  Braneworld scenarios consider our observable universe as a brane embedded in
a five-dimensional bulk. In this work, we consider thick braneworld systems in
the recently proposed dynamically equivalent scalar-tensor representation of
$f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ the trace of the
stress-energy tensor. In the general $f\left(R,T\right)$ case we consider two
different models: a brane model without matter fields where the geometry is
supported solely by the gravitational fields, and a second model where matter
is described by a scalar field with a potential. The particular cases for which
the function $f\left(R,T\right)$ is separable in the forms $F\left(R\right)+T$
and $R+G\left(T\right)$, which give rise to scalar-tensor representations with
a single auxiliary scalar field, are studied separately. The stability of the
gravitational sector is investigated and the models are shown to be stable
against small perturbations of the metric. Furthermore, we show that in the
$f\left(R,T\right)$ model in the presence of an extra matter field, the shape
of the graviton zero-mode develops internal structure under appropriate choices
of the parameters of the model.
