed $\mathbf{k}·\mathbf{p}$ model analyses have been performed to investigate the electronic and spin properties near the $\mathrm{\Gamma }$ and $K$ points of the Brillouin zone.

Figure 1

Side and top views of all the stacking orders with $Y\text{−}Y$ interface. Gray and blue spheres represent W and Mo, while magenta and cyan spheres denote $X$ and $Y$ chalcogen atoms, respectively. $Y$ atom is the chalcogen atom with the larger atomic number. The interlayer distance is denoted by $d$.

Figure 2

(a)–(c) Band structures of MoSSe/WSSe, MoSTe/WSTe, and MoSeTe/WSeTe J-HSs with (red) and without SOC (black) using PBE XC functional. Arrows in the figures represent the system band gaps ($E_g^{\text{SOC}}$) with SOC. As seen, MoSTe/WSTe and MoSeTe/WSeTe have indirect band gaps with and without SOC, while it changes from indirect to direct with the inclusion of SOC in the case of MoSSe/WSSe. $M$ (1/2,0,0), $K$ (1/3,1/3,0), and $\mathrm{\Gamma }$(0,0,0) are the high symmetry points in the Brillouin zone. (d)–(f) Orbital-projected band structures of all three J-HSs. The size of the circles is proportional to the corresponding orbital weight. The orbitals weights at the VBM and CBM show that all the three HSs have type-II band alignments.

Figure 3

(a) Schematics of calculating the Rashba parameter ${\alpha }_R$, where $E_R$ is the Rashba energy and $k_R$ is the momentum offset. (b) Directions of the intrinsic and external electric fields. ${\mathrm{E}}_{\text{int}}$, with the red arrow, represents the net intrinsic electric field, and olive green arrows denote the intrinsic fields due to the chalcogen atoms in each layer. $+\mathrm{EEF}$ is the positive external electric field applied in the downward direction from the $\mathrm{Mo}XY$ layer to the $\mathrm{W}XY$ layer. (c) Local electrostatic potential profile for $\mathrm{Mo}XY/\mathrm{W}XY$ HS without any external effect. The red arrow represents the direction of $E_{\text{int}}$. Here the dashed lines indicate the potential drop across the interface, confirming the existence of an intrinsic electric field in the direction from Mo to W atom. (d) A general band structure showing different types of gaps between the valence band and the conduction band. $D_1$ is the direct gap at the $K$ point, $D_2$ is the gap between VBM at $\mathrm{\Gamma }$ and CBM at the $K$ point, and $D_3$ ($D_4$) is the gap between VBM at $K$ ($\mathrm{\Gamma }$) and CBM in the $\mathrm{\Gamma }\text{−}K$ direction. Out of these four gaps, one of them turns out to be the energy band gap of the system.

Figure 4

(a)–(c) Bands around $\mathrm{\Gamma }$ point fitted along $\frac{2\pi }{a}(0.08,0.08,0)-(0,0,0)-\frac{2\pi }{a}(0.12,0,0)$ direction (along $K\text{−}\mathrm{\Gamma }\text{−}M$) of the Brillouin zone, where $a$ is the in-plane lattice constant. The dashed lines and scattered points represent the DFT bands and the $k·p$ bands, respectively. (d)–(f) Spin textures for the inner branch (red bands in the upper panel). (g)–(i) Spin textures for the outer branch (blue bands in the upper panel). The color bar on the right shows the magnitude of the out-of-plane spin projection ($S_z$). The almost colorless backgrounds in the middle and bottom panels depict the 2D nature of Rashba spin splitting around the $\mathrm{\Gamma }$ point.

Figure 5

Band structures for (a) MoSSe/WSSe, (b) MosTe/WSTe, and (c) MoSeTe/WSeTe around the $K$ point are fitted along the $\frac{2\pi }{a}(0.38,0.25,0)-\frac{2\pi }{a}(0.33,0.33,0)-\frac{2\pi }{a}(0.25,0.25,0)$ direction (along $M\text{−}K\text{−}\mathrm{\Gamma }$) of the Brillouin zone, where $a$ is the in-plane lattice constant. Right triangles represent the DFT bands and circles denote the $k·p$ bands in Eq. (5).

