Relativistic effects in complex compounds containing heavy elements: ternary Cu-Tl-X (X = S/Se/Te)

Cu-chalcogenides are a large group of multifunctional compounds traditionally used in photovoltaics and optoelectronics. Their bandgap sizes usually decrease with the element masses, e.g., 2.68, 1.68, and 1.04 eV for CuAlSe₂, CuGaSe₂, and CuInSe₂, respectively. Cu-Tl-X (X = S/Se/Te) with even heavier thallium (Tl) has received recent attention in topological insulators and high-performance thermoelectric converters. While the novel applications may be related to Tl relativistic effects, first-principles investigations are scarce for these complex compounds. Here, we reveal the relativistic effects in Cu-Tl-X using a tailored density-functional-theory approach. Three relativistic terms of mass-velocity, Darwin, and spin-orbit-coupling play distinct roles. In diamond-like CuTlX₂, the mass-velocity correction reduces the conduction band position and contributes to minimizing the bandgaps. The relativistic bandgap of CuTlS₂ of 0.11 eV is substantially smaller than 1.7 eV without considering the relativistic effects. In CuTlTe₂, the spin-orbit-coupling splits the valence bands, resulting in an exotic band inversion. CuTlSe₂ lies on the boundary of normal and inverted band topologies. Interestingly, the relativistic core contraction is so strong that it may favor non-centrosymmetric defective structures with stereoactive lone-pair electrons. The bandgap of the defective structure is much larger, leaving the system little chance to develop an inverted band topology. Our work provides deep insights into understanding the relativistic band topologies of the complex Cu-Tl-X compounds.