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Importance of exact exchange to the geometric and electronic structures of Cs$_2$$B$$B'$$X_6$ double perovskites

We investigate the lead-free halide double perovskites (HDPs) Cs$ _2BB'X_6$ ($B$=Ag, Na; $B'$=In, Bi; $X$=Cl, Br) via first-principles calculations. We find that both the geometric and electric structures of the HDPs obtained by the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional are much better than those of the Perdew-Burke-Ernzerhof (PBE) functional. Importantly, we find that the electronic structures of DHPs are very sensitive to their geometries, especially the $B$-$X$ bond lengths. As a consequence, the electronic structures calculated by the HSE functional using the PBE optimized geometries may still significantly underestimate the band gaps, whereas the calculations on the HSE optimized geometries provide much more satisfactory results. The sensitivity of the band gaps of the DHPs to their geometries opens a promising path for the band structure engineering via doping and alloying. This work therefore provides an useful guideline for further improvement of HDPs materials.