High-resolution magnetostriction measurements of the Pauli-limited superconductor Sr2RuO4

TL;DR

This work investigates whether the Pauli-limited superconductor Sr$_2$RuO$_4$ hosts an FFLO state by performing high-resolution, field-angle-resolved magnetostriction and thermal-expansion measurements on high-quality single crystals. Using a capacitively detected dilatometer with vector-magnet control, the authors identify a clear first-order superconducting transition with hysteresis under in-plane fields and an exceptionally small lattice response ($rac{ riangle L}{L} obreak \simreak 10^{-8}$). They report a hump-like anomaly in the magnetostriction coefficient and a double-peak structure in the field-angle derivative near the Pauli-limited field, which could be related to FFLO lattice effects, but no unambiguous thermodynamic FFLO signature is found, and the observed phase boundaries disagree with FFLO boundaries inferred from NMR. Overall, the results suggest a nuanced and probe-dependent manifestation of the FFLO state in Sr$_2$RuO$_4$ and call for further experiments and theory to resolve its existence and nature.

Abstract

We performed high-resolution magnetostriction measurements on the Pauli-limited superconductor Sr$_2$RuO$_4$ using high-quality single crystals. A first-order superconducting transition, accompanied by pronounced hysteresis, was observed under in-plane magnetic fields, where the relative length change of the sample, $ΔL/L$, was on the order of $10^{-8}$. To ensure the reliability of the measurements, particular attention was paid to minimizing the influence of magnetic torque, which can significantly affect data under in-plane field configurations, via field-angle-resolved magnetostriction. Within the hysteresis regime, slightly below the Pauli-limited upper critical field, a hump-like anomaly in the magnetostriction coefficient was identified. Furthermore, a characteristic double-peak structure in the field-angle derivative of the magnetostriction provides additional support for this anomaly. Although these findings may reflect a lattice response associated with the emergence of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in Sr$_2$RuO$_4$, the possibility of a broadened first-order transition cannot be excluded. Notably, this magnetostriction anomaly qualitatively deviates from the FFLO phase boundary suggested by previous NMR measurements, highlighting the necessity for further experimental and theoretical investigations to elucidate the nature of the FFLO state in this material.

Figures (9)

• Figure 1: (Color online) (a) Temperature dependence of the zero-field specific heat of the $\mathrm{Sr_2RuO_4}$ component, $c_{\rm 214}$, divided by temperature. (b) Magnetic field dependence of $c_{\rm 214}/T$ at several temperatures, measured at $\phi=27^\circ$ and $\theta=90^\circ$. Each dataset at the same temperature is vertically shifted by 10 mJ/(mol K$^2$) for clarity. The blue open and red closed circles represent data obtained during decreasing and increasing field sweeps, respectively. (c) Field-angle $\phi$ dependence of the upper critical field $B_\mathrm{c2}$ at $\theta=90^\circ$ (i.e., within the $ab$ plane) at 0.3 K, determined from specific-heat and magnetostriction $\Delta L_c$ measurements during increasing field sweeps. The inset in (a) depicts the definition of the field angles, $\phi$ and $\theta$.
• Figure 2: (Color online) Temperature dependence of the relative length change, $\Delta L_i/L_i$, along the (a) $c$ and (b) $a$ axes. The red circles represent zero-field data. The blue squares in (a) indicate normal-state data measured under an in-plane magnetic field of 2 T at $\phi=27^\circ$. The open and closed squares in (b) correspond to normal-state data measured under in-plane fields of 1.45 and 1.8 T, respectively, at $\phi=90^\circ$. The dashed (dotted) lines represent linear fits to the zero-field data just below (above) $T_\mathrm{c}$.
• Figure 3: (Color online) (a) Polar-angle $\theta$ dependence of the capacitance change relative to the normal-state value (taken at $\theta \sim 95^\circ$) for $L \parallel c$, measured at 0.3 K and 1 T for various azimuthal angles $\phi$. (b), (c) Magnetic-field dependence of the capacitance change relative to the value at 1.7 T, measured at low temperatures for two different sample orientations: (b) $L \parallel c$ at 0.3 K with $\phi = 27^\circ$ and $\theta = 90^\circ$, (c) $L \parallel a$ at 0.12 K with $\phi = 90^\circ$ and $\theta = 90^\circ$. The blue open and red closed circles denote data obtained during decreasing and increasing field sweep. The green triangles show normal-state response measured at $\theta = 85^\circ$ for each configuration, where $B_\mathrm{c2}$ is below 0.8 T. The dashed lines are fits to the normal-state data using a cubic polynomial function. (d) Superconducting contribution to the normalized magnetostriction $\Delta L_c^{\rm sc}/L_c$ for $L \parallel c$, obtained by subtracting the background [dashed line in (b)]. The characteristic field $B_{\rm 1st}$, at which the hysteresis in $\Delta L_c^{\rm sc}$ vanishes, is indicated by an arrow.
• Figure 4: (Color online) Magnetic-field dependence of the normalized magnetostriction $\Delta L_i^{\rm sc} / L_i$ at several temperatures for (a) $i=c$ ($\phi=27^\circ$ and $\theta=90^\circ$) and (b) $i=a$ ($\phi=90^\circ$ and $\theta=90^\circ$). Each dataset is vertically shifted by $1 \times 10^{-8}$ for clarity. The blue open and red closed circles represent data obtained during decreasing and increasing field sweeps, respectively. The arrows indicate the characteristic field $B_{\rm 1st}$, at which the hysteresis loop closes. The numbers labeling each dataset indicate the temperature in K.
• Figure 5: (Color online) Magnetic-field dependence of the magnetostriction coefficient $\lambda_i=(\partial \Delta L_i^{\rm sc} / \partial B) /L_i$ at several temperatures for (a) $i=c$ ($\phi=27^\circ$ and $\theta=90^\circ$) and (b) $i=a$ ($\phi=90^\circ$ and $\theta=90^\circ$). Each dataset is vertically shifted by $4 \times 10^{-7}$ T$^{-1}$ for clarity. The blue open and red closed circles represent data obtained during decreasing and increasing field sweeps, respectively. The arrows indicate a possible hump-like anomaly at $B_{\rm K}$. The numbers labeling each dataset indicate the temperature in K.