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 \textit{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 \textit{dc} field and the Campbell penetration depth was measured with the external \textit{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, $\rho_s=(\lambda(0)/\lambda(T))^2$ was found to follow the clean isotropic \textit{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 $\Delta (0)/k_BT_c = 1.71$. The temperature dependence of the upper critical field, \textit{$H_{c2}$}, was found to be isotropic with a slope at \textit{T$_c$} of -2.63 T/K and \textit{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.