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Two-dimensional semiconductors (2DSCs) with appropriate band gaps and high mobilities are highly desired for future-generation electronic and optoelectronic applications. Here, using first-principles calculations, we report a novel class of 2DSCs, group-11-chalcogenide monolayers (M2X, M = Cu, Ag, Au; X = S, Se, Te), featuring with a broad range of energy band gaps and high carrier mobilities. Their energy band gaps extend from 0.49 to 3.76 eV at a hybrid density functional level, covering from ultraviolet-A, visible light to near-infrared region, which are crucial for broadband photoresponse. Significantly, the calculated room-temperature carrier mobilities of the M2X monolayers are as high as thousands of cm2·V-1·s-1. Particularly, the carrier mobilities of η-Au2Se and ε-Au2Te are up to 104 cm2·V-1·s-1, which is very attracitive for electronic devices. Benefitting from the broad range of energy band gaps and superior carrier mobilities, the group-11-chalcogenide M2X monolayers are promising candidates for future-generation nanoelectronics and optoelectronics.