  We study the thick brane world system constructed in the recently proposed
$f(R,T)$ theories of gravity, with $R$ the Ricci scalar and $T$ the trace of
the energy-momentum tensor. We try to get the analytic background solutions and
discuss the full linear perturbations, especially the scalar perturbations. We
compare how the brane world model is modified with that of general relativity
coupled to a canonical scalar field. It is found that some more interesting
background solutions are allowed, and only the scalar perturbation mode is
modified. There is no tachyon state exists in this model and only the massless
tensor mode can be localized on the brane, which recovers the effective
four-dimensional gravity. These conclusions hold provided that two constraints
on the original formalism of the action are satisfied.
