Entropy mapping under uniaxial pressure utilizing the elastocaloric effect

Abstract: Uniaxial pressure is a powerful tuning parameter for quantum materials, but conventional thermodynamic probes such as specific heat are difficult to realize in the constrained geometries of strain apparatus. We develop a quantitative analysis framework for a.c. elastocaloric effect measurements that enable the reconstruction of the absolute entropy and hence specific heat across complex phase diagrams. The absolute accuracy is achieved by combining measurements in the strong coupling regime at low frequencies with high signal-to-noise measurements in the quasi-adiabatic regime at high frequencies. Applying the approach to the correlated superconductor Sr$_2$RuO$_4$, we obtain an absolute entropy map across the phase diagram including across phase transitions deep into the superconducting state. We demonstrate that from such data one can derive the absolute specific heat which is currently not possible through other approaches. This data reinforces the finding that the quenching of entropy within the superconductor Sr$_2$RuO$_4$ is strongest at the critical strain consistent with the superconducting gap being maximized at the Van Hove singularity (VHs). Furthermore, we demonstrate that, although $Δc /(γT)$ does increase at the VH strain, this increase is much weaker than previously inferred from more indirect caloric experiments.

• The paper develops a two-frequency elastocaloric framework that combines low-frequency absolute calibration with high-frequency sensitivity, improving signal-to-noise by roughly tenfold while enabling entropy and specific-heat mapping under strain.
• The method reconstructs entropy in Sr₂RuO₄ with approximately 30% absolute uncertainty, finds an entropy maximum near the Van Hove strain, and identifies a small offset between the entropy maximum and the strain that maximizes superconducting T_c.
• The reconstructed thermodynamics shows no additional transition inside the superconducting dome and finds a much smaller Van Hove enhancement of the superconducting specific-heat jump than earlier a.c. calorimetry reported, highlighting possible strain-dependent thermal conductance effects.

Motivation and context

Uniaxial strain has become a standard tuning parameter for quantum materials, but the constrained geometry of piezo-driven strain cells makes conventional absolute thermodynamic measurements—most notably specific heat—extremely difficult to perform quantitatively. The a.c. elastocaloric effect, in which an oscillatory strain produces a temperature oscillation proportional to the isothermal strain derivative of the entropy, offers an alternative route. Prior implementations, however, have relied almost exclusively on the high-frequency quasi-adiabatic regime, where the signal is largest but the extraction of absolute values suffers from systematic uncertainties analogous to those in a.c. specific heat calorimetry: finite thermal length effects decouple parts of the sample, the thermometer thermally decouples, and the signal amplitude depends explicitly on the a priori unknown heat capacity $C_{\mathrm{tot}}$. This paper introduces an analysis framework that combines measurements in two frequency regimes to reconstruct absolute entropy, and hence specific heat, across the uniaxial-strain phase diagram of Sr$_2$RuO$_4$ (2608.17777).

Two-regime thermodynamic model

The experiment uses a lumped-element thermal model: the sample, with an Au/AuFe thermocouple and resistive heater attached via silver epoxy, is weakly coupled to the bath through a thermal conductance $k$ dominated by the Stycast epoxy mounting layer. An analytical solution of the model gives the temperature oscillation amplitude

$$\left|\tilde{T}_s\right|=\frac{\omega}{\sqrt{1+\omega^2(C_{\mathrm{tot}}/k)^2}}\times\frac{T_0}{k}\left|\left(\frac{\partial S}{\partial \varepsilon}\right)_T\tilde{\varepsilon}\right|,$$

with a characteristic frequency $\omega_1 = k/C_{\mathrm{tot}}$ of order 100 Hz for this setup. At $\omega \ll \omega_1$ (strong-coupling regime) the amplitude is linear in frequency and independent of $C_{\mathrm{tot}}$, yielding the absolute elastocaloric coefficient directly; at $\omega \gg \omega_1$ (quasi-adiabatic regime) the signal is maximal but depends on $C_{\mathrm{tot}}$ and is corrupted by sample-internal and thermometer decoupling. Numerical modeling including finite sample length and imperfect thermometer coupling shows that the quasi-adiabatic regime is only valid over a bounded window, and that $_2$0 shifts by up to a factor of two across phase transitions—so no single frequency is safely quasi-adiabatic across the whole phase diagram. This motivates the central methodological claim: absolute calibration must come from the strong-coupling regime, while sensitivity must come from the quasi-adiabatic regime.

