Multipolar Fermi Surface Deformations in Sr$_2$RuO$_4$ Probed by Resistivity and Sound Attenuation: A Window into Electron Viscosity and the Collision Operator

Abstract

Recent developments in electron hydrodynamics have demonstrated the importance of considering the full structure of the electron-electron scattering operator, which encodes a sequence of lifetimes, one for each component of the Fermi surface deformation in a multipolar expansion. In this context, the dipolar lifetime is measured by resistivity, whereas the quadrupolar component probes the viscosity and can be measured in the bulk via sound attenuation. We introduce a framework to extract the collision operator of an arbitrary metal by combining resistivity and sound attenuation measurements with a realistic calculation of the scattering operator that includes multiband and Umklapp effects. The collision operator allows for the prediction of a plethora of properties, including the non-local conductivity, and can be used to predict hydrodynamic behavior for bulk metals. As a first application, we apply this framework to Sr$_2$RuO$_4$ in a temperature range where electron-electron scattering is dominant. We find quantitative agreement between our model and the temperature dependence of both the resistivity and the sound attenuation, we find the quadrupolar (B1g) relaxation rate to be 30% higher than the dipolar one due to the presence of hot spots on the $γ$ band, and we predict a strongly anisotropic viscosity arising from the $α$ and $β$ bands.

Figures (5)

• Figure 1: Comparison of electrical conductivity with sound attenuation. The Fermi surface deforms under an electric field for conductivity, and a strain field for sound attenuation (red and gray regions correspond to population and depopulation, respectively). The Fermi surface deformation relevant to conductivity, $\sigma$, is proportional to the Fermi velocity $v_\alpha(\mathbf{k})$ and is thus dipolar; in contrast, the Fermi surface deformation relevant to sound attenuation, $\eta$, is determined by the deformation potential, $D_{\alpha\beta}(\mathbf{k})$, and is quadrupolar. Both conductivity and sound attenuation are calculated by evaluating the expectation value of the inverse collision operator, $L^{-1}$. The final column shows measurements of the resistivity (reproduced from Lupien2002) and the inverse sound attenuation of Sr$_2$RuO$_4$: both quantities show Fermi liquid $T^2$ scaling. Data shown in pink are reproduced from LupienEtAl2001 and taken in a 1.5 T magnetic field to suppress the superconducting transition.
• Figure 2: (a),(b) Comparison of experiment and theory for the resistivity and inverse viscosity as a function of temperature. Two free parameters were adjusted to match $\rho(T)$; those same two parameters are then used to calculate the normalized inverse viscosity $\eta^{-1}/\eta_0^{-1}$, which shows excellent agreement with experiment. Above 10 K, the signal of the electronic contribution to sound attenuation becomes small compared to background, as shown in the inset, and is less reliable. Resistivity data from Lupien2002. Data shown in pink are reproduced from LupienEtAl2001 and taken in a 1.5 T magnetic field to suppress the superconducting transition. (c) Effective lifetimes extracted from the conductivity and viscosity. Lines are fits to $A + B T^2$.
• Figure 3: Non-local conductivity calculated at $T = 14$ K. Top: schematic showing the $\beta$ FS of Sr$_2$RuO$_4$ and selected Fermi velocities within an effective "channel", for $\mathbf{q}$ along 100 and 110.
• Figure 4: Resistivity data taken in zero field (red points) and in a 2.6 T field applied along the $c$ axis (blue points). The main text uses the zero-field resistivity data above 2.5 K, and the 2.6 T data below 2.5 K with a 8 n$\Omega\cdot$cm offset subtracted.
• Figure 5: Transport viscosities calculated as a function of $T$. Lines are fitted to $(A+B T^2)^{-1}$.