Using full-potential linear augmented plane-wave method (FP-LAPW) we have studied the stability and electronic properties of the chalcogenide CrTe in three competing structures: rocksalt (RS), zinc blende (ZB), and the NiAs-type (NA) hexagonal. Although the ground state is of NA structure, RS and ZB are interesting in that these fcc-based structures, which can perhaps be grown on various semiconductor substrates, exhibit half-metallic (HM) phases above some critical values of the lattice parameter. We find that the NA structure is not HM at its equilibrium volume while both ZB and RS structures are. The RS structure is more stable than the ZB with an energy that is lower by 0.25 eV/atom. While confirming previous results on the HM phase in ZB structure, we provide hitherto unreported results on the HM RS phase, with a gap in the minority channel and a magnetic moment of $4.0\text{ }{\ensuremath{\mu}}_{B}/\text{f}\text{.u}\text{.}$ A comparison of total energies for the ferromagnetic (FM), nonmagnetic, and antiferromagnetic (AFM) configurations shows the lowest energy configuration to be FM for CrTe in all the three structures. The exchange interactions in the RS and ZB structures are studied for a wide range of the lattice parameter using the linear-response method and a mapping of the total energy to the classical Heisenberg model. These linear-response calculations are performed in the linear muffin-tin orbitals (LMTOs) basis, using the atomic sphere approximation (ASA). We have verified that the results of the electronic structure obtained via the LMTO-ASA method under local-density approximation (LDA) and $\text{LDA}+U$ schemes are in close agreement with those obtained via the more accurate FP-LAPW method. The results show that the exchange interactions in the RS structure are much more short ranged than in the ZB structure. Hence, for the RS structure the exchange interactions are also studied by using a nearest- and next-nearest-neighbor $({J}_{1}\text{\ensuremath{-}}{J}_{2})$ model and the energy differences between FM and two AFM states. These ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model results are obtained by using both the FP-LAPW and LMTO-ASA methods and compared with the linear-response results. The calculated Curie temperatures for the RS phase are consistently higher than those for the ZB phase.
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