Stiffnessometer, a magnetic-field-free superconducting stiffness meter and its application

We provide a detailed account for a new method to measure superconducting stiffness $\rho_{s}$, critical current density $j_c$, and coherence length $\xi$, in one apparatus, without subjecting the sample to magnetic field or attaching leads. The method is based on the London equation $\mathbf{j}=-\rho_{s}\mathbf{A}$, where ${\bf j}$ is the current density and ${\bf A}$ is the vector potential. Using a rotor free $\bf{A}$ and a measurement of $\bf{j}$ via the magnetic moment of a superconducting ring, we determine $\rho_{s}$. By increasing $\mathbf{A}$ until the London equation fails we determine $j_c$ and $\xi$. The method is sensitive to very small stiffness, which translates to penetration depth $\lambda \lesssim 1$~mm. It is also sensitive to low critical current density $j_c \sim 10^3$ Amm$^{-2}$ or long coherence length $\xi \sim 1$~$\mu$m. Naturally, the method does not suffer from demagnetization factor complications, the presence of vortices, or out-of-equilibrium conditions. Therefore, the absolute values of the different parameters can be determined. We demonstrate the application of this method to La$_{2-x}$Sr$_{x}$CuO$_{4}$ with $x=0.17$.