Quantum spin Hall effect in ${\mathrm{Ta}}_2{M}_3{\mathrm{Te}}_5$ $(M=\mathrm{Pd},\mathrm{Ni})$

Quantum spin Hall (QSH) effect with great promise for the potential application in spintronics and quantum computing has attracted extensive research interest from both theoretical and experimental researchers. Here, we predict monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ can be a QSH insulator based on first-principles calculations. The interlayer binding energy in the layered van der Waals compound ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ is $19.6\mathrm{meV}/{\AA }^2$; thus, its monolayer/thin-film structures could be readily obtained by exfoliation. The band inversion near the Fermi level ($E_F$) is an intrinsic characteristic, which happens between $\mathrm{Ta}\text{−}5d$ and $\mathrm{Pd}\text{−}4d$ orbitals without spin-orbit coupling (SOC). The SOC effect opens a global gap and makes the system a QSH insulator. With the $d\text{−}d$ band-inverted feature, the nontrivial topology in monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ is characterized by the time-reversal topological invariant ${\mathbb{Z}}_2=1$, which is computed by the one-dimensional (1D) Wilson loop method as implemented in our first-principles calculations. The helical edge modes are also obtained using surface Green's function method. Our calculations show that the QSH state in ${\mathrm{Ta}}_2{M}_3{\mathrm{Te}}_5$ ($M=\mathrm{Pd}$, Ni) can be tuned by external strain. These monolayers and thin films provide feasible platforms for realizing QSH effect as well as related devices.

Figure 1

(a) Side view of ${\mathrm{Ta}}_2{X}_5$ ($X=\mathrm{Se}$, Te) layer. Magenta and green balls represent Ta and $X$ atoms, respectively. The positions of A site and B site in ${\mathrm{Ta}}_2{X}_5$ layer are marked by arrows. (b) Side view of ${\mathrm{Ta}}_2{\mathrm{NiSe}}_5$ layer. The A sites are filled by Ni atoms (cyan balls). (c) Side view of ${\mathrm{Ta}}_2{M}_3{\mathrm{Te}}_5$ ($M=\mathrm{Ni}$, Pd) layer. (d) Top view of monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$. It has three symmetry operations: ${\widehat{g}}_x\equiv \left\{M_x\right|0,1/2,1/2\}$, ${\widehat{M}}_y$, and ${\widehat{C}}_{2z}\equiv \left\{C_{2z}\right|0,1/2,1/2\}$. (e) The interlayer binding energies of ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ and other materials such as ${\mathrm{ZrTe}}_5$, graphene, ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, and Bi. (f) Phonon dispersion of monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$. (g) The orbital-resolved PBE band structure of monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$. The characters of $\mathrm{Ta}\text{−}5d$ and $\mathrm{Pd}\text{−}4d$ orbitals are marked by green and red dots, respectively. The irreps for four low-energy bands on the $\mathrm{\Lambda }$ line are denoted. (h) The $\mathrm{PBE}+\mathrm{SOC}$ band structure of monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$. A band gap of 4.6 meV is opened due to the SOC effect. (i) The calculated total DOS and partial DOS of monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$. The black, red, green, and blue lines represent the total DOS and partial DOS of $\mathrm{Ta}\text{−}5d$, $\mathrm{Pd}\text{−}4d$, and $\mathrm{Te}\text{−}5p$ orbitals, respectively. (j) The evolution of WCCs for monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$.

Figure 2

(a) COHP diagram of monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$. (b) The $k$-dependent COHP of Pd-Ta bonds. (c) The $k$-dependent COHP of ${\mathrm{Pd}}_A\text{−}{\mathrm{Te}}_V$ bonds. The inset shows the positions of ${\mathrm{Pd}}_A$ and ${\mathrm{Te}}_V$ atoms. ${\mathrm{Pd}}_A$ and ${\mathrm{Te}}_V$ represent the Pd atoms on the tetrahedral void A sites and the Te atoms on the vertexes of Te square pyramids, respectively.

Figure 3

(a)–(c) The band structures of monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ with uniaxial strain along the $z$ direction: $\eta c_0$ with $\eta =0.98,1,1.02$, respectively. Since there is only one doubly-valued irrep at $\mathrm{\Gamma }$ (the double point group $C_{2v}$), we prefer to label the bands of these band structure plots by the irreps without SOC. (d)–(f) The band structures of monolayer ${\mathrm{Ta}}_2{\mathrm{Ni}}_3{\mathrm{Te}}_5$ with uniaxial strain along the $y$ direction: $\eta b_0$ with $\eta =0.98,1,1.02$, respectively.

Figure 4

The edge states of monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ under 2% tensile strain. (a) The edge states along the $y$ direction. (b) The zoom-in plot of the dashed box in (a). (c) The edge states along the $z$ direction.

Figure 5

The XRD pattern of a flat surface of ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ single crystal. The inset shows a photograph of a typical ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ single crystal.

Figure 6

(a) The crystalline structure of ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ obtained by our single-crystal XRD study. (b) The schematic diagram of 2D BZ of monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ and high symmetry points. (c) The schematic diagram of 3D BZ of bulk material ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ and high symmetry points.

Figure 7

The $\mathrm{PBE}+\mathrm{SOC}$ band structures of bulk ${\mathrm{Ta}}_2{M}_3{\mathrm{Te}}_5$. (a), (c) The $\mathrm{PBE}+\mathrm{SOC}$ band structures of bulk ${\mathrm{Ta}}_2{\mathrm{Ni}}_3{\mathrm{Te}}_5$ and zoom-in plot of the shadowed area. (b), (d) The $\mathrm{PBE}+\mathrm{SOC}$ band structures of bulk ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ and zoom-in plot.

Figure 8

The evolution of WCCs for bulk (a) ${\mathrm{Ta}}_2{\mathrm{Ni}}_3{\mathrm{Te}}_5$ and (b) ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ at $k_y=0$ plane. The blue and red circles mark the $+i$ and $-i$ subspaces of the ${\widehat{M}}_y$ eigenvalues, respectively.

Figure 9

The PBE band structures of monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ with ${\mathrm{Pd}}_A$ atoms moving along the $x$ direction (perpendicular to ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ monolayer). The numbers of the left upper corner mark the distances of ${\mathrm{Pd}}_A$ atoms moving away from the ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ monolayer.

Figure 10

The PBE band structures for the slabs with different layers of ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$. As the thickness of the slab increases, the strength of the band inversion is enhanced.

Figure 11

The $\mathrm{PBE}+\mathrm{SOC}$ band structures of monolayer ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ under uniaxial strain along (a), (b) the $y$ direction and (c), (d) the $z$ direction, respectively.

Figure 12

The $\mathrm{PBE}+\mathrm{SOC}$ band structures of monolayer ${\mathrm{Ta}}_2{\mathrm{Ni}}_3{\mathrm{Te}}_5$ under uniaxial strain along (a), (b) the $y$ direction and (c), (d) the $z$ direction, respectively.