Electronic correlation effects and orbital density wave in the layered compound $1T{\text{-TaS}}_2$

In this paper, we present the electronic structures and orbital-resolved electronic properties of structurally distorted $1T\text{−}{\mathrm{TaS}}_2$ bulk and monolayer within density functional theory approaches. The relaxed commensurate-charge-density-wave (CCDW) structure shows that 13 Ta atoms condense into a star-of-David cluster, accompanied by a buckling of neighboring $S$ planes. Through detailed analyses to the orbital characters near the Fermi level, we show that there exists an orbital-density-wave (ODW) order, which is predominantly contributed by Ta-$5d_{3z^2-r^2}$ orbital in the central Ta of the star-of-David cluster. We further demonstrate that the structural distortion, together with the Coulomb interaction, stabilizes the CCDW insulating ground state with an ODW order. The results obtained from dynamical mean-field theory confirm the role of the electronic correlation. Moreover, such an ODW ground state favors an intralayer ferromagnetic order in bulk and monolayer $1T\text{−}{\mathrm{TaS}}_2$, and an interlayer antiferromagnetic order in bulk. We propose that $1T\text{−}{\mathrm{TaS}}_2$ monolayer may pave new ways to study exciton physics, flat-band physics, and potential applications in luminescence.

Figure 1

(a) The distortion mode in the Ta-atom layer. Red stars show the star-of-David cluster, and arrows illustrate the distortion directions of Ta atoms. Orange and purple circles correspond to the first-ring and the second-ring Ta atoms in the star-of-David cluster, respectively. ${\mathbf{a}}_{\mathbf{1},\mathbf{2}}$ and ${\mathbf{R}}_{\mathbf{1},\mathbf{2}}$ are the lattice vectors of the high-$T$ normal phase and the low-$T$ CCDW phase, respectively. $a=12.348$ Å is the optimized lattice constant in the CCDW phase. (b) The hexagonal Brillouin zones of the normal phase (green region) and the distorted phase (black lines). The Brillouin zone of the distorted phase is rotated by $13.9^{\circ }$ with respect to that of the normal phase.

Figure 2

Electronic structures of the bulk $1T\text{−}{\mathrm{TaS}}_2$ in the high-$T$ normal phase: (a) the band structure along high symmetry directions, (b) total and partial DOS, (c) the elevation view, and (d) the top view of Fermi surfaces.

Figure 3

Real part of static electronic susceptibility of the bulk $1T\text{−}{\mathrm{TaS}}_2$ with arbitrary units vs. wave vector (a) in the cross section of $q_z=\frac{{\mathbf{c}}^{*}}{13}$, (b) along $\mathrm{\Gamma }\text{−}M$ direction in the cross section of $q_z=\frac{{\mathbf{c}}^{*}}{13}$, and (c) along $q_z$ direction, with $q_x=0.295{\mathbf{a}}^{*}$ and $q_y=0$ in the high-$T$ normal phase. The dotted line frame in (a) corresponds to the first Brillouin zone. ${\mathbf{a}}^{*}$ and ${\mathbf{c}}^{*}$ are the reciprocal lattice vectors along the $q_x$ and $q_z$ directions, respectively.

Figure 4

Real part of static electronic susceptibility of the monolayer $1T\text{−}{\mathrm{TaS}}_2$ with arbitrary units vs. wave vector along $\mathrm{\Gamma }\text{−}M$ direction with $q_y=0,q_z=0$. Inset shows its distribution in momentum space. The dotted line frame corresponds to the first Brillouin zone of the monolayer in the high-$T$ normal phase.

Figure 5

Electronic structures of the bulk $1T\text{−}{\mathrm{TaS}}_2$ in the CCDW phase: (a) band structure, (b) atomic orbital-resolved partial DOS for $5d_{3z^2-r^2}$ and the other $5d$ orbitals of 13 Ta atoms in a star-of-David cluster.

Figure 6

Schematic diagram of the ODW order in the Ta layer. Each yellow star represents a cluster of thirteen Ta atoms. The right panel (b) shows the character of one-dimensional metal along the $c$ direction.

Figure 7

Charge density distribution of Ta-atom cross section in the CCDW phase of bulk $1T\text{−}{\mathrm{TaS}}_2$. The top blue and right green curves correspond to the charge density in the blue and green boxes, respectively.

Figure 8

The evolution of energy bands from undistorted to distorted bulk structure by changing the values $(x,y)$, including (a) $(1,0)$, (b) $(0.75,0.25)$, (c) $(0.5,0.5)$, (d) $(0.25,0.75)$, and (e) $(0,1)$. $(0,1)$ corresponds to the CCDW state. Red lines show the formation process of the band which crosses the Fermi level in the CCDW state.

Figure 9

(a) Band structure and orbital-resolved DOS for (b) the central, (c) the first-ring, and (d) the second-ring Ta atoms in the interlayer AFM phase with electronic correlation $U=2$ eV.

Figure 10

The comparisons of bulk CCDW-state energy bands between our calculated results with electronic correlations of $U=$ (a) 0.7 eV, (b) 2 eV, and (c) 4 eV and the ARPES data by Ang et al. [11]. Red points are artificially added to highlight occupied states in ARPES, and the curves are the energy bands obtained from our first-principles calculations.

Figure 11

Local DOS for different values of $U$. From top to bottom, $U=0.2,0.4,0.5,0.6,0.65,0.7$ eV.

Figure 12

(a) Quasiparticle weight $Z$ as a function of the Coulomb interaction $U$. Within the region 0.64 eV $<U<0.66$ eV, the metallic and insulating solutions coexist. (b) Local squared moments of spin and charge as functions of $U$ at $\beta =400$ and $200{\mathrm{eV}}^{-1}$. The coexistence regions are 0.64 eV $<U<0.66$ eV and $U=0.63$ eV for $\beta =400$ and $200{\mathrm{eV}}^{-1}$, respectively.

Figure 13

(a) Local spin-spin correlation function for different values of $U$. (b) Local spin susceptibility ${\chi }_S$ as a function of the Coulomb interaction $U$. The hysteresis corresponds to a MIT between 0.64 and 0.66 eV. The inset shows local spin susceptibility ${\chi }_S$ as a function of the inverse temperature $\beta$ in the insulator with $U=0.7$ and 0.9 eV. A good linear behavior is obtained, ${\chi }_S\propto \beta$.

Figure 14

(a) Frequency dependence of optical conductivity with different interactions $U$. (b) Drude peaks from the “free” quasiparticles at the Fermi level. (c) DC conductivity as a function of $U$ for $\beta =400{\mathrm{eV}}^{-1}$. The unit of the conductivity is $\frac{\pi e^2{a}^2{t}^2}{\hslash Vd}$.

Figure 15

A comparison between the calculated optical conductivity with $U=0.7$ eV in arbitrary units ($T=29$ K, DMFT, red dots) and the experimental measurements by Gasparov et al. ($T=30$ K, black curves) [45].

Figure 16

Band structure of the monolayer $1T\text{−}{\mathrm{TaS}}_2$ in the CCDW phase without electronic correlation.

Figure 17

Charge density distribution of Ta-atom cross section in the CCDW phase of the monolayer $1T\text{−}{\mathrm{TaS}}_2$. The top blue and right green lines correspond to the charge density in the blue and green boxes, respectively.

Figure 18

Electronic structures for the distorted monolayer $1T\text{−}{\mathrm{TaS}}_2$ in the GGA $+U$ scheme with $U=2$ eV. (a) Band structure. Black dotted and red solid lines correspond to up and down spins, respectively. (b) Spin- and orbital-resolved DOS.