We perform a lattice simulation to investigate the doubly charmed tetraquark $T^+_{cc}$ observed by the LHCb collaboration, slightly below the $D^{*+}D^0$ threshold, with flavor content $cc\bar{u}\bar{d}$ and isospin-$0$. Two-meson interpolators are implemented to explore the isospin quantum numbers $I=0$ and $I=1$.
We observe attraction near the $DD^*$ threshold for $I=0$ and repulsion for $I=1$.
Moreover, we also include diquark-antidiquark interpolators to study their effect on the energy spectrum.
There is no significant shift in the ground state energy when adding diquark-antidiquark interpolators to the interpolator basis
when the heavy quark mass is close to the physical charm quark mass.
However, we observe a non-negligible shift in the
second energy level. This effect has to be
taken into account to extract the scattering amplitude of the $T^+_{cc}$.
Finally, with a higher mass (close to the bottom quark), the ground state is shifted down significantly. The simulation is performed on $N_f=2+1$ CLS ensembles with $m_\pi \simeq 280$ MeV.