Observation of the spontaneous vortex phase in the weakly ferromagnetic superconductor ErNi$_{2}$B$_{2}$C: A penetration depth study

The coexistence of weak ferromagnetism and superconductivity in ErNi$_{2}$B$ _{2}$C suggests the possibility of a spontaneous vortex phase (SVP) in which vortices appear in the absence of an external field. We report evidence for the long-sought SVP from the in-plane magnetic penetration depth $\Delta \lambda (T)$ of high-quality single crystals of ErNi$_{2}$B$_{2}$C, using a high-precision tunnel-diode based, self-inductive technique at 21~MHz.. In addition to a slight depression of superconductivity at the N\'{e}el temperature $T_{N}$ = 6.0~K and at the weak ferromagnetic onset at $T_{WFM}$=2.3~K, $\Delta \lambda (T)$ rises to a maximum at $T_{m}=0.45$ K before dropping sharply down to $\sim $0.1~K. We assign the 0.45 K-maximum to the proliferation of spontaneous vortices. A model proposed by Koshelev and Vinokur explains the increasing $\Delta \lambda (T)$ as the vortex density increases, and its subsequent decrease below $T_{m}$ as defect pinning suppresses vortex hopping.