Giant Anomalous Hall Effect in (Ba,Sr)T$_{2+x}$Ru$_{4-x}$O$_{11}$ (T=Fe,Co,Mn) Ferrites*

Hexagonal R-type ferrites (Ba,Sr)T$_{2+x}$Ru$_{4-x}$O$_{11}$ are promising spintronic materials that exhibit collinear ferrimagnetic order at unusually high critical temperatures T$_{C} \quad \le $ 490 K for Fe-bearing compositions, and an in-plane, ``all-in/all-out'' order at T$_{C}$'s $<<$ 300 K due to frustrated antiferromagnetic interactions within the Kagome basal plane in metallic Co or Mn compositions. A strong, nonmonotonic field dependence of the anomalous Hall effect is observed in metallic ferrites, which is generated by non-zero scalar spin chirality and the Berry phase acquired by carriers moving in the ``topologically nontrivial'' spin background of the Kagome plane. The FM semiconductor BaFe$_{3.4}$Ru$_{2.6}$O$_{11}$ (T$_{C}$ = 440 K) exhibits a giant Hall resistivity = 77 $\mu \Omega $-cm at 300 K, with a low-temperature sign change and monotonic field dependence that are consistent with a strong Berry phase curvature (gauge field) acquired by carriers in momentum space.