Nonanalytic Fermi-liquid correction to the specific heat of RuO$_2$

TL;DR

This study reveals nonanalytic Fermi-liquid corrections in a bulk transition-metal, RuO$_2$, by demonstrating a negative $\delta T^3\ln(T/T^*)$ term in the low-temperature specific heat and a $T^2\ln T$ contribution to the magnetic susceptibility, both modulated by magnetic field in an isotropic manner. Through ultra-clean single crystals (RRR up to 1200) and precise measurements of resistivity, specific heat, and magnetization, the authors extract a small mass enhancement ($m^*/m\approx6.5$) and place RuO$_2$ in the weakly correlated 3D Fermi-liquid regime via Wilson and Kadowaki–Woods analyses. The field-dependent nonanalytic corrections point to an intrinsic Fermi-liquid origin, potentially tied to electron-phonon scattering, rather than spin fluctuations. Overall, the work provides rare, systematic experimental evidence of nonanalytic Fermi-liquid behavior in a bulk transition-metal and underscores the critical role of crystal purity in revealing such effects.

Abstract

The magnetic nature of the altermagnet candidate RuO$_2$ remains under debate. It has been recently shown from quantum oscillations and angle-resolved photoemission spectroscopy (ARPES) that the high-quality RuO$_2$ bulk single crystal is a paramagnetic metal. However, the low-temperature specific heat exhibits a clear deviation from the conventional $C(T)$=$γT$ + $βT^3$ dependence; it is well described with nonanalytic Fermi-liquid correction for a clean paramagnetic metal: $C(T)$ = $γT$ + $βT^3$ + $δT^3 \textrm{ln}(T/T^*)$. Correspondingly, the magnetic susceptibility is well fitted with the inclusion of $T^2\textrm{ln}T$ term as well as $H^2\mathrm{ln}H$ term. In contrast to the spin fluctuation mechanism applicable to some heavy-electron compounds with positive $δ$, RuO$_2$ shows negative $δ$ suggesting a different origin. The observation of such nonanalytic Fermi liquid corrections is attributable to the availability of an ultra-clean sample. The electronic specific heat, the magnetic susceptibility, and the $T^2$ coefficient in resistivity point to a weakly-correlated 3D Fermi-liquid state with a modest electron correlation, as supported by the Wilson and Kadowaki-Woods ratios.

Figures (9)

• Figure 1: Characterization of RuO$_2$ single crystal: (a) Schematic of the rutile crystal structure momma2008vesta. In the anticipated antiferromagnetic phase, the Ru sites shown in red (blue) would have magnetic moments up (down) along the [001] direction. The bottom figure shows that the [101]* direction, perpendicular the (101) plane, is about 20 degrees different from the [101] direction. (b) Optical image of a typical RuO$_2$ single crystal with the size 5$\times$3$\times$1.3 mm$^3$. Crystal orientation is confirmed by x-ray Laue photos. (c) Resistivity with a current along the [001] axis. Needle-shaped crystal elongated along the [001] direction is used paul2025growth. The residual resistivity ratio (RRR) is as large as 1200. The current directions is along the [100] axis for Wenzel $et$$al.$wenzel2025fermi; but not specified in Huang $et\ al.$huang1982growth.
• Figure 2: (a) Variations of the zero-field specific heat divided by temperature, $C/T$, of a RuO$_2$ single crystal measured from 1.1 to 10 K, plotted against $T^2$. Experimental data was fitted with both Eq. (\ref{['eq3']})-(\ref{['eq6']}). The results of the present studies (black sphere) and those of Passenheim and McCollum passenheim1969heat (open circle) are shown for comparison. (b) $C/T$ vs $T^2$ plotted for various fields along with the fitting with Eq. (\ref{['eq6']}). (c) The electronic contribution to the specific heat divided by $T$, $C_\mathrm{ele}/T$, for various fields. The inset presents the coefficient $\delta$ and $T^*$ in Eq. (\ref{['eq8']}) as a function of magnetic field. (d) The temperature dependence of the isothermal entropy change given by Eqs. (\ref{['eq9']}) and (\ref{['eq10']}). Note the negative sign of the vertical axis. The inset shows the entropy increased above $T_1$.
• Figure 3: (a) Variation of the DC magnetic susceptibility, $M/H$, of RuO$_2$ with temperature from 1.8 K to 200 K under various magnetic fields $\mu_{0}H$ applied perpendicular to the (101) plane. The inset shows the susceptibility over a broader temperature range (1.8-400 K), revealing no signs of a magnetic transition. (b) $M/H$ at 0.5 and 5 T fitted by Eqs. (\ref{['eq12']}) and (\ref{['eq14']}). (c) Anisotropy of $M/H$ at 2 T along different field directions, [101]*, [010], and [-101]. The susceptibility is well fitted with $T\ \mathrm{ln} (T/T_0)$ over a wide temperature range. (d) Variation of the DC susceptibility with field at various temperatures up to 10 K. The susceptibility is fitted with $\Delta\chi (H)$ = $\eta$$H^2\mathrm{ln}(H/H_0)$ (Eq. \ref{['eq16b']}), predicted for the Fermi-liquid correction. The data for $H \parallel [101]^*$ is taken from $M/H$ vs $T$ in Fig. \ref{['fig3']} (a); $H \parallel [010]$ from $M$ vs $H$ in Fig. \ref{['figA3']}
• Figure 4: (a) Pauli susceptibility $\chi_0$ in Eq. \ref{['eq16']} plotted on a logarithmic scale against the Sommerfeld coefficient $\gamma$ in the specific heat for various compounds including RuO$_2$. The solid and dashed lines represent the Wilson ratio $R_\mathrm{W}$ of 2 and 1, respectively. (b) The electrical resistivity of RuO$_2$ for current along the [001] direction plotted against $T^2$. The solid line represents the fitting below 25 K using $\rho (T)= \rho_0 + AT^2$ with $A$ = 0.053 n$\Omega$-cm/K$^2$. (c) The $T^2$ resistivity coefficient $A$ plotted on a logarithmic scale against $\gamma$. Solid and dashed lines represents the Kadowaki–Woods ratio (KWR) $A/\gamma^2 = a_0$=1.0$\times$10$^{-5}$$\mu \Omega$-cm/(mJ/mol-K)$^2$ and a$_0$/25, respectively kadowaki1986universal. (d) Plot of the Kadowaki–Woods-Jacko ratio ($R_\mathrm{{KWJR}}$) given by Eq. \ref{['eq18']}Jacko2009 including RuO$_2$.
• Figure A1: (a)–(c) The thermodynamic analysis of a crystal with RRR = 1200, corresponding to Fig. \ref{['fig2']}(b)-(d) for a crystal with RRR = 400. (a) $C/T$ is plotted against $T^2$ for different magnetic fields, together with the corresponding fits to Eq. \ref{['eq6']}. (b) The electronic contribution to the specific heat, $C_{\text{ele}}/T$, under various fields. The inset shows the field dependence of the parameters $\delta$ and $T^{*}$ obtained from Eq. \ref{['eq6']}. (c) The temperature dependence of the isothermal entropy change calculated using Eq. (\ref{['eq9']})-(\ref{['eq10']}). The inset shows the temperature-dependent entropy above $T_1$.