Charge inhomogeneities and transport in semiconductor heterostructures with a manganese $delta$-layer

We study experimentally and theoretically the effects of disorder, nonlinear screening, and magnetism in semiconductor heterostructures containing a $delta$-layer of Mn, where the charge carriers are confined within a quantum well and hence both ferromagnetism and transport are two-dimensional (2D) and differ qualitatively from their bulk counterparts. Anomalies in the electrical resistance observed in both metallic and insulating structures can be interpreted as a signature of significant ferromagnetic correlations. The insulating samples turn out to be the most interesting as they can give us valuable insights into the mechanisms of ferromagnetism in these heterostructures. At low charge carrier densities, we show how the interplay of disorder and nonlinear screening can result in the organization of the carriers in the 2D transport channel into charge droplets separated by insulating barriers. Based on such a droplet picture and including the effect of magnetic correlations, we analyze the transport properties of this set of droplets, compare it with experimental data, and find a good agreement between the model calculations and experiment. Our analysis shows that the peak or shoulder-like features observed in temperature dependence of resistance of 2D heterostructures $delta$-doped by Mn lie significantly below the Curie temperature $T_{C}$ unlike the three-dimensional case, where it lies above and close to $T_{C}$. We also discuss the consequences of our description for understanding the mechanisms of ferromagnetism in the heterostructures under study.