The tunneling characteristics of metal contacts on n-type CdTe and p-type InAs have been measured. Both the forward and reverse bias characteristics on CdTe are in good agreement with the two-band model for the energy vs. complex momentum relationship. The presence of trapping states increased the magnitude of the tunneling current at low voltage levels by providing a two-step transition. The slope of the forward bias log J vs. V curves for tunneling through the intermediate states was reduced by a factor of 2. The approximate density and energy of the trapping states was calculated from the observed J-V characteristics. The E-k dispersion relation for InAs was also determined and found to be in excellent agreement with the two-band model.