  Within a spin wave analysis and a fermionic large-n limit, it is shown that
the antiferromagnetic Heisenberg model on the checkerboard lattice may have
different ground states, depending on the spin size $S$. Through an additional
exchange interaction that corresponds to an inter-tetrahedra coupling, the
stability of the N\'eel state has been explored for all cases from the square
lattice to the isotropic checkerboard lattice. Away from the isotropic limit
and within the linear spin wave approximation, it is shown that there exists a
critical coupling for which the local magnetization of the N\'eel state
vanishes for any value of the spin $S$. One the other hand, using the
Dyson-Maleev approximation, this result is valid only in the case $S= \frac12$
and the limit between a stable and an unstable N\'eel state is at S=1. For $S=
\frac12$, the fermionic large-n limit suggests that the ground state is a
valence bond solid build with disconnected 4-spins singlets. This analysis
indicates that for low spin and in the isotropic limit, the checkerboard
antiferromagnet may be close to an instability between an ordered S=0 ground
state and a magnetized ground state.
