The structure in density and temperature of protoplanetary disks surrounding low-mass stars is not yet well known. The protoplanetary disks mid-planes are expected to be very cold and thus depleted in molecules in gas phase, especially CO. Recent observations of molecules at very low apparent temperature (~ 6 K) challenge this current picture of the protoplanetary disk structures. We aim at constraining the physical conditions, and in particular the gas-phase CO abundance in the mid-plane of protoplanetary disks. The light molecule H2D+,is a tracer of cold and CO-depleted environment. It is therefore a good candidate to explore the disks mid-planes. We performed a deep search for H2D+ in the two well-known disks surrounding TW Hya and DM Tau using the APEX and JCMT telescopes. The analysis of the observations are done with DISKFIT, a radiative transfer code dedicated to disks. In addition, we used a chemical model describing deuterium chemistry to infer the implications of our observations on the level of CO depletion and on the ionization rate in the disks mid-plane. The ortho-H2D+(1(1,0))-1(1,1)) line at 372 GHz was not detected. Although our limit is three times better than previous observations, comparison with the chemical modeling indicates that it remains insufficient to put valuable constraints on the CO abundance in the disk mid-plane. Even with ALMA, the detection of H2D+ may not be straightforward, and H2D+ may not be a sufficiently sensitive tracer of the protoplanetary disks mid-plane< Leer menos