Fermi surface investigation of the noncentrosymmetric superconductor $\alpha$-PdBi

The noncentrosymmetric superconductor $\alpha$-PdBi is a candidate material for the realization of topological superconductivity. Here, we present a detailed de Haas–van Alphen (dHvA) study together with band-structure calculations within the framework of density functional theory. The rich dHvA spectra are a manifestation of the 13 bands that cross the Fermi energy $E_F$. We find excellent agreement between calculated and experimentally observed dHvA frequencies with moderately enhanced effective masses. One of the bands crossing $E_F$, the so-called $\alpha$ band, exhibits topological character with Weyl nodes lying 43 meV below $E_F$.

Figure 1

Field-dependent torque signal of $\alpha$-PdBi measured at 40 mK and the applied magnetic field $B\parallel c$. Inset: Zoom into the low-field region of the main panel.

Figure 2

dHvA frequency spectra obtained from the torque data shown in Fig. 1 for different field ranges. Note the different scaling of the frequency axes between the top panel and the two bottom panels.

Figure 3

Effective masses determined by fits of the Lifshitz-Kosevich formula (lines) to the temperature-dependent dHvA oscillation amplitudes (symbols).

Figure 4

Angular dependence of the experimentally observed dHvA frequencies (symbols) and of the calculated extremal Fermi-surface cross sections (lines) in $\alpha$-PdBi. The line colors correspond to the band and FS colors shown in Fig. 5. Open symbols denote frequencies which could not clearly be assigned to a calculated frequency branch. Since spin-split bands in $\alpha$-PdBi produce similar Fermi-surface sheets and, hence, similar dHvA frequencies, we assigned and color-coordinated the experimental data only to the pair of bands. $\alpha$ and $\delta$ label the frequency branches which were also observed by Khan et al. [26].

Figure 5

(a) Calculated band structure of $\alpha$-PdBi along lines of the Brillouin zone indicated in the sketch. Along the symmetry lines $E\text{−}{M}_1\text{−}A$ and $M\text{−}D\text{−}Z$, SOC is zero, leading to degeneracy of the bands 321/322, 323/324, etc. Note that band 333 crosses the Fermi energy only outside the lines shown here. A detailed investigation of the Fermi-energy crossing of bands 331 to 333 is shown in Fig. 6. (b) Corresponding density of states. The shaded areas mark contributions of different atomic orbitals. (c) Calculated FSs. The colors correspond to those of the conduction bands in panel (a).

Figure 6

Band structure of $\alpha$-PdBi in the vicinity of the Weyl node at ${\mathbf{k}}_{\mathbf{0}}=(k_{x0},k_{y0},k_{z0})=(0.399706,0.056504,0)2\pi /a$. The cuts are running parallel to the (a) $k_x$, (b) $k_y$, and (c) $k_z$ directions. The band colors are identical to those used in Fig. 5. Panels (d), (e), and (f) are zooms into panels (a), (b), and (c), respectively. In these zoomed images, we used a fine $k$ spacing of steps in the range $|\mathbf{k}-{\mathbf{k}}_{\mathbf{0}}|\le {10}^{-3}$ in order to show that the crossing is indeed nearly linear in all directions.