Topological phase transition under pressure in the topological nodal-line superconductor ${\mathrm{PbTaSe}}_2$

A first-order-like resistivity hysteresis is induced by a subtle structural transition under hydrostatic pressure in the topological nodal-line superconductor ${\mathrm{PbTaSe}}_2$. This structure transition is quickly suppressed to zero at pressure $\sim 0.25$ GPa. As a result, superconductivity shows a marked suppression, accompanied with pronounced changes in the magnetoresistance and Hall resistivity. The first-principles calculations show that the spin-orbit interactions partially gap out the Dirac nodal line around $K$ point in the bulk Brillouin zone upon applying a small pressure, whereas the Dirac states around $H$ point are completely destroyed. The calculations further reveal a second structural phase transition under a pressure as high as $\sim 30$ GPa, through which a transition from a topologically nontrivial bulk phase to a trivial phase is uncovered, with a superconducting dome emerging under this high-pressure phase. Our calculations also reveal how the bulk Fermi surfaces and the surface bands evolve with pressure. This theoretical study shall inspire in-depth experimental investigations on the electronic structure of this novel topological superconductor under higher pressures.

Figure 1

(a) Temperature dependence of resistivity under different pressures. (b) An enlarged view of the resistivity curves at low temperatures.

Figure 2

(a) Magnetic field dependence of the magnetoresistance ($▵\rho /{\rho }_0$, $T=5$ K) under different pressures. In our measurements, we changed the polarity of the field and the even signal in field was defined as MR. The odd signal in field was determined as Hall resistivity ${\rho }_{xy}$ as shown in (b).

Figure 3

Phase diagram for the pressure dependence of the structural transition temperature $T_s$, the superconducting transition temperature $T_c$, and the resistivity at 5 K (${\rho }_{\text{5K}}$).

Figure 4

Calculated enthalpies per atom as a function of pressure from 0–60 GPa with respect to $P\overline{6}m2$ structure. Inset shows the enlarged enthalpy comparison between $P\overline{6}m2$ and $P\overline{6}m2\left(1e\right)$ from 0–10 GPa.

Figure 5

The schematic unit cell for (a) noncentrosymmetric hexagonal $P\overline{6}m2$, (b) $P\overline{6}m2\left(1e\right)$, and (c) tetragonal $P4/mmm$ phases. The band structures of ${\mathrm{PbTaSe}}_2$ calculated (d)–(f) without and (g)–(i) with SOC for three different structures. The weights of the Pb $p$ and Ta $d$ orbital contribution are color-coded by green and red, respectively. The rectangles in (g) indicate the DNL structure near $K$ and $H$ points. In (h), the DNL structure is gapped by SOC on the left-hand side of $K$ but still intact on the right-hand side, indicated by the circles.

Figure 6

Top: Calculated Fermi surfaces (FSs) of (a) $P\overline{6}m2$, (b) $P\overline{6}m2\left(1e\right)$, and (c) $P4/mmm$ phases. The blue boundary lines denote the first Brillouin zones (BZs). In $P\overline{6}m2$ and $P\overline{6}m2\left(1e\right)$ phases, there are two bands crossing the Fermi level, which construct the Fermi surfaces, while three bands cross the Fermi level in the high-pressure $P4/mmm$ phase. It is clear to see the Fermi surface nesting at the corners in the hexagonal phases, and the shape of these FSs changes from a cylinder to a dumbbell accompanied with the phase transition from $P\overline{6}m2$ to $P\overline{6}m2\left(1e\right)$. The FS pockets at the corner become more three dimensional from $P\overline{6}m2$ to $P\overline{6}m2\left(1e\right)$. Bottom: Surface state spectrum of (001) surfaces along $\overline{\mathrm{\Gamma }}\text{−}\overline{K}\text{−}\overline{M}$ for (d) $P\overline{6}m2$, (e) $P\overline{6}m2\left(1e\right)$ phases, and bands along $\overline{\mathrm{\Gamma }}\text{−}\overline{M}\text{−}\overline{X}$ for (f) $P4/mmm$ phase. The black arrows indicate the positions of Dirac nodes along the $k$ path of the first BZ.

Figure 7

(a) Electron-phonon coupling constant $\lambda$, and (b) superconducting $T_c^{\mathrm{Cal}}$ as a function of pressure. Inset shows the blowup of the superconducting dome in $P4/mmm$ phase.