Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides

We present a three-band tight-binding (TB) model for describing the low-energy physics in monolayers of group-VIB transition metal dichalcogenides $M{X}_2$ ($M=\text{Mo}$, W; $X=\text{S}$, Se, Te). As the conduction- and valence-band edges are predominantly contributed by the $d_{z^2}$, $d_{xy}$, and $d_{x^2-y^2}$ orbitals of $M$ atoms, the TB model is constructed using these three orbitals based on the symmetries of the monolayers. Parameters of the TB model are fitted from the first-principles energy bands for all $M{X}_2$ monolayers. The TB model involving only the nearest-neighbor $M$-$M$ hoppings is sufficient to capture the band-edge properties in the $\pm K$ valleys, including the energy dispersions as well as the Berry curvatures. The TB model involving up to the third-nearest-neighbor $M$-$M$ hoppings can well reproduce the energy bands in the entire Brillouin zone. Spin-orbit coupling in valence bands is well accounted for by including the on-site spin-orbit interactions of $M$ atoms. The conduction band also exhibits a small valley-dependent spin splitting which has an overall sign difference between Mo$X_2$ and W$X_2$. We discuss the origins of these corrections to the three-band model. The three-band TB model developed here is efficient to account for low-energy physics in $M{X}_2$ monolayers, and its simplicity can be particularly useful in the study of many-body physics and physics of edge states.

Figure 1

(a) Top view of monolayer $M{X}_2$. The large sphere is $M$ and the small sphere is $X$. ${\mathit{R}}_1$ through ${\mathit{R}}_6$ show the $M$-$M$ nearest neighbors. The shadowed diamond region shows the two-dimensional (2D) unit cell with lattice constant $a$. (b) Schematic for the structure of trigonal prismatic coordination, corresponding to the blue triangle in (a). (c) The 2D first Brillouin zone with special $\mathit{k}$ points. ${\mathit{b}}_1$ and ${\mathit{b}}_2$ are the reciprocal basis vectors. The two inequivalent valleys $K$ and $-K$ are shown in black and their equivalent counterparts in gray.

Figure 2

Orbital projected band structures for monolayer MoS${}_2$ from FP calculations. Fermi energy is set to zero. Symbol size is proportional to its population in corresponding state. (a) Contributions from Mo $d$ orbitals: blue dots for $d_{xy}$ and $d_{x^2-y^2}$, red open circles for $d_{z^2},$ and green open diamonds for $d_{xz}$ and $d_{yz}$. (b) Total $p$ orbitals, dominated by S atoms. (c) Total $s$ orbitals.

Figure 3

The NN TB band structures (blue or dark curves) of $M{X}_2$ monolayers compared with the FP ones (red or gray curves and dots). VBMs are shifted to zero. The dots show the band components from $d_{z^2}$, $d_{xy}$, and $d_{x^2-y^2}$ orbitals, with which the TB bands should compare. (a–f) GGA and (g–l) LDA.

Figure 4

Energy bands from the TNN TB model (blue or dark curves) of $M{X}_2$ monolayers compared with the FP ones in the GGA case (red or gray curves and dots). The dots show the band components from $d_{z^2}$, $d_{xy}$, and $d_{x^2-y^2}$ orbitals, with which the TB bands should compare.

Figure 5

Quantities from the TNN TB for monolayer MoS${}_2$ under GGA parameters: (a) degree of circular polarization, $\eta \left(\mathit{k}\right)$, and (b) Berry curvature $\Omega \left(\mathit{k}\right)$ in units of Å${}^2$ along $\mathit{k}$ path $-M\to -K\to \Gamma \to K\to M$. (c) Color map of $\eta \left(\mathit{k}\right)$, where the hexagon shows the BZ.

Figure 6

Energy bands of monolayers Mo$X_2$ with SOC. Thick blue dashed curves are the TB bands: (a–c) from the NN TB model and (d–f) from the TNN TB model. Thin red solid curves are FP results with GGA. VBMs are shifted to zero.

Figure 7

Contour maps in the $\mathit{k}$ space for monolayer MoS${}_2$ from the NN TB model (using the GGA parameters): (a) the valence-band SOC splittings in units of eV, (b) the Berry curvatures, and (c) the spin Berry curvatures in units of Å${}^2$. The hexagons show the BZ. The gray thin curves are the contour lines corresponding to their tick values on the color bars.

Figure 8

Schematic for the conduction- and valence-band spin splittings in the $\pm K$ valleys for Mo$X_2$ (left) and W$X_2$ (right). Red dashed curves are spin-up states and blue solid ones spin-down states. The conduction-band spin splitting has an overall sign change between Mo$X_2$ and W$X_2$. Crossings exist for the spin-split conduction bands of Mo$X_2$.

Figure 9

The energy bands for zigzag MoS${}_2$ nanoribbon with width $W=8$. Red dots are bands from the TB model using the GGA parameters. Curves are the FP bands, in which blue shows the contributions from the $d_{z^2},$ $d_{xy}$, and $d_{x^2-y^2}$ orbitals and green for other orbitals. For the bands labeled 1 through 4, see the text.

Figure 10

(a) Valence and (b) conduction bands in the $K$ valley of monolayer MoS${}_2$, within the range of 0.1$\times \frac{2\pi }{a}$ in $\Gamma$ and $M$ directions. Open circles are FP results (GGA case). Blue dashed, red solid, and black solid curves are the bands from the two-band $\mathit{k}·\mathit{p}$ model of $H_{kp}^{\left(1\right)}$, $H_{kp}^{\left(2\right)}$, and $H_{kp}^{\left(3\right)}$, respectively. CBMs and VBMs are both shifted to 0. $a=3.190$ Å and $\Delta =1.663$ eV for all. Other fitted parameters are the following: $t=1.105$ eV for $H_{kp}^{\left(1\right)}$; $t=1.059$ eV, ${\gamma }_1=0.055$ eV, ${\gamma }_2=0.077$ eV, and ${\gamma }_3=-0.123$ eV for $H_{kp}^{\left(2\right)}$; $t=1.003$ eV, ${\gamma }_1=0.196$ eV, ${\gamma }_2=-0.065$ eV, ${\gamma }_3=-0.248$ eV, ${\gamma }_4=0.163$ eV, ${\gamma }_5=-0.094$ eV, and ${\gamma }_6=-0.232$ eV for $H_{kp}^{\left(3\right)}$.