Quantum oscillations of the $j=3/2$ Fermi surface in the topological semimetal YPtBi

The bismuth-based half-Heusler materials host a nontrivial topological band structure, unconventional superconductivity, and large spin-orbit coupling in a system with very low electron density. In particular, the inversion of $p$-orbital-derived bands with an effective angular momentum $j$ of up to 3/2 is thought to play a central role in anomalous Cooper pairing in the cubic half-Heusler semimetal YPtBi, which is thought to be the first “high-spin” superconductor. Here, we report an extensive study of the angular dependence of quantum oscillations (QOs) in the electrical conductivity of YPtBi, revealing an anomalous Shubnikov–de Haas effect consistent with the presence of a coherent $j=3/2$ Fermi surface. The QO signal in YPtBi manifests an extreme anisotropy upon rotation of the magnetic field from the [100] to [110] crystallographic direction, where the QO amplitude vanishes. This radical anisotropy for such a highly isotropic system cannot be explained by trivial scenarios involving changes in effective mass or impurity scattering, but rather is naturally explained by the warping feature of the $j=3/2$ Fermi surface of YPtBi, providing direct proof of active high angular momentum quasiparticles in the half-Heusler compounds.

Figure 1

Angle-dependent Shubnikov–de Haas quantum oscillations at $T=2$ K in YPtBi. The oscillatory components $\mathrm{\Delta }R\left(T\right)$ are presented with various field orientations (a) from [001] to [100] and (b) from [100] to [010]. Corresponding contour plots of $\mathrm{\Delta }R$ are shown in (c) and (d), respectively, where the schematic of field rotation is shown.

Figure 2

Angle-dependent Shubnikov–de Haas quantum oscillations at $T=2$ K in YPtBi with magnetic fields rotating around [111]. (a) A contour plot of $\mathrm{\Delta }R$ with the schematic of field configuration. (b) A polar plot of $\mathrm{\Delta }R$ for magnetic fields of 12.3 and 13.4 T [depicted in (a) with dashed lines]. Both plots clearly show a sixfold symmetry confirming the vanishing quantum oscillation amplitude in the crystallographic [110] direction of YPtBi.

Figure 3

Angle-dependent cyclotron mass $m^{*}$ and impurity scattering time $\tau$ in YPtBi. (a) Temperature dependence of the normalized QO amplitude. Symbols represent experimental values, and the solid lines represent the best theoretical fit with the LK formula $A_T(T,H)$ [Eq. (1)]. (b) Angle-dependent $m^{*}$ obtained from the experimental results (solid symbols) by using Eq. (1) and the theoretical investigation (open symbols) within the the $\mathbf{k}·\mathbf{p}$ Hamiltonian model. (c) Field-dependent QO amplitude at $T=2$ K. The symbols represent experimental values between $\theta =0^{\circ }$ and ${30}^{\circ }$ and the solid straight lines represent a linear fit $(-a/{\mu }_{0}H+b)$ to the LK formula $A_D\left(H\right)$ [Eq. (2)]. (d) Angle-dependent $T_D$ determined from the linear fit in (c).

Figure 4

Angular variation of the prefactor ${\left[{\partial }_{k_{\parallel }}^{2}S\left({\tilde{k}}_{\parallel }\right)\right]}^{-1/2}$ of the QO amplitude. (a) A schematic cross section of the $j=3/2$ Fermi surface, where the warping sensitively depends on $D$ [see Eq. (3)]. The red lines and gray band represent the cyclotron orbits and the enhanced density of states on the Fermi surface perpendicular to the applied field $H$, when the quantum oscillation amplitude is maximum. (b) The angle-dependent QO amplitude calculated from fast Fourier transform. Note the maximum amplitude around ${15}^{\circ }$. (c) The angular variation of the outer Fermi surface's ${\left[{\partial }_{k_{\parallel }}^{2}S\left({\tilde{k}}_{\parallel }\right)\right]}^{-1/2}$ with different choices of $D$. The gray vertical bars represent the maximum amplitude of experimental QO shown in (b). (d) Angular variation of ${\left[{\partial }_{k_{\parallel }}^{2}S\left({\tilde{k}}_{\parallel }\right)\right]}^{-1/2}$ of $j=1/2$ and $j=3/2$ outer Fermi surfaces. The position of the maximum and overall tendency of $j=3/2$ theory show reasonable agreement with the experimental QO amplitude results.