Phenomenological model for the third-harmonic magnetic response of superconductors: application to Sr$_{2}$RuO$_{4}$.

We employ the phenomenological Lawrence-Doniach model to compute the contributions of the superconducting fluctuations to the third-harmonic magnetic response, denoted here by $\overline{M_{3}}$, which can be measured in a precise way using ac magnetic fields and lock-in techniques. We show that, in an intermediate temperature regime, this quantity behaves as the third-order nonlinear susceptibility, which shows a power-law dependence with the reduced temperature $\epsilon=\frac{T-T_{c}}{T_{c}}$ as $\epsilon^{-5/2}$. Very close to $T_{c}$, however, $\overline{M_{3}}$ saturates due to the nonzero amplitude of the ac field. We compare our theoretical results with experimental data for three conventional superconductors -- lead, niobium, and vanadium -- and for the unconventional superconductor Sr$_{2}$RuO$_{4}$ (SRO). We find good agreement between theory and experiment for the elemental superconductors, although the theoretical values for the critical field systematically deviate from the experimental ones. In the case of SRO, however, the phenomenological model completely fails to describe the data, as the third-harmonic response remains sizable over a much wider reduced temperature range compared to Pb, Nb, and V. We show that an inhomogeneous distribution of $T_{c}$ can partially account for this discrepancy, since regions with a locally higher $T_{c}$ contribute to the fluctuation $\overline{M_{3}}$ significantly more than regions with the "nominal" $T_{c}$ of the clean system. However, the exponential temperature dependence of $\overline{M_{3}}$ first reported in Ref. [3] is not captured by the model with inhomogeneity. We conclude that, while inhomogeneity is an important ingredient to understand the superconducting fluctuations of SRO and other perovskite superconductors, additional effects may be at play, such as non-Gaussian fluctuations or rare-region effects.