Tuning the Parity Mixing of Singlet-Septet Pairing in a Half-Heusler Superconductor

In superconductors, electrons with spin $s=1/2$ form Cooper pairs whose spin structure is usually singlet ($S=0$) or triplet ($S=1$). When the electronic structure near the Fermi level is characterized by fermions with angular momentum $j=3/2$ due to strong spin-orbit interactions, novel pairing states such as even-parity quintet ($J=2$) and odd-parity septet ($J=3$) states are allowed. Prime candidates for such exotic states are half-Heusler superconductors, which exhibit unconventional superconducting properties, but their pairing nature remains unsettled. Here, we show that the superconductivity in the noncentrosymmetric half-Heusler LuPdBi can be consistently described by the admixture of isotropic even-parity singlet and anisotropic odd-parity septet pairing, whose ratio can be tuned by electron irradiation. From magnetotransport and penetration depth measurements, we find that carrier concentrations and impurity scattering both increase with irradiation, resulting in a nonmonotonic change of the superconducting gap structure. Our findings shed new light on our fundamental understanding of unconventional superconducting states in topological materials.

Popular Summary

In a superconductor, electrons bind together in pairs known as Cooper pairs, and knowing how the spins of these electrons align is important to understanding the nature of unconventional superconductivity. In most superconductors, the spins arrange themselves so that the pair has zero angular momentum (a spin singlet); a few superconductors exhibit a spin triplet, in which the angular momentum is 1. Here, we experimentally demonstrate an unprecedented high-momentum pairing that may provide opportunities for so-called topological superconducting states, which may host Majorana quasiparticles sought after for robust quantum computing.

In our experiments, we study a half-Heusler superconductor (LuPdBi), a crystalline material already known to exhibit unusual superconducting properties. The peculiar electronic structure of these materials can be described by fermions with a spin of $3/2$, as opposed to the spin $1/2$ of electrons. We find unusual nodes in the superconducting gap—the energy range in which quasiparticle excitations are prohibited—that can be explained by a mixture of singlet pairings (as seen in other superconductors) and a novel septet pairing, where the spin-$3/2$ fermions pair up for a total angular momentum of 3. Furthermore, we can tune the exact ratio of these two pairing states via electron irradiation.

Our results show that the half-Heusler system offers a controllable platform for this new class of unconventional superconducting pairing with high angular momentum.

Figure 1

Crystal and electronic structures of LuPdBi. (a) Crystal structure of the half-Heusler superconductor LuPdBi. (b) Ab initio calculations of the band structure near the Fermi energy $E_{\mathrm{F}}$. The green and red lines are ${\mathrm{\Gamma }}_8$ and ${\mathrm{\Gamma }}_6$ bands, respectively. (c) Enlarged view of the calculated band structure near the $\mathrm{\Gamma }$ point [dashed square in (b)]. The horizontal purple, light-blue, green, red, and brown dashed lines represent chemical potentials obtained from the carrier densities in the pristine, 3.13, 5.28, 8.44, and $11.61\mathrm{C}/{\mathrm{cm}}^2$ irradiated samples, respectively. The red arrow illustrates the shift of the chemical potential caused by electron irradiation. (d) Fermi surfaces calculated from the $\mathbf{k}·\mathbf{p}$ Hamiltonian [Eq. (1)] with chemical potential $u=0.1\mathrm{eV}$. Orange and blue surfaces are the outer and inner spin-split FSs, respectively. Black dots represent the points where the split size of two FSs becomes zero.

Figure 2

Physical properties of pristine and electron-irradiated LuPdBi. (a) Upper and lower panels depict the carrier density $n_{\mathrm{H}}$ (blue diamonds) and normalized mobility $\mu (\varphi )/\mu (0)$ (green and red diamonds) extracted from magnetotransport measurements at $T=10\mathrm{K}$ as a function of the irradiation dose $\varphi$, respectively. The green and red diamonds represent the $\mu (\varphi )/\mu (0)$ values calculated from the magnetoconductance ${\sigma }_{xx}(H)$ and resistivity $\rho$, respectively (see Appendix pp4). (b) Normalized change of penetration depth as a function of temperature $\mathrm{\Delta }\lambda (T)$ in the pristine (purple), 3.13 (light-blue), 5.28 (green), 8.44 (red), and $11.61\mathrm{C}/{\mathrm{cm}}^2$ (brown) irradiated samples. The inset shows $T_{\mathrm{c}}$ [defined at the onset of $\mathrm{\Delta }\lambda (T)$] as a function of the irradiation dose. (c) Temperature dependence of resistivity at low temperatures in the pristine and irradiated samples. (d) Reduced upper critical field as a function of $T/T_{\mathrm{c}}$ in the pristine (purple and blue), 2.15 (light-blue), 4.80 (green), and $11.61\mathrm{C}/{\mathrm{cm}}^2$ (brown) irradiated samples. The black dashed line represents the standard temperature dependence of $H_{\mathrm{c}2}$ in the WHH model.

