Effects of structural distortions on the electronic structure of $T$-type transition metal dichalcogenides

Single-layer transition metal dichalcogenides (TMDCs) can adopt two distinct structures corresponding to different coordination of the metal atoms. TMDCs adopting the $T$-type structure exhibit a rich and diverse set of phenomena, including charge density waves (CDWs) in a $\sqrt{13}\times \sqrt{13}$ supercell pattern in ${\mathrm{TaS}}_2$ and ${\mathrm{TaSe}}_2$, and a possible excitonic insulating phase in ${\mathrm{TiSe}}_2$. These properties make the $T$-TMDCs desirable components of layered heterostructure devices. In order to predict the emergent properties of combinations of different layered materials, one needs simple and accurate models for the constituent layers which can take into account potential effects of lattice mismatch, relaxation, strain, and structural distortion. Previous studies have developed ab initio tight-binding Hamiltonians for $H$-type TMDCs [S. Fang et al., Phys. Rev. B 98, 075106 (2018)]. Here we extend this work to include $T$-type TMDCs. We demonstrate the capabilities and limitations of our model using three example systems: a one-dimensional sinusoidal ripple, which represents a longitudinal acoustic phonon; the $2\times 2$ CDW in ${\mathrm{TiSe}}_2$; and the $\sqrt{13}\times \sqrt{13}$ CDW in ${\mathrm{TaS}}_2$. Using the technique of band unfolding we compare the electronic structure of the distorted crystals to the pristine band structure and find our tight-binding model reproduces many features revealed by direct density functional theory calculations, provided the magnitude of the distortions remains in the linear regime. This model of the strain response of single layers is a necessary ingredient for the construction of models of van der Waals heterostructures with multiple layers, because the deformation and strain from mechanical relaxations in a twisted bilayer have important effects on the electronic structure.

Figure 1

$T$-type TMDC crystal structure seen from the (a) side and (b) top. For comparison we show the $H$-type TMDC crystal from the (c) side and (d) top, respectively. ${\mathit{a}}_1$ and ${\mathit{a}}_2$ are the primitive lattice vectors of the two-dimensional (2D) crystal (green arrows).

Figure 2

DFT band structure and density of states (DOS) for two $T$-type TMDCs, (a) ${\mathrm{TaS}}_2$ and (b) ${\mathrm{TiSe}}_2$, showing the 11 bands nearest the Fermi level. The valence-band character is dominated by the chalcogen $p$ orbitals, while the transition metal $d$ orbitals dominate the conduction-band character. A notable feature is the large peak in the conduction-band DOS for ${\mathrm{TiSe}}_2$ which is absent for ${\mathrm{TaS}}_2$.

Figure 3

(a) Comparison of the band structure calculated with DFT (black) and the full Wannier basis (cyan) for unstrained ${\mathrm{TaS}}_2$. Also shown are the energy windows used to define the Wannier transformation and disentangle the bands arising from other orbitals [62, 64]. (b) Comparison of the band structure computed directly with DFT (thin black), 5NN Wannier basis (dashed green), and the 3NN TBH parametrization (thick red) for unstrained ${\mathrm{TaS}}_2$.

Figure 4

Comparison of the band structure calculated using the 3NN TBH (red) and DFT (black) for a ${\mathrm{TaSe}}_2$ crystal with three values of isotropic strain: (a) 2% compression, (b) unstrained, and (c) 2% expansion. The TBH bands are labeled by the irreducible representation of the symmetry group of the wave vector at each high-symmetry point. ($\mathrm{\Gamma }$: ${\mathrm{D}}_{3d}$, M: ${\mathrm{C}}_{2h}$, K: ${\mathrm{D}}_3$).

Figure 5

Change in band energy due to spin-orbit coupling for three bands near the Fermi level in unstrained ${\mathrm{TaSe}}_2$. These bands can be identified as an $A_{1g}$ singlet and $E_g$ doublet at $\mathrm{\Gamma }$. The difference in DFT calculated band energies with and without SOC, $\mathrm{\Delta }{E}_{\mathrm{soc}}=E_{\mathrm{with}-\mathrm{soc}}-E_{\mathrm{no}-\mathrm{soc}}$, is shown in black, whereas the differences in tight-binding eigenvalues with and without SOC is shown in red.

Figure 6

The displacements of the metal atoms used in the applications of the strain-dependent TBH. (a) Sinusoidal strain in the $x$ direction with a period of $N=4$ rectangular units. (b) $2\times 2$ CDW pattern for ${\mathrm{TiSe}}_2$. (c) $\sqrt{13}\times \sqrt{13}$ CDW pattern for ${\mathrm{TaS}}_2$. The supercells are shown with dashed black lines, and the chalcogen atoms are omitted for clarity.

Figure 7

Band structures for an $N\times 1$ rectangular supercell of ${\mathrm{TaS}}_2$, unfolded to a path in the Brillouin zone of the three-atom primitive unit cell. (a) TBH bands for $N=4$ without strain. (b) TBH bands for $N=4$ subject to 1D sinusoidal strain with amplitude $B=0.02$ and wavelength $\lambda =23$ Å. (c) TBH bands for $N=16$, subject to 1D sinusoidal strain with amplitude $B=0.02$ and wavelength $\lambda =93$ Å.

Figure 8

Band structures for the $2\times 2$ CDW supercell of ${\mathrm{TiSe}}_2$, unfolded to a path in the Brillouin zone of the three-atom primitive unit cell. (a) TBH bands without the CDW distortion. (b) TBH bands for a strain pattern fit to the CDW displacements calculated with DFT. (c) DFT band structure for the relaxed CDW, for comparison.

Figure 9

Magnitude of the 1NN matrix element $H_{74}^{1\text{NN}}$, namely, the $d_{z^2}\text{−}{p}_z$ coupling in ${\mathrm{TaS}}_2$, as a function of isotropic strain, $u_{xx}+u_{yy}$. The red points represent strain in the range $\pm 2\%$ used to produce the linear parametrization (red dashed line) for the strain-dependent TBH. The vertical dotted black lines indicate the range of isotropic strain occurring in the ${\mathrm{TaS}}_2$ CDW pattern.

Figure 10

Band structures for the $\sqrt{13}\times \sqrt{13}$ CDW supercell of ${\mathrm{TaS}}_2$, unfolded to a path in the Brillouin zone of the three-atom primitive unit cell. (a) TBH bands without the CDW distortion, (b) TBH bands for a strain pattern fit to the CDW displacements calculated with DFT, and (c) DFT band structure for the relaxed CDW.