Strong increase in ultrasound attenuation below $T_c$ in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$: Possible evidence for domains

Recent experiments suggest that ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ has a two-component superconducting order parameter (OP). A two-component OP has multiple degrees of freedom in the superconducting state that can result in low-energy collective modes or the formation of domain walls—a possibility that would explain a number of experimental observations including the smallness of the signature of time reversal symmetry breaking at $T_{\mathrm{c}}$ and telegraph noise in critical current experiments. We use resonant ultrasound spectroscopy to perform ultrasound attenuation measurements across the superconducting $T_{\mathrm{c}}$ of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. We find that compressional sound attenuation increases by a factor of 7 immediately below $T_{\mathrm{c}}$, in sharp contrast with what is found in both conventional ($s$-wave) and high-$T_{\mathrm{c}}$ ($d$-wave) superconductors. Our observations are most consistent with the presence of domain walls that separate different configurations of the superconducting OP. The fact that we only observe an increase in sound attenuation for compressional strains, and not for shear strains, suggests an inhomogeneous superconducting state formed of two distinct, accidentally degenerate superconducting OPs that are not related to each other by symmetry. Whatever the mechanism, a factor of 7 increase in sound attenuation is a singular characteristic that must be reconciled with any potential theory of superconductivity in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$.

Figure 1

Measuring ultrasonic attenuation with resonant ultrasound spectroscopy. (a) The ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ unit cell under a deformation corresponding to the longitudinal strain ${\epsilon }_{xx}$, associated with the elastic constant $c_{11}$. This mode is a superposition of pure compression ${\epsilon }_{xx}+{\epsilon }_{yy}$ and pure shear ${\epsilon }_{xx}-{\epsilon }_{yy}$, associated with the elastic constants $(c_{11}+c_{12})/2$ and $(c_{11}-c_{12})/2$, respectively. (b) Resonant ultrasound spectrum of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ between 2.2 and 2.8 MHz. $X\left(\omega \right)$ and $Y\left(\omega \right)$ are the real and the imaginary parts of the response. The boxed resonance is shown in detail in (c). (c) Zoom-in on the resonance near 2.34 MHz. The center of the resonance and the linewidth are indicated. Inset shows the same resonance plotted in a complex plane and fit to a circle—$z_c$ denotes the center of the circle.

Figure 2

Symmetry-resolved sound viscosity in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. (a)  Compressional and (b) shear viscosities through $T_{\mathrm{c}}$. The irreducible strain corresponding to each viscosity is shown—${\eta }_{13}$ arises due to coupling between the two $A_{1g}$ strains. The compressional viscosities increase immediately below $T_{\mathrm{c}}$, whereas no such features are observed in the shear viscosities.

Figure 3

Comparison of different mechanisms for sound attenuation in the superconducting state. (a) Normalized viscosity [$\eta \left(T\right)/\eta$($T_{\mathrm{c}}$)] for an isotropic $s$-wave gap and a $d_{x^2-y^2}$ gap, calculated within the BCS framework. (b) $\eta \left(T\right)/\eta$($T_{\mathrm{c}}$) for a time reversal symmetry breaking gap below $T_{\mathrm{c}}$. A peak is seen at high enough frequencies (approximately terahertz) but not at our experimental frequencies (approximately megahertz). (c) Attenuation peak at different frequencies due to pair-breaking effects in a $d_{x^2-y^2}$ gap. The inset shows the plot at our experimental frequency in detail—a tiny peak is seen about 0.01 nK below $T_{\mathrm{c}}$ [$\delta \eta \left(T\right)=\eta \left(T\right)-\eta$($T_{\mathrm{c}}$) and $\delta T=T-T_{\mathrm{c}}$ ]. (d) Normalized viscosity in the $A_{1g}$ channels of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ through $T_{\mathrm{c}}$, fit to the viscosity expected from domain-wall motion below $T_{\mathrm{c}}$.