We report a significant improvement of the spectral properties of cadmium telluride (CdTe) detectors, fabricated in the form of a Schottky CdTe diode. With the use of high quality CdTe wafer, we formed a Schottky junction by evaporating indium on the Te-face and operated the detector as a diode. This allows us to apply much higher bias voltage than was possible with the previous CdTe detectors. A /2 mm/×2 mm detector of thickness 0.5 mm, when operated at a temperature of /5°C, shows leakage current of only 0.2 and 0.4 nA for an operating voltage of 400 and 800 V, respectively. We found that, at a high-electric field of several kV cm-1, the Schottky CdTe diode has very good energy resolution and stability, suitable for astronomical applications. The broad low-energy tail, often observed in CdTe detectors due to the low mobility and short lifetime of holes, was significantly reduced by the application of a higher bias voltage which improves the charge collection efficiency. We achieved very good FWHM energy resolution of /1.1% and /0.8% at energies 122 and 511 keV, respectively, without any rise time discrimination or pulse height correction electronics. For the detection of hard X-rays and gamma-rays above 100 keV, we have improved the detection efficiency by stacking a number of thin CdTe diodes. Using individual readout electronics for each layer, we obtained high detection efficiency without sacrificing the energy resolution. In this paper, we report the performance of the new CdTe diode and discuss its proposed applications in future hard X-ray and gamma-ray astronomy missions.