  The chirality associated with broken time reversal symmetry in magnetically
doped topological insulators has important implications to the quantum
transport phenomena. Here we report the anomalous Hall effect studies in Mn-
and Cr-doped Bi$_2$Te$_3$ topological insulators with varied thickness and
doping content. By tracing the chirality of the Hall loops, we find that the
Mn-type anomalous Hall effect with clockwise chirality is strengthened by the
reduction of film thickness, which is opposite to that of the Cr-type anomalous
Hall effect with counterclockwise chirality. We provide a phenomenological
model to explain the evolution of magnetic order and anomalous Hall effect
chirality in magnetically doped topological insulators.
