Topological phases in the ${\mathrm{TaSe}}_3$ compound

Based on first-principles calculations, we show that stoichiometric ${\mathrm{TaSe}}_3$, synthesized in space group $P{2}_1/m$, belongs to a three-dimensional strong topological insulator (TI) phase with $Z_2$ invariants ($1;100$). The calculated surface spectrum shows clearly a single Dirac cone on surfaces, with helical spin texture at a constant-energy contour. To check the stability of the topological phase, strain effects have been systematically investigated, showing that many topological phases survive in a wide range of the strains along both the $a$ and $c$ axes, such as strong TI, weak TI, and Dirac semimetal phases. ${\mathrm{TaSe}}_3$ provides us an ideal platform for experimental study of topological phase transitions. More interestingly, since superconductivity in ${\mathrm{TaSe}}_3$ has been reported for a long time, the coexistence of topological phases and a superconducting phase suggests that ${\mathrm{TaSe}}_3$ is a realistic system to study the interplay between topological and superconducting phases in the future.

Figure 1

Crystal structure and BZs of ${\mathrm{TaSe}}_3$. (a) Side views of the type-I and type-II chains, colored in blue and red, respectively. Two neighbor prisms differ by $b/2$ in the $b$ direction, making the Ta atoms in the planes of the ${\mathrm{Se}}_3$ triangles of the neighbor prism. (b) The projection view along the chain direction (i.e., the $b$ direction). The primitive unit cell is shown by the black parallelogram. The nonequivalent Ta atoms (large balls) and Se atoms (small balls) in the primitive unit cell are labeled by yellow and light blue numbers, respectively. The bonds (green dashed lines) of Se5-Ta1-Se3 and Se5-Ta2-Se6 make the prisms form a layer [spanned by the $b$ axis and $(a+c)$ axis]. (c) Bulk BZ, projected surface BZs, and high-symmetry points.

Figure 2

Electronic structures of ${\mathrm{TaSe}}_3$ (a) without and (b) with SOC. The calculated parity eigenvalues of the highest valence band (HVB) and the lowest conduction band (LCB) at the $B$ and $D$ points are given explicitly. The size of the red and blue circles in (a) represents the weights of Se3 $p_z$ and Ta1 $d_{z^2}$, respectively. The inset in (b) shows the zoomed-in band structure around the band crossing point located on the $A\text{−}B$ line. The red dashed line in (b) indicates the existence of the continuous direct gap. (c) The calculated total DOS and PDOS of ${\mathrm{TaSe}}_3$. The inset in (c) shows the DOS around $E_F$. (d) The Fermi surface of ${\mathrm{TaSe}}_3$.

Figure 3

The evolution of the crystal structures and the schematic diagram of the band-inversion mechanism. (a) The upper pattern is a layer of the ${\mathrm{ZrSe}}_3$ structure, which is a semiconductor and contains only a type-I chain with ${\mathrm{Zr}}^{4+}$, ${\mathrm{Se}}^{2-}$, and ${\left(\mathrm{Se}\right)}_2^{2-}$ states. The middle pattern is the intermediate structure generated by a shift in the $a$ direction (the black dashed line), which breaks the previous bonds of Se3 atoms, denoted by crosses. The lower pattern is a layer of the ${\mathrm{TaSe}}_3$ structure. The distortion happens in the chains in the box, changing the type-I chains to the type-II chains. The new bonds between Se6 and Ta2 are built. (b) Schematic diagram of the band evolution in ${\mathrm{TaSe}}_3$, starting from the atomic orbitals $d_{z^2}$ of Ta1 and Ta2 and $p_z$ of Se3, Se5, and Se6.

Figure 4

Surface states of ${\mathrm{TaSe}}_3$. (a) and (b) The surface band structures of ${\mathrm{TaSe}}_3$ on the (010) and $\left(10\overline{1}\right)$ surfaces, respectively. The chemical potentials at the Fermi level $E_F$ and 40 meV below the Fermi level are represented by ${\mu }_1$ and ${\mu }_2$, respectively. (c) The constant-energy contour ($E=E_F$) of the topological surface states and corresponding spin texture on the (010) surface of ${\mathrm{TaSe}}_3$. The inset shows the zoomed-in Fermi surface around the top left corner. (d) Zoomed-in surface band structures around the Dirac point on the $\left(10\overline{1}\right)$ plane.

Figure 5

Phase diagram of ${\mathrm{TaSe}}_3$ with strain along both the $a$ and $c$ directions. (a) The blue, green, and red regions represent the NI, STI, and WTI phases, respectively. (b)–(d) Band structures of ${\text{TaSe}}_3$ with SOC for $c=0.95c_0$ $\left({\mathrm{P}}_1\right)$, $c=0.97c_0$ $\left({\mathrm{P}}_2\right)$, and $c=1.05c_0$ $\left({\mathrm{P}}_3\right)$, respectively. The parities of HVB and LCB at three TRIMs ($B$, $D$, and $Z$) are given.

Figure 6

Phonon dispersion of ${\mathrm{TaSe}}_3$ with (a) $c=1.0c_0$, (b) $c=0.95c_0$ (P1), (c) $c=0.97c_0$ (P2), and (d) $c=1.05c_0$ (P3).