  Engineering heterostructures with various types of quantum materials can
provide an intriguing playground for studying exotic physics induced by the
proximity effect. Here, we report the successful synthesis of iron-based
superconductor FeSe$_{x}$Te$_{1-x}$ (FST) thin films across the entire
composition range of $0 \leq x \leq 1$ and its heterostructure with a magnetic
topological insulator by using molecular beam epitaxy. Superconductivity is
observed in the FST films with an optimal superconducting transition
temperature $T_c$ $\sim$ 12 K at around x = 0.1. We found that
superconductivity survives in the very Te-rich films ($x \leq 0.05$), showing
stark contrast to bulk crystals with suppression of superconductivity due to an
appearance of bicollinear antiferromagnetism accompanied by a monoclinic
structural transition. By examining thickness $t$ dependence of magnetic
susceptibility and electrical transport properties, we observed a trend where
anomalies associated with the first order structural transition broaden in
films with below $t \sim$ 100 nm. We infer this observation suggests a
suppression of the structural instability near substrates. Furthermore, we
fabricated an all chalcogenide-based heterointerface between FST and a magnetic
topological insulator (Cr,Bi,Sb)$_{2}$Te$_{3}$ for the first time, observing
both superconductivity and a large anomalous Hall conductivity. The anomalous
Hall conductivity increases with decreasing temperature, approaching the
quantized value of $e^2/h$ down to the measurable minimum temperature at $T_c$.
The result suggests coexistence of magnetic and superconducting gaps at low
temperatures opening at the top and bottom surfaces, respectively. Our novel
magnetic topological insulator/superconductor heterostructure could be an ideal
platform to explore chiral Majorana edge mode.