Figure 6

(a)–(c) Variation of gaps with EEF for MoSSe/WSSe, MoSTe/WSTe, and MoSeTe/WSeTe HSs, respectively. $D_1$ is the direct gap at the $K$ point, $D_2$ is the gap between VBM at $\mathrm{\Gamma }$ and CBM at the $K$ point, and $D_3$ ($D_4$) is the gap between VBM at $K$ ($\mathrm{\Gamma }$) and CBM in the $\mathrm{\Gamma }\text{−}K$ direction, as shown in Fig. 3. $E_g$ is the true energy band gap indicated by black dashed lines. (d) System band gaps ($E_g$) with HSE06 (solid lines) and GGA- PBE (dashed lines) functionals.

Figure 7

Layer projected band structures and partial charge density plots (band-decomposed charge density plots) under different external electric field strengths for MoSSe/WSSe (upper row), MoSTe/WSTe (middle row), and MoSeTe/WSeTe (bottom row) HSs. The magenta and blue bands denote $\mathrm{Mo}XY$ and $\text{W}XY$ layers, respectively. For all the cases, we plot the charge densities at the $\mathrm{\Gamma }$ point for the valence band (bottom unit cell), while for the conduction band (upper unit cell), we plot charge densities at the $K$ point under zero and positive EEF, and at CBM, which is located in between $\mathrm{\Gamma }$ and $K$ points, under reverse EEF. Isosurface value is set at $0.002e$/Å${}^3$ for charge density plots. Depending on the relative orientations of EEF and $E_{\text{int}}$, charges (yellow) accumulate or deplete from the $\mathrm{W}XY$ bottom layer. Moreover, the layer-projected band structures indicate that the type-II band alignment in the pristine HSs remains robust under a positive electric field, whereas the nature of the band alignment changes by the application of a reverse electric field.

Figure 8

Variation of the Rashba parameter ${\alpha }_R$ with EEF. As seen, ${\alpha }_R$ increases with increasing EEF. In the AB-stacked HSs with $Y\text{−}Y$ interface, the small intrinsic electric field superimposes with positive EEF and amplifies ${\alpha }_R$.

Figure 9

(a) Schematics for applying an in-plane biaxial strain in our AB-stacked HSs. The pink inward (green outward) arrow depicts the compressive (tensile) strain. $a$ and $b$ are the lattice parameters in the $x$ and $y$ directions, respectively, which are equally compressed and stretched when applying a biaxial strain. $\mu$ is the strength of the biaxial strain, $a_0$ and $a$ are the in-plane lattice parameters of the strained and unstrained structures, respectively. (b) Variation of the distance between the chalcogen atoms at the interface ($d_{\text{int}}$) as a function of biaxial strain. Negative and positive values of the strain represent the compressive and the tensile strains, respectively. $d_{\text{int}}$ decreases (increases) linearly with tensile (compressive) strain.

Figure 10

Phonon dispersions for all three HSs at the largest (a)–(c) compressive and (d)–(f) tensile strains. The phonon frequencies remain positive for the HSs at the maximum strains showing their dynamical stability.

Figure 11

Evolution of (a) band gap and (b) Rashba parameter as a function of strain. Different symbols represent different types of gaps shown in Fig. 3 and different colors correspond to different vdW HSs. MoSTe/WSTe HS shows the highest ${\alpha }_R$ at 4% compressive strain.

Figure 12

Projected band structures under $-4$% (upper row) and 5% (lower row) biaxial strains for MoSSe/WSSe, MoSTe/WSTe, and MoSeTe/WSeTe, respectively, from left to right. The weight of the orbitals is defined by the circles of different colors shown in the upper panel. In all cases, the CBM and VBM are located at the $\mathrm{Mo}XY$ and $\mathrm{W}XY$ layers, respectively, confirming that the type-II band alignment remains robust under biaxial strains.