Detection of a first-order superconducting transition in Sr$2$RuO$4$ by ultra-high-resolution magnetostriction measurements

Sr$_2$RuO$_4$ has been intensively investigated as one of novel unconventional superconductors for the last two decades [1]. The gap symmetry of the superconducting (SC) state, however, has still been an open problem. Recently,
there have been several reports indicating a first-order SC transition occurring under a magnetic field precisely controlled parallel to the $ab$ plane: a sharp entropy change observed by the magnetocaloric effect [2] and a sharp magnetization jump [3], involving hysteresis at the upper critical field $H_{c2}$. Recent improvements of NMR measurements have revealed a significant change of the Knight shift below the $H_{c2}$ for the magnetic field along the $ab$ plane [4,5] in contrast to the previous study [6]. These notable features imply pair-breaking mechanism derived from a strong Pauli-paramagnetic effect.
Motivated by these previous studies, we performed ultra-high-resolution magnetostriction measurements on a single crystal sample of Sr$_2$RuO$_4$ along the $c$ axis with the magnetic field parallel to the $ab$ plane. We have succeeded in detecting a small first-order SC transition signal on the magnetostriction of the sample (~0.1 Å) with improving a home-made capacitively-detected diameter for a resolution far better than 0.01 Å. The magnetization and specific-heat measurements were also conducted on the same sample, the results of which are in agreement with the magnetostriction data. Although a weak anomaly was detected in the magnetostriction slightly below the first-order transition field, a clear evidence for the occurrence of a Fulde-Ferrell-Larkin-Ovchinnikov transition was not provided in the present measurements.

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