Conventional superconductivity and hysteretic Campbell penetration depth in single crystals MgCNi${}_3$

Single crystals of MgCNi${}_3$, with areas sized up to 1 mm${}^2$, were grown by the self flux method using a cubic anvil high pressure technique. The first critical field $\mathit{\text{H}}{}_{c1}$, determined from a zero temperature extrapolation, is around 18 mT. Using the tunnel - diode resonator technique, the London penetration depth was measured with no applied $\mathit{\text{dc}}$ field and the Campbell penetration depth was measured with the external $\mathit{\text{dc}}$ fields up to 9T for two different sample orientations with respect to the direction of applied magnetic field. The absolute value of the London penetration depth, $\lambda (0)=245\pm 10$ nm was determined from the thermodynamic Rutgers formula. The superfluid density, ${\rho }_s=(\lambda (0)\mathrm{/}\lambda (T){)}^2$ was found to follow the clean isotropic $\mathit{\text{s}}$-wave behavior predicted by the weak - coupling BCS theory in the whole temperature range. The low - temperature behavior of the London penetration depth fits the BCS analytic form as well and produces close to the weak - coupling value of $\mathrm{\Delta }(0)\mathrm{/}{k}_B{T}_c=1.71$. The temperature dependence of the upper critical field, $H_{c2}$, was found to be isotropic with a slope at $\mathit{\text{T}}{}_c$ of -2.63 T/K and $\mathit{\text{H}}{}_{c2}$(0) $\approx$ 12.3 T at zero temperature. The Campbell penetration depth probes the vortex lattice response in the mixed state and is sensitive to the details of the pinning potential. For MgCNi${}_3$, an irreversible feature has been observed in the TDR response when the sample is field-cooled and warmed versus zero-field-cooled and warmed. This feature possesses a non-monotonic field dependence and has commonly been referred to as the peak effect and is most likely related to a field - dependent non - parabolic pinning potential.