Investigation of CeRh$_2$As$_2$ order parameters via ultrasound propagation anomalies

TL;DR

This work addresses the microscopic nature of order parameters in CeRh2As2, a heavy-fermion superconductor with two superconducting phases (SC1, SC2) and a coexisting phase I, probed by ultrasound to test symmetry of the order parameter under high magnetic fields and hydrostatic pressure, with transitions at $T_0\approx0.5$ K and $T_c\approx0.3$ K and a field-driven line at $B^*\approx4$ T. Speed-of-sound measurements of elastic constants $C_{11}$, $C_{33}$, $C_{66}$, $C_{44}$ and $(C_{11}-C_{12})/2$ reveal an inverse-lambda anomaly at $T_0$ and a largely featureless $T_c$ response when phase I is suppressed by hydrostatic pressure of $0.7$ GPa, indicating a single-component SC order parameter in both SC1 and SC2. A Landau free-energy analysis of magneto-elastic couplings shows that discontinuities in shear moduli would require a two-component AFM order parameter at a time-reversal-invariant momentum (TRIM), whereas the observed anomalies are consistent with an incommensurate AFM order parameter in phase I with an ordering vector away from TRIM points, e.g., along $Y(k_x,\pi,0)$ or $F(k_x,\pi,k_z)$ or general $(k_x,k_y,k_z)$ with $k_x\neq k_y$. Together with the pressure data that suppresses phase I while preserving superconductivity, the results rule out a multicomponent SC order parameter and place stringent constraints on phase I, advancing understanding of multi-phase superconductivity in CeRh2As2.

Abstract

Unconventional superconductors with nearly degenerate pairing states are rare. CeRh$_2$As$_2$ has recently emerged as one of the few existing multi-phase superconductors. It exhibits a first-order phase transition between two distinct superconducting states when a magnetic field is applied along the crystallographic $c$-axis. While this behavior has been linked to locally broken inversion symmetry, a phase diagram based on a multi-component superconducting order parameter remains a possibility. Furthermore, superconductivity appears to coexist with an ordered state (phase I). Despite being the subject of many studies, little is known about the nature of the order parameters in both superconducting phases and phase I. Here, we use ultrasound-propagation measurements at low temperatures, in high magnetic fields and under hydrostatic pressure to address this issue. Our results strongly suggest that the superconducting order parameter in both phases is single-component, corroborating the role of local non-centrosymmetry in the development of multi-phase superconductivity in CeRh$_2$As$_2$. In addition, analysis of the elastic anomalies within the Landau framework of phase transitions strongly suggests the presence of an incommensurate magnetic order parameter in phase I.

Figures (3)

• Figure 1: Phase diagram of CeRh$_{2}$As$_{2}$, crystal structure, and experimental set-up: a) Schematic phase diagram of CeRh2As2 for magnetic fields applied along the crystallographic c-axis, according to Ref. khanenko2025. Phases SC1 and SC2 represent the superconducting phases found at low temperatures, phase I is the magnetic phase. The inset shows a sketch of the experimental setup used in ultrasound experiments: LiNb transducers attached to the sample produce sound waves, which travel in the direction k with polarization u. b) Schematic phase diagram of CeRh2As2 under applied pressure at $B = 0$, according to Ref. pfeiffer2024. The dashed line represents the pressure at which ultrasonic experiments in this work were conducted. c) Locally non-centrosymmetric crystal structure of CeRh2As2 with a body-centered-tetragonal (bct) Ce sublattice.
• Figure 2: Temperature dependence of the elastic constants of CeRh$_{2}$As$_{2}$ across the phase transitions. (a) Comparison of the temperature dependence of the specific heat, $c$-axis thermal expansion and the speed of sound representing the elastic modes $C_{11}$, $C_{33}$, and$(C_{11}-C_{12})/2$ across the zero-field transitions. Dashed lines serve as guides to the eye. (b) Temperature dependence of the speed of sound of the shear modes $C_{66}$ (blue) and $C_{44}$ (orange). The $C_{66}$ mode exhibits a clear discontinuity at the superconducting transition at ambient pressure. The application of 0.7 GPa (violet) suppresses both transitions, indicating a single component OP for the SC1 phase. The insets represent the lattice distortion associated with each elastic mode.
• Figure 3: Summary of theoretical results:(a) The two antiferromagnetic orders with $\vec{Q}=\Gamma(0,0,0)$ corresponding to the $(m,0)$ (left panel) and $(m,m)$ states (right panel). Green and blue arrows represent the local magnetic moments at cerium atoms. (b) Phase-transition signatures of the order parameters and elastic constants expected from the Landau free energy for the AFM transition with $\vec{Q}=\Gamma(0,0,0)$ and the SC transition. $(\beta_1,\beta_2)=(1,2.5)$ and $(\beta_1,\beta_2)=(1,0.45)$ are used for the left and the right panels, respectively, with the common parameters $(\alpha_0,C^{(0)}_{O},C^{(0)}_{66},\gamma_{1},\gamma_{2})=(1,1,1,0.13,0.2)$. (c) Ordering vector $\vec{Q}$ compatible with anomalies at the $T=T_0$ transition. Left panel: The high-symmetry lines $Y(k_{x},\pi,0)$ and $T(k_{x},\pi,\pi)$ with $0<k_{x}<\pi$ are marked in blue, and the high-symmetry plane $F(k_{x},\pi,k_{z})$ with $0<k_{x},k_{z}<\pi$ is shown in red. The high-symmetry points (black dots) are not included in these regions. Right panel: The red triangular pillar includes general momentum $(k_{x},k_{x},k_{z})$ with $0<k_{x}<k_{y}<\pi$ and $0<k_{z}<\pi$. Black lines represent high-symmetry lines whose irreducible representations are incompatible with the elastic-moduli anomalies at $T=T_0$. (d) Table of elastic constants that can show discontinuous jumps at the magnetic transition at $T_{\textrm{0}}$ and at the superconducting transition at $T_{\textrm{c}}$ for various time-reversal-invariant ordering vectors $\vec{Q}$ of the magnetic state emerging at $T_{\textrm{0}}$ . The first column in the "at $T_{\textrm{c}}$" sector shows the direction of the magnetic order in the representation basis, along with the crystallographic point-groups of the corresponding magnetic states. $B_{1g}$, $B_{2g}$, and $E_{g}$ denote the symmetries of the associated strains. $\checkmark$ indicates allowed strain-order parameter coupling to shear strains, yielding discontinuous elastic jumps; $\times$ indicates cases forbidden by symmetry.