# Investigation of Superconducting Gap Structure in HfIrSi using muon spin relaxation/rotation

## TL;DR

This study investigates the superconducting gap structure of HfIrSi, revealing an isotropic s-wave gap, preserved time-reversal symmetry, and strong spin-orbit coupling effects, using muon spin relaxation/rotation and other measurements.

## Contribution

It provides the first detailed muon spin relaxation/rotation analysis of HfIrSi, demonstrating its isotropic s-wave superconductivity and the role of spin-orbit coupling in its properties.

## Key findings

- Superconducting transition at 3.6 K confirmed by specific heat and magnetization.
- Superfluid density fits an isotropic BCS s-wave gap model.
- No evidence of spontaneous internal magnetic fields below Tc, indicating preserved time-reversal symmetry.

## Abstract

Appearance of strong spin-orbit coupling (SOC) is apparent in ternary equiatomic compounds with 5$d$-electrons due to the large atomic radii of transition metals. SOC plays a significant role in the emergence of unconventional superconductivity. Here we examined the superconducting state of HfIrSi using magnetization, specific heat, zero and transverse-field (ZF/TF) muon spin relaxation/rotation ($\mu$SR) measurements. Superconductivity is observed at $T_\mathrm{C}$ = 3.6 K as revealed by specific heat and magnetization measurements. From the TF$-\mu$SR analysis it is clear that superfluid density well described by an isotropic BCS type $s$-wave gap structure. Furthermore, from TF$-\mu$SR data we have also estimated the superconducting carrier density $n_\mathrm{s}$ = 6.6 $\times$10$^{26}$m$^{-3}$, London penetration depth $\lambda_{L}(0)$ = 259.59 nm and effective mass $m^{*}$ = 1.57 $m_{e}$. Our zero-field muon spin relaxation data indicate no clear sign of spontaneous internal field below $T_\mathrm{C}$, which implies that the time-reversal symmetry is preserved in HfIrSi. Theoretical investigation suggests Hf and Ir atoms hybridize strongly along the $c$-axis of the lattice, which is responsible for the strong three-dimensionality of this system which screens the Coulomb interaction. As a result despite the presence of correlated $d$-electrons in this system, the correlation effect is weakened, promoting electron-phonon coupling to gain importance.