Nontrivial Berry phase and type-II Dirac transport in the layered material $\mathrm{PdT}{\mathrm{e}}_2$

We report on a systematic study of type-II Dirac fermions in a layered crystal of $\mathrm{PdT}{\mathrm{e}}_2$. De Haas–van Alphen oscillations show a small Fermi-surface pocket with a cross section of $0.077\mathrm{n}{\mathrm{m}}^{-2}$ with a nontrivial Berry phase. First-principles calculation reveals that the nontrivial Berry phase originates from a hole pocket formed by the tilted Dirac cone. In addition, the band dispersion measured with angle-resolved photoemission spectroscopy is found to be consistent with that of a type-II Dirac cone dispersion. We propose that $\mathrm{PdT}{\mathrm{e}}_2$ is an improved platform to host topological superconductors.

Figure 1

$\mathrm{PdT}{\mathrm{e}}_2$ crystal growth and characterization. (a) The $\mathrm{Cd}{\mathrm{I}}_2$-type crystal structure of $\mathrm{PdT}{\mathrm{e}}_2$ and the corresponding Brillouin zone (lower right panel) (b) The x-ray-diffraction data of the $\mathrm{PdT}{\mathrm{e}}_2$ sample. Strong ($00n$) peaks can be seen. The inset is the optical micrograph of several flakes. (c) The EDS spectrum indicates the stoichiometric ratio of the sample. The insets show the EDS mapping of Pd and Te, respectively, of a typical flake. (d) Raman spectrum of the $\mathrm{PdT}{\mathrm{e}}_2$ crystal.

Figure 2

Electrical transport under magnetic fields and low temperatures. (a) The magnetoresistance of $\mathrm{PdT}{\mathrm{e}}_2$ under 1.7 K. The inset shows SdH oscillation after background subtraction. (b) Hall resistance versus magnetic field under various temperatures. (c) and (d) are temperature dependence of resistance under different fields of configurations 1 ($I\perp c$, $B\parallel c$) and 2 ($I\parallel c$, $B\parallel c$), respectively, indicating anisotropic superconductivity in $\mathrm{PdT}{\mathrm{e}}_2$.

Figure 3

The dHvA oscillations and nontrivial Berry phase. (a) The magnetization curve ($B\parallel c$) for $\mathrm{PdT}{\mathrm{e}}_2$ at 1.8 K. Beautiful dHvA oscillations can be seen. (b) The magnetization strength versus $1/B$ under various temperatures after background subtraction. (c) The fits of effective mass for all six oscillation modes ($\alpha$ to $\zeta$). The inset is the FFT amplitudes versus temperature. The peaks corresponding to each oscillation mode are marked with greek letters. (d) The Landau fan diagram for the dHvA oscillation of low-frequency mode $\alpha$. The inset is the fit of Dingle temperature $\left(T_D\right)$. (e) Oscillatory component of other frequencies ($\beta$ to $\zeta$) and the multiband LK fit of it. The inset shows a part of the curve, indicating the perfect fit of the experimental data.

Figure 4

Matching the dHvA components in the calculations of the type-II Dirac cone. (a) The calculated electronic structures plotted along the directions shown in the Brillouin zone [Fig. 1]. The Dirac point and the small hole pocket with the $\alpha$ mode are marked in the panel. (b) and (c) show the contour of the Fermi surface along the $k_x\text{−}{k}_y$ and $k_x\text{−}{k}_z$ planes, respectively. Pockets irrelevant to the hole and electron pockets formed by the tilted Dirac cone in (c) are concealed for clearness. (d) and (e) show the projection of the Dirac cone along $k_x\text{−}{k}_y$ and $k_x\text{−}{k}_z$, respectively. The Dirac cone with tilted properties along the $k_z$ direction can clearly be seen.

Figure 5

ARPES measurement of $\mathrm{PdT}{\mathrm{e}}_2$. (a) The dispersion along the $T\text{−}D$ direction measured by ARPES. The linear dispersion of the Dirac cone is labeled by the red dashed lines. The Dirac point also is marked by the red arrow. (b)–(d) are dispersions along the $k_z$ directions. The black dashed lines are band calculations for comparison.

Figure 6

Optimizing the TSC by type-II Dirac semimetals. This figure displays the band-structure diagrams of the possible topological superconductors based on the mother materials of the type-II Dirac semimetal when $T<T_c$. The red solid curves in the SC gap (gray zone) represent the Majorana chiral mode. The Fermi arc states are killed in the Dirac semimetals when the superconductivity occurs.