Combining the two regimes

The authors exploit the observation that the strong-coupling ($_2$1, measured at 10 Hz) and quasi-adiabatic ($_2$2, 613 Hz) datasets remain closely correlated even as the signal itself varies by more than an order of magnitude across phase boundaries. The two are related by a smoothly varying linear mapping $_2$3, with a residual $_2$4 that varies weakly with strain and is amenable to Savitzky–Golay filtering without distorting features. The filtered residual is then used to reconstruct $_2$5, which combines the absolute accuracy of the low-frequency data with the signal-to-noise of the high-frequency data—an improvement in signal-to-noise of roughly an order of magnitude over the raw strong-coupling data.

Two calibrations anchor the absolute scale. The bath conductance $_2$6 is measured directly with the heater, and is dominated by the epoxy layer and essentially strain-independent. The molar amount of strained sample $_2$7 is determined by fitting the strong-coupling response in the low-strain Fermi-liquid regime, where Maxwell relations applied to the known compliance tensor yield a closed expression involving only $_2$8 and $_2$9. The fit gives $_4$0 μmol and $_4$1 Jm$_4$2K$_4$3, in reasonable agreement with the geometric estimate $_4$4 μmol. The authors state that the overall absolute uncertainty is of order 30%, dominated by the strained-volume determination—an important caveat, since the geometric estimate alone already carries at least 10% systematic error from clamping and strain inhomogeneity.

Entropy reconstruction across the phase diagram

With the calibrated $_4$5 surface in hand, the absolute molar entropy follows from an isothermal strain integral anchored at zero strain, where the entropy is fixed by an independent specific-heat measurement on a sample of the same batch. The reconstruction extends through and below the superconducting transitions, which was not possible in the authors' previous elastocaloric study.

Three results stand out. First, above the superconducting dome the entropy peaks near the Van Hove strain $_4$6, consistent with enhanced density of states. Second, the strain at which $_4$7 is maximized does not coincide with the entropy maximum: the Ehrenfest relation requires $_4$8 where $_4$9, and the data show this point lies approximately 0.01% below the entropy-maximum strain. The authors attribute this offset to Fermi-surface anisotropy—momentum-space regions away from the Van Hove singularity contribute a linear-in-strain entropy background. Third, within the superconducting phase the entropy falls below the normal-state extrapolation, consistent with gap opening, and the entropy surface shows no signature of any additional phase transition inside the superconducting dome, in tension with $k$0SR evidence for spontaneous fields in Sr$k$1RuO$k$2 [Grinenko et al.]. The authors argue that any bulk superconducting property generating the $k$3SR signal should leave a thermodynamic imprint, so this absence poses a significant challenge to such interpretations. The magnetic transition at compressive strains beyond $k$4 is tracked down to 2 K, with the temperature-normalized entropy jump showing no significant enhancement or reduction as it approaches the superconducting dome—leaving the coexistence-versus-competition question open for lower-temperature work.

Specific heat and revision of the jump anomaly

Numerical differentiation of the reconstructed entropy at constant strain yields the specific heat $k$5, a quantity the authors note is currently unobtainable by other means in strain apparatus. The reconstruction confirms that the normal-state specific heat increases with compressive strain—a non-trivial result, since it concerns the temperature dependence of the entropy rather than its magnitude. The normalized superconducting jump $k$6 is nearly strain-independent once broadening is accounted for, and increases near $k$7, qualitatively consistent with the superconducting gap being maximized at the Van Hove singularity.

The quantitative discrepancy with the direct a.c. specific heat measurements of Li et al. is the most consequential claim: the elastocaloric-derived increase in $k$8 at $k$9 is much smaller than previously reported. The authors identify a likely culprit in the earlier analysis—namely, the assumption that the thermal conductance of Sr$$\left|\tilde{T}_s\right|=\frac{\omega}{\sqrt{1+\omega^2(C_{\mathrm{tot}}/k)^2}}\times\frac{T_0}{k}\left|\left(\frac{\partial S}{\partial \varepsilon}\right)_T\tilde{\varepsilon}\right|,$$0RuO$$\left|\tilde{T}_s\right|=\frac{\omega}{\sqrt{1+\omega^2(C_{\mathrm{tot}}/k)^2}}\times\frac{T_0}{k}\left|\left(\frac{\partial S}{\partial \varepsilon}\right)_T\tilde{\varepsilon}\right|,$$1 is strain-independent—and note preliminary evidence of significant strain dependence of the thermal conductance. A dedicated study of this quantity is deferred to a future publication, so the resolution of the discrepancy rests on a measurement not yet reported.

