Tunable band gaps in bilayer transition-metal dichalcogenides

We investigate band-gap tuning in bilayer transition-metal dichalcogenides by external electric fields applied perpendicular to the layers. Using density functional theory, we show that the fundamental band gap of MoS${}_2$, MoSe${}_2$, MoTe${}_2$, and WS${}_2$ bilayer structures continuously decreases with increasing applied electric fields, eventually rendering them metallic. We interpret our results in the light of the giant Stark effect and obtain a robust relationship, which is essentially characterized by the interlayer spacing, for the rate of change of band gap with applied external field. Our study expands the known space of layered materials with widely tunable band gaps beyond the classic example of bilayer graphene and suggests potential directions for fabrication of novel electronic and photonic devices.

Figure 1

Schematic of the 2H${}_b$ MX${}_2$ (M $=$ Mo, W; X $=$ S, Se, Te) bilayer structure: (a) top and (b) side views. The metal and chalcogen atoms are represented by large purple and small yellow spheres, respectively. The unit cell is enclosed by black lines. Electric fields are applied normal to the sheets along the positive $c$ axis.

Figure 2

Band structure along $\Gamma -K-M-\Gamma$ direction in reciprocal space as a function of applied external electric field. For MoX${}_2$ compounds, the fundamental band gap at zero field is indirect between the VB maximum at $\Gamma$ and the CB minimum, which lies between $\Gamma$ and $K$. Application of an external field alters the positions of the VB maximum and CB minimum, the details being material specific. For MoS${}_2$, the VB maxima at $\Gamma$ and $K$ are nearly equal in value (to within the error of the calculation) and thus it is not possible to clearly identify whether the gap closing is indirect between $\Gamma$ and $K$ or direct at $K$. For MoSe${}_2$ and MoTe${}_2$, the gap closing is clearly direct at $K$. For WS${}_2$, the zero-field gap is initially between the VB maximum at $\Gamma$ and the CB minimum, which lies between $\Gamma$ and $K$. Upon application of an external field, the CB minimum still remains between $\Gamma$ and $K$, but the VB minimum shifts from $\Gamma$ to $K$; the gap closes eventually between these points.

Figure 3

Band structure along $\Gamma -K-M-\Gamma$ direction in reciprocal space and $lm$-decomposed atom-projected density of states (PDOS) as a function of applied external electric field for bilayer MoS${}_2$. The atom labels for the PDOS plots are indicated in Fig. 1. The two layers are degenerate, as expected, at zero fields. Application of an external electric field breaks inversion symmetry between the layers localizing the HOMO and LUMO on the lower and upper layers, respectively.

Figure 4

Partial charge density from the HOMO at $\Gamma$ [(a), (c)] and the LUMO at $K$ [(b), (d)] at external fields of 0 V/nm (upper row) and 2 V/nm (lower row). All isosurfaces are at $0.05$ $e$/Å${}^3$. At zero electric field inversion symmetry of the layers is preserved. The HOMO at $\Gamma$ is primarily of Mo $d_{z^2}$ and S $p_z$ character with smaller contributions from Mo $d_{xy}$ and $d_{x^2-y^2}$. The LUMO at $K$ is also primarily of Mo $d_{z^2}$ in character with smaller contributions from Mo $d_{xy}$ and $d_{x^2-y^2}$. An external field of 2 V/nm external fields clearly breaks symmetry between the MoS${}_2$ layers, localizing the HOMO and LUMO on different layers.

Figure 5

Charge density difference between bilayer MoS${}_2$ at nonzero and zero external field. Orange and blue isosurfaces correspond to positive and negative values of $5.4\times {10}^{-4}$ $e$/Å${}^3$, respectively. As the electric field increases in going from left to right, we see progressive depletion of charge density in the S $p_z$ orbitals and in the bonding region between the Mo and S atoms. Correspondingly, there is an accumulation of charge density in the S $p_x$ and $p_y$ orbitals as well as the Mo $d_{xy}$, $d_{x^2-y^2}$, and $d_{z^2}$ orbitals.

Figure 6

Band gap $E_g$ versus applied electric field $E$ for MoS${}_2$, MoSe${}_2$, MoTe${}_2$, and WS${}_2$. The lines are fits to the linear portion of the curve indicated by solid symbols. Hollow symbols are within the region of nonlinear response and are excluded from the fits. The GSE coefficients (magnitudes of the slopes of the linear fits) are indicated; interlayer spacings are in parentheses.

Figure 7

Band structure for bulk MoS${}_2$ at (a) experimental lattice parameters, (b) after relaxation with PBE-D2 method, and (c) after relaxation with PBE.

Figure 8

Band structure as a function of applied field for bilayer MoS${}_2$ at (a) experimental lattice parameters (chalcogen atoms relaxed at zero field and kept fixed thereafter), (b) relaxation of entire structure with PBE-D2 method at each applied field, and (c) relaxation of entire structure with PBE-D2 method at zero field (atoms kept fixed thereafter).