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This paper introduces a simple method to increase the bandwidth of slot-fed dielectric resonator antennas (DRAs) for millimetre-wave applications. Bandwidth augmentation is achieved by decreasing the dielectric constant of DRAs while the critical coupling is obtained by properly adjusting the slot size and position. Two numerical methods, based on finite element method and finite integral technique, are used to verify the approach. A 10-dB impedance bandwidth of 27 percent is attained for DRAs with dielectric constant as low as 4 while the radiation properties remain similar to those of higher dielectric constant DRAs.