Lead sulfide nanoparticles (PbS NPs) are used in the short-wavelength infrared photodetectors because of their excellent photosensitivity, band gap tunability, and solution processability. It has been a challenge to synthesize high-quality PbS NPs with an absorption peak beyond 2000 nm. In this work, using PbS seed crystals with an absorption peak at 1960 nm, we report a successful synthesis of very large monodispersed PbS NPs having a diameter up to 16 nm by multiple injections. The resulting NPs have an absorption peak over 2500 nm with a small full width at half-maximum of 24 meV. To demonstrate the applications of such large quantum dots (QDs), broadband heterojunction photodetectors are fabricated with the large PbS QDs of an absorption peak at 2100 nm. The resulting devices have an external quantum efficiency (EQE) of 25% (over 50% internal quantum efficiency) at 2100 nm corresponding to a responsivity of 0.385 A/W and an EQE of ∼60% in the visible range.