Limitations and open questions

The framework carries a stated overall uncertainty of order 30%, dominated by the strained sample volume; the absolute anchoring via the Fermi-liquid regime is specific to Sr$$\left|\tilde{T}_s\right|=\frac{\omega}{\sqrt{1+\omega^2(C_{\mathrm{tot}}/k)^2}}\times\frac{T_0}{k}\left|\left(\frac{\partial S}{\partial \varepsilon}\right)_T\tilde{\varepsilon}\right|,$$2RuO$$\left|\tilde{T}_s\right|=\frac{\omega}{\sqrt{1+\omega^2(C_{\mathrm{tot}}/k)^2}}\times\frac{T_0}{k}\left|\left(\frac{\partial S}{\partial \varepsilon}\right)_T\tilde{\varepsilon}\right|,$$3, though the authors argue that geometric volume estimation alone suffices to comparable accuracy for any material. The lumped-element model requires the thermal length to exceed sample dimensions and linear temperature gradients—conditions satisfied in the strong-coupling regime but violated at high frequency, which is precisely why the mapping procedure rather than direct quasi-adiabatic analysis is used. The Savitzky–Golay filtering of the residual $$\left|\tilde{T}_s\right|=\frac{\omega}{\sqrt{1+\omega^2(C_{\mathrm{tot}}/k)^2}}\times\frac{T_0}{k}\left|\left(\frac{\partial S}{\partial \varepsilon}\right)_T\tilde{\varepsilon}\right|,$$4 assumes that the residual is featureless on the filtered scale; any genuine fine structure in the mapping function at that scale would be suppressed. The question of superconductivity–magnetism coexistence near $$\left|\tilde{T}_s\right|=\frac{\omega}{\sqrt{1+\omega^2(C_{\mathrm{tot}}/k)^2}}\times\frac{T_0}{k}\left|\left(\frac{\partial S}{\partial \varepsilon}\right)_T\tilde{\varepsilon}\right|,$$5 strain, the microscopic origin of the $$\left|\tilde{T}_s\right|=\frac{\omega}{\sqrt{1+\omega^2(C_{\mathrm{tot}}/k)^2}}\times\frac{T_0}{k}\left|\left(\frac{\partial S}{\partial \varepsilon}\right)_T\tilde{\varepsilon}\right|,$$6SR signal given its thermodynamic silence, and the strain dependence of the thermal conductance all remain open.

Conclusion

This work establishes that combining strong-coupling and quasi-adiabatic a.c. elastocaloric measurements yields absolute entropy and specific heat maps across uniaxial-strain phase diagrams, with accuracy limited primarily by strained-volume determination rather than by the thermal modeling. Applied to Sr$$\left|\tilde{T}_s\right|=\frac{\omega}{\sqrt{1+\omega^2(C_{\mathrm{tot}}/k)^2}}\times\frac{T_0}{k}\left|\left(\frac{\partial S}{\partial \varepsilon}\right)_T\tilde{\varepsilon}\right|,$$7RuO$$\left|\tilde{T}_s\right|=\frac{\omega}{\sqrt{1+\omega^2(C_{\mathrm{tot}}/k)^2}}\times\frac{T_0}{k}\left|\left(\frac{\partial S}{\partial \varepsilon}\right)_T\tilde{\varepsilon}\right|,$$8, the method confirms entropy quenching in the superconducting state that is strongest near the Van Hove strain, reveals a small offset between the $$\left|\tilde{T}_s\right|=\frac{\omega}{\sqrt{1+\omega^2(C_{\mathrm{tot}}/k)^2}}\times\frac{T_0}{k}\left|\left(\frac{\partial S}{\partial \varepsilon}\right)_T\tilde{\varepsilon}\right|,$$9 maximum and the entropy maximum attributable to Fermi-surface anisotropy, finds no thermodynamic evidence for a transition inside the superconducting dome, and substantially revises downward the strain-induced enhancement of the superconducting specific-heat jump relative to earlier indirect analyses. The approach is material-agnostic in principle and provides a quantitative thermodynamic probe in geometries where conventional calorimetry is not feasible.