We develop a simple and low-cost technique based on chemical vapor deposition from which large-size graphene films with 5-10 graphene layers can be produced reliably and the graphene films can be transferred easily onto HgCdTe (MCT) thin wafers at room temperature. The proposed technique does not cause any thermal and mechanical damages to the MCT wafers. It is found that the averaged light transmittance of the graphene film on MCT thin wafer is about 80% in the mid-infrared bandwidth at room temperature and 77 K. Moreover, we find that the electrical conductance of the graphene film on the MCT substrate is about 25 times larger than that of the MCT substrate at room temperature and 77 K. These experimental findings suggest that, from a physics point of view, graphene can be utilized as transparent electrodes as a replacement for metal electrodes while producing better and cheaper MCT infrared detectors.