Drumhead surface states and topological nodal-line fermions in ${\mathrm{TlTaSe}}_2$

A topological nodal-line semimetal is a state of matter with one-dimensional bulk nodal lines and two-dimensional so-called drumhead surface bands. Based on first-principles calculations and an effective $\mathit{k}·\mathit{p}$ model, we theoretically propose the existence of topological nodal-line fermions in the ternary transition-metal chalcogenide ${\mathrm{TlTaSe}}_2$. The noncentrosymmetric structure and strong spin-orbit coupling give rise to spinful nodal-line bulk states which are protected by a mirror reflection symmetry of this compound. This is remarkably distinguished from other proposed nodal-line semimetals such as ${\mathrm{Cu}}_3\mathrm{NPb}$(Zn) in which the nodal line exists only in the limit of vanishing spin-orbit coupling and thus is not as robust. In addition, we show that the drumhead surface states in ${\mathrm{TlTaSe}}_2$, which are associated with the topological nodal lines, exhibit an unconventional chiral spin texture and an exotic Lifshitz transition as a consequence of the linkage among multiple drumhead surface-state pockets.

Figure 1

(a) Lattice structure of ${\mathrm{TlTaSe}}_2$: top view (left) and side view (right). (b) Bulk and (001)-projected surface Brillouin zones. (c) Calculated bulk band structure of ${\mathrm{TlTaSe}}_2$ without (left) and with (right) the inclusion of spin-orbit coupling. The color shows the atomic orbital decomposition. (d) Closeup of band structure around H point as marked in panel (c).

Figure 2

(a) Orbital composition, space-group representation, and spin polarization of nodal-line (NL) bands in ${\mathrm{TlTaSe}}_2$. The mirror parity of each band is given in parentheses. (b) Isoenergy bulk-band contours at $E=-0.18$ eV of ${\mathrm{TlTaSe}}_2$ bulk bands. (c) Band structure of the effective $\mathit{k}·\mathit{p}$ model that approximates the low-energy bands around the H point for parameters $m_1=0.7,m_2=0.8,\mu =2.2,{\mathrm{\Delta }}_{\mathrm{SOC}}=2.5$. See Ref. [38] for details. (d) Variation of the Berry phase along high-symmetry lines of the surface Brillouin zone of ${\mathrm{TlTaSe}}_2$, taking into account the two bands that constitute the nodal ring. $\overline{\text{K}}$ is inside the nodal ring whereas $\overline{\mathrm{\Gamma }}$ and $\overline{\text{M}}$ outside the ring. (e) Calculated bulk band structure of ${\mathrm{TlTaSe}}_2$ with the Ta atom shifted slightly away from the equilibrium position in the unit cell. The shift breaks the mirror symmetry of the system and reduces the symmetry of the nodal-line states.

Figure 3

(a) The nodal-line band structure of ${\mathrm{TlTaSe}}_2$ with various SOC (in the scale relative to the SOC of the material). (b) The energy difference between the two bands b2 and b3 as a function of SOC, showing nodal-line phases at different SOC.

Figure 4

(a) (001)-projected bulk bands of ${\mathrm{TlTaSe}}_2$. (b) Bulk and surface band structure of Se-terminated surface of ${\mathrm{TlTaSe}}_2$. The surface bands are marked by the arrows. (c) Same as panel (b) but for Tl-terminated surface of ${\mathrm{TlTaSe}}_2$. (d) Isoenergy band contour at $E=-0.25$ eV of Se-terminated surface. The drumhead surface states overlap with the bulk nodal rings. (e) The spin polarization of surface bands of Tl-terminated surface. (f) Isoenergy band contour at $E=-0.25$ eV (top), $-0.22$ eV (middle), and $-0.18$ eV (bottom) of Tl-terminated surface. The arrows indicate the spin texture of the surface bands.