Figure 3

Evolution of penetration depth and superconducting gap structure with electron irradiation. (a)–(e) $\mathrm{\Delta }\lambda (T)$ normalized by the value at $0.2T_{\mathrm{c}}$ or $0.35T_{\mathrm{c}}$ for (a) pristine, (b) 3.13, (c) 5.28, (d) 8.44, and (e) $11.61\mathrm{C}/{\mathrm{cm}}^2$ irradiated samples. Open circles are experimental data, and black solid and dashed lines in panels (a), (d), and (e) are fitting curves in the fully gapped and power-law models, respectively. Black solid and broken lines in panels (b) and (c) are the fully gapped and power-law components of the two-gap model described by Eq. (2), respectively. (f)–(j) Momentum direction dependence of the normalized superconducting gap in the pristine and irradiated samples derived from the effective $\mathbf{k}·\mathbf{p}$ model in Eq. (1) and ${\mathrm{\Delta }}_{p,\max }/{\mathrm{\Delta }}_s$ values calculated from the simple model described by Eq. (3). Red and blue lines correspond to the gap size on the outer and inner FSs, respectively. Black circles represent the positions of nodes. (k)–(o) Corresponding three-dimensional superconducting gap structure in the pristine and irradiated samples. White lines in panels (l) and (m) represent the nodal lines.

Figure 4

Doping dependence of the penetration depth exponent and the septet gap component. (a) Exponent $n$ in the power-law $T^n$ dependence of $\mathrm{\Delta }\lambda (T)$ as a function of the effective Fermi wave vector $k_{\mathrm{av}}=(3{\pi }^2{n}_{\mathrm{H}}{)}^{1/3}$ for the pristine (purple), 3.13 (light-blue), 5.28 (green), 8.44 (red), and 11.61 (brown) $\mathrm{C}/{\mathrm{cm}}^2$ irradiated samples. The exponent $n\le 2$ (shaded region) indicates a nodal gap structure, while $n>2$ implies a fully gapped state. (b) Ratio of the $p$-wave septet component $|{\mathrm{\Delta }}_{p,\max }|$ to the $s$-wave component $|{\mathrm{\Delta }}_s|$ as a function of $k_{\mathrm{av}}$. The black dashed line represents the relation ${\mathrm{\Delta }}_{p,\max }\propto k_{\mathrm{av}}$. The lengths of the colored vertical arrows are proportional to $1/\mu$ values [see Eq. (3)] in each sample. In the shaded region, a nodal gap structure is expected.

Figure 5

Ab initio calculations and effective $\mathbf{k}·\mathbf{p}$ model at low energies. First-principle results of energy dispersions (black crosses) along the (a) $\mathrm{\Gamma }\text{−}\mathrm{X}$, (b) $\mathrm{\Gamma }\text{−}\mathrm{K}$, and (c) $\mathrm{\Gamma }\text{−}\mathrm{L}$ directions calculated for LuPdBi with a modified Becke-Johnson (mBJ) potential are fitted with the effective $\mathbf{k}·\mathbf{p}$ model (red lines).

Figure 6

Distributions of Lu, Pd, and Bi elements. (a) Scanning electron microscopy (SEM) image of the crystal that is irradiated to 3.13 and then to the $5.28\mathrm{C}/{\mathrm{cm}}^2$ dose, taken after the penetration depth measurements. (b) Overlay of EDX analysis of Lu (green), Pd (red), and Bi (yellow) distributions on the SEM image. (c)–(e) Distribution of each (c) Lu, (d) Pd, and (e) Bi element.

Figure 7

Temperature dependence of resistivity in LuPdBi before and after irradiation. Zero-field resistivity is shown as a function of temperature in a wide temperature range for pristine (purple), 3.13 (light-blue), 5.28 (green), 8.44 (red), and $11.61\mathrm{C}/{\mathrm{cm}}^2$ (brown) irradiated samples.

Figure 8

Magnetotransport in LuPdBi before and after irradiation. (a,b) Hall resistivity as a function of magnetic field measured at 10 K in pristine and electron-irradiated samples up to 3 T (a) and 9 T (b). (c) Field dependence of the longitudinal magnetoconductivity normalized by the value at zero field measured at 10 K. The black dashed lines are fitting curves represented as ${\sigma }_{xx}(H)\propto 1/[1+{\mu }^2({\mu }_{0}H{)}^2]$. (d) Mobility calculated from the ${\sigma }_{xx}(H)$ (green diamonds) and the resistivity $\rho$ (red diamonds) as a function of the irradiation dose $\varphi$ (upper panel). Normalized mobility by the value in the pristine sample is also shown (lower panel).

Figure 9

Temperature dependence of upper critical field determined from resistive transitions in LuPdBi before and after irradiation. (a)–(e) Temperature dependence of the resistivity measured under various applied magnetic fields, normalized by the zero-field value at 2.5 K in (a) pristine No. 1, (b) pristine No. 2, (c) 2.15, (d) 4.60, and (e) $11.61\mathrm{C}/{\mathrm{cm}}^2$ samples. The data are plotted for different fields with 0.2-T steps. (f) Temperature dependence of the upper critical field $H_{\mathrm{c}2}$ in pristine and irradiated samples. The Pauli limits estimated in BCS theory, ${\mu }_0{H}_{\mathrm{P}}(\mathrm{Tesla})=1.85T_{\mathrm{c}}(\mathrm{Kelvin})$, of all measured samples are within the shaded region.