Electronic phase separation due to magnetic polaron formation in the semimetallic ferromagnet EuB${}_6$ - A weakly-nonlinear-transport study

We report measurements of weakly nonlinear electronic transport, as measured by third-harmonic voltage generation $V_{3\omega }$, in the low-carrier density semimetallic ferromagnet EuB${}_6$, which exhibits an unusual magnetic ordering with two consecutive transitions at $T_{c_1}=15.6$,K and $T_{c_2} = 12.5$,K. Upon cooling in zero magnetic field through the ferromagnetic transition, the dramatic drop in the linear resistivity at the upper transition $T_{c_1}$ coincides with the onset of nonlinearity, and upon further cooling is followed by a pronounced peak in $V_{3\omega }$ at the lower transition $T_{c_2}$. Likewise, in the paramagnetic regime, a drop of the material’s magnetoresistance $R(H)$ precedes a magnetic-field-induced peak in nonlinear transport. A striking observation is a linear temperature dependence of $V_{3\omega }^{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}(H)$. We suggest a picture where at the upper transition $T_{c_1}$ the coalescing MP form a conducting path giving rise to a strong decrease in the resistance. The MP formation sets in at around $T^\ast \sim 35$,K below which these entities are isolated and strongly fluctuating, while growing in number. The MP then start to form links at $T_{c_1}$, where percolative electronic transport is observed. The MP merge and start forming a continuum at the threshold $T_{c_2}$. In the paramagnetic temperature regime $T_{c_1}<T<T^{\ast }$, MP percolation is induced by a magnetic field, and the threshold accompanied by charge carrier delocalization occurs at a single critical magnetization.